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+{"lines":[{"page":2,"text":"Structure and Properties of Liquid Crystals","rect":[53.812843322753909,73.54093933105469,330.33689512165099,59.24553298950195]},{"page":3,"text":".","rect":[53.812843322753909,580.2719116210938,56.799101542947628,579.0049438476563]},{"page":4,"text":"Lev M. Blinov","rect":[53.812843322753909,70.46880340576172,149.0922072510661,59.47233963012695]},{"page":4,"text":"Structure and Properties","rect":[53.812843322753909,166.00665283203126,325.58997562410999,140.98973083496095]},{"page":4,"text":"of Liquid Crystals","rect":[53.812843322753909,195.8763427734375,258.83652713778187,170.8594207763672]},{"page":5,"text":"Dr. Lev M. Blinov","rect":[53.812843322753909,65.70849609375,121.58578961611107,59.523067474365237]},{"page":5,"text":"Russian Academy of Sciences","rect":[53.812843322753909,77.48319244384766,162.99669551661257,69.44213104248047]},{"page":5,"text":"Inst. Crystallography","rect":[53.812843322753909,87.4588394165039,129.81619733095483,79.41777801513672]},{"page":5,"text":"Leninskii prospect 69","rect":[53.812843322753909,97.43436431884766,132.1624802899392,89.39330291748047]},{"page":5,"text":"119333 Moscow","rect":[53.812843322753909,105.60543823242188,113.98518528277848,99.36621856689453]},{"page":5,"text":"Russia","rect":[53.812843322753909,115.473388671875,77.67895379033846,109.28795623779297]},{"page":5,"text":"lev39blinov@gmail.com","rect":[53.812843322753909,127.30460357666016,143.1674072499944,119.26354217529297]},{"page":5,"text":"ISBN 978-90-481-8828-4","rect":[53.812843322753909,446.8249816894531,146.80556195497827,440.58575439453127]},{"page":5,"text":"e-ISBN 978-90-481-8829-1","rect":[182.68087768554688,446.8249816894531,282.64169781923609,440.58575439453127]},{"page":5,"text":"DOI 10.1007/978-90-481-8829-1","rect":[53.812843322753909,457.5715026855469,173.96402447939233,450.4896240234375]},{"page":5,"text":"Springer Dordrecht Heidelberg London New York","rect":[53.812843322753909,468.5242004394531,236.74338238954858,460.483154296875]},{"page":5,"text":"Library of Congress Control Number: 2010937563","rect":[53.812843322753909,486.2212219238281,217.34310282586473,479.0256652832031]},{"page":5,"text":"# Springer ScienceþBusiness Media B.V. 2011","rect":[53.812843322753909,504.1328430175781,210.67365397332567,496.985107421875]},{"page":5,"text":"No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any","rect":[53.812843322753909,513.088623046875,385.2118467223491,505.94085693359377]},{"page":5,"text":"means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written","rect":[53.812843322753909,522.1010131835938,385.21242655633349,514.9532470703125]},{"page":5,"text":"permission from the Publisher, with the exception of any material supplied specifically for the purpose","rect":[53.812843322753909,531.0568237304688,385.2428152861863,523.9090576171875]},{"page":5,"text":"of being entered and executed on a computer system, for exclusive use by the purchaser of the work.","rect":[53.812843322753909,540.0126342773438,378.36315602719017,532.8648681640625]},{"page":5,"text":"Cover design: eStudio Calamar S.L.","rect":[53.812843322753909,557.9552612304688,170.23188466488549,550.7597045898438]},{"page":5,"text":"Printed on acid-free paper","rect":[53.812843322753909,575.8350219726563,137.54016254892043,568.687255859375]},{"page":5,"text":"Springer is part of Springer Science+Business Media (www.springer.com)","rect":[53.812843322753909,593.8031616210938,292.5535139893501,586.6553955078125]},{"page":6,"text":"To my family: Galina, Anastasia and Timothy","rect":[125.86552429199219,246.47299194335938,313.1141742534813,237.51852416992188]},{"page":7,"text":".","rect":[53.812843322753909,580.2719116210938,56.799101542947628,579.0049438476563]},{"page":8,"text":"Epigraph","rect":[53.812843322753909,73.8952407836914,118.31669084814573,59.344844818115237]},{"page":8,"text":"Ego plus quam feci, facere non possum","rect":[53.812843322753909,235.66690063476563,212.87912701249679,226.7323455810547]},{"page":8,"text":"Marcus Tillius Cicero","rect":[89.66889953613281,245.7239990234375,178.3758477311469,238.67198181152345]},{"page":8,"text":"English, translation close to the original","rect":[53.812843322753909,259.58599853515627,214.87394578525614,250.6514434814453]},{"page":8,"text":"More than I have done, I cannot do","rect":[53.812843322753909,269.51361083984377,196.3153008561469,262.61102294921877]},{"page":8,"text":"or maybe it better sounds like this in standard","rect":[53.812843322753909,283.44830322265627,238.31914607099066,274.51373291015627]},{"page":8,"text":"I cannot do more than I have done","rect":[53.812843322753909,293.37591552734377,192.77161443902816,286.47332763671877]},{"page":8,"text":"English","rect":[241.19293212890626,283.44830322265627,271.66277400067818,274.51373291015627]},{"page":9,"text":".","rect":[53.812843322753909,580.2719116210938,56.799101542947628,579.0049438476563]},{"page":10,"text":"Foreword","rect":[53.812843322753909,70.58036041259766,121.10389940527463,59.488277435302737]},{"page":10,"text":"Liquid crystals have found an important place in modern life. Just look around: we","rect":[53.812843322753909,235.66690063476563,385.1845172710594,226.7323455810547]},{"page":10,"text":"see them in our clocks, computer displays, TV screens, telephones and calcula-","rect":[53.812843322753909,247.62646484375,385.20418678132668,238.69190979003907]},{"page":10,"text":"tors, car dashboards, photo-cameras, etc. Other applications include slide projec-","rect":[53.812843322753909,259.58599853515627,385.14470802156105,250.6315155029297]},{"page":10,"text":"tion systems, spatial light modulators, temperature sensors and even liquid crystal","rect":[53.812843322753909,271.5455627441406,385.2681718594749,262.61102294921877]},{"page":10,"text":"lasers. In all these technical innovations, which appeared over the life of onlya","rect":[53.812843322753909,283.44830322265627,385.2114642925438,274.51373291015627]},{"page":10,"text":"single generation, liquid crystals occupy a key position. This is because they","rect":[53.812843322753909,295.4078674316406,385.2094658952094,286.47332763671877]},{"page":10,"text":"consume a barely perceptible amount of energy when they change their state","rect":[53.812843322753909,307.3674011230469,385.16553533746568,298.432861328125]},{"page":10,"text":"under external influences such as temperature, electric field, mechanical stress","rect":[53.812843322753909,319.3269348144531,385.2057229434128,310.39239501953127]},{"page":10,"text":"or whatever. In addition, there are very important biological aspects of liquid","rect":[53.812843322753909,331.2864685058594,385.2439812760688,322.3519287109375]},{"page":10,"text":"crystals.","rect":[53.812843322753909,343.24603271484377,87.62530179526095,334.31146240234377]},{"page":10,"text":"The army of people working in the liquid crystal field continues to grow. The","rect":[65.76486206054688,355.20556640625,385.14475286676255,346.27099609375]},{"page":10,"text":"first conferences held during the early part of the last century involved only tens of","rect":[53.812843322753909,367.16510009765627,385.14678321687355,358.23052978515627]},{"page":10,"text":"participants; then, later, a few hundreds. More recently a wide river of principal","rect":[53.812843322753909,379.06787109375,385.1736284024436,370.13330078125]},{"page":10,"text":"liquid crystal conferences has given rise to several subsidiary, but also quite broad","rect":[53.812843322753909,391.0274353027344,385.10588923505318,382.0928955078125]},{"page":10,"text":"streams of meetings: Worldwide Conferences, European conferences, conferences","rect":[53.812843322753909,402.9869689941406,385.10992826567846,394.05242919921877]},{"page":10,"text":"of National Liquid Crystal societies, separate conferences on chemistry (sometimes","rect":[53.812843322753909,414.9465026855469,385.0710793887253,406.011962890625]},{"page":10,"text":"only on chirality problems), optics, photonics and ferroelectricity of liquid crystals.","rect":[53.812843322753909,426.9060363769531,385.0969509651828,417.97149658203127]},{"page":10,"text":"Each of such meetings attracts hundreds of participants, but of different profiles:","rect":[53.812843322753909,438.8656311035156,385.1596513516624,429.93109130859377]},{"page":10,"text":"chemists, physicists, engineers for radio- and optoelectronics, biologists and phy-","rect":[53.812843322753909,450.8251647949219,385.1288083633579,441.890625]},{"page":10,"text":"sicians.","rect":[53.812843322753909,460.7228698730469,83.52814908530002,453.85015869140627]},{"page":10,"text":"In recent years a group of several excellent top-level books have been published","rect":[65.76486206054688,474.6874694824219,385.17556086591255,465.7529296875]},{"page":10,"text":"on the physics of liquid crystals and many others, dealing with particular problems","rect":[53.812843322753909,486.64703369140627,385.15167631255346,477.71246337890627]},{"page":10,"text":"related to physics of liquid crystals. Popular books on liquid crystals are very","rect":[53.812843322753909,498.6065979003906,385.11489192060005,489.67205810546877]},{"page":10,"text":"scarce; only three of them are mentioned in the list presented in Chapter 1.","rect":[53.812843322753909,510.5661315917969,385.1716579964328,501.631591796875]},{"page":10,"text":"Evidently, there is a huge gap between the first group of books and the second.","rect":[53.812843322753909,522.5256958007813,385.1726040413547,513.5911865234375]},{"page":10,"text":"The monographs have been written by theoreticians at a very high level using the","rect":[53.812843322753909,534.4852294921875,385.1706012554344,525.5507202148438]},{"page":10,"text":"advanced","rect":[53.812843322753909,545.0,91.61794367841253,537.51025390625]},{"page":10,"text":"mathematical","rect":[96.91856384277344,545.0,151.005751205178,537.51025390625]},{"page":10,"text":"apparatus","rect":[156.33822631835938,546.4447631835938,194.65594877349094,538.5261840820313]},{"page":10,"text":"of","rect":[199.95458984375,545.0,208.24642310945166,537.51025390625]},{"page":10,"text":"modern","rect":[213.54904174804688,545.0,244.10250178388129,537.51025390625]},{"page":10,"text":"physics.","rect":[249.40512084960938,546.4447631835938,281.81402249838598,537.51025390625]},{"page":10,"text":"The","rect":[287.1305847167969,545.0,302.6691058941063,537.51025390625]},{"page":10,"text":"popular","rect":[307.9757080078125,546.4447631835938,338.5251706680454,537.51025390625]},{"page":10,"text":"books","rect":[343.8317565917969,545.0,367.5725442324753,537.51025390625]},{"page":10,"text":"are","rect":[372.94683837890627,545.0,385.16065252496568,539.0]},{"page":10,"text":"written vividly without a single formula. If we consider the books as training","rect":[53.812843322753909,558.404296875,385.14772883466255,549.4697875976563]},{"page":10,"text":"devices, the second group is designed for children’s school sports, the first for","rect":[53.812843322753909,570.3070678710938,385.1776059707798,561.37255859375]},{"page":10,"text":"Olympians. But what about the intermediate levels?","rect":[53.812843322753909,582.2666015625,263.25937689020005,573.3121337890625]},{"page":10,"text":"ix","rect":[378.55963134765627,621.958984375,385.1677297988407,616.1848754882813]},{"page":11,"text":"x","rect":[53.812843322753909,42.454345703125,58.04337829737588,38.661399841308597]},{"page":11,"text":"Foreword","rect":[352.33209228515627,42.54747772216797,385.1864370742313,36.68026351928711]},{"page":11,"text":"This is the gap I would like to try to fill. The book proposed to bridge the gap has","rect":[65.76496887207031,68.2883529663086,385.1289407168503,59.35380554199219]},{"page":11,"text":"been written by an experimentalist who, through all his life, has tried to understand","rect":[53.812950134277347,80.24788665771485,385.14589777997505,71.31333923339844]},{"page":11,"text":"and explain to his students the complexity of liquid crystal physics using either","rect":[53.812950134277347,92.20748138427735,385.0841306289829,83.27293395996094]},{"page":11,"text":"simple analogies or going back to the very first principles we have studied in middle","rect":[53.812950134277347,104.11019134521485,385.1647113628563,95.17564392089844]},{"page":11,"text":"and high schools. In this book there is no sentence starting with “It is easy to","rect":[53.812950134277347,116.0697250366211,385.1398858170844,107.13517761230469]},{"page":11,"text":"show....”; either it has been shown, or explained by simple analogy. In fact I only","rect":[53.812950134277347,128.02932739257813,385.1259392838813,119.09477233886719]},{"page":11,"text":"use mathematics at the level of engineering high school. In those cases when I need","rect":[53.81294631958008,139.98886108398438,385.11797419599068,131.05430603027345]},{"page":11,"text":"something more (for example, the Fourier transform, tensor algebra or variation","rect":[53.81294631958008,151.94839477539063,385.0900811295844,143.0138397216797]},{"page":11,"text":"calculus) I carefully explain all the details. In addition there are about 300 drawings","rect":[53.81294631958008,163.907958984375,385.16977323638158,154.97340393066407]},{"page":11,"text":"clarifying the text. The aim of the book is modest: it is to introduce to a reader the","rect":[53.81294631958008,175.86749267578126,385.16977728082505,166.9329376220703]},{"page":11,"text":"most important ideas related to the structure and physical properties of liquid","rect":[53.81294631958008,187.8270263671875,385.11199275067818,178.89247131347657]},{"page":11,"text":"crystals, including some of the theoretical aspects. The book is intended for a","rect":[53.81294631958008,199.72976684570313,385.15686834527818,190.7952117919922]},{"page":11,"text":"wide spectrum of scientists, including experimental physicists, physical chemists,","rect":[53.81294631958008,211.6893310546875,385.1816677620578,202.75477600097657]},{"page":11,"text":"engineers, and especially, for undergraduate students and Ph.D. students.","rect":[53.81294631958008,223.64886474609376,347.88628812338598,214.7143096923828]},{"page":11,"text":"The book consists of three parts: Structure, Physical Properties, and Electro-","rect":[65.76496887207031,235.60842895507813,385.1060727676548,226.6738739013672]},{"page":11,"text":"Optics of liquid crystals. Of course, I am aware that electro-optical properties may","rect":[53.81294631958008,247.56796264648438,385.1637810807563,238.6134796142578]},{"page":11,"text":"be regarded as physical properties. However they are particularly relevant for","rect":[53.81294631958008,259.52752685546877,385.1190122207798,250.5929718017578]},{"page":11,"text":"modern technology and correspond more to the author’s own interests. For these","rect":[53.81294631958008,271.4870910644531,385.12595403863755,262.55255126953127]},{"page":11,"text":"reasons, electro-optic properties deserve a more honorable position. In the Part I,","rect":[53.81294631958008,283.4466247558594,385.1518215706516,274.5120849609375]},{"page":11,"text":"after a brief introduction, there is a short first chapter devoted to symmetry, the","rect":[53.81294631958008,295.3493957519531,385.1707233257469,286.41485595703127]},{"page":11,"text":"concept used throughout the book. In Chapter 2 we discuss the molecular aspects","rect":[53.81294631958008,307.3089599609375,385.18466581450658,298.3743896484375]},{"page":11,"text":"and the fundamental issue for all liquid crystal phases (or mesophases), the problem","rect":[53.81294631958008,319.26849365234377,385.1737131941374,310.33392333984377]},{"page":11,"text":"of the orientational distribution of molecules. In Chapter 3 there is a general","rect":[53.81294631958008,331.2280578613281,385.0871415860374,322.29351806640627]},{"page":11,"text":"description of the most important liquid crystal phases, beginning with the nematic","rect":[53.81294631958008,343.1875915527344,385.0990680523094,334.2530517578125]},{"page":11,"text":"phase and ending with chiral and achiral ferroelectric phases. After reading that","rect":[53.81294631958008,355.1471862792969,385.13691575595927,346.212646484375]},{"page":11,"text":"chapter, the reader who only wishes to make a slight acquaintance with liquid","rect":[53.81294631958008,367.1067199707031,385.1110467057563,358.17218017578127]},{"page":11,"text":"crystals may quit or, at least, have a rest.","rect":[53.81294631958008,379.0662536621094,218.70637174155002,370.1317138671875]},{"page":11,"text":"Chapter 4 will introduce the reader to the basic concepts of the X-ray analysis of","rect":[65.76496887207031,390.9690246582031,385.14788184968605,382.03448486328127]},{"page":11,"text":"crystals and its applications to particular liquid crystal phases. It should be noted","rect":[53.81294631958008,402.9285583496094,385.16372004560005,393.9940185546875]},{"page":11,"text":"that in the present literature this problem is not adequately dealt with anywhere, and","rect":[53.81294631958008,414.8880920410156,385.14186945966255,405.95355224609377]},{"page":11,"text":"this chapter attempts to rectify this deficiency. Chapter 5 covers phase transitions,","rect":[53.81294631958008,426.8476257324219,385.1249966194797,417.8533020019531]},{"page":11,"text":"one of the key problems of the liquid crystal physics, and which has been widely","rect":[53.81294631958008,438.8071594238281,385.1647576432563,429.87261962890627]},{"page":11,"text":"discussed in other texts at very different levels. In this chapter I give only a detailed","rect":[53.81294631958008,450.7666931152344,385.09706965497505,441.8321533203125]},{"page":11,"text":"explanation of the basic concepts of the phase transitions between most important","rect":[53.81294631958008,462.7262268066406,385.1746660000999,453.79168701171877]},{"page":11,"text":"mesophases.","rect":[53.81294631958008,474.6857604980469,103.95130582358127,465.751220703125]},{"page":11,"text":"Chapter 6 heralds the second part of the book and introduces the reader to","rect":[65.76496887207031,486.5885314941406,385.1398858170844,477.65399169921877]},{"page":11,"text":"anisotropy of the magnetic and electric properties of mesophases. Following in","rect":[53.81294631958008,498.5480651855469,385.13884821942818,489.613525390625]},{"page":11,"text":"Chapter 7 there is a focus on the anisotropy of transport properties, especially of","rect":[53.81294631958008,510.507568359375,385.14788184968605,501.572998046875]},{"page":11,"text":"electrical conductivity. Without these two chapters (Chapters 6 and 7), it would be","rect":[53.81294631958008,522.4671020507813,385.1488727398094,513.5325927734375]},{"page":11,"text":"impossible to discuss electro-optical properties in the third section of the book.","rect":[53.81294631958008,534.4266357421875,385.1488308479953,525.4921264648438]},{"page":11,"text":"Further, Chapters 7 and 8 deal with the anisotropy of the properties of elasticity and","rect":[53.81294631958008,546.3861694335938,385.14092341474068,537.45166015625]},{"page":11,"text":"viscosity. Chapter 8 is more difficult than the others, and in order to present the","rect":[53.81294631958008,558.345703125,385.17273748590318,549.4111938476563]},{"page":11,"text":"theoretical results as clearly as possible, the focus is on the experimental methods","rect":[53.81294631958008,570.3052368164063,385.0931435977097,561.3707275390625]},{"page":11,"text":"for the determination of Leslie viscosity coefficients from the viscous stress tensor","rect":[53.81294631958008,582.2079467773438,385.11699806062355,573.2734375]},{"page":11,"text":"of the nematic phase. Chapter 9 terminates the discussion of the anisotropy of","rect":[53.81294631958008,594.16748046875,385.1468442520298,585.2329711914063]},{"page":12,"text":"Foreword","rect":[53.812843322753909,42.54747772216797,86.66718048243448,36.68026351928711]},{"page":12,"text":"xi","rect":[378.5588073730469,42.454345703125,385.1668978139407,36.68026351928711]},{"page":12,"text":"physical properties. Here, the case in point is the interaction of liquid crystals,","rect":[53.812843322753909,68.2883529663086,385.0978970101047,59.35380554199219]},{"page":12,"text":"mostly nematics, with a solid substrate. The problems of interfaces, especially,","rect":[53.812843322753909,80.24788665771485,385.1567349007297,71.31333923339844]},{"page":12,"text":"surface polarization and anchoring conditions occupies the central place here and","rect":[53.812843322753909,92.20748138427735,385.14077082685005,83.27293395996094]},{"page":12,"text":"the chapter is, in fact, a bridge between the second and third parts of the book.","rect":[53.812843322753909,104.11019134521485,370.9878811409641,95.17564392089844]},{"page":12,"text":"Finally the three remaining Chapters 10–12 are devoted to optics and electro-","rect":[65.76486206054688,116.0697250366211,385.1786130508579,107.13517761230469]},{"page":12,"text":"optics of, respectively, nematic, cholesteric and smectic (ferroelectric and antiferro-","rect":[53.812843322753909,128.02932739257813,385.14873634187355,119.09477233886719]},{"page":12,"text":"electric) phases. In contrast to my earlier book published by Wiley in 1983, only the","rect":[53.812843322753909,139.98886108398438,385.17050970270005,131.05430603027345]},{"page":12,"text":"most principal effects have been considered and the discussion of the underlying","rect":[53.812843322753909,151.94839477539063,385.14772883466255,143.0138397216797]},{"page":12,"text":"principles is much more detailed.","rect":[53.812843322753909,163.907958984375,188.08614774252659,154.97340393066407]},{"page":12,"text":"Throughout the book the Gauss system of units is used, although all numerical","rect":[65.76486206054688,175.86749267578126,385.1477494961936,166.9329376220703]},{"page":12,"text":"estimates of quantities have been made in both systems, Gauss and International","rect":[53.812843322753909,187.8270263671875,385.1626115567405,178.89247131347657]},{"page":12,"text":"(SI). The referenced bibliography is rather small, because I deliberately included","rect":[53.812843322753909,199.72976684570313,385.17568293622505,190.7952117919922]},{"page":12,"text":"only books, review articles and the seminal papers that paved the way for further","rect":[53.812843322753909,211.6893310546875,385.0820249160923,202.75477600097657]},{"page":12,"text":"investigations. All these literature sources are presented with their titles.","rect":[53.812843322753909,223.64886474609376,345.2293362190891,214.7143096923828]},{"page":12,"text":"This book was written over a long period of 10 years before and during my","rect":[65.76486206054688,235.60842895507813,385.1337823014594,226.6738739013672]},{"page":12,"text":"teaching course (2003–2009) of liquid crystal physics to Ph.D. students in Calabria","rect":[53.812843322753909,247.56796264648438,385.1427387066063,238.63340759277345]},{"page":12,"text":"University (CU) (Italy). Among the students there were not only physicists but","rect":[53.812843322753909,259.52752685546877,385.1338029629905,250.5929718017578]},{"page":12,"text":"chemists and engineers and even biologists. I have tried to make my course serious,","rect":[53.812843322753909,271.4870910644531,385.1785854866672,262.55255126953127]},{"page":12,"text":"simple and interesting, but it is for others to decide if I have succeeded. I am","rect":[53.812843322753909,283.4466247558594,385.1716379988249,274.5120849609375]},{"page":12,"text":"indebted to Prof. Roberto Bartolino for his invitation to work in Italy and to his","rect":[53.812843322753909,295.3493957519531,385.17856229888158,286.41485595703127]},{"page":12,"text":"co-workers (Profs. G. Cipparrone, R. Barberi, C. Umeton, C. Versace, G. Strangi","rect":[53.812843322753909,307.3089599609375,385.1019731290061,298.3743896484375]},{"page":12,"text":"and Drs. M. de Santo, A. Mazzulla, P. Pagliusi, F. Ciuchi, M. Giocondo and many","rect":[53.812843322753909,319.26849365234377,385.15569392255318,310.33392333984377]},{"page":12,"text":"others) who were always friendly and attentive to any of my problems and from","rect":[53.812843322753909,331.2280578613281,385.13480328202805,322.29351806640627]},{"page":12,"text":"whom I learned a lot of new things concerning both science and life. I would like to","rect":[53.812843322753909,343.1875915527344,385.13979426435005,334.2530517578125]},{"page":12,"text":"express also many thanks to my coworkers from the Institute of Crystallography,","rect":[53.812843322753909,355.1471862792969,385.1098904183078,346.212646484375]},{"page":12,"text":"Russian Academy of Sciences Drs. M.I. Barnik, V.V. Lazarev, S.P. Palto, B.I.","rect":[53.812843322753909,367.1067199707031,385.1487087776828,358.17218017578127]},{"page":12,"text":"Ostrovsky, N.M. Shtykov, B.A. Umansky, S.V. Yablonsky and S.G. Yudin with","rect":[53.812843322753909,379.0662536621094,385.1138543229438,370.1117858886719]},{"page":12,"text":"whom I had the pleasure to work on liquid crystals for many years and have this","rect":[53.812843322753909,390.9690246582031,385.1397744570847,382.03448486328127]},{"page":12,"text":"pleasure now. I am always thankful to my friends-colleagues Guram Chilaya,","rect":[53.812843322753909,402.9285583496094,385.1019253304172,393.9940185546875]},{"page":12,"text":"Dietrich","rect":[53.812843322753909,413.0,86.63386622479925,405.95355224609377]},{"page":12,"text":"Demus,","rect":[92.5008773803711,413.0,123.25540586020236,406.1527404785156]},{"page":12,"text":"Elizabeth","rect":[129.0935516357422,413.0,167.00215998700629,405.95355224609377]},{"page":12,"text":"Dubois-Violette,","rect":[172.87911987304688,413.0,239.76845975424534,405.95355224609377]},{"page":12,"text":"George","rect":[245.610595703125,414.8880920410156,275.00537908746568,406.0133056640625]},{"page":12,"text":"Durand,","rect":[280.9002685546875,413.0,313.2315640022922,405.95355224609377]},{"page":12,"text":"David","rect":[319.1353759765625,413.0,343.56301203778755,405.95355224609377]},{"page":12,"text":"Dunmur,","rect":[349.44000244140627,413.0,385.1785854866672,406.1527404785156]},{"page":12,"text":"George Gray, Etienne Guyon, Wolfgang Haase, Wim de Jeu, Efim Kats, Mikhail","rect":[53.812843322753909,426.8476257324219,385.1228166348655,417.9130859375]},{"page":12,"text":"Osipov, Alexander Petrov, Sergei Pikin, Ludwig Pohl, Jacques Prost, and Katsumi","rect":[53.812843322753909,438.8071594238281,385.08296067783427,429.8526916503906]},{"page":12,"text":"Yoshino for fruitful discussions of many topics related and more frequently not","rect":[53.812843322753909,450.7666931152344,385.13178880283427,441.8321533203125]},{"page":12,"text":"related to liquid crystals but making our life in science more colorful.","rect":[53.812843322753909,462.7262268066406,334.9884914925266,453.79168701171877]},{"page":13,"text":".","rect":[53.812843322753909,580.2719116210938,56.799101542947628,579.0049438476563]},{"page":14,"text":"Contents","rect":[53.812843322753909,70.80348205566406,114.65351499469785,59.32891082763672]},{"page":14,"text":"1 Introductory Notes ............................................................1","rect":[53.812843322753909,230.00042724609376,385.19158259442818,220.58778381347657]},{"page":14,"text":"References.......................................................................4","rect":[68.76704406738281,239.49972534179688,385.19463435224068,232.6269989013672]},{"page":14,"text":"Part I Structure of Liquid Crystals","rect":[53.81283950805664,271.8304138183594,211.15208829985813,262.36798095703127]},{"page":14,"text":"2","rect":[53.81283950805664,293.2095947265625,58.789944563422299,286.426513671875]},{"page":14,"text":"Symmetry ......................................................................7","rect":[68.76705169677735,295.74951171875,385.1946648698188,286.326904296875]},{"page":14,"text":"2.1 Point Group Symmetry.....................................................7","rect":[68.76704406738281,307.3106689453125,385.1905449967719,298.3760986328125]},{"page":14,"text":"2.1.1 Symmetry Elements and Operations................................7","rect":[86.21377563476563,319.27020263671877,385.18547907880318,310.31573486328127]},{"page":14,"text":"2.1.2 Groups.............................................................. 10","rect":[86.21379089355469,331.2297668457031,385.18462458661568,322.35498046875]},{"page":14,"text":"2.1.3 Point Groups........................................................ 12","rect":[86.21378326416016,343.1893005371094,385.1835869889594,334.2547607421875]},{"page":14,"text":"2.1.4 Continuous Point Groups........................................... 14","rect":[86.21377563476563,355.1488342285156,385.18047419599068,346.21429443359377]},{"page":14,"text":"2.2 Translational Symmetry.................................................. 15","rect":[68.76702117919922,367.05157470703127,385.18254939130318,358.0572509765625]},{"page":14,"text":"References..................................................................... 18","rect":[68.76702117919922,376.9891662597656,385.1866387467719,370.07659912109377]},{"page":14,"text":"3","rect":[53.81282043457031,401.1074523925781,58.78992548993597,393.67694091796877]},{"page":14,"text":"Mesogenic Molecules and Orientational Order ..........................","rect":[68.76702880859375,403.0099182128906,370.4602322151828,393.9060363769531]},{"page":14,"text":"3.1 Molecular Shape and Properties .........................................","rect":[68.76701354980469,414.8897705078125,370.46319242026098,405.9552001953125]},{"page":14,"text":"3.1.1 Shape, Conformational Mobility and Isomerization...............","rect":[86.21379089355469,426.8493347167969,372.9497952034641,417.914794921875]},{"page":14,"text":"3.1.2 Symmetry and Chirality............................................","rect":[86.21379089355469,438.8088684082031,372.9567532112766,429.87432861328127]},{"page":14,"text":"3.1.3 Electric and Magnetic Properties ..................................","rect":[86.21379089355469,450.7684326171875,372.9547390511203,441.8338623046875]},{"page":14,"text":"3.2 Intermolecular Interactions...............................................","rect":[68.76703643798828,460.64923095703127,370.4652065804172,453.73663330078127]},{"page":14,"text":"3.3 Orientational Distribution Functions for Molecules.....................","rect":[68.76703643798828,472.6087951660156,370.4592556526828,465.6763000488281]},{"page":14,"text":"3.3.1 Molecules with Axial Symmetry...................................","rect":[86.21379089355469,486.5903015136719,372.9547390511203,477.65576171875]},{"page":14,"text":"3.3.2 Lath-Like Molecules ...............................................","rect":[86.21379089355469,496.5278625488281,372.9577297737766,489.61529541015627]},{"page":14,"text":"3.4 Principal Orientational Order Parameter (Microscopic Approach)......","rect":[68.76703643798828,510.5093994140625,370.45525784994848,501.554931640625]},{"page":14,"text":"3.5 Macroscopic Definition of the Orientational Order Parameter..........","rect":[68.76703643798828,522.4689331054688,370.4562649300266,513.4746704101563]},{"page":14,"text":"3.5.1 Tensor Properties...................................................","rect":[86.2137680053711,534.428466796875,372.9587673714328,525.4342041015625]},{"page":14,"text":"3.5.2 Uniaxial Order......................................................","rect":[86.2137680053711,544.3660278320313,372.95879788901098,537.3937377929688]},{"page":14,"text":"3.5.3 Microscopic Biaxiality .............................................","rect":[86.2137680053711,558.2908325195313,372.9566921761203,549.2965698242188]},{"page":14,"text":"3.6 Apparent Order Parameters for Flexible Chains.........................","rect":[68.76703643798828,570.2503662109375,370.4592556526828,561.2958984375]},{"page":14,"text":"References.....................................................................","rect":[68.76703643798828,580.1481323242188,370.4703640511203,573.275390625]},{"page":14,"text":"19","rect":[375.2223205566406,400.9879455566406,385.17653742841255,394.0554504394531]},{"page":14,"text":"19","rect":[375.22528076171877,412.9474792480469,385.17949763349068,406.0149841308594]},{"page":14,"text":"19","rect":[375.2193298339844,424.90704345703127,385.1735467057563,417.97454833984377]},{"page":14,"text":"21","rect":[375.226318359375,436.6673583984375,385.1805047135688,429.93408203125]},{"page":14,"text":"22","rect":[375.22430419921877,448.6269226074219,385.17852107099068,441.8936462402344]},{"page":14,"text":"24","rect":[375.2273254394531,460.5296936035156,385.1815117936469,453.7964172363281]},{"page":14,"text":"28","rect":[375.2213439941406,472.6087951660156,385.17556086591255,465.7559814453125]},{"page":14,"text":"29","rect":[375.22430419921877,484.64801025390627,385.17852107099068,477.71551513671877]},{"page":14,"text":"32","rect":[375.227294921875,496.5278625488281,385.1815117936469,489.675048828125]},{"page":14,"text":"33","rect":[375.21734619140627,508.4874267578125,385.17156306317818,501.6346130371094]},{"page":14,"text":"35","rect":[375.2183532714844,520.4469604492188,385.17253962567818,513.4746704101563]},{"page":14,"text":"35","rect":[375.22833251953127,532.406494140625,385.18254939130318,525.4342041015625]},{"page":14,"text":"36","rect":[375.2283630371094,544.3660278320313,385.18254939130318,537.4534912109375]},{"page":14,"text":"38","rect":[375.22625732421877,556.2688598632813,385.18047419599068,549.4160766601563]},{"page":14,"text":"39","rect":[375.2213439941406,568.30810546875,385.17556086591255,561.3756103515625]},{"page":14,"text":"39","rect":[375.2324523925781,580.2676391601563,385.18666926435005,573.3351440429688]},{"page":14,"text":"xiii","rect":[373.8014831542969,621.958984375,385.1647005483157,616.1848754882813]},{"page":15,"text":"xiv","rect":[53.812843322753909,42.52207565307617,64.6514753737919,36.68026351928711]},{"page":15,"text":"4","rect":[53.812843322753909,66.26025390625,58.78994837811956,59.477169036865237]},{"page":15,"text":"5","rect":[53.8138542175293,448.90789794921877,58.79095927289495,441.9554748535156]},{"page":15,"text":"Contents","rect":[355.05145263671877,42.55594253540039,385.17285616874019,36.73106384277344]},{"page":15,"text":"Liquid Crystal Phases ......................................................","rect":[68.76705169677735,68.80017852783203,370.4672512581516,59.3775634765625]},{"page":15,"text":"4.1 Polymorphism Studies ..................................................","rect":[68.76805114746094,80.36128997802735,370.4672512581516,71.42674255371094]},{"page":15,"text":"4.1.1 Polarized Light Microscopy......................................","rect":[91.19986724853516,92.3208236694336,370.4641995003391,83.38627624511719]},{"page":15,"text":"4.1.2 Differential Scanning and Adiabatic Calorimetry","rect":[91.19986724853516,104.2804183959961,313.0403985123969,95.34587097167969]},{"page":15,"text":"(DSC and AC)....................................................","rect":[116.12321472167969,115.8415298461914,370.4663967659641,107.30540466308594]},{"page":15,"text":"4.1.3 X-Ray Analysis...................................................","rect":[91.19986724853516,128.19949340820313,370.4672512581516,119.26493835449219]},{"page":15,"text":"4.2 Main Calamitic Phases..................................................","rect":[68.76805114746094,138.1370391845703,370.4672512581516,131.22447204589845]},{"page":15,"text":"4.2.1 Nematic Phase....................................................","rect":[91.19986724853516,150.09657287597657,370.4672512581516,143.1840057373047]},{"page":15,"text":"4.2.2 Classical Smectic A Phase .......................................","rect":[91.19986724853516,161.9993438720703,370.4641995003391,155.08677673339845]},{"page":15,"text":"4.2.3 Special SmA Phases..............................................","rect":[91.1998519897461,175.98086547851563,370.4662136604953,167.0463104248047]},{"page":15,"text":"4.2.4 Smectic C Phase..................................................","rect":[91.1998519897461,185.91844177246095,370.46722074057348,179.00587463378907]},{"page":15,"text":"4.2.5 Smectic B.........................................................","rect":[91.1998519897461,197.8779754638672,370.4692654183078,190.9056396484375]},{"page":15,"text":"4.3 Discotic, Bowl-Type and Polyphilic Phases............................","rect":[68.76903533935547,211.85952758789063,370.4612698128391,202.9249725341797]},{"page":15,"text":"4.4 Role of Polymerization .................................................","rect":[68.76902770996094,223.81903076171876,370.46722074057348,214.8844757080078]},{"page":15,"text":"4.5 Lyotropic Phases........................................................","rect":[68.76902770996094,235.778564453125,370.4692654183078,226.78424072265626]},{"page":15,"text":"4.6 General Remarks on the Role of Chirality .............................","rect":[68.76902770996094,247.73812866210938,370.46319242026098,238.80357360839845]},{"page":15,"text":"4.7 Cholesterics .............................................................","rect":[68.76902770996094,257.6189270019531,370.4703030159641,250.7063446044922]},{"page":15,"text":"4.7.1 Intermolecular Potential..........................................","rect":[91.1998519897461,269.5784606933594,370.4651760628391,262.6658935546875]},{"page":15,"text":"4.7.2 Cholesteric Helix and Tensor of Orientational Order ...........","rect":[91.1998519897461,281.53802490234377,370.4582180550266,274.60552978515627]},{"page":15,"text":"4.7.3 Tensor of Dielectric Anisotropy .................................","rect":[91.1998519897461,295.51953125,370.4632229378391,286.5849609375]},{"page":15,"text":"4.7.4 Grandjean 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III Electro-Optics","rect":[53.81377029418945,432.89947509765627,159.19301237212376,423.94500732421877]},{"page":18,"text":"11","rect":[53.81377029418945,454.7467956542969,63.76797953656683,447.8740539550781]},{"page":18,"text":"Optics and Electric Field Effects in Nematic and SmecticA","rect":[73.7530517578125,456.8185729980469,327.47399663880199,447.82427978515627]},{"page":18,"text":"Liquid Crystals ...........................................................","rect":[73.75304412841797,469.2462158203125,366.7304348519016,459.8236083984375]},{"page":18,"text":"11.1 Optical Properties of Uniaxial Phases ................................","rect":[73.75304412841797,480.8073425292969,367.9706081917453,471.8528747558594]},{"page":18,"text":"11.1.1 Dielectric Ellipsoid, Birefringence and","rect":[96.18485260009766,492.7668762207031,282.16436091474068,483.83233642578127]},{"page":18,"text":"Light Transmission............................................","rect":[126.15000915527344,504.6696472167969,367.9735989144016,495.735107421875]},{"page":18,"text":"11.1.2 Light Absorption and Linear Dichroism......................","rect":[96.18485260009766,516.629150390625,367.9685940315891,507.69464111328127]},{"page":18,"text":"11.1.3 Light Scattering in Nematics and Smectic A.................","rect":[96.1848373413086,528.5886840820313,367.9666714241672,519.6541748046875]},{"page":18,"text":"11.2 Frederiks Transition and Related Phenomena........................","rect":[73.75304412841797,538.5162963867188,367.9685940315891,531.6137084960938]},{"page":18,"text":"11.2.1 Field Free Energy and Torques...............................","rect":[96.1848373413086,552.5077514648438,367.9705776741672,543.5732421875]},{"page":18,"text":"11.2.2 Experiments on Field Alignment of a Nematic ..............","rect":[96.1848373413086,564.46728515625,367.9666409065891,555.5327758789063]},{"page":18,"text":"11.2.3 Theory of Frederiks Transition ...............................","rect":[96.1848373413086,576.4268188476563,367.9705776741672,567.4923095703125]},{"page":18,"text":"11.2.4 Generalizations of the Simplest Model.......................","rect":[96.1848373413086,588.3863525390625,367.9685940315891,579.4518432617188]},{"page":18,"text":"285","rect":[370.24627685546877,466.8258361816406,385.1775750260688,459.8534851074219]},{"page":18,"text":"285","rect":[370.24017333984377,478.7853698730469,385.1714715104438,471.8130187988281]},{"page":18,"text":"285","rect":[370.2431640625,502.6476745605469,385.17446223310005,495.6753234863281]},{"page":18,"text":"294","rect":[370.2381591796875,514.6868896484375,385.16948786786568,507.75439453125]},{"page":18,"text":"299","rect":[370.2362060546875,526.6464233398438,385.16753474286568,519.7139282226563]},{"page":18,"text":"304","rect":[370.2381591796875,538.5262451171875,385.16948786786568,531.6734619140625]},{"page":18,"text":"304","rect":[370.2401428222656,550.4857788085938,385.17144099286568,543.6329956054688]},{"page":18,"text":"306","rect":[370.2361755371094,562.4453125,385.16750422528755,555.5327758789063]},{"page":18,"text":"307","rect":[370.2401428222656,574.4048461914063,385.17144099286568,567.5520629882813]},{"page":18,"text":"312","rect":[370.2381591796875,586.3643798828125,385.16948786786568,579.5115966796875]},{"page":19,"text":"xviii","rect":[53.812843322753909,42.52207565307617,69.40659813132349,36.68026351928711]},{"page":19,"text":"Contents","rect":[355.05145263671877,42.55594253540039,385.17285616874019,36.73106384277344]},{"page":19,"text":"11.2.5 Dynamics of Frederiks Transition ............................","rect":[96.18343353271485,68.4017562866211,367.96819730307348,59.407447814941409]},{"page":19,"text":"11.2.6 Backflow Effect...............................................","rect":[96.18343353271485,78.33930969238281,367.9731716683078,71.42674255371094]},{"page":19,"text":"11.2.7 Electrooptical Response ......................................","rect":[96.18343353271485,92.3208236694336,367.9711880257297,83.38627624511719]},{"page":19,"text":"11.3 Flexoelectricity........................................................","rect":[73.75263214111328,104.2804183959961,367.9752468636203,95.34587097167969]},{"page":19,"text":"11.3.1 Flexoelectric Polarization.....................................","rect":[96.1834487915039,114.21797180175781,367.9701809456516,107.30540466308594]},{"page":19,"text":"11.3.2 Converse Flexoelectric Effect ................................","rect":[96.1834487915039,126.17750549316406,367.9692043831516,119.26493835449219]},{"page":19,"text":"11.3.3 Flexoelectric Domains ........................................","rect":[96.1834487915039,138.1370391845703,367.9711880257297,131.22447204589845]},{"page":19,"text":"11.4 Electrohydrodynamic Instability......................................","rect":[73.75163269042969,152.11856079101563,367.9701809456516,143.1840057373047]},{"page":19,"text":"11.4.1 The Reasons for Instabilities..................................","rect":[96.18243408203125,161.95950317382813,367.9692043831516,155.08677673339845]},{"page":19,"text":"11.4.2 Carr-Helfrich Mode...........................................","rect":[96.18243408203125,173.95887756347657,367.9711575081516,167.0463104248047]},{"page":19,"text":"References...................................................................","rect":[73.75162506103516,185.87860107421876,367.9762844612766,179.00587463378907]},{"page":19,"text":"315","rect":[370.23773193359377,66.37977600097656,385.1690606217719,59.407447814941409]},{"page":19,"text":"316","rect":[370.24273681640627,78.33930969238281,385.1740655045844,71.42674255371094]},{"page":19,"text":"318","rect":[370.24072265625,90.29884338378906,385.17205134442818,83.44603729248047]},{"page":19,"text":"322","rect":[370.24481201171877,102.25843811035156,385.1761406998969,95.40563201904297]},{"page":19,"text":"322","rect":[370.23974609375,114.21797180175781,385.17104426435005,107.36516571044922]},{"page":19,"text":"327","rect":[370.23876953125,126.17750549316406,385.17006770185005,119.32469940185547]},{"page":19,"text":"332","rect":[370.24072265625,138.1370391845703,385.17205134442818,131.28424072265626]},{"page":19,"text":"334","rect":[370.23974609375,150.09657287597657,385.17104426435005,143.2437744140625]},{"page":19,"text":"334","rect":[370.23876953125,161.9993438720703,385.17006770185005,155.14654541015626]},{"page":19,"text":"336","rect":[370.24072265625,173.95887756347657,385.17202082685005,167.0463104248047]},{"page":19,"text":"340","rect":[370.245849609375,185.91844177246095,385.17717829755318,179.06564331054688]},{"page":19,"text":"12","rect":[53.81234359741211,209.718017578125,63.76655283978948,202.8452911376953]},{"page":19,"text":"Electro-Optical Effects in Cholesteric Phase ...........................","rect":[73.75161743164063,211.789794921875,366.7209438851047,202.7954864501953]},{"page":19,"text":"12.1 Cholesteric as One-Dimensional Photonic Crystal...................","rect":[73.7506332397461,223.81903076171876,367.9652370979953,214.8645477294922]},{"page":19,"text":"12.1.1 Bragg Reflection ..............................................","rect":[96.18244171142578,235.778564453125,367.9711880257297,226.84400939941407]},{"page":19,"text":"12.1.2 Waves in Layered Medium and Photonic Crystals...........","rect":[96.18244171142578,247.73812866210938,367.9632229378391,238.80357360839845]},{"page":19,"text":"12.1.3 Simple Analytical Solution for Light Incident Parallel","rect":[96.18244171142578,259.6408996582031,343.6580644375999,250.7063446044922]},{"page":19,"text":"to the Helical Axis ............................................","rect":[126.1476058959961,269.5386047363281,367.9711575081516,262.6658935546875]},{"page":19,"text":"12.1.4 Other Important Cases ........................................","rect":[96.18244171142578,283.55999755859377,367.9701809456516,274.60552978515627]},{"page":19,"text":"12.2 Dielectric Instability of Cholesterics .................................","rect":[73.75062561035156,295.51953125,367.96819730307348,286.5849609375]},{"page":19,"text":"12.2.1 Untwisting of the Cholesteric Helix..........................","rect":[96.18244171142578,307.4790954589844,367.9662136604953,298.5445556640625]},{"page":19,"text":"12.2.2 Field Induced Anharmonicity and Dynamics of the Helix...","rect":[96.18244171142578,319.4386291503906,367.9613003304172,310.50408935546877]},{"page":19,"text":"12.2.3 Instability of the Planar Cholesteric Texture .................","rect":[96.18244171142578,331.398193359375,367.9642605354953,322.463623046875]},{"page":19,"text":"12.3 Bistability and Memory...............................................","rect":[73.75064849853516,343.3577575683594,367.9712185433078,334.4232177734375]},{"page":19,"text":"12.3.1 Naive Idea.....................................................","rect":[96.18147277832031,353.2385559082031,367.9722561409641,346.32598876953127]},{"page":19,"text":"12.3.2 Berreman–Heffner Model.....................................","rect":[96.18147277832031,365.1980895996094,367.9682278206516,358.2855224609375]},{"page":19,"text":"12.3.3 Bistability and Field-Induced Break of Anchoring...........","rect":[96.18147277832031,379.17962646484377,367.9623074104953,370.24505615234377]},{"page":19,"text":"12.4 Flexoelectricity in Cholesterics.......................................","rect":[73.74966430664063,391.13916015625,367.9682278206516,382.20458984375]},{"page":19,"text":"References...................................................................","rect":[73.74966430664063,401.0368957519531,367.9743313362766,394.1641845703125]},{"page":19,"text":"343","rect":[370.23675537109377,209.83753967285157,385.1680840592719,202.9847412109375]},{"page":19,"text":"343","rect":[370.2347717285156,221.7970428466797,385.1661004166938,214.94424438476563]},{"page":19,"text":"343","rect":[370.2407531738281,233.75657653808595,385.17205134442818,226.90377807617188]},{"page":19,"text":"347","rect":[370.2327880859375,245.7161407470703,385.16411677411568,238.86334228515626]},{"page":19,"text":"351","rect":[370.24072265625,269.5784606933594,385.17205134442818,262.6061096191406]},{"page":19,"text":"356","rect":[370.23974609375,281.53802490234377,385.17104426435005,274.565673828125]},{"page":19,"text":"358","rect":[370.2377624511719,293.49755859375,385.1690606217719,286.52520751953127]},{"page":19,"text":"358","rect":[370.2357482910156,305.4571228027344,385.1670769791938,298.4847717285156]},{"page":19,"text":"364","rect":[370.2308654785156,317.4166564941406,385.1621941666938,310.50408935546877]},{"page":19,"text":"366","rect":[370.2337951660156,329.376220703125,385.1651238541938,322.463623046875]},{"page":19,"text":"370","rect":[370.24078369140627,341.3357849121094,385.1721123795844,334.48297119140627]},{"page":19,"text":"370","rect":[370.2418212890625,353.2385559082031,385.17314997724068,346.3857421875]},{"page":19,"text":"371","rect":[370.23779296875,365.1980895996094,385.16909113935005,358.34527587890627]},{"page":19,"text":"375","rect":[370.2318420410156,377.15765380859377,385.1631707291938,370.185302734375]},{"page":19,"text":"376","rect":[370.23779296875,389.1171875,385.16909113935005,382.20458984375]},{"page":19,"text":"378","rect":[370.243896484375,401.0767517089844,385.17522517255318,394.22393798828127]},{"page":19,"text":"13","rect":[53.81038284301758,425.19500732421877,63.76459208539495,417.7644958496094]},{"page":19,"text":"Ferroelectricity and Antiferroelectricity in Smectics ..................","rect":[73.74966430664063,427.41619873046877,366.7170071175266,417.9537353515625]},{"page":19,"text":"13.1 Ferroelectrics..........................................................","rect":[73.74968719482422,436.95538330078127,367.9732937386203,430.04278564453127]},{"page":19,"text":"13.1.1 Crystalline Pyro-, Piezo- and Ferroelectrics..................","rect":[96.18150329589844,450.880126953125,367.9642605354953,441.945556640625]},{"page":19,"text":"13.1.2 Ferroelectric Cells with Non-ferroelectric Liquid Crystal ...","rect":[96.1814956665039,462.8396301269531,367.9593777229953,453.90509033203127]},{"page":19,"text":"13.1.3 Phase Transition SmA*–SmC*...............................","rect":[96.1814956665039,472.7771911621094,367.9672512581516,465.8048400878906]},{"page":19,"text":"13.1.4 Electro-Optic Effects in Ferroelectric Cells ..................","rect":[96.1814956665039,486.7586975097656,367.9642605354953,477.8042297363281]},{"page":19,"text":"13.1.5 Criteria for Bistability and Hysteresis-Free Switching.......","rect":[96.1814956665039,498.7182312011719,367.9613003304172,489.7239074707031]},{"page":19,"text":"13.2 Introduction to Antiferroelectrics.....................................","rect":[73.74970245361328,508.6557922363281,367.9682583382297,501.74322509765627]},{"page":19,"text":"13.2.1 Background: Crystalline Antiferroelectrics","rect":[96.18151092529297,522.6372680664063,297.90558256255346,513.7027587890625]},{"page":19,"text":"and Ferrielectrics..............................................","rect":[126.14667510986328,532.5648803710938,367.9702419808078,525.6622924804688]},{"page":19,"text":"13.2.2 Chiral Liquid Crystalline Antiferroelectrics..................","rect":[96.18151092529297,546.4995727539063,367.96429105307348,537.5650634765625]},{"page":19,"text":"13.2.3 Polar Achiral Systems ........................................","rect":[96.18151092529297,558.4591064453125,367.9692654183078,549.5245971679688]},{"page":19,"text":"References...................................................................","rect":[73.74969482421875,568.3568725585938,367.9743618538547,561.484130859375]},{"page":19,"text":"381","rect":[370.2328186035156,424.9958190917969,385.1641472916938,418.14300537109377]},{"page":19,"text":"381","rect":[370.24285888671877,436.95538330078127,385.1741875748969,430.1025695800781]},{"page":19,"text":"381","rect":[370.2337951660156,448.858154296875,385.1651238541938,442.0053405761719]},{"page":19,"text":"386","rect":[370.22894287109377,460.8176574707031,385.1602410416938,453.90509033203127]},{"page":19,"text":"392","rect":[370.23681640625,472.85687255859377,385.16811457685005,465.92437744140627]},{"page":19,"text":"398","rect":[370.2337951660156,484.81640625,385.1651238541938,477.8839111328125]},{"page":19,"text":"407","rect":[370.2308654785156,496.6962585449219,385.1621941666938,489.84344482421877]},{"page":19,"text":"410","rect":[370.23779296875,508.6557922363281,385.16912165692818,501.802978515625]},{"page":19,"text":"410","rect":[370.23980712890627,532.5748291015625,385.1711358170844,525.7220458984375]},{"page":19,"text":"413","rect":[370.23382568359377,544.4776000976563,385.1651543717719,537.6248168945313]},{"page":19,"text":"423","rect":[370.23883056640627,556.4371337890625,385.1701287370063,549.5843505859375]},{"page":19,"text":"429","rect":[370.2439270019531,568.4763793945313,385.17522517255318,561.5438842773438]},{"page":19,"text":"Index","rect":[53.8104133605957,592.345703125,78.05886927655706,585.4132080078125]},{"page":19,"text":"..........................................................................","rect":[87.45763397216797,591.0,362.99236722494848,587.0]},{"page":19,"text":"433","rect":[370.24700927734377,592.3157958984375,385.1783379655219,585.4630126953125]},{"page":20,"text":"Chapter1","rect":[53.812843322753909,72.10812377929688,114.14115996551633,59.571903228759769]},{"page":20,"text":"Introductory Notes","rect":[53.812843322753909,91.18268585205078,184.55179990680723,76.2657470703125]},{"page":20,"text":"First my middle school teachers and then my high school teachers told me that","rect":[53.812843322753909,211.74758911132813,385.1655717618186,202.8130340576172]},{"page":20,"text":"substances could be in the form of gases, liquids and crystalline solids, between","rect":[53.812843322753909,223.65036010742188,385.2569207291938,214.71580505371095]},{"page":20,"text":"which, on cooling, transitions could occur in the following sequence: gas!","rect":[53.812843322753909,235.60992431640626,385.16957265955946,226.6753692626953]},{"page":20,"text":"liquid ! crystal. And I believed them, although even then liquid crystals had","rect":[53.81181716918945,247.5694580078125,385.2611016373969,238.63490295410157]},{"page":20,"text":"already been around for a respectable time. Today we tell students that if an","rect":[53.81183624267578,259.5290222167969,385.2383660416938,250.59446716308595]},{"page":20,"text":"organic substance consists of rod-like molecules, it may, on cooling, change","rect":[53.81183624267578,271.4885559082031,385.18845403863755,262.55401611328127]},{"page":20,"text":"from a gas to a normal (isotropic) liquid, then into a strange anisotropic liquid","rect":[53.81183624267578,283.44805908203127,385.15163508466255,274.51348876953127]},{"page":20,"text":"(called a nematic liquid crystal), see Fig. 1.1.","rect":[53.81183624267578,295.4076232910156,245.1546597054172,286.47308349609377]},{"page":20,"text":"Further cooling may cause the anisotropic liquid to change into a lamellar","rect":[65.76384735107422,307.3671569824219,385.15670142976418,298.4326171875]},{"page":20,"text":"structure, like a stack of paper, but with thin liquid sheets. Something that is a one-","rect":[53.811824798583987,319.2699279785156,385.2432797989048,310.33538818359377]},{"page":20,"text":"dimensional crystal, but within the stack is a two-dimensional liquid. This is","rect":[53.811824798583987,331.2294921875,385.24228300200658,322.294921875]},{"page":20,"text":"the smectic A phase with molecules standing upright within the layer. Such layers","rect":[53.811824798583987,343.1889953613281,385.2482949648972,334.25445556640627]},{"page":20,"text":"easily slide on each other. These three phases have been identified by Friedel [1]. On","rect":[53.811824798583987,355.1485595703125,385.2432488541938,346.194091796875]},{"page":20,"text":"further cooling the molecules may decide to tilt a little giving rise to the smectic C","rect":[53.81181716918945,367.10809326171877,385.18455764468458,358.17352294921877]},{"page":20,"text":"phase, the tilt angle of which increases with decreasing temperature.","rect":[53.81181716918945,379.067626953125,323.9193996956516,370.133056640625]},{"page":20,"text":"But this is not all. In other substances, further cooling the smectic A phase results","rect":[65.76383972167969,391.0271911621094,385.12485136138158,382.0926513671875]},{"page":20,"text":"in the layers breaking up into hexagons but still sliding easily over each other;this is","rect":[53.81181716918945,402.9867248535156,385.1825295840378,394.05218505859377]},{"page":20,"text":"the smectic Bhex phase. Only at even lower temperatures does the sample acquire a","rect":[53.81181716918945,414.8915710449219,385.15796697809068,405.9371032714844]},{"page":20,"text":"normal crystalline structure. Thus instead of two phase transitions gas-liquid and","rect":[53.813106536865237,426.85113525390627,385.14101496747505,417.91656494140627]},{"page":20,"text":"liquid–crystal we have found four or five transitions between different phases.","rect":[53.813106536865237,438.8106994628906,368.9953274300266,429.87615966796877]},{"page":20,"text":"Other substances manifest other sequences. For instance, in organic compounds","rect":[65.76512908935547,450.7702331542969,385.1071816836472,441.8157653808594]},{"page":20,"text":"having disc-like molecules we find a columnar phase built of liquid molecular","rect":[53.813106536865237,462.7297668457031,385.1500180801548,453.79522705078127]},{"page":20,"text":"columns packed in a two-dimensional crystalline structure. It is a one-dimensional","rect":[53.813106536865237,474.6893005371094,385.2047563321311,465.7547607421875]},{"page":20,"text":"liquid along the columns, and, at the same time, a two-dimensional crystal. An","rect":[53.813106536865237,486.64886474609377,385.1270684342719,477.71429443359377]},{"page":20,"text":"ancient Greek temple with liquid columns would be a good model of the columnar","rect":[53.813106536865237,498.6083984375,385.13701759187355,489.673828125]},{"page":20,"text":"phase. Today we define liquid crystals as fluids with a certain long-range order in","rect":[53.813106536865237,510.5111389160156,385.14101496747505,501.57659912109377]},{"page":20,"text":"their molecular arrangement (i.e. they are anisotropic liquids). Each mesophase is","rect":[53.813106536865237,522.4706420898438,385.18381132231908,513.5361328125]},{"page":20,"text":"a macroscopically uniform intermediate state between an isotropic liquid and a","rect":[53.813106536865237,534.43017578125,385.15598333551255,525.4956665039063]},{"page":20,"text":"crystalline solid. The history of liquid crystals began with the observation by","rect":[53.813106536865237,546.3897705078125,385.1679314713813,537.4552612304688]},{"page":20,"text":"Reinitzer [2] of a strange phase intermediate between the liquid melt and the","rect":[53.813106536865237,558.3493041992188,385.11411321832505,549.414794921875]},{"page":20,"text":"crystalline phase upon heating and cooling cholesteryl benzoate. The samples of","rect":[53.813106536865237,570.308837890625,385.14800391999855,561.3743286132813]},{"page":20,"text":"this compound were sent by Reinitzer to O. Lehmann (Karlsruhe) who was an","rect":[53.813106536865237,582.2683715820313,385.14898005536568,573.3139038085938]},{"page":20,"text":"expert in polarizing microscopy. In Fig. 1.2 we can see the nice photos of the two","rect":[53.813106536865237,594.2279052734375,385.1529473405219,585.2933959960938]},{"page":20,"text":"L.M. Blinov, Structure and Properties of Liquid Crystals,","rect":[53.812843322753909,623.7623291015625,251.63688919141254,616.1510620117188]},{"page":20,"text":"DOI 10.1007/978-90-481-8829-1_1, # Springer ScienceþBusiness Media B.V. 2011","rect":[53.812843322753909,633.698486328125,347.38995880274697,625.4920043945313]},{"page":20,"text":"1","rect":[380.9389343261719,621.958984375,385.1694693007938,616.2357177734375]},{"page":21,"text":"2","rect":[53.81285858154297,42.4549560546875,58.04339355616494,36.73167419433594]},{"page":21,"text":"Fig. 1.1","rect":[53.812843322753909,172.38433837890626,81.04310363917276,164.65452575683595]},{"page":21,"text":"phases","rect":[53.81200408935547,182.29257202148438,76.35229952811517,174.69821166992188]},{"page":21,"text":"From","rect":[87.06315612792969,170.56407165527345,105.38983391160953,164.89158630371095]},{"page":21,"text":"left","rect":[108.13544464111328,172.2996826171875,119.05022148337427,164.68838500976563]},{"page":21,"text":"to","rect":[121.78652954101563,170.56407165527345,128.3946279922001,165.58583068847657]},{"page":21,"text":"right:","rect":[131.13262939453126,172.35047912597657,149.99406151022974,164.705322265625]},{"page":21,"text":"molecular","rect":[152.77098083496095,171.0,186.75909513098888,164.72225952148438]},{"page":21,"text":"structure","rect":[189.5334930419922,171.0,219.20646044992925,165.58583068847657]},{"page":21,"text":"of","rect":[221.99099731445313,171.0,229.03906339270763,164.72225952148438]},{"page":21,"text":"isotropic,","rect":[231.7906036376953,172.31661987304688,263.5873133857485,164.72225952148438]},{"page":21,"text":"nematic,","rect":[266.34393310546877,171.0,295.3992640693422,164.72225952148438]},{"page":21,"text":"1 Introductory Notes","rect":[314.0965270996094,44.275230407714847,385.1398361492089,36.68087387084961]},{"page":21,"text":"smectic A and smecticC","rect":[298.2353820800781,171.0,385.15686213697776,164.72225952148438]},{"page":21,"text":"Fig. 1.2 Photos of Friedrich Reinitzer (left) and Otto Lehmann (right)","rect":[53.812843322753909,404.2655029296875,296.12603849036386,396.5356750488281]},{"page":21,"text":"founding-fathers of the liquid crystal community taken from the book on liquid","rect":[53.812843322753909,438.8108825683594,385.11089411786568,429.8763427734375]},{"page":21,"text":"crystal history written by Prof. Sonin [3].","rect":[53.812843322753909,450.7704162597656,220.3179134895969,441.83587646484377]},{"page":21,"text":"It was Lehmann who, having investigated the gift of Reinitzer, understood that","rect":[65.7648696899414,462.72998046875,385.13679368564677,453.79541015625]},{"page":21,"text":"he was dealing with a new state of matter. Lehmann also observed such intermedi-","rect":[53.81285095214844,474.68951416015627,385.17962013093605,465.75494384765627]},{"page":21,"text":"ate phases in other substances and, at first, gave them the name fliebende Kristalle","rect":[53.81285095214844,486.6490783691406,385.18152654840318,477.40576171875]},{"page":21,"text":"(crystals showing fluidity) [4]. Later he decided that the term flu€ssige Kristalle","rect":[53.81282424926758,498.6484375,385.1804889507469,489.0]},{"page":21,"text":"(liquid crystals) corresponds better to the essence of mesophases and used it asa","rect":[53.81282424926758,510.5681457519531,385.15869939996568,501.63360595703127]},{"page":21,"text":"title of the very first book on liquid crystals [5] (for more details about history of","rect":[53.81282424926758,522.4708862304688,385.1506589492954,513.4766235351563]},{"page":21,"text":"liquid crystals see [6, 7]).","rect":[53.81280517578125,534.430419921875,156.56513638998752,525.4959106445313]},{"page":21,"text":"Today we know that the cholesterol esters consist of helical (chiral) molecules,","rect":[65.76482391357422,546.3899536132813,385.1546902229953,537.4554443359375]},{"page":21,"text":"and on cooling from the isotropic phase they undergo a transition into another phase","rect":[53.812801361083987,558.3494873046875,385.09690130426255,549.4149780273438]},{"page":21,"text":"called a cholesteric phase. This shows unique optical properties. In Fig. 1.3a we see","rect":[53.812801361083987,570.30908203125,385.1147846050438,561.3546142578125]},{"page":21,"text":"a photo-image of a 20 mm thick polycrystalline layer of cholesteryl acetate viewed","rect":[53.81280517578125,582.2686157226563,385.09392634442818,573.3341064453125]},{"page":21,"text":"in a polarizing microscope. Upon heating the substance melts, that is it becomes","rect":[53.81279754638672,594.2281494140625,385.0859719668503,585.2936401367188]},{"page":22,"text":"1 Introductory Notes","rect":[53.814517974853519,44.275108337402347,124.9145859050683,36.68075180053711]},{"page":22,"text":"3","rect":[380.9397888183594,42.55643081665039,385.1703237929813,36.73155212402344]},{"page":22,"text":"Fig. 1.3 Photo-image of a 20 mm thick polycrystalline layer of cholesteryl acetate placed between","rect":[53.812843322753909,171.19400024414063,385.12555450587197,163.260986328125]},{"page":22,"text":"two cover glasses in crystalline phase (a), cholesteric phase (b) and at the transition from the","rect":[53.813690185546878,181.10223388671876,385.1559385993433,173.50787353515626]},{"page":22,"text":"cholesteric to isotropic phase (c)","rect":[53.813682556152347,191.02145385742188,165.16389554602794,183.42709350585938]},{"page":22,"text":"fluid but optically anisotropic and shows bright diffraction colours, Fig. 1.3b. With","rect":[53.812843322753909,235.61026000976563,385.13176814130318,226.6757049560547]},{"page":22,"text":"increasing temperature we observe a phase transition to the isotropic phase. The","rect":[53.812843322753909,247.56979370117188,385.14478338434068,238.63523864746095]},{"page":22,"text":"latter is not birefringent, and therefore looks black between crossed polarizers.","rect":[53.812843322753909,259.52935791015627,385.1417507698703,250.5948028564453]},{"page":22,"text":"In Fig. 1.3c we see black drops on the bright background of the superheated","rect":[53.812843322753909,271.4888916015625,385.1148614030219,262.5543212890625]},{"page":22,"text":"cholesteric phase.","rect":[53.812835693359378,283.44842529296877,125.20841641928439,274.51385498046877]},{"page":22,"text":"It should be noted that the appearance of the “cholesteric” phase of Reinitzer was","rect":[65.76485443115235,295.407958984375,385.14777006255346,286.473388671875]},{"page":22,"text":"different from the appearance of the classical cholesteric phase shown in Fig. 1.3b.","rect":[53.812835693359378,307.36749267578127,385.1815456917453,298.43292236328127]},{"page":22,"text":"The phase was opaque and had blue tint. It took a century to decipher its structure:","rect":[53.81288146972656,319.3270568847656,385.1338029629905,310.39251708984377]},{"page":22,"text":"it appears to be a blue phase (see Chapter 4) with a structure of liquid lattice","rect":[53.81288146972656,331.2298278808594,385.1557086773094,322.2753601074219]},{"page":22,"text":"consisting exclusively of defects of an initially ideal helical structure. This phase is","rect":[53.81288146972656,343.1893615722656,385.1846047793503,334.25482177734377]},{"page":22,"text":"periodic and shows Bragg diffraction of light in all the three principal directions.","rect":[53.81288146972656,355.14892578125,385.1657375862766,346.21435546875]},{"page":22,"text":"Therefore, Reinitzer has discovered the first generic photonic crystal! At present, a","rect":[53.81288146972656,367.10845947265627,385.15869939996568,358.17388916015627]},{"page":22,"text":"study of photonic crystals, mostly artificial, is one of the hot topics in physics [8].","rect":[53.81288146972656,379.0679931640625,384.1354031136203,370.1334228515625]},{"page":22,"text":"The timing of the discovery of liquid crystals was unlucky. It coincided with the","rect":[65.7658920288086,391.02752685546877,385.17368353082505,382.09295654296877]},{"page":22,"text":"period when the beautiful foundations of modern physics were being laid, but the","rect":[53.81387710571289,402.987060546875,385.17466009332505,394.052490234375]},{"page":22,"text":"stone with the mark “liquid crystals” was somehow lost in controversy. Only now,","rect":[53.81387710571289,414.9466247558594,385.1179165413547,405.9921569824219]},{"page":22,"text":"through the enormous efforts of several generations of scientists, has the missing","rect":[53.81387710571289,426.8493957519531,385.12981501630318,417.91485595703127]},{"page":22,"text":"stone of liquid crystals been inserted in its legitimate place in the foundation","rect":[53.81387710571289,438.8089294433594,385.1557244401313,429.8743896484375]},{"page":22,"text":"of Science. And among those who put liquid crystals into the mainstream of physics","rect":[53.81387710571289,450.76849365234377,385.11886991606908,441.83392333984377]},{"page":22,"text":"there were such giants as F. Leslie, A. Saupe and especially P.G. de Gennes","rect":[53.81387710571289,462.7280578613281,385.10992826567846,453.79351806640627]},{"page":22,"text":"(The Nobel Prize in Physics, 1991).","rect":[53.81387710571289,474.6875915527344,198.08224149252659,465.7530517578125]},{"page":22,"text":"The early book of de Gennes [9] and the subsequent one written together with","rect":[65.7658920288086,486.6471252441406,385.11687556317818,477.71258544921877]},{"page":22,"text":"Prost [10] may be highly recommended to physicists. During the work on the","rect":[53.81387710571289,498.6067199707031,385.1159137554344,489.67218017578127]},{"page":22,"text":"present book I used them frequently as well as the other excellent books on liquid","rect":[53.81387710571289,510.5662536621094,385.11391535810005,501.6317138671875]},{"page":22,"text":"crystal physics [11–14]. The reader can also find a great deal of interesting","rect":[53.81387710571289,522.468994140625,385.1368340592719,513.5344848632813]},{"page":22,"text":"information on particular problems related to the physical properties of mesophases","rect":[53.81387710571289,534.4285278320313,385.14377225981908,525.4940185546875]},{"page":22,"text":"in monographs [15–21]. For newcomers I would recommend a nice, philosophically","rect":[53.81387710571289,546.3880615234375,385.1238946061469,537.393798828125]},{"page":22,"text":"tinted book by P. Collings [22], a piece of art prepared by A.S. Sonin in Russian","rect":[53.81290054321289,558.34765625,385.15374079755318,549.4131469726563]},{"page":22,"text":"[23], and a slightly more scientific book written for schoolboys by S.A. Pikin and","rect":[53.81290054321289,570.3071899414063,385.14275446942818,561.3726806640625]},{"page":22,"text":"myself [24] (in Russian and Spanish). The literature for further reading is given at","rect":[53.81190490722656,582.2667236328125,385.1776567227561,573.3322143554688]},{"page":22,"text":"the ends of relevant chapters.","rect":[53.810909271240237,594.2262573242188,171.94747586752659,585.291748046875]},{"page":23,"text":"4","rect":[53.81205368041992,42.45953369140625,58.0425886550419,36.73625183105469]},{"page":23,"text":"References","rect":[53.812843322753909,68.09864807128906,109.59614448282879,59.31352233886719]},{"page":23,"text":"1 Introductory Notes","rect":[314.095703125,44.279808044433597,385.13901217459957,36.68545150756836]},{"page":23,"text":"1.","rect":[58.06126022338867,93.52374267578125,64.40706131055318,87.80046081542969]},{"page":23,"text":"2.","rect":[58.06124496459961,103.49969482421875,64.40704605176411,97.77641296386719]},{"page":23,"text":"3.","rect":[58.061195373535159,114.0,64.40699646069966,107.75236511230469]},{"page":23,"text":"4.","rect":[58.061195373535159,123.39483642578125,64.40699646069966,117.67155456542969]},{"page":23,"text":"5.","rect":[58.06121063232422,133.47238159179688,64.40701171948872,127.54590606689453]},{"page":23,"text":"6.","rect":[58.06121063232422,144.0,64.40701171948872,137.57272338867188]},{"page":23,"text":"7.","rect":[58.06121063232422,154.0,64.40701171948872,147.71800231933595]},{"page":23,"text":"8.","rect":[58.06121063232422,174.0,64.40701171948872,167.49464416503907]},{"page":23,"text":"9.","rect":[58.06121063232422,193.3391571044922,64.40701171948872,187.44654846191407]},{"page":23,"text":"10.","rect":[53.81290817260742,203.19064331054688,64.3892466743227,197.3657684326172]},{"page":23,"text":"11.","rect":[53.81290817260742,223.04095458984376,64.3892466743227,217.3176727294922]},{"page":23,"text":"12.","rect":[53.81290817260742,233.01693725585938,64.3892466743227,227.2936553955078]},{"page":23,"text":"13.","rect":[53.81290817260742,253.01370239257813,64.3892466743227,247.18882751464845]},{"page":23,"text":"14.","rect":[53.81290817260742,273.0,64.3892466743227,267.083984375]},{"page":23,"text":"15.","rect":[53.81290817260742,293.0,64.3892466743227,286.9342956542969]},{"page":23,"text":"16.","rect":[53.81290817260742,313.0,64.3892466743227,306.8803405761719]},{"page":23,"text":"17.","rect":[53.81290817260742,343.0,64.3892466743227,336.8590087890625]},{"page":23,"text":"18.","rect":[53.81290817260742,363.0,64.3892466743227,356.7541809082031]},{"page":23,"text":"19.","rect":[53.81290817260742,383.0,64.3892466743227,376.6493835449219]},{"page":23,"text":"20.","rect":[53.81290817260742,393.0,64.3892466743227,386.6253356933594]},{"page":23,"text":"21.","rect":[53.81290817260742,412.300537109375,64.3892466743227,406.5772399902344]},{"page":23,"text":"22.","rect":[53.81205368041992,433.0,64.3883921821352,426.4723815917969]},{"page":23,"text":"23.","rect":[53.81205368041992,452.2491760253906,64.3883921821352,446.4242858886719]},{"page":23,"text":"24.","rect":[53.81205368041992,462.06683349609377,64.3883921821352,456.3435363769531]},{"page":23,"text":"Friedel, G.: Les e´tats mesomorphic da la matie´r. Ann. Phys. 18, 273–474 (1922)","rect":[68.59698486328125,95.3440170288086,343.8482674942701,87.44319152832031]},{"page":23,"text":"Reinitzer, F.: Beitr€age zur Kenntniss des Cholesterins. Monathefte f€ur Chemie 9, 421 (1888)","rect":[68.59696960449219,105.3199691772461,385.19830411536386,97.0]},{"page":23,"text":"Sonin, A.S.: The Road a Century Long. Nauka, Moscow (1988) (in Russian)","rect":[68.596923828125,115.2959213256836,331.05426114661386,107.70156860351563]},{"page":23,"text":"€","rect":[119.06804656982422,117.0,123.2985815444462,115.0]},{"page":23,"text":"Lehmann, O.: Uber fliebende Kristalle. Zs. Phys. Chem. 4, 510–514 (1889)","rect":[68.596923828125,125.2151107788086,326.9683846817701,117.3582992553711]},{"page":23,"text":"Lehmann, O.: Die Flu€ssige Kristalle. Leipzig, Wilhelm Engelmann (1904)","rect":[68.59693908691406,135.19107055664063,323.34283536536386,127.0]},{"page":23,"text":"Kelker, H.: History of liquid crystals. Mol. Cryst. Liq. Cryst. 21, 1–48 (1973)","rect":[68.59693908691406,145.16708374023438,334.27450650794199,137.50498962402345]},{"page":23,"text":"Sluckin, T.J., Dunmur, D.A., Stegemeyer, H.: Crystals That Flow: Classic Papers in the","rect":[68.59693908691406,155.14303588867188,385.1551146247339,147.54867553710938]},{"page":23,"text":"History of Liquid Crystals. Taylor & Francis, London (2004)","rect":[68.59693908691406,165.06222534179688,277.8011788712232,157.46786499023438]},{"page":23,"text":"Joannopoulos, J.D., Meade, R.D., Winn, J.N.: Photonic Crystals: Molding the Flow of Light.","rect":[68.59693908691406,175.0382080078125,385.1500575263735,167.44384765625]},{"page":23,"text":"Princeton University Press, Princeton, NJ (1995)","rect":[68.59693908691406,185.01416015625,235.25972074134044,177.36900329589845]},{"page":23,"text":"De Gennes, P.G.: The Physics of Liquid Crystals. Clarendon, Oxford (1975)","rect":[68.59693908691406,194.9901123046875,330.14132779700449,187.34495544433595]},{"page":23,"text":"de Gennes, P.G., Prost, J.: The Physics of Liquid Crystals, 2nd edn. Oxford, Science Publica-","rect":[68.59693908691406,204.90933227539063,385.1330575333326,197.29803466796876]},{"page":23,"text":"tions (1995)","rect":[68.59693908691406,214.54661560058595,110.01725858313729,207.24012756347657]},{"page":23,"text":"Chandrasekhar, S.: Liquid Crystals, 2nd edn. Cambridge University Press, Cambridge (1992)","rect":[68.59693908691406,224.86123657226563,385.20086759192636,217.26687622070313]},{"page":23,"text":"Vertogen, G., de Jeu, V.H.: Thermotropic Liquid Crystals. Fundamentals. Springer-Verlag,","rect":[68.59693908691406,234.83721923828126,385.1314112861391,227.24285888671876]},{"page":23,"text":"Berlin (1987)","rect":[68.59693908691406,244.41773986816407,114.77491849524667,237.16204833984376]},{"page":23,"text":"Demus, D., Goodby, J., Gray, G.W., Spiess H.-W., Vill V. (eds): Physical Properties of Liquid","rect":[68.59693908691406,254.73239135742188,385.17120880274697,247.13803100585938]},{"page":23,"text":"Crystals. Wiley-VCH, Weinheim (1999)","rect":[68.59693908691406,264.7083435058594,207.16302579505138,257.1139831542969]},{"page":23,"text":"Dunmur, D., Fukuda A., Luckhurst, G. (eds): Physical Properties of Liquid crystals: Nematics.","rect":[68.59693908691406,274.6275634765625,385.13220474317037,267.033203125]},{"page":23,"text":"INSPEC, London (2001)","rect":[68.59693908691406,284.2648620605469,153.46402066809825,277.0091552734375]},{"page":23,"text":"Priestley, E.B., Wojtowicz, P., Sheng, P. (eds): Introduction to Liquid Crystals. Plenum Press,","rect":[68.59693908691406,294.5794677734375,385.1482569892641,286.985107421875]},{"page":23,"text":"New York (1975)","rect":[68.59693908691406,304.216796875,129.21965879065685,296.9102783203125]},{"page":23,"text":"Blinov, L.M.: Electro-Optical and Magneto-Optical Properties of Liquid Crystals. Wiley,","rect":[68.59693908691406,314.4747009277344,385.1415431220766,306.8634033203125]},{"page":23,"text":"Chichester (1983); Blinov, L.M., Chigrinov, V.G.: Electrooptic Effects in Liquid Crystal","rect":[68.59693908691406,324.45068359375,385.1382418080813,316.8563232421875]},{"page":23,"text":"Materials. Springer-Verlag, New York (1993)","rect":[68.59693908691406,334.4266357421875,225.4042672501295,326.832275390625]},{"page":23,"text":"Pikin, S.A.: Structural Transformations in Liquid Crystals. Gordon & Breach, New York","rect":[68.59693908691406,344.402587890625,385.14999908595009,336.8082275390625]},{"page":23,"text":"(1991)","rect":[68.59693908691406,353.983154296875,91.15415280921151,346.7782287597656]},{"page":23,"text":"Kats, E.I., Lebedev, V.V.: Fluctuation effects in the Dynamics of Liquid Crystals. Springer-","rect":[68.59693908691406,364.2977600097656,385.15680020911386,356.7033996582031]},{"page":23,"text":"Verlag, New York (1993)","rect":[68.59693908691406,374.27374267578127,156.8061227188795,366.67938232421877]},{"page":23,"text":"Kleman, M., Lavrentovich, O.: Soft Matter Physics. Springer-Verlag, New York (2003)","rect":[68.59693908691406,384.1929626464844,368.8287362442701,376.5816650390625]},{"page":23,"text":"Lagerwall, S.T.: Ferroelectric and Antiferroelectric Liquid Crystals. Wiley-VCH, Weinheim,","rect":[68.59693908691406,394.1689147949219,385.1483485419985,386.5745544433594]},{"page":23,"text":"NY (1999)","rect":[68.59693908691406,403.80621337890627,105.71226590735604,396.6012878417969]},{"page":23,"text":"ˇ","rect":[151.2421112060547,405.9676818847656,154.05964749915294,404.55377197265627]},{"page":23,"text":"Musˇevicˇ, I., Blinc, R., Zeksˇ, B.: The Physics of Ferroelectric and Antiferroelectric Liquid","rect":[68.59693908691406,414.1208190917969,385.1695913711063,406.19708251953127]},{"page":23,"text":"Crystals. World Scientific, Singapore (2000)","rect":[68.59607696533203,424.0400390625,220.81411832434825,416.4456787109375]},{"page":23,"text":"Collings, P.: Liquid Crystals: Nature’s Delicate Phase of Matter. Princeton University Press,","rect":[68.59608459472656,434.0159606933594,385.1475245673891,426.4216003417969]},{"page":23,"text":"Princeton, NJ (1990)","rect":[68.59607696533203,443.65325927734377,139.92459195716075,436.3975524902344]},{"page":23,"text":"Sonin, A.S.: Centaurs of Nature. Atomizdat, Moscow (1980) (in Russian)","rect":[68.59608459472656,453.62921142578127,320.2334298477857,446.3735046386719]},{"page":23,"text":"Pikin, S.A., Blinov, L.M.: Liquid Crystals, Kvant Series 20. Nauka, Moscow, (1982)","rect":[68.59608459472656,463.8871154785156,385.1424874649732,456.2927551269531]},{"page":23,"text":"(in Russian); Spanish translation “Cristales Liquidos”, Mir, Moscow, 1985","rect":[68.59607696533203,473.863037109375,324.1644339004032,466.2178649902344]},{"page":24,"text":"Part I","rect":[344.4591979980469,70.58036041259766,385.20173276813537,59.488277435302737]},{"page":24,"text":"Structure of Liquid Crystals","rect":[166.19692993164063,94.86486053466797,385.1655567123365,77.83232879638672]},{"page":25,"text":"Chapter2","rect":[53.812843322753909,72.10812377929688,114.14115996551633,59.571903228759769]},{"page":25,"text":"Symmetry","rect":[53.812843322753909,91.18268585205078,124.6078093628825,76.10637664794922]},{"page":25,"text":"The concept of symmetry is equally important for understanding properties of","rect":[53.812843322753909,211.74758911132813,385.14873634187355,202.8130340576172]},{"page":25,"text":"individual molecules, crystals and liquid crystals [1]. The symmetry is of special","rect":[53.812843322753909,223.65036010742188,385.18049485752177,214.71580505371095]},{"page":25,"text":"importance in physics of liquid crystal because it allows us to distinguish numerous","rect":[53.8138313293457,235.60992431640626,385.0950051699753,226.6753692626953]},{"page":25,"text":"liquid crystalline phases from each other. In fact, all properties of mesophases are","rect":[53.8138313293457,247.5694580078125,385.1616901226219,238.63490295410157]},{"page":25,"text":"determined by their symmetry [2]. In the first section we consider the so-called","rect":[53.8138313293457,259.5290222167969,385.0909356217719,250.59446716308595]},{"page":25,"text":"point group symmetry very often used for discussion of the most important liquid","rect":[53.8138427734375,271.52838134765627,385.1138848405219,262.55401611328127]},{"page":25,"text":"crystalline","rect":[53.8138427734375,283.44805908203127,95.5488513775047,274.51348876953127]},{"page":25,"text":"phases.","rect":[101.11126708984375,283.44805908203127,130.11783261801487,274.51348876953127]},{"page":25,"text":"A","rect":[135.72103881835938,281.3065490722656,142.86816167786447,274.5732727050781]},{"page":25,"text":"brief","rect":[148.40968322753907,282.0,167.2629636367954,274.51348876953127]},{"page":25,"text":"discussion","rect":[172.8233642578125,282.0,214.38520136884223,274.51348876953127]},{"page":25,"text":"of","rect":[219.89483642578126,282.0,228.18670020906104,274.51348876953127]},{"page":25,"text":"the","rect":[233.7162628173828,282.0,245.94004095270004,274.51348876953127]},{"page":25,"text":"space","rect":[251.50445556640626,283.3385009765625,274.1901019878563,276.78448486328127]},{"page":25,"text":"group","rect":[279.77044677734377,283.4878845214844,303.5112237077094,276.7048034667969]},{"page":25,"text":"symmetry","rect":[309.1124572753906,283.44805908203127,349.0905694108344,275.52947998046877]},{"page":25,"text":"will","rect":[354.59820556640627,282.0,370.1367326504905,274.51348876953127]},{"page":25,"text":"be","rect":[375.72705078125,282.0,385.15366399957505,274.51348876953127]},{"page":25,"text":"presented in Section 2.2.","rect":[53.8138313293457,295.4076232910156,153.27828641440159,286.47308349609377]},{"page":25,"text":"2.1 Point Group Symmetry","rect":[53.812843322753909,346.0805358886719,201.61948909663745,334.7734069824219]},{"page":25,"text":"2.1.1 Symmetry Elements and Operations","rect":[53.812843322753909,375.44677734375,267.8344028446452,364.8089904785156]},{"page":25,"text":"There are only few symmetry elements, which generates a number of symmetry","rect":[53.812843322753909,402.9886169433594,385.1706170182563,394.0540771484375]},{"page":25,"text":"operations [3, 4]. We may illustrate them by their applications to simple geometri-","rect":[53.812843322753909,414.94818115234377,385.14873634187355,406.01361083984377]},{"page":25,"text":"cal objects.","rect":[53.813846588134769,426.8509216308594,99.1552700569797,417.9163818359375]},{"page":25,"text":"(a) Proper rotation axis of nth-order, Cn","rect":[53.813846588134769,444.7987060546875,219.3242610010276,435.84423828125]},{"page":25,"text":"Consider first rotational symmetry. Let us take an equilateral triangle and rotate it","rect":[53.812843322753909,462.7301940917969,385.17060716220927,453.7757263183594]},{"page":25,"text":"clockwise about its center by 360\u0001/3 ¼ 120\u0001, Fig. 2.1a. The new triangle would be","rect":[53.812862396240237,474.6898498535156,385.15247381402818,465.75518798828127]},{"page":25,"text":"undistinguishable from the original one (but not identical). The symmetry element","rect":[53.81464385986328,486.6494140625,385.2082353360374,477.6949462890625]},{"page":25,"text":"we used is the proper rotation axis of order 3 (C3-axis). The same triangle can be","rect":[53.81464385986328,498.6091613769531,385.1526264019188,489.6644592285156]},{"page":25,"text":"rotated by a half of the full turn about one of the three other axes (medians going out","rect":[53.81382369995117,510.5686950683594,385.19746263095927,501.6341552734375]},{"page":25,"text":"of each vertex), Fig. 2.1b. The corresponding symmetry element is a C2 axis","rect":[53.81382369995117,522.471435546875,385.09930814849096,513.5269775390625]},{"page":25,"text":"(rotation angle 360\u0001/2, n ¼ 2). For a square, we can find one C4 axis and four C2","rect":[53.814231872558597,534.431396484375,385.18130140630105,525.4869384765625]},{"page":25,"text":"axes, for a hexagonal benzene molecule one C6 axis and six C2 axes.","rect":[53.812843322753909,546.3910522460938,331.9720120003391,537.4465942382813]},{"page":25,"text":"The symmetry element C3 may generate two other symmetry operations. For","rect":[65.76557159423828,558.3507080078125,385.14946876374855,549.4061279296875]},{"page":25,"text":"instance, applying C3 rotation twice we again obtain an indistinguishable triangle.","rect":[53.81363296508789,570.310302734375,385.1761135628391,561.3657836914063]},{"page":25,"text":"Symbolically, C32 means rotation by 2 \u0003 2p/3 ¼ 240\u0001. The same C3 rotation","rect":[53.813350677490237,582.2699584960938,385.1192254166938,571.2192993164063]},{"page":25,"text":"applied three times result in the exactly the same triangle. Therefore C33 is one of","rect":[53.814231872558597,594.2294921875,385.15166602937355,583.178955078125]},{"page":25,"text":"L.M. Blinov, Structure and Properties of Liquid Crystals,","rect":[53.812843322753909,623.7623291015625,251.63688919141254,616.1510620117188]},{"page":25,"text":"DOI 10.1007/978-90-481-8829-1_2, # Springer ScienceþBusiness Media B.V. 2011","rect":[53.812843322753909,633.698486328125,347.38995880274697,625.4920043945313]},{"page":25,"text":"7","rect":[380.9389343261719,622.0606079101563,385.1694693007938,616.354248046875]},{"page":26,"text":"8","rect":[53.812843322753909,42.55594253540039,58.04337829737588,36.73106384277344]},{"page":26,"text":"Fig. 2.1 Rotational","rect":[53.812843322753909,67.58130645751953,122.0682955190188,59.85148620605469]},{"page":26,"text":"symmetry. The illustration of","rect":[53.812843322753909,77.4895248413086,154.09936612708263,69.89517211914063]},{"page":26,"text":"operations made by proper","rect":[53.812843322753909,87.4087142944336,145.25500577552013,79.81436157226563]},{"page":26,"text":"rotation axes of third (a) and","rect":[53.812843322753909,97.04601287841797,152.38260406641886,89.79031372070313]},{"page":26,"text":"second (b) order","rect":[53.81285095214844,107.02202606201172,110.26679319007089,99.76632690429688]},{"page":26,"text":"a","rect":[232.5220184326172,68.23306274414063,238.07728082985103,62.64430618286133]},{"page":26,"text":"1","rect":[232.15594482421876,117.05023193359375,236.1525765060294,111.46742248535156]},{"page":26,"text":"b","rect":[231.8485870361328,133.21853637695313,237.95338078561103,125.91016387939453]},{"page":26,"text":"1","rect":[234.70608520507813,183.63375854492188,238.70271688688877,178.0509490966797]},{"page":26,"text":"C2","rect":[250.60809326171876,207.0986328125,259.35263733557675,199.47625732421876]},{"page":26,"text":"3","rect":[321.1397399902344,116.13870239257813,325.136371672045,110.46791076660156]},{"page":26,"text":"1","rect":[323.9683532714844,182.57803344726563,327.964984953295,176.99522399902345]},{"page":26,"text":"2 Symmetry","rect":[343.21270751953127,44.274620056152347,385.16522735743447,36.73106384277344]},{"page":26,"text":"1","rect":[349.4319152832031,71.80413818359375,353.42854696501379,66.22132873535156]},{"page":26,"text":"2","rect":[351.9967346191406,136.6766357421875,355.99336630095129,131.0938262939453]},{"page":26,"text":"C2","rect":[340.3436584472656,207.14163208007813,349.0881720035455,199.51922607421876]},{"page":26,"text":"Fig. 2.2 Bilateral symmetry.","rect":[53.812843322753909,270.6120910644531,149.85275528027973,262.88226318359377]},{"page":26,"text":"Plane sv (vertical) is plane","rect":[53.812843322753909,280.46356201171877,139.73116442942144,272.8522644042969]},{"page":26,"text":"of reflection that contains","rect":[53.813236236572269,288.68695068359377,134.75099642752924,282.84515380859377]},{"page":26,"text":"the axis of the highest order","rect":[53.813236236572269,300.4154968261719,142.430268226692,292.8211364746094]},{"page":26,"text":"C3 for the equilateral triangle.","rect":[53.813236236572269,310.3910827636719,148.5662257392641,302.7967224121094]},{"page":26,"text":"After applying this element","rect":[53.8132209777832,320.3103332519531,141.16276268210474,312.7159729003906]},{"page":26,"text":"points 1 and 3 exchange","rect":[53.8132209777832,330.2862548828125,131.22353503489019,322.69189453125]},{"page":26,"text":"their positions. Plane of","rect":[53.8132209777832,340.26220703125,128.99830716712169,332.6678466796875]},{"page":26,"text":"the triangle is reflection","rect":[53.8132209777832,350.2381896972656,128.96275848536417,342.6438293457031]},{"page":26,"text":"plane sh (horizontal)","rect":[53.8132209777832,360.15740966796877,121.43250363928964,352.54547119140627]},{"page":26,"text":"perpendicular to C3","rect":[53.81332015991211,370.13275146484377,116.1232351095889,362.53839111328127]},{"page":26,"text":"1","rect":[267.3085021972656,330.9234313964844,271.30513387907629,325.3406066894531]},{"page":26,"text":"C3","rect":[317.46905517578127,273.8973693847656,326.2136602847955,266.2090148925781]},{"page":26,"text":"2","rect":[354.341552734375,291.7117919921875,358.33818441618566,286.12896728515627]},{"page":26,"text":"the identity operations, C33 ¼ E. Generally, the identity operation corresponds to","rect":[53.812843322753909,414.94830322265627,385.14153376630318,403.8975524902344]},{"page":26,"text":"doing nothing with any figure and can also be obtained with C2 or C4 axes or with","rect":[53.81362533569336,426.8510437011719,385.2010125260688,417.9065246582031]},{"page":26,"text":"any axis of order n: C22 ¼ C44 ¼ Cnn ¼ E.","rect":[53.814414978027347,438.810791015625,226.23751493002659,427.7597351074219]},{"page":26,"text":"(b) A plane of symmetry, s","rect":[53.814674377441409,456.7586669921875,165.06692763980176,447.80419921875]},{"page":26,"text":"This element generates only one operation, a reflection in the plane as in a mirror,","rect":[53.81470489501953,474.68951416015627,385.1266750862766,465.75494384765627]},{"page":26,"text":"Fig.2.2. Repeated twice this operation results in the initial structure that is s2 ¼ E.","rect":[53.81470489501953,486.6490783691406,385.1836208870578,475.59857177734377]},{"page":26,"text":"Taking again our triangle we can see that plane s interchanges points 1 and 3","rect":[53.81493377685547,498.6090393066406,385.17864314130318,489.67449951171877]},{"page":26,"text":"leaving point 2 at the same place. Such symmetry is called bilateral symmetry.","rect":[53.81493377685547,510.568603515625,385.1836208870578,501.6141357421875]},{"page":26,"text":"There may be several symmetry planes and they designated either as sh (the plane","rect":[53.81493377685547,522.4718017578125,385.14173162652818,513.5368041992188]},{"page":26,"text":"perpendicular to the axis Cn with highest number n) or sv (plane containing the","rect":[53.813838958740237,534.431396484375,385.17444647027818,525.4869384765625]},{"page":26,"text":"Cn axis). By convention, the Cn axis with highest number n is taken as a vertical,","rect":[53.81370162963867,546.3910522460938,385.17913480307348,537.4464721679688]},{"page":26,"text":"therefore, indices h and v mean “horizontal” and “vertical”. In our figure we see","rect":[53.81342697143555,558.3505859375,385.1134723491844,549.3961181640625]},{"page":26,"text":"the sv plane, and the plane of the triangle is sh. Note that a chiral object, for","rect":[53.81342697143555,570.310302734375,385.18084083406105,561.3756103515625]},{"page":26,"text":"instance a hand, has no mirror plane (however, two hands in praying position have a","rect":[53.81417465209961,582.2698974609375,385.1580280132469,573.3353881835938]},{"page":26,"text":"mirror plane between them [4]).","rect":[53.81417465209961,594.2294311523438,183.0765804817844,585.294921875]},{"page":27,"text":"2.1 Point Group Symmetry","rect":[53.813838958740237,44.276084899902347,146.29334015040323,36.68172836303711]},{"page":27,"text":"9","rect":[380.9391174316406,42.62513732910156,385.16965240626259,36.73252868652344]},{"page":27,"text":"(c) Inversion center,I","rect":[53.812843322753909,67.88993072509766,144.01989112702979,59.41356658935547]},{"page":27,"text":"This symmetry element I generates an operation of inversion through a point called","rect":[53.812843322753909,86.19930267333985,385.1726006608344,77.26475524902344]},{"page":27,"text":"the inversion center, Fig. 2.3. Therefore now we deal with inversion symmetry. We","rect":[53.812843322753909,98.1588363647461,385.12778509332505,89.22428894042969]},{"page":27,"text":"can take any point of an object and connect it by a straight line with the center O.","rect":[53.81288146972656,110.11837005615235,385.1327786019016,101.16390228271485]},{"page":27,"text":"Then, along the same line behind the center and at the equal distance from it we","rect":[53.81288146972656,122.0779037475586,385.16565740777818,113.14335632324219]},{"page":27,"text":"must find the point equivalent to the first one. A good example is a parallelogram.","rect":[53.81288146972656,134.03750610351563,385.1249050667453,125.10295104980469]},{"page":27,"text":"Note, that two inversions result in the identical object, I2 ¼ E.","rect":[53.81288146972656,145.99703979492188,304.6900906136203,134.8900604248047]},{"page":27,"text":"(d) Improper rotation, Sn","rect":[53.813167572021487,163.79917907714845,157.0145686182151,154.9741973876953]},{"page":27,"text":"This element is also called a rotation–reflection axis or mirror–rotation axis. It","rect":[53.812843322753909,179.78814697265626,385.17658860752177,172.8855438232422]},{"page":27,"text":"consists of two steps, a rotation through 1/n of the full turn followed by reflection in","rect":[53.812843322753909,193.7796630859375,385.1974724870063,184.84510803222657]},{"page":27,"text":"a plane perpendicular to the rotation axis, Fig. 2.4. A molecule of ethane in the","rect":[53.812843322753909,205.73919677734376,385.1735309429344,196.8046417236328]},{"page":27,"text":"staggered configuration is a good illustration of S6 rotation–reflection axis, see the","rect":[53.81282424926758,217.69876098632813,385.1737750835594,208.7642059326172]},{"page":27,"text":"figure. Note, that this object has neither C6 axis nor s plane on their own. But after","rect":[53.81403732299805,229.65866088867188,385.1896909317173,220.71408081054688]},{"page":27,"text":"combined operations C6 (60\u0001 clockwise) and s we obtain an indistinguishable","rect":[53.8139762878418,241.61834716796876,385.14856756402818,232.6737060546875]},{"page":27,"text":"object with interchanged positions of all hydrogen atoms. Therefore S6, and,","rect":[53.81375503540039,253.52108764648438,385.1421780159641,244.58653259277345]},{"page":27,"text":"more generally, Sn is independent symmetry operation. Like element Cn, element","rect":[53.814292907714847,265.4809265136719,385.10273606845927,256.5364074707031]},{"page":27,"text":"S6 may generate several operations, for instance, S62 ¼ C3 because this operation","rect":[53.813716888427737,277.4405822753906,385.1583184342719,266.3898620605469]},{"page":27,"text":"consists of rotation by 2 \u0003 2p/6 ¼ 120\u0001 ¼ 2p/3 and identity operation s2 ¼ E.","rect":[53.813472747802737,289.4002685546875,385.1831936409641,278.34930419921877]},{"page":27,"text":"Totally, S62 ¼ C3E. Other examples are: S32 ¼ C32; S33 ¼ s; S34 ¼ C3; S36 ¼ E.","rect":[53.81450653076172,301.35992431640627,379.00359769369848,290.2693176269531]},{"page":27,"text":"Finally, we have five independent symmetry elements: identity E, proper rotation","rect":[65.7665023803711,313.3194885253906,385.19313899091255,304.38494873046877]},{"page":27,"text":"axis Cn, symmetry plane s, inversion center I and rotation–reflection axis Sn, generating","rect":[53.814476013183597,325.2792053222656,385.1765069108344,316.3345031738281]},{"page":27,"text":"single (elements E, s, I) or multiple (elements Cn, Sn) symmetry operations.","rect":[53.81374740600586,337.2388610839844,345.7099880745578,328.29425048828127]},{"page":27,"text":"Fig. 2.3 Inversion symmetry.","rect":[53.812843322753909,378.30572509765627,155.34060928418598,370.3727111816406]},{"page":27,"text":"Point O is inversion center and","rect":[53.812843322753909,386.4952697753906,155.35919708399698,380.6026611328125]},{"page":27,"text":"the inversion operation","rect":[53.812843322753909,398.18994140625,130.47352356104777,390.5955810546875]},{"page":27,"text":"exchanges positions of points","rect":[53.812843322753909,408.1658630371094,151.72433932303705,400.5715026855469]},{"page":27,"text":"1–10, 2–20, 3–30 etc.","rect":[53.812843322753909,416.3659362792969,119.75374099805318,409.896728515625]},{"page":27,"text":"1","rect":[261.2472839355469,414.1208801269531,265.2439156173575,408.5380554199219]},{"page":27,"text":"H5","rect":[73.2762451171875,473.767578125,82.85217194251033,466.207275390625]},{"page":27,"text":"H4","rect":[75.80191040039063,501.08184814453127,85.37785248450251,493.5874328613281]},{"page":27,"text":"H1","rect":[64.74503326416016,535.0397338867188,74.32126526526423,527.545654296875]},{"page":27,"text":"C","rect":[96.50386047363281,516.583984375,102.25101683207652,510.857177734375]},{"page":27,"text":"H6","rect":[133.6937255859375,473.57232666015627,143.2696447818658,466.01202392578127]},{"page":27,"text":"H2","rect":[125.35714721679688,506.25555419921877,134.9329748599908,498.7611389160156]},{"page":27,"text":"H3","rect":[124.38566589355469,540.2409057617188,133.96160034827205,532.6805419921875]},{"page":27,"text":"C6","rect":[161.31935119628907,514.4779052734375,170.06406311682674,506.79754638671877]},{"page":27,"text":"30o","rect":[161.31935119628907,525.61669921875,172.3101721256158,519.6800537109375]},{"page":27,"text":"H4","rect":[184.04917907714845,479.5059814453125,193.6251135318658,472.0115661621094]},{"page":27,"text":"H3","rect":[206.16644287109376,542.5055541992188,215.74236206702205,534.9451293945313]},{"page":27,"text":"H5","rect":[241.34959411621095,460.38873291015627,250.9255285709283,452.8283386230469]},{"page":27,"text":"σ","rect":[286.6811828613281,489.8365478515625,291.5011206695918,486.14935302734377]},{"page":27,"text":"H1","rect":[311.7439880371094,465.87664794921877,321.31980042151425,458.3822326660156]},{"page":27,"text":"H4","rect":[300.34234619140627,527.74072265625,309.9185247867486,520.2459716796875]},{"page":27,"text":"H2","rect":[364.69000244140627,467.24359130859377,374.26596741370175,459.7491760253906]},{"page":27,"text":"H5","rect":[356.8336486816406,505.142578125,366.4096136539361,497.5821838378906]},{"page":27,"text":"H6","rect":[358.1435546875,539.4696044921875,367.71945862463925,531.9091796875]},{"page":27,"text":"Fig. 2.4 Improper rotation. Axis connecting points C–C is a rotation–reflection axis S6. An ethane","rect":[53.812843322753909,564.1054077148438,385.1668028571558,556.3756103515625]},{"page":27,"text":"molecule has a symmetry element including two subsequent operations, the rotation of the whole","rect":[53.813594818115237,573.9569091796875,385.15075061106207,566.362548828125]},{"page":27,"text":"structure through an angle of 30\u0001 with a subsequent reflection by plane s. After this the left and","rect":[53.813594818115237,583.932861328125,385.19501251368447,576.3382568359375]},{"page":27,"text":"right sketch become identical","rect":[53.813838958740237,593.9085693359375,154.3914156362063,586.314208984375]},{"page":28,"text":"10","rect":[53.813594818115237,42.55740737915039,62.274663392590728,36.73252868652344]},{"page":28,"text":"2.1.2 Groups","rect":[53.812843322753909,69.85308837890625,125.98234443400067,59.370697021484378]},{"page":28,"text":"2 Symmetry","rect":[343.2135009765625,44.276084899902347,385.1660208144657,36.73252868652344]},{"page":28,"text":"Now, for each geometrical object or a molecule we can write a set of symmetry","rect":[53.812843322753909,97.47896575927735,385.1605914898094,88.50457763671875]},{"page":28,"text":"operations,which transform the object into its equivalent formal representation. Let","rect":[53.812843322753909,109.43856048583985,385.11689622470927,100.50401306152344]},{"page":28,"text":"consider three examples.","rect":[53.812843322753909,121.3980941772461,153.58587308432346,112.46354675292969]},{"page":28,"text":"(i)","rect":[53.812843322753909,138.9105682373047,63.23948036042822,130.3744354248047]},{"page":28,"text":"(ii)","rect":[53.81475067138672,174.78990173339845,66.03852210847509,166.25376892089845]},{"page":28,"text":"(iii)","rect":[53.81356430053711,198.7091827392578,68.83446632234228,190.1730499267578]},{"page":28,"text":"Water molecule, Fig. 2.5a. It has the C2 axis, two symmetry planes and","rect":[73.75212097167969,139.30941772460938,385.1447381120063,130.31466674804688]},{"page":28,"text":"together with identity element we have a full set of symmetry operations E,","rect":[73.75316619873047,151.26895141601563,385.1835598519016,142.3343963623047]},{"page":28,"text":"C2, s, s0. As we shall see soon this set corresponds to symmetry group C2v.","rect":[73.75313568115235,163.228759765625,380.8782620003391,153.5691680908203]},{"page":28,"text":"The next is an ion [Co(NH3)4ClBr]+3 shown in Fig. 2.5b. Its set of symmetry","rect":[73.86651611328125,175.18841552734376,385.17314997724068,164.1374969482422]},{"page":28,"text":"operations is E, C4, C42, C43, 4sv (group C4v).","rect":[73.86613464355469,187.14804077148438,261.1030239632297,176.0973663330078]},{"page":28,"text":"Finally we take a flat borate molecule BCl3 having the symmetry of an","rect":[73.86532592773438,199.10772705078126,385.1524590592719,190.1730499267578]},{"page":28,"text":"equilateral triangle, therefore allowing the following operations:","rect":[73.86639404296875,211.0672607421875,332.819227767678,202.13270568847657]},{"page":28,"text":"E, C3, C32, C2, C20, C200, sh, S3, S32, sv, sv0, sv00 (group D3h).","rect":[73.86639404296875,222.97030639648438,319.5608791878391,211.43382263183595]},{"page":28,"text":"Some operations belong to the same classes (see below), therefore we may write","rect":[65.7663345336914,240.93798828125,385.1273578472313,232.00343322753907]},{"page":28,"text":"the set in a more compact way: E, 2C3, 3C2, sh, 3S3, 3sv.","rect":[53.814292907714847,252.89755249023438,288.1501125862766,243.95303344726563]},{"page":28,"text":"It is essential that any element of each set of operations can be obtained by a","rect":[65.76607513427735,264.85736083984377,385.1568988628563,255.9228057861328]},{"page":28,"text":"combination of other elements from the same set. Application of the subsequent","rect":[53.81405258178711,276.7601318359375,385.15791184970927,267.8255615234375]},{"page":28,"text":"symmetry operations is called multiplication. For a water molecule we can write the","rect":[53.81405258178711,288.7196960449219,385.1718219585594,279.7652282714844]},{"page":28,"text":"corresponding multiplication table, Table 2.1. For instance, the multiplication of","rect":[53.81405258178711,300.6792297363281,385.1499265274204,291.7247619628906]},{"page":28,"text":"operations C2 from the first row and the first column corresponds to the identity","rect":[53.81406784057617,312.639404296875,385.1096123795844,303.6942443847656]},{"page":28,"text":"operation, C2 \u0004 C2 ¼ E, shown in the table. Further, C2s ¼ s0 and sC2 ¼ s0, so","rect":[53.81352996826172,324.5989685058594,385.1595086198188,314.9394226074219]},{"page":28,"text":"these operations commute in our particular case. Generally they may not commute","rect":[53.81468963623047,336.5585632324219,385.1117633648094,327.6240234375]},{"page":28,"text":"and the order of operations is important (by convention, multiplication operation","rect":[53.81468963623047,348.51812744140627,385.15554133466255,339.58355712890627]},{"page":28,"text":"C2s means that first we apply operation s and then C2).","rect":[53.81468963623047,360.4779968261719,281.0421413948703,351.5334777832031]},{"page":28,"text":"H","rect":[105.87906646728516,448.01776123046877,112.45752221554547,442.5229187011719]},{"page":28,"text":"C2","rect":[109.79557037353516,474.7391662597656,118.54009155920955,467.1247863769531]},{"page":28,"text":"z","rect":[152.52685546875,404.3481750488281,156.02790482201613,400.07708740234377]},{"page":28,"text":"x","rect":[142.18756103515626,420.7120056152344,146.5998424118752,416.96881103515627]},{"page":28,"text":"b","rect":[211.93348693847657,396.3589172363281,218.03828068795478,389.050537109375]},{"page":28,"text":"H3N","rect":[211.89552307128907,466.06671142578127,227.63415004749263,458.5063171386719]},{"page":28,"text":"H3N","rect":[231.54385375976563,431.1451416015625,247.28249599475826,423.5847473144531]},{"page":28,"text":"Cl","rect":[257.6337890625,417.5390319824219,265.54712529927579,411.812255859375]},{"page":28,"text":"Br","rect":[259.0246276855469,471.76910400390627,267.4335465317129,466.2742614746094]},{"page":28,"text":"C4","rect":[280.2864685058594,410.55877685546877,289.03113465002988,402.9443664550781]},{"page":28,"text":"Co","rect":[275.031005859375,437.6127014160156,284.77479940642,431.88592529296877]},{"page":28,"text":"NH3","rect":[321.8171691894531,435.0296325683594,337.55490784338925,427.46893310546877]},{"page":28,"text":"Fig. 2.5 Set of symmetry operations and the structure for water molecule belonging","rect":[53.812843322753909,496.3718566894531,340.5991568007938,488.64202880859377]},{"page":28,"text":"(a) and the structure of ion Co(NH3)4ClBr]+3 belonging to group C4v (b)","rect":[53.812843322753909,506.2799377441406,302.6969003067701,496.90362548828127]},{"page":28,"text":"Table 2.1","rect":[53.812843322753909,535.2716064453125,87.50059265284463,528.8286743164063]},{"page":28,"text":"E","rect":[53.812843322753909,558.8732299804688,59.00794027158969,553.2684936523438]},{"page":28,"text":"C2","rect":[53.812843322753909,570.26611328125,62.65019738986234,563.1258544921875]},{"page":28,"text":"s","rect":[53.813594818115237,578.892822265625,59.448667404311709,574.91357421875]},{"page":28,"text":"0","rect":[59.47737121582031,585.422607421875,61.07644854513558,582.3765258789063]},{"page":28,"text":"s","rect":[53.812843322753909,588.8685302734375,59.44791590895038,584.8892822265625]},{"page":28,"text":"Multiplication table","rect":[93.46479797363281,536.49072265625,161.05182788156987,528.8963623046875]},{"page":28,"text":"E","rect":[134.47560119628907,546.7435302734375,139.67069814512485,541.1387939453125]},{"page":28,"text":"E","rect":[134.47561645507813,558.8732299804688,139.6707134039139,553.2684936523438]},{"page":28,"text":"C2","rect":[134.47555541992188,570.26611328125,143.25627649142485,563.1258544921875]},{"page":28,"text":"s","rect":[134.4763641357422,578.892822265625,140.11143672193868,574.91357421875]},{"page":28,"text":"0","rect":[140.0834503173828,585.422607421875,141.68252764669809,582.3765258789063]},{"page":28,"text":"s","rect":[134.47555541992188,588.8685302734375,140.11062800611834,584.8892822265625]},{"page":28,"text":"for","rect":[163.42176818847657,535.0,173.28738492591075,528.8963623046875]},{"page":28,"text":"symmetry","rect":[175.65731811523438,536.49072265625,209.56929535059855,529.7599487304688]},{"page":28,"text":"elements","rect":[211.9671630859375,535.0,242.16472323416986,528.8963623046875]},{"page":28,"text":"C2","rect":[215.08168029785157,548.16064453125,223.91896570040923,541.020263671875]},{"page":28,"text":"C2","rect":[215.0816650390625,560.2902221679688,223.91896570040923,553.1499633789063]},{"page":28,"text":"E","rect":[215.0816192626953,568.84912109375,220.2767162115311,563.244384765625]},{"page":28,"text":"0","rect":[220.7461395263672,575.50341796875,222.34521685568246,572.4573364257813]},{"page":28,"text":"s","rect":[215.0824432373047,578.892822265625,220.71751582350118,574.91357421875]},{"page":28,"text":"s","rect":[215.0816192626953,588.8685302734375,220.71669184889178,584.8892822265625]},{"page":28,"text":"of","rect":[244.59474182128907,535.0,251.64282315833263,528.8963623046875]},{"page":28,"text":"water","rect":[253.99752807617188,535.0,272.83187955481699,529.7599487304688]},{"page":28,"text":"molecule","rect":[275.2398986816406,535.0,306.41051623606207,528.8963623046875]},{"page":28,"text":"s","rect":[295.74432373046877,546.811279296875,301.37939631666526,542.83203125]},{"page":28,"text":"s","rect":[295.74432373046877,558.9409790039063,301.37939631666526,554.9617309570313]},{"page":28,"text":"0","rect":[301.3522644042969,565.5276489257813,302.95134173361216,562.4815673828125]},{"page":28,"text":"s","rect":[295.744384765625,568.9168701171875,301.3794573518215,564.9376220703125]},{"page":28,"text":"E","rect":[295.74432373046877,578.8248901367188,300.93942067930456,573.2201538085938]},{"page":28,"text":"C2","rect":[295.744384765625,590.2177734375,304.5250905783389,583.0775146484375]},{"page":28,"text":"to","rect":[342.7051086425781,495.0,349.31320709376259,489.5733337402344]},{"page":28,"text":"group","rect":[351.3717956542969,496.30413818359377,371.1114553847782,490.0]},{"page":28,"text":"C2v","rect":[373.18017578125,495.9460144042969,385.18845117404205,488.76055908203127]},{"page":28,"text":"0","rect":[382.0149230957031,543.4220581054688,383.6140004250184,540.3759765625]},{"page":28,"text":"s","rect":[376.35040283203127,546.811279296875,381.98547541822776,542.83203125]},{"page":28,"text":"0","rect":[382.0149230957031,555.5517578125,383.6140004250184,552.5056762695313]},{"page":28,"text":"s","rect":[376.35040283203127,558.9409790039063,381.98547541822776,554.9617309570313]},{"page":28,"text":"s","rect":[376.35040283203127,568.9168701171875,381.98547541822776,564.9376220703125]},{"page":28,"text":"C2","rect":[376.35040283203127,580.2418823242188,385.18774926974518,573.1016235351563]},{"page":28,"text":"E","rect":[376.35040283203127,588.80078125,381.54549978086706,583.196044921875]},{"page":29,"text":"2.1 Point Group Symmetry","rect":[53.812843322753909,44.274620056152347,146.2923330703251,36.68026351928711]},{"page":29,"text":"11","rect":[376.74566650390627,42.454345703125,385.20673126368447,36.73106384277344]},{"page":29,"text":"In the mathematical sense, each set of the operations shown above and, more","rect":[65.76496887207031,68.2883529663086,385.1378864116844,59.35380554199219]},{"page":29,"text":"generally, characterizing the symmetry of objects of different shape forms a group.","rect":[53.812950134277347,80.28772735595703,385.1816372444797,71.31333923339844]},{"page":29,"text":"So, what is a group? A set of elements A, B, C, etc. form a group if there is a rule for","rect":[53.81296157836914,92.20748138427735,385.17864356843605,83.26296997070313]},{"page":29,"text":"combining any two elements to form their “product” AB (or A \u0004 B) such that:","rect":[53.81296157836914,104.11019134521485,365.1089921719749,95.17564392089844]},{"page":29,"text":"–","rect":[53.81298065185547,121.0,58.790085707221127,115.0]},{"page":29,"text":"–","rect":[53.81298065185547,131.0,58.790085707221127,126.0]},{"page":29,"text":"–","rect":[53.813987731933597,144.0,58.79109278729925,139.0]},{"page":29,"text":"–","rect":[53.81595993041992,154.0,58.79306498578558,150.0]},{"page":29,"text":"Every “product” of two elements is an element of the same set.","rect":[67.29496765136719,122.0779037475586,322.8703579476047,113.14335632324219]},{"page":29,"text":"There is a unit element E such that EA ¼ AE ¼ A.","rect":[67.29496765136719,131.97567749023438,270.5280727913547,125.10295104980469]},{"page":29,"text":"The associative law is valid, that is A(BC) ¼ (AB)C, etc.","rect":[67.29597473144531,145.59861755371095,295.48626370932348,137.05252075195313]},{"page":29,"text":"Each element A has its inverse element A\u00051 belonging to the same","rect":[67.29794311523438,157.90078735351563,333.04660833551255,146.8498992919922]},{"page":29,"text":"AA\u00051 ¼ A\u00051A ¼ E.","rect":[67.29508972167969,167.71896362304688,145.90853543783909,158.80943298339845]},{"page":29,"text":"set","rect":[335.6207580566406,156.0,346.6998124844749,149.98220825195313]},{"page":29,"text":"such","rect":[349.27197265625,156.0,367.50808039716255,148.9662322998047]},{"page":29,"text":"that","rect":[370.1170654296875,156.0,385.1379838711936,148.9662322998047]},{"page":29,"text":"The group “product” might be different (multiplication, addition, permutation,","rect":[65.76519012451172,187.82815551757813,385.1251492073703,178.8936004638672]},{"page":29,"text":"space rotation, etc).","rect":[53.813167572021487,199.73092651367188,132.77195401694065,190.79637145996095]},{"page":29,"text":"For example, a set of integers including negative, zero and positive numbers","rect":[65.76519012451172,211.69049072265626,385.1679726992722,202.7559356689453]},{"page":29,"text":"\u00051, ..., \u0005n, ..., \u00053, \u00052, \u00051, 0, 1, 2, 3, ..., n, ..., 1 form a group under addition","rect":[53.813167572021487,223.6500244140625,385.17189875653755,214.69554138183595]},{"page":29,"text":"as a group operation. Indeed, all the group rules are fulfilled:","rect":[53.81415939331055,235.60958862304688,299.5109697110374,226.67503356933595]},{"page":29,"text":"A þ B ¼ C (e.g. 2 þ 18 ¼ 20) belongs to the same set,","rect":[53.81415939331055,253.52047729492188,276.57244534994848,244.57595825195313]},{"page":29,"text":"evidently A þ (B þ C) ¼ (A þ B) þ C","rect":[65.7681655883789,265.48004150390627,223.24476882632519,256.5355224609375]},{"page":29,"text":"E ¼ 0 (e.g., 0 þ A ¼ A, that is addition of zero changes nothing)","rect":[53.818153381347659,277.4395751953125,315.8906186660923,268.5050048828125]},{"page":29,"text":"A þ (\u0005A)¼ 0 ¼ E(underadditionasgroupoperation(\u0005A)symbolicallymeansA\u00051)","rect":[53.82014083862305,289.3991394042969,385.15975318757668,278.34930419921877]},{"page":29,"text":"This group contains infinite number of elements, therefore it is an infinite group","rect":[65.76595306396485,307.35101318359377,385.1358574967719,298.3367919921875]},{"page":29,"text":"or group of infinite order.","rect":[53.81393051147461,319.2707214355469,156.75042386557346,310.336181640625]},{"page":29,"text":"Another example is a set of four 2 \u0003 2 matrices that form a finite group of order","rect":[65.76595306396485,331.23028564453127,385.14775977937355,322.2558898925781]},{"page":29,"text":"4 (because we have four elements in the group) under multiplication operation (it","rect":[53.81393051147461,343.1898193359375,385.1457353360374,334.2353515625]},{"page":29,"text":"may directly be checked using the rule for matrix multiplication):","rect":[53.81393051147461,355.1493835449219,318.69050462314677,346.21484375]},{"page":29,"text":"A ¼ \u000110 10\u0003;B ¼ \u0001\u000501 01\u0003;C ¼ \u0001\u000501 \u000501\u0003;D ¼ \u000110 \u000501\u0003","rect":[81.28652954101563,395.1984558105469,338.58689482398656,371.313232421875]},{"page":29,"text":"(2.1)","rect":[366.0943908691406,387.5119323730469,385.1666501602329,379.03558349609377]},{"page":29,"text":"Here the identity element is matrix A, the associative law for matrices is always","rect":[65.76293182373047,420.7281494140625,385.08005155669408,411.7935791015625]},{"page":29,"text":"valid, the inverse matrices are B for D and vice versa (BD ¼ DB ¼ A), elements","rect":[53.81185531616211,432.28924560546877,385.17856229888158,423.75311279296877]},{"page":29,"text":"A and C are inverses on their own (AA ¼ A, CC ¼ A). It is interesting that there is a","rect":[53.810848236083987,444.6472473144531,385.1586078472313,435.7027282714844]},{"page":29,"text":"one-to-one correspondence between the elements of the group of our four matrices","rect":[53.81185531616211,456.6067810058594,385.1616555606003,447.6722412109375]},{"page":29,"text":"and the elements of the group of complex numbers 1, i, \u00051, \u0005i. The two groups","rect":[53.81185531616211,468.5663146972656,385.10593046294408,459.63177490234377]},{"page":29,"text":"have the same multiplication table. For example, the product of the elements of the","rect":[53.81185531616211,480.4690856933594,385.16968572809068,471.5345458984375]},{"page":29,"text":"first group (matrices) B1C1 ¼ D1 corresponds to the product B2C2 ¼ D2 of the","rect":[53.81185531616211,492.4309997558594,385.1758502788719,483.4864807128906]},{"page":29,"text":"second group (numbers). Therefore, we say that these two groups are isomorphic.","rect":[53.814144134521487,504.3905334472656,383.2447170784641,495.4360656738281]},{"page":29,"text":"The concept of isomorphism is very important. Here, we are interested in the","rect":[65.7661361694336,516.3500366210938,385.17200506402818,507.41552734375]},{"page":29,"text":"groups of elements whose products are symmetry operations. Each set of symmetry","rect":[53.81411361694336,528.3095703125,385.17189875653755,519.3750610351563]},{"page":29,"text":"operations is a group, which may be represented by a group of matrices isomorphic","rect":[53.81411361694336,540.2691040039063,385.16590154840318,531.3146362304688]},{"page":29,"text":"to our group. To demonstrate this, let us go back to group C2v in Fig. 2.5a and pay","rect":[53.813106536865237,552.2291870117188,385.14544001630318,543.2349243164063]},{"page":29,"text":"attention to the coordinate system: this is the basis for representation of the selected","rect":[53.81356430053711,564.188720703125,385.10060969403755,555.2542114257813]},{"page":29,"text":"group in the matrix form. Since the water molecule is two-dimensional the group","rect":[53.81356430053711,576.0914916992188,385.1265801530219,567.156982421875]},{"page":29,"text":"will be represented by 4 \u0003 4 matrices. We shall consider the hydrogen–oxygen","rect":[53.81356430053711,588.051025390625,385.1503838639594,579.1165161132813]},{"page":30,"text":"12","rect":[53.812843322753909,42.454345703125,62.2739118972294,36.73106384277344]},{"page":30,"text":"2 Symmetry","rect":[343.21270751953127,44.274620056152347,385.16522735743447,36.73106384277344]},{"page":30,"text":"bonds as vectors x, y and the symmetry operation will transform them into vectors","rect":[53.812843322753909,68.2883529663086,385.09201444731908,59.35380554199219]},{"page":30,"text":"x’and y’. The identity operation E does not change anything, and this operation is","rect":[53.8138313293457,80.24788665771485,385.1885110293503,71.31333923339844]},{"page":30,"text":"represented by the unit matrix E:","rect":[53.81382369995117,92.20748138427735,186.7732072598655,83.27293395996094]},{"page":30,"text":"E ¼ \u000101 01\u0003;C2 ¼ \u000110 \u000501\u0003;sv ¼ \u000101 \u000501\u0003;s0v ¼ \u000110 10\u0003","rect":[94.03181457519531,130.27342224121095,344.9336355466428,106.38813781738281]},{"page":30,"text":"(2.2)","rect":[366.0972595214844,122.58695220947266,385.16951881257668,114.11058807373047]},{"page":30,"text":"The C2 operation exchanges only y-projections of the two vectors, and this is","rect":[65.76583099365235,153.76303100585938,385.1859475527878,144.82847595214845]},{"page":30,"text":"given by matrix C2. In our particular case, the result of sv operation is the same and","rect":[53.813289642333987,165.72274780273438,385.14565363935005,156.77804565429688]},{"page":30,"text":"the corresponding matrices C2 and sv are equal. Operation sv0 does not change","rect":[53.813777923583987,177.682373046875,385.0877155132469,168.0227813720703]},{"page":30,"text":"positions of the two vectors and is matrices E and sv0 are identical. The sums of the","rect":[53.81356430053711,189.64193725585938,385.17444647027818,179.98243713378907]},{"page":30,"text":"diagonal elements of a matrix is a trace or a character of the matrix. Our matrices","rect":[53.813716888427737,201.6015625,385.16354765044408,192.64707946777345]},{"page":30,"text":"have the following characters 2, 0, 0, 2 corresponding to one of possible representa-","rect":[53.813716888427737,213.56112670898438,385.1435788711704,204.62657165527345]},{"page":30,"text":"tions of group C2v.","rect":[53.813716888427737,225.52066040039063,129.7701839974094,216.5861053466797]},{"page":30,"text":"Therefore, we can operate withthe matricesusing a powerfulapparatus of modern","rect":[65.76561737060547,237.48052978515626,385.1754082780219,228.5459747314453]},{"page":30,"text":"mathematics and obtain important results not at all evident from the beginning. Such","rect":[53.813594818115237,249.38327026367188,385.26699152997505,240.44871520996095]},{"page":30,"text":"a theory of group representations will not be considered here although that theory is","rect":[53.813594818115237,261.3826599121094,385.2419778262253,252.36843872070313]},{"page":30,"text":"very powerful and widely used not only in crystallography [2] but in many areas of","rect":[53.813594818115237,273.3023681640625,385.2649167617954,264.3677978515625]},{"page":30,"text":"physics [1, 5, 6]. In this section we shall only list the sets of symmetry operations","rect":[53.813594818115237,285.2619323730469,385.27191557036596,276.2676086425781]},{"page":30,"text":"corresponding to the most important point symmetry groups. The term “point”","rect":[53.81259536743164,297.2612609863281,385.1574176616844,288.286865234375]},{"page":30,"text":"reflects the fact, that under any operation of the groups listed, at least, one point of","rect":[53.81161880493164,309.1809997558594,385.2619870742954,300.2464599609375]},{"page":30,"text":"the object is not changed. For comparison, when an object is translated in space we","rect":[53.81161880493164,321.1405334472656,385.2261432476219,312.20599365234377]},{"page":30,"text":"should","rect":[53.81161880493164,332.0,79.99617091229925,324.16552734375]},{"page":30,"text":"discussed","rect":[85.02404022216797,332.0,122.45784083417425,324.16552734375]},{"page":30,"text":"its","rect":[127.5633544921875,332.0,136.77698148833469,324.16552734375]},{"page":30,"text":"translational","rect":[141.89442443847657,332.0,190.2898088223655,324.16552734375]},{"page":30,"text":"symmetry","rect":[195.366455078125,333.10009765625,234.6696709733344,325.1815185546875]},{"page":30,"text":"and","rect":[239.77517700195313,332.0,254.04155818036566,324.16552734375]},{"page":30,"text":"corresponding","rect":[259.1470642089844,333.10009765625,315.0479668717719,324.16552734375]},{"page":30,"text":"space","rect":[320.2131652832031,332.99053955078127,342.50659979059068,326.4365234375]},{"page":30,"text":"symmetry","rect":[347.6300354003906,333.0801696777344,385.2529681987938,325.44049072265627]},{"page":30,"text":"groups [7].","rect":[53.81161880493164,345.04266357421877,97.56733365561252,336.2674865722656]},{"page":30,"text":"2.1.3 Point Groups","rect":[53.812843322753909,387.3222961425781,155.9476535770671,376.83990478515627]},{"page":30,"text":"Generally, there is infinite number of point groups, but not all of them correspond to","rect":[53.812843322753909,414.94830322265627,385.13970271161568,406.01373291015627]},{"page":30,"text":"real physical objects such as molecules or crystals. For example, only 32 point","rect":[53.812843322753909,426.9078369140625,385.14668138095927,417.9732666015625]},{"page":30,"text":"groups are compatible with crystal lattices. Each of them is labeled by a certain","rect":[53.812843322753909,438.81060791015627,385.1138543229438,429.87603759765627]},{"page":30,"text":"symbolaccordingto Scho€nflies or according tothe International classifications. The","rect":[53.812843322753909,450.7701721191406,385.1437457866844,441.0]},{"page":30,"text":"Sch€onflies symbols are vivid and more often used in scientific literature. Here we","rect":[53.812843322753909,462.7297058105469,385.1666950054344,453.0]},{"page":30,"text":"present only those point groups we may encounter in the literature on liquid","rect":[53.812843322753909,474.6892395019531,385.10891047528755,465.75469970703127]},{"page":30,"text":"crystals.","rect":[53.812843322753909,486.6487731933594,86.75132413412814,477.7142333984375]},{"page":30,"text":"In the Table 2.2 the Scho€nflies symbols are given on the left side, symmetry","rect":[65.76486206054688,498.6083068847656,385.2631463151313,489.0]},{"page":30,"text":"operations (not symmetry elements) are given on the right side. As we see a number","rect":[53.8138427734375,510.56787109375,385.2423337539829,501.63330078125]},{"page":30,"text":"of symmetry operations increases from top to bottom. Therefore, we say that D6h is","rect":[53.8138427734375,522.52734375,385.24570097075658,513.5928344726563]},{"page":30,"text":"more symmetric phase than, say, C2h. The capital letter D (with index n ¼ 2, 3,6, ...)","rect":[53.81432342529297,534.431396484375,385.15947852937355,525.4966430664063]},{"page":30,"text":"is usedwhen n number ofC2 axes appear, which are perpendicular tothe principal Cn","rect":[53.813655853271487,546.3910522460938,385.18130140630105,537.4464721679688]},{"page":30,"text":"axis. All symmetry operations except sd have been discussed above. The operation","rect":[53.812843322753909,558.3507080078125,385.17949763349068,549.4161987304688]},{"page":30,"text":"sd appears in D4h, D6h etc. to distinguish between vertical reflection planes sv","rect":[53.81382369995117,570.310302734375,385.18130140630105,561.3757934570313]},{"page":30,"text":"containing the C2 axes and additional planes (also vertical) passing along bisectors","rect":[53.812843322753909,582.2699584960938,385.1601907168503,573.3255004882813]},{"page":30,"text":"between already available pairs of the C2 axes.","rect":[53.8133659362793,594.2294921875,239.20760770346409,585.2850341796875]},{"page":31,"text":"2.1 Point Group Symmetry","rect":[53.812843322753909,44.274620056152347,146.2923330703251,36.68026351928711]},{"page":31,"text":"Table 2.2","rect":[53.812843322753909,65.89774322509766,87.50059265284463,59.53101348876953]},{"page":31,"text":"Sch€onflies","rect":[53.813682556152347,78.0,89.05065615653314,71.0]},{"page":31,"text":"C1","rect":[53.813682556152347,90.91644287109375,62.65019738986234,83.77629089355469]},{"page":31,"text":"Cs or C1h","rect":[53.8131217956543,100.93730163574219,85.70556390597563,93.75193786621094]},{"page":31,"text":"Ci or S2","rect":[53.812843322753909,110.8681640625,80.88992822482328,103.72776794433594]},{"page":31,"text":"C2","rect":[53.8131217956543,121.41064453125,62.65019738986234,114.27024841308594]},{"page":31,"text":"C2h","rect":[53.812843322753909,131.329833984375,65.82302224948148,124.18955993652344]},{"page":31,"text":"C2v","rect":[53.812843322753909,141.3508758544922,65.82302224948148,134.16539001464845]},{"page":31,"text":"C3v","rect":[53.812843322753909,151.34927368164063,65.82302224948148,144.1412811279297]},{"page":31,"text":"D2h","rect":[53.812843322753909,161.20062255859376,66.21959436251859,154.2355499267578]},{"page":31,"text":"D3h","rect":[53.812843322753909,171.24423217773438,66.21959436251859,164.15464782714845]},{"page":31,"text":"D6h","rect":[53.812843322753909,181.90011596679688,66.21959436251859,174.81068420410157]},{"page":31,"text":"Some point","rect":[93.46479797363281,67.11690521240235,132.5498629018313,59.52254867553711]},{"page":31,"text":"symbol","rect":[91.4256820678711,79.19002532958985,116.36045555319849,71.59567260742188]},{"page":31,"text":"groups with Scho€nflies and international symbols","rect":[134.9291229248047,67.11690521240235,303.6022384929589,59.0]},{"page":31,"text":"International","rect":[137.4217529296875,78.0,180.83549217429224,71.59567260742188]},{"page":31,"text":"Symmetry operations","rect":[201.9407958984375,79.19002532958985,274.98352511405269,71.59567260742188]},{"page":31,"text":"1","rect":[137.4211883544922,89.49945068359375,141.65172332911417,83.77616882324219]},{"page":31,"text":"m","rect":[137.4211883544922,99.62761688232422,143.49623657804936,95.69920349121094]},{"page":31,"text":"\u0002","rect":[137.4211883544922,104.0,141.65172332911417,103.0]},{"page":31,"text":"1","rect":[137.4211883544922,110.017578125,141.65172332911417,104.29429626464844]},{"page":31,"text":"2","rect":[137.4211883544922,119.99371337890625,141.65172332911417,114.27043151855469]},{"page":31,"text":"2/m","rect":[137.4211883544922,130.81027221679688,150.12380005461186,124.13876342773438]},{"page":31,"text":"mm","rect":[137.4211883544922,140.0410614013672,149.57127930265873,136.11265563964845]},{"page":31,"text":"3m","rect":[137.4211883544922,150.01695251464845,147.74452759367436,144.1412811279297]},{"page":31,"text":"mmm","rect":[137.4211883544922,159.99269104003907,155.64633728605717,156.0642852783203]},{"page":31,"text":"\u0002","rect":[137.4211883544922,164.0,141.65172332911417,163.0]},{"page":31,"text":"6m2","rect":[137.4211883544922,170.59190368652345,152.01736969141886,164.66543579101563]},{"page":31,"text":"6/mmm","rect":[137.4211883544922,181.3128662109375,162.27390076261967,174.641357421875]},{"page":31,"text":"Fig. 2.6 The procedure","rect":[53.812843322753909,232.74935913085938,136.14835498117925,225.01954650878907]},{"page":31,"text":"illustrating that, in NH3","rect":[53.812843322753909,242.6575927734375,134.58957422091704,235.063232421875]},{"page":31,"text":"molecule (group C3v),","rect":[53.812843322753909,252.57659912109376,129.26891586377583,244.98223876953126]},{"page":31,"text":"operations C3 and C32, belong","rect":[53.8130989074707,262.55255126953127,155.3480429091923,253.17628479003907]},{"page":31,"text":"to the same class and","rect":[53.81266403198242,270.8013000488281,126.49664825587198,264.93408203125]},{"page":31,"text":"operations s, s0, s00 belong to","rect":[53.81266403198242,282.5044250488281,155.3582968154423,274.2595520019531]},{"page":31,"text":"another class","rect":[53.8127555847168,290.6708068847656,98.13860781668939,284.8290100097656]},{"page":31,"text":"s","rect":[281.30517578125,263.284912109375,286.12511358951368,259.59771728515627]},{"page":31,"text":"2S3, 2S6,","rect":[308.20709228515627,181.90011596679688,338.9274012763735,174.69215393066407]},{"page":31,"text":"P¢","rect":[306.0885314941406,227.53216552734376,312.5071119297207,222.11732482910157]},{"page":31,"text":"s¢¢","rect":[324.0739440917969,305.9790344238281,332.8425611484707,301.0]},{"page":31,"text":"C3","rect":[315.8310241699219,329.5879211425781,324.15976624182675,322.103515625]},{"page":31,"text":"sh,","rect":[341.2872009277344,181.8323974609375,352.2389857490297,176.0]},{"page":31,"text":"13","rect":[376.74566650390627,42.55594253540039,385.20673126368447,36.73106384277344]},{"page":31,"text":"3sd, 3sv","rect":[354.5987854003906,181.8944854736328,385.18774926974518,174.69215393066407]},{"page":31,"text":"P","rect":[366.2881774902344,286.7691345214844,370.7324319204078,281.4742736816406]},{"page":31,"text":"For the sake of brevity, some operations form classes consisted of conjugate","rect":[65.76496887207031,391.0289611816406,385.1408160991844,382.0744934082031]},{"page":31,"text":"symmetry operations. For example, consider the C3v group: operation 2C3 includes","rect":[53.81296157836914,402.9886169433594,385.1667519961472,394.0440979003906]},{"page":31,"text":"two conjugate operations C3 and C32 (not two C3 axes!). The definition of the","rect":[53.813961029052737,414.94818115234377,385.1744769878563,403.8975524902344]},{"page":31,"text":"conjugate elements of a group is as follows: we say operations A and B of a group","rect":[53.81374740600586,426.8510437011719,385.12770930341255,417.91650390625]},{"page":31,"text":"are conjugate and belong to the same class if XAX\u00051 ¼ B where X is any of the same","rect":[53.81376266479492,438.8108825683594,385.12973821832505,427.75994873046877]},{"page":31,"text":"group","rect":[53.81379318237305,450.7704162597656,77.03696528485784,443.0]},{"page":31,"text":"operations","rect":[82.36247253417969,450.7704162597656,123.91135038481906,441.83587646484377]},{"page":31,"text":"and","rect":[129.26373291015626,449.0,143.6674737077094,441.83587646484377]},{"page":31,"text":"X\u00051","rect":[148.97703552246095,448.62890625,164.60479383794167,439.7196044921875]},{"page":31,"text":"is","rect":[169.8788299560547,449.0,176.5083352481003,441.83599853515627]},{"page":31,"text":"its","rect":[181.77410888671876,449.0,191.20074857817844,441.83599853515627]},{"page":31,"text":"inverse","rect":[196.50137329101563,449.0,225.2988971538719,441.83599853515627]},{"page":31,"text":"operation.","rect":[230.60150146484376,450.7705383300781,270.9001431038547,441.83599853515627]},{"page":31,"text":"Therefore,","rect":[276.20074462890627,449.0,318.1448025276828,441.83599853515627]},{"page":31,"text":"B","rect":[323.44342041015627,448.6290283203125,329.5553054181453,442.05511474609377]},{"page":31,"text":"is","rect":[334.8290710449219,449.0,341.45856107817846,441.83599853515627]},{"page":31,"text":"similarity","rect":[346.724365234375,450.7705383300781,385.1645135026313,441.83599853515627]},{"page":31,"text":"transform of A (note, that XX\u00051 ¼ E). Note that single operation E forms the","rect":[53.813724517822269,462.7301940917969,385.11829412652818,451.6792297363281]},{"page":31,"text":"class on its own because, for any X, XEX\u00051 ¼ EXX\u00051 ¼ E.","rect":[53.81325912475586,474.6897277832031,291.8832974007297,463.6389465332031]},{"page":31,"text":"Since operation s is equal to its own inverse, it is convenient to use it as X and","rect":[65.7664566040039,486.6494140625,385.14629450849068,477.71484375]},{"page":31,"text":"analyze whether operations Cn. form a class or not. For example, consider NH3","rect":[53.814430236816409,498.6091613769531,385.18130140630105,489.6644592285156]},{"page":31,"text":"molecule (group C3v) whose projection along the C3 axis is shown in Fig. 2.6. Let us","rect":[53.812843322753909,510.5688171386719,385.15976347075658,501.63427734375]},{"page":31,"text":"take point P and, at first, make twice C3 operation to arrive at point P0. Then, we","rect":[53.8139762878418,522.4718017578125,385.17160833551255,512.8688354492188]},{"page":31,"text":"start again from point P and, guided by arrows in the figure, make operation","rect":[53.813838958740237,534.431396484375,385.15667048505318,525.4968872070313]},{"page":31,"text":"sC3s\u00051. We again arrive at point P0. Therefore sC3s\u00051 ¼ C32, and operations","rect":[53.813838958740237,546.3910522460938,385.1015664492722,535.3399658203125]},{"page":31,"text":"C3 and C32 belong to the same class. Now, since from the C3 symmetry is evident","rect":[53.81350326538086,558.3507080078125,385.174208236428,547.2999877929688]},{"page":31,"text":"that C3 is inverse of C32, we may find a conjugate of s: C3sC32 ¼ s00. Finally, all","rect":[53.8134880065918,570.310302734375,385.1477494961936,559.2596435546875]},{"page":31,"text":"symmetry operations for the C3v group, i.e. E, C3, C32, s, s0, s00 can be combined in","rect":[53.813899993896487,582.2699584960938,385.14324275067818,571.2192993164063]},{"page":31,"text":"three classes E, 2C3, 3s, as shown in Table 2.2.","rect":[53.814369201660159,593.8115844726563,247.1949429085422,585.2850341796875]},{"page":32,"text":"14","rect":[53.8127555847168,42.45513916015625,62.27382415919229,36.73185729980469]},{"page":32,"text":"2.1.4 Continuous Point Groups","rect":[53.812843322753909,69.85308837890625,216.95455054972335,59.35874557495117]},{"page":32,"text":"2 Symmetry","rect":[343.2126159667969,44.275413513183597,385.16513580470009,36.73185729980469]},{"page":32,"text":"There are also so-called continuous point groups, which include rotations of","rect":[53.812843322753909,97.51880645751953,385.15068946687355,88.54441833496094]},{"page":32,"text":"objects (or coordinate systems) by infinitesimal angles. Therefore the number of","rect":[53.812862396240237,109.43856048583985,385.1457761367954,100.50401306152344]},{"page":32,"text":"their elements tends to a limit that is infinity and the groups themselves are infinite.","rect":[53.812862396240237,121.3980941772461,385.1586575081516,112.46354675292969]},{"page":32,"text":"The continuous point groups were introduced by P. Curie andcan be representedby","rect":[53.812862396240237,133.35763549804688,385.1746758561469,124.42308044433594]},{"page":32,"text":"physical objects [3]. Totally there are seven continuous groups. They are important","rect":[53.812862396240237,145.31716918945313,385.115858627053,136.3826141357422]},{"page":32,"text":"for","rect":[53.812862396240237,156.0,65.51104865876806,148.34214782714845]},{"page":32,"text":"description","rect":[70.74995422363281,157.27670288085938,115.54986659101019,148.34214782714845]},{"page":32,"text":"of","rect":[120.82559204101563,156.0,129.1174558242954,148.34214782714845]},{"page":32,"text":"very","rect":[134.42005920410157,157.27670288085938,152.2281426530219,150.0]},{"page":32,"text":"symmetric","rect":[157.53074645996095,157.27670288085938,199.9834674663719,148.34214782714845]},{"page":32,"text":"liquid","rect":[205.3388214111328,157.27670288085938,228.8028954606391,148.34214782714845]},{"page":32,"text":"crystalline","rect":[234.11346435546876,157.27670288085938,276.1749957866844,148.34214782714845]},{"page":32,"text":"phases","rect":[281.4118957519531,157.27670288085938,308.18576444731908,148.34214782714845]},{"page":32,"text":"such","rect":[313.4167175292969,156.0,331.8180779557563,148.34214782714845]},{"page":32,"text":"as","rect":[337.0938415527344,156.0,345.3757363711472,150.0]},{"page":32,"text":"nematic,","rect":[350.6883239746094,156.0,385.18157620932348,148.34214782714845]},{"page":32,"text":"smectic A and polar nematic phase the existence of which is still under discussion.","rect":[53.812862396240237,169.23623657226563,385.1776394417453,160.3016815185547]},{"page":32,"text":"There are also continuous space groups describing helical (chiral) phases, see","rect":[53.812862396240237,181.13900756835938,385.13184393121568,172.20445251464845]},{"page":32,"text":"below. For the beginning consider the groups of cones. The symmetry of an","rect":[53.812862396240237,193.09857177734376,385.1736992936469,184.1640167236328]},{"page":32,"text":"immobile cone, see Fig. 2.7a, is C1v (or 1m according to the International","rect":[53.812862396240237,205.05923461914063,385.1251054532249,196.12355041503907]},{"page":32,"text":"classification). It includes an infinite order axis C1, an infinite number of symme-","rect":[53.81315231323242,217.01885986328126,385.1335691055454,208.07424926757813]},{"page":32,"text":"try planes sv like the ABC plane but has no sh plane. Therefore, the C1-axis has","rect":[53.81364059448242,228.978515625,385.1427041445847,220.03399658203126]},{"page":32,"text":"properties of a genuine vector. We say this axis is a polar axis and the phase is also","rect":[53.813838958740237,240.93804931640626,385.12380305341255,231.9835662841797]},{"page":32,"text":"polar, in particular, it may possess spontaneous polarization. In the liquid crystal","rect":[53.81283187866211,252.89761352539063,385.1487260586936,243.9630584716797]},{"page":32,"text":"physics this group would describe the polar nematic phase, the very existence of","rect":[53.81283187866211,264.8571472167969,385.14971290437355,255.9026641845703]},{"page":32,"text":"which is still questionable. The rotating cone, see Fig. 2.7b, has polar symmetry","rect":[53.81185531616211,276.7599182128906,385.11284724286568,267.82537841796877]},{"page":32,"text":"reduced to C1 (or 1) because the only symmetry element is a rotation axis Cn with","rect":[53.81185531616211,288.7200927734375,385.1439141373969,279.6952819824219]},{"page":32,"text":"n ¼ 1. Due to rotation there is no symmetry plane. The cone may rotate either","rect":[53.814022064208987,300.67962646484377,385.13106666413918,291.74505615234377]},{"page":32,"text":"clockwise or anti-clockwise and we can say that it has two enantiomorphic","rect":[53.814022064208987,312.63916015625,385.1229938335594,303.70458984375]},{"page":32,"text":"modifications.","rect":[53.814022064208987,322.5667724609375,111.05769773031955,315.6641845703125]},{"page":32,"text":"The next is a series of cylinders. The immobile cylinder has symmetry elements","rect":[65.76604461669922,336.5582580566406,385.17880643950658,327.62371826171877]},{"page":32,"text":"shown in Fig. 2.8a: a rotation axis of infinite order C1, an infinite number of C2","rect":[53.814022064208987,348.5177917480469,385.18130140630105,339.5732727050781]},{"page":32,"text":"axes perpendicular to C1, a horizontal symmetry plane sh and infinite number of","rect":[53.812843322753909,360.4779968261719,385.1517270645298,351.5334777832031]},{"page":32,"text":"vertical symmetry planes sv. Its point group D1h (or 1/mm) corresponds","rect":[53.81388473510742,372.3809814453125,385.15131010161596,363.44622802734377]},{"page":32,"text":"to symmetry of the conventional (non-chiral) nematic or smectic A phase.A","rect":[53.81350326538086,384.3405456542969,385.13644170716136,375.406005859375]},{"page":32,"text":"rotating cylinder, Fig. 2.8b, has no C2 axis perpendicular to the C1 rotation axis","rect":[53.81350326538086,396.3002624511719,385.1553079043503,387.3555603027344]},{"page":32,"text":"but has a horizontal symmetry plane sh (no chirality). Its point group symmetry is","rect":[53.81350326538086,408.2599182128906,385.1866799746628,399.32525634765627]},{"page":32,"text":"C1h (or 1/m). A twisted cylinder, Fig. 2.8c, is chiral therefore has lost all","rect":[53.813961029052737,420.2195739746094,385.14362962314677,411.2850341796875]},{"page":32,"text":"symmetry planes but still has the C1 axis and infinite number of C2 axes perpen-","rect":[53.81177520751953,432.1792297363281,385.1147702774204,423.2345886230469]},{"page":32,"text":"dicular to C1. Therefore, according to Scho€nflies, it keeps the D letter and its","rect":[53.81374740600586,444.1388854980469,385.1532327090378,435.0]},{"page":32,"text":"symmetry","rect":[53.81341552734375,456.0984191894531,93.79151240399847,448.1798400878906]},{"page":32,"text":"group","rect":[98.78952026367188,456.0984191894531,122.01270381024847,449.0]},{"page":32,"text":"is","rect":[126.99876403808594,455.0,133.62826933013157,447.16387939453127]},{"page":32,"text":"D","rect":[138.61134338378907,453.9569091796875,145.75846624329416,447.3630676269531]},{"page":32,"text":"1","rect":[145.7479705810547,455.66717529296877,153.21347866709426,452.6726989746094]},{"page":32,"text":"(or","rect":[158.2099609375,455.7001037597656,169.81656776277198,447.2237548828125]},{"page":32,"text":"12)","rect":[174.80662536621095,455.7001037597656,193.06762061921729,447.2237548828125]},{"page":32,"text":"corresponding","rect":[198.03077697753907,456.0985412597656,255.11916438153754,447.16400146484377]},{"page":32,"text":"to","rect":[260.1131896972656,455.0,267.8874444108344,448.1799621582031]},{"page":32,"text":"chiral","rect":[272.85858154296877,455.0,295.64375169345927,447.16400146484377]},{"page":32,"text":"cholesteric","rect":[300.6706237792969,455.0,344.06003606988755,447.16400146484377]},{"page":32,"text":"or","rect":[349.04412841796877,455.0,357.3359616836704,449.0]},{"page":32,"text":"chiral","rect":[362.29913330078127,455.0,385.0843339688499,447.16400146484377]},{"page":32,"text":"smectic A* phase. Both the twisted and rotating cylinders may be encountered in","rect":[53.81320571899414,468.00128173828127,385.1391533952094,459.06671142578127]},{"page":32,"text":"Fig. 2.7 Continuous groups","rect":[53.812843322753909,551.522216796875,151.07199557303705,543.7924194335938]},{"page":32,"text":"of cones: symmetry elements","rect":[53.812843322753909,561.3737182617188,153.79730685233393,553.7793579101563]},{"page":32,"text":"of an immobile cone C1v","rect":[53.812843322753909,571.0027465820313,141.84387659471077,563.7553100585938]},{"page":32,"text":"(a) and group of a rotating","rect":[53.812843322753909,581.3253173828125,144.81165069727823,573.73095703125]},{"page":32,"text":"cone C1 (b)","rect":[53.812843322753909,590.9624633789063,96.90498441321542,583.698486328125]},{"page":32,"text":"a","rect":[256.71331787109377,509.2099304199219,262.2685802683276,503.6211853027344]},{"page":32,"text":"C","rect":[256.6591796875,574.2177734375,262.43031583603456,568.6829223632813]},{"page":32,"text":"A","rect":[291.2729187011719,525.5187377929688,297.4357247545239,519.783935546875]},{"page":32,"text":"B","rect":[297.12396240234377,559.6986694335938,302.45546906587915,554.3958129882813]},{"page":33,"text":"2.2 Translational Symmetry","rect":[53.812870025634769,44.274497985839847,149.29520172266886,36.68014144897461]},{"page":33,"text":"Fig. 2.8 Continuous groups","rect":[53.812843322753909,67.58130645751953,151.07199557303705,59.6313591003418]},{"page":33,"text":"of cylinders: symmetry","rect":[53.812843322753909,77.4895248413086,132.98392242579386,69.89517211914063]},{"page":33,"text":"elements of an immobile","rect":[53.812843322753909,85.65617370605469,138.80089709543706,79.81436157226563]},{"page":33,"text":"cylinder, group D1h (a), the","rect":[53.812843322753909,97.3846664428711,150.9827971198511,89.79006958007813]},{"page":33,"text":"group of a rotating cylinder","rect":[53.8124885559082,107.36043548583985,148.142426429817,99.76608276367188]},{"page":33,"text":"C1h (b) and chiral group ofa","rect":[53.8124885559082,117.33626556396485,155.36437365793706,109.73356628417969]},{"page":33,"text":"twisted cylinder D1 (c)","rect":[53.812923431396487,127.25545501708985,135.14065641028575,119.66110229492188]},{"page":33,"text":"Fig. 2.9 Continuous groups","rect":[53.812843322753909,209.56692504882813,151.07199557303705,201.53231811523438]},{"page":33,"text":"of spheres: group of a chiral","rect":[53.812843322753909,219.47515869140626,150.44332603659692,211.88079833984376]},{"page":33,"text":"sphere K or R (a) and group","rect":[53.812843322753909,229.45111083984376,150.81647247462198,221.85675048828126]},{"page":33,"text":"of achiral (having mirror","rect":[53.812843322753909,239.42706298828126,138.7543649063795,231.83270263671876]},{"page":33,"text":"symmetry) sphere Kh or O (b)","rect":[53.812843322753909,249.34628295898438,155.3634957657545,241.7344970703125]},{"page":33,"text":"a","rect":[196.50209045410157,68.23361206054688,202.0573528513354,62.64485549926758]},{"page":33,"text":"C2","rect":[200.40740966796876,113.84857177734375,209.17742066077205,106.42615509033203]},{"page":33,"text":"σ","rect":[221.01882934570313,150.04354858398438,225.83876715396677,146.35633850097657]},{"page":33,"text":"v","rect":[225.8384552001953,152.00314331054688,228.8359289615533,149.1897430419922]},{"page":33,"text":"C","rect":[250.5944061279297,151.27859497070313,256.3655422764642,145.7437744140625]},{"page":33,"text":"∞","rect":[256.3668212890625,152.8542022705078,260.641218872759,150.40072631835938]},{"page":33,"text":"σh","rect":[277.44757080078127,106.2672119140625,285.2646856754205,100.69239807128906]},{"page":33,"text":"b","rect":[305.62451171875,68.23361206054688,311.7293054682282,60.92523956298828]},{"page":33,"text":"c","rect":[357.00091552734377,68.23361206054688,362.5561779245776,62.64485549926758]},{"page":33,"text":"15","rect":[376.74652099609377,42.55582046508789,385.20758575587197,36.62934494018555]},{"page":33,"text":"two enantiomorphic modifications. Note that all cylinders and cones are optically","rect":[53.812843322753909,308.9551696777344,385.17461482099068,300.0206298828125]},{"page":33,"text":"uniaxial.","rect":[53.812843322753909,318.8529052734375,88.60977597738986,311.98016357421877]},{"page":33,"text":"The two objects shown in Fig. 2.9 are spheres made of different materials. The","rect":[65.76486206054688,332.874267578125,385.1456989116844,323.939697265625]},{"page":33,"text":"sphere (b) is made of non-chiral material. Such a sphere has full orthogonal","rect":[53.81285095214844,344.8338317871094,385.1626115567405,335.8992919921875]},{"page":33,"text":"symmetry group O(3) or Kh (i.e. 11m): infinite number of C1 axes, infinite","rect":[53.81285095214844,356.7933654785156,385.18308294488755,347.8388977050781]},{"page":33,"text":"number of reflection planes passing through the center of the sphere, an inversion","rect":[53.81438446044922,368.7533264160156,385.14629450849068,359.81878662109377]},{"page":33,"text":"center. Any isotropic achiral liquid has this point symmetry group. However,","rect":[53.81438446044922,380.6560974121094,385.1542629769016,371.7215576171875]},{"page":33,"text":"liquids consisting of chiral molecules, which rotate the light polarization plane","rect":[53.81438446044922,392.61566162109377,385.1433795757469,383.68109130859377]},{"page":33,"text":"(like some sugar solutions), have lower symmetry, Fig. 2.9a; they belong to the full","rect":[53.81438446044922,404.5751953125,385.13728196689677,395.640625]},{"page":33,"text":"rotational group R(3) or K (or 11) because they have lost all symmetry planes and","rect":[53.81438446044922,416.53472900390627,385.14531794599068,407.60015869140627]},{"page":33,"text":"the inversion center. To conclude, the full list of seven continuous point groups","rect":[53.81438446044922,428.4942932128906,385.1045266543503,419.55975341796877]},{"page":33,"text":"includes (in the order of reducing symmetry): spheres (Kh, K), cylinders (D1h,","rect":[53.81438446044922,440.45465087890627,385.17860236330679,431.519287109375]},{"page":33,"text":"C1h, D1,), and cones (C1v and C1).","rect":[53.812843322753909,452.0158386230469,206.61037869955784,443.479736328125]},{"page":33,"text":"2.2 Translational Symmetry","rect":[53.812843322753909,501.49945068359377,206.26490535640307,490.19232177734377]},{"page":33,"text":"(a) Crystals made of atoms with spherical symmetry","rect":[53.812843322753909,528.4600219726563,265.6792682476219,519.5055541992188]},{"page":33,"text":"Such crystals have only translational (i.e., positional) order and no orientational","rect":[53.812843322753909,546.390869140625,385.12678392002177,537.4364013671875]},{"page":33,"text":"order. Their structure is characterized by (i) the point group symmetry of an","rect":[53.81187438964844,558.3504028320313,385.14873591474068,549.4158935546875]},{"page":33,"text":"elementary cell which includes rotations, reflections and inversion as group opera-","rect":[53.81187438964844,570.3099365234375,385.1168149551548,561.3754272460938]},{"page":33,"text":"tion and (ii) the group of translations which includes vectors with their addition as a","rect":[53.81187438964844,582.26953125,385.1547321148094,573.3350219726563]},{"page":33,"text":"group operation. The translation vector is T ¼ n1a + n2b + n3c where a, b, c are unit","rect":[53.81187438964844,594.2290649414063,385.1586137540061,585.2945556640625]},{"page":34,"text":"16","rect":[53.813697814941409,42.55667495727539,62.2747663894169,36.68099594116211]},{"page":34,"text":"Fig. 2.10 The translation","rect":[53.812843322753909,67.58130645751953,142.33426422266886,59.85148620605469]},{"page":34,"text":"vector T ¼ n1a þ n2b þ n3c","rect":[53.812843322753909,77.15409088134766,151.73920581125737,69.90363311767578]},{"page":34,"text":"with n1 ¼ 2, n2 ¼ 1, n3 ¼ 1;","rect":[53.81332015991211,87.07315826416016,151.27998069968286,79.81448364257813]},{"page":34,"text":"(a), (b) and (c) are unit basis","rect":[53.81354904174805,97.04595184326172,152.5711563396386,89.79025268554688]},{"page":34,"text":"vectors","rect":[53.813541412353519,105.60807800292969,78.2914169597558,100.62983703613281]},{"page":34,"text":"c","rect":[232.64163208007813,114.69204711914063,236.19064101352599,110.83686828613281]},{"page":34,"text":"a","rect":[266.1861877441406,147.3459014892578,270.18281942595129,143.5467071533203]},{"page":34,"text":"2 Symmetry","rect":[343.2135925292969,44.275352478027347,385.16611236720009,36.73179626464844]},{"page":34,"text":"T","rect":[340.86358642578127,100.95916748046875,346.19509308931665,95.65629577636719]},{"page":34,"text":"Fig. 2.11 Molecular crystal","rect":[53.812843322753909,193.80966186523438,150.81309227194849,186.07984924316407]},{"page":34,"text":"with rigidly fixed molecules","rect":[53.812843322753909,203.66119384765626,149.98389132499018,196.06683349609376]},{"page":34,"text":"(a) and Euler angles W and F","rect":[53.812843322753909,213.63711547851563,152.84288574797123,205.78875732421876]},{"page":34,"text":"(b) for a particular molecule","rect":[53.813697814941409,223.61306762695313,150.63627002024175,216.01870727539063]},{"page":34,"text":"a","rect":[193.2166748046875,190.66775512695313,198.77193720192134,185.07899475097657]},{"page":34,"text":"c","rect":[242.16819763183595,239.06649780273438,245.7172065652838,235.21131896972657]},{"page":34,"text":"a","rect":[227.257568359375,264.88818359375,231.25420004118565,261.0890197753906]},{"page":34,"text":"b","rect":[303.9427185058594,190.66387939453126,310.04751225533757,183.35549926757813]},{"page":34,"text":"z","rect":[307.76922607421877,208.04421997070313,310.87860552266747,204.59695434570313]},{"page":34,"text":"q","rect":[307.84832763671877,238.50050354003907,312.01281784916548,232.85369873046876]},{"page":34,"text":"y","rect":[318.73016357421877,223.75738525390626,322.2791725076666,218.50250244140626]},{"page":34,"text":"Φ","rect":[347.1136779785156,251.80850219726563,353.21253792495869,246.5136260986328]},{"page":34,"text":"x","rect":[351.7761535644531,262.3726501464844,355.325162497901,258.7574157714844]},{"page":34,"text":"vectors and ni are integers. For example, n1 ¼ 2, n2 ¼ 1, n3 ¼ 1, see Fig. 2.10. The","rect":[53.812843322753909,307.3681335449219,385.14832342340318,298.43359375]},{"page":34,"text":"overall symmetry (the crystal group) is determined by combination of all these","rect":[53.81345748901367,319.32769775390627,385.1264118023094,310.39312744140627]},{"page":34,"text":"elements.","rect":[53.81345748901367,329.16864013671877,91.92613645102267,322.2958984375]},{"page":34,"text":"(b) Molecular crystals","rect":[53.81345748901367,349.17822265625,145.93268217192844,340.2436828613281]},{"page":34,"text":"Due to anisometric (particularly elongated) shape of molecules, these crystals","rect":[53.81345748901367,367.1091003417969,385.1025430117722,358.174560546875]},{"page":34,"text":"possess both the translational and orientational order. The latter is determined by","rect":[53.81345748901367,379.0686340332031,385.16921320966255,370.1141662597656]},{"page":34,"text":"Euler angles W, F such molecules form with respect to selected coordinate frame as","rect":[53.81344223022461,391.0281677246094,385.13840116606908,381.7947998046875]},{"page":34,"text":"shown in the right part of Fig. 2.11. The third Euler angle C describing rotation of a","rect":[53.81344223022461,402.9877014160156,385.1602252788719,393.9435729980469]},{"page":34,"text":"molecule about its longest axis is not shown for simplicity. The point group","rect":[53.814414978027347,414.947265625,385.1264275651313,406.0126953125]},{"page":34,"text":"symmetry includes this orientational order.","rect":[53.814414978027347,426.8500061035156,226.99081082846409,417.91546630859377]},{"page":34,"text":"(c) Plastic crystals and liquid crystals","rect":[53.814414978027347,444.79779052734377,209.28625883208469,435.8632507324219]},{"page":34,"text":"A loss of the orientational order of a molecular crystal due to free rotation of","rect":[53.814414978027347,462.7286682128906,385.1503842910923,453.79412841796877]},{"page":34,"text":"molecules around x, y and z-axes with the positional order remained results in","rect":[53.814414978027347,474.6882019042969,385.14531794599068,465.753662109375]},{"page":34,"text":"plastic crystals. The point group symmetry increases to that characteristic of","rect":[53.81641387939453,486.64776611328127,385.15129981843605,477.69329833984377]},{"page":34,"text":"crystals with spherical atoms. However, such crystals are much softer. An example","rect":[53.81641387939453,498.6073303222656,385.1761859722313,489.67279052734377]},{"page":34,"text":"is solid methane CH4 at low temperature.","rect":[53.81641387939453,510.5688171386719,220.7373013069797,501.63232421875]},{"page":34,"text":"A loss of the translational order (at least, partially) results in liquid crystals","rect":[53.81314468383789,522.4715576171875,385.14401640044408,513.51708984375]},{"page":34,"text":"of different rotational and translational symmetry. On heating, one can observe","rect":[53.81315994262695,534.4310913085938,385.1182025737938,525.4766235351563]},{"page":34,"text":"step-by-step melting and separate phase transitions to less ordered phases of","rect":[53.81315994262695,546.390625,385.1490720352329,537.4561157226563]},{"page":34,"text":"enhanced symmetry. On cooling, correspondingly one observes step-by-step “crys-","rect":[53.81315994262695,558.3501586914063,385.10723243562355,549.3956909179688]},{"page":34,"text":"tallization”. An isotropic liquid is the most symmetric phase, it has full translational","rect":[53.81315994262695,570.3096923828125,385.1251664883811,561.3751831054688]},{"page":34,"text":"and orientational freedom, and this can be written as a product of group multipli-","rect":[53.81315994262695,582.269287109375,385.17888770906105,573.3347778320313]},{"page":34,"text":"cation, O(3) \u0003 T(3), where O(3) is the full orthogonal symmetry (infinite and","rect":[53.81315994262695,594.228759765625,385.14406672528755,585.2742919921875]},{"page":35,"text":"2.2 Translational Symmetry","rect":[53.81199645996094,44.274986267089847,149.2943319716923,36.68062973022461]},{"page":35,"text":"Fig. 2.12 One-dimensional","rect":[53.812843322753909,67.58130645751953,149.30534080710474,59.85148620605469]},{"page":35,"text":"(a) and two-dimensional (b)","rect":[53.812843322753909,77.15087127685547,150.37733548743419,69.89517211914063]},{"page":35,"text":"periodicity in the three-","rect":[53.81199645996094,87.4087142944336,134.56951993567638,79.81436157226563]},{"page":35,"text":"dimensional space","rect":[53.81199645996094,97.3846664428711,116.54660937571049,89.79031372070313]},{"page":35,"text":"∞","rect":[202.22467041015626,124.07731628417969,207.92386718841824,120.8060073852539]},{"page":35,"text":"∞","rect":[341.9310607910156,76.34751892089844,347.6302575692776,73.07621002197266]},{"page":35,"text":"17","rect":[376.74566650390627,42.55630874633789,385.20673126368447,36.73143005371094]},{"page":35,"text":"continuous) group and T is the full infinite continuous group of translations [8].","rect":[53.812843322753909,193.77975463867188,385.1546597054172,184.84519958496095]},{"page":35,"text":"Upon cooling, a series of phase transitions occurs, and each time the symmetry is","rect":[53.81288146972656,205.73928833007813,385.1846047793503,196.8047332763672]},{"page":35,"text":"reduced. Each new point symmetry group is a subgroup of O(3) and a new group of","rect":[53.81288146972656,217.6988525390625,385.1487973770298,208.74436950683595]},{"page":35,"text":"translations is a subgroup of T(3). For example, on transition to the nematic phase","rect":[53.81288146972656,229.65838623046876,385.0969928569969,220.7238311767578]},{"page":35,"text":"the translational freedom is not confined but the rotational symmetry is lowered and","rect":[53.81288146972656,241.61795043945313,385.1428155045844,232.6833953857422]},{"page":35,"text":"the full symmetry is reduced to D1h \u0003 T(3).","rect":[53.81288146972656,253.57745361328126,234.66604276205784,244.6428985595703]},{"page":35,"text":"InFig.2.12a one-dimensionalperiodicstructure isshowninthe three-dimensional","rect":[65.76599884033203,265.5375671386719,385.23847825595927,256.60302734375]},{"page":35,"text":"space. The curve line having point group C1h (or Cs) is repeated with a certain period","rect":[53.8139762878418,277.497314453125,385.28566828778755,268.56256103515627]},{"page":35,"text":"along the horizontal axis. Such translation may be symbolically written as T13. It is","rect":[53.81345748901367,289.40008544921877,385.2453957949753,278.3490905761719]},{"page":35,"text":"the same space group the smectic A phase has, see Fig. Int.1 in Chapter 1. In","rect":[53.81399154663086,301.35980224609377,385.2653435807563,292.42523193359377]},{"page":35,"text":"Fig. 2.12b the two-dimensional hexagonal lattice in the three-dimensional space","rect":[53.81499481201172,313.3193664550781,385.16187322809068,304.38482666015627]},{"page":35,"text":"(T23) is presented. We shall discuss later the smectic B phase having this type of","rect":[53.81499481201172,325.2792053222656,385.26278053132668,314.22845458984377]},{"page":35,"text":"the hexagonal order of molecules.","rect":[53.81339645385742,337.23876953125,188.11361356283909,328.30419921875]},{"page":35,"text":"It should be noted that cholesteric liquid crystals (chiral nematics) having point","rect":[65.76541900634766,349.19830322265627,385.14729173252177,340.26373291015627]},{"page":35,"text":"group symmetry D1 are also periodic with the pitch considerably exceeding a","rect":[53.81339645385742,361.1581726074219,385.1571735210594,352.2236328125]},{"page":35,"text":"molecular size. The preferable direction of the local molecular orientation, i.e. the","rect":[53.81332015991211,373.1177062988281,385.17310369684068,364.18316650390627]},{"page":35,"text":"director oriented along the C1 axis, rotates additionally through subsequent infini-","rect":[53.81332015991211,385.0207824707031,385.1290219864048,376.0859375]},{"page":35,"text":"tesimal angles in the direction perpendicular to that axis. Hence a helical structure","rect":[53.81306076049805,396.9803161621094,385.08725774957505,388.0457763671875]},{"page":35,"text":"forms with a screw axis and continuous translation group.","rect":[53.81306076049805,408.93988037109377,287.2392849495578,400.00531005859377]},{"page":35,"text":"(d) Classification of liquid crystals","rect":[53.81306076049805,426.83087158203127,195.9492532168503,417.87640380859377]},{"page":35,"text":"Liquid crystals can be classified according to","rect":[65.76507568359375,444.8185119628906,247.55181208661566,435.88397216796877]},{"page":35,"text":"(i)","rect":[53.81306076049805,462.33099365234377,63.23969779817236,453.79486083984377]},{"page":35,"text":"(ii)","rect":[53.81306838989258,510.1691589355469,66.03683601228369,501.633056640625]},{"page":35,"text":"(iii)","rect":[53.8140754699707,534.031494140625,68.83497749177588,525.495361328125]},{"page":35,"text":"(iv)","rect":[53.8140754699707,545.9910278320313,68.217820509354,537.4548950195313]},{"page":35,"text":"(v)","rect":[53.8140754699707,557.9506225585938,65.42068610993994,549.4742431640625]},{"page":35,"text":"Their mean of formation: thermotropic (change of temperature and pressure),","rect":[73.75233459472656,462.72943115234377,385.1290554573703,453.79486083984377]},{"page":35,"text":"lyotropic (change of the molecular concentration in water and some other","rect":[73.7523422241211,474.6889953613281,385.13802467195168,465.75445556640627]},{"page":35,"text":"solvents), carbonized (change of polymerization degree), some rare special","rect":[73.7523422241211,486.6485290527344,385.178846908303,477.7139892578125]},{"page":35,"text":"mechanisms (e.g., formation of chain structures in some inorganic substances).","rect":[73.7523422241211,498.6080627441406,385.1051907112766,489.67352294921877]},{"page":35,"text":"Molecular shape, as discussed in Chapter 2, like rod-like or calamitic (from","rect":[74.03504943847656,510.5675964355469,385.1419443953093,501.633056640625]},{"page":35,"text":"Greek kalamoz that means “cane”), discotic, banana- or bent-like, dendrites, etc.","rect":[74.03604888916016,522.3807373046875,385.2684292366672,513.2171020507813]},{"page":35,"text":"Optical properties (uniaxial, biaxial, helical).","rect":[73.86583709716797,534.4298706054688,255.06135221030002,525.4754028320313]},{"page":35,"text":"Chemical classes (biphenyls, Schiff bases, pyrimidines, tolanes, etc).","rect":[73.24270629882813,546.389404296875,350.7392849495578,537.4548950195313]},{"page":35,"text":"Thesymmetryofa liquidcrystallinephase whichdetermines physicalproperties","rect":[73.41291809082031,558.3489990234375,385.1628762637253,549.4144897460938]},{"page":35,"text":"of the phase. This classification is a generalization of the earlier one suggested","rect":[73.41291809082031,570.3085327148438,385.1738213639594,561.3740234375]},{"page":35,"text":"by G. Friedel. In Chapter 3 we consider symmetry and structure of the most","rect":[73.41291809082031,582.26806640625,385.1369767911155,573.3335571289063]},{"page":35,"text":"important liquid crystalline phases.","rect":[73.41291809082031,594.2276000976563,214.91500516440159,585.2930908203125]},{"page":36,"text":"18","rect":[53.81285858154297,42.55771255493164,62.27392715601846,36.73283386230469]},{"page":36,"text":"References","rect":[53.812843322753909,68.09864807128906,109.59614448282879,59.31352233886719]},{"page":36,"text":"2 Symmetry","rect":[343.2127380371094,44.276390075683597,385.16525787501259,36.73283386230469]},{"page":36,"text":"1.","rect":[53.812843322753909,94.0,60.15864440991841,87.80046081542969]},{"page":36,"text":"2.","rect":[53.812843322753909,114.0,60.15864440991841,107.75236511230469]},{"page":36,"text":"3.","rect":[53.812843322753909,134.0,60.15864440991841,127.64750671386719]},{"page":36,"text":"4.","rect":[53.812843322753909,154.0,60.15864440991841,147.59947204589845]},{"page":36,"text":"5.","rect":[53.812843322753909,174.0,60.15864440991841,167.39305114746095]},{"page":36,"text":"6.","rect":[53.812843322753909,193.27142333984376,60.15864440991841,187.395751953125]},{"page":36,"text":"7.","rect":[53.812843322753909,203.19064331054688,60.15864440991841,197.4842987060547]},{"page":36,"text":"8.","rect":[53.812843322753909,213.16659545898438,60.15864440991841,207.3417205810547]},{"page":36,"text":"Landau, L.D., Lifshits, E.M.: Quantum Mechanics, 3rd edn. Nauka, Moscow (1974). Ch.12","rect":[64.40525817871094,95.00536346435547,385.18222564845009,87.74966430664063]},{"page":36,"text":"(in Russian)","rect":[64.4052505493164,104.98131561279297,105.77565091712167,97.72561645507813]},{"page":36,"text":"Pikin, S.A.: Structural Transformations in Liquid Crystals. Gordon & Breach, New York","rect":[64.40525817871094,115.2959213256836,385.15085357813759,107.70156860351563]},{"page":36,"text":"(1991)","rect":[64.4052505493164,124.87645721435547,86.96246427161386,117.67155456542969]},{"page":36,"text":"Sonin, A.S.: Besedy o kristallofizike (Talks on Crystal Physics). Atomizdat, Moscow (1976)","rect":[64.40525817871094,135.19107055664063,385.2016610489576,127.59671020507813]},{"page":36,"text":"(in Russian)","rect":[64.4052505493164,144.8284149169922,105.77565091712167,137.57272338867188]},{"page":36,"text":"Hargittai I., Hargittai, M.: Symmetry Through the Eyes of a Chemist. VCH Verlaggesellschaft,","rect":[64.40525817871094,155.14303588867188,385.1118800361391,147.54867553710938]},{"page":36,"text":"New York (1986)","rect":[64.4052505493164,164.7235565185547,125.02797025305917,157.46786499023438]},{"page":36,"text":"Bhagavantam, S., Venkatarayudu, T.: Theory of Groups and Its Applications to Physical","rect":[64.40525817871094,175.0382080078125,385.17696861472197,167.44384765625]},{"page":36,"text":"Problems, 2nd edn. Andhra University, Waltair (1951)","rect":[64.4052505493164,185.01416015625,251.17749112708263,177.36900329589845]},{"page":36,"text":"Weyl H.: Symmetry. Princeton University Press, Princeton (1952)","rect":[64.40525817871094,194.9901123046875,290.77276700598886,187.34495544433595]},{"page":36,"text":"Kittel, Ch: Introduction to Solid State Physics, 4th edn. Wiley, New York (1971)","rect":[64.40525817871094,204.90933227539063,342.03585904700449,197.31497192382813]},{"page":36,"text":"Kats, E.I.: New types of ordering in liquid crystals. Uspekhi Fis. Nauk 142, 99–129 (1984)","rect":[64.40525817871094,214.88528442382813,375.68295377356699,207.2231903076172]},{"page":37,"text":"Chapter3","rect":[53.812843322753909,72.10812377929688,114.14115996551633,59.25117874145508]},{"page":37,"text":"Mesogenic Molecules and Orientational Order","rect":[53.812843322753909,90.6727066040039,371.94911733396676,76.10637664794922]},{"page":37,"text":"3.1 Molecular Shape and Properties","rect":[53.812843322753909,212.0922088623047,246.9186008426921,201.07196044921876]},{"page":37,"text":"A great variety of organic molecules can form liquid crystalline phases. They are","rect":[53.812843322753909,239.63455200195313,385.1636432476219,230.6999969482422]},{"page":37,"text":"called mesogenic molecules and belong to different chemical classes, see the","rect":[53.812843322753909,251.5941162109375,385.1158832378563,242.65956115722657]},{"page":37,"text":"comprehensive book by G. Gray on chemical aspects [1] and his review articles","rect":[53.812843322753909,263.49688720703127,385.1626321231003,254.5623321533203]},{"page":37,"text":"[2, 3]. The discussion of more recent achievements in the chemistry of liquid","rect":[53.81385040283203,275.4564208984375,385.1138543229438,266.5218505859375]},{"page":37,"text":"crystals may be found in beautifully illustrated article by Hall et al. [4].","rect":[53.81385040283203,287.4159851074219,342.6153530647922,278.4814453125]},{"page":37,"text":"3.1.1 Shape, Conformational Mobility and Isomerization","rect":[53.812843322753909,337.58349609375,343.2207333455002,326.9457092285156]},{"page":37,"text":"Figure 3.1 represents the characteristic types of mesogenic molecules. Among them","rect":[53.812843322753909,365.1258239746094,385.12982891679368,356.1912841796875]},{"page":37,"text":"are rods, laths, discs, helices which are more popular for physical investigations and","rect":[53.812843322753909,377.0853576660156,385.14272395185005,368.15081787109377]},{"page":37,"text":"technological applications and also main-chain and side-chain polymers. We may","rect":[53.812843322753909,389.044921875,385.1646355729438,380.1103515625]},{"page":37,"text":"add to this list banana- or bent-shape molecules and dendrimers [4] that recently","rect":[53.812843322753909,400.94769287109377,385.1098565202094,392.01312255859377]},{"page":37,"text":"become very popular.","rect":[53.812843322753909,412.9072265625,141.32928128744846,403.97265625]},{"page":37,"text":"Rigid rods (a), laths (b) and disks (c) have no conformational degree of freedom.","rect":[65.76486206054688,424.86676025390627,385.1547512581516,415.93218994140627]},{"page":37,"text":"They are very convenient for theoretical discussions and computer simulations of","rect":[53.812843322753909,436.8263244628906,385.1477292617954,427.89178466796877]},{"page":37,"text":"the mesophase structure. Closer to reality are rods (or disks) with flexible tails","rect":[53.812843322753909,448.7858581542969,385.1477090273972,439.851318359375]},{"page":37,"text":"(hydrocarbon chains) shown in Fig. 3.2a, which facilitate formation of layered","rect":[53.812843322753909,460.7453918457031,385.0879754166938,451.81085205078127]},{"page":37,"text":"liquid crystal phases. As an example of conformational degrees of freedom of","rect":[53.8138313293457,472.7049255371094,385.14971290437355,463.7703857421875]},{"page":37,"text":"flexible molecular fragments is the trans–cis isomerization. In sketch 3.2b trans-","rect":[53.8138313293457,484.6644592285156,385.15960059968605,475.72991943359377]},{"page":37,"text":"form is on the left, cis-form in the middle, a combination of the two on the right.","rect":[53.81379318237305,496.5672302246094,385.1238064339328,487.6326904296875]},{"page":37,"text":"The rotational isomerization is another example: in sketch 3.2c the internal rotation","rect":[53.813804626464847,508.5267639160156,385.1158074479438,499.59222412109377]},{"page":37,"text":"of phenyl rings about the single bond in a biphenyl moiety is sketched.","rect":[53.813804626464847,520.486328125,340.3179592659641,511.55181884765627]},{"page":37,"text":"A molecule having the same chemical structure can exist in different atomic","rect":[65.76582336425781,532.4458618164063,385.1258014507469,523.5113525390625]},{"page":37,"text":"configurations [5]. It forms different stereoisomers either mesogenic or not. One","rect":[53.813804626464847,544.4053955078125,385.16260564996568,535.4111328125]},{"page":37,"text":"important example is a molecule of cyclohexane (CH in Fig. 3.3) having all the","rect":[53.813804626464847,556.3649291992188,385.1745075054344,547.430419921875]},{"page":37,"text":"bonds single. The cyclohexane can acquire a form of either chair or trough to be","rect":[53.81377410888672,568.3245239257813,385.15161932184068,559.3900146484375]},{"page":37,"text":"compared with a flat form of the benzene molecule having conjugated single and","rect":[53.81377410888672,580.2840576171875,385.1426934342719,571.3495483398438]},{"page":37,"text":"double","rect":[53.81377410888672,591.0,81.03455389215314,583.2523193359375]},{"page":37,"text":"bonds.","rect":[86.89459991455078,591.0,113.12394376303439,583.2523193359375]},{"page":37,"text":"Moreover,","rect":[119.06859588623047,591.0,160.91408963705784,583.4515380859375]},{"page":37,"text":"the","rect":[166.81991577148438,591.0,179.0436939068016,583.2523193359375]},{"page":37,"text":"cyclohexane","rect":[184.946533203125,592.1868286132813,235.0520709819969,583.2523193359375]},{"page":37,"text":"molecule","rect":[240.912109375,591.0,277.67002142145005,583.2523193359375]},{"page":37,"text":"reveals","rect":[283.5649108886719,591.0,311.92047514067846,583.2523193359375]},{"page":37,"text":"another","rect":[317.8352966308594,591.0,347.77755103913918,583.2523193359375]},{"page":37,"text":"type","rect":[353.74810791015627,592.1868286132813,370.94898260309068,584.2682495117188]},{"page":37,"text":"of","rect":[376.8587951660156,591.0,385.1506284317173,583.2523193359375]},{"page":37,"text":"L.M. Blinov, Structure and Properties of Liquid Crystals,","rect":[53.812843322753909,623.7623291015625,251.63688919141254,616.1510620117188]},{"page":37,"text":"DOI 10.1007/978-90-481-8829-1_3, # Springer ScienceþBusiness Media B.V. 2011","rect":[53.812843322753909,633.698486328125,347.38995880274697,625.4920043945313]},{"page":37,"text":"19","rect":[376.7464599609375,622.1282958984375,385.2075247207157,616.2357177734375]},{"page":38,"text":"20","rect":[53.81368637084961,42.55667495727539,62.2747549453251,36.73179626464844]},{"page":38,"text":"3 Mesogenic Molecules and Orientational Order","rect":[219.67013549804688,44.275352478027347,385.1415109024732,36.6640625]},{"page":38,"text":"a","rect":[120.96314239501953,122.09378051757813,126.51840479225338,116.5050277709961]},{"page":38,"text":"b","rect":[153.60231018066407,122.09378051757813,159.70710393014228,114.78540802001953]},{"page":38,"text":"c","rect":[194.56736755371095,122.09378051757813,200.12262995094478,116.5050277709961]},{"page":38,"text":"d","rect":[233.5341339111328,122.09378051757813,239.63892766061103,114.78540802001953]},{"page":38,"text":"e","rect":[271.1690368652344,122.09378051757813,276.7242992624682,116.5050277709961]},{"page":38,"text":"f","rect":[312.466796875,121.96380615234375,315.79395942586128,114.69542694091797]},{"page":38,"text":"g","rect":[125.29244995117188,198.773681640625,131.39724370065009,191.15538024902345]},{"page":38,"text":"h","rect":[190.237060546875,196.61416625976563,196.3418542963532,189.43576049804688]},{"page":38,"text":"i","rect":[254.84896850585938,196.61416625976563,257.6265997044763,189.36578369140626]},{"page":38,"text":"j","rect":[304.80633544921877,198.7436981201172,307.58396664783569,189.36578369140626]},{"page":38,"text":"Fig. 3.1 Different forms of mesogenic molecules: rods (a), lath-like (b), discs (c), swallow-tail (d),","rect":[53.812843322753909,218.7491455078125,385.2481410224672,210.81613159179688]},{"page":38,"text":"bowls (e) double swallow-tails (f), main-chain (g) and comb-like (h) polymers, propellers (i) and","rect":[53.81365203857422,228.66836547851563,385.1948294082157,220.89622497558595]},{"page":38,"text":"spirals (j)","rect":[53.813682556152347,238.6527862548828,86.76615995032479,230.9145050048828]},{"page":38,"text":"Fig. 3.2 Different degrees","rect":[53.812843322753909,266.5311584472656,144.53073580741205,258.59814453125]},{"page":38,"text":"of freedom for non-rigid","rect":[53.812843322753909,276.43939208984377,135.25064605860636,268.84503173828127]},{"page":38,"text":"mesogenic molecules:","rect":[53.812843322753909,286.41534423828127,127.56376365866724,278.82098388671877]},{"page":38,"text":"molecules with flexible tails","rect":[53.812843322753909,294.5820007324219,147.41597445487299,288.7402038574219]},{"page":38,"text":"(a), trans-, cis- and combined","rect":[53.812843322753909,305.97186279296877,152.4723104873173,298.7161560058594]},{"page":38,"text":"trans-cis isomerisation of the","rect":[53.813690185546878,314.6440124511719,150.76571032785894,308.692138671875]},{"page":38,"text":"flexible chains (b); rotational","rect":[53.813690185546878,325.8670654296875,150.44924644675317,318.6113586425781]},{"page":38,"text":"isomerism of biphenyl","rect":[53.81368637084961,336.1816711425781,128.73562339987817,328.5873107910156]},{"page":38,"text":"moiety (c)","rect":[53.81368637084961,346.1576232910156,88.2959222183912,338.5632629394531]},{"page":38,"text":"a","rect":[248.33282470703126,263.3852233886719,253.88808710426509,257.7964782714844]},{"page":38,"text":"b","rect":[248.4577178955078,291.57293701171877,254.56251164498603,284.2645568847656]},{"page":38,"text":"c","rect":[248.4577178955078,316.47137451171877,254.01298029274165,310.88262939453127]},{"page":38,"text":"a","rect":[106.83275604248047,381.6214294433594,112.38801843971432,376.0326843261719]},{"page":38,"text":"b","rect":[183.64224243164063,381.6214294433594,189.74703618111884,374.31304931640627]},{"page":38,"text":"c","rect":[186.01573181152345,392.7092590332031,189.5647407449713,388.9180603027344]},{"page":38,"text":"t","rect":[209.49594116210938,392.6772766113281,211.7180683771961,387.8623046875]},{"page":38,"text":"c","rect":[261.20111083984377,381.6214294433594,266.7563732370776,376.0326843261719]},{"page":38,"text":"c","rect":[239.22048950195313,442.9489440917969,242.76949843540099,439.1577453613281]},{"page":38,"text":"t","rect":[260.4530029296875,439.1673889160156,262.67513014477427,434.3524169921875]},{"page":38,"text":"Fig. 3.3 Rigid benzene molecule (a) and chair (b) and trough (c) isomeric forms of a cyclohexane","rect":[53.812843322753909,464.4605712890625,385.1449522712183,456.5275573730469]},{"page":38,"text":"molecule","rect":[53.813697814941409,472.6162414550781,84.98427722239018,466.7744445800781]},{"page":38,"text":"isomerization. The hydrogen atoms marked by t and c letters are in nonequivalent","rect":[53.812843322753909,498.2691345214844,385.1735978848655,489.3345947265625]},{"page":38,"text":"positions with respect to the longest molecular axes: only the trans-position is","rect":[53.81385040283203,510.1719055175781,385.1885110293503,501.23736572265627]},{"page":38,"text":"compatible with that axis. It is known that atoms in different positions have","rect":[53.81385040283203,522.1314086914063,385.11789739801255,513.1968994140625]},{"page":38,"text":"different chemical reactivity. For instance, the –COOH group can be attached to","rect":[53.81385040283203,534.0909423828125,385.13979426435005,525.136474609375]},{"page":38,"text":"the cyclohexane moiety in the trans-position. Then, a combination of the chair CH","rect":[53.81385040283203,546.050537109375,385.13976812317699,537.1160278320313]},{"page":38,"text":"structure with the trans-position of that group, due to a chemical reaction, results in","rect":[53.81385803222656,558.0100708007813,385.14177790692818,549.0755615234375]},{"page":38,"text":"an elongated overall structure of the new-synthesized molecule, which is more","rect":[53.81385803222656,569.9696044921875,385.1377643413719,561.0350952148438]},{"page":38,"text":"appropriate for liquid crystal formation. In addition, elongated dimers can form due","rect":[53.81385803222656,581.9291381835938,385.14185369684068,572.99462890625]},{"page":38,"text":"to H-bonds between –COOH groups, see below.","rect":[53.81385803222656,593.888671875,248.39182706381565,584.9342041015625]},{"page":39,"text":"3.1 Molecular Shape and Properties","rect":[53.813499450683597,44.276329040527347,176.07088168143549,36.68197250366211]},{"page":39,"text":"3.1.2 Symmetry and Chirality","rect":[53.812843322753909,69.93675231933594,208.10087378950326,59.298980712890628]},{"page":39,"text":"21","rect":[376.7480163574219,42.4560546875,385.20908111720009,36.73277282714844]},{"page":39,"text":"The word chirality originates from Greek wiros (hand). Chiral objects (and","rect":[53.812843322753909,97.47896575927735,385.1177910905219,88.52449798583985]},{"page":39,"text":"molecules) have no mirror symmetry (no one mirror plane). Examples of such","rect":[53.81280517578125,109.43856048583985,385.12282649091255,100.50401306152344]},{"page":39,"text":"objects are spirals, propellers, screws, hands. Note that the symmetry of a liquid","rect":[53.81280517578125,121.3980941772461,385.11382380536568,112.46354675292969]},{"page":39,"text":"crystal phase is not the same as the symmetry of constituent molecules but they","rect":[53.81280517578125,133.35763549804688,385.16359797528755,124.42308044433594]},{"page":39,"text":"often share some symmetry elements. As an example let us look at the symmetry of","rect":[53.81280517578125,145.31716918945313,385.1476987442173,136.3826141357422]},{"page":39,"text":"a “brick” and a “building” in Fig. 3.4. They are different although it is not a","rect":[53.81280517578125,157.27670288085938,385.1596149273094,148.34214782714845]},{"page":39,"text":"convincing example, because our tower has not been erected by self-assembling","rect":[53.81280517578125,169.23623657226563,385.09694758466255,160.3016815185547]},{"page":39,"text":"of bricks.","rect":[53.81280517578125,179.07717895507813,91.75128598715549,172.20445251464845]},{"page":39,"text":"Chiral molecules, only left or only right, form chiral phases, left and right chiral","rect":[65.76482391357422,193.09857177734376,385.139784408303,184.1640167236328]},{"page":39,"text":"molecules in equal amount form achiral (enantiomorphic) phases [6]. Consider a","rect":[53.81280517578125,205.05810546875,385.1605609722313,196.12355041503907]},{"page":39,"text":"chiral molecule of a popular compound DOBAMBC (D(or L)-p-decyloxybenzyli-","rect":[53.81280517578125,217.01763916015626,385.1446469864048,208.0631561279297]},{"page":39,"text":"dene-p0-amino-2methylbutyl cinnamate). It has an asymmetric carbon in its tail and","rect":[53.81280517578125,228.978515625,385.14150324872505,219.31898498535157]},{"page":39,"text":"form a chiral SmC* phase in the range of 95–117\u0002C, Fig. 3.5a. A molecule witha","rect":[53.812564849853519,240.93817138671876,385.16098821832505,231.9437255859375]},{"page":39,"text":"chiral tail looks like an ice-hockey stick and forms a helical liquid crystal phase.","rect":[53.81417465209961,252.89773559570313,385.17986722494848,243.9631805419922]},{"page":39,"text":"Left and right forms of a chiral tail result in the left and right handedness of a","rect":[53.81417465209961,264.8572692871094,385.15903509332505,255.92271423339845]},{"page":39,"text":"molecule Fig. 3.6. On the other hand, chirality of cholesterol esters is exclusively","rect":[53.81417465209961,276.7600402832031,385.1729668717719,267.8055725097656]},{"page":39,"text":"due to a curvature of the molecular skeleton Fig. 3.5b.","rect":[53.81417465209961,288.7196044921875,273.1392483284641,279.72528076171877]},{"page":39,"text":"The synthesis of chiral molecules is a real challenge. There are, at least, three","rect":[65.76619720458985,300.67913818359377,385.1182330913719,291.74456787109377]},{"page":39,"text":"different approaches.","rect":[53.81417465209961,312.638671875,138.50359769369846,303.7041015625]},{"page":39,"text":"(i) A chemist needs simple chiral molecules as initial or intermediary reagents.","rect":[53.81417465209961,330.5495910644531,385.15008206869848,321.61505126953127]},{"page":39,"text":"They can be found among natural substances because the Nature selects left or","rect":[68.25872802734375,342.5091552734375,385.15108619538918,333.5745849609375]},{"page":39,"text":"right forms. For example, left (or right) amino acids can be used. Then the","rect":[68.25872802734375,354.46868896484377,385.17197454645005,345.53411865234377]},{"page":39,"text":"synthesis can be continued until the left (or right) form of the final chiral product","rect":[68.25872802734375,366.4282531738281,385.2048173672874,357.49371337890627]},{"page":39,"text":"is obtained.","rect":[68.25872802734375,376.3558349609375,114.6085171272922,369.4532470703125]},{"page":39,"text":"a","rect":[213.0735321044922,433.42767333984377,218.62879450172603,427.83892822265627]},{"page":39,"text":"b","rect":[309.9908752441406,433.42767333984377,316.0956689936188,426.1192932128906]},{"page":39,"text":"Fig. 3.4 Illustration of","rect":[53.812843322753909,552.939208984375,133.19714444739513,545.0062255859375]},{"page":39,"text":"different symmetry (D2h and","rect":[53.812843322753909,562.847412109375,151.1371054091923,555.2529907226563]},{"page":39,"text":"D1h) of a brick (a) and an","rect":[53.812808990478519,572.4845581054688,143.84773773341105,565.2288208007813]},{"page":39,"text":"architectural “masterpiece”","rect":[53.81349563598633,582.742431640625,146.79471728586675,575.1480712890625]},{"page":39,"text":"made from the same bricks (b)","rect":[53.81349563598633,592.3797607421875,155.3632516251295,585.1240234375]},{"page":39,"text":"D2h","rect":[237.2337646484375,512.3575439453125,250.11878134503923,504.86322021484377]},{"page":39,"text":"D¥h","rect":[289.50872802734377,550.7722778320313,303.6707527805861,543.2778930664063]},{"page":40,"text":"22","rect":[53.812843322753909,42.454833984375,62.2739118972294,36.73155212402344]},{"page":40,"text":"a","rect":[84.51457214355469,68.23348999023438,90.06977887770649,62.64473342895508]},{"page":40,"text":"3 Mesogenic Molecules and Orientational Order","rect":[219.66928100585938,44.275108337402347,385.1406564102857,36.663818359375]},{"page":40,"text":"asymmetric carbon","rect":[215.7133026123047,71.44844818115235,289.01138622180027,63.946048736572269]},{"page":40,"text":"C10H21","rect":[85.49423217773438,100.97498321533203,110.368169338678,92.97238159179688]},{"page":40,"text":"b","rect":[84.51457214355469,144.83889770507813,90.6193047237107,137.530517578125]},{"page":40,"text":"Cholesteryl acetate","rect":[105.18057250976563,162.65750122070313,173.59405533391669,155.05911254882813]},{"page":40,"text":"Cr-26oC-Ch-41oC-Iso","rect":[105.18070983886719,173.80470275878907,181.1236268671198,167.43406677246095]},{"page":40,"text":"C8H17-CH=CH-(CH2)11)OCO","rect":[116.94793701171875,200.51083374023438,219.2991100849962,192.508544921875]},{"page":40,"text":"Fig. 3.5 Chemical formulas of two important chiral","rect":[53.812843322753909,222.26336669921876,233.93888572897974,214.33035278320313]},{"page":40,"text":"acetate (b)","rect":[53.812843322753909,231.8329315185547,90.2208260880201,224.62803649902345]},{"page":40,"text":"Fig. 3.6 Asymmetric","rect":[53.812843322753909,259.6161804199219,128.58331439280034,251.68316650390626]},{"page":40,"text":"carbon and its left","rect":[53.812843322753909,267.7972717285156,115.6387300893313,261.9300537109375]},{"page":40,"text":"or right surrounding","rect":[53.812843322753909,279.4436340332031,122.50912231593057,271.8492736816406]},{"page":40,"text":"CH=CH-COO-CH2-C*","rect":[242.07498168945313,101.33453369140625,319.7765768320489,93.44021606445313]},{"page":40,"text":"CH3","rect":[313.9407653808594,123.53833770751953,328.8143058132874,115.53628540039063]},{"page":40,"text":"molecules: DOBAMBC (a)","rect":[236.4365997314453,221.8569793701172,330.73602384192636,214.5843505859375]},{"page":40,"text":"Right","rect":[236.62716674804688,262.24627685546877,255.28343702953985,254.7518768310547]},{"page":40,"text":"H","rect":[243.0509490966797,274.40570068359377,248.82208524521426,268.6629333496094]},{"page":40,"text":"C2H5","rect":[336.2591247558594,101.93073272705078,354.4665885281311,93.92843627929688]},{"page":40,"text":"and cholesteryl","rect":[333.18634033203127,222.19564819335938,385.1593294545657,214.60128784179688]},{"page":40,"text":"Left","rect":[344.0763244628906,258.5324401855469,357.4090778986805,252.59771728515626]},{"page":40,"text":"H","rect":[347.99102783203127,273.36737060546877,353.7621639805658,267.6246032714844]},{"page":40,"text":"CH3","rect":[221.7089385986328,314.47119140625,236.58430471423166,306.46923828125]},{"page":40,"text":"C2H5","rect":[257.70989990234377,315.46234130859377,275.9185484886457,307.46044921875]},{"page":40,"text":"C2H5","rect":[310.6756286621094,313.9538269042969,328.88418569567696,305.9515380859375]},{"page":40,"text":"CH3","rect":[370.3321838378906,309.28485107421877,385.20758047106758,301.28302001953127]},{"page":40,"text":"(ii) As shown by Pasteur, chiral solutes can crystallize from a solution in the form","rect":[53.812843322753909,367.10968017578127,385.1337656843718,358.17510986328127]},{"page":40,"text":"of left and right optically active crystals and left and right chiral isomers can be","rect":[71.03263092041016,379.0692138671875,385.15073431207505,370.1346435546875]},{"page":40,"text":"separated.","rect":[71.03263092041016,391.0287780761719,111.32329221274142,382.09423828125]},{"page":40,"text":"(iii) The synthesis can be made in the chiral conditions (e.g., in a chiral solution,","rect":[53.812843322753909,402.9883117675781,385.1496853401828,394.05377197265627]},{"page":40,"text":"like a cholesteric liquid crystal, or on special substrates, or using a chiral, one","rect":[70.86341094970703,414.9478454589844,385.14279974176255,406.0133056640625]},{"page":40,"text":"directional stirring, etc.).","rect":[70.86341094970703,426.8506164550781,170.7210659554172,417.91607666015627]},{"page":40,"text":"(iv) Chirality can be created optically by circularly polarized light.","rect":[53.812843322753909,438.8101806640625,322.1843227913547,429.8756103515625]},{"page":40,"text":"3.1.3 Electric and Magnetic Properties","rect":[53.812843322753909,488.89434814453127,253.99605445597335,478.3402404785156]},{"page":40,"text":"(a) Polarizability","rect":[53.812843322753909,516.5003662109375,124.88690984918439,507.5658264160156]},{"page":40,"text":"All atoms and molecules can be polarized by an electric field. The polarization","rect":[53.812843322753909,534.4312133789063,385.0870293717719,525.4967041015625]},{"page":40,"text":"(induced dipole of a unit volume) is P ¼ aE where a is molecular polarizability. For","rect":[53.812843322753909,546.3907470703125,385.14971290437355,537.4562377929688]},{"page":40,"text":"spherically symmetric atoms or molecules (like C60 fullerenes) the polarizability is","rect":[53.812862396240237,558.3502807617188,385.1826211367722,549.415771484375]},{"page":40,"text":"a scalar quantity (tensor of zero rank) and P||E. In general case of lath-like mole-","rect":[53.812862396240237,570.309814453125,385.1028989395298,560.9967651367188]},{"page":40,"text":"cules, aij is a second rank tensor (9 components) and Pj ¼ aijEi. By a proper choice of","rect":[53.812862396240237,583.1865234375,385.15242896882668,573.3348999023438]},{"page":40,"text":"the reference frame the tensor can be diagonalized","rect":[53.813594818115237,594.2294921875,255.35552302411566,585.2949829101563]},{"page":41,"text":"3.1 Molecular Shape and Properties","rect":[53.812843322753909,44.276573181152347,176.0702255535058,36.68221664428711]},{"page":41,"text":"23","rect":[376.7473449707031,42.55789566040039,385.2084097304813,36.73301696777344]},{"page":41,"text":"\u0001\u0001 axx","rect":[198.82452392578126,71.58540344238281,214.91005844965359,59.306087493896487]},{"page":41,"text":"0","rect":[228.73312377929688,68.02351379394531,233.71022883466254,61.17070770263672]},{"page":41,"text":"0 \u0001\u0001","rect":[250.03115844726563,71.58442687988281,262.81017508924927,59.305110931396487]},{"page":41,"text":"aij ¼ \u0001\u0001 0 ayy 0 \u0001\u0001","rect":[176.1664276123047,83.43135070800781,262.81017508924927,71.151123046875]},{"page":41,"text":"\u0001\u0001 0","rect":[198.82452392578126,95.27735900878906,212.01563349774848,82.94134521484375]},{"page":41,"text":"0 azz \u0001\u0001","rect":[228.73373413085938,95.27638244628906,262.81017508924927,82.94036865234375]},{"page":41,"text":"and components axx, ayy and azz represent three principal molecular polarizabilities.","rect":[53.81357955932617,119.70750427246094,385.1584133675266,109.79777526855469]},{"page":41,"text":"For molecules having cylindrical symmetry (rods or disks) with the symmetry axis","rect":[53.813594818115237,130.69381713867188,385.1504250918503,121.75926208496094]},{"page":41,"text":"z, only two different components remain axx ¼ ayy ¼ a⊥ and azz ¼ a||.","rect":[53.813594818115237,143.5701141357422,338.1110806038547,133.7187957763672]},{"page":41,"text":"(b) Permanent dipole moments","rect":[53.81386947631836,160.45497131347657,180.22536100493626,151.6100616455078]},{"page":41,"text":"If a molecule has an inversion center it is non-polar and its dipole moment (a vector,","rect":[53.81386947631836,178.53225708007813,385.1119045784641,169.5977020263672]},{"page":41,"text":"i.e., a tensor of rank 1) pe ¼ 0. In a less symmetric case pe is finite. It is measured in","rect":[53.812862396240237,190.49224853515626,385.1432732682563,181.55726623535157]},{"page":41,"text":"units Debye and in the Gauss system 1D ¼ 10\u000318 CGSQ\u0004cm (3.3 \u0004 10\u000330 C\u0004m in SI","rect":[53.813411712646487,202.39501953125,385.1606381973423,191.3442840576172]},{"page":41,"text":"system). More vividly, 1D corresponds to one electron positive and one electron","rect":[53.81481170654297,214.354736328125,385.10991755536568,205.42018127441407]},{"page":41,"text":"˚","rect":[269.17913818359377,216.69741821289063,272.4938901504673,214.7152862548828]},{"page":41,"text":"negative charges separated by a distance of \u00050.2 A. The dipole moment of a","rect":[53.81481170654297,226.31430053710938,385.16159856988755,217.37974548339845]},{"page":41,"text":"complex molecule can be estimated as a vector sum of the moments of all intra-","rect":[53.81481170654297,238.27383422851563,385.1417172989048,229.3392791748047]},{"page":41,"text":"molecular chemical bonds, pe ¼ S pi. Consider two classical examples shown in","rect":[53.81481170654297,250.23379516601563,385.14235774091255,241.2593994140625]},{"page":41,"text":"Fig. 3.7.","rect":[53.813472747802737,262.193359375,87.34322019125705,253.25880432128907]},{"page":41,"text":"(i) A molecule of 5CB (4-pentyl-40-cyanobiphenyl) has a longitudinal electric","rect":[53.813472747802737,280.1046142578125,385.14917791559068,270.4449157714844]},{"page":41,"text":"dipole moment about 3D due to a triple –C\u0006N bond.","rect":[68.25785827636719,292.06414794921877,282.3977932503391,283.12957763671877]},{"page":41,"text":"(ii) A molecule of MBBA (4-methoxy-benzylidene-40-butylaniline) has a trans-","rect":[53.813289642333987,304.02392578125,385.10006080476418,294.3643798828125]},{"page":41,"text":"verse dipole moment due to the methoxy-group and, of course, both molecules","rect":[68.08830261230469,315.98345947265627,385.15878690825658,307.04888916015627]},{"page":41,"text":"have anisotropic polarizabilities.","rect":[68.08830261230469,327.9430236816406,199.16036649252659,319.00848388671877]},{"page":41,"text":"The vector of a permanent dipole moment pe and polarizability tensor aij","rect":[65.76599884033203,346.7707824707031,385.1910084557709,336.91937255859377]},{"page":41,"text":"describe the linear (in field) electric and optical properties. The nonlinear properties","rect":[53.812843322753909,357.8140869140625,385.1536904727097,348.8795166015625]},{"page":41,"text":"are described by tensors of higher ranks (this depends on the number of fields","rect":[53.812843322753909,369.77362060546877,385.15365995513158,360.83905029296877]},{"page":41,"text":"included). For instance, the efficiency of mixing two optical waves of frequencies","rect":[53.812843322753909,381.7331848144531,385.15866483794408,372.79864501953127]},{"page":41,"text":"o1 and o2 is determined by polarization Pk(o3) ¼ gijk Ei(o1) · Ej(o2) where Ei(o1)","rect":[53.812843322753909,394.60943603515627,385.15975318757668,384.7017822265625]},{"page":41,"text":"and Ej(o2)j are amplitudes of two interacting fields. Here gijk is a third rank tensor of","rect":[53.81393051147461,406.58233642578127,385.15191016999855,396.66131591796877]},{"page":41,"text":"the electric hyperpolarizability.","rect":[53.814083099365237,417.5555114746094,180.0065884163547,408.6209716796875]},{"page":41,"text":"(c) Magnetic moments","rect":[53.814083099365237,435.56304931640627,145.95422758208469,426.6484375]},{"page":41,"text":"A magnetic field induces magnetic moments in a molecule: pmi ¼ mikHk. The","rect":[53.814083099365237,453.4341735839844,385.16159856988755,444.4996337890625]},{"page":41,"text":"diamagnetic susceptibility tensor mik has the same structure as the tensor of","rect":[53.813777923583987,465.3941345214844,385.1498960098423,456.4595947265625]},{"page":41,"text":"Fig. 3.7 The most popular","rect":[53.812843322753909,506.1209716796875,147.0072945938795,498.1879577636719]},{"page":41,"text":"among physicists molecules","rect":[53.812843322753909,516.0291748046875,149.47455294608393,508.4348449707031]},{"page":41,"text":"5CB (a) and MBBA (b)","rect":[53.812843322753909,525.66650390625,136.10351651770763,518.3599853515625]},{"page":41,"text":"forming liquid crystals at","rect":[53.812843322753909,535.9243774414063,140.18259910788599,528.3300170898438]},{"page":41,"text":"room temperature. Note","rect":[53.812843322753909,545.9003295898438,135.64663073801519,538.475341796875]},{"page":41,"text":"strictly longitudinal and","rect":[53.812843322753909,555.8762817382813,135.5027822890751,548.2819213867188]},{"page":41,"text":"almost transverse direction of","rect":[53.812843322753909,564.1251220703125,155.23144620520763,558.2578735351563]},{"page":41,"text":"the dipole moments of the two","rect":[53.812843322753909,575.771484375,155.31607574610636,568.1771240234375]},{"page":41,"text":"molecules with respect to","rect":[53.812843322753909,585.7474365234375,141.14038604884073,578.153076171875]},{"page":41,"text":"their long molecular axes","rect":[53.812843322753909,595.723388671875,140.69196017264643,588.1290283203125]},{"page":41,"text":"a","rect":[208.6636505126953,493.2260437011719,214.21891290992915,487.6372985839844]},{"page":41,"text":"5CB    Cr-22oC-N-35oC-Iso","rect":[227.6630096435547,499.734130859375,323.5972695668922,493.36309814453127]},{"page":42,"text":"24","rect":[53.812618255615237,42.45556640625,62.273686830090728,36.73228454589844]},{"page":42,"text":"3 Mesogenic Molecules and Orientational Order","rect":[219.6690673828125,44.275840759277347,385.14044278723886,36.66455078125]},{"page":42,"text":"molecular polarizability with three or two different principal components. Some","rect":[53.812843322753909,68.2883529663086,385.1695941753563,59.35380554199219]},{"page":42,"text":"molecules possess permanent magnetic moments. For instance, the moments origi-","rect":[53.812843322753909,80.24788665771485,385.15264259187355,71.31333923339844]},{"page":42,"text":"nate from unpaired electron spins in the inner shells of such metal atoms as M ¼ Ni,","rect":[53.812843322753909,92.20748138427735,385.1566128304172,83.27293395996094]},{"page":42,"text":"Co, Fe, etc. in metal-mesogenic compounds (Fig. 3.8).","rect":[53.812843322753909,104.11019134521485,273.7899441292453,95.17564392089844]},{"page":42,"text":"Another case is free radicals with permanent magnetic moments of such molec-","rect":[65.76387786865235,116.0697250366211,385.16167579499855,107.13517761230469]},{"page":42,"text":"ular groups as –NO, in which unpaired electron spins are located on oxygen atoms.","rect":[53.81186294555664,128.02932739257813,385.1059231331516,119.0748519897461]},{"page":42,"text":"Stability of such radicals is provided by sterical screening of a reaction center from","rect":[53.81186294555664,139.98886108398438,385.13385723710618,131.05430603027345]},{"page":42,"text":"the surrounding medium by bulky chemical groups (like methyl one). Such a","rect":[53.81186294555664,151.94839477539063,385.15671575738755,143.0138397216797]},{"page":42,"text":"radical can be a fragment of an elongated mesogenic molecule. It should be","rect":[53.81186294555664,163.907958984375,385.14673650934068,154.97340393066407]},{"page":42,"text":"noted, however, that the field orientation of spin moments is almost decoupled","rect":[53.81186294555664,175.86749267578126,385.1715935807563,166.9329376220703]},{"page":42,"text":"from the molecular skeleton motion (in contrast to electric moments of molecular","rect":[53.81186294555664,187.42860412597657,385.14876685945168,178.89247131347657]},{"page":42,"text":"groups).","rect":[53.81186294555664,199.72976684570313,86.74735684897189,190.85498046875]},{"page":42,"text":"The","rect":[91.93350219726563,198.0,107.47202337457502,190.7952117919922]},{"page":42,"text":"simultaneous","rect":[112.66513061523438,198.0,165.38959135161594,190.7952117919922]},{"page":42,"text":"orientation","rect":[170.611572265625,198.0,214.00396052411566,190.7952117919922]},{"page":42,"text":"of","rect":[219.21299743652345,198.0,227.50486121980323,190.7952117919922]},{"page":42,"text":"spins","rect":[232.69398498535157,199.72976684570313,253.11008085356907,190.7952117919922]},{"page":42,"text":"and","rect":[258.2972106933594,198.0,272.70095149091255,190.7952117919922]},{"page":42,"text":"molecular","rect":[277.9527893066406,198.0,317.9388669571079,190.7952117919922]},{"page":42,"text":"skeletons","rect":[323.21160888671877,198.0,360.30099882231908,190.7952117919922]},{"page":42,"text":"by","rect":[365.58172607421877,199.72976684570313,375.53594294599068,190.7952117919922]},{"page":42,"text":"a","rect":[380.7061767578125,198.0,385.1557086773094,192.0]},{"page":42,"text":"magnetic field takes place only if the so-called spin-orbital interaction is significant.","rect":[53.81186294555664,211.6893310546875,385.0939907601047,202.75477600097657]},{"page":42,"text":"3.2 Intermolecular Interactions","rect":[53.812843322753909,259.8498840332031,222.94017677042647,250.83767700195313]},{"page":42,"text":"Atoms in an organic molecule are mostly bound by covalent bonds with high intra-","rect":[53.812843322753909,289.4002685546875,385.1377194961704,280.4656982421875]},{"page":42,"text":"molecular interaction energy W ~ 1012 erg/M (or 100 kJ/M in SI units). In units","rect":[53.812843322753909,301.35992431640627,385.1366311465378,290.3089599609375]},{"page":42,"text":"more convenient for a physicist: W ¼ 105 J/(1.6 \u0004 10\u000319 \u0004 6.02 \u0004 1023) \u0005 1 eV/","rect":[53.813716888427737,313.3194885253906,385.135634017678,302.18896484375]},{"page":42,"text":"molecule.","rect":[53.81374740600586,324.0,93.06020780112033,316.34454345703127]},{"page":42,"text":"Intermolecular","rect":[98.67538452148438,324.0,157.62320838777198,316.34454345703127]},{"page":42,"text":"interactions","rect":[163.3060760498047,324.0,210.02415861235813,316.34454345703127]},{"page":42,"text":"are","rect":[215.6453094482422,324.0,227.85913885309066,318.0]},{"page":42,"text":"essentially","rect":[233.54498291015626,325.27911376953127,275.70107356122505,316.34454345703127]},{"page":42,"text":"weaker,","rect":[281.4098205566406,324.0,312.7864956429172,316.34454345703127]},{"page":42,"text":"of","rect":[318.3976745605469,324.0,326.68953834382668,316.34454345703127]},{"page":42,"text":"the","rect":[332.2758483886719,324.0,344.49961126520005,316.34454345703127]},{"page":42,"text":"order","rect":[350.1755065917969,324.0,371.2087644180454,316.34454345703127]},{"page":42,"text":"of","rect":[376.85479736328127,324.0,385.14666114656105,316.34454345703127]},{"page":42,"text":"0.01–0.1 eV. Their nature can be quite different. A good example is 5CB forming","rect":[53.81374740600586,337.2386779785156,385.16851130536568,328.2443542480469]},{"page":42,"text":"molecular dimers, Fig. 3.9, due to interaction between two dipoles located on the","rect":[53.81374740600586,349.14141845703127,385.1735309429344,340.20684814453127]},{"page":42,"text":"two cyano-groups. Below we shall briefly consider the most important mechanisms","rect":[53.81374740600586,361.1009826660156,385.1585732852097,352.16644287109377]},{"page":42,"text":"of interactions between liquid crystal molecules. For the more advanced discussion","rect":[53.81374740600586,373.0605163574219,385.1426629166938,364.1259765625]},{"page":42,"text":"of intermolecular potentials see [7].","rect":[53.81374740600586,385.0200500488281,197.60430570151096,376.08551025390627]},{"page":42,"text":"Fig. 3.8","rect":[53.812843322753909,486.39599609375,80.70296997218057,478.446044921875]},{"page":42,"text":"CH3(CH2)n","rect":[108.70207977294922,444.21148681640627,147.66854004553768,436.4111633300781]},{"page":42,"text":"S","rect":[209.57147216796876,437.4866943359375,213.9837535446877,431.7519226074219]},{"page":42,"text":"S","rect":[226.85130310058595,437.3459167480469,231.26358447730488,431.61114501953127]},{"page":42,"text":"M","rect":[217.41204833984376,449.87286376953127,225.3253790698288,444.3780212402344]},{"page":42,"text":"S","rect":[210.41395568847657,462.6501159667969,214.8262370651955,456.91534423828127]},{"page":42,"text":"S","rect":[227.83447265625,461.9471130371094,232.24675403296895,456.21234130859377]},{"page":42,"text":"(CH2)nCH3.","rect":[289.204345703125,458.1498107910156,330.3360962465414,450.3495788574219]},{"page":42,"text":"An example of a molecule of metal-mesogenic compounds","rect":[86.66717529296875,486.3282775878906,288.60837252616207,478.7339172363281]},{"page":42,"text":"Fig. 3.9 A structure of a dimer formed by two molecules of compound 5CB due to dipole-dipole","rect":[53.812843322753909,579.8626708984375,385.1466307380152,571.8280639648438]},{"page":42,"text":"interaction. Pe and aij are molecular dipole and polarizability","rect":[53.812843322753909,590.512939453125,263.0954641738407,582.1764526367188]},{"page":43,"text":"3.2 Intermolecular Interactions","rect":[53.812843322753909,42.55765151977539,159.2445419597558,36.68197250366211]},{"page":43,"text":"25","rect":[376.7464904785156,42.55765151977539,385.2075552382938,36.63117599487305]},{"page":43,"text":"(a) Electrostatic interaction","rect":[53.812843322753909,67.88993072509766,167.76665583661566,59.333885192871097]},{"page":43,"text":"At a large distance from a system of charges, the electric field around the system","rect":[65.76486206054688,86.19930267333985,385.17362164140305,77.26475524902344]},{"page":43,"text":"can be expanded in a series of multipoles, see Fig. 3.10. Correspondingly the","rect":[53.812843322753909,98.1588363647461,385.11682928277818,89.22428894042969]},{"page":43,"text":"electrostatic","rect":[53.812862396240237,109.0,102.26000250055158,101.18382263183594]},{"page":43,"text":"molecular","rect":[107.56858825683594,109.0,147.6412595352329,101.18382263183594]},{"page":43,"text":"interactions","rect":[152.88414001464845,109.0,199.60222257720188,101.18382263183594]},{"page":43,"text":"can","rect":[204.82620239257813,109.0,218.70237055829535,103.0]},{"page":43,"text":"be","rect":[223.9721221923828,109.0,233.46346319391098,101.18382263183594]},{"page":43,"text":"classified","rect":[238.7003936767578,109.0,275.77983943036568,101.18382263183594]},{"page":43,"text":"by","rect":[281.07049560546877,110.11837005615235,291.02471247724068,101.18382263183594]},{"page":43,"text":"interaction","rect":[296.30841064453127,109.0,339.19414607099068,101.18382263183594]},{"page":43,"text":"energy","rect":[344.3992004394531,110.11837005615235,371.6299676041938,103.0]},{"page":43,"text":"as","rect":[376.85693359375,109.0,385.1387978945847,103.0]},{"page":43,"text":"follows [8]:","rect":[53.812862396240237,121.4703140258789,100.93609483310769,113.14335632324219]},{"page":43,"text":"Monopole (q) – monopole (q) (Coulomb energy):","rect":[53.812843322753909,142.78585815429688,224.3160982533938,135.19149780273438]},{"page":43,"text":"Monopole (q) – electric dipole (pe):","rect":[53.81330490112305,152.76181030273438,176.9432497426516,145.16744995117188]},{"page":43,"text":"Dipole (pe) – dipole (pe) (Keesom energy):","rect":[53.81364059448242,172.71343994140626,201.7725191518313,165.11895751953126]},{"page":43,"text":"Monopole (q) – induced dipole:","rect":[53.81364059448242,192.60836791992188,163.6340914174563,185.01400756835938]},{"page":43,"text":"Dipole (pe) – induced dipole (Debye energy):","rect":[53.812843322753909,202.58416748046876,210.32619957175317,194.98980712890626]},{"page":43,"text":"Dipole–quadrupole, quadrupole–quadrupole, etc.","rect":[53.812843322753909,212.50326538085938,220.90799209668598,204.90890502929688]},{"page":43,"text":"W ~ q2/r","rect":[247.36827087402345,142.69271850585938,276.74048312186519,133.4094696044922]},{"page":43,"text":"qpe/r2","rect":[247.3689422607422,152.66854858398438,267.4791616232608,143.38523864746095]},{"page":43,"text":"q2pe2/r4","rect":[247.36993408203126,162.64450073242188,273.8233999045108,153.3043670654297]},{"page":43,"text":"p1p2/r3","rect":[247.3692169189453,172.62017822265626,270.9911245138858,163.2803192138672]},{"page":43,"text":"p12p22/kTr6","rect":[247.36993408203126,182.53939819335938,285.77551782443268,173.22225952148438]},{"page":43,"text":"q2a/r4","rect":[247.36907958984376,192.51522827148438,268.21555078341705,183.2319793701172]},{"page":43,"text":"p12a/r6","rect":[247.36875915527345,202.49102783203126,270.9911245138858,193.11721801757813]},{"page":43,"text":"(Fixed dipole)","rect":[308.77276611328127,152.76168823242188,357.3469247208326,145.16732788085938]},{"page":43,"text":"(Free rotating dipole)","rect":[308.77276611328127,162.73745727539063,381.9305734024732,155.14309692382813]},{"page":43,"text":"(Fixed dipoles)","rect":[308.77276611328127,172.71331787109376,360.60443204505136,165.11895751953126]},{"page":43,"text":"(Free rotating dipoles)","rect":[308.77276611328127,182.63238525390626,385.18808072669199,175.03802490234376]},{"page":43,"text":"These general formulas can be used in the molecular theory of formation of","rect":[65.76496887207031,234.76004028320313,385.14885841218605,225.8254852294922]},{"page":43,"text":"mesophases.","rect":[53.812950134277347,246.71957397460938,103.95130582358127,237.78501892089845]},{"page":43,"text":"(b) Dispersion interaction","rect":[53.812950134277347,264.57769775390627,160.91228571942816,255.75270080566407]},{"page":43,"text":"This","rect":[53.812950134277347,281.0,71.53144468413547,273.66363525390627]},{"page":43,"text":"is","rect":[76.86689758300781,281.0,83.49640287505344,273.66363525390627]},{"page":43,"text":"also","rect":[88.76217651367188,281.0,104.81832209638128,273.66363525390627]},{"page":43,"text":"dipole–dipole","rect":[110.116943359375,282.59820556640627,165.13684881402816,273.66363525390627]},{"page":43,"text":"interaction","rect":[170.44244384765626,281.0,213.30327693036566,273.66363525390627]},{"page":43,"text":"but","rect":[218.64669799804688,281.0,231.39805467197489,273.66363525390627]},{"page":43,"text":"between","rect":[236.71658325195313,281.0,269.96367732099068,273.66363525390627]},{"page":43,"text":"oscillating,","rect":[275.2911682128906,282.59820556640627,319.5157742073703,273.66363525390627]},{"page":43,"text":"not","rect":[324.8551940917969,281.0,337.6065507657249,274.67962646484377]},{"page":43,"text":"permanent","rect":[342.86834716796877,282.59820556640627,385.1389299161155,274.67962646484377]},{"page":43,"text":"dipoles. It is a pure quantum-mechanical effect of oscillatory motion of electrons","rect":[53.812950134277347,294.5577392578125,385.0891457949753,285.6231689453125]},{"page":43,"text":"in the ground state. It is described by the London formula (here n is a frequency of a","rect":[53.812950134277347,306.51727294921877,385.1597369976219,297.58270263671877]},{"page":43,"text":"single oscillator considered):","rect":[53.81296157836914,318.4768371582031,170.007460189553,309.54229736328127]},{"page":43,"text":"3ðhnÞa2","rect":[222.72877502441407,343.47235107421877,253.23246834358535,332.7080383300781]},{"page":43,"text":"U1;2 ¼ \u0003 4r6","rect":[183.6427459716797,354.5955810546875,244.2825324304994,341.2195739746094]},{"page":43,"text":"In a more general case one has a sum of different oscillators. The dispersion","rect":[65.76496887207031,378.1624450683594,385.10405818036568,369.2279052734375]},{"page":43,"text":"interactions are partially responsible for the well known attractive term a/V2","rect":[53.812950134277347,390.1219787597656,385.18130140630105,379.071044921875]},{"page":43,"text":"between neutral molecules in the Van der Waals equation of state","rect":[53.812843322753909,402.08172607421877,385.0989459819969,393.14715576171877]},{"page":43,"text":"ðp þ a=V2ÞðV \u0003 bÞ ¼ RT. The corresponding energy is of the order of 0.1 kJ/M","rect":[53.812843322753909,415.17376708984377,385.1516390339347,404.3525085449219]},{"page":43,"text":"or 10\u00034 eV/mol. By the way, the repulsive term (V \u0003 b) in the same equation that","rect":[53.81275177001953,426.7945861816406,385.1392045743186,415.7436218261719]},{"page":43,"text":"takes into account the excluded volume effect b is due to the steric interaction","rect":[53.81331253051758,436.76202392578127,385.08843318036568,429.819580078125]},{"page":43,"text":"discussed next.","rect":[53.81331253051758,448.62493896484377,114.5887417366672,441.72235107421877]},{"page":43,"text":"(c) Steric interaction and intermolecular potential","rect":[53.81331253051758,468.5150451660156,258.32954270908427,459.6701354980469]},{"page":43,"text":"Classically, one can consider atoms or molecules as non-penetrable for other atoms","rect":[53.81331253051758,486.5354919433594,385.1093789492722,477.6009521484375]},{"page":43,"text":"or molecules, Fig. 3.11. In fact it is a quantum-mechanical effect related to the Pauli","rect":[53.81331253051758,498.49505615234377,385.1362138516624,489.56048583984377]},{"page":43,"text":"principle. For spherical molecules, the Lennard-Jones (or 6–12) potential is often","rect":[53.81330108642578,510.4546203613281,385.1371697526313,501.52008056640627]},{"page":43,"text":"used [8, 9]:","rect":[53.81330108642578,521.8065795898438,99.57678849765847,513.4796142578125]},{"page":43,"text":"Fig. 3.10 Structures of","rect":[53.812843322753909,550.0486450195313,134.66937345130138,542.1156616210938]},{"page":43,"text":"different molecular","rect":[53.812843322753909,558.1729736328125,119.40898221594979,552.3057250976563]},{"page":43,"text":"multipoles, which could be","rect":[53.812843322753909,569.8760375976563,146.51064440989019,562.2816772460938]},{"page":43,"text":"responsible for the interaction","rect":[53.812843322753909,579.8519897460938,155.3211416640751,572.2576293945313]},{"page":43,"text":"of mesogenic molecules","rect":[53.812843322753909,589.8279418945313,136.33281405448236,582.2335815429688]},{"page":43,"text":"monopole","rect":[198.71189880371095,569.7142944335938,230.5418666140083,562.5398559570313]},{"page":43,"text":"dipole","rect":[250.96786499023438,569.7142944335938,270.8031428591255,562.411865234375]},{"page":43,"text":"quadrupole","rect":[295.85723876953127,569.7142944335938,332.5998652712349,562.5398559570313]},{"page":43,"text":"octupole","rect":[349.5887145996094,569.7142944335938,377.4220698610786,562.5398559570313]},{"page":44,"text":"26","rect":[53.812469482421878,42.55667495727539,62.273538056897368,36.68099594116211]},{"page":44,"text":"Fig. 3.11 The form of the","rect":[53.812843322753909,67.58130645751953,145.14925524973394,59.648292541503909]},{"page":44,"text":"Lennard-Jones potential for","rect":[53.812843322753909,77.4895248413086,148.080170570442,69.89517211914063]},{"page":44,"text":"interaction of two spherical","rect":[53.812843322753909,87.4087142944336,147.72225670065942,79.81436157226563]},{"page":44,"text":"molecules located ata","rect":[53.812843322753909,95.65752410888672,129.93032214426519,89.79031372070313]},{"page":44,"text":"distance r from each other","rect":[53.812843322753909,105.63353729248047,143.9274606095045,99.76632690429688]},{"page":44,"text":"3 Mesogenic Molecules and Orientational Order","rect":[219.66891479492188,44.275352478027347,385.1402901993482,36.6640625]},{"page":44,"text":"U","rect":[278.4676208496094,68.8676528930664,284.6304269029614,63.26084899902344]},{"page":44,"text":"R","rect":[307.17620849609377,101.2039794921875,312.92336485453748,95.70915222167969]},{"page":44,"text":"ε","rect":[316.29931640625,115.88065338134766,319.80835902287978,112.08145904541016]},{"page":44,"text":"V-d-W attraction","rect":[325.5262451171875,136.40025329589845,385.19475207984399,130.23866271972657]},{"page":44,"text":"Fig. 3.12 Gay–Berne potential U(r) as a function of intermolecular distance r between elongated","rect":[54.379371643066409,352.9127197265625,385.69455475001259,344.9797058105469]},{"page":44,"text":"molecules. Black ellipsoids mimic the pairs of interacting molecules in different geometry of","rect":[54.378517150878909,362.82098388671877,385.77658170325449,355.22662353515627]},{"page":44,"text":"interaction. Both scales are arbitrary","rect":[54.378517150878909,372.79693603515627,178.77570099024698,365.20257568359377]},{"page":44,"text":"UðxÞ ¼ eðx\u000312 \u0003 x\u00036Þ","rect":[175.6566619873047,403.3272705078125,263.32391752349096,391.95428466796877]},{"page":44,"text":"where x ¼ r/R, R and e are the equilibrium distance and interaction energy shown in","rect":[53.813655853271487,426.8509216308594,385.1425408463813,417.9163818359375]},{"page":44,"text":"the figure. For elongated molecules consisting of several atoms, the Gay–Berne","rect":[53.81367111206055,438.81048583984377,385.1535114116844,429.87591552734377]},{"page":44,"text":"potential shown in Fig. 3.12 is more realistic. This potential takes into account a","rect":[53.81367111206055,450.7700500488281,385.1594623394188,441.83551025390627]},{"page":44,"text":"mutual orientation of elongated molecules. From the same figure one may see how","rect":[53.81367111206055,462.7295837402344,385.12167119934886,453.7950439453125]},{"page":44,"text":"the equilibrium distance and the depth of the energy minimum differ for differently","rect":[53.81367111206055,474.6891174316406,385.09478083661568,465.75457763671877]},{"page":44,"text":"oriented molecules.","rect":[53.81367111206055,484.61669921875,132.52261014487034,477.714111328125]},{"page":44,"text":"(d) Hydrogen bonds","rect":[53.81367111206055,504.59942626953127,136.8606301699753,495.6051330566406]},{"page":44,"text":"This either intra- or intermolecular bond arises between strongly electro-negative","rect":[53.81367111206055,522.470458984375,385.1584857769188,513.5359497070313]},{"page":44,"text":"atoms, such as oxygen or nitrogen, chlorine and fluorine. These atoms can be bound","rect":[53.81367111206055,534.4300537109375,385.14360896161568,525.4955444335938]},{"page":44,"text":"by a mediator, a proton that is partially forms a covalent bond with one of such","rect":[53.81367111206055,546.3895874023438,385.1236504655219,537.455078125]},{"page":44,"text":"atoms but also strongly interacts with the other electronegative atom. In this","rect":[53.81367111206055,558.34912109375,385.14255155669408,549.4146118164063]},{"page":44,"text":"situation, an electrostatic interaction plays the dominant role but with some admix-","rect":[53.81367111206055,570.3086547851563,385.1704038223423,561.3741455078125]},{"page":44,"text":"ture of the covalent bond. To some extent, a proton has common orbital for the","rect":[53.81367111206055,582.2681884765625,385.1724323101219,573.3336791992188]},{"page":44,"text":"two connected atoms. A well-known example is water where oxygen atoms form","rect":[53.81367111206055,594.2277221679688,385.13468121171555,585.293212890625]},{"page":45,"text":"3.2 Intermolecular Interactions","rect":[53.812828063964847,42.55740737915039,159.24452670096674,36.68172836303711]},{"page":45,"text":"27","rect":[376.7464904785156,42.55740737915039,385.2075552382938,36.73252868652344]},{"page":45,"text":"a network using the hydrogen atoms as bridges between them. A typical energy of","rect":[53.812843322753909,68.2883529663086,385.14974342195168,59.35380554199219]},{"page":45,"text":"the hydrogen bond is rather high, 10–50 kJ/M, i.e. 0.1–0.5 eV/molecule.","rect":[53.812843322753909,80.24788665771485,345.5926479866672,71.25357818603516]},{"page":45,"text":"Hydrogen bonds can be responsible for formation of molecular dimers which, in","rect":[65.76486206054688,92.20748138427735,385.13979426435005,83.27293395996094]},{"page":45,"text":"turn, become building blocks for liquid crystal phases as shown Fig. 3.13a. Without","rect":[53.812843322753909,104.11019134521485,385.2054277188499,95.17564392089844]},{"page":45,"text":"O...H bonding, short molecules of benzoic acids would never form the nematic or","rect":[53.81183624267578,116.0697250366211,385.14571510163918,107.1152572631836]},{"page":45,"text":"SmC phases as they, in fact, do. Another example is the derivative of the cyclohexane-","rect":[53.81183624267578,128.02932739257813,385.1137631973423,119.09477233886719]},{"page":45,"text":"carboxylic acid (CHCA) shown in Fig. 3.13b Such cyclohexane-type dimers form","rect":[53.81183624267578,139.98886108398438,385.1347422468718,131.05430603027345]},{"page":45,"text":"the nematic phase with very low optical anisotropy. In the molecule (monomer) the","rect":[53.81181716918945,151.94839477539063,385.1696246929344,143.0138397216797]},{"page":45,"text":"cyclohexane moiety is in the chair-form and the –C4H9 and –COOH groups are in","rect":[53.81181716918945,163.908935546875,385.14348689130318,154.95445251464845]},{"page":45,"text":"the trans-positions (t) as explained in Fig. 3.3. Such a dimer (trans-isomer) may be","rect":[53.81364059448242,175.86846923828126,385.15149725152818,166.9339141845703]},{"page":45,"text":"considered as rod-like. The corresponding cis-isomer would have a strongly bent-","rect":[53.81266403198242,187.8280029296875,385.1375669082798,178.89344787597657]},{"page":45,"text":"shape structure hardly compatible with liquid crystal phase. By the way, similar but","rect":[53.81264877319336,199.73077392578126,385.20429856845927,190.7962188720703]},{"page":45,"text":"reversible trans–cis–trans photo-isomerization is observed in compounds with azo-","rect":[53.81264877319336,211.69033813476563,385.15349708406105,202.7557830810547]},{"page":45,"text":"(–N¼N–) or azoxy-(–N¼NO–) bridges between phenyl rings. Such compounds","rect":[53.81264877319336,223.64987182617188,385.12460722075658,214.6953887939453]},{"page":45,"text":"may be used for the light control of the liquid crystal structure and properties.","rect":[53.81162643432617,235.60940551757813,368.2124294808078,226.6748504638672]},{"page":45,"text":"(e) Hydrophilic and hydrophobic interactions","rect":[53.81162643432617,253.50039672851563,239.7821389590378,244.5658416748047]},{"page":45,"text":"These interaction, although very important, are not as fundamental as the others.","rect":[53.81162643432617,271.488037109375,385.1136440804172,262.553466796875]},{"page":45,"text":"They are related to the affinity to water. Hydrophilic interactions include the same","rect":[53.81162643432617,283.44757080078127,385.12558782770005,274.51300048828127]},{"page":45,"text":"electrostatic, steric and H-bond interactions and all of them are, generally speaking,","rect":[53.81162643432617,295.350341796875,385.09576077963598,286.415771484375]},{"page":45,"text":"electromagnetic. The hydrophobic “interaction” is an entropy effect; there is no","rect":[53.81162643432617,307.3099060058594,385.16741267255318,298.3753662109375]},{"page":45,"text":"special repulsive force. For example, oil and water are immiscible. Merely water","rect":[53.81162643432617,319.2694091796875,385.07674537507668,310.3348388671875]},{"page":45,"text":"molecules feel more comfortable among the same neighbors, to which they form a","rect":[53.81162643432617,331.2289733886719,385.1554645366844,322.29443359375]},{"page":45,"text":"network of H-bonds. If an oil molecule with its long hydrocarbon tail were","rect":[53.81162643432617,343.1885070800781,385.11365545465318,334.25396728515627]},{"page":45,"text":"incorporated into water, it would destroy the network and reduce the entropy of","rect":[53.81162643432617,355.1480712890625,385.1474851211704,346.2135009765625]},{"page":45,"text":"the mixture.","rect":[53.81162643432617,365.0458068847656,102.45486875082736,358.173095703125]},{"page":45,"text":"a","rect":[107.37779998779297,398.22943115234377,112.93306238502682,392.64068603515627]},{"page":45,"text":"C5H11","rect":[106.65240478515625,423.070068359375,128.19385519030588,415.0679016113281]},{"page":45,"text":"b","rect":[105.92604064941406,455.00775146484377,112.03083439889227,447.6993713378906]},{"page":45,"text":"O...H-O","rect":[196.05555725097657,408.31787109375,223.5923550883036,402.2791442871094]},{"page":45,"text":"C","rect":[176.90330505371095,420.49493408203127,182.6744412022455,414.4562072753906]},{"page":45,"text":"C","rect":[233.36831665039063,422.85223388671877,239.1394527989252,416.8135070800781]},{"page":45,"text":"O-H...O","rect":[194.11158752441407,432.7686767578125,221.6483853617411,426.7299499511719]},{"page":45,"text":"a hydrogen bond","rect":[236.2798614501953,446.6864318847656,296.2692971547828,439.1920471191406]},{"page":45,"text":"Fig. 3.13 The role of the hydrogen bond in formation of dimers of benzoic acid molecules (a) and","rect":[53.812843322753909,584.0003662109375,385.1939749160282,576.0673828125]},{"page":45,"text":"cyclohexane-carboxylic acid molecules (b)","rect":[53.81281280517578,593.9085693359375,201.07519620520763,586.314208984375]},{"page":46,"text":"28","rect":[53.813690185546878,42.55734634399414,62.274758760022368,36.73246765136719]},{"page":46,"text":"3 Mesogenic Molecules and Orientational Order","rect":[219.67013549804688,44.276023864746097,385.1415109024732,36.66473388671875]},{"page":46,"text":"3.3 Orientational Distribution Functions for Molecules","rect":[53.812843322753909,68.46917724609375,342.64858131144208,59.086421966552737]},{"page":46,"text":"The translational and orientational degrees of freedom can be treated separately","rect":[53.812843322753909,97.64900970458985,385.1725701432563,88.71446228027344]},{"page":46,"text":"(this follows from fundamentals of group theory which states that groups of translations","rect":[53.812843322753909,109.6085433959961,385.13812650786596,100.67399597167969]},{"page":46,"text":"and rotations are subgroups of the crystalline space groups: P(r, O) ¼ P(r) \u0007 P(O).","rect":[53.812843322753909,121.56807708740235,385.1556057503391,112.45423889160156]},{"page":46,"text":"Here \u0007 is a symbol of the group product. In particular case of the isotropic liquid or","rect":[53.8138313293457,133.5674591064453,385.15068946687355,124.5731430053711]},{"page":46,"text":"nematic phase (no positional order) P(r, O) ¼ rP(O) where r ¼ constant is density.","rect":[53.814815521240237,145.48721313476563,385.1844753792453,136.3733673095703]},{"page":46,"text":"Generally, a one-particle distribution function P(r, O) represents a probability to","rect":[65.76883697509766,157.44671630859376,385.1466912370063,148.33287048339845]},{"page":46,"text":"find a molecule with orientation O at position r. Here O includes three Euler angles","rect":[53.81783676147461,169.40625,385.1247598086472,160.2924041748047]},{"page":46,"text":"C, F and # as shown in Fig. 3.14. This probability is assumed to be independent of","rect":[53.81783676147461,181.36581420898438,385.1527036270298,172.21212768554688]},{"page":46,"text":"other particles. In the figure, x, y, z is a Cartesian laboratory frame, the z-axis is","rect":[53.81782913208008,193.2685546875,385.19345487700658,184.33399963378907]},{"page":46,"text":"taken as a reference: usually it coincides with one of the symmetry axis of a","rect":[53.81880187988281,205.22811889648438,385.1636432476219,196.29356384277345]},{"page":46,"text":"molecular system. For a nematic phase discussed in this chapter, such a symmetry","rect":[53.81880187988281,217.18765258789063,385.17656794599068,208.23316955566407]},{"page":46,"text":"axis coincides with the preferable axis of orientation of molecules. This axis is","rect":[53.81880187988281,229.147216796875,385.1894875918503,220.21266174316407]},{"page":46,"text":"called the director, a unit axial vector with head and tail indistinguishable, n ¼ \u0003n.","rect":[53.81880187988281,241.10675048828126,385.18743558432348,232.1522674560547]},{"page":46,"text":"We say the director has head-to-tail symmetry. If there is no interaction with","rect":[53.818763732910159,253.0662841796875,385.12175837567818,244.13172912597657]},{"page":46,"text":"surrounding, the director may take any direction and its realignment cost no energy","rect":[53.818763732910159,265.02581787109377,385.1735467057563,256.09124755859377]},{"page":46,"text":"(no energy gap to overcome). Such a gapless orientational motion that restores the","rect":[53.818763732910159,276.9853820800781,385.1785358257469,268.05084228515627]},{"page":46,"text":"spherical symmetry of the isotropic phase is called a Goldstone mode. Evidently,","rect":[53.818763732910159,288.88812255859377,385.1586575081516,279.95355224609377]},{"page":46,"text":"that the direction of the director may be fixed by a weak magnetic field or by","rect":[53.818763732910159,300.8476867675781,385.1745537858344,291.91314697265627]},{"page":46,"text":"interaction with the surfaces, and our z-axis is assumed to be fixed by some external","rect":[53.818763732910159,312.8072204589844,385.11579759189677,303.8726806640625]},{"page":46,"text":"factor.","rect":[53.818756103515628,322.7049560546875,79.70169492270236,315.83221435546877]},{"page":46,"text":"The frame x, Z, z is attached to a molecule. Then Euler angles correspond to","rect":[65.7707748413086,336.726318359375,375.68816462567818,327.4730224609375]},{"page":46,"text":"– Deflection of the longitudinal molecular z-axis from axis z (angle #)","rect":[53.819732666015628,354.63726806640627,343.8714841446079,345.38397216796877]},{"page":46,"text":"– Rotation of the molecular shortest Z- axis about its own longitudinal z-axis","rect":[53.819732666015628,366.5995788574219,385.12860502349096,357.3462829589844]},{"page":46,"text":"(angle C)","rect":[67.29566955566406,378.5591125488281,106.63669715730322,369.5149841308594]},{"page":46,"text":"– Precession of the longitudinal z-axis within a cone surface around z (angle F)","rect":[53.814674377441409,390.5186462402344,381.08321510163918,381.2653503417969]},{"page":46,"text":"In this chapter we consider only an orientational distribution function f(O) [10,","rect":[65.76866912841797,408.4295654296875,385.18536038901098,399.31573486328127]},{"page":46,"text":"11]. Why do we need it? Because it is a kind of a bridge between the microscopic","rect":[53.81665802001953,420.3891296386719,385.1236957378563,411.45458984375]},{"page":46,"text":"and macroscopic descriptions of the nematic phase. We define a value","rect":[53.81665802001953,432.3486633300781,337.10657537652818,423.41412353515627]},{"page":46,"text":"ζ","rect":[259.204345703125,463.8063049316406,263.1530178047539,456.5278625488281]},{"page":46,"text":"z","rect":[290.3516845703125,456.91656494140627,294.34831625212316,452.73345947265627]},{"page":46,"text":"θ","rect":[279.7398376464844,476.1428527832031,283.9043278589311,470.4800720214844]},{"page":46,"text":"Euler angels","rect":[315.4145812988281,469.4691162109375,357.82684771016747,462.422607421875]},{"page":46,"text":"Fig. 3.14 Euler angles of the","rect":[53.812843322753909,562.3483276367188,155.34484240793706,554.4153442382813]},{"page":46,"text":"molecular frame x, Z, z with","rect":[53.812843322753909,572.1803588867188,152.36823028712198,564.3912963867188]},{"page":46,"text":"respect to the Cartesian","rect":[53.812835693359378,582.2324829101563,134.0601553359501,574.6381225585938]},{"page":46,"text":"laboratory frame x, y, z","rect":[53.812835693359378,592.1516723632813,133.08882601737299,584.5573120117188]},{"page":46,"text":"x","rect":[233.1079559326172,539.882080078125,237.10458761442784,535.698974609375]},{"page":46,"text":"Ψ","rect":[335.87652587890627,542.6358642578125,342.2311702529852,537.2769775390625]},{"page":46,"text":"φ","rect":[333.90301513671877,563.433837890625,338.06750534916548,556.1633911132813]},{"page":46,"text":"η","rect":[353.6671447753906,532.353271484375,358.4870825836543,526.9544677734375]},{"page":46,"text":"y","rect":[348.62017822265627,586.2777709960938,352.6168099044669,580.3910522460938]},{"page":47,"text":"3.3 Orientational Distribution Functions for Molecules","rect":[53.812843322753909,42.55594253540039,240.7035263347558,36.663330078125]},{"page":47,"text":"29","rect":[376.7465515136719,42.62367248535156,385.20761627345009,36.73106384277344]},{"page":47,"text":"fðOÞdO ¼ fðF;#;CÞ sin#dFd#dC","rect":[143.93528747558595,69.47722625732422,295.03221422664617,59.52671432495117]},{"page":47,"text":"(3.1)","rect":[366.0963134765625,68.74015045166016,385.1685727676548,60.26378631591797]},{"page":47,"text":"as a fraction of molecules in the “angular volume” with Euler angles between F and","rect":[53.81283950805664,93.05770111083985,385.14467707685005,84.09326934814453]},{"page":47,"text":"dF, # and d#, C and dC. Function f(O) is a single-particle function because","rect":[53.81283950805664,104.96041107177735,385.1387409038719,95.80673217773438]},{"page":47,"text":"molecules are considered to be independent, i.e. any correlation in their motion is","rect":[53.81184005737305,116.9199447631836,385.18451322661596,107.98539733886719]},{"page":47,"text":"disregarded. The total probability to find a molecule with any orientation equals 1:","rect":[53.81184005737305,128.87954711914063,385.2034135586936,119.94499206542969]},{"page":47,"text":"ð fðOÞdO ¼ ð dFð sin#d#ð dC \u0004 fðF;#;CÞ ¼1","rect":[110.85344696044922,167.28268432617188,329.78142634442818,145.158935546875]},{"page":47,"text":"(3.2)","rect":[366.0983581542969,160.39305114746095,385.1706479629673,151.91668701171876]},{"page":47,"text":"We can use this normalization condition to find the f(O) function, at least, for","rect":[65.76689910888672,191.20880126953126,385.1795896133579,182.11488342285157]},{"page":47,"text":"the isotropic liquid or isotropic liquid crystal phase. Indeed, in this case there is","rect":[53.8138542175293,203.18826293945313,385.18756498442846,194.2537078857422]},{"page":47,"text":"no angular dependence of f(O) i.e. f(F,#,C) ¼ const. After integrating we find:","rect":[53.8138542175293,215.09103393554688,385.1338029629905,205.93734741210938]},{"page":47,"text":"f(F,#,C)iso ¼ 1/8p2.","rect":[53.8119010925293,227.0306396484375,135.9445766976047,216.00038146972657]},{"page":47,"text":"And what about optically uniaxial phases? In the case of a nematic, the molecu-","rect":[65.76539611816406,239.01101684570313,385.15325294343605,230.0764617919922]},{"page":47,"text":"lar distribution is independent of the precession angle (F ¼ const) but may depend","rect":[53.81337356567383,250.97055053710938,385.1034783463813,242.00611877441407]},{"page":47,"text":"on angle C. For a smectic A, in the first approximation, the orientational distribu-","rect":[53.81339645385742,262.93011474609377,385.0965512832798,253.88600158691407]},{"page":47,"text":"tion function is the same as for the nematic. However, there is some interaction","rect":[53.81339645385742,273.0,385.08657160810005,265.955078125]},{"page":47,"text":"between the translational and orientational degrees of freedom that can be taken","rect":[53.81339645385742,286.84918212890627,385.13930598310005,277.91461181640627]},{"page":47,"text":"into account as a correction to f(#,F). At first, consider a distribution function for a","rect":[53.81339645385742,298.7887878417969,385.15720403863755,289.655029296875]},{"page":47,"text":"uniaxial phase consisting of axially symmetric molecules [12].","rect":[53.81241989135742,310.71148681640627,306.53094144369848,301.77691650390627]},{"page":47,"text":"3.3.1 Molecules with Axial Symmetry","rect":[53.812843322753909,346.9924621582031,247.42750342817514,336.35467529296877]},{"page":47,"text":"The molecules either have a generic infinite rotation axis (cones, rods, rotational","rect":[53.812843322753909,374.5347900390625,385.1248002774436,365.6002197265625]},{"page":47,"text":"ellipsoids, spherocylinders or discs) or acquire this average uniaxial form due","rect":[53.812843322753909,386.4943542480469,385.14075506402818,377.559814453125]},{"page":47,"text":"to free rotation around the longitudinal molecular axis z. Then f(O) becomes","rect":[53.812843322753909,398.4538879394531,385.0869485293503,389.2005920410156]},{"page":47,"text":"C-independent [13]: f ¼ f(#)/4p2 with f(#) ¼ f(p \u0003 #), see Fig. 3.15. This figure","rect":[53.81185531616211,410.413818359375,385.1173785991844,399.36279296875]},{"page":47,"text":"shows that angle 0 and p are equally and the most populated by molecules and these","rect":[53.8123893737793,422.3733825683594,385.12635076715318,413.4388427734375]},{"page":47,"text":"two angles correspond to the condition n ¼ \u0003n. The angles close to p/2 are the less","rect":[53.81240463256836,434.3329162597656,385.14227689849096,425.39837646484377]},{"page":47,"text":"populated. Now our task is to find the form of f(#) and relate it to experimentally","rect":[53.8114128112793,446.2356872558594,385.1044854264594,437.0820007324219]},{"page":47,"text":"measured parameters.","rect":[53.810420989990237,458.1952209472656,141.32784696127659,449.26068115234377]},{"page":47,"text":"As any axially symmetric function, f(#) can be expanded in series of the","rect":[65.76243591308594,470.1547546386719,385.1154254741844,461.0010681152344]},{"page":47,"text":"Legendre polynomials Pi(cos#)","rect":[53.810420989990237,482.11431884765627,180.6134275039829,472.9613952636719]},{"page":47,"text":"fð#Þ ¼ ð1=2Þ½1 þ a1P1ðcos#Þ þ a2P2ðcos#Þ þ a3P3ðcos#Þ þ a4P4ðcos#Þ þ :::\b","rect":[58.06117248535156,507.33697509765627,380.90635621231936,497.3860778808594]},{"page":47,"text":"(3.3)","rect":[366.09649658203127,521.7334594726563,385.16875587312355,513.257080078125]},{"page":47,"text":"Recall","rect":[65.76500701904297,544.0,91.33737046787332,536.2667846679688]},{"page":47,"text":"that","rect":[107.05905151367188,544.0,122.07995469638894,536.2667846679688]},{"page":47,"text":"the","rect":[137.81756591796876,544.0,150.0413287944969,536.2667846679688]},{"page":47,"text":"Legendre","rect":[165.74310302734376,545.2012939453125,203.55616033746566,536.2667846679688]},{"page":47,"text":"polynomials","rect":[219.21511840820313,545.2012939453125,268.6138955508347,536.2667846679688]},{"page":47,"text":"1 dn½ðx2 \u0003 1Þn\b","rect":[91.14208984375,558.6327514648438,161.0107873646631,547.4880981445313]},{"page":47,"text":"PðxÞ ¼ 2nn!\u0004","rect":[53.812984466552737,569.9351806640625,106.85850464981933,555.4845581054688]},{"page":47,"text":"dxn","rect":[128.3579864501953,569.9348754882813,141.30160591682754,562.942626953125]},{"page":47,"text":", n ¼ 0, 1, 2, ... are solutions","rect":[162.6848907470703,564.3587036132813,281.4635049258347,556.1612548828125]},{"page":47,"text":"ð1 \u0003 x2Þy00\u0003 2xy0 þ nðn þ 1Þy ¼ 0, which are orthogonal","rect":[53.81271743774414,582.608642578125,281.993544173928,571.7877197265625]},{"page":47,"text":"tion for orthogonality reads:","rect":[53.81416702270508,594.2294921875,167.13288743564676,585.2949829101563]},{"page":47,"text":"of","rect":[284.35650634765627,544.0,292.6483396133579,536.2667846679688]},{"page":47,"text":"of","rect":[284.24371337890627,564.0,292.53557716218605,556.1612548828125]},{"page":47,"text":"to","rect":[284.5845947265625,581.0,292.3588494401313,574.3513793945313]},{"page":47,"text":"general","rect":[308.3172912597656,545.2012939453125,337.8255449063499,536.2667846679688]},{"page":47,"text":"formula","rect":[353.5203552246094,544.0,385.12497747613755,536.2667846679688]},{"page":47,"text":"the Legendre equation","rect":[295.233154296875,565.0957641601563,385.17644587567818,556.1612548828125]},{"page":47,"text":"each other. The condi-","rect":[294.950927734375,581.0,385.1320737442173,573.33544921875]},{"page":48,"text":"30","rect":[53.812843322753909,42.55630874633789,62.2739118972294,36.73143005371094]},{"page":48,"text":"Fig. 3.15 Form of the","rect":[53.812843322753909,67.58130645751953,131.38393542551519,59.648292541503909]},{"page":48,"text":"molecular distribution","rect":[53.812843322753909,75.76238250732422,129.35159057764933,69.89517211914063]},{"page":48,"text":"function over the polar","rect":[53.812843322753909,87.4087142944336,132.1445779190748,79.81436157226563]},{"page":48,"text":"Euler angles #","rect":[53.812843322753909,97.3846664428711,103.6087757652725,89.60404968261719]},{"page":48,"text":"3 Mesogenic Molecules and Orientational Order","rect":[219.66928100585938,44.274986267089847,385.1406564102857,36.6636962890625]},{"page":48,"text":"1","rect":[85.59085845947266,209.6060791015625,89.07482216438612,204.89279174804688]},{"page":48,"text":"ð PmðxÞPnðxÞdx ¼ 0 for m 6¼ n and ð ½PmðxÞ\b2dx ¼ 2m2þ1","rect":[84.45791625976563,233.71258544921876,339.63591853192818,211.58987426757813]},{"page":48,"text":"\u00031","rect":[82.41905212402344,239.98724365234376,91.28435585579237,235.27395629882813]},{"page":48,"text":"(3.4)","rect":[366.09716796875,226.87904357910157,385.1694272598423,218.40267944335938]},{"page":48,"text":"The Legendre polynomials Pm(x) are tabulated for x ¼ 0–1. In our case, the","rect":[65.76570892333985,262.5325927734375,385.17420232965318,253.59803771972657]},{"page":48,"text":"polynomials depend on angle # with x ¼ cos# and the integration should be made","rect":[53.81446075439453,274.4931335449219,385.14331854059068,265.3394470214844]},{"page":48,"text":"from p to 0. For even or odd m the polynomials are even or odd functions of cos#,","rect":[53.816444396972659,286.4526672363281,385.1851467659641,277.2989807128906]},{"page":48,"text":"respectively:","rect":[53.81645965576172,298.4122314453125,104.99005754062722,289.4776611328125]},{"page":48,"text":"P0(cos","rect":[53.81645965576172,315.92529296875,80.24290098296359,307.4489440917969]},{"page":48,"text":"P1(cos","rect":[53.81348419189453,327.88482666015627,80.24290098296359,319.4084777832031]},{"page":48,"text":"P2(cos","rect":[53.813480377197269,339.8446044921875,80.24290098296359,331.3682556152344]},{"page":48,"text":"P3(cos","rect":[53.81437683105469,351.8041076660156,80.24290098296359,343.3277587890625]},{"page":48,"text":"P4(cos","rect":[53.81437683105469,363.7638854980469,80.24290098296359,355.28753662109377]},{"page":48,"text":"#)","rect":[83.09876251220703,315.92529296875,92.24768195954931,307.1700439453125]},{"page":48,"text":"#)","rect":[83.09876251220703,327.88482666015627,92.24768195954931,319.12957763671877]},{"page":48,"text":"#)","rect":[83.09876251220703,339.8446044921875,92.24768195954931,331.08935546875]},{"page":48,"text":"#)","rect":[83.09876251220703,351.8041076660156,92.24768195954931,343.0488586425781]},{"page":48,"text":"#)","rect":[83.09876251220703,363.7638854980469,92.24768195954931,355.0086364746094]},{"page":48,"text":"¼","rect":[94.99404907226563,312.85748291015627,102.65879085752873,310.5267333984375]},{"page":48,"text":"¼","rect":[94.99404907226563,324.8170166015625,102.65879085752873,322.48626708984377]},{"page":48,"text":"¼","rect":[94.99404907226563,336.77679443359377,102.65879085752873,334.446044921875]},{"page":48,"text":"¼","rect":[94.99404907226563,348.7362976074219,102.65879085752873,346.4055480957031]},{"page":48,"text":"¼","rect":[94.99404907226563,360.6960754394531,102.65879085752873,358.3653259277344]},{"page":48,"text":"1","rect":[105.47384643554688,314.1822204589844,110.45095149091253,307.4489440917969]},{"page":48,"text":"cos#","rect":[105.47384643554688,326.2413635253906,124.5889108788337,319.12957763671877]},{"page":48,"text":"(1/2)(3cos2# \u0003 1)","rect":[105.47384643554688,339.8446044921875,177.6616910537876,329.1920166015625]},{"page":48,"text":"(1/2)(5cos3# \u0003 3cos#)","rect":[105.47384643554688,351.8041076660156,196.75882850495948,341.1516418457031]},{"page":48,"text":"(1/8)(35cos4# \u0003 30cos2#","rect":[105.47384643554688,363.7638854980469,207.12021428459543,353.11126708984377]},{"page":48,"text":"þ","rect":[209.92730712890626,362.83758544921877,217.59204891416938,356.2138671875]},{"page":48,"text":"3),","rect":[220.4071044921875,363.7638854980469,231.18751187826877,355.28753662109377]},{"page":48,"text":"etc.","rect":[234.00157165527345,362.1004943847656,248.1863216927219,356.2437438964844]},{"page":48,"text":"Each function has a particular symmetry (like electron shells in atoms have their","rect":[65.7664566040039,382.0732116699219,385.11153541413918,373.138671875]},{"page":48,"text":"own symmetry s, p, d, etc.). The angular dependencies of the first two polynomials","rect":[53.814430236816409,394.03277587890627,385.13632597075658,385.07830810546877]},{"page":48,"text":"are plotted in Fig. 3.16.","rect":[53.81641387939453,405.9923095703125,148.5795101692844,397.0577392578125]},{"page":48,"text":"In order to find numerical coefficients aL we multiply both sides of Eq. (3.3) by","rect":[65.76844024658203,417.9523620605469,385.17116633466255,409.017333984375]},{"page":48,"text":"PL(cos#) and integrate over #, using the orthogonality of Legendre polynomials,","rect":[53.813411712646487,429.8553161621094,385.12798734213598,420.7016296386719]},{"page":48,"text":"Eq.(3.4):","rect":[53.81205368041992,441.8148498535156,89.32270677158425,432.9400634765625]},{"page":48,"text":"0","rect":[65.42516326904297,461.86273193359377,68.90912697395643,457.0657653808594]},{"page":48,"text":"0","rect":[181.60433959960938,461.86273193359377,185.08830330452285,457.0657653808594]},{"page":48,"text":"ðPLðcos#Þfð#Þdðcos#Þ ¼ 12 ðaL½PLðcos#Þ\b2dðcos#Þ ¼ 2LaþL 1L ¼ 0;1;2;...","rect":[64.2918472290039,485.8849792480469,373.3720239369287,463.76263427734377]},{"page":48,"text":"p","rect":[64.7450942993164,491.8687438964844,68.59139022954088,488.7172546386719]},{"page":48,"text":"p","rect":[180.9242706298828,491.8687438964844,184.77056656010729,488.7172546386719]},{"page":48,"text":"Now we obtain","rect":[65.7650375366211,514.3841552734375,127.27608576825628,507.47161865234377]},{"page":48,"text":"0","rect":[202.3364715576172,536.39794921875,205.82043526253066,531.6010131835938]},{"page":48,"text":"Ð PLðcos#Þfð#Þsin#d#","rect":[200.80706787109376,549.354248046875,296.2232568871345,538.298095703125]},{"page":48,"text":"aL ¼ ð2L þ 1Þp","rect":[141.21595764160157,562.6570434570313,204.53973159916979,552.2000122070313]},{"page":48,"text":"0","rect":[223.86163330078126,564.7950439453125,227.3455970056947,559.9981079101563]},{"page":48,"text":"Ð fð#Þdcos#","rect":[222.388916015625,577.751220703125,274.6980188500251,566.695068359375]},{"page":48,"text":"p","rect":[222.27574157714845,583.6914672851563,226.1220375073729,580.5399780273438]},{"page":48,"text":"(3.5)","rect":[366.0975646972656,561.9198608398438,385.1698239883579,553.323974609375]},{"page":49,"text":"3.3 Orientational Distribution Functions for Molecules","rect":[53.812843322753909,42.55636978149414,240.7035263347558,36.66375732421875]},{"page":49,"text":"31","rect":[376.7465515136719,42.55636978149414,385.20761627345009,36.73149108886719]},{"page":49,"text":"a","rect":[72.63838195800781,68.23361206054688,78.19364435524166,62.64485549926758]},{"page":49,"text":"1","rect":[73.10908508300781,79.31622314453125,76.99780770940957,74.34728240966797]},{"page":49,"text":"0","rect":[73.10908508300781,106.9771728515625,76.99780770940957,101.88225555419922]},{"page":49,"text":"150","rect":[87.7782211303711,103.17560577392578,97.77779378893867,98.80853271484375]},{"page":49,"text":"120","rect":[110.7226791381836,80.66663360595703,120.72225179675117,76.299560546875]},{"page":49,"text":"90","rect":[144.08995056152345,72.11304473876953,150.75633230212228,67.7459716796875]},{"page":49,"text":"–1 180","rect":[69.2203598022461,133.78826904296876,91.32063283923164,128.63616943359376]},{"page":49,"text":"Fig.","rect":[53.812843322753909,225.77764892578126,67.73130294873677,218.04783630371095]},{"page":49,"text":"0","rect":[72.05867004394531,161.89862060546876,75.94739267034707,156.8037109375]},{"page":49,"text":"1","rect":[73.10918426513672,189.30966186523438,76.99790689153848,184.34072875976563]},{"page":49,"text":"3.16","rect":[70.12663269042969,224.41456604003907,84.93350738673135,217.84463500976563]},{"page":49,"text":"210","rect":[87.7782211303711,164.55032348632813,97.77779378893867,160.18325805664063]},{"page":49,"text":"240","rect":[110.79761505126953,186.98431396484376,120.7971877098371,182.61724853515626]},{"page":49,"text":"P =cosθ","rect":[87.55249786376953,198.8153076171875,111.44638096499908,194.4242401123047]},{"page":49,"text":"1","rect":[91.55104064941406,200.73089599609376,93.77316786450078,197.89149475097657]},{"page":49,"text":"270","rect":[142.36582946777345,194.8624267578125,152.36540212634103,190.495361328125]},{"page":49,"text":"Angular dependencies","rect":[90.91548919677735,225.70993041992188,166.74866183280268,218.11557006835938]},{"page":49,"text":"60","rect":[173.5587158203125,80.66663360595703,180.22509756091135,76.299560546875]},{"page":49,"text":"300","rect":[173.55873107910157,186.98431396484376,183.55830373766916,182.61724853515626]},{"page":49,"text":"of the","rect":[169.14230346679688,224.0,188.93529650949956,218.11557006835938]},{"page":49,"text":"30","rect":[196.5775146484375,103.09943389892578,203.24389638903635,98.73236083984375]},{"page":49,"text":"0","rect":[204.97523498535157,133.78704833984376,208.30842580798166,129.41998291015626]},{"page":49,"text":"b","rect":[225.30618286132813,68.23361206054688,231.41097661080634,60.92523956298828]},{"page":49,"text":"1.0","rect":[224.6775360107422,84.3162841796875,234.39934274358925,79.22136688232422]},{"page":49,"text":"0.5","rect":[224.6775360107422,102.02325439453125,234.39934274358925,96.92833709716797]},{"page":49,"text":"0.0","rect":[224.6775360107422,119.656005859375,234.39934274358925,114.56108856201172]},{"page":49,"text":"–0.5","rect":[220.788818359375,137.36407470703126,234.39934274358925,132.2691650390625]},{"page":49,"text":"150","rect":[246.4503173828125,107.07654571533203,256.44989004138008,102.70947265625]},{"page":49,"text":"180","rect":[239.76895141601563,136.71380615234376,249.7685240745832,132.34674072265626]},{"page":49,"text":"120","rect":[269.9217529296875,85.39234161376953,279.92132558825508,81.0252685546875]},{"page":49,"text":"330","rect":[196.578125,164.47531127929688,206.5776976585676,160.10824584960938]},{"page":49,"text":"0.0","rect":[224.6775360107422,155.06887817382813,234.39934274358925,149.97396850585938]},{"page":49,"text":"0.5","rect":[224.6775360107422,172.77581787109376,234.39934274358925,167.680908203125]},{"page":49,"text":"210","rect":[246.4503173828125,166.35107421875,256.44989004138008,161.9840087890625]},{"page":49,"text":"1.0","rect":[224.6775360107422,190.40927124023438,234.39934274358925,185.31436157226563]},{"page":49,"text":"P2=(1/2)(3cos2θ−1)","rect":[222.9819793701172,204.55606079101563,274.7971292926425,196.7177276611328]},{"page":49,"text":"first two Legendre polynomials","rect":[191.34669494628907,225.70993041992188,298.3090256023339,218.11557006835938]},{"page":49,"text":"90","rect":[304.0389709472656,77.13928985595703,310.70535268786446,72.772216796875]},{"page":49,"text":"270","rect":[302.390380859375,195.61224365234376,312.38995351794258,191.24517822265626]},{"page":49,"text":"in polar","rect":[300.6730651855469,225.70993041992188,327.11813443762949,218.11557006835938]},{"page":49,"text":"60","rect":[334.18157958984377,85.39234161376953,340.84796133044258,81.0252685546875]},{"page":49,"text":"300","rect":[334.25653076171877,187.95965576171876,344.2561034202863,183.59259033203126]},{"page":49,"text":"30","rect":[357.8016662597656,107.00092315673828,364.46804800036446,102.63385009765625]},{"page":49,"text":"0","rect":[366.4253845214844,136.71319580078126,369.75857534411446,132.34613037109376]},{"page":49,"text":"330","rect":[357.80169677734377,166.276123046875,367.8012694359113,161.9090576171875]},{"page":49,"text":"coordinates","rect":[329.505859375,224.0,368.60446627616207,218.11557006835938]},{"page":49,"text":"with the normalization condition (3.2) in denominator. As to the numerator, it is just","rect":[53.812843322753909,250.57391357421876,385.14375169345927,241.6393585205078]},{"page":49,"text":"the average value of polynomial PL(cos#) written in the form of the theorem of","rect":[53.812843322753909,262.5334777832031,385.1423173672874,253.37991333007813]},{"page":49,"text":"average. Finally we obtain numerical coefficients:","rect":[53.81246566772461,274.532958984375,256.502089095803,265.55859375]},{"page":49,"text":"aL ¼ ð2L þ 1Þ \u0004 hPLðcos#Þi","rect":[163.70297241210938,298.751220703125,275.2563516055222,288.8006896972656]},{"page":49,"text":"(3.6)","rect":[366.0974426269531,298.0141296386719,385.1697019180454,289.47802734375]},{"page":49,"text":"Note that <PL(cos#)> are #-dependent numbers, not functions. Finally we can","rect":[65.7660140991211,322.275146484375,385.1275567155219,313.1214599609375]},{"page":49,"text":"write the orientational distribution function for a uniaxial medium composed of","rect":[53.81357955932617,334.21478271484377,385.142469955178,325.26031494140627]},{"page":49,"text":"uniaxial molecules:","rect":[53.81357955932617,344.26190185546877,132.23185594150614,337.2397766113281]},{"page":49,"text":"fð#Þ ¼ ð1=2Þ\"17þhP33hðPco1ðsc#oÞsi#PÞ3iðPco1ðsc#oÞsþ#Þ9þhP54hðPco2ðsc#oÞsi#PÞ4iðPco2ðsc#oÞsþ#Þ:þ::#","rect":[67.46477508544922,390.29376220703127,352.3618684860727,360.4322814941406]},{"page":49,"text":"(3.7)","rect":[366.0967102050781,379.63507080078127,385.1689694961704,371.1587219238281]},{"page":49,"text":"The set of amplitudes aL may be considered as a set of order parameters for the","rect":[65.7652359008789,413.8154602050781,385.1738971538719,404.88006591796877]},{"page":49,"text":"medium discussed. All of them together provide a complete description of f(#).","rect":[53.81313705444336,425.7741394042969,374.3346828987766,416.6204528808594]},{"page":49,"text":"For uniaxial molecules with inversion center (i.e. having head-to tail symmetry","rect":[65.7641372680664,437.73370361328127,385.1718682389594,428.77923583984377]},{"page":49,"text":"of a cylinder) the odd terms disappear:","rect":[53.81312942504883,449.6932373046875,210.03152329990457,440.7586669921875]},{"page":49,"text":"fð#Þ ¼ ð1=2Þ½1 þ 5hP2ðcos#ÞiP2ðcos#Þ þ 9hP4ðcos#ÞiP4ðcos#Þ þ :::\b (3.8)","rect":[65.87863159179688,473.9515380859375,385.1695798477329,464.0004577636719]},{"page":49,"text":"or briefly:","rect":[53.8138542175293,497.4751892089844,93.72326524326394,488.5406494140625]},{"page":49,"text":"11","rect":[172.9943084716797,520.5003051757813,190.26343623212348,512.7777099609375]},{"page":49,"text":"fð#Þ ¼ fðcos#Þ ¼ X2ð4l þ 1ÞS2lP2lðcos#Þ with l ¼ 0;1;2:::","rect":[92.44615173339844,534.1033325195313,346.51979005020999,517.8363647460938]},{"page":49,"text":"0","rect":[174.75022888183595,538.0983276367188,178.2341925867494,533.3013916015625]},{"page":49,"text":"(3.9)","rect":[366.0997009277344,528.9884643554688,385.17196021882668,520.5120849609375]},{"page":49,"text":"As mentioned above, coefficients S2L are unknown numbers: S0 \u0006 1, S1 ¼","rect":[65.76827239990235,562.2404174804688,385.14807918760689,553.7239379882813]},{"page":49,"text":"<P1(cos#)> ¼ <cos#>, S2 ¼ <P2(cos#)> \u0006 (1/2) <3cos2# \u0003 1>, S4","rect":[53.814231872558597,574.2197265625,385.18130140630105,563.5672607421875]},{"page":49,"text":"¼<P4(cos#)>","rect":[53.812843322753909,586.1793823242188,112.45809200498967,577.4241333007813]},{"page":49,"text":"\u0006","rect":[117.595458984375,584.0975952148438,125.2602007696381,579.794677734375]},{"page":49,"text":"(1/8)<35cos4#","rect":[130.3975830078125,586.1793823242188,190.29660588615793,575.5269165039063]},{"page":49,"text":"\u0003","rect":[195.3692626953125,583.0,203.03400448057563,581.0]},{"page":49,"text":"30cos2#","rect":[208.17138671875,584.5557861328125,240.99397832268137,575.5269165039063]},{"page":49,"text":"+","rect":[246.066650390625,584.3665771484375,251.660916472856,579.3364868164063]},{"page":49,"text":"3>,","rect":[256.77239990234377,584.8147583007813,271.9495815804172,577.7030029296875]},{"page":49,"text":"etc.","rect":[277.0511169433594,584.5159912109375,291.2358669808078,578.6591796875]},{"page":49,"text":"Therefore,","rect":[296.36724853515627,584.5159912109375,338.3113064339328,577.6432495117188]},{"page":49,"text":"instead","rect":[343.438720703125,584.5458374023438,371.71865168622505,577.6432495117188]},{"page":49,"text":"of","rect":[376.8590087890625,584.5159912109375,385.1508725723423,577.6432495117188]},{"page":50,"text":"32","rect":[53.81438446044922,42.55740737915039,62.27545303492471,36.73252868652344]},{"page":50,"text":"3 Mesogenic Molecules and Orientational Order","rect":[219.67083740234376,44.276084899902347,385.1422128067701,36.664794921875]},{"page":50,"text":"unknown function f(#) we can operate with unknown numbers that can easier be","rect":[53.812843322753909,68.2883529663086,385.1487506694969,59.134674072265628]},{"page":50,"text":"found from experiment.","rect":[53.810855865478519,80.24788665771485,149.61713834311252,71.31333923339844]},{"page":50,"text":"3.3.2 Lath-Like Molecules","rect":[53.812843322753909,122.9268569946289,192.84227637980147,114.39273071289063]},{"page":50,"text":"The phase is still uniaxial with head-to-tail symmetry, i.e., its distribution function","rect":[53.812843322753909,152.57284545898438,385.16668025067818,143.63829040527345]},{"page":50,"text":"is independent of the precession angle F that is fðOÞ ¼ fð#;CÞ=2p . However,","rect":[53.812843322753909,164.87103271484376,385.15557523276098,154.92051696777345]},{"page":50,"text":"nuclear magnetic resonance (NMR) shows that the free rotation of molecules about","rect":[53.81477737426758,176.491943359375,385.15669114658427,167.55738830566407]},{"page":50,"text":"their long axes (angle C) is, to some extent, hindered as shown in Fig. 3.17a. In","rect":[53.81477737426758,188.39468383789063,385.1516045670844,179.35057067871095]},{"page":50,"text":"the figure the preferable direction of the longest molecular axes (director) is parallel","rect":[53.81379318237305,200.354248046875,385.1655717618186,191.41969299316407]},{"page":50,"text":"to z. Then we can distinguish among two different cases of local molecular","rect":[53.81379318237305,212.31378173828126,385.15364967195168,203.3792266845703]},{"page":50,"text":"orientation with two projections Sz of a short molecular axis onto the director,","rect":[53.81378936767578,224.2733154296875,385.1042751839328,215.33876037597657]},{"page":50,"text":"either large as in Fig. 3.17b or, in fact, zero (Fig. 3.17c).","rect":[53.81321334838867,236.23358154296876,281.94781156088598,227.2990264892578]},{"page":50,"text":"As a consequence, the refraction index component perpendicular to the director","rect":[65.7652359008789,248.19314575195313,385.10433326570168,239.2585906982422]},{"page":50,"text":"n⊥ is larger in case b than in case c, and the component n|| is smaller. Therefore, the","rect":[53.81321334838867,260.1529846191406,385.1742328472313,251.2184295654297]},{"page":50,"text":"optical anisotropy Dn ¼ n|| \u0003 n⊥ in case b is smaller. To take the new situation into","rect":[53.8134880065918,272.112548828125,385.1559990983344,262.84930419921877]},{"page":50,"text":"account, two local order parameters are introduced for the uniaxial nematic phase,","rect":[53.81313705444336,284.015380859375,385.1818508675266,275.0609130859375]},{"page":50,"text":"one is the same as discussed above for the longitudinal molecular axes (S ¼ Szz),","rect":[53.813167572021487,296.62969970703127,385.1559109261203,287.0404052734375]},{"page":50,"text":"and the other describes the local order of the shortest molecular axes that is local","rect":[53.81411361694336,306.0024108886719,385.1330705411155,298.98028564453127]},{"page":50,"text":"biaxiality (D):","rect":[53.81417465209961,319.8743896484375,111.13347489902566,310.9398498535156]},{"page":50,"text":"SBB ¼ 12\u00033cos2# \u0003 1\u0004","rect":[152.14979553222657,354.4251403808594,238.15010170389258,334.0888366699219]},{"page":50,"text":"D ¼ Sxx \u0003 S\u0002\u0002 ¼ 23\u0003sin2#cos2C\u0004","rect":[148.8070831298828,378.6845703125,286.809388202916,358.3482666015625]},{"page":50,"text":"(3.10)","rect":[361.056884765625,360.6468200683594,385.10625587312355,352.17047119140627]},{"page":50,"text":"For the ideal nematic with sin# ¼ 0 and Szz ¼ 1 there is no difference between","rect":[65.7672348022461,402.84954833984377,385.1773614030219,393.0412902832031]},{"page":50,"text":"cases b and c. The locally (microscopically) biaxial nematic phase should not be","rect":[53.81367111206055,414.1546936035156,385.15052068902818,405.22015380859377]},{"page":50,"text":"confused with macroscopically biaxial phases to be discussed in the next section.","rect":[53.81367111206055,426.09429931640627,382.8729519417453,417.1597595214844]},{"page":50,"text":"a","rect":[73.69539642333985,444.6507873535156,79.25065882057369,439.0620422363281]},{"page":50,"text":"b","rect":[189.84634399414063,444.6507873535156,195.95113774361884,437.3424072265625]},{"page":50,"text":"short","rect":[106.0473403930664,469.8923034667969,122.03386427563359,464.35748291015627]},{"page":50,"text":"axis","rect":[107.6020278930664,476.8908386230469,120.47917620137837,471.35601806640627]},{"page":50,"text":"Z","rect":[198.37660217285157,461.66015625,203.26048608802416,456.36529541015627]},{"page":50,"text":"S","rect":[200.16310119628907,500.08465576171877,204.1597328780997,494.61383056640627]},{"page":50,"text":"z","rect":[204.15972900390626,502.42816162109377,206.49176359024276,499.3808288574219]},{"page":50,"text":"l","rect":[255.2887420654297,465.3157043457031,257.51086928051645,459.77288818359377]},{"page":50,"text":"s","rect":[247.50091552734376,507.6626281738281,251.3876009095811,503.1236572265625]},{"page":50,"text":"S = 0","rect":[293.39703369140627,481.8825378417969,310.22883993376379,476.3397216796875]},{"page":50,"text":"z","rect":[297.3936767578125,484.2259521484375,299.72571134414906,481.1786193847656]},{"page":50,"text":"long","rect":[78.07891082763672,527.209716796875,92.29093557585361,520.01123046875]},{"page":50,"text":"axis","rect":[78.74634552001953,532.5445556640625,91.62349382833149,527.009765625]},{"page":50,"text":"Fig. 3.17 Local packing of lath-like molecules that hinders rotation of individual molecules about","rect":[53.812843322753909,574.0244750976563,385.17712120261259,566.0914916992188]},{"page":50,"text":"their longest axes (a) and illustration of a large (b) or zero (c) projections Sz of a short molecular","rect":[53.812843322753909,583.9326782226563,385.1303719864576,576.3382568359375]},{"page":50,"text":"axis onto the director axis z. s and l are the shortest and longest axes of a lath-like molecule","rect":[53.81350326538086,593.9085693359375,368.3800291755152,586.2973022460938]},{"page":51,"text":"3.4 Principal Orientational Order Parameter (Microscopic Approach)","rect":[53.812843322753909,44.274620056152347,289.79721158606699,36.663330078125]},{"page":51,"text":"3.4 Principal Orientational Order Parameter","rect":[53.812843322753909,70.10667419433594,293.83515723676887,59.086421966552737]},{"page":51,"text":"(Microscopic Approach)","rect":[80.71903228759766,84.05052185058594,205.63541079304805,73.20955657958985]},{"page":51,"text":"33","rect":[376.74737548828127,42.55594253540039,385.20844024805947,36.73106384277344]},{"page":51,"text":"We discuss nematics, therefore n ¼ \u0003n (no polarity) and f(#) is cylindrically","rect":[53.812843322753909,111.5925521850586,385.13173762372505,102.43887329101563]},{"page":51,"text":"symmetric function (point group D1h). We would like to know this function","rect":[53.810848236083987,123.55208587646485,385.17083064130318,114.61753845214844]},{"page":51,"text":"for each particular substance at variable temperature but, unfortunately, f(#) (that","rect":[53.81405258178711,135.51168823242188,385.1807695157249,126.35800170898438]},{"page":51,"text":"is all amplitudes S2l in expansion (3.9)) is difficult to measure. We may, however,","rect":[53.812068939208987,147.47152709960938,385.1789211800266,138.53672790527345]},{"page":51,"text":"limit ourselves with one or few leading terms of the expansion and find approximate","rect":[53.812191009521487,159.43106079101563,385.1251605816063,150.4965057373047]},{"page":51,"text":"form of f(#).","rect":[53.812191009521487,171.37066650390626,104.25017209555392,162.23690795898438]},{"page":51,"text":"For instance, why not to take S0 or S1? For conventional nematics they are","rect":[65.76321411132813,183.35049438476563,385.1632770366844,174.4156036376953]},{"page":51,"text":"useless because S0 is angle independent and S1 ¼ <cos#> is an odd function","rect":[53.8134880065918,195.31008911132813,385.1712273698188,186.15640258789063]},{"page":51,"text":"incompatible with n ¼ \u0003n condition. By the way, S1 is very useful for discussion of","rect":[53.813472747802737,207.21307373046876,385.1524594864048,198.27830505371095]},{"page":51,"text":"phases with polar order, in which the head-to-tail molecular symmetry is broken","rect":[53.81364059448242,219.172607421875,385.17742243817818,210.23805236816407]},{"page":51,"text":"(e.g., in phases with the conical symmetry C1v instead of cylindrical symmetry","rect":[53.81364059448242,231.13235473632813,385.1734551530219,222.19761657714845]},{"page":51,"text":"D1h).","rect":[53.813716888427737,242.69358825683595,77.96927304770236,234.21722412109376]},{"page":51,"text":"The next is coefficient S2 ¼ (1/2) <3cos2#\u00031> introduced by Tsvetkov [14] that","rect":[65.76541137695313,255.05169677734376,385.13911302158427,244.0007781982422]},{"page":51,"text":"describes the quadrupolar order. It looks suitable, at least, when we consider","rect":[53.8132438659668,266.90167236328127,385.08440528718605,258.0567626953125]},{"page":51,"text":"important particular cases:","rect":[53.81223678588867,278.9707946777344,160.60697801181864,270.0362548828125]},{"page":51,"text":"(i) For the ideal nematic with all rod-like molecules parallel to each other","rect":[53.81223678588867,296.8817138671875,385.1361325821079,287.9471435546875]},{"page":51,"text":"<cos2#> ¼ 1 and S2 ¼ 1.","rect":[70.23966217041016,308.3440246582031,176.21107144858127,297.7908630371094]},{"page":51,"text":"(ii) For complete orientational disorder <cos2#> ¼ 1/3, S2 ¼ 0 and this corre-","rect":[53.8130989074707,320.80133056640627,385.10320411531105,309.7503967285156]},{"page":51,"text":"sponds to the isotropic phase.","rect":[71.03394317626953,332.7610168457031,190.1539730599094,323.82647705078127]},{"page":51,"text":"(iii) There is another possible molecular orientation also corresponding to <cos2#>","rect":[53.81415939331055,344.7205505371094,385.17359188291939,333.6131591796875]},{"page":51,"text":"¼ 1/3 and S2 ¼ 0: it is a “magic” orientation (see below), that would correspond","rect":[70.8643798828125,356.62371826171877,385.1679619889594,347.68896484375]},{"page":51,"text":"to the nematic phase with finite higher S2l coefficients.","rect":[70.86373901367188,368.5832824707031,291.08941312338598,359.64874267578127]},{"page":51,"text":"(iv) One can put all molecules in the plane perpendicular to the principal axis","rect":[53.813899993896487,380.54290771484377,385.1517678652878,371.58843994140627]},{"page":51,"text":"and then everywhere # ¼ p/2, <cos2#> ¼ 0 and S2 ¼ \u00031/2. The phase with","rect":[70.24134063720703,392.502685546875,385.11834040692818,381.45159912109377]},{"page":51,"text":"S2 ¼ \u00031/2 would still be conventional nematic phase, but such nematics have","rect":[70.24077606201172,404.4623107910156,385.1182330913719,395.52777099609377]},{"page":51,"text":"not been found yet. However, by evaporation of organic compounds consisted","rect":[70.2406234741211,416.4218444824219,385.1530694108344,407.4873046875]},{"page":51,"text":"of rod-like molecules onto a solid substrate, one can prepare amorphous solid","rect":[70.2406234741211,428.38140869140627,385.1371697526313,419.44683837890627]},{"page":51,"text":"films of the D1h symmetry which would mimic the nematic phase with S2 \u0005","rect":[70.2406234741211,440.3412780761719,385.14807918760689,431.4063720703125]},{"page":51,"text":"\u00031/2, see Fig. 3.18.","rect":[70.2416763305664,452.2440490722656,150.1072964241672,443.30950927734377]},{"page":51,"text":"From (i) to (iv) we conclude that, as the first approximation to the microscopic","rect":[65.76726531982422,470.2117614746094,385.1212848491844,461.2772216796875]},{"page":51,"text":"orientational distribution function of a nematic, we can take from Eq. (3.9) only one","rect":[53.81523895263672,482.1712951660156,385.14710272027818,473.23675537109377]},{"page":51,"text":"term with l ¼ 1 and S2 ¼ (1/2) <3cos2# \u0003 1>:","rect":[53.81523895263672,493.67626953125,245.0693498379905,483.023681640625]},{"page":51,"text":"fð#Þ ¼ fðcos#Þ \u0005 ð1=2Þð4L þ 1ÞS2P2ðcos#Þ ¼ ð5=2ÞS2P2ðcos#Þ","rect":[74.54676055908203,518.3326416015625,340.3612405215378,508.3819274902344]},{"page":51,"text":"(3.11)","rect":[361.0560607910156,517.5955810546875,385.10543189851418,509.11920166015627]},{"page":51,"text":"The function (3.11) with coefficient 5/2 ignored is shown in Fig. 3.19 for two","rect":[65.76641082763672,541.9130859375,385.15422907880318,532.9188232421875]},{"page":51,"text":"different values of S2. The curves marked as S2 ¼ 1 and S2 ¼ 0.6 correspond to the","rect":[53.81438446044922,553.8729248046875,385.17456854059068,544.9381103515625]},{"page":51,"text":"ideal and typical nematics, respectively. For the isotropic phase the corresponding","rect":[53.81386947631836,565.83251953125,385.11489192060005,556.8980102539063]},{"page":51,"text":"curve would coincide with zero line.","rect":[53.81386947631836,575.7601318359375,201.4497494270969,568.8575439453125]},{"page":52,"text":"34","rect":[53.812843322753909,42.55594253540039,62.2739118972294,36.73106384277344]},{"page":52,"text":"Fig. 3.18 Illustration ofa","rect":[53.812843322753909,68.37506866455078,143.7523283698511,60.42512130737305]},{"page":52,"text":"virtual nematic phase with","rect":[53.812843322753909,78.28328704833985,144.4952139297001,70.68893432617188]},{"page":52,"text":"order parameter S ¼ \u00031/2","rect":[53.812843322753909,88.25923919677735,143.3318075576298,80.66488647460938]},{"page":52,"text":"3 Mesogenic Molecules and Orientational Order","rect":[219.66928100585938,44.274620056152347,385.1406564102857,36.663330078125]},{"page":52,"text":"S = –1/2","rect":[252.93162536621095,72.5392837524414,281.3796578048575,66.80450439453125]},{"page":52,"text":"y","rect":[381.1747131347656,106.5616683959961,385.17134481657629,101.20281219482422]},{"page":52,"text":"x","rect":[262.93280029296877,145.45181274414063,267.3450816696877,141.70860290527345]},{"page":52,"text":"Fig. 3.19","rect":[53.529685974121097,367.25286865234377,86.34933227686807,359.21826171875]},{"page":52,"text":"parameter","rect":[53.529685974121097,377.1611328125,87.50933927161386,370.4303283691406]},{"page":52,"text":"1.0","rect":[130.81011962890626,214.6475830078125,142.03518201962985,208.43907165527345]},{"page":52,"text":"0.5","rect":[130.81011962890626,251.29229736328126,142.03518201962985,245.0837860107422]},{"page":52,"text":"0.0","rect":[130.81011962890626,287.8142395019531,142.03518201962985,281.605712890625]},{"page":52,"text":"S2=1","rect":[177.1476593017578,233.38970947265626,195.08515455380954,225.105224609375]},{"page":52,"text":"S2=0.6","rect":[154.06785583496095,276.67572021484377,179.14820387021579,268.390869140625]},{"page":52,"text":"–0.5","rect":[125.57708740234375,324.458984375,142.0359754766611,318.2504577636719]},{"page":52,"text":"0","rect":[145.1222686767578,333.5832824707031,149.56435682431735,327.374755859375]},{"page":52,"text":"30","rect":[171.7883758544922,333.5832824707031,180.67255689755954,327.374755859375]},{"page":52,"text":"Orientational","rect":[92.27462005615235,365.4664611816406,137.0929079457766,359.5738525390625]},{"page":52,"text":"S2","rect":[89.95205688476563,376.7578125,97.31707513644438,369.6175537109375]},{"page":52,"text":"distribution","rect":[141.2159881591797,365.4580078125,180.3315176406376,359.5907897949219]},{"page":52,"text":"60","rect":[200.67633056640626,333.5832824707031,209.56049635068454,327.374755859375]},{"page":52,"text":"function","rect":[184.4359893798828,365.4325866699219,212.71288055567667,359.5907897949219]},{"page":52,"text":"90","rect":[229.5642852783203,333.5832824707031,238.4484510625986,327.374755859375]},{"page":52,"text":"120","rect":[256.17047119140627,333.5832824707031,269.61576368466896,327.374755859375]},{"page":52,"text":"θ, deg","rect":[222.6069793701172,347.30322265625,245.6091492160358,339.337890625]},{"page":52,"text":"of molecules for","rect":[216.78012084960938,365.4325866699219,276.26739591223886,359.5907897949219]},{"page":52,"text":"150","rect":[284.9976806640625,333.5832824707031,298.5636091436533,327.374755859375]},{"page":52,"text":"180","rect":[313.88482666015627,333.5832824707031,327.45234205380958,327.374755859375]},{"page":52,"text":"two different","rect":[280.3363037109375,365.4580078125,326.33663658347197,359.5907897949219]},{"page":52,"text":"values","rect":[330.4114990234375,365.4325866699219,352.0718429851464,359.5907897949219]},{"page":52,"text":"of","rect":[356.1280822753906,365.4325866699219,363.1761483536451,359.5907897949219]},{"page":52,"text":"order","rect":[367.2873840332031,365.4580078125,385.16561978919199,359.5907897949219]},{"page":52,"text":"Parameter S2 can be found from the anisotropy of magnetic susceptibility,","rect":[65.76496887207031,425.9443664550781,385.1228298714328,417.00982666015627]},{"page":52,"text":"optical dichroism and birefringence, NMR, etc. The determination of higher order","rect":[53.81282424926758,437.9039001464844,385.1417478164829,428.9693603515625]},{"page":52,"text":"parameters requires for more sophisticated techniques. For instance, S4 can be","rect":[53.81282424926758,449.8066711425781,385.1535114116844,440.87213134765627]},{"page":52,"text":"found from Raman light scattering [15], luminescence or other two-wave interac-","rect":[53.813716888427737,461.7664794921875,385.0898374160923,452.77215576171877]},{"page":52,"text":"tion optical experiments. Data on S6, S8 are not available at present. In some cases,","rect":[53.813716888427737,473.72625732421877,385.1222195198703,464.79150390625]},{"page":52,"text":"the X-ray scattering can even provide f(#) as a whole but with limited accuracy.","rect":[53.813289642333987,485.6858215332031,377.47539945151098,476.5321350097656]},{"page":52,"text":"To illustrate the importance of higher order terms, particularly S4, consider two","rect":[65.7643051147461,497.6453552246094,385.15349665692818,488.7108154296875]},{"page":52,"text":"molecular distributions shown in Fig. 3.20. On the left side, all molecules of a","rect":[53.81362533569336,509.6051025390625,385.15943182184068,500.650634765625]},{"page":52,"text":"virtual nematic phase are at the same angle # ¼ 54.73 deg, therefore f (#) / d(# \u0003","rect":[53.81362533569336,521.5646362304688,385.14551571104439,512.331298828125]},{"page":52,"text":"54.73). For this “magic” orientation, cos2# ¼ 1/3, cos4# ¼ 1/9 and S2 ¼ 0, S4 ¼ \u00037/18","rect":[53.81167221069336,533.524169921875,385.1800469498969,522.4169311523438]},{"page":52,"text":"(see Legendre polynomials P2 and P4 written above). On the right side, the","rect":[53.814353942871097,545.4275512695313,385.11768377496568,536.4730834960938]},{"page":52,"text":"molecules of another virtual nematic are scattered over angles around # ¼ 54.73","rect":[53.81367111206055,557.3870849609375,385.15252009442818,548.2334594726563]},{"page":52,"text":"deg in such a special way that the average <cos2#> calculated with new f(#)","rect":[53.81367111206055,569.3466186523438,385.15828834382668,558.2958374023438]},{"page":52,"text":"function is again equal to 1/3 and, as before, S2 ¼0. However, <cos4#> calculated","rect":[53.812435150146487,581.306396484375,385.14959040692818,570.2554931640625]},{"page":52,"text":"with new f(#) and new S4 is different from \u00037/18. Therefore to distinguish between","rect":[53.813777923583987,593.2660522460938,385.17656794599068,584.1124267578125]},{"page":53,"text":"3.5 Macroscopic Definition of","rect":[53.81290054321289,44.275108337402347,156.7612771378248,36.62995529174805]},{"page":53,"text":"Fig. 3.20 Illustration of","rect":[53.812843322753909,67.58130645751953,137.61550992591075,59.648292541503909]},{"page":53,"text":"importance of higher order","rect":[53.812843322753909,77.4895248413086,145.78297513587169,69.89517211914063]},{"page":53,"text":"terms: the two very different,","rect":[53.812843322753909,87.4087142944336,154.04098770215473,79.81436157226563]},{"page":53,"text":"virtual molecular","rect":[53.812843322753909,95.63212585449219,112.38545316321542,89.79031372070313]},{"page":53,"text":"distributions have the same S2","rect":[53.812843322753909,106.95709228515625,155.32170617892485,99.76632690429688]},{"page":53,"text":"order parameter but differ by","rect":[53.812843322753909,117.33626556396485,153.35986847071573,109.74191284179688]},{"page":53,"text":"the values of higher order","rect":[53.812843322753909,127.25545501708985,141.98733609778575,119.66110229492188]},{"page":53,"text":"parameters S4, S6, etc.","rect":[53.812843322753909,137.23141479492188,130.16221878125629,129.68748474121095]},{"page":53,"text":"the","rect":[159.11599731445313,43.0,169.50618884348394,36.68075180053711]},{"page":53,"text":"Orientational","rect":[171.86090087890626,43.0,216.67919639792505,36.663818359375]},{"page":53,"text":"Order","rect":[219.10330200195313,43.0,238.82606595618419,36.663818359375]},{"page":53,"text":"Parameter","rect":[241.25184631347657,43.0,275.6715096817701,36.85007858276367]},{"page":53,"text":"n","rect":[342.2060546875,66.70843505859375,346.2007382734348,62.65754699707031]},{"page":53,"text":"35","rect":[376.7466125488281,42.55643081665039,385.2076773086063,36.62995529174805]},{"page":53,"text":"two molecular distributions we have to take into account at least S4 or as many S2l","rect":[53.812843322753909,196.61383056640626,385.18914688350528,187.6792755126953]},{"page":53,"text":"coefficients as possible.","rect":[53.812843322753909,208.57345581054688,148.99998136069065,199.63890075683595]},{"page":53,"text":"3.5 Macroscopic Definition of the Orientational","rect":[53.812843322753909,252.7323455810547,304.78279325316967,241.71209716796876]},{"page":53,"text":"Order Parameter","rect":[80.71903228759766,264.3693542480469,170.63050635786264,255.93084716796876]},{"page":53,"text":"3.5.1 Tensor Properties","rect":[53.812843322753909,296.5205383300781,177.97667579386397,286.03814697265627]},{"page":53,"text":"Generally, properties of liquid crystals depend on direction, they are tensorial.","rect":[53.812843322753909,324.1456298828125,385.1775784065891,315.2110595703125]},{"page":53,"text":"Some of them (like density in nematics) may be scalar. A scalar is a tensor of","rect":[53.812843322753909,336.1051940917969,385.14775977937355,327.170654296875]},{"page":53,"text":"rank 0. It has one component in a space of any dimensionality, 10 ¼ 20 ¼ 30 ... ¼ 1.","rect":[53.812843322753909,348.0079345703125,385.17547269369848,336.95721435546877]},{"page":53,"text":"Other","rect":[53.81374740600586,357.9458312988281,76.49939094392431,351.0133361816406]},{"page":53,"text":"properties,","rect":[81.68254852294922,359.9678039550781,124.13328214194064,351.03326416015627]},{"page":53,"text":"like","rect":[129.26368713378907,357.9059753417969,144.35526312066879,351.03326416015627]},{"page":53,"text":"spontaneous","rect":[149.4866485595703,359.9678039550781,198.796833172905,352.0492248535156]},{"page":53,"text":"polarization","rect":[203.92225646972657,359.9678039550781,251.75823298505316,351.03326416015627]},{"page":53,"text":"P","rect":[256.9423522949219,357.8262939453125,263.0542373029109,351.043212890625]},{"page":53,"text":"(e.g.,","rect":[268.2145080566406,359.9678039550781,288.42156644369848,351.093017578125]},{"page":53,"text":"in","rect":[293.5907897949219,358.0,301.36501399091255,351.03326416015627]},{"page":53,"text":"chiral","rect":[306.50537109375,358.0,329.2905412442405,351.03326416015627]},{"page":53,"text":"smectic","rect":[334.430908203125,358.0,365.06797064020005,351.03326416015627]},{"page":53,"text":"C*)","rect":[370.230224609375,359.5693664550781,385.1516049942173,351.093017578125]},{"page":53,"text":"are vectors, i.e., the tensors of rank 1. In the two-dimensional space they have 21","rect":[53.813716888427737,371.9273376464844,385.18130140630105,360.8764953613281]},{"page":53,"text":"¼ 2 components, in the 3D space there are 31 ¼ 3 components. For instance in","rect":[53.812843322753909,383.8872375488281,385.1427849870063,372.8362731933594]},{"page":53,"text":"the Cartesian system P ¼ iPx + jPy + kPz. Such a vector can be written as a row","rect":[53.81388473510742,396.776611328125,385.1591162677082,386.83251953125]},{"page":53,"text":"(Px, Py, Pz) or as a column. Properties described by tensor of rank 2 have 22 ¼4","rect":[53.813289642333987,408.7362365722656,385.17934504560005,396.7554626464844]},{"page":53,"text":"components in 2D space and 32 ¼ 9 components in the 3D-space. They relate two","rect":[53.814598083496097,419.7661437988281,385.1519707780219,408.7152099609375]},{"page":53,"text":"vector quantities, such as magnetization M and magnetic induction B, M ¼ wB,","rect":[53.814144134521487,431.7257080078125,385.1848110726047,422.7911376953125]},{"page":53,"text":"where w is magnetic susceptibility. Each of the two vectors has three components","rect":[53.81415939331055,443.62847900390627,385.1301614199753,434.69390869140627]},{"page":53,"text":"and, generally, each component (projection) Ma (a ¼ x,y,z) may depend on each of","rect":[53.81415939331055,455.58843994140627,385.15096412507668,446.65350341796877]},{"page":53,"text":"Bb components (b ¼ x,y,z):","rect":[53.81513214111328,468.4180908203125,163.92666490146707,458.30474853515627]},{"page":53,"text":"Mx ¼ wxxBx þ wxyBy þ wxzBz; My ¼ wyxBx þ wyyBy + wyzBz; Mz ¼ wzxBx þ wzyBy þ","rect":[65.76720428466797,480.4241943359375,385.14807918760689,470.7922058105469]},{"page":53,"text":"wzzBz or in the matrix form:","rect":[53.814231872558597,491.3775939941406,164.24238450595926,482.5328369140625]},{"page":53,"text":"\u0001\u0001M1 \u0001\u0001 \u0001\u0001w11 w12 w13 \u0001\u0001 \u0001\u0001B1 \u0001\u0001","rect":[154.3013458251953,504.9669189453125,284.6746465247961,492.6866455078125]},{"page":53,"text":"\u0001\u0001M2 \u0001\u0001 ¼ \u0001\u0001w21 w22 w23 \u0001\u0001 \u0004 \u0001\u0001B2 \u0001\u0001","rect":[154.3013458251953,516.8128662109375,284.6746465247961,504.5326232910156]},{"page":53,"text":"\u0001\u0001M3 \u0001\u0001 \u0001\u0001w31 w32 w33 \u0001\u0001 \u0001\u0001B3 \u0001\u0001","rect":[154.3013458251953,528.6588745117188,284.6746465247961,516.3218383789063]},{"page":53,"text":"The matrix representation can be written in a more compact form","rect":[65.7656021118164,544.2350463867188,330.02005718827805,535.300537109375]},{"page":53,"text":"Ma ¼ XwabBb ¼ wabBb:","rect":[166.93324279785157,572.4681396484375,272.0353234974756,558.533447265625]},{"page":53,"text":"b","rect":[198.031494140625,580.4879150390625,201.87779007084948,574.0803833007813]},{"page":53,"text":"(3.12)","rect":[361.05584716796877,569.7420043945313,385.1052182754673,561.265625]},{"page":54,"text":"36","rect":[53.812843322753909,42.55594253540039,62.2739118972294,36.68026351928711]},{"page":54,"text":"3 Mesogenic Molecules and Orientational Order","rect":[219.66928100585938,44.274620056152347,385.1406564102857,36.663330078125]},{"page":54,"text":"Here a, b ¼ 1, 2, 3 and, following Einstein, the repeated index b means","rect":[65.76496887207031,68.2883529663086,385.1219521914597,59.04502868652344]},{"page":54,"text":"summation over b.","rect":[53.81296157836914,80.15824127197266,129.6003384163547,71.00456237792969]},{"page":54,"text":"The magnetization was only taken as an example. Many other properties","rect":[65.76498413085938,92.20748138427735,385.07818998442846,83.27293395996094]},{"page":54,"text":"(dielectric susceptibility, electric and thermal conductivity, molecular diffusion,","rect":[53.81296920776367,104.11019134521485,385.1149868538547,95.17564392089844]},{"page":54,"text":"etc.) are also described by second rank tensors of the same (quadrupolar) type wab.","rect":[53.81296920776367,116.99702453613281,385.1832241585422,107.13517761230469]},{"page":54,"text":"Microscopically, such properties can be described by single-particle distribution","rect":[53.814537048339847,128.02999877929688,385.17632380536568,119.09544372558594]},{"page":54,"text":"functions, when intermolecular interaction is neglected. There are also properties","rect":[53.814537048339847,139.98953247070313,385.15536893950658,131.0549774169922]},{"page":54,"text":"described by tensors of rank 3 with 33 ¼ 27 components (e.g., molecular hyperpo-","rect":[53.814537048339847,151.94937133789063,385.12383399812355,140.8982391357422]},{"page":54,"text":"larizability gijk) and even of rank 4 (e.g., elasticity in nematics, Kijkl) with 34 ¼ 81","rect":[53.81388473510742,164.83863830566407,385.17192927411568,152.8579559326172]},{"page":54,"text":"components. Microscopically, such elastic properties must be described by many-","rect":[53.81417465209961,175.86846923828126,385.1291745742954,166.9339141845703]},{"page":54,"text":"particle distribution functions.","rect":[53.81417465209961,187.8280029296875,175.25356717612034,178.89344787597657]},{"page":54,"text":"As physical properties of the matter are independent of the chosen frame, suffixes","rect":[65.76619720458985,199.73077392578126,385.16400541411596,190.7962188720703]},{"page":54,"text":"a and b can be interchanged. Therefore, wab ¼ wba and only 6 components of wab are","rect":[53.81417465209961,212.6175079345703,385.1655963726219,202.44700622558595]},{"page":54,"text":"different, three diagonal and the other three off-diagonal. Such a symmetric tensor","rect":[53.813838958740237,223.65048217773438,385.11687599031105,214.71592712402345]},{"page":54,"text":"(or matrix) can always be diagonalized by a proper choice of the Cartesian frame","rect":[53.813838958740237,235.61004638671876,385.1706012554344,226.6754913330078]},{"page":54,"text":"whose axes would coincide with the symmetry axes of the LC phase. In that","rect":[53.813838958740237,247.569580078125,385.13777024814677,238.63502502441407]},{"page":54,"text":"reference system only three diagonal components w11, w22 and w33 are finite.","rect":[53.813838958740237,259.5291442871094,359.0795559456516,250.59458923339845]},{"page":54,"text":"3.5.2 Uniaxial Order","rect":[53.812843322753909,304.3975830078125,165.3185337040202,296.0546875]},{"page":54,"text":"For a uniaxial phase (nematic, discotic nematic, SmA, SmB, etc.) with the symme-","rect":[53.812843322753909,334.2348327636719,385.1426938614048,325.30029296875]},{"page":54,"text":"try axis along z, all properties along x and y are the same and w11¼ w22 6¼ w33. The","rect":[53.812843322753909,346.1943664550781,385.1477741069969,336.9311828613281]},{"page":54,"text":"corresponding matrix","rect":[53.813899993896487,358.15399169921877,139.85311213544379,349.21942138671877]},{"page":54,"text":"\u0001\u0001w?0","rect":[201.82667541503907,384.52020263671877,235.46625605634223,372.24090576171877]},{"page":54,"text":"0 \u0001\u0001","rect":[251.0515594482422,384.5191955566406,263.54577079237427,372.2398986816406]},{"page":54,"text":"wab ¼ \u0001\u0001 0 w? 0 \u0001\u0001","rect":[175.43045043945313,396.5019836425781,263.54577079237427,384.02911376953127]},{"page":54,"text":"\u0001\u0001 0","rect":[201.82667541503907,408.1553955078125,214.6208886247016,395.8761291503906]},{"page":54,"text":"0 wII \u0001\u0001","rect":[230.4888916015625,408.1543884277344,263.54577079237427,395.8751220703125]},{"page":54,"text":"(3.13)","rect":[361.0552673339844,394.4826354980469,385.1046384414829,386.00628662109377]},{"page":54,"text":"has only two different components and the relevant physical quantity can be","rect":[53.81357955932617,431.6670837402344,385.15049017145005,422.7325439453125]},{"page":54,"text":"decomposed into two parts, the mean value <w> ¼ (1/3)( w|| þ 2w⊥) and the","rect":[53.81357955932617,443.6266174316406,385.17517889215318,434.69207763671877]},{"page":54,"text":"anisotropic part Dw ¼ wa ¼ w|| \u0003 w⊥.","rect":[53.81444549560547,455.5883483886719,202.61587186362034,446.3251037597656]},{"page":54,"text":"The anisotropic part of tensor (3.13) is","rect":[65.76538848876953,467.5480041503906,221.5975686465378,458.61346435546877]},{"page":54,"text":"wðaabÞ ¼ wab \u0003 hwidab","rect":[180.13131713867188,495.7501220703125,258.34744033940418,480.8091125488281]},{"page":54,"text":"where dab is second rank unit tensor with trace dxx þ dyy þ dzz ¼ 3. Hence, the","rect":[53.812843322753909,519.19384765625,385.11814153863755,509.04400634765627]},{"page":54,"text":"anisotropy tensor is traceless, has dimension of the w value and becomes zero in the","rect":[53.813167572021487,530.2169799804688,385.1728900737938,521.3023681640625]},{"page":54,"text":"isotropic phase:","rect":[53.813167572021487,542.1396484375,117.16375596592019,533.2051391601563]},{"page":54,"text":"waab ¼ \u0001\u0001\u0001\u0001\u0001\u0001w00? w00? w00II \u0001\u0001\u0001\u0001\u0001\u0001 \u0003 \u0001\u0001\u0001\u0001\u0001\u0001hw00i h0w0i h00wi\u0001\u0001\u0001\u0001\u0001\u0001 ¼ \u0001\u0001\u0001\u0001\u0001\u0001\u0001\u00031=003wa \u00031=003wa 2=300wa \u0001\u0001\u0001\u0001\u0001\u0001\u0001","rect":[75.62184143066406,597.1290283203125,363.35473075331177,555.26220703125]},{"page":55,"text":"3.5 Macroscopic Definition of the Orientational Order Parameter","rect":[53.812843322753909,44.274620056152347,275.67144864661386,36.62946701049805]},{"page":55,"text":"37","rect":[376.7465515136719,42.55594253540039,385.20761627345009,36.73106384277344]},{"page":55,"text":"In principle, this tensor might be used as orientational order parameter for a","rect":[65.76496887207031,68.2883529663086,385.15882147027818,59.35380554199219]},{"page":55,"text":"uniaxial phase, however, its dimensionless form would be more preferable. Therefore,","rect":[53.812950134277347,80.24788665771485,385.15081449057348,71.31333923339844]},{"page":55,"text":"we normalize anisotropy wa to the maximum possible anisotropy corresponding to","rect":[53.81295394897461,92.20772552490235,385.1405877213813,83.27293395996094]},{"page":55,"text":"the ideal molecular alignment as in a solid crystal at absolute zero temperature.","rect":[53.81368637084961,104.1104965209961,385.1147732308078,95.17594909667969]},{"page":55,"text":"Then we arrive at the order parameter tensor [16]:","rect":[53.81368637084961,115.96040344238281,259.8021482071079,107.1155014038086]},{"page":55,"text":"Qab ¼ wwamðabaabÞx ¼ wwamaax \u0004 \u0001\u0001\u0001\u0001\u0001\u0001\u0003100=3 \u0003100=3 200=3\u0001\u0001\u0001\u0001\u0001\u0001","rect":[132.49176025390626,165.1082305908203,306.48497977674927,129.1927490234375]},{"page":55,"text":"(3.14)","rect":[361.0579528808594,151.4354705810547,385.1073239883579,142.9591064453125]},{"page":55,"text":"Here S ¼ wa=wamax is a scalar modulus of the order parameter dependent on the","rect":[65.76831817626953,188.30014038085938,385.1733783550438,178.0563507080078]},{"page":55,"text":"degree of molecular (statistical) order whereas the tensor shows the orientational","rect":[53.8136100769043,199.56100463867188,385.1255937344749,190.62644958496095]},{"page":55,"text":"part of the order parameter. With such an approach, the macroscopic and micro-","rect":[53.8136100769043,211.52056884765626,385.1315854629673,202.5860137939453]},{"page":55,"text":"scopic definitions of the order parameter would coincide if we assume","rect":[53.8136100769043,223.4801025390625,337.1771625347313,214.54554748535157]},{"page":55,"text":"S ¼ wa=wamax ¼ S2 ¼ <P2ðcos#Þ> ¼ 21\u00033cos2# \u0003 1\u0004","rect":[112.7813491821289,257.0474853515625,326.233216327916,236.711181640625]},{"page":55,"text":"(3.15)","rect":[361.0560302734375,251.9326629638672,385.10540138093605,243.33676147460938]},{"page":55,"text":"The experimental values of the orientational order parameter found macroscop-","rect":[65.7663803100586,279.5948791503906,385.08257423249855,270.66033935546877]},{"page":55,"text":"ically for conventional nematics from the magnetic or optical anisotropy are","rect":[53.81433868408203,291.554443359375,385.1063922710594,282.619873046875]},{"page":55,"text":"in good agreement with those calculated from microscopic data (NMR, Raman","rect":[53.81433868408203,303.51397705078127,385.1641472916938,294.57940673828127]},{"page":55,"text":"spectroscopy).","rect":[53.81433868408203,315.4735412597656,111.7199673226047,306.5987548828125]},{"page":55,"text":"Order parameter tensor can be written using the director components na (a¼x,y,z).","rect":[65.7663803100586,327.3763122558594,385.2131619026828,318.4218444824219]},{"page":55,"text":"Qab ¼ wwmaaax ðnanb \u0003 13dabÞ ¼ Sðnanb \u0003 31dabÞ","rect":[129.09628295898438,363.78369140625,309.8861428652878,340.607177734375]},{"page":55,"text":"(3.16)","rect":[361.0557556152344,355.8285827636719,385.1051267227329,347.29248046875]},{"page":55,"text":"For example, for n||z the director components are (0, 0, 1) and from (3.16) we","rect":[65.76612091064453,386.267822265625,385.1708453960594,377.333251953125]},{"page":55,"text":"immediately get the form (3.14). Here we clearly see the two components of the","rect":[53.8140983581543,398.2273864746094,385.17380560113755,389.2928466796875]},{"page":55,"text":"order parameter, the scalar amplitude S and the orientational part (in parentheses).","rect":[53.8140983581543,410.1869201660156,385.1469692757297,401.25238037109377]},{"page":55,"text":"Fig. 3.21 Packing of","rect":[53.812843322753909,547.327880859375,127.64584439856698,539.3948974609375]},{"page":55,"text":"molecules in a macroscopic","rect":[53.812843322753909,557.1793823242188,148.69359729074956,549.5850219726563]},{"page":55,"text":"nematic biaxial phase of","rect":[53.812843322753909,567.1553344726563,137.44543546302013,559.5609741210938]},{"page":55,"text":"symmetry D2h","rect":[53.812843322753909,577.1312866210938,102.52908838107328,569.706298828125]},{"page":55,"text":"n1","rect":[348.1629638671875,573.5004272460938,356.50629424445227,567.35400390625]},{"page":55,"text":"n3","rect":[368.2084655761719,546.304931640625,376.5516738831241,540.0465698242188]},{"page":56,"text":"38","rect":[53.812843322753909,42.55746841430664,62.2739118972294,36.73258972167969]},{"page":56,"text":"Fig. 3.22 A molecule of","rect":[53.812843322753909,67.58130645751953,139.99392789466075,59.648292541503909]},{"page":56,"text":"potassium (K) laurate with","rect":[53.812843322753909,77.4895248413086,145.2880148330204,69.89517211914063]},{"page":56,"text":"deuterium (D) label (a) anda","rect":[53.812843322753909,87.07006072998047,154.34472033762456,79.81436157226563]},{"page":56,"text":"structure of the lyotropic","rect":[53.812835693359378,97.3846664428711,138.7332091071558,89.79031372070313]},{"page":56,"text":"lamellar phase (b)","rect":[53.812835693359378,107.36067962646485,116.10830777747323,99.76632690429688]},{"page":56,"text":"3 Mesogenic Molecules and Orientational Order","rect":[219.66928100585938,44.276145935058597,385.1406564102857,36.66485595703125]},{"page":56,"text":"Fig. 3.23 Lamellar lyotropic","rect":[53.812843322753909,247.20297241210938,154.70772692942144,239.26995849609376]},{"page":56,"text":"phase of potassium laurate:","rect":[53.812843322753909,257.1112060546875,147.20019249167505,249.516845703125]},{"page":56,"text":"orientational order parameter","rect":[53.812843322753909,267.087158203125,153.44447415930919,259.4927978515625]},{"page":56,"text":"for individual links of the","rect":[53.812843322753909,275.27923583984377,141.5803770759058,269.4120178222656]},{"page":56,"text":"molecular chain as a function","rect":[53.812843322753909,285.2297668457031,154.8786367812626,279.3879699707031]},{"page":56,"text":"of the distance of the link","rect":[53.812843322753909,295.2311706542969,141.6294454970829,289.36395263671877]},{"page":56,"text":"from the potassium atom","rect":[53.812843322753909,306.93426513671877,139.18164299364077,299.33990478515627]},{"page":56,"text":"0.2","rect":[269.79150390625,255.49119567871095,280.90214017236095,249.72442626953126]},{"page":56,"text":"0.1","rect":[269.79150390625,276.4251403808594,280.90214017236095,270.6583557128906]},{"page":56,"text":"0.05","rect":[265.3472595214844,299.45062255859377,280.90214017236095,293.683837890625]},{"page":56,"text":"0.02","rect":[265.3472595214844,327.48065185546877,280.90214017236095,321.7138671875]},{"page":56,"text":"2","rect":[290.45489501953127,343.70440673828127,294.8991494497047,338.08160400390627]},{"page":56,"text":"4","rect":[307.8746032714844,343.70440673828127,312.3188577016578,338.08160400390627]},{"page":56,"text":"6","rect":[325.2943115234375,343.848388671875,329.73856595361095,338.08160400390627]},{"page":56,"text":"8 10 12","rect":[341.8699645996094,343.848388671875,381.7379250844703,338.08160400390627]},{"page":56,"text":"No. of carbon atoms","rect":[295.70489501953127,354.76519775390627,372.9837746938453,348.8464660644531]},{"page":56,"text":"3.5.3 Macroscopic Biaxiality","rect":[53.812843322753909,391.7136535644531,204.19364722700326,381.07586669921877]},{"page":56,"text":"In contrast to quite common microscopically biaxial nematics belonging to point group","rect":[53.812843322753909,419.2560729980469,385.2123650651313,410.321533203125]},{"page":56,"text":"D1h and discussed in the previous section, the macroscopically biaxial phase (group","rect":[53.812843322753909,431.1590270996094,385.20281306317818,422.2244873046875]},{"page":56,"text":"D2h) shown in Fig. 3.21 has unequivocally been found only in lyotropic nematics [17]","rect":[53.81308364868164,443.1186828613281,385.1580747207798,434.18414306640627]},{"page":56,"text":"formed by some biphilic (or amphiphilic) molecules in water solutions [18]. Some","rect":[53.812252044677737,455.0782165527344,385.16004217340318,446.1436767578125]},{"page":56,"text":"other cases are still under discussion (nematics formed by metallomesogens, banana-","rect":[53.81223678588867,467.03778076171877,385.16304908601418,458.10321044921877]},{"page":56,"text":"like [19] or polymer molecules [20]). Strictly speaking, cholesteric liquid crystals (or,","rect":[53.81223678588867,478.997314453125,385.1760220101047,470.062744140625]},{"page":56,"text":"more generally, chiral nematics) may be regarded as weakly biaxial. Less symmetric","rect":[53.81223678588867,490.9568786621094,385.14426458551255,482.0223388671875]},{"page":56,"text":"phases such as smectic C, smectic E, etc. are, of course, macroscopically biaxial.","rect":[53.81223678588867,502.9164123535156,366.89031644369848,493.98187255859377]},{"page":56,"text":"In macroscopically biaxial phase all the three components of a physical property","rect":[65.76425170898438,514.8759155273438,385.16802302411568,505.94140625]},{"page":56,"text":"are different, e.g., w11 ¼6 w22 ¼6 w33, the trace of tensor wab is w11 þ w22 þ w33 ¼","rect":[53.81223678588867,527.7061767578125,385.14807918760689,517.5162963867188]},{"page":56,"text":"3<w> and the tensor itself can be written as wab ¼ 3<w>Qab with the traceless","rect":[53.814231872558597,539.6658325195313,385.06219877349096,529.7249145507813]},{"page":56,"text":"order parameter tensor","rect":[53.813961029052737,550.69873046875,144.94576392976416,541.7642211914063]},{"page":56,"text":"(3.17)","rect":[361.0555725097656,587.0841064453125,385.10494361726418,578.6077270507813]},{"page":57,"text":"References","rect":[53.81211853027344,42.52524948120117,91.48080142020501,36.68343734741211]},{"page":57,"text":"39","rect":[376.7457580566406,42.62684631347656,385.2068228164188,36.73423767089844]},{"page":57,"text":"Now the biaxial nematic phase has two order parameters Q1 and Q2 and, in","rect":[65.76496887207031,68.2883529663086,385.1440362077094,59.27412033081055]},{"page":57,"text":"general,three different phasescan bedistinguished, namely, isotropic (Q1 ¼ Q2 ¼ 0),","rect":[53.81417465209961,80.24788665771485,385.18316312338598,71.23365783691406]},{"page":57,"text":"uniaxial nematic (Q1,Q2 ¼ 0) and biaxial nematic (Q1,Q2) phases. Note that biaxial","rect":[53.814476013183597,92.20772552490235,385.1278520352561,83.19337463378906]},{"page":57,"text":"molecules may form both biaxial and uniaxial phases; the latter appear due, for","rect":[53.81393051147461,104.1104965209961,385.1787351211704,95.17594909667969]},{"page":57,"text":"instance, to free rotation of biaxial molecules around their long molecular axes. As","rect":[53.81393051147461,116.0699691772461,385.1737100039597,107.13542175292969]},{"page":57,"text":"to the uniaxial molecules, they may also form either uniaxial (as a rule) or biaxial","rect":[53.81393051147461,128.02957153320313,385.124891830178,119.09501647949219]},{"page":57,"text":"phases; the latter may be formed by biaxial dimers or other “building blocks”","rect":[53.81393051147461,139.98910522460938,385.13086736871568,131.05455017089845]},{"page":57,"text":"formed by uniaxial molecules.","rect":[53.81393051147461,151.94863891601563,176.42193265463596,143.0140838623047]},{"page":57,"text":"3.6 Apparent Order Parameters for Flexible Chains","rect":[53.812843322753909,190.15672302246095,329.4540622684733,179.136474609375]},{"page":57,"text":"When molecules are not so simple as rigid rods or discs, one may introduce","rect":[53.812843322753909,217.69906616210938,385.13181341363755,208.76451110839845]},{"page":57,"text":"apparent partial order parameters different for different molecular moieties. This","rect":[53.812843322753909,229.65859985351563,385.18454374419408,220.7240447998047]},{"page":57,"text":"is especially evident for lyotropic liquid crystals [21], such as, for instance, the","rect":[53.812843322753909,241.6181640625,385.1735309429344,232.68360900878907]},{"page":57,"text":"lamellar phase formed by surfactants in water, see Fig. 3.22b. A good example is a","rect":[53.81282424926758,253.52090454101563,385.16062200738755,244.5863494873047]},{"page":57,"text":"water","rect":[53.81280517578125,264.0,75.97087989656103,257.5618896484375]},{"page":57,"text":"solution","rect":[82.07878112792969,264.0,114.21097651532659,256.5458984375]},{"page":57,"text":"of","rect":[120.31390380859375,264.0,128.605759962479,256.5458984375]},{"page":57,"text":"potassium","rect":[134.64498901367188,265.48046875,175.05907391191085,256.5458984375]},{"page":57,"text":"laurate.","rect":[181.15005493164063,264.0,210.97286649252659,256.5458984375]},{"page":57,"text":"A","rect":[217.006103515625,263.3389587402344,224.1532263751301,256.6056823730469]},{"page":57,"text":"flexible","rect":[230.20440673828126,264.0,260.1466754741844,256.5458984375]},{"page":57,"text":"hydrocarbon","rect":[266.230712890625,265.48046875,316.6816338639594,256.5458984375]},{"page":57,"text":"chain","rect":[322.7616882324219,264.0,344.4121026139594,256.5458984375]},{"page":57,"text":"K–CH2–","rect":[350.4612731933594,265.03985595703127,385.17937556317818,256.6056823730469]},{"page":57,"text":"CH2–CD2–CH2–... can be deuterated with a position of deuterium label varied","rect":[53.81462860107422,277.4405822753906,385.11272517255318,268.50604248046877]},{"page":57,"text":"along the chain, as shown by in Fig. 3.22a. Then, by the NMR technique sensitive","rect":[53.81264114379883,289.400146484375,385.17234075738755,280.465576171875]},{"page":57,"text":"only to deuterium nuclei, the apparent order parameter of the corresponding chain","rect":[53.81162643432617,301.35968017578127,385.1375359635688,292.42510986328127]},{"page":57,"text":"link can be determined. As shown in Fig. 3.23, it decreases with increasing the","rect":[53.81162643432617,313.3192443847656,385.17237127496568,304.38470458984377]},{"page":57,"text":"distance from the potassium atom due to flexibility of the hydrocarbon chain. Thus,","rect":[53.8116340637207,325.2787780761719,385.1395535042453,316.34423828125]},{"page":57,"text":"we can say that the hydrocarbon tail is “solid” at the left end and “liquid” at the right","rect":[53.8116340637207,337.2383117675781,385.20027024814677,328.30377197265627]},{"page":57,"text":"one [22].","rect":[53.8116340637207,348.5334777832031,90.14649625326877,340.26629638671877]},{"page":57,"text":"References","rect":[53.812843322753909,394.2967834472656,109.59614448282879,385.51165771484377]},{"page":57,"text":"1.","rect":[58.06126022338867,420.0,64.40706131055318,413.9985046386719]},{"page":57,"text":"2.","rect":[58.06126022338867,440.0,64.40706131055318,433.9504089355469]},{"page":57,"text":"3.","rect":[58.06126022338867,470.0,64.40706131055318,463.8215637207031]},{"page":57,"text":"4.","rect":[58.06126022338867,500.0,64.40706131055318,493.69268798828127]},{"page":57,"text":"5.","rect":[58.06126022338867,530.0,64.40706131055318,523.4622192382813]},{"page":57,"text":"6.","rect":[58.06126022338867,550.0,64.40706131055318,543.4649047851563]},{"page":57,"text":"7.","rect":[58.060420989990237,569.23583984375,64.40622207715474,563.5294799804688]},{"page":57,"text":"Gray, G.W.: Molecular Structure and the Properties of Liquid Crystals. Academic, London","rect":[68.59698486328125,421.5420837402344,385.1949514785282,413.9477233886719]},{"page":57,"text":"(1962)","rect":[68.59698486328125,431.17938232421877,91.1541985855787,423.9236755371094]},{"page":57,"text":"Gray, G.W.: Liquid crystals and molecular structure: nematics and cholesterics. In: Luckhurst,","rect":[68.59698486328125,441.4939880371094,385.17553970410787,433.8996276855469]},{"page":57,"text":"G.R., Gray, G.W. (eds.) The Molecular Physics of Liquid Crystals, pp. 1–29. Academic,","rect":[68.59698486328125,451.4132385253906,385.1280848701235,443.8188781738281]},{"page":57,"text":"London (1979). Chapter1","rect":[68.59698486328125,461.3891906738281,157.9086660781376,453.7948303222656]},{"page":57,"text":"Gray, G.W.: Liquid crystals and molecular structure: smectics. In: Luckhurst, G.R., Gray, G.W.","rect":[68.59698486328125,471.3651428222656,385.28723404004537,463.7707824707031]},{"page":57,"text":"(eds.) The Molecular Physics of Liquid Crystals, pp. 263–284. Academic, London (1979).","rect":[68.59698486328125,481.3410949707031,385.162874909186,473.7467346191406]},{"page":57,"text":"Chapter 12","rect":[68.59698486328125,491.2603454589844,106.34351104884073,483.6659851074219]},{"page":57,"text":"Hall, A.W., Hollinghurst, J., Goodby, J.W.: Chiral and achiral calamitic liquid crystals for","rect":[68.59698486328125,501.23626708984377,385.1966866837232,493.64190673828127]},{"page":57,"text":"display applications. In: Collings, P., Patel, J. (eds.) Handbook of Liquid Crystal Research, pp.","rect":[68.59698486328125,511.21221923828127,385.1712977607485,503.61785888671877]},{"page":57,"text":"17–71. Oxford University Press, New York (1997)","rect":[68.59698486328125,521.1881713867188,242.11409085852794,513.576904296875]},{"page":57,"text":"Blinov, L.M., Chigrinov, V.G.: Electrooptic Effects in Liquid Crystal Materials. Springer-","rect":[68.59698486328125,531.1073608398438,385.1577157364576,523.5130004882813]},{"page":57,"text":"Verlag, New York (1993)","rect":[68.59698486328125,541.0833129882813,156.80618375403575,533.4889526367188]},{"page":57,"text":"Kuball, H.-G.: From chiral molecules to chiral phases: comment on the chirality of liquid","rect":[68.59698486328125,551.0592651367188,385.18478912501259,543.4649047851563]},{"page":57,"text":"crystal phases. Liquid Crystals Today 9(1), 1–7 (1999)","rect":[68.59698486328125,560.978515625,256.10446256262949,553.0116577148438]},{"page":57,"text":"Osipov, M.A.: Molecular theories of liquid crystals. In: Demus, D., Goodby, J., Gray, G.W.,","rect":[68.59614562988281,570.9544677734375,385.1704737861391,563.3432006835938]},{"page":57,"text":"Spiess, H.-W., Vill, V. (eds.) Physical Properties of Liquid Crystals, pp. 40–71. Wiley-VCH,","rect":[68.59614562988281,580.930419921875,385.1247279365297,573.3360595703125]},{"page":57,"text":"Weinheim (1999)","rect":[68.59614562988281,590.5677490234375,128.9921120987623,583.31201171875]},{"page":58,"text":"40","rect":[53.812110900878909,42.56039810180664,62.2731794753544,36.73551940917969]},{"page":58,"text":"8.","rect":[58.06126022338867,65.22824096679688,64.40706131055318,59.40336608886719]},{"page":58,"text":"9.","rect":[58.06126022338867,85.24787902832031,64.40706131055318,79.35527038574219]},{"page":58,"text":"10.","rect":[53.812957763671878,105.07534790039063,64.38929626538716,99.25047302246094]},{"page":58,"text":"11.","rect":[53.812957763671878,135.0,64.38929626538716,129.12156677246095]},{"page":58,"text":"12.","rect":[53.812957763671878,155.0,64.38929626538716,149.07350158691407]},{"page":58,"text":"13.","rect":[53.812957763671878,185.0,64.38929626538716,178.9446563720703]},{"page":58,"text":"14.","rect":[53.812957763671878,204.61984252929688,64.38929626538716,198.8965606689453]},{"page":58,"text":"15.","rect":[53.81293869018555,225.0,64.38927719190083,218.6901397705078]},{"page":58,"text":"16.","rect":[53.81293869018555,255.0,64.38927719190083,248.612060546875]},{"page":58,"text":"17.","rect":[53.81294250488281,275.0,64.3892810065981,268.6147766113281]},{"page":58,"text":"18.","rect":[53.81294250488281,285.0,64.3892810065981,278.5340576171875]},{"page":58,"text":"19.","rect":[53.812110900878909,305.0,64.38844940259419,298.4859619140625]},{"page":58,"text":"20.","rect":[53.81126403808594,324.2060241699219,64.38760253980122,318.3811340332031]},{"page":58,"text":"21.","rect":[53.812110900878909,344.0563659667969,64.38844940259419,338.33306884765627]},{"page":58,"text":"22.","rect":[53.812110900878909,354.03228759765627,64.38844940259419,348.3089904785156]},{"page":58,"text":"3 Mesogenic Molecules and Orientational Order","rect":[219.66854858398438,44.279075622558597,385.1399239884107,36.66778564453125]},{"page":58,"text":"Israelachvili, J.N.: Intermolecular and Surface Forces, 2nd edn. Academic Press, London","rect":[68.59698486328125,65.22824096679688,385.1949819961063,59.35256576538086]},{"page":58,"text":"(1992)","rect":[68.59698486328125,76.58422088623047,91.1541985855787,69.37931823730469]},{"page":58,"text":"Stone, A.J.: Intermolecular forces. In: Luckhurst, G.R., Gray, G.W. (eds.) The Molecular","rect":[68.59698486328125,86.8988265991211,385.16107267005136,79.30447387695313]},{"page":58,"text":"Physics of Liquid Crystals, pp. 31–50. Academic, London (1979). Chapter2","rect":[68.59698486328125,96.8747787475586,330.33600372462197,89.22962188720703]},{"page":58,"text":"Zannoni, C.: Distribution functions and order parameters. In: Luckhurst, G.R., Gray, G.W.","rect":[68.59698486328125,106.79402923583985,385.15271255567037,99.19967651367188]},{"page":58,"text":"(eds.) The Molecular Physics of Liquid Crystals, pp. 51–84. Academic, London (1979).","rect":[68.59698486328125,116.76998138427735,385.1620204169985,109.12482452392578]},{"page":58,"text":"Chapter3","rect":[68.59698486328125,126.74593353271485,102.11297363428995,119.15158081054688]},{"page":58,"text":"Vertogen, G., de Jeu, V.H.: Thermotropic Liquid Crystals. Fundamentals. Springer-Verlag,","rect":[68.59698486328125,136.66513061523438,385.13147232129537,129.07077026367188]},{"page":58,"text":"Berlin (1987)","rect":[68.59698486328125,146.3024139404297,114.77496427161386,139.04672241210938]},{"page":58,"text":"Wojtowicz,","rect":[68.59698486328125,156.6170654296875,107.95788833692036,149.022705078125]},{"page":58,"text":"P.:","rect":[112.61062622070313,155.0,121.77396873679224,149.19203186035157]},{"page":58,"text":"Introduction","rect":[126.31840515136719,155.0,168.31830352930948,149.022705078125]},{"page":58,"text":"to","rect":[172.88052368164063,155.0,179.4886221328251,149.8862762451172]},{"page":58,"text":"molecular","rect":[184.0965118408203,155.0,218.0846413956373,149.022705078125]},{"page":58,"text":"theory","rect":[222.6722412109375,156.6170654296875,244.3410391250126,149.022705078125]},{"page":58,"text":"of","rect":[248.95571899414063,155.0,256.0037850723951,149.022705078125]},{"page":58,"text":"liquid","rect":[260.56768798828127,156.6170654296875,280.3919729629032,149.022705078125]},{"page":58,"text":"crystals.","rect":[284.98211669921877,156.6170654296875,312.9797999580141,149.022705078125]},{"page":58,"text":"In:","rect":[317.60968017578127,155.0,327.0353365346438,149.19203186035157]},{"page":58,"text":"Priestley,","rect":[331.6009216308594,156.6170654296875,363.38904068067037,149.022705078125]},{"page":58,"text":"E.B.,","rect":[367.96734619140627,155.0,385.1432826240297,149.19203186035157]},{"page":58,"text":"Wojtowicz, P., Sheng, P. (eds.) Introduction to Liquid Crystals, pp. 31–44. Plenum Press,","rect":[68.59698486328125,166.59304809570313,385.1492335517641,158.99868774414063]},{"page":58,"text":"N-Y (1975)","rect":[68.59698486328125,176.17356872558595,108.54423612219979,168.86708068847657]},{"page":58,"text":"de Gennes, P.G., Prost, J.: The Physics of Liquid Crystals, 2nd edn. Oxford Publishers","rect":[68.59698486328125,186.48822021484376,385.1298263835839,178.87692260742188]},{"page":58,"text":"University Press, Oxford (1995)","rect":[68.59698486328125,196.46417236328126,178.44453519446544,188.8190155029297]},{"page":58,"text":"Zwetkoff, W.: U€ber die Molecu€lanordnung in der anisotrop-fl€ussigen phase. Acta Physico-","rect":[68.59698486328125,206.44012451171876,385.1586007462232,197.0288848876953]},{"page":58,"text":"chim. URSS 16, 132 (1942)","rect":[68.59696197509766,216.0206756591797,164.113129554817,208.69725036621095]},{"page":58,"text":"Pershan, P.S.: Raman studies of orientational order in liquid crystals. In: Luckhurst, G.R.,","rect":[68.59696960449219,226.33529663085938,385.1755091865297,218.74093627929688]},{"page":58,"text":"Gray, G.W. (eds.) The Molecular Physics of Liquid Crystals, pp. 385–410. Academic, London","rect":[68.59696197509766,236.31124877929688,385.19574493555947,228.6660919189453]},{"page":58,"text":"(1979). Chapter 17","rect":[68.59696197509766,246.23046875,133.36307281641886,238.6361083984375]},{"page":58,"text":"de Gennes, P.G.: Short range order effects in the isotropic phase of nematics and cholesterics.","rect":[68.59696960449219,256.2064208984375,385.1568324287172,248.612060546875]},{"page":58,"text":"Mol. Cryst. Liq. Cryst. 12, 193–314 (1971)","rect":[68.59696197509766,266.1824035644531,216.623383461067,258.5202941894531]},{"page":58,"text":"Sonin, A.S.: Lyotropic nematics. Usp. Fiz. Nauk. 153, 273–310 (1987). in Russian","rect":[68.59696960449219,276.1583557128906,351.3473257461063,268.2930603027344]},{"page":58,"text":"Yu, L.J., Saupe, A.: Observation of a biaxial nematic phase in potassium-laurate -1-decanol-","rect":[68.59696960449219,286.07763671875,385.14328092200449,278.4663391113281]},{"page":58,"text":"water mixtures. Phys. Rev. Lett. 45, 1000 (1980)","rect":[68.59696960449219,296.0535888671875,236.10839933020763,288.459228515625]},{"page":58,"text":"Acharya, B.R., Primak, A., Kumar, S.: Biaxial nematic phase in bent-core thermotropic","rect":[68.59613800048828,306.029541015625,385.14843127512457,298.4351806640625]},{"page":58,"text":"mesogens. Phys. Rev. Lett. 92(14), 145506 (2004)","rect":[68.59613800048828,316.0055236816406,241.149506507942,308.03863525390627]},{"page":58,"text":"Severing, K., Saalw€achter, K.: Biaxial nematic phase in a thermotropic liquid-crystalline side-","rect":[68.59529113769531,325.9247131347656,385.1679696427076,318.0]},{"page":58,"text":"chain polymers. Phys. Rev. Lett. 92(14), 125501 (2004)","rect":[68.59613800048828,335.90069580078127,260.06930631262949,327.9338073730469]},{"page":58,"text":"Pershan, P.S.: Lyotropic Liquid crystals, pp. 34–39. Physics Today, May (1982)","rect":[68.59613800048828,345.87664794921877,342.9413155899732,338.28228759765627]},{"page":58,"text":"Charvolin, J., Tardieu, A.: Lyotropic liquid crystals; structures and molecular motions. In:","rect":[68.59613800048828,355.8525695800781,385.15096763815947,348.2582092285156]},{"page":58,"text":"Liebert, L., Ehrenreich, H., Seitz, F., Turnbull, D. (eds.) Liquid Crystals. Solid State Physics,","rect":[68.59613800048828,365.7718200683594,385.1822230537172,358.1774597167969]},{"page":58,"text":"pp. 209–258. Academic, New York (1978)","rect":[68.59613800048828,375.7477722167969,215.26287931067638,368.10260009765627]},{"page":59,"text":"Chapter4","rect":[53.812843322753909,72.10812377929688,114.14115996551633,59.571903228759769]},{"page":59,"text":"Liquid Crystal Phases","rect":[53.812843322753909,91.18268585205078,204.9206658736041,76.10637664794922]},{"page":59,"text":"This chapter presents a review of different liquid crystal phases. The main attention","rect":[53.812843322753909,211.74758911132813,385.1656426530219,202.8130340576172]},{"page":59,"text":"is paid to the thermotropic liquid crystals, which manifest rich polymorphism upon","rect":[53.812843322753909,223.65036010742188,385.15273371747505,214.71580505371095]},{"page":59,"text":"variation of temperature. Moreover, the thermotropic phases are subdivided into","rect":[53.812843322753909,235.60992431640626,385.15468684247505,226.6753692626953]},{"page":59,"text":"rod-like or calamitic and discotic ones; the latter are discussed only briefly. At first,","rect":[53.812843322753909,247.5694580078125,385.15472074057348,238.63490295410157]},{"page":59,"text":"we discuss achiral media with lyotropic phases included and then consider the role","rect":[53.812843322753909,259.5290222167969,385.1427387066063,250.59446716308595]},{"page":59,"text":"of chirality.","rect":[53.812843322753909,271.4885559082031,100.7598843147922,262.55401611328127]},{"page":59,"text":"4.1 Polymorphism Studies","rect":[53.812843322753909,322.2179870605469,196.17015723917647,310.9108581542969]},{"page":59,"text":"The polymorphic transformations can be studied by different techniques that are","rect":[53.812843322753909,349.198486328125,385.1606830425438,340.263916015625]},{"page":59,"text":"illustrated below by some characteristic examples.","rect":[53.812843322753909,361.1580505371094,257.11663480307348,352.2235107421875]},{"page":59,"text":"4.1.1 Polarized Light Microscopy","rect":[53.812843322753909,411.2685546875,226.702863535597,400.6307678222656]},{"page":59,"text":"It is very simple and vivid method [1]. One can observe characteristic streaks","rect":[53.812843322753909,438.8108825683594,385.08200468169408,429.8564147949219]},{"page":59,"text":"(Schlieren-textures)","rect":[53.8138313293457,450.3719787597656,133.2703336319126,441.83587646484377]},{"page":59,"text":"showing","rect":[138.38381958007813,450.7704162597656,172.15049067548285,441.83587646484377]},{"page":59,"text":"particular","rect":[177.29881286621095,450.7704162597656,215.73301063386573,441.83587646484377]},{"page":59,"text":"macroscopic","rect":[220.85842895507813,450.7704162597656,271.39698064996568,441.83587646484377]},{"page":59,"text":"defects,","rect":[276.5393371582031,449.0,307.38348050619848,441.83587646484377]},{"page":59,"text":"e.g.,","rect":[312.5089111328125,450.7704162597656,329.4012112190891,443.0]},{"page":59,"text":"disclinations","rect":[334.60028076171877,449.0,385.1706582461472,441.83587646484377]},{"page":59,"text":"and establish the phase symmetry. In Fig. 4.1a the characteristic defects of the","rect":[53.8138313293457,462.72998046875,385.17456854059068,453.79541015625]},{"page":59,"text":"nematic phase (disclinations), are well seen. Fan-shape texture of the smectic C","rect":[53.8138313293457,474.68951416015627,385.1855952423408,465.75494384765627]},{"page":59,"text":"phase is shown in Fig. 4.1b. One can also distinguish between different types of","rect":[53.8138313293457,486.6490783691406,385.15068946687355,477.6946105957031]},{"page":59,"text":"uniform molecular orientation in different liquid crystal preparations using a cono-","rect":[53.81482696533203,498.6086120605469,385.10991798249855,489.674072265625]},{"page":59,"text":"scopy technique (microscopic observations in the convergent light beam): in this","rect":[53.81482696533203,510.5681457519531,385.1436807070847,501.63360595703127]},{"page":59,"text":"case symmetry of the pattern corresponds to the texture symmetry.","rect":[53.81482696533203,522.4708862304688,323.1479153206516,513.536376953125]},{"page":59,"text":"A very useful technique is a study of miscibility of different substances [2]. As a","rect":[65.766845703125,534.430419921875,385.1626361675438,525.4759521484375]},{"page":59,"text":"rule, only identical phases are mixed with each other (nematic with nematic,","rect":[53.81483459472656,546.3899536132813,385.0978664925266,537.4554443359375]},{"page":59,"text":"smectic A (SmA) with SmA, SmC with SmC etc.). Therefore, using a well inves-","rect":[53.81483459472656,558.3494873046875,385.1208432754673,549.4149780273438]},{"page":59,"text":"tigated substance as a reference, one can make a preliminary conclusion about a","rect":[53.81483459472656,570.30908203125,385.15967596246568,561.3745727539063]},{"page":59,"text":"structure of a new compound not doing X-ray and other cumbersome structural","rect":[53.81483459472656,582.2686157226563,385.14570481845927,573.3341064453125]},{"page":59,"text":"studies. For instance, by mixing with a reference liquid crystal, it was concluded","rect":[53.81483459472656,594.2281494140625,385.09099665692818,585.2936401367188]},{"page":59,"text":"L.M. Blinov, Structure and Properties of Liquid Crystals,","rect":[53.812843322753909,623.7623291015625,251.63688919141254,616.1510620117188]},{"page":59,"text":"DOI 10.1007/978-90-481-8829-1_4, # Springer ScienceþBusiness Media B.V. 2011","rect":[53.812843322753909,633.698486328125,347.38995880274697,625.4920043945313]},{"page":59,"text":"41","rect":[376.7464599609375,621.958984375,385.2075247207157,616.2357177734375]},{"page":60,"text":"42","rect":[53.812843322753909,42.4564208984375,62.2739118972294,36.73313903808594]},{"page":60,"text":"4 Liquid Crystal Phases","rect":[303.6741638183594,44.276695251464847,385.1685226726464,36.68233871459961]},{"page":60,"text":"Fig. 4.1 Textures of the nematic (a) and smectic C (b) phases observed with a polarization","rect":[53.812843322753909,182.01995849609376,385.16864532618447,174.29014587402345]},{"page":60,"text":"microscope","rect":[53.812843322753909,191.87149047851563,92.91144702219487,184.27713012695313]},{"page":60,"text":"Fig. 4.2 Miscibility","rect":[53.812843322753909,221.41317749023438,123.9212622085087,213.68336486816407]},{"page":60,"text":"diagram: Ref and Inv mean","rect":[53.812843322753909,231.3214111328125,150.04565948634073,223.6169891357422]},{"page":60,"text":"the reference compound with","rect":[53.812835693359378,241.29739379882813,154.5782675185673,233.70303344726563]},{"page":60,"text":"well known phase sequence","rect":[53.812835693359378,251.2733154296875,149.69283435129644,243.678955078125]},{"page":60,"text":"and unknown compound to","rect":[53.812835693359378,261.19256591796877,147.8246359267704,253.59820556640626]},{"page":60,"text":"be investigated. Starting with","rect":[53.812835693359378,271.16851806640627,154.8608602920048,263.57415771484377]},{"page":60,"text":"molar content c ¼ 1 and","rect":[53.812835693359378,279.451171875,138.5623068252079,273.55010986328127]},{"page":60,"text":"proceeding to the left while","rect":[53.81283950805664,291.12042236328127,149.24018237375737,283.52606201171877]},{"page":60,"text":"measuring phase transition","rect":[53.81283950805664,301.0396423339844,145.8252920547001,293.4452819824219]},{"page":60,"text":"temperatures one finally","rect":[53.81283950805664,311.015625,136.88448089747355,303.4212646484375]},{"page":60,"text":"arrives at c ¼ 0 with","rect":[53.81283950805664,319.29827880859377,125.80215973048135,313.397216796875]},{"page":60,"text":"complete phase diagram,","rect":[53.812843322753909,330.9107971191406,139.49556228711567,323.3164367675781]},{"page":60,"text":"therefore, having information","rect":[53.812843322753909,340.8867492675781,155.0131582656376,333.2923889160156]},{"page":60,"text":"about the unknown","rect":[53.812843322753909,349.1101379394531,119.76772827296182,343.2683410644531]},{"page":60,"text":"compound","rect":[53.812843322753909,360.83868408203127,89.58624786524698,353.24432373046877]},{"page":60,"text":"Inv","rect":[269.86663818359377,324.5429992675781,280.5296517013419,318.80023193359377]},{"page":60,"text":"0","rect":[272.9784240722656,341.608154296875,277.42267850243908,335.84136962890627]},{"page":60,"text":"I","rect":[297.61688232421877,247.50982666015626,299.8390095393055,241.7670440673828]},{"page":60,"text":"N","rect":[310.3301696777344,285.7520446777344,316.1013058262689,280.00927734375]},{"page":60,"text":"that substance p-methoxy-p0-pentylstilbene (MOPS), see Fig. 4.2, has the SmB","rect":[53.812843322753909,392.3326416015625,385.1458003204658,382.6730041503906]},{"page":60,"text":"(below 110\u0002C), Nematic (110–125\u0002C) and Isotropic (above 125\u0002C) phases.","rect":[53.812931060791019,404.2355651855469,355.8214382698703,395.2412414550781]},{"page":60,"text":"4.1.2 Differential Scanning and Adiabatic Calorimetry","rect":[53.812843322753909,445.16363525390627,333.0579050395032,434.5258483886719]},{"page":60,"text":"(DSC and AC)","rect":[89.66943359375,458.7369384765625,161.9548077657043,448.4696960449219]},{"page":60,"text":"These techniques are widely used in investigations of phase transitions. DSC allows","rect":[53.812843322753909,486.6495056152344,385.1258584414597,477.7149658203125]},{"page":60,"text":"the express measurements of the transition enthalpy and determination of the phase","rect":[53.812843322753909,498.6090393066406,385.0979694194969,489.67449951171877]},{"page":60,"text":"transition type. For example, at the SmB-Cr (crystal) transition a great amount of","rect":[53.812843322753909,510.568603515625,385.14678321687355,501.634033203125]},{"page":60,"text":"enthalpy is released as evident from Fig. 4.3. Therefore, with high probability, this","rect":[53.812843322753909,522.528076171875,385.1446877871628,513.5935668945313]},{"page":60,"text":"transition is of strong first order, the others shown in the figure (Isotropic phase-","rect":[53.81282424926758,534.430908203125,385.12581764070168,525.4963989257813]},{"page":60,"text":"SmA and SmA–SmB) are not as strong and may be referred to as weak first order","rect":[53.81282424926758,546.390380859375,385.14275489656105,537.4558715820313]},{"page":60,"text":"transitions. True second order transitions may not be seen in DSC plots due to","rect":[53.81282424926758,558.3499755859375,385.1397637467719,549.4154663085938]},{"page":60,"text":"negligibly small transition enthalpy. Specific features of such transitions are studied","rect":[53.81282424926758,570.3095092773438,385.1715935807563,561.375]},{"page":60,"text":"by","rect":[53.81282424926758,582.26904296875,63.76703349164495,573.3345336914063]},{"page":60,"text":"adiabatic","rect":[69.21995544433594,581.0,105.36368597223127,573.3345336914063]},{"page":60,"text":"calorimetry","rect":[110.91017150878906,582.26904296875,157.12459651044379,573.3345336914063]},{"page":60,"text":"(e.g.,","rect":[162.5137939453125,582.26904296875,182.72083707358127,573.394287109375]},{"page":60,"text":"anomaly","rect":[188.23049926757813,582.26904296875,222.71885005048285,573.3345336914063]},{"page":60,"text":"in","rect":[228.10806274414063,581.0,235.88230219892035,573.3345336914063]},{"page":60,"text":"heat","rect":[241.36309814453126,581.0,258.03640611240459,573.3345336914063]},{"page":60,"text":"capacitance)","rect":[263.4544982910156,582.26904296875,313.6715329727329,573.3345336914063]},{"page":60,"text":"and","rect":[319.07965087890627,581.0,333.48342219403755,573.3345336914063]},{"page":60,"text":"dilatometry","rect":[338.96221923828127,582.26904296875,385.1706475358344,573.3345336914063]},{"page":60,"text":"(density changes at transitions).","rect":[53.81282424926758,594.2285766601563,181.29641385580784,585.2940673828125]},{"page":61,"text":"4.1 Polymorphism Studies","rect":[53.812843322753909,44.275230407714847,143.93762667655268,36.68087387084961]},{"page":61,"text":"Fig. 4.3 Qualitative example","rect":[53.812843322753909,67.58130645751953,155.3380675055933,59.648292541503909]},{"page":61,"text":"of a DSC spectrum: latent","rect":[53.812843322753909,77.4895248413086,142.97982506003442,69.89517211914063]},{"page":61,"text":"heat of transitions asa","rect":[53.812843322753909,85.65617370605469,130.72313067698003,79.81436157226563]},{"page":61,"text":"function of temperature","rect":[53.812843322753909,97.3846664428711,134.50521228098394,89.79031372070313]},{"page":61,"text":"43","rect":[376.7464904785156,42.55655288696289,385.2075552382938,36.73167419433594]},{"page":61,"text":"IPS","rect":[256.9403381347656,268.54412841796877,270.1736458741046,262.8154296875]},{"page":61,"text":"Shutter","rect":[168.85919189453126,315.4704284667969,193.59069762481406,309.7417297363281]},{"page":61,"text":"BDR","rect":[244.21617126464845,322.1930236816406,261.4634511143722,316.69635009765627]},{"page":61,"text":"Fig. 4.4 Scheme of an X-ray diffractogram for a smectic A phase. The beam is impinged on the","rect":[53.812843322753909,346.3943176269531,385.15514514231207,338.66448974609377]},{"page":61,"text":"sample perpendicularly to the figure plane and forms a cone of diffraction. The directly transmitted","rect":[53.812843322753909,356.30255126953127,385.17108673243447,348.70819091796877]},{"page":61,"text":"beam is blocked by a shutter. The sharp ring BDR means the small-angle Bragg diffraction ring","rect":[53.812843322753909,366.255615234375,385.17795318751259,358.5935363769531]},{"page":61,"text":"while IPS means diffuse wide-angle in-plane scattering halo","rect":[53.812843322753909,376.2315673828125,261.25290435938759,368.58642578125]},{"page":61,"text":"4.1.3 X-Ray Analysis","rect":[53.812843322753909,411.2685546875,166.05196265909835,400.6307678222656]},{"page":61,"text":"This is a very powerful method [3, 4] and later we shall discuss it in detail in","rect":[53.812843322753909,438.8108825683594,385.14272395185005,429.8763427734375]},{"page":61,"text":"Chapter 5. Here, only a schematic picture is presented, Fig. 4.4. An X-ray beam","rect":[53.812843322753909,450.7704162597656,385.14069317460618,441.7760925292969]},{"page":61,"text":"passes through a liquid crystal preparation and the diffracted beams form a cone","rect":[53.81282424926758,462.72998046875,385.11789739801255,453.79541015625]},{"page":61,"text":"with 2# angle at the apex and are registered by a photodetector. In this particular","rect":[53.81282424926758,474.68951416015627,385.14473853913918,465.53582763671877]},{"page":61,"text":"example a smectic A liquid crystal is not oriented and the presented pattern is","rect":[53.81282424926758,486.6490783691406,385.1845742617722,477.71453857421877]},{"page":61,"text":"an analog of a Lauegram observed on crystalline powders. First we see a very","rect":[53.81282424926758,498.6086120605469,385.11489192060005,489.674072265625]},{"page":61,"text":"sharp ring at small angles. It is a fingerprint of a lamellar structure. Like in solid","rect":[53.81282424926758,510.5681457519531,385.1327752213813,501.63360595703127]},{"page":61,"text":"crystals, the X-ray beam of wavelength l can be reflected from stacks of parallel","rect":[53.81282424926758,522.4708862304688,385.16657884189677,513.2176513671875]},{"page":61,"text":"molecular layers according to the Bragg law to be discussed in Section 5.2.2:","rect":[53.812843322753909,534.430419921875,385.1506791836936,525.4361572265625]},{"page":61,"text":"2dhkl sin# ¼ m;lm ¼ 1;2;3;..., where h, k, l are Miller indices (001, 002, etc.","rect":[53.812843322753909,546.1719360351563,385.1562466194797,537.1477661132813]},{"page":61,"text":"in our case), d is interlayer distance and # is the diffraction angle. From this formula","rect":[53.817420959472659,558.3505859375,385.1273883648094,549.1969604492188]},{"page":61,"text":"d can be found from the #-angle measured: for instance, if 2# \u0003 3\u0002 for m¼1 (first","rect":[53.817420959472659,570.3101196289063,385.13435227939677,561.156494140625]},{"page":61,"text":"order reflection), then sin# \u0003 0.026 and l \u0003 0.1 nm, d \u0003 1.9 nm. Thus, the","rect":[53.813411712646487,581.8715209960938,385.1183551616844,573.0166625976563]},{"page":61,"text":"interlayer distance corresponds to the length of the molecule and the phase is an","rect":[53.813411712646487,594.2294311523438,385.1492852311469,585.294921875]},{"page":62,"text":"44","rect":[53.812835693359378,42.45599365234375,62.273904267834868,36.73271179199219]},{"page":62,"text":"4 Liquid Crystal Phases","rect":[303.67413330078127,44.276268005371097,385.1685226726464,36.68191146850586]},{"page":62,"text":"orthogonal smectic. A diffuse ring at wide angle shows that the in-plane structure is","rect":[53.812843322753909,68.2883529663086,385.1865579043503,59.35380554199219]},{"page":62,"text":"liquid like therefore the phase is most probably SmA. The average intermolecular","rect":[53.812843322753909,80.24788665771485,385.1098874649204,71.31333923339844]},{"page":62,"text":"distance in the transverse direction may be estimated from the radius of the diffuse","rect":[53.812843322753909,92.20748138427735,385.15772283746568,83.27293395996094]},{"page":62,"text":"ring using the same formula.","rect":[53.812843322753909,104.11019134521485,169.67886014487034,95.17564392089844]},{"page":62,"text":"4.2 Main Calamitic Phases","rect":[53.812843322753909,152.2697296142578,199.04891090128585,143.53240966796876]},{"page":62,"text":"4.2.1 Nematic Phase","rect":[53.812843322753909,181.85350036621095,164.2016886088392,173.5106201171875]},{"page":62,"text":"The isotropic phase formed by achiral molecules has continuous point group","rect":[53.812843322753909,211.69094848632813,385.12383357099068,202.7563934326172]},{"page":62,"text":"symmetry Kh (spherical). According to the group representations [5], upon cooling,","rect":[53.812843322753909,223.65057373046876,385.11370511557348,214.65625]},{"page":62,"text":"the symmetry Kh lowers, at first, retaining its overall translation symmetry T(3) but","rect":[53.81367492675781,235.61026000976563,385.13410813877177,226.6755828857422]},{"page":62,"text":"reduces the orientational symmetry down to either conical or cylindrical. The cone","rect":[53.81313705444336,247.56979370117188,385.11524236871568,238.63523864746095]},{"page":62,"text":"has a polar symmetry C1v and the cylinder has a quadrupolar one D1h. The","rect":[53.81313705444336,259.529541015625,385.14780462457505,250.5948028564453]},{"page":62,"text":"absence of polarity of the nematic phase has been established experimentally. At","rect":[53.813961029052737,271.48907470703127,385.159803939553,262.55450439453127]},{"page":62,"text":"least, polar nematic phases have not been found yet. In other words, there is a head-","rect":[53.813961029052737,283.4486083984375,385.1487973770298,274.5140380859375]},{"page":62,"text":"to-tail symmetry taken into account by introduction of the director n(r), a unit axial","rect":[53.813961029052737,295.35137939453127,385.20857102939677,286.41680908203127]},{"page":62,"text":"vector coinciding with the preferred direction of molecular axes dependent on","rect":[53.814964294433597,307.3109436035156,385.1707696061469,298.37640380859377]},{"page":62,"text":"coordinate (r is radius-vector).","rect":[53.814964294433597,318.8720397949219,177.42336698080784,310.3359375]},{"page":62,"text":"The nematic phase is characterized by the following properties:","rect":[65.7669906616211,331.2300109863281,322.19343431064677,322.29547119140627]},{"page":62,"text":"(i)","rect":[59.76258850097656,348.74249267578127,69.18922553865088,340.20635986328127]},{"page":62,"text":"(ii)","rect":[56.98578643798828,372.6625671386719,69.20955787507666,364.12646484375]},{"page":62,"text":"n(r)¼ \u0004n(r) (absenceofpolarity)and,intheCartesiansystemshowninFig.4.5a","rect":[74.32063293457031,349.14093017578127,385.16370428277818,340.1466064453125]},{"page":62,"text":"the director has components (nx, ny, nz) ¼ (0, 0, 1).","rect":[74.32160949707031,362.0879211425781,281.43829007651098,352.16595458984377]},{"page":62,"text":"€","rect":[270.7066650390625,365.0,275.68377009442818,363.0]},{"page":62,"text":"Point group symmetry is D1h (according to Schonflies) or 1/mm (interna-","rect":[74.31905364990235,373.0611877441406,384.01641212312355,364.12646484375]},{"page":62,"text":"tional). There are one 1-fold rotation axis, i.e., the director axis, the infinite","rect":[74.31805419921875,384.622314453125,385.1800922222313,376.086181640625]},{"page":62,"text":"number of vertical symmetry planes containing n and one mirror plane","rect":[74.31705474853516,396.98028564453127,385.13929022027818,388.04571533203127]},{"page":62,"text":"perpendicular to n. The same symmetry has a discotic nematic phase. The","rect":[74.3170394897461,408.9398498535156,385.1412738628563,400.00531005859377]},{"page":62,"text":"a","rect":[105.13338470458985,437.4522399902344,110.68864710182369,431.8634948730469]},{"page":62,"text":"ϑ","rect":[123.79618835449219,443.2777099609375,128.84145033354839,437.61322021484377]},{"page":62,"text":"z","rect":[110.71849060058594,456.125244140625,114.22059004088169,452.55694580078127]},{"page":62,"text":"b","rect":[190.507568359375,437.4522399902344,196.6123621088532,430.14385986328127]},{"page":62,"text":"n","rect":[232.4015350341797,461.61529541015627,236.81458872154384,457.8154296875]},{"page":62,"text":"n","rect":[106.91319274902344,502.228515625,110.91102316031996,498.4281921386719]},{"page":62,"text":"n^","rect":[245.13870239257813,515.3773803710938,253.49756744374589,509.57806396484377]},{"page":62,"text":"Fig. 4.5 The nematic phase: molecular orientation (a), optical indicatrix (b) and characteristic","rect":[53.812843322753909,583.7169189453125,385.1398558356714,575.9871215820313]},{"page":62,"text":"microscopic texture (c)","rect":[53.812843322753909,593.6251220703125,133.10151761634044,586.03076171875]},{"page":63,"text":"4.2 Main Calamitic Phases","rect":[53.812843322753909,42.55728530883789,145.50293429374018,36.68160629272461]},{"page":63,"text":"45","rect":[376.74566650390627,42.55728530883789,385.20673126368447,36.63080978393555]},{"page":63,"text":"(iii)","rect":[54.15444564819336,103.7118911743164,69.2510007461704,95.17576599121094]},{"page":63,"text":"(iv)","rect":[54.77759552001953,151.5500946044922,69.25101600495947,143.0139617919922]},{"page":63,"text":"(v)","rect":[57.60869598388672,211.2919158935547,69.21530280915869,202.8155517578125]},{"page":63,"text":"orientational order is characterized by a tensor discussed in Chapter 3 whose","rect":[74.31846618652344,68.2883529663086,385.1356891460594,59.35380554199219]},{"page":63,"text":"amplitude (order parameter) is S ¼ (1/2)<3cos2Y \u0004 1> (here Y is an angle","rect":[74.31845092773438,80.24800872802735,385.1419147319969,69.19712829589844]},{"page":63,"text":"the individual molecule forms with the direction n).","rect":[74.3197021484375,91.8091812133789,284.4411282112766,83.27305603027344]},{"page":63,"text":"Translational symmetry is T(3), the translational motion of molecules is","rect":[74.31968688964844,104.11031341552735,385.1748391543503,95.17576599121094]},{"page":63,"text":"possible in any direction, therefore, the density is independent of coordinates,","rect":[74.3197021484375,116.0698471069336,385.15789456869848,107.13529968261719]},{"page":63,"text":"r ¼ const, and the nematic phase is the most fluid one. For this reason it is the","rect":[74.3197021484375,128.02944946289063,385.1737750835594,119.09489440917969]},{"page":63,"text":"most interesting for applications to displays.","rect":[74.3197021484375,139.98898315429688,252.99879117514377,131.05442810058595]},{"page":63,"text":"It is optically uniaxial phase, as a rule positively uniaxial, nz ¼ n|| > nx ¼ ny ¼","rect":[74.3197021484375,152.8791046142578,384.4116900274506,143.0139617919922]},{"page":63,"text":"n⊥. The optical indicatrix presented in Fig. 4.5b has a form of the prolate","rect":[74.32014465332031,163.908935546875,385.15598333551255,154.91461181640626]},{"page":63,"text":"ellipsoid contrary to oblate optical ellipsoid typical of discotic nematics","rect":[74.3188247680664,175.86846923828126,385.0803262148972,166.9339141845703]},{"page":63,"text":"which, as a rule, are optically negative. The dielectric ellipsoid is discussed","rect":[74.3188247680664,187.8280029296875,385.17495051435005,178.89344787597657]},{"page":63,"text":"in more detail in Section 11.1.1.","rect":[74.3188247680664,197.7087860107422,204.20235868002659,190.7962188720703]},{"page":63,"text":"The nematic phase has very characteristic microscopic texture observed with","rect":[74.31883239746094,211.69033813476563,384.66225520185005,202.7557830810547]},{"page":63,"text":"crossed polarizers. In Fig. 4.5c we can see typical point disclinations, the","rect":[74.31883239746094,223.64987182617188,385.1738971538719,214.65554809570313]},{"page":63,"text":"nuclei of divergent brushes or threads. The threads (Greek nema) have given","rect":[74.31982421875,235.60940551757813,385.17885676435005,226.6748504638672]},{"page":63,"text":"the name “nematic” to the phase considered. The structure of disclinations is","rect":[74.31983184814453,247.56893920898438,385.18588651763158,238.63438415527345]},{"page":63,"text":"accounted for by modern theory of elasticity, Section 8.4.","rect":[74.31983184814453,259.52850341796877,307.0134243538547,250.5939483642578]},{"page":63,"text":"4.2.2 Classical Smectic A Phase","rect":[53.812843322753909,307.4016418457031,220.79008277387826,299.0587463378906]},{"page":63,"text":"The classical SmA phase can form on cooling the nematic phase or directly from","rect":[53.812843322753909,337.2388610839844,385.13480328202805,328.3043212890625]},{"page":63,"text":"the isotropic phase. Now we meet a new feature: the phase becomes periodic in one","rect":[53.812843322753909,349.1416015625,385.14377630426255,340.20703125]},{"page":63,"text":"direction. In Fig. 4.6 the interlayer distance equal in this case to period is marked by","rect":[53.812843322753909,361.1011657714844,385.1685723405219,352.1666259765625]},{"page":63,"text":"letter d. Thus, the SmA phase is simultaneously a one-dimensional solid and a two-","rect":[53.8138427734375,373.0606994628906,385.11980567781105,364.1062316894531]},{"page":63,"text":"dimensional liquid. There is no correlation between molecular positions in the","rect":[53.813812255859378,385.020263671875,385.11582220270005,376.0657958984375]},{"page":63,"text":"neighbor layers. Such a phase predominantly forms by more or less symmetric","rect":[53.813812255859378,396.97979736328127,385.1795123882469,388.04522705078127]},{"page":63,"text":"molecules with long alkyl chains.","rect":[53.813812255859378,408.9393310546875,189.1273617317844,400.0047607421875]},{"page":63,"text":"The SmA phase is characterized by the following properties:","rect":[65.76583099365235,420.89886474609377,310.8584123379905,411.96429443359377]},{"page":63,"text":"(i) As in the nematic phase,n(r)¼ \u0004n(r).Inthe figure the directorhas components","rect":[59.76145553588867,438.8097839355469,385.2462502871628,429.875244140625]},{"page":63,"text":"(nx, ny, nz) ¼ (0, 0, 1).","rect":[74.32148742675781,451.7002868652344,165.19540067221409,441.89453125]},{"page":63,"text":"z","rect":[245.5132598876953,491.37823486328127,249.50989156950596,487.19512939453127]},{"page":63,"text":"Smectic A","rect":[290.8063049316406,495.1350402832031,329.4537228184564,489.0963134765625]},{"page":63,"text":"Fig. 4.6 A lamellar structure","rect":[53.812843322753909,571.1338500976563,155.08507678293706,563.404052734375]},{"page":63,"text":"of the thermotropic smectic A","rect":[53.812843322753909,581.0420532226563,155.3355310116431,573.4476928710938]},{"page":63,"text":"phase","rect":[53.812843322753909,591.0180053710938,73.09562060617924,583.4236450195313]},{"page":63,"text":"y","rect":[262.923828125,589.1040649414063,266.92045980681066,583.2173461914063]},{"page":63,"text":"d","rect":[377.3861389160156,536.4638061523438,381.83039334618908,530.6090087890625]},{"page":63,"text":"x","rect":[373.0257873535156,584.9158935546875,377.02241903532629,580.7327880859375]},{"page":64,"text":"46","rect":[53.813682556152347,42.55722427368164,62.274751130627837,36.68154525756836]},{"page":64,"text":"(ii)","rect":[56.98497009277344,67.88993072509766,69.20874152986181,59.35380554199219]},{"page":64,"text":"(iii)","rect":[54.15337371826172,103.71207427978516,69.24993263093603,95.17594909667969]},{"page":64,"text":"(iv)","rect":[54.774810791015628,272.22412109375,69.24823127595556,263.68798828125]},{"page":64,"text":"(v)","rect":[57.60850524902344,296.0867004394531,69.2151120742954,287.6103515625]},{"page":64,"text":"4 Liquid Crystal Phases","rect":[303.67498779296877,44.275901794433597,385.1693771648339,36.68154525756836]},{"page":64,"text":"The point group symmetry is also D1h. However, the translational invariance","rect":[74.3182373046875,68.2883529663086,385.07593572809068,59.35380554199219]},{"page":64,"text":"retains only in two directions and the symmetry group is different from that of","rect":[74.31942749023438,80.24788665771485,385.1506284317173,71.31333923339844]},{"page":64,"text":"nematics: D1h \u0005 T(2).","rect":[74.31942749023438,91.8093032836914,167.42829556967502,83.27293395996094]},{"page":64,"text":"The density is independent of x and y, rx, ry ¼ const, however, rz is a periodic","rect":[74.31861114501953,105.0970687866211,385.1366657085594,95.17594909667969]},{"page":64,"text":"function along the normal to smectic layers z. It has to be even function","rect":[74.31944274902344,116.0702133178711,385.1705254655219,107.13566589355469]},{"page":64,"text":"because there is a symmetry plane perpendicular to the director (e.g. it can","rect":[74.31947326660156,128.02981567382813,385.1287774186469,119.09526062011719]},{"page":64,"text":"be the middle plane of the three layer system shown in the Figure):","rect":[74.31947326660156,139.98934936523438,385.1516252286155,131.05479431152345]},{"page":64,"text":"1","rect":[127.96134948730469,146.47657775878907,134.92927689713162,143.3320770263672]},{"page":64,"text":"rðzÞ ¼ r0 þ Prn cos nqz. Here n ¼ 1, 2, 3, ... and q ¼ 2p/d is the wave-","rect":[74.31947326660156,158.37876892089845,385.10860572663918,148.3451690673828]},{"page":64,"text":"n","rect":[129.66082763671876,164.4435577392578,133.1447913416322,161.20840454101563]},{"page":64,"text":"vector of the periodic structure. The modulation of density is not very strong,","rect":[74.31726837158203,177.00189208984376,385.1295132210422,168.0673370361328]},{"page":64,"text":"rn < r0 and, to the first approximation, the density wave may be represented","rect":[74.31726837158203,188.90521240234376,385.16506281903755,179.9706573486328]},{"page":64,"text":"by a single harmonic (n ¼ 1):","rect":[74.31888580322266,200.86474609375,194.28007371494364,191.93019104003907]},{"page":64,"text":"rðzÞ ¼ r0 þ r1 cosqz","rect":[176.11065673828126,225.16355895996095,262.86288847075658,215.17198181152345]},{"page":64,"text":"(4.1)","rect":[366.09698486328127,224.38572692871095,385.16924415437355,215.90936279296876]},{"page":64,"text":"The value of r1 is usually taken as translational order parameter, see","rect":[74.31919860839844,248.70346069335938,385.1152728862938,239.7489776611328]},{"page":64,"text":"Section 6.3.","rect":[74.31791687011719,258.64105224609377,122.12201352621799,251.7284698486328]},{"page":64,"text":"The orientational order parameter has the same form as in nematics, but its","rect":[74.3169174194336,272.62255859375,377.21860136138158,263.68798828125]},{"page":64,"text":"absolute value is larger SA > SN. The phase is optically positive.","rect":[74.3169174194336,284.5823669433594,335.2503628304172,275.64752197265627]},{"page":64,"text":"Typical texture of the SmA is shown in Fig. 4.7a. We see here the so-called","rect":[74.31863403320313,296.4851379394531,380.80776301435005,287.55059814453127]},{"page":64,"text":"“fans” consisted of “focal-conic” domains. Such domains are originated from","rect":[74.31866455078125,308.4447021484375,385.13590191484055,299.5101318359375]},{"page":64,"text":"a layered structure [1, 6]. Although layers are more or less rigid, they can be","rect":[74.31866455078125,320.40423583984377,385.15186346246568,311.46966552734377]},{"page":64,"text":"bent and may form cylinders and tori with central disclination lines (G1) or","rect":[74.31965637207031,332.3638000488281,385.15230689851418,323.389404296875]},{"page":64,"text":"more complex structures with disclinations of the G2 type. The sketches in","rect":[74.32014465332031,344.3240661621094,385.1437920670844,335.3494567871094]},{"page":64,"text":"Fig. 4.7b represent projections of the tori on the x,z-plane perpendicular to G1","rect":[74.31961059570313,356.28363037109377,385.18130140630105,347.3092346191406]},{"page":64,"text":"(upper sketches) and on the y,z-plane including G1 (lower sketches).","rect":[74.31846618652344,368.24322509765627,349.1080593636203,359.2688293457031]},{"page":64,"text":"4.2.3 Special SmA Phases","rect":[53.812843322753909,412.3185729980469,190.66588477823897,401.76446533203127]},{"page":64,"text":"The structure of the so-called de Vries phase is shown in Fig. 4.8. It is a uniaxial","rect":[53.812843322753909,439.9445495605469,385.11982591220927,431.010009765625]},{"page":64,"text":"smectic A phase (group D1h) with very strong molecular tilt (about \u000620\u0002) in any","rect":[53.81183624267578,451.9042053222656,385.1464470963813,442.96954345703127]},{"page":64,"text":"Fig. 4.7 Smectic A: fan-shape texture (a) and the structure of typical defects (b)","rect":[53.812843322753909,593.9761352539063,332.03903287512949,586.246337890625]},{"page":65,"text":"4.2 Main Calamitic Phases","rect":[53.812843322753909,42.55594253540039,145.50293429374018,36.68026351928711]},{"page":65,"text":"Fig. 4.8 Structure ofa","rect":[53.812843322753909,67.58130645751953,133.32912585520269,59.6313591003418]},{"page":65,"text":"uniaxial smectic A phase with","rect":[53.812843322753909,77.4895248413086,155.3143820205204,69.89517211914063]},{"page":65,"text":"very strong molecular tilt (de","rect":[53.812843322753909,87.4087142944336,153.91576525949956,79.81436157226563]},{"page":65,"text":"Vries phase). x is tilt","rect":[53.812843322753909,97.3846664428711,125.23611931052271,89.5193862915039]},{"page":65,"text":"correlation length","rect":[53.812843322753909,107.36067962646485,114.24011749415323,99.76632690429688]},{"page":65,"text":"Fig. 4.9 Structure of polar","rect":[53.812843322753909,169.83367919921876,145.90228360755138,161.799072265625]},{"page":65,"text":"smectic A phases A1, A2 and","rect":[53.812843322753909,179.741943359375,151.84802765040323,172.14736938476563]},{"page":65,"text":"Ad and a frustrated phase Amod","rect":[53.81300735473633,189.71749877929688,155.32153833224516,182.12313842773438]},{"page":65,"text":"ξ","rect":[288.0454406738281,123.47769165039063,291.9861195120934,116.20723724365235]},{"page":65,"text":"Amod","rect":[290.8403625488281,258.8747863769531,307.83197011950508,251.04852294921876]},{"page":65,"text":"z","rect":[318.378173828125,66.26190185546875,321.8792231813911,62.55069351196289]},{"page":65,"text":"47","rect":[376.74566650390627,42.55594253540039,385.20673126368447,36.73106384277344]},{"page":65,"text":"azimuthal direction. The local molecular tilt is correlated along a certain distancex","rect":[53.812843322753909,379.0693359375,385.1775750260688,369.8160400390625]},{"page":65,"text":"within a smectic layer but on average the tilt is zero. Properties of this phase are","rect":[53.812843322753909,391.0289001464844,385.15964544488755,382.0943603515625]},{"page":65,"text":"different from those of the classical SmA, for example, the birefringence is smaller,","rect":[53.812843322753909,402.9884338378906,385.1795925667453,394.05389404296877]},{"page":65,"text":"i.e., nz is closer to nx, ny than in SmA. The dielectric response is also spectacular. De","rect":[53.812843322753909,415.86474609375,385.16907537652818,406.013427734375]},{"page":65,"text":"Vries phase formed by chiral molecules manifests very interesting electrooptical","rect":[53.813289642333987,426.8510437011719,385.17903001377177,417.91650390625]},{"page":65,"text":"effects.","rect":[53.813289642333987,436.748779296875,82.9094433357883,429.87603759765627]},{"page":65,"text":"Some compounds consisting of molecules with longitudinal permanent dipoles","rect":[65.76530456542969,450.7701721191406,385.10141386138158,441.83563232421877]},{"page":65,"text":"form locally polar smectic A phases and also so-called frustrated phases. In Fig. 4.9","rect":[53.813289642333987,462.7297058105469,385.16704646161568,453.795166015625]},{"page":65,"text":"are shown three structure A1, A2 and Ad which have the same point group symmetry","rect":[53.813297271728519,474.6898498535156,385.1717461686469,465.75469970703127]},{"page":65,"text":"but differ by translational symmetry due to specific packing of the molecules. The","rect":[53.81400680541992,486.6494140625,385.14392889215318,477.71484375]},{"page":65,"text":"A1 phase is the classical SmA discussed above: its interlayer distance, i.e., the","rect":[53.81400680541992,498.6091613769531,385.17176092340318,489.67462158203127]},{"page":65,"text":"structure period, is equal to molecular length. Dipoles are antiparallel within each","rect":[53.812950134277347,510.5686950683594,385.0981072526313,501.6341552734375]},{"page":65,"text":"nonpolar layer. A2 is a smectic with polar layers and antiparallel (sometimes-called","rect":[53.812950134277347,522.4718017578125,385.17156306317818,513.5169677734375]},{"page":65,"text":"antiferroelectric) packing of molecular dipoles in the neighbor layers. Phase Ad","rect":[53.81283950805664,534.431396484375,385.18130140630105,525.4968872070313]},{"page":65,"text":"represents a more general intermediate case. The spectacular orientation of dipoles","rect":[53.812843322753909,546.3910522460938,385.09991850005346,537.45654296875]},{"page":65,"text":"results in modulation of charge density along the smectic normal and the period of","rect":[53.812843322753909,558.3505859375,385.14775977937355,549.4160766601563]},{"page":65,"text":"the charge density wave may be different from the period of the mass density wave.","rect":[53.812843322753909,570.3499755859375,385.13372464682348,561.3556518554688]},{"page":65,"text":"Therefore, there are two waves along the smectic normal, a density one and the","rect":[53.813812255859378,582.2696533203125,385.17359197809068,573.3351440429688]},{"page":65,"text":"electric polarization one. These waves can be incommensurate that is the ratio of","rect":[53.813812255859378,594.229248046875,385.1516355117954,585.2947387695313]},{"page":66,"text":"48","rect":[53.812843322753909,42.55698013305664,62.2739118972294,36.73210144042969]},{"page":66,"text":"4 Liquid Crystal Phases","rect":[303.6741638183594,44.275657653808597,385.1685226726464,36.68130111694336]},{"page":66,"text":"their periods is not an integer, e.g., 1 < l0/l < 2. In some cases, two tendencies,","rect":[53.812843322753909,68.2883529663086,385.1004910042453,58.62885665893555]},{"page":66,"text":"namely, a formation of either monolayer or bilayer structure are in conflict and the","rect":[53.813411712646487,80.24788665771485,385.1712116069969,71.31333923339844]},{"page":66,"text":"resulting phase is “frustrated” or, in other words, is modulated not only along the","rect":[53.813411712646487,92.20748138427735,385.1752094097313,83.27293395996094]},{"page":66,"text":"normal to the layers but also along the smectic plane like the phase Amod shown in","rect":[53.813411712646487,104.11019134521485,385.14382258466255,95.17564392089844]},{"page":66,"text":"the same figure.","rect":[53.81393051147461,116.0702133178711,117.9210400154758,107.13566589355469]},{"page":66,"text":"4.2.4 Smectic C Phase","rect":[53.812843322753909,163.9425506591797,172.8690012309095,155.59967041015626]},{"page":66,"text":"In the SmC phase the longitudinal molecular axes are tilted from the smectic layer","rect":[53.812843322753909,193.77975463867188,385.11885963288918,184.84519958496095]},{"page":66,"text":"normal by an angle #, Fig. 4.10. The phase has the following properties:","rect":[53.812843322753909,205.73928833007813,346.317274642678,196.58560180664063]},{"page":66,"text":"(i)","rect":[59.75949478149414,223.2517852783203,69.18612800447119,214.7156524658203]},{"page":66,"text":"(ii)","rect":[56.98572540283203,306.91290283203127,69.2094968399204,298.37677001953127]},{"page":66,"text":"(iii)","rect":[54.15412521362305,330.83221435546877,69.25068031160009,322.29608154296877]},{"page":66,"text":"(iv)","rect":[54.776268005371097,342.791748046875,69.24968849031103,334.255615234375]},{"page":66,"text":"(v)","rect":[57.60823059082031,366.7108459472656,69.21483741609228,358.2344970703125]},{"page":66,"text":"(vi)","rect":[54.77745819091797,390.57366943359377,69.2508786758579,382.03753662109377]},{"page":66,"text":"The director n coincides with the direction of molecular axes and, as before,","rect":[74.3175277709961,222.0,382.9060329964328,214.7156524658203]},{"page":66,"text":"n ¼ \u0004n. Its components are (nx, ny, nz) ¼ (sin#cosF, sin#sinF, cos#). The","rect":[74.3165283203125,236.5266876220703,385.1478656597313,226.45657348632813]},{"page":66,"text":"projection of n onto the smectic layer plane is called c-director, c ¼ sin#exp","rect":[74.31869506835938,247.56979370117188,385.1448906998969,238.41610717773438]},{"page":66,"text":"(\u0006iF). The c-director is taken as a two-component order parameter of the C-","rect":[74.31869506835938,259.52935791015627,385.15886817781105,250.56492614746095]},{"page":66,"text":"phase. Sin# and F may be considered as the amplitude and phase of the tilt","rect":[74.31869506835938,271.4888916015625,385.157850814553,262.335205078125]},{"page":66,"text":"angle (sign \u0006 determines a sign of rotation). In experiment, angle # varies","rect":[74.3197021484375,283.44842529296877,385.1030618106003,274.29473876953127]},{"page":66,"text":"from 0\u0002 to 45\u0002.","rect":[74.31971740722656,293.3298034667969,135.32137723471409,286.3574523925781]},{"page":66,"text":"The point group symmetryis C2hor2/m (atwofold axis x and a symmetryplane","rect":[74.3189926147461,307.31146240234377,385.15967596246568,298.37677001953127]},{"page":66,"text":"zy). The symmetry group is C2h \u0005 T(2).","rect":[74.31855773925781,319.27099609375,232.72369046713596,310.33642578125]},{"page":66,"text":"The density wave has the same form (4.1) as that of the SmA phase.","rect":[74.31936645507813,331.23065185546877,350.4581265022922,322.29608154296877]},{"page":66,"text":"The spatial positions of molecules in neighbor layers are uncorrelated but their","rect":[74.31837463378906,343.190185546875,385.1087583145298,334.255615234375]},{"page":66,"text":"tilt is correlated.","rect":[74.31837463378906,353.1177978515625,140.32574124838596,346.2152099609375]},{"page":66,"text":"The phase is optically biaxial, Fig. 4.11a, there is no rotation axis coinciding","rect":[74.318359375,367.1092834472656,384.11330500653755,358.17474365234377]},{"page":66,"text":"with the director and n1 6¼ n2 6¼ n3 (z is the smectic normal).","rect":[74.31736755371094,379.0494079589844,318.6536526253391,369.80609130859377]},{"page":66,"text":"In SmC the director is freetorotate alongtheconicalsurfacewithanapexangle","rect":[74.31956481933594,390.97210693359377,385.15674627496568,382.03753662109377]},{"page":66,"text":"2#, therefore, as in a nematic, the Schlieren-texture is observed seen in the","rect":[74.31956481933594,401.0,385.1577838726219,393.7779846191406]},{"page":66,"text":"central part of Fig. 4.11b. On the other hand, the smectic structure reveals the","rect":[74.31956481933594,414.8912048339844,385.1597369976219,405.9367370605469]},{"page":66,"text":"fan-shape texture seen in the left-bottom corner of the same figure.","rect":[74.31857299804688,426.8507385253906,338.76116605307348,417.91619873046877]},{"page":66,"text":"Fig. 4.10 Smectic C.","rect":[53.812843322753909,569.2633056640625,128.4048030097719,561.5335083007813]},{"page":66,"text":"Molecular structure (a) and","rect":[53.812843322753909,578.7761840820313,147.62495941309855,571.5204467773438]},{"page":66,"text":"definition of the c-director (b)","rect":[53.812843322753909,588.7521362304688,155.3625954971998,581.4963989257813]},{"page":66,"text":"Smectic C","rect":[219.1792755126953,491.3747253417969,253.9099981553187,485.4960021972656]},{"page":66,"text":"x","rect":[298.28692626953127,560.00341796875,302.7013588571441,556.2584228515625]},{"page":66,"text":"z","rect":[334.07904052734377,494.357421875,337.58179681968877,490.0842590332031]},{"page":66,"text":"n","rect":[356.405029296875,522.3021240234375,360.4008618258088,518.5037231445313]},{"page":66,"text":"ϑ","rect":[339.4404296875,528.573974609375,344.48317033901449,522.912353515625]},{"page":67,"text":"4.2 Main","rect":[53.813716888427737,42.52274703979492,84.66362518458291,36.68093490600586]},{"page":67,"text":"Fig. 4.11","rect":[53.812843322753909,211.21063232421876,84.93350738673135,203.48081970214845]},{"page":67,"text":"Calamitic","rect":[87.00733947753906,42.55661392211914,120.10704943918705,36.68093490600586]},{"page":67,"text":"a","rect":[97.82575225830078,66.48978424072266,102.27000668847421,62.018733978271487]},{"page":67,"text":"x","rect":[97.82613372802735,162.54721069335938,101.82276540983799,158.36410522460938]},{"page":67,"text":"Phases","rect":[122.52352905273438,42.52274703979492,145.50380404471674,36.68093490600586]},{"page":67,"text":"n'","rect":[131.62525939941407,107.801025390625,137.59623234542043,103.0]},{"page":67,"text":"n1","rect":[124.93020629882813,150.4554443359375,132.7078551292707,144.1527862548828]},{"page":67,"text":"n\"","rect":[129.64051818847657,163.12759399414063,136.9223863261168,158.0]},{"page":67,"text":"z","rect":[152.8653564453125,70.3140869140625,156.86198812712315,66.1309814453125]},{"page":67,"text":"The","rect":[90.91548919677735,210.0,104.1232237555933,203.54855346679688]},{"page":67,"text":"a","rect":[88.989501953125,242.186279296875,94.54476435035885,236.59751892089845]},{"page":67,"text":"optical indicatrix","rect":[106.4931640625,211.14291381835938,164.6241201797001,203.54855346679688]},{"page":67,"text":"SmB","rect":[116.88778686523438,260.1284484863281,135.10007977598918,254.08851623535157]},{"page":67,"text":"(a)","rect":[167.04736328125,210.8042449951172,176.88755887610606,203.59934997558595]},{"page":67,"text":"and","rect":[179.28118896484376,210.0,191.52435821680948,203.54855346679688]},{"page":67,"text":"the","rect":[193.89599609375,210.0,204.2861876227808,203.54855346679688]},{"page":67,"text":"microscopic","rect":[206.6409149169922,211.14291381835938,248.11708972239019,203.54855346679688]},{"page":67,"text":"b","rect":[257.3968505859375,117.7676773071289,262.2807345011101,111.9209213256836]},{"page":67,"text":"texture","rect":[250.54116821289063,210.0,274.1390928962183,204.41212463378907]},{"page":67,"text":"(b)","rect":[276.54119873046877,210.8042449951172,286.83575528723886,203.59934997558595]},{"page":67,"text":"of","rect":[289.2294006347656,210.0,296.2774667130201,203.54855346679688]},{"page":67,"text":"the","rect":[298.6321716308594,210.0,309.0223631598902,203.54855346679688]},{"page":67,"text":"SmC","rect":[311.43377685546877,210.0,328.3474749299465,203.59934997558595]},{"page":67,"text":"biaxial","rect":[330.69287109375,210.0,353.85082725730009,203.54855346679688]},{"page":67,"text":"49","rect":[376.7465515136719,42.62434387207031,385.20761627345009,36.73173522949219]},{"page":67,"text":"phase","rect":[356.240234375,211.14291381835938,375.5230040290308,203.54855346679688]},{"page":67,"text":"6–fold axis","rect":[89.45101928710938,332.5506896972656,130.7764574400412,326.59075927734377]},{"page":67,"text":"symmetry","rect":[172.60557556152345,326.59716796875,210.3733049742209,319.4852600097656]},{"page":67,"text":"plane","rect":[172.60557556152345,336.14105224609377,193.48815787949435,328.7491455078125]},{"page":67,"text":"Fig. 4.12 Structure (a) and a microscopic texture (b) of the smectic B (SmB) phase","rect":[53.812843322753909,374.79150390625,342.78295275949957,367.0616760253906]},{"page":67,"text":"4.2.5 SmecticB","rect":[53.812843322753909,403.0221252441406,139.71317388101796,394.7509460449219]},{"page":67,"text":"In this phase we have:","rect":[53.812843322753909,432.8594665527344,143.776014877053,423.9249267578125]},{"page":67,"text":"(i)","rect":[59.76048278808594,450.3719787597656,69.18711982576025,441.83587646484377]},{"page":67,"text":"(ii)","rect":[56.98524856567383,462.3315124511719,69.20901618806494,453.79541015625]},{"page":67,"text":"Head-to-tail symmetry n ¼ \u0004n.","rect":[74.31851196289063,450.7704162597656,202.78558011557346,441.83587646484377]},{"page":67,"text":"One sixfold rotation z-axis, one mirror plane perpendicular to that axis and 12","rect":[74.31851196289063,462.7299499511719,385.16957942060005,453.7754821777344]},{"page":67,"text":"mirror planes including the sixfold axis. Six of them connect the hexagon","rect":[74.31851196289063,474.6894836425781,385.1765984635688,465.75494384765627]},{"page":67,"text":"angles as shown in Fig. 4.12a and, the other six bisect the angles between those","rect":[74.31851196289063,486.6490173339844,385.1447223491844,477.7144775390625]},{"page":67,"text":"planes. The point group symmetry is D6h (or 6/mmm) and the phase has the","rect":[74.31849670410156,498.6091613769531,385.17322576715318,489.67401123046877]},{"page":67,"text":"following properties:","rect":[74.31913757324219,510.5686950683594,158.7119279629905,501.6341552734375]},{"page":67,"text":"(a) Optical uniaxiality n|| 6¼n⊥ and, as a rule, nz > nx ¼ ny.","rect":[78.62435150146485,523.4581909179688,318.6817593147922,513.2085571289063]},{"page":67,"text":"(b) Three-dimensional density wave along x, y and z axes:","rect":[78.11437225341797,534.431396484375,315.1738725430686,525.4968872070313]},{"page":67,"text":"rðx;y;zÞ ¼ rII cosðqIIzÞ \u0007 r? cosðq?xÞ \u0007 r? cosðq?yÞ","rect":[115.2166748046875,558.6893920898438,323.7645303164597,548.7386474609375]},{"page":67,"text":"(4.2)","rect":[366.0978088378906,557.9523315429688,385.1700681289829,549.4759521484375]},{"page":67,"text":"with different density modulation depth parallel and perpendicular to the","rect":[75.33927154541016,582.269775390625,385.17404974176255,573.3352661132813]},{"page":67,"text":"director. In this respect Smectic B should be referred to as a three dimensional","rect":[75.33927154541016,594.2293090820313,385.130262923928,585.2947998046875]},{"page":68,"text":"50","rect":[53.812843322753909,42.55722427368164,62.2739118972294,36.6307487487793]},{"page":68,"text":"4 Liquid Crystal Phases","rect":[303.6741638183594,44.275901794433597,385.1685226726464,36.68154525756836]},{"page":68,"text":"crystal. However, the situation is not as simple and dependent on correlation","rect":[75.3380126953125,68.2883529663086,385.1250847916938,59.35380554199219]},{"page":68,"text":"in molecular positions in neighbor layers. If such correlations do exist, we deal","rect":[75.3380126953125,80.24788665771485,385.14692552158427,71.31333923339844]},{"page":68,"text":"with a normal 3D crystal having a very small shear modulus corresponding to","rect":[75.3380126953125,92.20748138427735,385.1409539323188,83.27293395996094]},{"page":68,"text":"the velocity gradient ∂vx,y/∂z. If there is no interlayer molecular correlations,","rect":[75.3380126953125,105.08378601074219,385.1806911995578,94.16963195800781]},{"page":68,"text":"the phase is called hexatic and will be considered in Section 5.7.3 in more","rect":[75.33797454833985,116.0702133178711,385.13990057184068,107.0759048461914]},{"page":68,"text":"detail.","rect":[75.33899688720703,125.99787139892578,100.09511991049533,119.09526062011719]},{"page":68,"text":"(iii) A typical, so-called mosaic texture of the SmB is shown in Fig. 4.12b.","rect":[74.71585845947266,139.98934936523438,379.29180570151098,131.05479431152345]},{"page":68,"text":"4.3 Discotic, Bowl-Type and Polyphilic Phases","rect":[53.812843322753909,178.7588348388672,298.34801368448896,167.17681884765626]},{"page":68,"text":"One should distinguish between the discotic nematic, ND phase shown in Fig. 4.13b","rect":[53.812843322753909,205.7393798828125,385.15236750653755,196.78489685058595]},{"page":68,"text":"and several discotic columnar phases, e.g. that shown in Fig. 4.13a. The discotic","rect":[53.81354904174805,217.69894409179688,385.10755193902818,208.76438903808595]},{"page":68,"text":"nematics form on cooling the isotropic phase consisting of disc-like molecules,","rect":[53.813533782958987,229.65847778320313,385.1573757698703,220.7239227294922]},{"page":68,"text":"e.g. of triphenylen type, see Fig. 4.14. The symmetry reduces from T(3) \u0005 O(3) to","rect":[53.813533782958987,241.6180419921875,385.1444024186469,232.66355895996095]},{"page":68,"text":"T(3)\u0005D1h.Thenewphase is not misciblewith calamitic nematics despitethe same","rect":[53.814537048339847,253.5780029296875,385.12769354059068,244.64344787597657]},{"page":68,"text":"symmetry: n ¼ \u0004n, point group D1h, r ¼ const, optical uniaxiality. However,","rect":[53.812705993652347,265.53765869140627,385.15227933432348,256.60302734375]},{"page":68,"text":"hydrodynamic properties of discotic nematics are quite different from those of","rect":[53.81342697143555,277.4971923828125,385.1483090957798,268.5626220703125]},{"page":68,"text":"calamitic nematics. A columnar phase is an example of a two-dimensional (2D)","rect":[53.81342697143555,289.39996337890627,385.1303647598423,280.46539306640627]},{"page":68,"text":"crystal and 1D liquid, a lattice of liquid threads. The translational motion of","rect":[53.81342697143555,301.3594970703125,385.1492856582798,292.3851013183594]},{"page":68,"text":"molecules is allowed only along their normals, the translation group is T(1) and","rect":[53.812435150146487,313.3190612792969,385.1413506608344,304.384521484375]},{"page":68,"text":"the point group can be different. For example it is D6h for an orthogonal hexagonal","rect":[53.812435150146487,325.2792053222656,385.174208236428,316.34405517578127]},{"page":68,"text":"phase or C2h for a tilted phase. We meet even more phases formed by disc-like","rect":[53.81350326538086,337.2388610839844,385.17014349176255,328.30419921875]},{"page":68,"text":"molecules,namelyIsotropicI,nematic ND,D0(columnar orthogonal),Dt(columnar","rect":[53.81337356567383,349.198486328125,385.09395728913918,340.26385498046877]},{"page":68,"text":"tilted) and K (crystalline) ones.","rect":[53.81386947631836,361.1580505371094,179.32949491049534,352.2235107421875]},{"page":68,"text":"Fig. 4.13 Structure of","rect":[53.812843322753909,451.7073974609375,131.44062894446544,443.7743835449219]},{"page":68,"text":"discotic columnar (a) and","rect":[53.812843322753909,461.2769775390625,141.5634512343876,454.0212707519531]},{"page":68,"text":"nematic (b) phases","rect":[53.812843322753909,471.53485107421877,118.38349612235345,463.94049072265627]},{"page":68,"text":"a","rect":[195.82505798339845,385.65997314453127,201.38032038063228,380.07122802734377]},{"page":68,"text":"z","rect":[256.0489196777344,388.7734069824219,260.045551359545,384.5903015136719]},{"page":68,"text":"y","rect":[271.4031677246094,411.03826904296877,275.39979940642,405.1515197753906]},{"page":68,"text":"x","rect":[297.68182373046877,431.74505615234377,301.6784554122794,427.56195068359377]},{"page":68,"text":"b","rect":[318.2016906738281,385.6459655761719,324.3064844233063,378.33758544921877]},{"page":68,"text":"n","rect":[364.6725769042969,401.902587890625,369.55646081946949,397.5355224609375]},{"page":68,"text":"Fig. 4.14 Molecular formula","rect":[53.812843322753909,570.1702270507813,155.052102539773,562.4404296875]},{"page":68,"text":"and a phase sequence ofa","rect":[53.812843322753909,580.0784301757813,142.9586729743433,572.4840698242188]},{"page":68,"text":"triphenylene compound","rect":[53.812843322753909,590.0543823242188,134.44937652735636,582.4600219726563]},{"page":68,"text":"C","rect":[205.3385467529297,584.7009887695313,211.10968290146426,578.6622924804688]},{"page":68,"text":"R","rect":[230.73953247070313,532.0521240234375,236.5106686192377,526.309326171875]},{"page":68,"text":"R","rect":[230.73953247070313,543.800048828125,236.5106686192377,538.0572509765625]},{"page":68,"text":"R","rect":[276.9273986816406,503.33819580078127,282.69853483017519,497.5954284667969]},{"page":68,"text":"R","rect":[292.3128356933594,513.2560424804688,298.0839718418939,507.5132751464844]},{"page":69,"text":"4.4","rect":[53.813690185546878,43.0,64.39002746729776,36.73204040527344]},{"page":69,"text":"Role","rect":[66.78535461425781,43.0,82.8106245734644,36.68124008178711]},{"page":69,"text":"of","rect":[85.19495391845703,43.0,92.24302762610604,36.68124008178711]},{"page":69,"text":"Polymerization","rect":[94.59774017333985,44.275596618652347,146.4725546279423,36.68124008178711]},{"page":69,"text":"a","rect":[99.24235534667969,68.23361206054688,104.79761774391354,62.64485549926758]},{"page":69,"text":"R","rect":[110.42762756347656,122.24127197265625,116.19876371201113,116.49849700927735]},{"page":69,"text":"d","rect":[99.5600814819336,136.79733276367188,105.6648752314118,129.48895263671876]},{"page":69,"text":"ferro-","rect":[117.38467407226563,152.0450439453125,136.4834451774782,146.10940551757813]},{"page":69,"text":"R= alkyl","rect":[159.26507568359376,76.43184661865235,189.7034199396961,68.98543548583985]},{"page":69,"text":"Packing","rect":[153.7274932861328,145.140625,182.16379339141722,137.6450958251953]},{"page":69,"text":"antiferro-","rect":[188.83119201660157,152.0450439453125,220.81703282396257,146.10940551757813]},{"page":69,"text":"b","rect":[261.42578125,68.45529174804688,267.5305749994782,61.14691925048828]},{"page":69,"text":"BOWLS","rect":[284.8739318847656,74.87217712402344,314.624872142051,68.83346557617188]},{"page":69,"text":"uniaxial","rect":[266.5195007324219,87.09630584716797,293.6207180089438,81.24067687988281]},{"page":69,"text":"biaxial","rect":[310.2532043457031,87.09630584716797,332.9095058507407,81.24067687988281]},{"page":69,"text":"c","rect":[261.23297119140627,123.69894409179688,266.7882335886401,118.11019134521485]},{"page":69,"text":"n π –n","rect":[265.3284606933594,132.13433837890626,288.37225671790699,127.21538543701172]},{"page":69,"text":"n = –n","rect":[308.31463623046877,131.7744140625,331.35843225501636,127.4073486328125]},{"page":69,"text":"51","rect":[376.7473449707031,42.55691909790039,385.2084097304813,36.63044357299805]},{"page":69,"text":"Fig. 4.15 Possible bowl phases: forms of molecules (a) and bowls (b), polar and non-polar","rect":[53.812843322753909,230.59555053710938,385.16693204505136,222.86573791503907]},{"page":69,"text":"columns consisting of bowl molecules (c), and two types of column packing, ferroelectric (left)","rect":[53.813697814941409,240.5037841796875,385.1720284805982,232.87554931640626]},{"page":69,"text":"and antiferroelectric (right) (d)","rect":[53.812843322753909,250.4568634033203,160.00857633216075,242.81170654296876]},{"page":69,"text":"A molecule that, in principle, may form a bowl phase should itself have the bowl","rect":[65.76496887207031,277.497314453125,385.14192063877177,268.562744140625]},{"page":69,"text":"form like that seen in Fig. 4.15a [7]. Molecular bowls may have different symmetry","rect":[53.812950134277347,289.40008544921877,385.1706475358344,280.40576171875]},{"page":69,"text":"as shown on the top of Fig. 4.15b and the corresponding phases could be either","rect":[53.81294250488281,301.359619140625,385.0851071914829,292.36529541015627]},{"page":69,"text":"uniaxial or biaxial. The packing of bowls into the columns may have specific","rect":[53.81295394897461,313.3191833496094,385.10300481988755,304.3846435546875]},{"page":69,"text":"features. For example, when all molecules in the column are oriented bottom","rect":[53.81295394897461,325.2787170410156,385.1060557234343,316.34417724609377]},{"page":69,"text":"down then the head-to-to tail symmetry is broken and the column has conical, i.e.","rect":[53.81295394897461,337.23828125,385.1727566292453,328.3037109375]},{"page":69,"text":"polar symmetry C1v, Fig. 4.15c. Only polar columns may form ferroelectric or","rect":[53.81295394897461,349.198486328125,385.14959083406105,340.20416259765627]},{"page":69,"text":"antiferroelectric phases shown in Fig. 4.15d.","rect":[53.8127326965332,361.1580505371094,232.9208187630344,352.1637268066406]},{"page":69,"text":"Actually, such bowl phases are still to be found. However, polar achiral phases","rect":[65.7657241821289,373.1175842285156,385.14560331450658,364.18304443359377]},{"page":69,"text":"have been observed in the so-called polyphilic compounds [8]. The rod-like mole-","rect":[53.81370162963867,385.0203552246094,385.1047605117954,376.0858154296875]},{"page":69,"text":"cules of these compounds consist of distinctly different chemical parts, a hydro-","rect":[53.81468963623047,396.9798889160156,385.1715329727329,388.04534912109377]},{"page":69,"text":"philic rigid core (a biphenyl moiety) and hydrophobic perfluoroalkyl- and alkyl-","rect":[53.81468963623047,408.939453125,385.1813901504673,400.0048828125]},{"page":69,"text":"chains at opposite edges. Such molecules form polar blocks that, in turn, form a","rect":[53.81468963623047,420.8989562988281,385.15854681207505,411.96441650390627]},{"page":69,"text":"polar phase manifesting pyroelectric and piezoelectric properties with a field-","rect":[53.81468963623047,432.8584899902344,385.1476071914829,423.9239501953125]},{"page":69,"text":"induced hysteresis characteristic of ferroelectric phases.","rect":[53.81468963623047,444.81805419921877,278.7112087776828,435.88348388671877]},{"page":69,"text":"4.4 Role of Polymerization","rect":[53.812843322753909,495.4912109375,198.48748749589087,484.1362609863281]},{"page":69,"text":"There are two types of polymers, which form thermotropic liquid crystals, the side-","rect":[53.812843322753909,522.4718017578125,385.12481056062355,513.5372924804688]},{"page":69,"text":"chain, Fig. 4.16a and the main-chain polymers, Fig. 4.16b. In the side-chain","rect":[53.812843322753909,534.431396484375,385.1138543229438,525.4968872070313]},{"page":69,"text":"polymers the mesogenic units are attached to a backbone by more or less flexible","rect":[53.812862396240237,546.3909301757813,385.1785358257469,537.4564208984375]},{"page":69,"text":"chains. In the main-chain polymers mesogenic units are incorporated into the","rect":[53.812862396240237,558.3504638671875,385.1159137554344,549.4159545898438]},{"page":69,"text":"polymer backbone and separated from each other by flexible chains [9,10]. Flexible","rect":[53.812862396240237,570.3099975585938,385.17359197809068,561.37548828125]},{"page":69,"text":"chains (spacers) are necessary to provide a certain freedom to mesogenic moieties","rect":[53.812843322753909,582.26953125,385.1606484805222,573.3350219726563]},{"page":69,"text":"to form an ordered state. For the side-chain polymer to be in the nematic or smectic","rect":[53.812843322753909,594.2291259765625,385.18054998590318,585.2946166992188]},{"page":70,"text":"52","rect":[53.81283950805664,42.55691909790039,62.27390808253213,36.63044357299805]},{"page":70,"text":"Fig. 4.16 Structure of","rect":[53.812843322753909,67.58130645751953,131.44062894446544,59.85148620605469]},{"page":70,"text":"polymer chains appropriate","rect":[53.812843322753909,77.4895248413086,147.19091174387456,69.89517211914063]},{"page":70,"text":"for side-chain (a) and main-","rect":[53.812843322753909,87.07006072998047,149.98643582923106,79.81436157226563]},{"page":70,"text":"chain (b) polymer liquid","rect":[53.812843322753909,97.3846664428711,137.75934356837198,89.79031372070313]},{"page":70,"text":"crystals","rect":[53.812843322753909,107.36067962646485,79.69525607108392,99.76632690429688]},{"page":70,"text":"4 Liquid Crystal Phases","rect":[303.6741638183594,44.275596618652347,385.1685226726464,36.68124008178711]},{"page":70,"text":"Fig. 4.17 Scheme of packing","rect":[53.812843322753909,261.1465148925781,155.33553070216105,253.4167022705078]},{"page":70,"text":"of main chain polymer","rect":[53.812843322753909,271.0547790527344,131.710541664192,263.4604187011719]},{"page":70,"text":"mesogenic groups in the","rect":[53.812843322753909,280.9739685058594,137.27620837473394,273.3796081542969]},{"page":70,"text":"nematic (a) and smectic A","rect":[53.812843322753909,290.6112976074219,144.45883728605717,283.3555908203125]},{"page":70,"text":"(b) phases","rect":[53.81283950805664,300.9259033203125,89.04134829520501,293.33154296875]},{"page":70,"text":"a","rect":[229.80943298339845,261.7987060546875,235.36469538063228,256.2099609375]},{"page":70,"text":"b","rect":[298.4757385253906,261.7987060546875,304.5805322748688,254.49032592773438]},{"page":70,"text":"phase is quite natural, because mesogenic units can easily be arranged parallel to","rect":[53.812843322753909,438.0740661621094,385.13979426435005,429.1395263671875]},{"page":70,"text":"each other, Fig. 4.17a. However, even in the main chain polymers with long enough","rect":[53.812843322753909,450.0335998535156,385.1208428483344,441.09906005859377]},{"page":70,"text":"flexible spacers between the mesogenic groups, the latter can form nematic and","rect":[53.812843322753909,461.9931335449219,385.1407403092719,453.05859375]},{"page":70,"text":"even smectic phases and the flexible backbones are forced to acquire the liquid","rect":[53.812843322753909,473.9526672363281,385.1128777604438,465.01812744140627]},{"page":70,"text":"crystalline structure, Fig. 4.17b.","rect":[53.812843322753909,485.9122314453125,182.1664242317844,476.9776611328125]},{"page":70,"text":"In the same way one can synthesized liquid crystalline copolymers in which","rect":[65.7648696899414,497.81500244140627,385.11980525067818,488.86053466796877]},{"page":70,"text":"mesogenic groups alternate with some functional groups like chiral, polar, photo-","rect":[53.81386947631836,509.7745666503906,385.14678321687355,500.84002685546877]},{"page":70,"text":"chromic, luminescent, etc., groups useful for various applications especially in","rect":[53.81386947631836,521.7340698242188,385.13881770185005,512.799560546875]},{"page":70,"text":"nonlinear optics. For example, incorporating chromophores, manifesting a light-","rect":[53.81386947631836,533.693603515625,385.14876685945168,524.7590942382813]},{"page":70,"text":"induced intramolecular charge transfer, one can develop materials with enhanced","rect":[53.81386947631836,545.6531372070313,385.1497429948188,536.7186279296875]},{"page":70,"text":"nonlinear susceptibility, so-called w(2)- or w(3)-materials capable of wave mixing,","rect":[53.81386947631836,557.6138305664063,385.1736111214328,546.5628662109375]},{"page":70,"text":"generation of light harmonics, etc. Polymer liquid crystals with photochromic","rect":[53.81386947631836,569.5733642578125,385.13681829645005,560.6388549804688]},{"page":70,"text":"moieties, showing reversible and multiple photo-induced cis–trans–cis isomeriza-","rect":[53.81386947631836,581.532958984375,385.08599220124855,572.5984497070313]},{"page":70,"text":"tion, are very perspective for holographic grating recording, polarimetry, optical","rect":[53.813838958740237,593.4356689453125,385.1238237149436,584.5011596679688]},{"page":71,"text":"4.5 Lyotropic Phases","rect":[53.812843322753909,44.276084899902347,125.62109072684564,36.63093185424805]},{"page":71,"text":"53","rect":[376.7464904785156,42.55740737915039,385.2075552382938,36.63093185424805]},{"page":71,"text":"information processing, lasers without mirrors and so on. The nematic phase formed","rect":[53.812843322753909,68.2883529663086,385.26421443036568,59.35380554199219]},{"page":71,"text":"by main-chain polymers can be used in a technological process of manufacturing","rect":[53.812843322753909,80.24788665771485,385.09002009442818,71.31333923339844]},{"page":71,"text":"extra strong polymer fibers, because the material goes through draw plates in the","rect":[53.812843322753909,92.20748138427735,385.1715778179344,83.27293395996094]},{"page":71,"text":"well-oriented nematic state and the fiber contains less defects.","rect":[53.812843322753909,102.07825469970703,303.8128628304172,95.17564392089844]},{"page":71,"text":"Polymers can form the same thermotropic phases as low-molecular mass com-","rect":[65.76486206054688,116.0697250366211,385.13680396882668,107.13517761230469]},{"page":71,"text":"pounds (nematic, smectic A, C, B, chiral phases as well). Despite the same","rect":[53.812843322753909,128.02932739257813,385.1278461284813,119.09477233886719]},{"page":71,"text":"symmetry, physical properties of polymer liquid crystals are very specific. They","rect":[53.812843322753909,139.98886108398438,385.13677302411568,131.05430603027345]},{"page":71,"text":"are very viscous due to the entangling of long polymer chains hindering the","rect":[53.812843322753909,151.94839477539063,385.11682928277818,143.0138397216797]},{"page":71,"text":"translational motion (flow). On cooling the polymer liquid crystal acquire a glassy","rect":[53.812843322753909,163.9477996826172,385.12778509332505,154.95347595214845]},{"page":71,"text":"state very useful for many applications. For example, one can create some macro-","rect":[53.812843322753909,175.86749267578126,385.1776059707798,166.9329376220703]},{"page":71,"text":"scopic structures in the nematic phase very sensitive to external fields (for instance,","rect":[53.812843322753909,187.8270263671875,385.1527065804172,178.89247131347657]},{"page":71,"text":"a grating, or a field induced polar, pyroelectric structure) and then froze it into the","rect":[53.812843322753909,199.72976684570313,385.1726459331688,190.7952117919922]},{"page":71,"text":"glassy state which is not crystalline but mechanically solid and use the latter for","rect":[53.812843322753909,211.6893310546875,385.17864356843605,202.75477600097657]},{"page":71,"text":"applications. You can also make cholesteric polymer doped with a proper lumines-","rect":[53.812843322753909,223.64886474609376,385.16658912507668,214.7143096923828]},{"page":71,"text":"cent dye for laser devices with distributed feedback (due to natural periodicity of","rect":[53.812843322753909,235.60842895507813,385.14873634187355,226.6738739013672]},{"page":71,"text":"the helical structure). Some polymer liquid crystals can be as elastic as rubber","rect":[53.812843322753909,247.56796264648438,385.13582740632668,238.63340759277345]},{"page":71,"text":"(elastomers). They have very good prospects as piezoelectric materials as well as","rect":[53.812843322753909,259.52752685546877,385.1378213320847,250.5730438232422]},{"page":71,"text":"materials having mechanically tunable optical properties.","rect":[53.812843322753909,271.4870910644531,284.57732816244848,262.55255126953127]},{"page":71,"text":"4.5 Lyotropic Phases","rect":[53.812843322753909,322.1612548828125,169.02628211710616,310.8660888671875]},{"page":71,"text":"Lyotropic liquid crystalline phases form by water solutions of amphiphilic (particu-","rect":[53.812843322753909,349.1418151855469,385.15862403718605,340.207275390625]},{"page":71,"text":"larly biphilic) molecules [11, 12]. The building blocks of those phases are either","rect":[53.812843322753909,361.10137939453127,385.24724708406105,352.16680908203127]},{"page":71,"text":"bilayers, Fig. 4.18, or micelles. The form of the micelles can be spherical or","rect":[53.81285095214844,373.0609130859375,385.2612241348423,364.1263427734375]},{"page":71,"text":"cylindrical, Fig. 4.19a, b. For low concentration of oil in water, the micelles are","rect":[53.81285095214844,385.0204772949219,385.15967596246568,376.0859375]},{"page":71,"text":"normal (sketch (a),tails inside, polar heads outside, inwater). Forhigh concentration,","rect":[53.81285095214844,396.9800109863281,385.1796536019016,388.04547119140627]},{"page":71,"text":"the structure is inversed ((b) and (c), water and polar heads inside, tails outside).","rect":[53.81285095214844,408.9395446777344,385.1766628792453,400.0050048828125]},{"page":71,"text":"Examples of the structure of some typical lyotropic phases (lamellar, cubic, hax-","rect":[53.81285095214844,420.8990783691406,385.25823341218605,411.96453857421877]},{"page":71,"text":"agonal) are showninFig. 4.20.Under a microscopetheyshowcharacteristic features,","rect":[53.81285095214844,432.8586120605469,385.1676907112766,423.924072265625]},{"page":71,"text":"Fig. 4.18 Bilayers formed by","rect":[53.812843322753909,563.2551879882813,155.34146637110636,555.3052368164063]},{"page":71,"text":"biphilic molecules having","rect":[53.812843322753909,573.106689453125,142.0931143203251,565.5123291015625]},{"page":71,"text":"polar hydrophilic heads and","rect":[53.812843322753909,583.0826416015625,149.04134124903605,575.48828125]},{"page":71,"text":"hydrophobic tails","rect":[53.812843322753909,593.05859375,113.18840487479486,585.4642333984375]},{"page":71,"text":"hydrophilic","rect":[211.56967163085938,506.6450500488281,254.652816953753,499.1427307128906]},{"page":71,"text":"heads","rect":[211.56967163085938,514.6424560546875,234.669906797503,508.7957458496094]},{"page":71,"text":"hydrophobic","rect":[211.56967163085938,534.1884155273438,259.97627276429986,526.6860961914063]},{"page":71,"text":"tails","rect":[211.56967163085938,542.1866455078125,227.56399554262019,536.283935546875]},{"page":71,"text":"hydrophilic","rect":[211.56967163085938,557.5335693359375,254.652816953753,550.03125]},{"page":71,"text":"heads","rect":[211.56967163085938,565.5317993164063,234.669906797503,559.6851196289063]},{"page":72,"text":"54","rect":[53.812843322753909,42.55667495727539,62.2739118972294,36.63019943237305]},{"page":72,"text":"Fig. 4.19","rect":[53.812843322753909,152.20587158203126,84.93350738673135,144.1712646484375]},{"page":72,"text":"Micelles: spherical normal (a),","rect":[90.91548919677735,152.13815307617188,196.3928858955141,144.54379272460938]},{"page":72,"text":"Fig. 4.20 Lamellar, cubic and hexagonal","rect":[53.812843322753909,286.1993408203125,196.4292574330813,278.4695129394531]},{"page":72,"text":"Fig. 4.21 Microphotograph","rect":[53.812843322753909,315.8434143066406,150.20221466212198,308.11358642578127]},{"page":72,"text":"of the hexagonal lyotropic","rect":[53.812843322753909,325.7516784667969,143.9443907477808,318.1573181152344]},{"page":72,"text":"phase texture","rect":[53.812843322753909,335.6708679199219,99.10579821848393,328.0765075683594]},{"page":72,"text":"cylindrical inverse","rect":[198.76705932617188,152.13815307617188,261.99071643137457,144.54379272460938]},{"page":72,"text":"lyotropic phases","rect":[198.825439453125,286.1316223144531,254.44684298514643,278.5372619628906]},{"page":72,"text":"(b)","rect":[264.4190368652344,151.7994842529297,274.71359342200449,144.59458923339845]},{"page":72,"text":"and","rect":[277.050537109375,151.0,289.2937063613407,144.54379272460938]},{"page":72,"text":"4 Liquid Crystal Phases","rect":[303.6741638183594,44.275352478027347,385.1685226726464,36.68099594116211]},{"page":72,"text":"spherical inverse (c)","rect":[291.7220458984375,152.13815307617188,361.04022306067636,144.54379272460938]},{"page":72,"text":"Hexagonal","rect":[313.9302978515625,320.1025085449219,354.3522236506336,312.6321105957031]},{"page":72,"text":"lyotropic phase","rect":[313.9302978515625,329.6764831542969,373.0092873891578,322.174072265625]},{"page":72,"text":"e.g., a fan-shape texture is typical of the hexagonal lyotropic phase presented in","rect":[53.812843322753909,486.6495056152344,385.25624934247505,477.7149658203125]},{"page":72,"text":"Fig. 4.21.","rect":[53.812843322753909,498.6090393066406,91.64780087973361,489.67449951171877]},{"page":72,"text":"There is also a group of the so-called lyotropic nematics. They are intermediate","rect":[65.76486206054688,510.568603515625,385.11890447809068,501.634033203125]},{"page":72,"text":"between the isotropic micellar phase and structured (lamellar or hexagonal) phases","rect":[53.812843322753909,522.4713134765625,385.1437112246628,513.5368041992188]},{"page":72,"text":"and can be formed byboth discotic and calamitic molecules. The lyotropic nematics","rect":[53.812843322753909,534.430908203125,385.07806791411596,525.4963989257813]},{"page":72,"text":"can be aligned by an electric or magnetic field and show Schlieren texture as","rect":[53.812843322753909,546.390380859375,385.1377297793503,537.4558715820313]},{"page":72,"text":"thermotropic nematics. The building blocks of these mesophases are vesicles or","rect":[53.812843322753909,558.3499755859375,385.1487973770298,549.4154663085938]},{"page":72,"text":"similar mesoscopic objects. From the symmetry point of view the nematic phases","rect":[53.812843322753909,570.3095092773438,385.1477090273972,561.375]},{"page":72,"text":"can be uniaxial or biaxial, as shown in Fig. 4.22. In fact, the biaxial nematics have","rect":[53.812843322753909,582.26904296875,385.1188129253563,573.3345336914063]},{"page":72,"text":"been found unequivocally only in the lyotropic systems [13].","rect":[53.812862396240237,594.2285766601563,299.6768765022922,585.2940673828125]},{"page":73,"text":"4.6 General Remarks on the Role of Chirality","rect":[53.812843322753909,44.274620056152347,210.16666168360636,36.68026351928711]},{"page":73,"text":"a","rect":[97.65629577636719,68.27377319335938,103.21155817360104,62.68501663208008]},{"page":73,"text":"Fig. 4.22 Structure of lyotropic nematics: a phase","rect":[53.812843322753909,164.90243530273438,225.64193103098394,157.17262268066407]},{"page":73,"text":"Ne by cylindrical rod-like blocks (b)","rect":[53.813594818115237,174.81048583984376,178.757172523567,167.21612548828126]},{"page":73,"text":"Nd","rect":[227.77072143554688,164.49351501464845,237.00406396701079,157.40968322753907]},{"page":73,"text":"is","rect":[239.0991973876953,163.01443481445313,244.73427279471674,157.2403564453125]},{"page":73,"text":"b","rect":[254.69602966308595,68.23318481445313,260.8008234125641,60.92481231689453]},{"page":73,"text":"formed","rect":[246.80300903320313,163.10755920410157,271.2893423476688,157.2403564453125]},{"page":73,"text":"by","rect":[273.4266052246094,164.834716796875,281.88766998438759,157.2403564453125]},{"page":73,"text":"disc-like","rect":[283.96234130859377,163.10755920410157,313.1953367927027,157.2403564453125]},{"page":73,"text":"blocks","rect":[315.3435974121094,163.08216857910157,337.45236666678707,157.2403564453125]},{"page":73,"text":"(a)","rect":[339.58795166015627,164.4960479736328,349.42901700598886,157.29115295410157]},{"page":73,"text":"and","rect":[351.5392150878906,163.10755920410157,363.7823843398563,157.2403564453125]},{"page":73,"text":"55","rect":[376.74566650390627,42.55594253540039,385.20673126368447,36.62946701049805]},{"page":73,"text":"phase","rect":[365.8705749511719,164.834716796875,385.1533446052027,157.2403564453125]},{"page":73,"text":"P0","rect":[217.45492553710938,315.29864501953127,226.11491588283062,307.452392578125]},{"page":73,"text":"Fig. 4.23 Structure of the cholesteric phase. Each sheet models a cross-section of the helical","rect":[53.812843322753909,337.4388122558594,385.1255159780032,329.50579833984377]},{"page":73,"text":"structure within one period of the helix P0. The helix axis is directed from the left to the right. The","rect":[53.812843322753909,347.3470764160156,385.19725940012457,339.7525329589844]},{"page":73,"text":"short bars show orientation of chiral molecules within each sheet","rect":[53.8134880065918,355.5702819824219,276.4674349233157,349.7284851074219]},{"page":73,"text":"Table 4.1 Point group","rect":[53.812843322753909,384.52960205078127,133.42642730860636,376.86749267578127]},{"page":73,"text":"symmetry of main achiral","rect":[53.812843322753909,394.50555419921877,142.0474214955813,386.91119384765627]},{"page":73,"text":"and chiral phases","rect":[53.812843322753909,404.4247741699219,112.71965487479486,396.8304138183594]},{"page":73,"text":"Phase/ chirality","rect":[167.10317993164063,387.87384033203127,219.9273428359501,380.27947998046877]},{"page":73,"text":"Isotropic","rect":[167.10317993164063,400.0036315917969,197.2245878913355,392.4092712402344]},{"page":73,"text":"Nematic or smectic A","rect":[167.1034393310547,408.2039489746094,242.1717584286353,402.3282775878906]},{"page":73,"text":"SmecticC","rect":[167.1031494140625,418.1796569824219,202.53387629713397,412.3039855957031]},{"page":73,"text":"Achiral","rect":[290.3065185546875,386.0535583496094,315.7489595815188,380.27947998046877]},{"page":73,"text":"Kh","rect":[290.3065185546875,399.6001892089844,299.59702050997955,392.5785827636719]},{"page":73,"text":"D1h","rect":[290.3059387207031,409.5757141113281,305.94138086154205,402.4975891113281]},{"page":73,"text":"C2h","rect":[290.3064880371094,419.4951477050781,302.3165944845889,412.3547668457031]},{"page":73,"text":"Chiral","rect":[363.94488525390627,386.1551513671875,385.16528038230009,380.27947998046877]},{"page":73,"text":"K","rect":[363.9451599121094,398.18310546875,370.0202081356665,392.5783386230469]},{"page":73,"text":"D","rect":[363.9451599121094,408.102294921875,370.0202081356665,402.4975280761719]},{"page":73,"text":"1","rect":[370.00616455078127,409.5757141113281,376.3518168988653,407.0304260253906]},{"page":73,"text":"C2","rect":[363.9451599121094,419.4951477050781,372.7823842795108,412.3547668457031]},{"page":73,"text":"4.6 General Remarks on the Role of Chirality","rect":[53.812843322753909,459.6689453125,296.8190130224187,448.3139953613281]},{"page":73,"text":"Chirality is lack of mirror symmetry. The name came from Greek word for “hand”.","rect":[53.812843322753909,486.6495056152344,385.18059964682348,477.7149658203125]},{"page":73,"text":"W.H. Thomson (Lord Kelvin) defined it as follows: “any geometrical figure has","rect":[53.812843322753909,498.6090393066406,385.1318398867722,489.67449951171877]},{"page":73,"text":"chirality if its image in a plane mirror cannot be brought into coincidence with","rect":[53.812843322753909,510.568603515625,385.1158379655219,501.634033203125]},{"page":73,"text":"itself”. Examples of chiral phases are the cholesteric, schematically shown in","rect":[53.812843322753909,522.4713134765625,385.1426934342719,513.516845703125]},{"page":73,"text":"Fig. 4.23, and smectic C* ones (the asterisk at letter C is used to distinguish this","rect":[53.81282424926758,534.430908203125,385.14072050200658,525.45654296875]},{"page":73,"text":"phase from the achiral smectic C). Unfortunately there is no quantitative definition","rect":[53.81282424926758,546.390380859375,385.15371027997505,537.4558715820313]},{"page":73,"text":"of chirality [14]. The chirality of a molecule results in a spatial modulation of liquid","rect":[53.81282424926758,558.3499755859375,385.1109246354438,549.4154663085938]},{"page":73,"text":"crystalline phases. Table 4.1 shows how the point group symmetry is changed when","rect":[53.81282424926758,570.3095092773438,385.15264216474068,561.375]},{"page":73,"text":"the achiral liquid crystal material is doped with a chiral compound. The isotropic","rect":[53.81282424926758,582.26904296875,385.1715778179344,573.3345336914063]},{"page":73,"text":"liquids formed by chiral molecules, e.g. sugar solutions in water, have continuous","rect":[53.81282424926758,594.2285766601563,385.15570463286596,585.2940673828125]},{"page":74,"text":"56","rect":[53.812843322753909,42.55765151977539,62.2739118972294,36.63117599487305]},{"page":74,"text":"4 Liquid Crystal Phases","rect":[303.6741638183594,44.276329040527347,385.1685226726464,36.68197250366211]},{"page":74,"text":"group symmetry K (no mirror plane). Experimentally, this can be recognized by a","rect":[53.812843322753909,68.2883529663086,385.1557086773094,59.35380554199219]},{"page":74,"text":"rotation of the polarization plane of transmitted light (optical activity). In case of","rect":[53.812843322753909,80.24788665771485,385.14873634187355,71.31333923339844]},{"page":74,"text":"the racemic solution with equal amount of the right and left isomers of the same","rect":[53.812843322753909,92.20748138427735,385.12680853082505,83.27293395996094]},{"page":74,"text":"molecule, the symmetry is Kh and the optical activity is absent.","rect":[53.812843322753909,104.11067962646485,309.5527004769016,95.17564392089844]},{"page":74,"text":"The nematic phase has point group symmetry D1h. If we add some amount of","rect":[65.7651138305664,116.0702133178711,385.15279517976418,107.13566589355469]},{"page":74,"text":"chiral, e.g., right-handed molecules, the symmetry is reduced from D1h to D1","rect":[53.81399154663086,128.02999877929688,385.17546902353959,119.09544372558594]},{"page":74,"text":"(symmetry of a twisted cylinder). Such a phase is called chiral nematic phase.","rect":[53.812843322753909,139.98959350585938,385.1815151741672,131.0351104736328]},{"page":74,"text":"Chiral molecules used as a dopant (solute) in nematic solvent considerably modify","rect":[53.812843322753909,151.94912719726563,385.11089411786568,143.0145721435547]},{"page":74,"text":"the nematic surrounding and the overall structure becomes twisted with a helical","rect":[53.812843322753909,163.90869140625,385.1089006192405,154.97413635253907]},{"page":74,"text":"pitch P0 incommensurate with a molecular size a, P0 6¼ na (n is an integer) and","rect":[53.812843322753909,175.86868286132813,385.14507380536568,166.60543823242188]},{"page":74,"text":"usually P0 \b a. Typically, a < 10 nm, P0 ¼ 0.1–10 mm.","rect":[53.81319808959961,187.82827758789063,283.0667995979953,178.89366149902345]},{"page":74,"text":"The pitch of the helix depends on concentration c of a dopant; for small c P0\u00041 \u0003","rect":[65.76636505126953,199.73104858398438,385.14807918760689,188.6801300048828]},{"page":74,"text":"ac and a is called helical twisting power of the dopant [15]. However, with","rect":[53.814231872558597,211.7306671142578,385.1182488541938,202.69650268554688]},{"page":74,"text":"increasing c the dependence becomes nonlinear and the helix handedness can","rect":[53.814231872558597,223.65036010742188,385.1271599870063,214.71580505371095]},{"page":74,"text":"even change sign (the case of cholesteryl chloride dopant in p-butoxybenzyli-","rect":[53.815208435058597,235.60992431640626,385.18193946687355,226.6753692626953]},{"page":74,"text":"dene-p0-butylaniline, BBBA, see Fig. 4.24). The same chiral, locally nematic","rect":[53.815208435058597,247.56985473632813,385.1016315288719,237.91017150878907]},{"page":74,"text":"phase with a short pitch in the range of 0.1–1 mm is traditionally called cholesteric","rect":[53.81255340576172,259.5294189453125,385.11353338434068,250.57493591308595]},{"page":74,"text":"phase because, at first, it has been found in cholesteryl esters. Such short-pitch","rect":[53.81354904174805,271.48895263671877,385.13750544599068,262.53448486328127]},{"page":74,"text":"phases manifest some properties of layered (smectic) phases.","rect":[53.81354904174805,283.448486328125,299.6826138069797,274.513916015625]},{"page":74,"text":"The smectic C* phase formed by chiral molecules (SmC* phase) has also a","rect":[65.76557159423828,295.35125732421877,385.15940130426255,286.41668701171877]},{"page":74,"text":"helical superstructure having a pitch incommensurate with the smectic layer thick-","rect":[53.81354904174805,307.3108215332031,385.10164771882668,298.37628173828127]},{"page":74,"text":"ness. Theoretically chiral phases can also be formed by achiral molecules due to","rect":[53.81354904174805,319.27032470703127,385.1404656510688,310.33575439453127]},{"page":74,"text":"very specific packing [16]. For instance, three achiral rod-like molecules of differ-","rect":[53.81354904174805,331.2298889160156,385.1135495742954,322.2554931640625]},{"page":74,"text":"ent length may form a chiral trimer or a tripod due to Van der Waals interactions","rect":[53.81254196166992,343.229248046875,385.1723672305222,334.2349548339844]},{"page":74,"text":"between their fragments, see Fig. 4.25a, and such trimers, in their turn, may form a","rect":[53.81254196166992,355.14898681640627,385.15833318902818,346.1546630859375]},{"page":74,"text":"kind of helical structure. Another example is bent-core or banana like-molecules [17]","rect":[53.81254196166992,367.1085205078125,385.1583493789829,358.1739501953125]},{"page":74,"text":"4.5","rect":[240.74392700195313,398.067138671875,251.85456326806406,392.30035400390627]},{"page":74,"text":"3.0","rect":[240.74392700195313,427.5616455078125,251.85456326806406,421.79486083984377]},{"page":74,"text":"1.5","rect":[240.74392700195313,455.95001220703127,251.85456326806406,450.1832275390625]},{"page":74,"text":"0","rect":[247.6453094482422,484.4967041015625,252.08956387841563,478.72991943359377]},{"page":74,"text":"–1.5","rect":[236.29966735839845,511.99005126953127,251.85456326806406,506.2232666015625]},{"page":74,"text":"Fig. 4.24 Helical twisting","rect":[53.812843322753909,547.8380126953125,145.18225616602823,540.1082153320313]},{"page":74,"text":"power of different cholesteric","rect":[53.812843322753909,557.7462158203125,155.18493029856206,550.15185546875]},{"page":74,"text":"dopants in the nematic phase","rect":[53.812843322753909,567.6654663085938,152.90889880442144,560.0711059570313]},{"page":74,"text":"as a function of the content of","rect":[53.812843322753909,575.888916015625,155.346527992317,570.0470581054688]},{"page":74,"text":"dopant","rect":[53.812843322753909,587.6173706054688,76.89464287009302,580.0230102539063]},{"page":74,"text":"–3.0","rect":[235.50033569335938,543.4025268554688,251.854548009275,537.6357421875]},{"page":74,"text":"–4.5","rect":[235.50033569335938,568.1029052734375,251.854548009275,562.3361206054688]},{"page":74,"text":"composition in BBBA","rect":[261.8616027832031,585.6671142578125,344.01637540634706,578.220703125]},{"page":75,"text":"4.7 Cholesterics","rect":[53.814517974853519,42.55667495727539,108.69386752127923,36.68099594116211]},{"page":75,"text":"Fig. 4.25 Hypothetical chiral","rect":[53.812843322753909,67.58130645751953,155.31522850241724,59.85148620605469]},{"page":75,"text":"trimers formed by rod like","rect":[53.812843322753909,77.4895248413086,144.35390612864019,69.89517211914063]},{"page":75,"text":"molecules due to specific Van","rect":[53.812843322753909,87.4087142944336,155.3042196670048,79.81436157226563]},{"page":75,"text":"der Waals interaction (a) and","rect":[53.812843322753909,97.04601287841797,153.9123739638798,89.79031372070313]},{"page":75,"text":"achiral bent-core molecules","rect":[53.812843322753909,105.60813903808594,148.22823031424799,99.76632690429688]},{"page":75,"text":"capable of formation chiral","rect":[53.812843322753909,117.33663177490235,146.8186159535891,109.74227905273438]},{"page":75,"text":"domains (b)","rect":[53.812843322753909,126.91716766357422,95.20608609778573,119.66146850585938]},{"page":75,"text":"a","rect":[76.1311264038086,170.48150634765626,81.68366632548754,164.89549255371095]},{"page":75,"text":"2","rect":[81.18316650390625,185.93478393554688,85.17844934718508,180.35385131835938]},{"page":75,"text":"c","rect":[211.31739807128907,170.48150634765626,216.869937992968,164.89549255371095]},{"page":75,"text":"a","rect":[223.23876953125,68.23225402832031,228.79347658347315,62.64405822753906]},{"page":75,"text":"w","rect":[232.59835815429688,187.01882934570313,238.37237958877663,182.83363342285157]},{"page":75,"text":"d","rect":[291.13616943359377,170.48150634765626,297.2379713978848,163.1767120361328]},{"page":75,"text":"b","rect":[301.632080078125,68.23225402832031,307.73626354738459,60.92461395263672]},{"page":75,"text":"y","rect":[309.0793151855469,135.87136840820313,313.0759468673575,130.5124969482422]},{"page":75,"text":"z","rect":[318.3770751953125,79.14659881591797,321.8781245485786,74.8755111694336]},{"page":75,"text":"57","rect":[376.7481689453125,42.55667495727539,385.2092337050907,36.63019943237305]},{"page":75,"text":"l","rect":[381.6681823730469,93.57638549804688,383.83435674458829,87.75362396240235]},{"page":75,"text":"φ","rect":[112.54804229736328,232.77146911621095,116.71159526831988,225.50265502929688]},{"page":75,"text":"0","rect":[167.49264526367188,242.1493377685547,171.93589948987509,236.3838653564453]},{"page":75,"text":"φ","rect":[195.4928436279297,242.55690002441407,199.6563965988863,235.2880859375]},{"page":75,"text":"0","rect":[240.1060333251953,242.1493377685547,244.54928755139853,236.3838653564453]},{"page":75,"text":"φ","rect":[267.9913635253906,242.55690002441407,272.1549164963472,235.2880859375]},{"page":75,"text":"0","rect":[307.83770751953127,242.1493377685547,312.28096174573445,236.3838653564453]},{"page":75,"text":"φ0","rect":[325.2949523925781,243.12364196777345,332.5660021207662,235.71670532226563]},{"page":75,"text":"φ","rect":[351.5948791503906,242.95750427246095,355.7584321213472,235.68869018554688]},{"page":75,"text":"Fig. 4.26 Interaction of two rod-like molecules, one molecule (1) on the top of the other (2) at an","rect":[53.812843322753909,264.6606750488281,385.15854400782509,256.93084716796877]},{"page":75,"text":"angle f (a). The forms of the interaction potential in different models: for achiral molecules","rect":[53.812843322753909,274.56890869140627,385.1187179851464,266.6951599121094]},{"page":75,"text":"harmonic (b), and anharmonic (c) and harmonic potential for chiral molecules (c)","rect":[53.812843322753909,284.4881286621094,333.45623868567636,276.8937683105469]},{"page":75,"text":"that can form a smectic multidomain system with right and left domains depending","rect":[53.812843322753909,309.4085998535156,385.10991755536568,300.47406005859377]},{"page":75,"text":"on the direction of the tilt of long molecular axes l with respect to the x,z-plane","rect":[53.812843322753909,321.3113708496094,385.1138385601219,312.3569030761719]},{"page":75,"text":"Fig. 4.25b. On the other hand, chiral molecules can be packed in such a way that the","rect":[53.812843322753909,333.2709045410156,385.17258489801255,324.2765808105469]},{"page":75,"text":"phase would lose their optical anisotropy, for example in the so-called blue phase","rect":[53.812843322753909,345.23046875,385.0969623394188,336.2958984375]},{"page":75,"text":"(see below) or optically isotropic SmC* phase.","rect":[53.812843322753909,357.19000244140627,243.00161405112034,348.25543212890627]},{"page":75,"text":"4.7 Cholesterics","rect":[53.812843322753909,400.81500244140627,143.1079318973796,392.07769775390627]},{"page":75,"text":"4.7.1 Intermolecular Potential","rect":[53.812843322753909,430.5899963378906,212.17653104613837,422.0558776855469]},{"page":75,"text":"Basically the structure of the molecules forming nematic and cholesteric phases is","rect":[53.812843322753909,460.2362365722656,385.18454374419408,451.30169677734377]},{"page":75,"text":"similar. However, chiral molecules possess a certain chiral asymmetry that results","rect":[53.812843322753909,472.19580078125,385.12582792388158,463.26123046875]},{"page":75,"text":"in asymmetry of intermolecular interactions. This asymmetry is weak and, there-","rect":[53.812843322753909,484.1553649902344,385.0919431289829,475.2208251953125]},{"page":75,"text":"fore, the helical pitch is much larger than a molecular size. Consider now an","rect":[53.812843322753909,496.1148986816406,385.1497429948188,487.18035888671877]},{"page":75,"text":"interaction potential V(f) between two rod-like molecules (1) and (2) as a function","rect":[53.812843322753909,508.0744323730469,385.16765681317818,498.8111877441406]},{"page":75,"text":"of the twist angle f between their long molecular axes, see Fig. 4.26a. Molecule (1)","rect":[53.81285095214844,520.033935546875,385.17852149812355,510.7707214355469]},{"page":75,"text":"is considered to be fixed. The twist corresponds to rotation of the longitudinal axis","rect":[53.81183624267578,531.9367065429688,385.15170683013158,523.002197265625]},{"page":75,"text":"of molecule (2) about the axis connecting gravity centers of the two molecules. For","rect":[53.81183624267578,543.896240234375,385.14571510163918,534.9617309570313]},{"page":75,"text":"achiral molecules, the two-particle potential curve W(f) is symmetric, Fig. 4.26b. It","rect":[53.81183624267578,555.8557739257813,385.1785112149436,546.592529296875]},{"page":75,"text":"may be described in terms of the Legendre polynomial P2 and order parameter S:","rect":[53.81181716918945,567.8164672851563,382.3764482266624,558.8807983398438]},{"page":75,"text":"W12ðfÞ ¼ \u0004vSP2ðcosfÞ","rect":[169.5395050048828,592.0743408203125,269.44186796294408,582.1238403320313]},{"page":75,"text":"(4.3)","rect":[366.0972595214844,591.3372802734375,385.16951881257668,582.8609008789063]},{"page":76,"text":"58","rect":[53.812843322753909,42.55594253540039,62.2739118972294,36.62946701049805]},{"page":76,"text":"4 Liquid Crystal Phases","rect":[303.6741638183594,44.274620056152347,385.1685226726464,36.68026351928711]},{"page":76,"text":"For chiral molecules the mirror symmetry is broken and such a curve cannot be","rect":[65.76496887207031,68.2883529663086,385.14783514215318,59.35380554199219]},{"page":76,"text":"symmetric. We can distinguish three cases:","rect":[53.812950134277347,80.24788665771485,228.07237107822489,71.31333923339844]},{"page":76,"text":"(i) The interaction is still harmonic but centered at a finite angle f0 6¼ 0 as in","rect":[59.760589599609378,98.1588363647461,382.63152400067818,88.89559173583985]},{"page":76,"text":"Fig. 4.26d:","rect":[74.32075500488281,110.1187973022461,118.12921769687722,101.18424987792969]},{"page":76,"text":"W12ðfÞ ¼ \u0004vSP2 cosðf \u0004 f0Þ:","rect":[156.85308837890626,133.35650634765626,282.17518555802249,123.4059829711914]},{"page":76,"text":"(4.4)","rect":[366.0981140136719,132.6194305419922,385.17037330476418,124.14305877685547]},{"page":76,"text":"This is a “classical” cholesteric with local nematic structure. The value of","rect":[74.32032775878906,153.85513305664063,385.14959083406105,146.98240661621095]},{"page":76,"text":"f0 determines the equilibrium pitch (a is the diameter of a rod-like molecule):","rect":[74.32032775878906,167.87655639648438,385.08882005283427,158.61331176757813]},{"page":76,"text":"P0 ¼ 2pa\u0002:f0","rect":[189.87478637695313,205.0354766845703,248.5875861414369,181.15025329589845]},{"page":76,"text":"(4.5)","rect":[366.09765625,197.34898376464845,385.1699155410923,188.75308227539063]},{"page":76,"text":"(ii) The potential is centered at f0 ¼ 0 but the interaction is anharmonic and cannot","rect":[56.986610412597659,227.56112670898438,385.25739915439677,218.29788208007813]},{"page":76,"text":"be described in terms of cylindrically symmetric functions. In this case, the","rect":[74.31871795654297,239.52108764648438,385.1578754253563,230.58653259277345]},{"page":76,"text":"equilibrium pitch is determined by an average fav shown in Fig. 4.26c.","rect":[74.31871795654297,251.4808349609375,356.1235012581516,242.2174072265625]},{"page":76,"text":"(iii) Both (i) and (ii) factors contribute to chirality together.","rect":[54.15357971191406,263.44036865234377,296.6172756722141,254.5058135986328]},{"page":76,"text":"Of course, in each case a particular form of the potential curve depends on","rect":[65.76615142822266,281.3512878417969,385.28237238935005,272.3968200683594]},{"page":76,"text":"chemical structure of constituting molecules. For instance, in nemato-cholesteric","rect":[53.81412887573242,293.31085205078127,385.2376178569969,284.37628173828127]},{"page":76,"text":"mixtures, V(f) depends on the structure of both a nematic matrix and a chiral dopant.","rect":[53.81412887573242,305.2703857421875,385.1838345101047,296.00714111328127]},{"page":76,"text":"4.7.2 Cholesteric Helix and Tensor of Orientational Order","rect":[53.812843322753909,347.1922302246094,351.56911353800458,336.63812255859377]},{"page":76,"text":"We can imagine a cholesteric as a stuck of nematic “quasi-layers” of molecular","rect":[53.812843322753909,374.8749694824219,385.1527036270298,365.9404296875]},{"page":76,"text":"thickness a with the director slightly turned by df from one layer to the next one. In","rect":[53.812843322753909,386.7777404785156,385.14968195966255,377.5144958496094]},{"page":76,"text":"fact it is Oseen model [18]. Such a structure is, to some extent, similar to lamellar","rect":[53.812862396240237,398.1296691894531,385.10295997468605,389.7828063964844]},{"page":76,"text":"phase. Indeed, the quasi-nematic layers behave like smectic layers in formation of","rect":[53.81187438964844,410.6968078613281,385.1457456192173,401.76226806640627]},{"page":76,"text":"defects, in flow experiments, etc. Then, according to the Landau–Peierls theorem,","rect":[53.81187438964844,422.6563720703125,385.1388210823703,413.7218017578125]},{"page":76,"text":"the fluctuations of molecular positions in the direction of the helical axis blur the","rect":[53.81187438964844,434.61590576171877,385.17163885309068,425.68133544921877]},{"page":76,"text":"one-dimensional, long-range, positional (smectic A phase like) helical order but in","rect":[53.81187438964844,446.5754699707031,385.1378106217719,437.64093017578127]},{"page":76,"text":"reality the corresponding scale for this effect is astronomic.","rect":[53.81187438964844,458.5350036621094,293.75017972494848,449.6004638671875]},{"page":76,"text":"In the first approximation, the parameter of the local orientational order of a","rect":[65.7638931274414,470.4945373535156,385.15967596246568,461.5400695800781]},{"page":76,"text":"cholesteric liquid crystal is the same uniaxial traceless tensor Qij ¼ Sðninj \u0004 dij=3Þ","rect":[53.81290054321289,483.27239990234377,385.16815580474096,472.7865295410156]},{"page":76,"text":"as in the nematic phase with the director axis always lying in the x,y-plane, e.g.","rect":[53.814353942871097,494.3579406738281,385.1472134163547,485.42340087890627]},{"page":76,"text":"along the x direction at a selected cross-section of the helix:","rect":[53.815330505371097,506.3175048828125,295.824110580178,497.3829345703125]},{"page":76,"text":"Q~ ¼ S0Bþ02=3 \u000401=3","rect":[157.47547912597657,546.5797119140625,241.8652733903266,519.5449829101563]},{"page":76,"text":"01","rect":[258.98052978515627,537.5535278320313,281.5098911222805,519.5449829101563]},{"page":76,"text":"0C","rect":[258.97955322265627,543.5756225585938,281.5098911222805,536.7228393554688]},{"page":76,"text":"@0","rect":[182.84774780273438,560.7963256835938,205.38608637860785,542.7877807617188]},{"page":76,"text":"0","rect":[229.69427490234376,557.292236328125,234.6713799577094,550.439453125]},{"page":76,"text":"\u00041=3 A","rect":[251.78562927246095,560.7963256835938,281.5098911222805,542.7877807617188]},{"page":76,"text":"In the helical structure this tensor, as well as the tensor of the dielectric","rect":[65.76232147216797,580.2381591796875,385.0725787944969,573.3355712890625]},{"page":76,"text":"anisotropy (ellipsoid) rotates upon the translation along the z-axis as shown in","rect":[53.810298919677737,594.2296142578125,385.14116755536568,585.2951049804688]},{"page":77,"text":"4.7 Cholesterics","rect":[53.81285095214844,42.55594253540039,108.69220431327142,36.68026351928711]},{"page":77,"text":"Fig. 4.27 Helical stricture of","rect":[53.812843322753909,67.58130645751953,155.1180886612623,59.85148620605469]},{"page":77,"text":"the ellipsoid of dielectric","rect":[53.812843322753909,77.4895248413086,139.31195971750737,69.89517211914063]},{"page":77,"text":"permittivity for a cholesteric","rect":[53.812843322753909,87.4087142944336,151.56020495676519,79.81436157226563]},{"page":77,"text":"liquid crystal (a very weak","rect":[53.812843322753909,97.3846664428711,145.32184356837198,89.79031372070313]},{"page":77,"text":"biaxiality is determined by","rect":[53.812843322753909,107.36067962646485,145.42931121973917,99.76632690429688]},{"page":77,"text":"component de3)","rect":[53.812843322753909,117.33663177490235,107.83763211829354,109.48828125]},{"page":77,"text":"z","rect":[279.3874816894531,89.05594635009766,282.8881373906546,84.78533172607422]},{"page":77,"text":"δε2","rect":[347.3797302246094,114.12471771240235,357.4141709245064,108.45972442626953]},{"page":77,"text":"59","rect":[376.74652099609377,42.62367248535156,385.20758575587197,36.62946701049805]},{"page":77,"text":"y","rect":[267.20550537109377,169.66297912597657,271.2016876784014,164.30471801757813]},{"page":77,"text":"n","rect":[381.2030029296875,150.96453857421876,385.19918523699519,147.165771484375]},{"page":77,"text":"x","rect":[374.21990966796877,167.6026153564453,378.6316949352364,163.85983276367188]},{"page":77,"text":"Fig. 4.27. Then, the components of the director are n ¼ (cosqz, sinqz, 0). In the","rect":[53.812843322753909,200.58148193359376,385.1735309429344,191.6469268798828]},{"page":77,"text":"uniaxial approximation, there are only two principal components of the local","rect":[53.812843322753909,212.54104614257813,385.1108842618186,203.6064910888672]},{"page":77,"text":"dielectric tensor, e|| and e⊥ and two refraction indices, n|| and n⊥ ¼ nz. As a rule","rect":[53.812843322753909,224.00302124023438,385.14771307184068,215.56602478027345]},{"page":77,"text":"n|| > n⊥, and a uniaxial cholesteric is locally optically positive. For the overall","rect":[53.813838958740237,236.44052124023438,385.15046556064677,227.50596618652345]},{"page":77,"text":"helical structure, one can introduce average refraction indices, one along the helical","rect":[53.81363296508789,248.41998291015626,385.205214095803,239.4854278564453]},{"page":77,"text":"axis nz ¼ n⊥, and the other perpendicular to it, nx,y2 ¼ (1/2)(n||2 + n⊥2). Thus the","rect":[53.81363296508789,261.30938720703127,385.1747211284813,249.3289031982422]},{"page":77,"text":"helical axis becomes the optical axis. As a rule, nz \u0003 1.5 and nx,y \u0003 1.6 and the","rect":[53.814022064208987,273.2691345214844,385.1741107769188,263.3450927734375]},{"page":77,"text":"overall helical structure is usually optically negative.","rect":[53.81338119506836,284.3387756347656,267.3042873909641,275.344482421875]},{"page":77,"text":"4.7.3 Tensor of Dielectric Anisotropy","rect":[53.812843322753909,322.4496765136719,245.915265879347,311.8118896484375]},{"page":77,"text":"~","rect":[168.12200927734376,340.0,173.0991143327094,338.0]},{"page":77,"text":"In general, however, tensor Qij is biaxial but the biaxiality is small, on the order of","rect":[53.812843322753909,350.8659973144531,385.14983497468605,340.977783203125]},{"page":77,"text":"x/P0 where x is the length corresponding to nematic correlations. This correlation","rect":[53.812984466552737,361.9516906738281,385.12661067060005,352.6982727050781]},{"page":77,"text":"length may be found, for example, from the light scattering in the isotropic phase","rect":[53.812618255615237,373.9112243652344,385.09473455621568,364.9766845703125]},{"page":77,"text":"close to the transition to the nematic phase. Then, at each point, that is locally, the","rect":[53.812618255615237,385.87078857421877,385.17334783746568,376.91632080078127]},{"page":77,"text":"anisotropic part of dielectric susceptibility tensor is biaxial and traceless de1 þ","rect":[53.812618255615237,397.830322265625,385.14807918760689,388.5969543457031]},{"page":77,"text":"de2 þ de3 ¼ 0 with de2 \u0003 de3.","rect":[53.814231872558597,409.3722839355469,176.7856411507297,400.5569152832031]},{"page":77,"text":"d~e ¼ 0@d00e1 d00e2 d00e3 1A ¼ d2e1 @0020 \u0004100þ Z \u0004100\u0004 Z1A","rect":[99.46929931640625,457.2973327636719,339.5158420500149,421.9971923828125]},{"page":77,"text":"(4.6)","rect":[366.0955810546875,443.91021728515627,385.1678403457798,435.37408447265627]},{"page":77,"text":"Here Z is a measure of biaxiality","rect":[65.7641372680664,478.8266906738281,199.21526423505316,469.89215087890627]},{"page":77,"text":"Z ¼ ð d e2 \u0004 d e3Þ= d e1 ¼ ð2 d e2 þ d e1Þ= d e1","rect":[115.78106689453125,502.1120300292969,322.7360999353822,492.1147155761719]},{"page":77,"text":"¼ ð2 d e2 þ d e1Þ þ 1:","rect":[125.18578338623047,514.02490234375,220.14862000626466,504.0743713378906]},{"page":77,"text":"(4.7)","rect":[366.09722900390627,513.287841796875,385.16948829499855,504.81146240234377]},{"page":77,"text":"Particularly, for Z ¼ 0 we return to the nematic tensor of dielectric anisotropy","rect":[65.76578521728516,536.5853271484375,385.1496514420844,527.6508178710938]},{"page":77,"text":"with factor 2/3 included in de1:","rect":[53.81376266479492,548.0472412109375,179.81326158115457,539.3115844726563]},{"page":77,"text":"d~^e ¼ de10@100 \u0004100=2 \u0004100=2A1","rect":[156.05752563476563,597.072265625,282.92975196212429,561.7723388671875]},{"page":77,"text":"(4.8)","rect":[366.09814453125,583.6286010742188,385.1704038223423,575.1522216796875]},{"page":78,"text":"60","rect":[53.812843322753909,42.55594253540039,62.2739118972294,36.68026351928711]},{"page":78,"text":"4 Liquid Crystal Phases","rect":[303.6741638183594,44.274620056152347,385.1685226726464,36.68026351928711]},{"page":78,"text":"To obtain the tensor of the cholesteric helical structure one should imagine that","rect":[65.76496887207031,68.2883529663086,385.1389299161155,59.35380554199219]},{"page":78,"text":"the local tensor rotates in the laboratory co-ordinate system, or, alternatively, to","rect":[53.812950134277347,80.24788665771485,385.1398858170844,71.31333923339844]},{"page":78,"text":"introduce a rotating co-ordinate system. In the latter case, one should make trans-","rect":[53.812950134277347,92.20748138427735,385.0970700821079,83.27293395996094]},{"page":78,"text":"formation","rect":[53.812950134277347,102.04837036132813,93.27840510419378,95.17564392089844]},{"page":78,"text":"d~e ¼ R~fd~eR~\u0004f1","rect":[190.1577606201172,128.51527404785157,248.36103890022597,115.0]},{"page":78,"text":"(4.9)","rect":[366.09765625,126.0444107055664,385.1699155410923,117.56804656982422]},{"page":78,"text":"~~","rect":[83.66587829589844,139.0,102.63378993085394,137.0]},{"page":78,"text":"where Rf;R\u0004f1 are the matrix of rotation about the z-axis and its inverse matrix,","rect":[53.81417465209961,151.2440948486328,385.18169827963598,138.57614135742188]},{"page":78,"text":"respectively. Both matrices are known from the textbooks on the matrix algebra:","rect":[53.81302261352539,161.01812744140626,379.4272599942405,152.0835723876953]},{"page":78,"text":"R~f ¼ 24csoin0sff \u0004cos0isnff 00135 R~f\u00041 ¼ 24\u0004cos0isnff csoin0sff 01035","rect":[82.81561279296875,208.52537536621095,332.0676510416241,173.2254638671875]},{"page":78,"text":"(4.10)","rect":[361.0542907714844,195.0817413330078,385.1036618789829,186.60537719726563]},{"page":78,"text":"Note that for our rotation matrix, which is antisymmetric, the inverse matrix is","rect":[65.76465606689453,226.37057495117188,385.18536771880346,217.43601989746095]},{"page":78,"text":"equal to the transposed one. Now using Eqs. (4.9) and (4.10) we write","rect":[53.8126335144043,238.27334594726563,336.8685687847313,229.3387908935547]},{"page":78,"text":"\u0003\u0003cosf","rect":[128.1309051513672,262.7133483886719,153.22645827945019,250.37728881835938]},{"page":78,"text":"d~e¼R~fd~eR~f\u00041 ¼\u0003\u0003sinf","rect":[60.610355377197269,275.1484680175781,152.43409988101269,262.0131530761719]},{"page":78,"text":"\u0003\u0003 0","rect":[128.1309051513672,286.34857177734377,145.2869805436469,274.01251220703127]},{"page":78,"text":"\u0004sinf","rect":[159.2288360595703,261.0270080566406,187.15738174136426,251.85342407226563]},{"page":78,"text":"cosf","rect":[162.74168395996095,272.986572265625,183.64552566226269,263.81298828125]},{"page":78,"text":"0","rect":[170.7279510498047,283.0137939453125,175.70505610517035,276.1609802246094]},{"page":78,"text":"001\u0003\u0003\u0003\u0003\u0003\u0003\u0007de0@010","rect":[193.15975952148438,286.3475646972656,231.78308955243598,250.37628173828126]},{"page":78,"text":"0","rect":[246.57406616210938,259.09271240234377,251.55117121747504,252.23989868164063]},{"page":78,"text":"\u00041=2","rect":[237.73770141601563,273.4029235839844,260.38852778485787,263.47235107421877]},{"page":78,"text":"0","rect":[246.57504272460938,283.0118103027344,251.55214777997504,276.15899658203127]},{"page":78,"text":"\u0004100=2A1\u0007\u0003\u0003\u0003\u0003\u0003\u0003\u0004cos0isnff","rect":[266.4008483886719,286.3465576171875,336.24754592593458,250.37530517578126]},{"page":78,"text":"sinf","rect":[343.0980529785156,261.0250244140625,362.3594233673408,251.8514404296875]},{"page":78,"text":"cosf","rect":[342.24993896484377,272.9845886230469,363.2104975860908,263.8110046386719]},{"page":78,"text":"0","rect":[350.2362060546875,283.0118103027344,355.21331111005318,276.15899658203127]},{"page":78,"text":"0\u0003\u0003","rect":[369.1551513671875,262.7103576660156,378.365778116593,250.37429809570313]},{"page":78,"text":"0\u0003\u0003","rect":[369.1561584472656,274.49957275390627,378.365778116593,262.2203063964844]},{"page":78,"text":"1\u0003\u0003","rect":[369.1551513671875,286.3455810546875,378.365778116593,274.009521484375]},{"page":78,"text":"and then multiply the dielectric tensor first by the inverse matrix on the right and","rect":[53.813716888427737,309.8582763671875,385.1426934342719,300.9237060546875]},{"page":78,"text":"then multiply the rotation matrix from the left side by the result of the first","rect":[53.813716888427737,321.76104736328127,385.15754563877177,312.82647705078127]},{"page":78,"text":"operation. Next, we obtain the tensor of dielectric anisotropy of a locally uniaxial","rect":[53.813716888427737,333.7205810546875,385.16352708408427,324.7461853027344]},{"page":78,"text":"cholesteric.","rect":[53.813716888427737,343.747802734375,100.25906796957736,336.7256774902344]},{"page":78,"text":"d~e ¼ d4e0@1 þ3s3i0nco2sf2f","rect":[125.69606018066406,393.2474060058594,219.95447036441113,357.94744873046877]},{"page":78,"text":"3 sin 2f","rect":[239.32537841796876,368.1529235839844,270.8782221954658,358.9793395996094]},{"page":78,"text":"1 \u0004 3cos2f","rect":[229.921630859375,380.11248779296877,280.28196975405958,370.93890380859377]},{"page":78,"text":"0","rect":[252.63711547851563,390.13970947265627,257.6142205338813,383.2868957519531]},{"page":78,"text":"00 1","rect":[294.10040283203127,378.18017578125,313.2911106047024,357.94744873046877]},{"page":78,"text":"\u00042 A","rect":[290.2500915527344,393.2474060058594,313.2911106047024,375.2388610839844]},{"page":78,"text":"(4.11)","rect":[361.056396484375,379.8037109375,385.10576759187355,371.3273620605469]},{"page":78,"text":"~","rect":[350.6900634765625,407.0,355.66716853192818,405.0]},{"page":78,"text":"Finally, we can write the tensors of the orientational order parameter Qij in the","rect":[65.7667465209961,417.57940673828127,385.17575872613755,407.6900634765625]},{"page":78,"text":"rotating frame for locally uniaxial and biaxial cholesteric liquid crystal (ChLC):","rect":[53.814022064208987,428.6649475097656,376.624159408303,419.73040771484377]},{"page":78,"text":"Uniaxial ChLC:","rect":[65.76604461669922,438.6025390625,130.18272264072489,431.68994140625]},{"page":78,"text":"Q~uijni ¼ 83S0@1 þ3s3i0nco2sf2f","rect":[121.73160552978516,487.16748046875,223.91968032534863,451.8675231933594]},{"page":78,"text":"3 sin 2f","rect":[243.29055786132813,462.0730285644531,274.8434321564033,452.8994445800781]},{"page":78,"text":"1 \u0004 3cos2f","rect":[233.88682556152345,474.0325927734375,284.18944045718458,464.8590087890625]},{"page":78,"text":"0","rect":[256.5445861816406,484.059814453125,261.5216912370063,477.2070007324219]},{"page":78,"text":"00 1","rect":[298.00885009765627,472.10028076171877,317.1985813078274,451.8675231933594]},{"page":78,"text":"\u00042 A","rect":[294.1575622558594,487.16748046875,317.1985813078274,469.158935546875]},{"page":78,"text":"(4.12)","rect":[361.05487060546877,473.72381591796877,385.1042417129673,465.2474670410156]},{"page":78,"text":"Biaxial ChLC","rect":[65.76521301269531,508.6024475097656,121.89102431948925,501.68988037109377]},{"page":78,"text":"Q~biji ¼ 83S@01 þ Zð3þ\u0004ð3Z0Þ\u0004siZnÞ2cfos2f","rect":[66.44508361816406,557.22509765625,210.3241908233955,521.868408203125]},{"page":78,"text":"ð3 \u0004 ZÞsin2f","rect":[239.0410919189453,532.5590209960938,297.0472895782783,522.6085205078125]},{"page":78,"text":"1 þ Z \u0004 ð3 \u0004 ZÞcos2f","rect":[220.2923583984375,544.5185546875,315.7980220001533,534.5680541992188]},{"page":78,"text":"0","rect":[265.5521240234375,554.117431640625,270.52922907880318,547.2646484375]},{"page":78,"text":"0","rect":[341.51171875,530.1983642578125,346.48882380536568,523.3455810546875]},{"page":78,"text":"0","rect":[341.5127258300781,542.1578979492188,346.4898308854438,535.3051147460938]},{"page":78,"text":"1","rect":[363.8310546875,539.876953125,372.5409885343899,521.868408203125]},{"page":78,"text":"\u00042 \u0004 2ZA","rect":[325.76617431640627,557.22509765625,372.5409885343899,539.216552734375]},{"page":78,"text":"(4.13)","rect":[361.05584716796877,570.4785766601563,385.1052182754673,562.002197265625]},{"page":78,"text":"with Z defined by Eq. (4.7)","rect":[53.81417465209961,593.9464111328125,164.24418006745948,585.0119018554688]},{"page":79,"text":"4.7 Cholesterics","rect":[53.81283950805664,42.55875015258789,108.69218905448236,36.68307113647461]},{"page":79,"text":"4.7.4 Grandjean Texture","rect":[53.812843322753909,69.85308837890625,184.56677039594858,59.298980712890628]},{"page":79,"text":"61","rect":[376.7464904785156,42.55875015258789,385.2075552382938,36.68307113647461]},{"page":79,"text":"This interesting texture is observed in the so-called Cano wedges formed by two","rect":[53.812843322753909,97.47896575927735,385.15068903974068,88.54441833496094]},{"page":79,"text":"optically polished glasses with a gap filled by a cholesteric liquid crystal (CLC). Let","rect":[53.812843322753909,109.43856048583985,385.1158891446311,100.50401306152344]},{"page":79,"text":"the equilibrium helical pitch of the CLC in a bulky sample is P0. In the wedge the","rect":[53.812843322753909,121.3984603881836,385.1756976909813,112.46354675292969]},{"page":79,"text":"molecules are oriented along its acute edge. Since the boundary condition are fixed","rect":[53.81399154663086,133.35800170898438,385.14287653974068,124.42344665527344]},{"page":79,"text":"the equilibrium pitch can only be undistorted when the layer thickness is exactly","rect":[53.81399154663086,145.31753540039063,385.1767205338813,136.3829803466797]},{"page":79,"text":"equal to d0 ¼ mP0/2 where m is an integer as shown in Fig. 4.28a. In the close","rect":[53.81399154663086,157.27734375,385.12964666559068,147.6177520751953]},{"page":79,"text":"proximity of each d’ value, the helix can still fit to the boundary conditions at the","rect":[53.812747955322269,169.23687744140626,385.17249334527818,160.2823944091797]},{"page":79,"text":"cost of some pitch compression or dilatation. Therefore rather large areas form with","rect":[53.812747955322269,181.1396484375,385.11480036786568,172.20509338378907]},{"page":79,"text":"the same number of half-turns within the gap, which are marked by numbers m ¼ 0,","rect":[53.812747955322269,193.09921264648438,385.17449613119848,184.16465759277345]},{"page":79,"text":"1, 2, 3 in Fig. 4.28b. These are Grandjean zones separated by the defects called","rect":[53.81278610229492,205.05874633789063,385.17250910810005,196.0943145751953]},{"page":79,"text":"disclinations (thin lines seen in the photo, Fig. 4.28c). At each disclination, the","rect":[53.81178665161133,217.01828002929688,385.1725543804344,208.0637969970703]},{"page":79,"text":"number of half-turns changes usually by one. In the zero zone, the cholesteric is","rect":[53.811763763427737,228.97781372070313,385.18246854888158,220.0432586669922]},{"page":79,"text":"unwound but its properties (e.g., elastic moduli) in this quasi-nematic area are","rect":[53.811763763427737,240.93734741210938,385.1595844097313,232.00279235839845]},{"page":79,"text":"different from the corresponding achiral nematic. Grandjean textures are very","rect":[53.811763763427737,252.89691162109376,385.11184016278755,243.9623565673828]},{"page":79,"text":"a","rect":[200.43287658691407,282.5994873046875,205.9881389841479,277.0107421875]},{"page":79,"text":"Fig. 4.28 A wedge type cell","rect":[53.812843322753909,462.4766845703125,153.1390505239016,454.5267333984375]},{"page":79,"text":"filled with a cholesteric (a)","rect":[53.812843322753909,472.0462646484375,146.29910367591075,464.7905578613281]},{"page":79,"text":"with Grandjean zones marked","rect":[53.812843322753909,482.3608703613281,155.3380636611454,474.7665100097656]},{"page":79,"text":"by numbers 0, 1, 2, 3, 4 ...","rect":[53.812843322753909,492.2801208496094,147.35335180133493,484.6857604980469]},{"page":79,"text":"and the disclination lines","rect":[53.812843322753909,500.5289306640625,139.28574069022455,494.6617126464844]},{"page":79,"text":"shown by arrows. The","rect":[53.812843322753909,512.2319946289063,129.72726580881597,504.6376647949219]},{"page":79,"text":"distance dependence of the","rect":[53.812843322753909,522.1511840820313,146.3380217292261,514.5568237304688]},{"page":79,"text":"helix pitch in different zones","rect":[53.812843322753909,532.1271362304688,152.45538790702143,524.5327758789063]},{"page":79,"text":"with numbers m is","rect":[53.812843322753909,540.435302734375,116.88588412284173,534.5087890625]},{"page":79,"text":"schematically shown in","rect":[53.812843322753909,552.0791015625,134.00348419337198,544.4847412109375]},{"page":79,"text":"sketch (b). Photo of","rect":[53.812843322753909,561.65966796875,121.81106656653573,554.4039306640625]},{"page":79,"text":"disclinations limiting the","rect":[53.81283950805664,571.9742431640625,138.86184069895269,564.3798828125]},{"page":79,"text":"Grandjean zones (c)","rect":[53.81283950805664,581.9502563476563,122.59373563391854,574.3558959960938]},{"page":79,"text":"b","rect":[200.43287658691407,357.8177185058594,206.53767033639228,350.50933837890627]},{"page":79,"text":"P","rect":[213.41543579101563,365.5833740234375,218.74694245455104,359.8406066894531]},{"page":79,"text":"3P0/2","rect":[200.36483764648438,394.6832275390625,220.7395577749,386.6810607910156]},{"page":79,"text":"P0","rect":[211.7752227783203,407.0105895996094,220.4399107933332,399.1602783203125]},{"page":79,"text":"P0/2","rect":[204.8079833984375,420.89654541015627,220.7395577749,412.8942565917969]},{"page":79,"text":"c","rect":[200.43287658691407,464.3859558105469,205.9881389841479,458.7972106933594]},{"page":79,"text":"m","rect":[223.4929656982422,372.5276794433594,229.71072543721909,368.70648193359377]},{"page":79,"text":"=0","rect":[231.65509033203126,372.6536560058594,239.62906381292519,367.60772705078127]},{"page":79,"text":"m = 1","rect":[248.736083984375,371.8236389160156,266.8158466498393,366.9036865234375]},{"page":79,"text":"P0/2","rect":[251.87171936035157,442.7817687988281,267.80167630517345,434.77947998046877]},{"page":79,"text":"m = 2","rect":[282.2784118652344,372.21343994140627,300.35817453069867,367.2934875488281]},{"page":79,"text":"P0","rect":[289.1972351074219,442.7817687988281,297.8620604515363,434.93145751953127]},{"page":79,"text":"m = 3","rect":[315.9186706542969,371.3631286621094,333.99843331976117,366.31719970703127]},{"page":79,"text":"3P0/2","rect":[316.4405212402344,442.7817687988281,336.81351712548595,434.77947998046877]},{"page":79,"text":"m = 4","rect":[355.8626708984375,371.23016357421877,373.9424640814799,366.3102111816406]},{"page":79,"text":"2P0 d","rect":[351.54449462890627,442.7817687988281,379.9749551137672,434.9304504394531]},{"page":80,"text":"62","rect":[53.81307601928711,42.55740737915039,62.2741445937626,36.68172836303711]},{"page":80,"text":"4 Liquid Crystal Phases","rect":[303.67437744140627,44.276084899902347,385.1687668132714,36.68172836303711]},{"page":80,"text":"useful as an experimental tool: one can study and compare different physical,","rect":[53.812843322753909,68.2883529663086,385.1805386116672,59.35380554199219]},{"page":80,"text":"especially electrooptical effects at different thickness under the same conditions","rect":[53.812843322753909,80.24788665771485,385.1079141055222,71.31333923339844]},{"page":80,"text":"(same material, alignment, ambient conditions, fields, etc.) One can also observe","rect":[53.812843322753909,92.20748138427735,385.11887396051255,83.25301361083985]},{"page":80,"text":"the Grandjean zones on preparations without cover slips because a drop of a CLC","rect":[53.812843322753909,104.11019134521485,385.12483474429396,95.17564392089844]},{"page":80,"text":"has an edge with decreasing thickness. Since the free surface can also align liquid","rect":[53.812843322753909,116.0697250366211,385.1109246354438,107.13517761230469]},{"page":80,"text":"crystals the Grandjean zones form in this natural wedge at the border of the","rect":[53.812843322753909,128.02932739257813,385.11484564020005,119.09477233886719]},{"page":80,"text":"preparation. By the way, in smectics such zones exist in a form of microscopic","rect":[53.812843322753909,139.98886108398438,385.1198200054344,131.05430603027345]},{"page":80,"text":"steps, which could be measured by an atomic force microscope (AFM).","rect":[53.812843322753909,151.94839477539063,342.6432766487766,143.0138397216797]},{"page":80,"text":"4.7.5 Methods of the Pitch Measurements","rect":[53.812843322753909,202.03268432617188,269.13402442667646,191.47857666015626]},{"page":80,"text":"Due to its periodic structure cholesteric liquid crystals manifest very interesting","rect":[53.812843322753909,229.65866088867188,385.0910271745063,220.72410583496095]},{"page":80,"text":"optical properties. In fact, a cholesteric is one-dimensional photonic crystal having","rect":[53.812843322753909,241.61822509765626,385.14971247724068,232.6836700439453]},{"page":80,"text":"forbidden frequency bands (stop-bands) for a particular circular polarization. This","rect":[53.812843322753909,253.52096557617188,385.18658842192846,244.58641052246095]},{"page":80,"text":"band appears due to the Bragg diffraction of light on the helical structure. In the","rect":[53.812843322753909,265.48052978515627,385.17261541559068,256.54595947265627]},{"page":80,"text":"vicinity of the stop band a giant optical rotation of light is observed. Since the pitch","rect":[53.812843322753909,277.4400634765625,385.12981501630318,268.5054931640625]},{"page":80,"text":"of the helix can easily be changes by external factors such as composition, temper-","rect":[53.812843322753909,289.3996276855469,385.10393653718605,280.465087890625]},{"page":80,"text":"ature, UV light, mechanical tension, electric and magnetic field, a variety of tunable","rect":[53.812843322753909,301.3591613769531,385.1705707378563,292.42462158203127]},{"page":80,"text":"optical devices (like filters and lasers) has been suggested. We shall discuss the","rect":[53.812843322753909,313.3186950683594,385.17160833551255,304.3841552734375]},{"page":80,"text":"optical properties of cholesterics in detail in Chapter 12.","rect":[53.812843322753909,325.2782287597656,281.0684475472141,316.34368896484377]},{"page":80,"text":"The key parameter for the tunability is the helical pitch P0, which can be found","rect":[65.7648696899414,337.23779296875,385.1287469010688,328.30322265625]},{"page":80,"text":"from the measurements of","rect":[53.81380844116211,347.07977294921877,159.42102943269385,340.20703125]},{"page":80,"text":"(i)","rect":[59.76144790649414,366.71087646484377,69.18808112947119,358.17474365234377]},{"page":80,"text":"(ii)","rect":[56.98566436767578,390.57354736328127,69.20943580476416,382.03741455078127]},{"page":80,"text":"The wedge thickness in the centers of Grandjean zones, P0 ¼ 2d0/m, as shown","rect":[74.3194808959961,367.10931396484377,385.17897883466255,357.4499816894531]},{"page":80,"text":"in Fig. 4.28.","rect":[74.31892395019531,379.0692138671875,123.42605252768283,370.1346435546875]},{"page":80,"text":"The angular position of diffraction spots for the light incident perpendicularly","rect":[74.31893157958985,390.97198486328127,385.16799250653755,382.03741455078127]},{"page":80,"text":"to the helical axis Fig. 4.29a. Such a texture is formed by the so-called","rect":[74.31895446777344,402.9315490722656,385.0904168229438,393.99700927734377]},{"page":80,"text":"homeotropic boundary conditions with liquid crystal molecules oriented per-","rect":[74.31794738769531,414.8910827636719,385.15117774812355,405.95654296875]},{"page":80,"text":"pendicularly to the plane glasses. Due to the head-to-tail symmetry the period","rect":[74.31794738769531,426.8506164550781,385.12725153974068,417.91607666015627]},{"page":80,"text":"of the optical properties is P0/2 and wavevector of the optical structure q0 ¼","rect":[74.31794738769531,438.8108825683594,385.14807918760689,429.8756103515625]},{"page":80,"text":"4p/P0. The diffraction spots are located at angles \u00062y symmetric with respect","rect":[74.31993103027344,450.7705383300781,385.1560502774436,441.5272216796875]},{"page":80,"text":"k0","rect":[126.62227630615235,519.565673828125,133.9522093773176,511.7153625488281]},{"page":80,"text":"IT’ k0","rect":[295.01678466796877,520.91552734375,312.82766714098946,513.065185546875]},{"page":80,"text":"Fig. 4.29 Measurements of the pitch of the helix in a cholesteric. (a) Geometry for monochro-","rect":[53.812843322753909,554.1295776367188,385.1483163223951,546.094970703125]},{"page":80,"text":"matic light diffraction on the focal-conic texture. The pitch is found from the angle 2# between the","rect":[53.81281280517578,564.0377807617188,385.1542601325464,556.2572021484375]},{"page":80,"text":"incident and diffracted beams with wavevectors k0 and k; (b) Spectral measurements of the light","rect":[53.812843322753909,573.9567260742188,385.2020540639407,566.345458984375]},{"page":80,"text":"transmission by a planar cholesteric texture (I0 and IT are intensities of the incident and transmitted","rect":[53.81240463256836,583.9326782226563,385.1705374160282,576.3382568359375]},{"page":80,"text":"beam)","rect":[53.81307601928711,593.5699462890625,75.03344052893807,586.314208984375]},{"page":81,"text":"4.8 Blue Phases","rect":[53.812843322753909,42.55630874633789,108.17437441825189,36.68062973022461]},{"page":81,"text":"Fig. 4.30 A fingerprint","rect":[53.812843322753909,67.58130645751953,135.12035850730005,59.648292541503909]},{"page":81,"text":"texture of a cholesteric liquid","rect":[53.812843322753909,77.4895248413086,154.41241974024698,69.89517211914063]},{"page":81,"text":"crystal seen in a polarization","rect":[53.812843322753909,87.4087142944336,152.01709503321573,79.81436157226563]},{"page":81,"text":"microscope (the distance","rect":[53.812843322753909,97.3846664428711,138.73153064035894,89.79031372070313]},{"page":81,"text":"between stripes equalsa","rect":[53.812843322753909,107.36067962646485,136.78548571848394,99.76632690429688]},{"page":81,"text":"half-pitch)","rect":[53.812843322753909,117.33663177490235,89.6539315567701,109.74227905273438]},{"page":81,"text":"63","rect":[376.7464904785156,42.55630874633789,385.2075552382938,36.68062973022461]},{"page":81,"text":"(iii)","rect":[54.15400314331055,283.0502624511719,69.25055824128759,274.51416015625]},{"page":81,"text":"(iv)","rect":[54.776756286621097,402.5900573730469,69.25017677156103,394.053955078125]},{"page":81,"text":"to the incident beam (zero order diffraction). The modulus of the scattering","rect":[74.31846618652344,235.61026000976563,385.16265193036568,226.6757049560547]},{"page":81,"text":"wavevector is q ¼ 2k0msinym=2 where incident light vector k0 ¼ o/c. The","rect":[74.31846618652344,247.89854431152345,385.14685857965318,237.9679412841797]},{"page":81,"text":"first order diffraction (m ¼ 1) is very intense and, using angle pm¼1 and the","rect":[74.31867980957031,259.5294189453125,385.17490423395005,250.59486389160157]},{"page":81,"text":"wavevector conservation law q ¼ q0, the pitch can be found P0 ¼ l0=msin#.","rect":[74.31986999511719,271.8178405761719,385.1832546761203,261.88726806640627]},{"page":81,"text":"The spectral position of the selective reflection or transmission band l0 in the","rect":[74.31924438476563,283.4486999511719,385.1756976909813,274.1954040527344]},{"page":81,"text":"planar texture formed by the homogeneous, planar boundary conditions with","rect":[74.31965637207031,295.4083557128906,385.11602107099068,286.47381591796877]},{"page":81,"text":"molecules oriented parallel to the glasses, Fig. 4.29b. In this case, we may use","rect":[74.31965637207031,307.3678894042969,385.13391912652818,298.433349609375]},{"page":81,"text":"unpolarised light and the Bragg condition for one of the circular polarizations","rect":[74.31962585449219,319.32745361328127,385.1727639590378,310.39288330078127]},{"page":81,"text":"ml ¼ ml0=hni ¼ 2ðP0=2Þsin#0;m ¼ 1;2;3:::","rect":[126.2626953125,343.529052734375,312.7021936146631,333.5785217285156]},{"page":81,"text":"(4.14)","rect":[361.0577392578125,342.7919616699219,385.10711036531105,334.31561279296877]},{"page":81,"text":"with incident angle #0 ¼ p/2 and m ¼ 1. Therefore, P0 ¼ l0=<n> where <n>","rect":[74.32075500488281,367.4383544921875,385.14890316221627,357.5077819824219]},{"page":81,"text":"is related to the two principal refraction indices n|| and n⊥ parallel and","rect":[74.32077026367188,379.0693359375,385.14592829755318,370.1346435546875]},{"page":81,"text":"perpendicular to the director: <n> ¼ (n|| + n⊥)/2.","rect":[74.31971740722656,391.0289001464844,275.10821195151098,382.0943603515625]},{"page":81,"text":"The distance between stripes observed under a polarization microscope in the","rect":[74.31886291503906,402.9884948730469,385.1740192241844,394.053955078125]},{"page":81,"text":"fingerprint texture, shown in Fig. 4.30. Again due to the head-to-tail symmetry","rect":[74.31884765625,414.9878845214844,385.17192927411568,405.9736633300781]},{"page":81,"text":"the distance between stripes equals a half-pitch.","rect":[74.31884765625,426.8507995605469,266.9626736214328,417.916259765625]},{"page":81,"text":"4.8 Blue Phases","rect":[53.812843322753909,474.9744567871094,141.77965430948897,465.9742126464844]},{"page":81,"text":"These phases were an enigma of the centuries. Since the experiments of Reinitzer","rect":[53.812843322753909,504.5606994628906,385.08297096101418,495.62615966796877]},{"page":81,"text":"[19] up to recent times it was not clear whether it was a special texture of the known","rect":[53.812843322753909,516.5202026367188,385.17461482099068,507.585693359375]},{"page":81,"text":"cholesteric phase or a thermodynamically new phase. The textures of the blue","rect":[53.81184768676758,528.479736328125,385.1059039898094,519.5452270507813]},{"page":81,"text":"phases are often of blue color, Fig. 4.31. Properties of the blue phases are very","rect":[53.81184768676758,540.4392700195313,385.11480036786568,531.5047607421875]},{"page":81,"text":"interesting from the fundamental point of view.","rect":[53.81185531616211,552.3988037109375,245.3209194710422,543.4642944335938]},{"page":81,"text":"(i) TherearethreebluephasesBPI,BPIIandBPIII(orfoggy)phase[20].Allblue","rect":[68.25640869140625,570.3097534179688,385.26511419488755,561.375244140625]},{"page":81,"text":"phases are usually observed in rather a narrow temperature interval between","rect":[82.81443786621094,582.269287109375,385.17363825849068,573.3347778320313]},{"page":81,"text":"the isotropic phase and cholesteric (Ch) phase. Recently, however, a wide","rect":[82.81443786621094,594.2288208007813,385.26026189996568,585.2943115234375]},{"page":82,"text":"64","rect":[53.81284713745117,42.55728530883789,62.27391571192666,36.68160629272461]},{"page":82,"text":"Fig. 4.31 A texture of blue","rect":[53.812843322753909,67.58130645751953,149.37894580149175,59.648292541503909]},{"page":82,"text":"phase BPI","rect":[53.812843322753909,77.4895248413086,88.63099759680917,69.89517211914063]},{"page":82,"text":"4 Liquid Crystal Phases","rect":[303.6741638183594,44.275962829589847,385.1685226726464,36.68160629272461]},{"page":82,"text":"Fig. 4.32 A phase diagram","rect":[53.812843322753909,257.2921447753906,148.82220695848452,249.359130859375]},{"page":82,"text":"showing phase transition lines","rect":[53.812843322753909,267.2004089355469,155.3160751628808,259.6060485839844]},{"page":82,"text":"between the isotropic, BPIII,","rect":[53.812843322753909,277.1763610839844,152.29040786817036,269.5820007324219]},{"page":82,"text":"BPII, BP1 and cholesteric","rect":[53.812843322753909,285.3684387207031,142.61092517160894,279.501220703125]},{"page":82,"text":"phase: x is percentage of the","rect":[53.812843322753909,297.071533203125,151.0398497321558,289.4771728515625]},{"page":82,"text":"racemic component ina","rect":[53.81284713745117,307.0474853515625,135.9918303229761,299.453125]},{"page":82,"text":"racemic-chiral mixture","rect":[53.81284713745117,315.21417236328127,131.7841582038355,309.37237548828127]},{"page":82,"text":"0","rect":[217.34352111816407,258.0893249511719,221.78760889184336,252.32278442382813]},{"page":82,"text":"0.2","rect":[210.7061767578125,276.32073974609377,221.81640222680429,270.55419921875]},{"page":82,"text":"0.4","rect":[211.02908325195313,294.5561828613281,222.1393087209449,288.7896423339844]},{"page":82,"text":"0.5","rect":[210.6750030517578,312.78839111328127,221.7852285207496,307.0218505859375]},{"page":82,"text":"0.6","rect":[210.7141571044922,331.0206298828125,221.82438257348398,325.25408935546877]},{"page":82,"text":"1.0","rect":[210.77731323242188,349.2552795410156,221.88753870141367,343.4887390136719]},{"page":82,"text":"1.2","rect":[210.7061767578125,367.34271240234377,221.81640222680429,361.7201232910156]},{"page":82,"text":"0","rect":[223.16720581054688,373.898681640625,227.61129358422617,368.13214111328127]},{"page":82,"text":"BPIII","rect":[242.62094116210938,273.3041076660156,259.95509917280989,267.55975341796877]},{"page":82,"text":"BPII","rect":[240.84274291992188,315.01947021484377,255.95416838667709,309.2751159667969]},{"page":82,"text":"20","rect":[257.80072021484377,373.898681640625,266.688899785398,368.13214111328127]},{"page":82,"text":"ISO","rect":[292.2311706542969,252.44522094726563,306.00736489960658,246.40484619140626]},{"page":82,"text":"BPI","rect":[292.53662109375,281.1797790527344,305.42531401655989,275.4354248046875]},{"page":82,"text":"40","rect":[294.65545654296877,373.898681640625,303.543636113523,368.13214111328127]},{"page":82,"text":"Ch","rect":[309.15838623046877,315.1634826660156,319.37659849210419,309.12310791015627]},{"page":82,"text":"(ii)","rect":[65.42716979980469,473.6107482910156,77.65094123689306,465.07464599609377]},{"page":82,"text":"(iii)","rect":[62.651939392089847,485.5135192871094,77.67284522859228,476.9774169921875]},{"page":82,"text":"(iv)","rect":[63.27505874633789,497.4730529785156,77.67880378572119,488.93695068359377]},{"page":82,"text":"(v)","rect":[66.05029296875,509.4325866699219,77.65689979402197,500.95623779296877]},{"page":82,"text":"(vi)","rect":[63.275054931640628,533.3517456054688,77.67879615632666,524.8156127929688]},{"page":82,"text":"(vii)","rect":[60.44209289550781,581.1331176757813,77.71264777986181,572.5969848632813]},{"page":82,"text":"temperature blue phases have been prepared using stabilization of the helical","rect":[82.8152084350586,414.2114562988281,385.2361284024436,405.27691650390627]},{"page":82,"text":"structure by a polymer network [21]. A typical example of a phase diagram is","rect":[82.8152084350586,426.1709899902344,385.2449990664597,417.2364501953125]},{"page":82,"text":"shown in Fig. 4.32 [20]. In the diagram, the abscissa is a percentage of the","rect":[82.81619262695313,438.13055419921877,385.15943182184068,429.19598388671877]},{"page":82,"text":"racemic component in a mixture with a chiral component of the same com-","rect":[82.81717681884766,450.090087890625,385.1604245742954,441.155517578125]},{"page":82,"text":"pound.","rect":[82.81717681884766,462.04962158203127,109.82593961264377,453.11505126953127]},{"page":82,"text":"X-ray diffraction shows that local order is liquid-like.","rect":[82.81717681884766,474.0091857910156,299.23467679526098,465.07464599609377]},{"page":82,"text":"Drops of a BP1 show facets typical of solid crystals seen in Fig. 4.31.","rect":[82.81717681884766,485.9119567871094,364.5093960335422,476.9774169921875]},{"page":82,"text":"Blue phases strongly rotate the light polarization plane.","rect":[82.81716918945313,497.8714904785156,306.0622829964328,488.93695068359377]},{"page":82,"text":"Despite properties (iii) and (iv) the blue phases do not show any birefrin-","rect":[82.81716918945313,509.8310241699219,378.28106056062355,500.896484375]},{"page":82,"text":"gence. Blue phases are optically isotropic.","rect":[82.8171615600586,521.79052734375,252.96249051596409,512.8560180664063]},{"page":82,"text":"BP1 and BPII show the optical reflections similar to the X-ray reflections","rect":[82.8171615600586,533.7501220703125,378.8494912539597,524.8156127929688]},{"page":82,"text":"from solid crystals. Bragg reflections correspond to the three-dimensional","rect":[82.8171615600586,545.7096557617188,385.13252122470927,536.775146484375]},{"page":82,"text":"periodicity at the micrometer scale like in three-dimensional photonic band-","rect":[82.8171615600586,557.669189453125,385.1106198867954,548.7346801757813]},{"page":82,"text":"gap crystals.","rect":[82.8171615600586,569.6287231445313,132.97542996908909,560.6942138671875]},{"page":82,"text":"The phase transition between the isotropic and a blue phase III is accom-","rect":[82.81616973876953,581.531494140625,377.94963966218605,572.5969848632813]},{"page":82,"text":"panied by a very blurred anomaly in specific heat (H), and there are also","rect":[82.8171615600586,593.4910278320313,385.15740290692818,584.5565185546875]},{"page":83,"text":"4.9 Smectic C* Phase","rect":[53.81196975708008,42.62416076660156,128.48092028879644,36.62995529174805]},{"page":83,"text":"65","rect":[376.7456359863281,42.55643081665039,385.2067007461063,36.62995529174805]},{"page":83,"text":"Fig. 4.33 Double-twist cylinder (a) and the structure of the body-centered cubic phase BP1 (b)","rect":[53.812843322753909,201.68829345703126,385.1720284805982,193.75527954101563]},{"page":83,"text":"and simple cubic phase BPII (c), both consisted of double-twist cylinders (adapted from [22])","rect":[53.812843322753909,211.59652709960938,374.62020963294199,204.00216674804688]},{"page":83,"text":"noticeable H anomalies between the BPIII and BPI phases [22] and also at","rect":[82.8152084350586,242.46856689453126,385.181288314553,233.5340118408203]},{"page":83,"text":"the BP-cholesteric phase transition. This means that we deal not with","rect":[82.81421661376953,254.4281005859375,385.11269465497505,245.49354553222657]},{"page":83,"text":"different textures of the cholesteric phase but with different phases.","rect":[82.81421661376953,266.33087158203127,354.91155667807348,257.39630126953127]},{"page":83,"text":"(viii) NMR spectra of BP are different from those of the Ch phase.","rect":[57.663909912109378,278.2904052734375,329.5811733772922,269.3558349609375]},{"page":83,"text":"It has been concluded that blue phases I and II are three-dimensional periodic","rect":[65.7636489868164,296.25811767578127,385.1614459819969,287.32354736328127]},{"page":83,"text":"structures, formed by pieces of the helix, a kind of regular lattice of defects having a","rect":[53.81162643432617,308.2574768066406,385.1583942241844,299.2432556152344]},{"page":83,"text":"period comparable with the wavelength of visible light. How such a phase can be","rect":[53.81161880493164,320.12042236328127,385.14853704645005,311.18585205078127]},{"page":83,"text":"modeled? One of the most interesting models is a defect structure made of double-","rect":[53.81161880493164,332.0799865722656,385.12261329499855,323.1255187988281]},{"page":83,"text":"twist cylinders as building blocks [23]. The helical structure forms in two direc-","rect":[53.81161880493164,344.0395202636719,385.1604245742954,335.0850524902344]},{"page":83,"text":"tions, Fig. 4.33a. Such cylinders can be packed either in the body-centered lattice","rect":[53.81161880493164,355.99908447265627,385.1534808941063,347.06451416015627]},{"page":83,"text":"forming the BPI phase as shown in Fig. 4.33b or in more symmetric simple cubic","rect":[53.81162643432617,367.9586181640625,385.1384662456688,359.0240478515625]},{"page":83,"text":"lattice, Fig. 4.33c that may correspond to the high temperature BPII blue phase. The","rect":[53.81160354614258,379.91815185546877,385.1435016460594,370.98358154296877]},{"page":83,"text":"foggy phase is, more probably, amorphous. It is important that the concept of a","rect":[53.81161117553711,391.877685546875,385.15744817926255,382.943115234375]},{"page":83,"text":"lattice of defects is quite general and can be used in other areas of physics of the","rect":[53.81161117553711,403.83721923828127,385.17334783746568,394.8628234863281]},{"page":83,"text":"condensed matter (theory of melting, theory of phase transitions, superfluidity and","rect":[53.81161117553711,415.739990234375,385.14052668622505,406.805419921875]},{"page":83,"text":"Abrikosov vortices, structure of amorphous medium, etc).","rect":[53.81161117553711,427.6995544433594,287.2965969612766,418.7650146484375]},{"page":83,"text":"4.9 Smectic C* Phase","rect":[53.812843322753909,475.01031494140627,171.79154761762826,465.85467529296877]},{"page":83,"text":"4.9.1 Symmetry, Polarization and Ferroelectricity","rect":[53.812843322753909,506.88897705078127,307.60066016645637,496.2511901855469]},{"page":83,"text":"The chirality of molecules breaks the mirror symmetry C2h of the achiral smecticC","rect":[53.812843322753909,534.431396484375,385.1882502716377,525.4968872070313]},{"page":83,"text":"phase. The only symmetry element left is a twofold rotation axis C2, and the point","rect":[53.81356430053711,546.3910522460938,385.1496415860374,537.4564208984375]},{"page":83,"text":"symmetry group becomes C2 instead of C2h. The structure of a single smectic C*","rect":[53.81380844116211,558.3507080078125,385.17961970380318,549.4160766601563]},{"page":83,"text":"layer is shown in Fig. 4.34. As in achiral smectic C, the molecules in the layer obey","rect":[53.813961029052737,570.3102416992188,385.1389092545844,561.375732421875]},{"page":83,"text":"head-to-tail","rect":[53.81494903564453,581.0,100.01940782138894,573.3352661132813]},{"page":83,"text":"symmetry,","rect":[106.15419006347656,582.269775390625,148.69051785971409,574.3511962890625]},{"page":83,"text":"the","rect":[154.81234741210938,581.0,167.03611028863754,573.3352661132813]},{"page":83,"text":"director","rect":[173.22166442871095,581.0,204.29869972077979,573.3352661132813]},{"page":83,"text":"n","rect":[210.49520874023438,580.1282958984375,215.98993272135807,575.2078247070313]},{"page":83,"text":"coincides","rect":[222.16453552246095,581.0,259.9576760684128,573.3352661132813]},{"page":83,"text":"with","rect":[266.1203308105469,581.0,283.8388299088813,573.3352661132813]},{"page":83,"text":"average","rect":[290.0243835449219,582.269775390625,321.0914691753563,575.0]},{"page":83,"text":"orientation","rect":[327.2958984375,581.0,370.68828669599068,573.3352661132813]},{"page":83,"text":"of","rect":[376.85992431640627,581.0,385.15178809968605,573.3352661132813]},{"page":83,"text":"molecular axes and form angle # with the smectic normal h.","rect":[53.81592559814453,594.2293090820313,298.0086636116672,585.07568359375]},{"page":84,"text":"66","rect":[53.812843322753909,42.55667495727539,62.2739118972294,36.68099594116211]},{"page":84,"text":"Fig. 4.34 Structure ofa","rect":[53.812843322753909,67.58130645751953,137.5774473884058,59.648292541503909]},{"page":84,"text":"single monolayer of chiral","rect":[53.812843322753909,77.4895248413086,143.93000511374536,69.89517211914063]},{"page":84,"text":"smectic C*. Rotation axis C2","rect":[53.812843322753909,87.00543212890625,152.99924218966704,79.76355743408203]},{"page":84,"text":"is polar axis. Chiral molecules","rect":[53.812843322753909,97.3844223022461,155.30844576834955,89.79006958007813]},{"page":84,"text":"are tilted through angle # the","rect":[53.812843322753909,107.36043548583985,153.70168444895269,99.57981872558594]},{"page":84,"text":"director n forms with layer","rect":[53.81199645996094,117.33638763427735,146.19673246009044,109.74203491210938]},{"page":84,"text":"normal h","rect":[53.81199645996094,125.50303649902344,84.93603642985376,119.66122436523438]},{"page":84,"text":"C2 axis","rect":[230.4525604248047,132.916748046875,255.51135109714265,125.29437255859375]},{"page":84,"text":"4","rect":[303.6741638183594,42.455078125,307.9046987929813,36.73179626464844]},{"page":84,"text":"Liquid Crystal","rect":[310.30169677734377,44.275352478027347,359.76000694480009,36.68099594116211]},{"page":84,"text":"h","rect":[329.6081848144531,70.31658935546875,333.6045167543678,64.82217407226563]},{"page":84,"text":"ϑ","rect":[351.11395263671877,72.1650161743164,356.1577018191638,66.5022201538086]},{"page":84,"text":"Phases","rect":[362.1883239746094,43.0,385.1685226726464,36.68099594116211]},{"page":84,"text":"n","rect":[365.2030944824219,87.26507568359375,369.19942642233658,83.46617126464844]},{"page":84,"text":"Fig. 4.35 Correlation of","rect":[53.812843322753909,165.92269897460938,139.0310525284498,157.98968505859376]},{"page":84,"text":"temperature dependencies of","rect":[53.812843322753909,175.8309326171875,152.00357144934825,168.236572265625]},{"page":84,"text":"the molecular tilt # and","rect":[53.812843322753909,184.0229949951172,134.42651886134073,177.9695281982422]},{"page":84,"text":"spontaneous polarization Ps","rect":[53.812843322753909,195.72610473632813,148.70093443037957,188.13174438476563]},{"page":84,"text":"for a ferroelectric compound","rect":[53.812843322753909,205.70156860351563,151.95279449610636,198.10720825195313]},{"page":84,"text":"DOBAMBC (for the formula","rect":[53.812843322753909,215.33888244628907,153.52233264231206,208.06625366210938]},{"page":84,"text":"see Fig. 3.5a)","rect":[53.812843322753909,225.59674072265626,99.99166959387948,217.9515838623047]},{"page":84,"text":"DOBAMBC","rect":[278.64404296875,164.8707733154297,320.37687955673769,158.83206176757813]},{"page":84,"text":"Ps , nC/cm2","rect":[237.50750732421876,181.82594299316407,280.5985106468488,172.48992919921876]},{"page":84,"text":"10","rect":[231.5087890625,194.11622619628907,240.39730313622813,188.29347229003907]},{"page":84,"text":"5","rect":[235.95303344726563,237.0350799560547,240.39728787743906,231.3083038330078]},{"page":84,"text":"q, deg","rect":[362.3641052246094,180.67855834960938,385.1856722452507,173.20814514160157]},{"page":84,"text":"30","rect":[361.1315612792969,199.00315856933595,370.02006009423595,193.18040466308595]},{"page":84,"text":"20","rect":[361.1315612792969,226.38291931152345,370.02006009423595,220.56016540527345]},{"page":84,"text":"10","rect":[361.1315612792969,253.7058868408203,370.02006009423595,247.8831329345703]},{"page":84,"text":"0","rect":[235.95303344726563,280.6393737792969,240.39728787743906,274.81658935546877]},{"page":84,"text":"30","rect":[258.59796142578127,287.5019226074219,267.4864602407203,281.67913818359377]},{"page":84,"text":"20","rect":[285.0572509765625,287.5019226074219,293.9457803090797,281.67913818359377]},{"page":84,"text":"10","rect":[312.3086853027344,287.5019226074219,321.19721463525158,281.67913818359377]},{"page":84,"text":"0","rect":[340.166015625,287.5019226074219,344.61027005517345,281.67913818359377]},{"page":84,"text":"0","rect":[361.1315612792969,280.6337890625,365.5758157094703,274.8110046386719]},{"page":84,"text":"Tc - T,K","rect":[327.2510986328125,302.1546630859375,357.67286954697206,294.3343505859375]},{"page":84,"text":"The smectic C and C* order parameters are the same, the two-component tilt","rect":[65.76496887207031,331.967529296875,385.15778977939677,323.032958984375]},{"page":84,"text":"#exp(ij). However, the plane of the figure is no longer a mirror plane and the C2","rect":[53.812950134277347,343.9270935058594,385.18130140630105,334.7734069824219]},{"page":84,"text":"axis is a polar axis directed forward or backward with respect to the tilt plane h,n.","rect":[53.812843322753909,355.8868713378906,385.1745876839328,346.9324035644531]},{"page":84,"text":"This depends on a sign of handedness. Such symmetry allows for the existence of","rect":[53.81288146972656,367.846435546875,385.1497739395298,358.911865234375]},{"page":84,"text":"the spontaneous polarization vector Ps (that is a dipole moment of a unit volume)","rect":[53.81288146972656,379.8061828613281,385.0878842910923,370.85150146484377]},{"page":84,"text":"directed along the polar axis. Thus, each SmC* layer is polar and possess pyroelec-","rect":[53.8137321472168,391.7657470703125,385.1276792129673,382.8311767578125]},{"page":84,"text":"tric properties. Moreover, the direction of Ps can be aligned by an electric field in","rect":[53.8137321472168,403.7254638671875,385.1435479264594,394.79071044921877]},{"page":84,"text":"any direction. At a certain boundary conditions provided by e.g. aligning glasses,","rect":[53.81368637084961,415.62823486328127,385.17937894369848,406.69366455078127]},{"page":84,"text":"the layer manifests two memory states, and, under this condition, each smectic layer","rect":[53.81368637084961,427.5877990722656,385.11773048249855,418.65325927734377]},{"page":84,"text":"may be considered ferroelectric, for details see Chapter 13. For small tilt angles, the","rect":[53.81368637084961,439.5473327636719,385.17343939020005,430.57293701171877]},{"page":84,"text":"value of the spontaneous polarization is proportional to # as illustrated in Fig. 4.35","rect":[53.81368637084961,451.5068664550781,385.15944758466255,442.3531799316406]},{"page":84,"text":"by experimental curves for the DOBAMBC (for formula see Fig. 3.5a), the first","rect":[53.813655853271487,463.4664001464844,385.16047532627177,454.4720764160156]},{"page":84,"text":"liquid crystal ferroelectric compound synthesized in Orsay (France) following ideas","rect":[53.813655853271487,475.42596435546877,385.12762846099096,466.47149658203127]},{"page":84,"text":"of Meyer [24]. The value of Ps in DOBAMBC is rather small, about 6 nC/cm2 at","rect":[53.813655853271487,487.3854675292969,385.1831498868186,476.3354187011719]},{"page":84,"text":"room temperature, however, nowadays there have been synthesized many com-","rect":[53.814414978027347,499.34588623046877,385.13936744538918,490.41131591796877]},{"page":84,"text":"pounds with Ps of several hundreds nC/cm2.","rect":[53.814414978027347,511.2486572265625,232.24129910971409,500.1980285644531]},{"page":84,"text":"4.9.2 Helical Structure","rect":[53.812843322753909,552.4328002929688,175.55067237348764,544.0899658203125]},{"page":84,"text":"Due","rect":[53.812843322753909,581.0,70.38660467340314,573.53466796875]},{"page":84,"text":"to","rect":[75.45130920410156,581.0,83.22554102948675,574.3513793945313]},{"page":84,"text":"chiral","rect":[88.31015014648438,581.0,111.095350814553,573.33544921875]},{"page":84,"text":"intermolecular","rect":[116.17896270751953,581.0,174.60320411531104,573.33544921875]},{"page":84,"text":"interaction","rect":[179.67684936523438,581.0,222.55959406903754,573.33544921875]},{"page":84,"text":"the","rect":[227.59841918945313,581.0,239.82218206598129,573.33544921875]},{"page":84,"text":"overall","rect":[244.93170166015626,581.0,272.6940141446311,573.33544921875]},{"page":84,"text":"multi-layered","rect":[277.78558349609377,582.2699584960938,331.7642754655219,573.33544921875]},{"page":84,"text":"structure","rect":[336.80908203125,581.0,371.8120807476219,574.3513793945313]},{"page":84,"text":"of","rect":[376.8569030761719,581.0,385.14873634187355,573.33544921875]},{"page":84,"text":"smectic C* becomes twisted, like in cholesterics. The twist angle of the tilt plane is","rect":[53.812843322753909,594.2294921875,385.18756498442846,585.2949829101563]},{"page":85,"text":"4.9 Smectic C* Phase","rect":[53.812992095947269,42.62440490722656,128.4819426276636,36.63019943237305]},{"page":85,"text":"67","rect":[376.74664306640627,42.55667495727539,385.20770782618447,36.68099594116211]},{"page":85,"text":"j ¼ qz ¼ 2pz=P0","rect":[185.45620727539063,73.71239471435547,253.00586006721816,59.41368865966797]},{"page":85,"text":"that is the tilt plane rotates about z upon translation along z, Fig. 4.36. The period","rect":[53.812843322753909,97.3089828491211,385.13173762372505,88.37443542480469]},{"page":85,"text":"(pitch) of the helical superstructure P0 is incommensurate to the thickness of a","rect":[53.812862396240237,109.21175384521485,385.1598285503563,100.27720642089844]},{"page":85,"text":"molecular layer. The helicity is a secondary phenomenon. By proper mixing left","rect":[53.814022064208987,121.17159271240235,385.1180253750999,112.23704528808594]},{"page":85,"text":"and right molecular isomers one can compensate for helicity. For racemic mixtures,","rect":[53.814022064208987,133.13113403320313,385.1648220589328,124.19657897949219]},{"page":85,"text":"this is trivial and results in the achiral SmC structure with unpolar layers. However,","rect":[53.814022064208987,145.09066772460938,385.15093656088598,136.15611267089845]},{"page":85,"text":"we can mix right and left isomers of chemically different molecules. In this case, the","rect":[53.814022064208987,157.05020141601563,385.1737445659813,148.0758056640625]},{"page":85,"text":"helicity is compensated for, but not the polarity of layers. Alternatively, one","rect":[53.814022064208987,169.00973510742188,385.14194524957505,160.07518005371095]},{"page":85,"text":"can compensate for the spontaneous polarization but keep the helical structure as","rect":[53.814022064208987,180.96926879882813,385.13898100005346,172.0347137451172]},{"page":85,"text":"it is [25].","rect":[53.814022064208987,192.3212432861328,90.77202268149142,183.93450927734376]},{"page":85,"text":"In the helical structure, the optical ellipsoid of the smectic C* phase rotates","rect":[65.76604461669922,204.83160400390626,385.1210061465378,195.8970489501953]},{"page":85,"text":"together with the tilt plane. Like in cholesterics, we can imagine that helical turns","rect":[53.814022064208987,216.7911376953125,385.14401640044408,207.85658264160157]},{"page":85,"text":"form a stuck of equidistant quasi-layers that results in optical Bragg reflections in","rect":[53.814022064208987,228.75070190429688,385.1419304948188,219.81614685058595]},{"page":85,"text":"the visible range. Therefore, like cholesterics, smectic C* liquid crystals are one-","rect":[53.814022064208987,240.71023559570313,385.11605201570168,231.7756805419922]},{"page":85,"text":"dimensional photonic crystals. However, in the case of SmC*, the distance between","rect":[53.814022064208987,252.66976928710938,385.1747674088813,243.73521423339845]},{"page":85,"text":"the reflecting “layers” is equal to the full pitch P0 and not to the half-pitch as","rect":[53.814022064208987,264.6307373046875,385.14081205474096,255.6947479248047]},{"page":85,"text":"in cholesterics, because at each half-pitch the molecules in the SmC* are tilted in","rect":[53.81391525268555,276.59027099609377,385.13881770185005,267.65570068359377]},{"page":85,"text":"opposite directions. Hence, we have a situation physically different from that in","rect":[53.81391525268555,288.5498352050781,385.1408318620063,279.61529541015627]},{"page":85,"text":"cholesterics.","rect":[53.81391525268555,298.3907775878906,103.52424283530002,291.51806640625]},{"page":85,"text":"In Fig. 4.37 the location of the Bragg reflections on the optical wavelength scale","rect":[65.76593780517578,312.4121398925781,385.10999334527818,303.47760009765627]},{"page":85,"text":"is compared for a cholesteric and smectic C* (ySmC* ¼ 25\u0002) liquid crystals. The","rect":[53.81391525268555,324.372314453125,385.14795721246568,315.1283874511719]},{"page":85,"text":"spectra have been calculated numerically using the Palto’s software [26] with the","rect":[53.814083099365237,336.33184814453127,385.17481268121568,327.39727783203127]},{"page":85,"text":"same parameters for both materials: P0 ¼ 0.25 mm, sample thickness d ¼ 4 mm and","rect":[53.814083099365237,348.2915954589844,385.14614192060005,339.2972717285156]},{"page":85,"text":"principle refraction indices 1.73 and 1.51. The calculations are made for normal and","rect":[53.814292907714847,360.25115966796877,385.14223567060005,351.2568359375]},{"page":85,"text":"oblique light incidence angles of a ¼ 0 (dash line), and 45\u0002 (solid lines). The Bragg","rect":[53.814292907714847,372.21099853515627,385.1203240495063,363.21636962890627]},{"page":85,"text":"formula (3.14) is valid for both materials. However, at the light incidence along the","rect":[53.81433868408203,384.1705627441406,385.1741107769188,375.23602294921877]},{"page":85,"text":"helical axis (a ¼ 0) , the left edge of the first order Bragg reflection (m ¼ 1) in","rect":[53.81433868408203,396.0733337402344,385.14324275067818,387.1387939453125]},{"page":85,"text":"thecholestericcorrespondstol0 ¼ P0<n> \u0003 380nmbut,inthesmecticC*phase,the","rect":[53.8133659362793,408.03326416015627,385.1746295757469,398.78955078125]},{"page":85,"text":"first order corresponds to the full pitch l0 ¼ 2P0<n> \u0003 730 nm. At this wavelength","rect":[53.81386947631836,419.9928894042969,385.1788262467719,410.74951171875]},{"page":85,"text":"Smectic C*","rect":[245.00161743164063,447.50909423828127,283.40126425017,441.8302917480469]},{"page":85,"text":"Fig. 4.36 Helical structure of","rect":[53.812843322753909,561.7815551757813,155.34483426673106,553.8485717773438]},{"page":85,"text":"the chiral smectic C* phase;","rect":[53.812843322753909,571.6897583007813,150.64132408347192,564.0953979492188]},{"page":85,"text":"P0 is a pitch of the helix","rect":[53.812843322753909,581.6087036132813,137.31189483790323,574.0143432617188]},{"page":85,"text":"P0","rect":[372.5158996582031,512.6016235351563,380.39719299963925,505.1172790527344]},{"page":86,"text":"68","rect":[53.812843322753909,42.55594253540039,62.2739118972294,36.68026351928711]},{"page":86,"text":"Fig. 4.37 Comparison of the","rect":[53.812843322753909,67.58130645751953,154.49535510324956,59.648292541503909]},{"page":86,"text":"calculated transmission","rect":[53.812843322753909,75.76238250732422,133.54999298243448,69.89517211914063]},{"page":86,"text":"spectra of a cholesteric (top","rect":[53.812843322753909,87.4087142944336,148.71304840235636,79.81436157226563]},{"page":86,"text":"panel) and smectic C*","rect":[53.812843322753909,97.29153442382813,130.74681610255167,89.77337646484375]},{"page":86,"text":"(bottom panel) for two angles","rect":[53.812843322753909,107.36067962646485,155.34228976249018,99.7493896484375]},{"page":86,"text":"of incoming light incidence:","rect":[53.81283950805664,117.33663177490235,150.62185386862817,109.74227905273438]},{"page":86,"text":"dashed curves for a ¼ 0","rect":[53.81283950805664,125.53713989257813,136.6103262343876,119.66146850585938]},{"page":86,"text":"(along the helical axis), solid","rect":[53.812843322753909,137.23178100585938,153.21519989161417,129.63742065429688]},{"page":86,"text":"curves for a ¼ 45\u0002 with","rect":[53.812843322753909,145.48904418945313,135.8282217422001,139.5625762939453]},{"page":86,"text":"respect to the helical axis.","rect":[53.81338119506836,157.182861328125,142.87544509961567,149.5885009765625]},{"page":86,"text":"Both materials have helical","rect":[53.81338119506836,165.3495330810547,147.3860445424563,159.50772094726563]},{"page":86,"text":"pitch 0.25 mm, refraction","rect":[53.81338119506836,177.07803344726563,139.16866058497355,169.43287658691407]},{"page":86,"text":"indices n|| ¼ 1.73 and n|⊥¼","rect":[53.81338882446289,186.65032958984376,149.4307648277147,179.459228515625]},{"page":86,"text":"1.51, cell thickness 4 mm. Tilt","rect":[53.81306838989258,196.8966064453125,155.35691551413599,189.32765197753907]},{"page":86,"text":"angle for the SmC* is 25\u0002","rect":[53.81306838989258,206.94876098632813,142.59156437214055,199.30360412597657]},{"page":86,"text":"1.0","rect":[252.23170471191407,67.02862548828125,263.34232564838347,60.790771484375]},{"page":86,"text":"0.8","rect":[252.23170471191407,99.47393798828125,263.34232564838347,93.236083984375]},{"page":86,"text":"0.6","rect":[252.23170471191407,131.96734619140626,263.34232564838347,125.7294921875]},{"page":86,"text":"1.0","rect":[252.23129272460938,159.95318603515626,263.3419289198678,153.71533203125]},{"page":86,"text":"0.8","rect":[252.23129272460938,197.30145263671876,263.3419289198678,191.0635986328125]},{"page":86,"text":"0.6","rect":[252.23129272460938,234.65057373046876,263.3419289198678,228.4127197265625]},{"page":86,"text":"200","rect":[262.81597900390627,253.40182495117188,276.1487526503366,247.16397094726563]},{"page":86,"text":"4 Liquid Crystal","rect":[303.6741638183594,44.274620056152347,359.76000694480009,36.68026351928711]},{"page":86,"text":"CLC","rect":[332.1502685546875,122.4776611328125,348.5764339080908,116.00846099853516]},{"page":86,"text":"SmC*","rect":[313.0419616699219,225.35791015625,334.3599897504399,218.88870239257813]},{"page":86,"text":"400","rect":[295.9288635253906,253.40182495117188,309.26163717182097,247.16397094726563]},{"page":86,"text":"600","rect":[329.140869140625,253.40182495117188,342.4736427870553,247.16397094726563]},{"page":86,"text":"Wavelength, nm","rect":[295.0440673828125,267.84442138671877,356.3364188451031,260.3740234375]},{"page":86,"text":"Phases","rect":[362.1883239746094,43.0,385.1685226726464,36.68026351928711]},{"page":86,"text":"and a ¼ 0, the band is invisible (forbidden) due only to the coincidence of effective","rect":[53.812843322753909,300.56640625,385.17163885309068,291.6318359375]},{"page":86,"text":"refraction indices for the same absolute value of the tilt |\u0006ysmC*|. The band at about","rect":[53.81283950805664,312.1079406738281,385.160048080178,303.2826232910156]},{"page":86,"text":"380 nm is the second diffraction order (m ¼ 2). In the smectic C* phase, the first","rect":[53.81427764892578,324.4855651855469,385.160048080178,315.551025390625]},{"page":86,"text":"order Bragg diffraction band appears only at an oblique light incidence, see the","rect":[53.814247131347659,336.4450988769531,385.1730121441063,327.51055908203127]},{"page":86,"text":"transmission minimum at about 680 nm for a ¼ 45\u0002 in the same figure. Such a shift","rect":[53.814247131347659,348.3482971191406,385.2089982754905,339.3535461425781]},{"page":86,"text":"to the shorter waves of both m ¼ 1 and m ¼ 2 bands in the smectic C* (as well as of","rect":[53.81342697143555,359.9093933105469,385.1492856582798,351.373291015625]},{"page":86,"text":"m ¼ 1 band in the cholesteric) increases with increasing angle of light incidence.","rect":[53.812435150146487,372.26739501953127,380.9914516976047,363.33282470703127]},{"page":86,"text":"4.10 Chiral Smectic A*","rect":[53.812843322753909,408.86370849609377,181.14451351069995,400.12640380859377]},{"page":86,"text":"4.10.1 Uniform Smectic A*","rect":[53.812843322753909,440.65869140625,197.4590276708562,430.1045837402344]},{"page":86,"text":"This is a chiral smectic A* with symmetry D1. Its properties are similar to those of","rect":[53.812843322753909,468.284912109375,385.14910255281105,459.350341796875]},{"page":86,"text":"the achiral SmA. However, close to the transition to the smectic C* phase, the chiral","rect":[53.813228607177737,480.2444763183594,385.1411576993186,471.3099365234375]},{"page":86,"text":"smectic A* phase shows interesting pretransitional phenomena in the dielectric and","rect":[53.813228607177737,492.2040100097656,385.1431817155219,483.26947021484377]},{"page":86,"text":"electrooptical effects (the so-called soft dielectric mode and electroclinic effect).","rect":[53.813228607177737,504.1635437011719,385.0983852913547,495.22900390625]},{"page":86,"text":"They will be discussed in Chapter 13.","rect":[53.813228607177737,516.123046875,206.29780240561252,507.18853759765627]},{"page":86,"text":"4.10.2 TGB Phase","rect":[53.812843322753909,552.4328002929688,152.7583292338392,544.0899658203125]},{"page":86,"text":"This phase consists of uniform SmA* blocks separated by defect walls [22]. At each","rect":[53.812843322753909,582.2699584960938,385.1018914323188,573.33544921875]},{"page":86,"text":"wall, the normal to smectic layers in one blocks turns through a small angle with","rect":[53.81282424926758,594.2294921875,385.1148308854438,585.2949829101563]},{"page":87,"text":"4.11 Spontaneous Break of Mirror Symmetry","rect":[53.813682556152347,44.276084899902347,208.94232696680948,36.68172836303711]},{"page":87,"text":"69","rect":[376.74652099609377,42.62513732910156,385.20758575587197,36.68172836303711]},{"page":87,"text":"respect to the layer normal in the preceding block as very schematically shown in","rect":[53.812843322753909,68.2883529663086,385.14174738935005,59.35380554199219]},{"page":87,"text":"Fig. 4.38. From such blocks, a helical structure forms. Thus, the phase is twisted,","rect":[53.812843322753909,80.24788665771485,385.1696438362766,71.31333923339844]},{"page":87,"text":"consists of grains and has defects in a form of grain boundaries. That is why it is","rect":[53.813838958740237,92.20748138427735,385.18454374419408,83.27293395996094]},{"page":87,"text":"called a twist-grain-boundary or TGB phase. We should distinguish among the","rect":[53.813838958740237,104.15003204345703,385.17456854059068,95.15572357177735]},{"page":87,"text":"TGBA and TGBC phases based, respectively, on the smectic A* or smectic C*","rect":[53.81382369995117,116.0697250366211,385.1765984635688,107.13517761230469]},{"page":87,"text":"structure of their blocks. The TGB phases having a helical pitch shorter than light","rect":[53.81382369995117,128.02932739257813,385.1198564297874,119.09477233886719]},{"page":87,"text":"wavelength are optically isotropic. Such substances, especially based on side-chain","rect":[53.81382369995117,139.98886108398438,385.17263117841255,131.05430603027345]},{"page":87,"text":"polymers","rect":[53.81382369995117,151.94839477539063,91.009723797905,143.0138397216797]},{"page":87,"text":"with","rect":[96.75032043457031,149.88656616210938,114.46881190351019,143.0138397216797]},{"page":87,"text":"photochromic","rect":[120.31492614746094,151.94839477539063,175.83652532770004,143.0138397216797]},{"page":87,"text":"moieties","rect":[181.60398864746095,149.88656616210938,215.56377042876438,143.0138397216797]},{"page":87,"text":"are","rect":[221.3690643310547,149.88656616210938,233.58289373590316,145.24497985839845]},{"page":87,"text":"interesting","rect":[239.38221740722657,151.94839477539063,281.6477898698188,143.0138397216797]},{"page":87,"text":"for","rect":[287.4172668457031,149.88656616210938,299.02385841218605,143.0138397216797]},{"page":87,"text":"optical","rect":[304.8072509765625,151.94839477539063,332.0519243008811,143.0138397216797]},{"page":87,"text":"information","rect":[337.8871154785156,149.88656616210938,385.1098565202094,143.0138397216797]},{"page":87,"text":"recording and applications to holography.","rect":[53.81382369995117,163.907958984375,221.9225277474094,154.97340393066407]},{"page":87,"text":"4.11 Spontaneous Break of Mirror Symmetry","rect":[53.812843322753909,214.6377716064453,296.02464046382496,203.28285217285157]},{"page":87,"text":"This phenomenon has been discovered in the liquid crystal phases consisting of","rect":[53.812843322753909,241.61834716796876,385.14766822663918,232.6837921142578]},{"page":87,"text":"so-called banana (or bent-core) shape molecules [17, 27]. A mechanical model in","rect":[53.812843322753909,253.52108764648438,385.14165583661568,244.58653259277345]},{"page":87,"text":"Fig. 4.39a illustrates the idea. Each of the two dumb-bells has symmetry D1h with","rect":[53.81280517578125,265.48065185546877,385.11895075849068,256.54608154296877]},{"page":87,"text":"infinite number of mirror planes containing the longitudinal rotation axis and one","rect":[53.813961029052737,277.4404602050781,385.1438678569969,268.50592041015627]},{"page":87,"text":"mirror plane perpendicular to that axis. Imagine now that one of the dumb-bells is","rect":[53.813961029052737,289.4000244140625,385.1856728945847,280.4654541015625]},{"page":87,"text":"lying on the table and we try to put another one on the top of the first one parallel to","rect":[53.813961029052737,301.35955810546877,385.1439141373969,292.42498779296877]},{"page":87,"text":"TGBA phase","rect":[264.930908203125,325.5205078125,309.12566908969787,318.3620300292969]},{"page":87,"text":"Fig. 4.38 Schematic picture","rect":[53.812843322753909,360.9046630859375,152.29548022531987,352.9547119140625]},{"page":87,"text":"of the block structure of the","rect":[53.812843322753909,369.0603332519531,149.11409899973394,363.2185363769531]},{"page":87,"text":"twist-grain-boundary smectic","rect":[53.812843322753909,380.7888488769531,153.7017149665308,373.1944885253906]},{"page":87,"text":"A* (TGBA) phase","rect":[53.812843322753909,390.70806884765627,116.76912829660893,383.11370849609377]},{"page":87,"text":"a","rect":[95.10724639892578,433.4272766113281,100.66250879615963,427.8385314941406]},{"page":87,"text":"Left","rect":[95.3934326171875,463.79583740234377,109.59728550962888,458.16510009765627]},{"page":87,"text":"b","rect":[232.26626586914063,433.4272766113281,238.37105961861884,426.118896484375]},{"page":87,"text":"W","rect":[270.9580383300781,439.5471496582031,278.0639604956084,434.0603942871094]},{"page":87,"text":"homogeneous","rect":[297.7846984863281,445.77215576171877,343.96920380137518,438.72576904296877]},{"page":87,"text":"achiral","rect":[319.9897155761719,453.8343200683594,343.9692002249234,448.2035827636719]},{"page":87,"text":"phase","rect":[324.4259033203125,464.9598083496094,343.969191551901,457.9214172363281]},{"page":87,"text":"Right","rect":[95.32426452636719,524.6112060546875,114.41993260435544,517.44482421875]},{"page":87,"text":"– q0","rect":[233.94142150878907,503.0325622558594,247.0983343570611,495.208251953125]},{"page":87,"text":"Left","rect":[241.55401611328126,522.35302734375,255.75786900572263,516.7222900390625]},{"page":87,"text":"domains","rect":[269.68792724609377,528.4615478515625,298.5592306568439,522.830810546875]},{"page":87,"text":"q","rect":[335.28851318359377,487.9547424316406,339.4529512581059,482.06805419921877]},{"page":87,"text":"q0","rect":[323.0779113769531,503.8892822265625,330.2397833316705,496.06500244140627]},{"page":87,"text":"right","rect":[310.853515625,522.8971557617188,327.72709202818359,515.7307739257813]},{"page":87,"text":"Fig. 4.39 A mechanical model of two interacting dumb-bells illustrating a break of the mirror","rect":[53.812843322753909,553.506103515625,385.16192716223886,545.4714965820313]},{"page":87,"text":"symmetry (a) and the potential curves with two minima corresponding to two possible azimuthal","rect":[53.812843322753909,563.414306640625,385.14501671042509,555.8199462890625]},{"page":87,"text":"angles between the dumb-bells (b). The same curves qualitatively illustrate the energy of the","rect":[53.81283950805664,573.3902587890625,385.15514514231207,565.7958984375]},{"page":87,"text":"achiral phase and two chiral domains (left- and right-handed) as functions of the tilt angle # of","rect":[53.812835693359378,583.3662109375,385.15423673255136,575.5856323242188]},{"page":87,"text":"molecules in the smectic layer","rect":[53.813682556152347,593.2854614257813,157.75369351966075,585.6911010742188]},{"page":88,"text":"70","rect":[53.813690185546878,42.55820083618164,62.274758760022368,36.73332214355469]},{"page":88,"text":"4 Liquid Crystal Phases","rect":[303.67498779296877,44.276878356933597,385.1693771648339,36.68252182006836]},{"page":88,"text":"the other. Such a construction, although unstable, would have mirror symmetry. In","rect":[53.812843322753909,68.2883529663086,385.14775935224068,59.35380554199219]},{"page":88,"text":"reality, the dumb-bells will form a kind of a chiral propeller, left or right, shown in","rect":[53.812843322753909,80.24788665771485,385.14165583661568,71.29341888427735]},{"page":88,"text":"the figure. The reason is that the gravitational potential energy of the upper dumb-","rect":[53.81280517578125,92.20748138427735,385.14568458406105,83.27293395996094]},{"page":88,"text":"bell is lower in a chiral construction. Due to this, the mirror symmetry is broken.","rect":[53.81280517578125,104.11019134521485,385.17351956869848,95.17564392089844]},{"page":88,"text":"Since the formation of right- and left-hand propellers is equally probable, the","rect":[53.81280517578125,116.0697250366211,385.1706012554344,107.13517761230469]},{"page":88,"text":"potential energy roughly has a shape of a two minima curve, see Fig. 4.39b, that","rect":[53.81280517578125,128.02932739257813,385.1386857754905,119.09477233886719]},{"page":88,"text":"will also be discussed below.","rect":[53.81280517578125,137.9569091796875,171.16695828940159,131.05430603027345]},{"page":88,"text":"Something similar happens with achiral banana or bent-shape molecules. Chem-","rect":[65.76482391357422,151.94839477539063,385.16961036531105,143.0138397216797]},{"page":88,"text":"ical formula of a typical compound is given in Fig. 4.40. In this particular case, the","rect":[53.81280517578125,163.907958984375,385.1745075054344,154.97340393066407]},{"page":88,"text":"dipole moment is approximately directed from up to down. The molecules have","rect":[53.81379318237305,175.86749267578126,385.11777532770005,166.9329376220703]},{"page":88,"text":"banana-like shape and located within the plane of the drawing forming a single","rect":[53.81379318237305,187.8270263671875,385.1804889507469,178.89247131347657]},{"page":88,"text":"layers with long molecular axes perpendicular to the smectic plane, Fig. 4.41. Such","rect":[53.81379318237305,199.72976684570313,385.14663020185005,190.7952117919922]},{"page":88,"text":"a monolayer is achiral and can be unpolar (a) or polar (b). Note that the polar achiral","rect":[53.813812255859378,211.6893310546875,385.1187577969749,202.75477600097657]},{"page":88,"text":"layer possesses spontaneous polarization Ps located within the figure plane and","rect":[53.814796447753909,223.65057373046876,385.1452569108344,214.7143096923828]},{"page":88,"text":"directed depending on the sign of molecular dipole moment (to the right in the","rect":[53.81338119506836,235.61013793945313,385.1721576519188,226.6755828857422]},{"page":88,"text":"figure). If the direction of Ps can be switched by an external electric field between","rect":[53.81338119506836,247.56985473632813,385.1777276139594,238.63511657714845]},{"page":88,"text":"two stable positions the monolayer is ferroelectric. A stuck of unpolar or polar","rect":[53.81300735473633,259.5294189453125,385.1349118789829,250.59486389160157]},{"page":88,"text":"layers may form either an unpolar or polar smectic phase. An example (a polar","rect":[53.81300735473633,271.48895263671877,385.1369565567173,262.55438232421877]},{"page":88,"text":"phase) is shown in Fig. 4.41c. Packing of polar layers with opposite in-plane","rect":[53.81300735473633,283.448486328125,385.1488727398094,274.513916015625]},{"page":88,"text":"directions of Ps results in the antiferroelectric phase, like in chiral antiferroelectrics","rect":[53.812007904052737,295.3517761230469,385.10495390044408,286.41668701171877]},{"page":88,"text":"(see Chapter 13).","rect":[53.81288528442383,307.31134033203127,122.94586606528049,298.37677001953127]},{"page":88,"text":"O","rect":[180.35409545898438,354.51220703125,186.57285435588174,348.4734802246094]},{"page":88,"text":"Cl","rect":[237.02603149414063,348.0633850097656,244.5716693837161,342.024658203125]},{"page":88,"text":"O","rect":[248.5782012939453,354.3422546386719,254.79696019084268,348.30352783203127]},{"page":88,"text":"H29C14","rect":[54.832489013671878,422.03619384765627,79.58711996081367,414.0339050292969]},{"page":88,"text":"N","rect":[123.12761688232422,387.8188781738281,128.89875303085879,382.07611083984377]},{"page":88,"text":"H","rect":[135.25210571289063,406.00433349609377,141.0232418614252,400.2615661621094]},{"page":88,"text":"O","rect":[191.6624755859375,373.9493713378906,197.88123448283486,367.91064453125]},{"page":88,"text":"O","rect":[237.27381896972657,373.7790222167969,243.49257786662393,367.74029541015627]},{"page":88,"text":"N","rect":[306.73187255859377,389.5177001953125,312.5030087071283,383.7749328613281]},{"page":88,"text":"H","rect":[294.06365966796877,405.83343505859377,299.8347958165033,400.0906677246094]},{"page":88,"text":"Fig. 4.40 Chemical formula of a typical bent-shape molecule. The electro-negative Cl","rect":[53.812843322753909,443.6020202636719,356.77758506980009,435.8721923828125]},{"page":88,"text":"responsible for the molecular dipole moment directed approximately down close to the","rect":[53.812843322753909,453.51025390625,356.7741331794214,445.9158935546875]},{"page":88,"text":"axis","rect":[53.812843322753909,461.733642578125,67.46054537772455,455.891845703125]},{"page":88,"text":"Fig. 4.41 Structure of single","rect":[53.812843322753909,502.32330322265627,153.2118010261011,494.5934753417969]},{"page":88,"text":"non-polar (a) and polar","rect":[53.812843322753909,512.2315063476563,132.69116300208263,504.6371765136719]},{"page":88,"text":"(b) smectic layers formed by","rect":[53.81199645996094,522.1506958007813,152.0001730361454,514.5563354492188]},{"page":88,"text":"bent-shape molecules: the","rect":[53.812843322753909,532.1266479492188,141.2030119636011,524.5322875976563]},{"page":88,"text":"longitudinal axes are aligned","rect":[53.812843322753909,542.1026611328125,151.16760772852823,534.50830078125]},{"page":88,"text":"upright and the plane of the","rect":[53.812843322753909,552.07861328125,148.11232898020269,544.4842529296875]},{"page":88,"text":"figure is mirror plane. Possible","rect":[53.812843322753909,561.997802734375,155.42015979074956,554.4034423828125]},{"page":88,"text":"polar three-dimensional","rect":[53.812843322753909,571.9737548828125,133.55337242331567,564.37939453125]},{"page":88,"text":"smectic biaxial phase (c)","rect":[53.812843322753909,581.9497680664063,137.80334561927013,574.3554077148438]},{"page":88,"text":"a","rect":[185.6828155517578,495.4300842285156,191.23807794899165,489.8413391113281]},{"page":88,"text":"b","rect":[185.1162872314453,547.3714599609375,191.22108098092353,540.0631103515625]},{"page":88,"text":"P =0","rect":[263.1678771972656,519.2830200195313,278.93584066618566,513.6842041015625]},{"page":88,"text":"s","rect":[268.0518798828125,521.2645874023438,270.38391446914906,518.3491821289063]},{"page":88,"text":"c","rect":[294.9825744628906,494.7562255859375,300.53783686012448,489.16748046875]},{"page":88,"text":"P > 0","rect":[273.5433654785156,568.2864379882813,292.80842977751379,562.6876220703125]},{"page":88,"text":"s","rect":[278.4273681640625,570.2680053710938,280.75940275039906,567.3526000976563]},{"page":88,"text":"C14H29","rect":[359.36419677734377,422.06048583984377,384.23916616442696,414.0583190917969]},{"page":88,"text":"atom is","rect":[359.697509765625,442.0,385.1585129070214,435.93994140625]},{"page":88,"text":"vertical","rect":[359.6390686035156,452.0,385.1660433217532,445.9158935546875]},{"page":89,"text":"4.11","rect":[53.81215286254883,43.0,68.61902374415323,36.73277282714844]},{"page":89,"text":"Spontaneous","rect":[70.97543334960938,44.276329040527347,114.2284286785058,36.73277282714844]},{"page":89,"text":"Break","rect":[116.63136291503906,43.0,136.87869781641886,36.68197250366211]},{"page":89,"text":"of","rect":[139.2892608642578,43.0,146.3373269425123,36.68197250366211]},{"page":89,"text":"Mirror","rect":[148.69204711914063,43.0,171.24079984290294,36.68197250366211]},{"page":89,"text":"n","rect":[161.6252899169922,86.52984619140625,166.06954434716563,82.77064514160156]},{"page":89,"text":"Symmetry","rect":[173.61582946777345,44.276329040527347,208.94078582911417,36.73277282714844]},{"page":89,"text":"h","rect":[190.5065460205078,66.161376953125,194.95080045068125,60.86650466918945]},{"page":89,"text":"J","rect":[186.1332244873047,88.06729888916016,191.1769736697497,82.40450286865235]},{"page":89,"text":"z","rect":[241.70260620117188,74.67558288574219,244.81198564962058,70.49246978759766]},{"page":89,"text":"y","rect":[261.6953430175781,82.84024047851563,265.244351951026,77.50537872314453]},{"page":89,"text":"x","rect":[270.040283203125,89.35648345947266,274.03691488493566,85.63727569580078]},{"page":89,"text":"71","rect":[376.7449951171875,42.55765151977539,385.2060598769657,36.73277282714844]},{"page":89,"text":"P","rect":[167.47386169433595,123.9215087890625,172.35774560950854,118.62664031982422]},{"page":89,"text":"s","rect":[172.35784912109376,125.99906158447266,174.68988370743026,123.1016845703125]},{"page":89,"text":"Fig. 4.42 Breaks of the mirror symmetry by the molecular tilt. Due to the collective tilt of","rect":[53.812843322753909,158.55416870117188,385.15597623450449,150.82435607910157]},{"page":89,"text":"molecules the zy plane is no longer a mirror plane. Three vectors, namely, molecular dipole","rect":[53.812843322753909,168.46240234375,385.1754698493433,160.8680419921875]},{"page":89,"text":"moment Ps, the normal to the layers h and the director n form the right-handed triple of vectors.","rect":[53.812843322753909,178.38140869140626,385.155306549811,170.78704833984376]},{"page":89,"text":"For the tilt angle equal to (\u0004#) the triple changes the sense of chirality, i.e. becomes left-handed.","rect":[53.81219482421875,188.35736083984376,385.1739833076235,180.5767364501953]},{"page":89,"text":"As a result, right-handed and left-handed domains are observed","rect":[53.81219482421875,198.33334350585938,270.9063772597782,190.73898315429688]},{"page":89,"text":"Table 4.2 Symmetry and structural features of the most popular thermotropic","rect":[53.812843322753909,241.07037353515626,335.13073107981207,233.47601318359376]},{"page":89,"text":"phases consisting of rod-like molecules (for the nomenclature we follow [28])","rect":[53.812843322753909,251.04635620117188,321.1479501114576,243.45199584960938]},{"page":89,"text":"liquid","rect":[338.90692138671877,241.07037353515626,358.7312063613407,233.47601318359376]},{"page":89,"text":"crystal","rect":[362.5276794433594,241.07037353515626,385.15255455222197,233.47601318359376]},{"page":89,"text":"Symbol","rect":[53.812843322753909,263.1761474609375,80.16061882224146,255.581787109375]},{"page":89,"text":"Symmetry","rect":[107.56941223144531,263.1761474609375,142.8943838515751,255.63258361816407]},{"page":89,"text":"Structural features","rect":[161.3259735107422,262.0,224.02930148124018,255.581787109375]},{"page":89,"text":"I or Iso","rect":[53.812843322753909,273.49664306640627,78.73238891749307,267.82415771484377]},{"page":89,"text":"I (chiral)","rect":[53.81291580200195,294.86224365234377,84.02655118567636,287.6065368652344]},{"page":89,"text":"N","rect":[53.81290817260742,313.37738037109377,59.887956396164579,307.6710205078125]},{"page":89,"text":"Kh \u0005 T(3)","rect":[107.56855773925781,274.9103088378906,143.135590492317,267.70538330078127]},{"page":89,"text":"K \u0005 T(3)","rect":[107.56863403320313,294.86224365234377,139.96353238684825,287.6573181152344]},{"page":89,"text":"D1h \u0005 T(3)","rect":[107.56947326660156,314.7569274902344,149.479828773567,307.552001953125]},{"page":89,"text":"Ordinary liquid phase with full rotational and translational","rect":[161.3251953125,275.24896240234377,361.36694053855009,267.6376647949219]},{"page":89,"text":"symmetry","rect":[173.2772979736328,285.2249450683594,207.1892599502079,278.494140625]},{"page":89,"text":"Liquid consisted of chiral molecules showing rotation of linearly","rect":[161.32603454589845,295.2008972167969,383.52057403712197,287.6065368652344]},{"page":89,"text":"polarized light","rect":[173.2772979736328,305.17681884765627,222.8583345815188,297.58245849609377]},{"page":89,"text":"Uniaxial nematic phase possessing long range orientational order","rect":[161.32533264160157,315.0955810546875,384.42941373450449,307.501220703125]},{"page":89,"text":"and no translational order","rect":[173.27743530273438,323.3443908691406,260.9992684708326,317.4771728515625]},{"page":89,"text":"Nb","rect":[53.813053131103519,334.6890869140625,63.04676950289945,327.6224670410156]},{"page":89,"text":"D2h \u0005 T(3)","rect":[107.56914520263672,334.7085876464844,146.30772489173106,327.503662109375]},{"page":89,"text":"Biaxial nematic phase possessing long range orientational order","rect":[161.32528686523438,335.0472412109375,379.7275399552076,327.452880859375]},{"page":89,"text":"and no translational order","rect":[173.27737426757813,343.2960510253906,260.99920743567636,337.4288330078125]},{"page":89,"text":"N* or Ch","rect":[53.812992095947269,353.2237243652344,85.7696051040165,347.2972412109375]},{"page":89,"text":"SmA","rect":[53.812992095947269,373.1754150390625,71.16664673429935,367.35052490234377]},{"page":89,"text":"SmA*","rect":[53.813053131103519,413.02227783203127,75.39723724512979,407.0957946777344]},{"page":89,"text":"SmC","rect":[53.812984466552737,422.9409484863281,70.72666346754413,417.1160583496094]},{"page":89,"text":"SmC*","rect":[53.81328201293945,462.7879333496094,74.9574942031376,456.8614501953125]},{"page":89,"text":"TGBA* or","rect":[53.813838958740237,532.5050659179688,90.48750394446542,526.5785522460938]},{"page":89,"text":"TGBC*","rect":[53.813053131103519,542.4808349609375,80.58388275294229,536.5543212890625]},{"page":89,"text":"SmBhex","rect":[53.81320571899414,563.79296875,79.93192804171781,556.6077880859375]},{"page":89,"text":"D1 \u0005 T(3)","rect":[107.56871032714844,354.6035461425781,146.30772489173106,347.39862060546877]},{"page":89,"text":"D1h \u0005 T(2)","rect":[107.5695571899414,374.5552062988281,149.479828773567,367.35028076171877]},{"page":89,"text":"D1 \u0005 T(2)","rect":[107.56961059570313,414.40179443359377,146.30772489173106,407.1968688964844]},{"page":89,"text":"C2h \u0005 T(2)","rect":[107.56954956054688,424.32086181640627,145.85449308020763,417.1159362792969]},{"page":89,"text":"C2 \u0005 T(2)","rect":[107.5698471069336,464.1675720214844,142.68237393958263,456.962646484375]},{"page":89,"text":"D1 \u0005 T(2) or","rect":[107.57040405273438,533.8848876953125,155.345063148567,526.6799926757813]},{"page":89,"text":"C2 \u0005 T(2)","rect":[107.56961822509766,543.8607788085938,142.68237393958263,536.6558837890625]},{"page":89,"text":"D6h \u0005 T(1)","rect":[107.56914520263672,563.8155517578125,146.30772489173106,556.6075439453125]},{"page":89,"text":"Chiral nematic or cholesteric phase with twist axis perpendicular","rect":[161.32528686523438,354.94219970703127,383.5959786759107,347.34783935546877]},{"page":89,"text":"to the director and macroscopic periodicity","rect":[173.27737426757813,364.91815185546877,320.55075592188759,357.32379150390627]},{"page":89,"text":"Uniaxial lamellar smectic A phase possessing one-dimensional","rect":[161.32533264160157,374.89385986328127,376.81698326315947,367.29949951171877]},{"page":89,"text":"periodicity along the director (i.e. layer normal). Quasi-long-","rect":[173.27743530273438,384.8698425292969,382.0543526993482,377.2754821777344]},{"page":89,"text":"range positional order along the layer normal and two-","rect":[173.27743530273438,394.7890319824219,360.5835275040357,387.1946716308594]},{"page":89,"text":"dimensional liquid-like order within the layer plane","rect":[173.27743530273438,404.7650146484375,349.8329405524683,397.170654296875]},{"page":89,"text":"Optically active, chiral version of SmA phase","rect":[161.32528686523438,414.7404479980469,317.6892332770777,407.129150390625]},{"page":89,"text":"Optically biaxial, tilted, lamellar phase: the director forms an","rect":[161.3255615234375,424.6595153808594,371.1093801894657,417.0482177734375]},{"page":89,"text":"angle with the normal to layers. Quasi-long-range positional","rect":[173.2776641845703,434.635498046875,379.63134483542509,427.0411376953125]},{"page":89,"text":"order along the layer normal and two-dimensional liquid-like","rect":[173.2776641845703,444.6114501953125,382.75853869699957,437.01708984375]},{"page":89,"text":"structure within the layer plane","rect":[173.2776641845703,454.58740234375,279.9336790778589,446.9930419921875]},{"page":89,"text":"Optically active chiral analogy of SmC phase showing","rect":[161.32550048828126,464.5062255859375,348.1103872695438,456.8949279785156]},{"page":89,"text":"macroscopic periodicity with twist axis perpendicular to","rect":[173.27760314941407,474.482177734375,365.79575103907509,466.8878173828125]},{"page":89,"text":"smectic layers. Quasi-long-range positional order along the","rect":[173.27760314941407,484.4581298828125,375.8652891852808,476.86376953125]},{"page":89,"text":"layer normal and two-dimensional liquid-like structure within","rect":[173.27760314941407,494.4341125488281,384.6521047988407,486.8397521972656]},{"page":89,"text":"the layer plane. Single layers of the same symmetry may form","rect":[173.27760314941407,504.3533020019531,385.1766686284064,496.7589416503906]},{"page":89,"text":"different phases in the bulk: ferroelectric (SmC*),","rect":[173.27760314941407,514.3292846679688,344.2953211982485,506.7349548339844]},{"page":89,"text":"antiferroelectric (SmCA*) and ferrielectric (SmCg*).","rect":[173.27760314941407,525.0637817382813,352.6029689033266,516.7100830078125]},{"page":89,"text":"Twist-grain-boundary (chiral) phases consisted of twisted grains","rect":[161.32534790039063,534.2235107421875,381.3555343913964,526.629150390625]},{"page":89,"text":"or blocks of the smectic A* (or C*) phases with defect walls","rect":[173.27745056152345,544.199462890625,381.1346786785058,536.6051025390625]},{"page":89,"text":"(boundaries) between them","rect":[173.27745056152345,553.8367919921875,266.12414787645329,546.5810546875]},{"page":89,"text":"A stack of interacting hexatic layers with three-dimensional, long-","rect":[161.32528686523438,564.1510620117188,385.17477506262949,556.5567016601563]},{"page":89,"text":"range, sixfold, bond orientational order and liquid-like","rect":[173.27737426757813,574.0703125,359.5901427009058,566.4759521484375]},{"page":89,"text":"positional correlations within the layers","rect":[173.27737426757813,584.0462646484375,308.6841171550683,576.451904296875]},{"page":89,"text":"(continued)","rect":[346.0447998046875,595.8941650390625,385.17132657630136,588.6215209960938]},{"page":90,"text":"72","rect":[53.81291580200195,42.55716323852539,62.27398437647744,36.73228454589844]},{"page":90,"text":"Table 4.2","rect":[53.812843322753909,65.89774322509766,87.50059265284463,59.53101348876953]},{"page":90,"text":"Symbol","rect":[53.812843322753909,79.19002532958985,80.16061882224146,71.59567260742188]},{"page":90,"text":"SmF","rect":[53.812843322753909,89.60116577148438,69.76195653483423,83.77629089355469]},{"page":90,"text":"BPI","rect":[53.81328201293945,129.34649658203126,66.9364022598951,123.74174499511719]},{"page":90,"text":"BPII","rect":[53.81291580200195,139.32281494140626,69.75357144934823,133.7180633544922]},{"page":90,"text":"BPIII","rect":[53.812923431396487,149.29779052734376,72.57111447913339,143.6930389404297]},{"page":90,"text":"B1, B2, ..., Bn","rect":[53.812923431396487,190.60626220703126,101.56550348117094,183.5392608642578]},{"page":90,"text":"SmBcr","rect":[53.81257629394531,230.39637756347657,75.6993830546659,223.2681121826172]},{"page":90,"text":"SmE","rect":[53.812992095947269,258.96356201171877,70.28669545957797,253.13868713378907]},{"page":90,"text":"Unknown","rect":[107.56914520263672,189.24588012695313,140.8719229140751,183.37020874023438]},{"page":90,"text":"4 Liquid Crystal Phases","rect":[303.6742248535156,44.275840759277347,385.1686142253808,36.68148422241211]},{"page":90,"text":"Structural features","rect":[161.3259735107422,78.0,224.02930148124018,71.59567260742188]},{"page":90,"text":"Tilted analogy of the hexatic phase. A stack of interacting hexatic","rect":[161.3255615234375,91.3197250366211,385.16576525949957,83.72537231445313]},{"page":90,"text":"layers with three-dimensional, long-range, sixfold, bond","rect":[173.2776641845703,101.2956771850586,365.4032034316532,93.70132446289063]},{"page":90,"text":"orientational order and liquid-like positional correlations","rect":[173.2776641845703,111.2716293334961,367.3340194988183,103.67727661132813]},{"page":90,"text":"within the layers","rect":[173.2776641845703,121.1908187866211,230.34419711112299,113.59646606445313]},{"page":90,"text":"Blue phases BPI (body-centered cubic) and BPII (simple cubic)","rect":[161.3251953125,131.16622924804688,379.5582589493482,123.57186889648438]},{"page":90,"text":"are chiral phases with three-dimensional macroscopic","rect":[173.2772979736328,141.14224243164063,356.83007190012457,133.54788208007813]},{"page":90,"text":"periodicity and liquid-like molecular correlations. Three-","rect":[173.2772979736328,151.11819458007813,368.50045865637949,143.52383422851563]},{"page":90,"text":"dimensional photonic bandgap crystals showing optical","rect":[173.2772979736328,161.03738403320313,362.80358604636259,153.44302368164063]},{"page":90,"text":"activity but no birefringence. BPIII is lower symmetry “foggy”","rect":[173.2772979736328,171.01336669921876,385.18822619699957,163.41900634765626]},{"page":90,"text":"phase strongly scattering light","rect":[173.2772979736328,180.98931884765626,276.1046724721438,173.39495849609376]},{"page":90,"text":"Series of achiral phases formed by banana- or bent-shape","rect":[161.32571411132813,190.96456909179688,357.47448107981207,183.37020874023438]},{"page":90,"text":"molecules. The phase symmetry reduces with suffix n.","rect":[173.27781677246095,200.88381958007813,359.4323451240297,193.28945922851563]},{"page":90,"text":"Manifest spontaneous brake of mirror symmetry and","rect":[173.27696228027345,210.8597412109375,353.30402130274697,203.265380859375]},{"page":90,"text":"interesting ferroelectric and antiferroelectric properties","rect":[173.27696228027345,220.83572387695313,360.7464035320214,213.24136352539063]},{"page":90,"text":"Crystalline lamellar phase with upright molecules and hexagonal","rect":[161.32528686523438,230.81112670898438,383.6407747670657,223.21676635742188]},{"page":90,"text":"lattice. True three-dimensional positional order. Soft crystal","rect":[173.27737426757813,240.73037719726563,378.1308870717532,233.13601684570313]},{"page":90,"text":"with small shear elastic modulus","rect":[173.27737426757813,248.97914123535157,285.08789523124019,243.1119384765625]},{"page":90,"text":"Biaxial crystal with upright molecules having true three-","rect":[161.3251953125,260.6819763183594,355.3164376602857,253.08761596679688]},{"page":90,"text":"dimensional positional order. Rectangular in-plane lattice and","rect":[173.2772979736328,270.6012268066406,384.5722708144657,263.0068664550781]},{"page":90,"text":"herringbone packing of molecules (orthorhombic syngony).","rect":[173.2772979736328,280.5771484375,377.0689418037172,272.9827880859375]},{"page":90,"text":"Soft crystal with small shear elastic modulus","rect":[173.2772979736328,290.5531311035156,326.55212862967769,282.9587707519531]},{"page":90,"text":"Now we are ready to discuss spontaneous break of mirror symmetry. An achiral","rect":[65.76496887207031,325.9026794433594,385.1150041348655,316.9681396484375]},{"page":90,"text":"phase is spatially uniform and has mirror symmetry, i.e. its potential energy has a","rect":[53.812950134277347,337.8622131347656,385.1568073101219,328.92767333984377]},{"page":90,"text":"minimum located at zero tilt angle #, see Fig. 4.39b. With decreasing temperature,","rect":[53.812950134277347,349.82177734375,385.1169399788547,340.6680908203125]},{"page":90,"text":"the same molecules can acquire a collective tilt, some of them become tilted to the","rect":[53.81296157836914,361.78131103515627,385.16968572809068,352.84674072265627]},{"page":90,"text":"left with respect to the smectic layer normal (positive # in Fig. 4.42), the others to","rect":[53.81296157836914,373.7408752441406,385.14385310224068,364.5871887207031]},{"page":90,"text":"the right (negative #) in equal amounts. In fact, due to the tilt a triple of non","rect":[53.81298065185547,385.70037841796877,385.16082087567818,376.54669189453127]},{"page":90,"text":"coplanar vectors occurs, the vector of layer normal h, the vector of polarization Ps","rect":[53.812984466552737,397.6599426269531,385.1724078689846,388.72540283203127]},{"page":90,"text":"and the director n, that is necessary condition for chirality. This results in a break of","rect":[53.812843322753909,409.6202087402344,385.14876685945168,400.6856689453125]},{"page":90,"text":"the uniform structure and formation of right-handed and left-handed ferroelectric","rect":[53.81284713745117,421.5229797363281,385.16260564996568,412.548583984375]},{"page":90,"text":"domains. Now the potential energy has two minima at the tilt angles +# and \u0004# for","rect":[53.812843322753909,433.4825134277344,385.1795590957798,424.3288269042969]},{"page":90,"text":"the two types of domains, like in Fig. 4.39b. The banana phases manifest remark-","rect":[53.81288146972656,445.4420471191406,385.1138852676548,436.50750732421877]},{"page":90,"text":"able electrooptical properties; for example, upon application of a d.c. voltage, the","rect":[53.81288146972656,457.401611328125,385.17066229059068,448.467041015625]},{"page":90,"text":"directors rotate in opposite direction in the domains of opposite chirality.","rect":[53.81288146972656,469.36114501953127,348.8049587776828,460.42657470703127]},{"page":90,"text":"In conclusion of this chapter we demonstrate Table 4.2, in which the most","rect":[65.76490020751953,481.3207092285156,385.13679368564677,472.38616943359377]},{"page":90,"text":"important liquid crystal phases and their structural properties are listed.","rect":[53.81288146972656,493.2802429199219,341.5752224495578,484.345703125]},{"page":90,"text":"References","rect":[53.812843322753909,536.7923583984375,109.59614448282879,528.0072631835938]},{"page":90,"text":"1. Demus, D., Richter, L.: Textures of Liquid Crystals. Verlag Chemie, Weinheim (1978)","rect":[58.06126022338867,564.0376586914063,366.90301602942636,556.4432983398438]},{"page":90,"text":"2. Arnold, H., Sackmann, H.: Isomorphiebeziechungen zwischen kristallin-fl€ussigen Phasen. 4.","rect":[58.06126022338867,573.9569091796875,385.188265534186,566.0]},{"page":90,"text":"Mitteilung: Mischbarkeit in bin€aren Systemen mit mehreren Phasen. Z. fu€r Elektrochemie 11,","rect":[68.59784698486328,583.932861328125,385.2059657294985,576.0]},{"page":90,"text":"1171–1177 (1959)","rect":[68.59783172607422,593.5701904296875,131.65649503333263,586.263671875]},{"page":91,"text":"References","rect":[53.81043243408203,42.52982711791992,91.4791153240136,36.68801498413086]},{"page":91,"text":"73","rect":[376.7440490722656,42.56369400024414,385.2051138320438,36.73881530761719]},{"page":91,"text":"3.","rect":[58.06126022338867,65.22824096679688,64.40706131055318,59.40336608886719]},{"page":91,"text":"4.","rect":[58.06126022338867,95.05450439453125,64.40706131055318,89.33122253417969]},{"page":91,"text":"5.","rect":[58.06126022338867,115.05130004882813,64.40706131055318,109.12482452392578]},{"page":91,"text":"6.","rect":[58.06126022338867,135.0,64.40706131055318,129.07077026367188]},{"page":91,"text":"7.","rect":[58.06126022338867,145.0,64.40706131055318,139.21604919433595]},{"page":91,"text":"8.","rect":[58.06126022338867,165.0,64.40706131055318,159.0494842529297]},{"page":91,"text":"9.","rect":[58.060420989990237,185.0,64.40622207715474,178.9446563720703]},{"page":91,"text":"10.","rect":[53.81211853027344,204.721435546875,64.38845703198872,198.8965606689453]},{"page":91,"text":"11.","rect":[53.81211853027344,245.0,64.38845703198872,238.68690490722657]},{"page":91,"text":"12.","rect":[53.812103271484378,255.0,64.38844177319966,248.66285705566407]},{"page":91,"text":"13.","rect":[53.812103271484378,285.0,64.38844177319966,278.5340576171875]},{"page":91,"text":"14.","rect":[53.81127166748047,305.0,64.38761016919576,298.4859619140625]},{"page":91,"text":"15.","rect":[53.811279296875,324.2060241699219,64.38761779859029,318.279541015625]},{"page":91,"text":"16.","rect":[53.81043243408203,344.157958984375,64.38677093579732,338.28228759765627]},{"page":91,"text":"17.","rect":[53.810447692871097,364.0531311035156,64.38678619458638,358.2282409667969]},{"page":91,"text":"18.","rect":[53.81045150756836,384.0050354003906,64.38679000928365,378.1801452636719]},{"page":91,"text":"19.","rect":[53.810455322265628,394.0,64.38679382398091,388.099365234375]},{"page":91,"text":"20.","rect":[53.810420989990237,404.0,64.38675949170552,398.0753173828125]},{"page":91,"text":"21.","rect":[53.811256408691409,424.0,64.38759491040669,418.0271911621094]},{"page":91,"text":"22.","rect":[53.812095642089847,444.0,64.38843414380513,437.9223937988281]},{"page":91,"text":"23.","rect":[53.812095642089847,474.0,64.38843414380513,467.7934875488281]},{"page":91,"text":"24.","rect":[53.812103271484378,494.0,64.38844177319966,487.7453918457031]},{"page":91,"text":"25.","rect":[53.81125259399414,514.0,64.38759109570943,507.53900146484377]},{"page":91,"text":"26.","rect":[53.81126403808594,534.0,64.38760253980122,527.5416870117188]},{"page":91,"text":"27.","rect":[53.81043243408203,553.3125610351563,64.38677093579732,547.4876708984375]},{"page":91,"text":"28.","rect":[53.81043243408203,583.1837158203125,64.38677093579732,577.3588256835938]},{"page":91,"text":"Leadbetter, A.J.: Structural studies of nematic, smectic A and smectic C phases. 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Science 278, 1924–1927 (1997)","rect":[68.5944595336914,574.6444702148438,216.5641793595045,567.1008911132813]},{"page":91,"text":"Goodby, J.W., Gray, G.W.: Guide to the nomenclature and classification of liquid crystals.","rect":[68.5944595336914,584.90234375,385.14245864942037,577.3079833984375]},{"page":91,"text":"In: Demus, D., Goodby, J., Gray, G.W., Spiess, H.-W., Vill, V. (eds.) Physical Properties","rect":[68.5944595336914,594.8782958984375,385.1450546550683,587.283935546875]},{"page":91,"text":"of Liquid Crystals, pp. 17–23. Wiley-VCH, Weinheim (1999)","rect":[68.5944595336914,604.8541870117188,280.17706388098886,597.2598266601563]},{"page":92,"text":"Chapter5","rect":[53.812843322753909,72.10812377929688,114.14115996551633,59.571903228759769]},{"page":92,"text":"Structure Analysis and X-Ray Diffraction","rect":[53.812843322753909,91.18268585205078,339.27665178564578,76.0426254272461]},{"page":92,"text":"5.1 Diffraction Studies and X-Ray Experiment","rect":[53.812843322753909,212.6539764404297,300.4240460469543,201.29905700683595]},{"page":92,"text":"5.1.1 General Consideration","rect":[53.812843322753909,239.85919189453126,202.56314057206274,231.3250732421875]},{"page":92,"text":"The diffraction of the electromagnetic waves or the de Broglie waves of electrons","rect":[53.812843322753909,269.5052795410156,385.08603300200658,260.57073974609377]},{"page":92,"text":"and neutrons on a liquid, liquid crystalline or crystalline structures results in a","rect":[53.812843322753909,281.4648132324219,385.15772283746568,272.5302734375]},{"page":92,"text":"characteristic pattern from that one can restore a distribution of density in space or","rect":[53.812843322753909,293.42437744140627,385.1467221817173,284.48980712890627]},{"page":92,"text":"density function r(r) [1, 2]. What kind of density we speak about?","rect":[53.812843322753909,305.3271484375,322.68103826715318,296.392578125]},{"page":92,"text":"The electron density is probed by electromagnetic waves, as in optics. In fact, the","rect":[65.76586151123047,317.28668212890627,385.17359197809068,308.33221435546877]},{"page":92,"text":"same theory of light diffraction and dispersion is relevant to the X-ray diffraction","rect":[53.8138427734375,329.2462158203125,385.0929802995063,320.3116455078125]},{"page":92,"text":"for wavelengths comparable to the size of atoms. For X-rays, the wavelength lX \u0001","rect":[53.8138427734375,341.2057800292969,385.14807918760689,331.9524841308594]},{"page":92,"text":"˚","rect":[81.5705337524414,343.493408203125,84.88528571931494,341.51129150390627]},{"page":92,"text":"0.5–1 A depends on material of the anticathode in an X-ray tube. In a synchrotron,","rect":[53.814231872558597,353.1660461425781,385.1520657112766,344.1717224121094]},{"page":92,"text":"the electromagnetic wave spectrum is very large and determined by the speed of","rect":[53.814231872558597,365.1256103515625,385.1491635879673,356.1910400390625]},{"page":92,"text":"moving electrons. From the experiment we can find the density (or number) of","rect":[53.814231872558597,377.08514404296877,385.15010963288918,368.15057373046877]},{"page":92,"text":"electrons in atomic shells.","rect":[53.814231872558597,386.9828796386719,158.63106961752659,380.11016845703127]},{"page":92,"text":"An electric potential of a substance is probed by charge particles emitted, for","rect":[65.7662582397461,400.9474792480469,385.1789487442173,391.9930114746094]},{"page":92,"text":"example, by an electronic gun or an accelerator. The electron beam is scattered by","rect":[53.814231872558597,412.9070129394531,385.16909113935005,403.97247314453127]},{"page":92,"text":"the electric potential of positive nuclei and negative electrons and the maximum","rect":[53.814231872558597,424.8665466308594,385.0924143660124,415.9320068359375]},{"page":92,"text":"positive potential corresponds to the center of an atom. The electrons in the beam","rect":[53.814231872558597,436.8260803222656,385.14118145585618,427.89154052734377]},{"page":92,"text":"have the de Broglie wavelength le dependent on their velocity v, i. e. on the","rect":[53.814231872558597,448.78668212890627,385.1183551616844,439.5323181152344]},{"page":92,"text":"accelerating voltage V, namely, eV ¼ meve2/2 ¼W","rect":[53.8133659362793,460.7462463378906,261.04240633864546,449.6957092285156]},{"page":92,"text":"le ¼ h.men ¼ h.ð2meWÞ1=2:","rect":[158.15370178222657,492.83587646484377,280.81532227677249,474.92694091796877]},{"page":92,"text":"(5.1)","rect":[366.0970153808594,488.1215515136719,385.16927467195168,479.5256652832031]},{"page":92,"text":"Here me is electron mass and h is Planck’s constant. Hence, for electron energy","rect":[65.76555633544922,516.3502197265625,385.14006892255318,507.3957824707031]},{"page":92,"text":"˚","rect":[283.4514465332031,518.6364135742188,286.76619850007668,516.654296875]},{"page":92,"text":"W ¼ 1 eV–10 keV, the wavelength is le \u0001 10–0.1 A. From this diffraction","rect":[53.81315612792969,528.2529907226563,385.10186091474068,518.999755859375]},{"page":92,"text":"experiment we can find the distribution of the electric potential correlated to","rect":[53.81281661987305,540.2127685546875,385.1407403092719,531.2782592773438]},{"page":92,"text":"some extent with the distribution of the mass density. Another technique for","rect":[53.81281661987305,552.1723022460938,385.1208432754673,543.23779296875]},{"page":92,"text":"mapping the local electric potential is Atomic Force Microscopy [3].","rect":[53.81281661987305,564.1318359375,331.6248745491672,555.1973266601563]},{"page":92,"text":"The distribution of the mass of nuclei almost equal to the full mass density is","rect":[65.76482391357422,576.0913696289063,385.18649686919408,567.1170043945313]},{"page":92,"text":"probed by neutron beams. To this effect, one can use the so-called thermal or cold","rect":[53.81281661987305,588.0509033203125,385.16460505536568,579.1163940429688]},{"page":92,"text":"L.M. Blinov, Structure and Properties of Liquid Crystals,","rect":[53.812843322753909,623.7623291015625,251.63688919141254,616.1510620117188]},{"page":92,"text":"DOI 10.1007/978-90-481-8829-1_5, # Springer ScienceþBusiness Media B.V. 2011","rect":[53.812843322753909,633.698486328125,347.38995880274697,625.4920043945313]},{"page":92,"text":"75","rect":[376.7464599609375,622.0606079101563,385.2075247207157,616.1340942382813]},{"page":93,"text":"76","rect":[53.812843322753909,42.55765151977539,62.2739118972294,36.68197250366211]},{"page":93,"text":"5 Structure Analysis and X-Ray Diffraction","rect":[235.81301879882813,44.276329040527347,385.1466421523563,36.63117599487305]},{"page":93,"text":"neutrons with energy W \u0001 0.05 eV provided by nuclear reactors. The corresponding","rect":[53.812843322753909,68.2883529663086,385.1148308854438,59.294044494628909]},{"page":93,"text":"˚","rect":[328.8255615234375,70.575439453125,332.14031349031105,68.59330749511719]},{"page":93,"text":"wavelength is in the proper range for structure analysis, ln ¼ 1–1.5 A, because the","rect":[53.8138313293457,80.24788665771485,385.1753619976219,70.9946060180664]},{"page":93,"text":"mass of a neutron is large, mn ¼ 1840me.","rect":[53.81462860107422,92.20760345458985,221.19553800131565,83.27305603027344]},{"page":93,"text":"All these techniques have certain advantages and disadvantages. The electron","rect":[65.7655258178711,104.1104965209961,385.10955134442818,95.17594909667969]},{"page":93,"text":"diffraction experiment requires for vacuum or low pressure gas, and thin films, very","rect":[53.8134880065918,116.0699691772461,385.11553278974068,107.13542175292969]},{"page":93,"text":"often on conductive substrates (otherwise the surface is charged by incoming","rect":[53.8134880065918,128.02957153320313,385.16631403974068,119.09501647949219]},{"page":93,"text":"electrons). On the other hand, the interaction between charges is very strong and","rect":[53.8134880065918,139.98910522460938,385.14238825849068,131.0346221923828]},{"page":93,"text":"one may operate with small samples and short expositions and, due to short","rect":[53.8134880065918,151.94863891601563,385.144423080178,143.0140838623047]},{"page":93,"text":"wavelengths, the spatial resolution can be very high. In addition, the data process-","rect":[53.8134880065918,163.908203125,385.1742185196079,154.97364807128907]},{"page":93,"text":"ing is sometimes simpler due to a small curvature of the Ewald sphere to be","rect":[53.8134880065918,175.86773681640626,385.1493610210594,166.9331817626953]},{"page":93,"text":"discussed later. Using electrons even light atoms like hydrogen are well seen.","rect":[53.8134880065918,187.8272705078125,366.6922573616672,178.89271545410157]},{"page":93,"text":"Neutron diffraction requires for larger samples (linear dimension about 1 cm)","rect":[65.76551055908203,199.73004150390626,385.1463864883579,190.7954864501953]},{"page":93,"text":"and the reactors producing short lifetime (minutes) cold neutrons are expensive. On","rect":[53.8134880065918,211.68960571289063,385.1294793229438,202.73512268066407]},{"page":93,"text":"the other hand, in contrast to X-rays, neutrons are sensitive to isotopes and atoms","rect":[53.8134880065918,223.64910888671876,385.1095315371628,214.7145538330078]},{"page":93,"text":"with slightly different atomic mass, such as Co and Ni. In addition, a neutron has an","rect":[53.8134880065918,235.60867309570313,385.1503838639594,226.6741180419922]},{"page":93,"text":"intrinsic magnetic moment about two Bohr magnetons, pm ¼ 1.9 mB. For this","rect":[53.8134880065918,247.56820678710938,385.1460305606003,238.63365173339845]},{"page":93,"text":"reason, neutrons strongly interact with magnetic moments of electrons and nuclei.","rect":[53.814205169677737,259.5294189453125,385.17693753744848,250.59486389160157]},{"page":93,"text":"Thus, a neutron experiment provides a unique possibility for studying different","rect":[53.814205169677737,271.48895263671877,385.1749711758811,262.55438232421877]},{"page":93,"text":"magnetic structures, spin effects, para- and ferro-magnetism. However, the X-ray","rect":[53.814205169677737,283.448486328125,385.10527888349068,274.513916015625]},{"page":93,"text":"technique is the most universal for the structure analysis. In fact, the majority of","rect":[53.814205169677737,295.35125732421877,385.15010963288918,286.41668701171877]},{"page":93,"text":"structures of crystals from the simplest ones to those formed by protein molecules","rect":[53.814205169677737,307.3108215332031,385.1610452090378,298.37628173828127]},{"page":93,"text":"were found by the X-ray diffraction.","rect":[53.814205169677737,319.27032470703127,200.69256253744846,310.33575439453127]},{"page":93,"text":"5.1.2 X-Ray Experiment","rect":[53.812843322753909,369.4380798339844,182.09413358520087,358.87200927734377]},{"page":93,"text":"One can use conventional low intensity sources (X-ray tubes) providing very narrow","rect":[53.812843322753909,396.9804992675781,385.24328374817699,388.0260314941406]},{"page":93,"text":"spectral lines, but low intensity. A set-up consists of an X-ray tube (X), beam","rect":[53.812843322753909,408.9400634765625,385.2332835066374,400.0054931640625]},{"page":93,"text":"collimators (C), one or several monochromators (M), a detector (D) and a data","rect":[53.812843322753909,420.50115966796877,385.2362750835594,411.96502685546877]},{"page":93,"text":"acquisition system (PC). A sample is installed in a camera with controllable tempera-","rect":[53.812843322753909,432.859130859375,385.24227271882668,423.924560546875]},{"page":93,"text":"ture, Fig. 5.1. In the case of a liquid crystal, a magnetic or electric field is necessary","rect":[53.812843322753909,444.7619323730469,385.1766289811469,435.7676086425781]},{"page":93,"text":"for the sample orientation. Historically, for a long time, fluorescent screens and","rect":[53.812843322753909,456.7214660644531,385.1715935807563,447.78692626953127]},{"page":93,"text":"X","rect":[146.9525146484375,494.9124755859375,152.70168224641086,489.41571044921877]},{"page":93,"text":"N","rect":[218.91700744628907,491.15509033203127,225.07981349964107,485.5882568359375]},{"page":93,"text":"M","rect":[172.56790161132813,537.64794921875,180.4840016419173,532.1511840820313]},{"page":93,"text":"S","rect":[221.0368194580078,538.1370849609375,225.86475052963508,532.4102783203125]},{"page":93,"text":"D","rect":[256.0307922363281,520.0662841796875,262.19575498742338,514.5695190429688]},{"page":93,"text":"Fig. 5.1 A set-up for a study of X-ray diffraction on liquid crystals: X-ray tube (X), beam","rect":[53.812843322753909,564.1054077148438,385.1813378178595,556.3756103515625]},{"page":93,"text":"collimators (C), mirrors (M), a detector (D) and a data acquisition system (PC). A sample is","rect":[53.812843322753909,573.9569091796875,385.15845187186519,566.362548828125]},{"page":93,"text":"represented by a stack of parallel layers placed in a camera with controllable temperature installed","rect":[53.812843322753909,583.932861328125,385.1482290664188,576.3385009765625]},{"page":93,"text":"between the poles of a magnet","rect":[53.812843322753909,593.9088134765625,158.12597374167505,586.314453125]},{"page":94,"text":"5.2 X-Ray Scattering","rect":[53.81283950805664,44.276573181152347,126.13638824610635,36.63142013549805]},{"page":94,"text":"77","rect":[376.7464904785156,42.55789566040039,385.2075552382938,36.85154342651367]},{"page":94,"text":"photographic films were primary tools for detecting X-rays. The latter are two-","rect":[53.812843322753909,68.2883529663086,385.2612241348423,59.35380554199219]},{"page":94,"text":"dimensional, very cheap and sensitive but their processing requires densitometers","rect":[53.812843322753909,80.24788665771485,385.25128568755346,71.31333923339844]},{"page":94,"text":"for the image digitizing. Since few decades, point detectors have been using every-","rect":[53.812843322753909,92.20748138427735,385.2641843399204,83.27293395996094]},{"page":94,"text":"where based on the proportional and scintillation counters both one- and two-dimen-","rect":[53.812843322753909,104.11019134521485,385.2651609024204,95.17564392089844]},{"page":94,"text":"sional. Automatic two-dimensional detectors are very convenient because they grasp","rect":[53.812843322753909,116.0697250366211,385.17360774091255,107.13517761230469]},{"page":94,"text":"the entire diffraction pattern and save a lot of time.","rect":[53.812843322753909,128.02932739257813,253.96514554526096,119.09477233886719]},{"page":94,"text":"Nowadays,","rect":[65.76486206054688,139.98886108398438,110.31293912436252,131.05430603027345]},{"page":94,"text":"however,","rect":[115.55482482910156,138.0,152.42620511557346,131.05430603027345]},{"page":94,"text":"synchrotrons","rect":[157.58447265625,139.98886108398438,209.08258451567844,131.05430603027345]},{"page":94,"text":"are","rect":[214.34140014648438,137.92703247070313,226.55521429254379,133.2854461669922]},{"page":94,"text":"available","rect":[231.78814697265626,137.92703247070313,268.0254596538719,131.05430603027345]},{"page":94,"text":"that","rect":[273.25244140625,137.92703247070313,288.273329330178,131.05430603027345]},{"page":94,"text":"provide","rect":[293.5311584472656,139.98886108398438,324.00098455621568,131.05430603027345]},{"page":94,"text":"million","rect":[329.27374267578127,137.92703247070313,358.26039973310005,131.05430603027345]},{"page":94,"text":"times","rect":[363.48736572265627,137.92703247070313,385.13779081450658,131.05430603027345]},{"page":94,"text":"higher intensity and wide spectrum of the polarized emission. One can use different","rect":[53.812843322753909,151.94839477539063,385.1696611172874,142.99391174316407]},{"page":94,"text":"wavelength ranges and short expositions when studying dynamic processes. Of","rect":[53.812843322753909,163.907958984375,385.1666196426548,154.95347595214845]},{"page":94,"text":"course, there are not so many synchrotron accelerators all over the world but they","rect":[53.812843322753909,175.86749267578126,385.1646660905219,166.9329376220703]},{"page":94,"text":"have many output beams, as shown in Fig. 5.2, and attached are many experimental","rect":[53.812843322753909,187.8270263671875,385.180555892678,178.83270263671876]},{"page":94,"text":"stations. Such a work is usually organized at the international level.","rect":[53.812862396240237,199.72976684570313,326.8200954964328,190.7952117919922]},{"page":94,"text":"What does an X-ray diffraction experiment bring about? In fact, a lot:","rect":[65.76488494873047,211.6893310546875,348.025526595803,202.75477600097657]},{"page":94,"text":"1.","rect":[53.812862396240237,228.0,61.27851911093478,220.78225708007813]},{"page":94,"text":"2.","rect":[53.812862396240237,252.0,61.27851911093478,244.64456176757813]},{"page":94,"text":"3.","rect":[53.81285858154297,276.0,61.278515296237518,268.5636901855469]},{"page":94,"text":"Number of diffraction peaks on a diffractogram, their precise positions and the","rect":[66.2745361328125,229.65704345703126,385.17163885309068,220.7224884033203]},{"page":94,"text":"symmetry of the pattern","rect":[66.27454376220703,241.61660766601563,162.97274104169379,232.6820526123047]},{"page":94,"text":"The peak amplitudes I and areas A under peaks as functions of temperature,","rect":[66.2745361328125,253.51934814453126,385.1158718636203,244.5847930908203]},{"page":94,"text":"pressure, external fields, etc.","rect":[66.2745361328125,265.4789123535156,181.00278897787815,256.54437255859377]},{"page":94,"text":"The peak profile that is the profile of the diffraction intensity I(q) within a","rect":[66.2745361328125,277.4384765625,385.1605914898094,268.50390625]},{"page":94,"text":"particular","rect":[66.27552032470703,289.3980407714844,104.69876991120947,280.4635009765625]},{"page":94,"text":"diffraction","rect":[109.94862365722656,288.0,152.2112206559516,280.4635009765625]},{"page":94,"text":"spot,","rect":[157.416259765625,289.3980407714844,176.5582241585422,281.4794616699219]},{"page":94,"text":"which","rect":[181.71649169921876,288.0,206.15407649091254,280.4635009765625]},{"page":94,"text":"is","rect":[211.34222412109376,288.0,217.9717141543503,280.4635009765625]},{"page":94,"text":"a","rect":[223.23748779296876,288.0,227.68701971246566,282.0]},{"page":94,"text":"function","rect":[232.9239501953125,288.0,266.1909112077094,280.4635009765625]},{"page":94,"text":"of","rect":[271.4417724609375,288.0,279.7336362442173,280.4635009765625]},{"page":94,"text":"the","rect":[284.9794921875,288.0,297.2032855816063,280.4635009765625]},{"page":94,"text":"diffraction","rect":[302.4262390136719,288.0,344.6619805436469,280.4635009765625]},{"page":94,"text":"angle","rect":[349.9506530761719,289.3980407714844,371.60105169488755,280.4635009765625]},{"page":94,"text":"or","rect":[376.85687255859377,288.0,385.14873634187355,282.0]},{"page":94,"text":"scattering wavevector q. The key problem of X-ray analysis is how to relate","rect":[66.27552032470703,301.3575744628906,385.12781561090318,292.42303466796877]},{"page":94,"text":"I(q) to the electron density function or density correlation function that takes into","rect":[66.27552032470703,313.317138671875,385.15471736005318,304.382568359375]},{"page":94,"text":"account thermal fluctuations.","rect":[66.27552032470703,323.21484375,182.78956265951877,316.34210205078127]},{"page":94,"text":"5.2 X-Ray Scattering","rect":[53.812843322753909,375.951416015625,169.6868413915593,364.644287109375]},{"page":94,"text":"5.2.1 Scattering by a Single Electron","rect":[53.812843322753909,405.317626953125,244.68014496659399,394.6798400878906]},{"page":94,"text":"Protons and electrons are charge particles interacting with electromagnetic waves","rect":[53.812843322753909,432.8594665527344,385.16464628325658,423.9249267578125]},{"page":94,"text":"andtheirnumberandparticular locationdeterminetheamplitude ofscatteredwaves.","rect":[53.812843322753909,444.7622375488281,385.21636624838598,435.82769775390627]},{"page":94,"text":"Asthe electronsarevery light they contribute muchstronger toX-rayscattering than","rect":[53.812843322753909,456.7217712402344,385.17559138349068,447.7872314453125]},{"page":94,"text":"protons (nuclei). In fact intensity of scattering is even measured in electron units.","rect":[53.812843322753909,468.68133544921877,385.2402615120578,459.74676513671877]},{"page":94,"text":"Therefore, scattering by a single electron deserves a brief consideration.","rect":[53.812843322753909,480.640869140625,339.69280667807348,471.706298828125]},{"page":94,"text":"Fig. 5.2 A geometry of","rect":[53.812843322753909,513.7161865234375,136.59510892493419,505.9863586425781]},{"page":94,"text":"electromagnetic wave","rect":[53.812843322753909,523.6243896484375,128.44117877268315,516.030029296875]},{"page":94,"text":"emission from an accelerator","rect":[53.812843322753909,531.8478393554688,152.8691110001295,526.0059814453125]},{"page":94,"text":"of relativistic particles","rect":[53.812843322753909,543.5762939453125,130.34914095877924,535.98193359375]},{"page":94,"text":"(synchrotron). R is radius of","rect":[53.812843322753909,553.4955444335938,150.13366788489513,545.9011840820313]},{"page":94,"text":"the synchrotron ring. X-ray","rect":[53.81283950805664,563.4714965820313,147.2873739638798,555.8771362304688]},{"page":94,"text":"emission in the form of the","rect":[53.81283950805664,571.6949462890625,146.73485705637456,565.8530883789063]},{"page":94,"text":"cone is delivered to one of the","rect":[53.81283950805664,581.6962890625,155.34484240793706,575.8290405273438]},{"page":94,"text":"many experimental stations","rect":[53.81283950805664,593.3425903320313,147.73495944022455,585.7482299804688]},{"page":95,"text":"78","rect":[53.813961029052737,42.55728530883789,62.275029603528228,36.73240661621094]},{"page":95,"text":"5 Structure Analysis and X-Ray Diffraction","rect":[235.8141326904297,44.275962829589847,385.14777130274697,36.63080978393555]},{"page":95,"text":"Let linearly polarized, plane electromagnetic wave of amplitude E0 is incident","rect":[65.76496887207031,68.2883529663086,385.12141282627177,59.35380554199219]},{"page":95,"text":"on a free electron, Fig. 5.3. The equation of oscillatory motion of the electron about","rect":[53.814414978027347,80.24788665771485,385.1582475430686,71.25357818603516]},{"page":95,"text":"the centre of coordinate is:","rect":[53.814414978027347,90.18550109863281,161.7449327237327,83.27293395996094]},{"page":95,"text":"meðd2r=dt2Þ ¼ ðme=eÞðd2d=dt2Þ ¼ eE0 cosðot þ aÞ","rect":[116.06703186035156,117.71263885498047,322.9162637148972,106.43820190429688]},{"page":95,"text":"(5.2)","rect":[366.0995788574219,116.9755630493164,385.17183814851418,108.37967681884766]},{"page":95,"text":"where vector r is displacement of the electron that creates a dipole moment d ¼ er.","rect":[53.816123962402347,141.29306030273438,385.1858181526828,132.35850524902345]},{"page":95,"text":"The current produced by moving electron is proportional to its velocity v, i.e.","rect":[53.81714630126953,153.25259399414063,385.12212796713598,144.3180389404297]},{"page":95,"text":"j ¼ ev ¼ dd/dt, which, in turn, is a source the electromagnetic field in point P [4].","rect":[53.81714630126953,165.30178833007813,385.1609158089328,156.1979217529297]},{"page":95,"text":"1€","rect":[149.42996215820313,187.72726440429688,174.62966242841254,180.99398803710938]},{"page":95,"text":"E ¼ c2R0 ½ðd \u0003 nÞ \u0003 n\u0004 ¼ mc2R0 ½ðE0 \u0003 nÞ \u0003 n\u0004","rect":[122.75598907470703,202.94468688964845,316.2188256947412,187.0586700439453]},{"page":95,"text":"(5.3)","rect":[366.0973815917969,196.27210998535157,385.16964088288918,187.67620849609376]},{"page":95,"text":"Vector ðE0 \u0003 nÞis perpendicular to vector n and has modulus E0sing where","rect":[65.76595306396485,226.82321166992188,385.1119769878563,216.8723602294922]},{"page":95,"text":"g¼(p/2)\u0005y. Therefore, the modulus of the scattered field amplitude is E0cos2y.","rect":[53.81493377685547,238.44412231445313,385.1839565804172,229.20079040527345]},{"page":95,"text":"Note that the angle between the wavevectors of incident and scattered wave is","rect":[53.815269470214847,250.40377807617188,385.1899758731003,241.46922302246095]},{"page":95,"text":"assumed to be 2y according to the convention adopted below (see Fig. 5.4) and used","rect":[53.815269470214847,262.3065490722656,385.12725153974068,253.06321716308595]},{"page":95,"text":"throughout the book.","rect":[53.81523895263672,274.26611328125,137.89846463705784,265.33154296875]},{"page":95,"text":"The energy flux is given by the Pointing vector","rect":[65.76726531982422,286.22564697265627,256.30379615632668,277.29107666015627]},{"page":95,"text":"S ¼ ðc=4pÞE2n","rect":[188.51661682128907,311.7872314453125,250.485034650069,300.4566650390625]},{"page":95,"text":"(5.4)","rect":[366.0971984863281,311.0506286621094,385.1694577774204,302.4547424316406]},{"page":95,"text":"and the dipolar emission energy incident on a small surface element df ¼ R02dO in a","rect":[53.8137321472168,336.0312805175781,385.16123235895005,324.8860168457031]},{"page":95,"text":"solid angle dO is given by dW ¼ Sdf ¼ ðc=4pÞE2R02dO. After substituting E2 �� E2","rect":[53.81444549560547,347.99090576171877,385.18130140630105,336.2203674316406]},{"page":95,"text":"from Eq. 5.3 we find the intensity of the scattered, polarized wave.","rect":[53.812843322753909,359.2310485839844,323.4126858284641,350.2367248535156]},{"page":95,"text":"dW ¼ e4E02 cos22ydO","rect":[170.27610778808595,388.72698974609377,268.7348914141926,373.46148681640627]},{"page":95,"text":"4pm2c3","rect":[197.2951202392578,395.5850830078125,226.89245674690566,387.8017272949219]},{"page":95,"text":"(5.5)","rect":[366.0979919433594,390.2904968261719,385.17025123445168,381.6946105957031]},{"page":95,"text":"Fig. 5.3 Geometry of","rect":[53.812843322753909,424.61395263671877,130.02508634192638,416.6809387207031]},{"page":95,"text":"scattering linearly polarized","rect":[53.812843322753909,434.5221862792969,149.20633453516886,426.9278259277344]},{"page":95,"text":"electromagnetic wave bya","rect":[53.812843322753909,444.4981384277344,145.45131823801519,436.9037780761719]},{"page":95,"text":"single electron. The incident","rect":[53.812843322753909,454.4740905761719,151.32836631980005,446.8797302246094]},{"page":95,"text":"wave field E0 causes","rect":[53.812843322753909,464.0572814941406,124.78949435233392,456.7989807128906]},{"page":95,"text":"oscillatory displacement r of","rect":[53.81296920776367,474.3688049316406,152.5121774063795,466.7744445800781]},{"page":95,"text":"an electron and the scattered","rect":[53.81211853027344,482.61761474609377,151.57892364649698,476.7503967285156]},{"page":95,"text":"wave is detected in pointP","rect":[53.81211853027344,494.2640075683594,146.33985265299828,486.6696472167969]},{"page":95,"text":"E0","rect":[261.57794189453127,440.6342468261719,269.907141730108,433.07403564453127]},{"page":95,"text":"n","rect":[343.6663513183594,454.6854248046875,347.66298300017,450.88623046875]},{"page":95,"text":"γ","rect":[321.65911865234377,462.4557800292969,324.9443498947921,457.08892822265627]},{"page":95,"text":"2θ","rect":[327.7559814453125,471.1833190917969,335.91711472416548,465.488525390625]},{"page":95,"text":"r","rect":[320.6448669433594,488.6553955078125,323.72227333835357,484.856201171875]},{"page":95,"text":"Fig. 5.4 Illustration of an","rect":[53.812843322753909,544.5504760742188,143.73456329493448,536.8206787109375]},{"page":95,"text":"electromagnetic wave","rect":[53.812843322753909,554.4586791992188,128.44117877268315,546.8643188476563]},{"page":95,"text":"scattering by two material","rect":[53.812843322753909,564.4346313476563,143.06443504538599,556.8402709960938]},{"page":95,"text":"points: k0 and k are vector of","rect":[53.812843322753909,574.3538818359375,154.60899442298106,566.7592163085938]},{"page":95,"text":"incident and scattered waves,","rect":[53.813961029052737,582.6024169921875,154.09457656934223,576.7351684570313]},{"page":95,"text":"q is vector of scattering","rect":[53.813961029052737,594.3054809570313,134.92093414454386,586.7111206054688]},{"page":95,"text":"k","rect":[284.1568908691406,537.4067993164063,288.60114529931408,531.6640014648438]},{"page":95,"text":"k0","rect":[273.5409851074219,577.6730346679688,281.31863393786446,569.8226928710938]},{"page":95,"text":"k0r","rect":[309.8856201171875,536.1820678710938,321.25156084493309,528.3319091796875]},{"page":95,"text":"′","rect":[335.2589416503906,530.9114379882813,337.23327770120508,528.6318969726563]},{"page":95,"text":"r","rect":[314.4313659667969,559.7772216796875,317.54074541524559,555.41015625]},{"page":95,"text":"kr","rect":[325.9468078613281,569.8167114257813,333.50043169454247,564.0739135742188]},{"page":95,"text":"k","rect":[349.9761657714844,527.2009887695313,354.4204202016578,521.4581909179688]},{"page":95,"text":"k","rect":[345.8548583984375,562.8181762695313,350.29911282861095,557.0753784179688]},{"page":95,"text":"q","rect":[363.4524841308594,532.936279296875,367.09641306675027,527.9953002929688]},{"page":95,"text":"qq","rect":[363.73223876953127,540.0542602539063,379.14004846595386,534.03955078125]},{"page":95,"text":"k0","rect":[364.588623046875,552.3035278320313,372.36602773669258,544.453125]},{"page":96,"text":"5.2 X-Ray Scattering","rect":[53.812843322753909,44.274620056152347,126.13638824610635,36.62946701049805]},{"page":96,"text":"79","rect":[376.7464904785156,42.62367248535156,385.2075552382938,36.73106384277344]},{"page":96,"text":"Now, normalizing Eq. 5.5 by the Pointing vector of the incident wave E0 we find","rect":[65.76496887207031,68.2883529663086,385.1719903092719,59.294044494628909]},{"page":96,"text":"the differential cross-section of one-electron scattering:","rect":[53.814231872558597,80.24788665771485,277.3957658536155,71.27349853515625]},{"page":96,"text":"dse ¼ \u0002e2=mec2\u00032 cos2ydO","rect":[162.79891967773438,108.68547821044922,276.21200323059886,94.47854614257813]},{"page":96,"text":"(5.6)","rect":[366.0975036621094,106.9997329711914,385.16976295320168,98.40384674072266]},{"page":96,"text":"The emission of the dipole is symmetric with respect to the dipole axis x,","rect":[65.7660903930664,132.22463989257813,385.1837429573703,123.29008483886719]},{"page":96,"text":"Fig. 5.3, and has the 1-form in the xz plane (no emission exactly along the","rect":[53.81505584716797,144.18417358398438,385.11902654840318,135.18984985351563]},{"page":96,"text":"x-axis). It is spectacular that the cross section is independent of frequency.","rect":[53.8140983581543,156.14373779296876,354.5835842659641,147.2091827392578]},{"page":96,"text":"In order to obtain the total cross-section of scattering we should integrate the","rect":[65.7671127319336,168.10324096679688,385.1758197612938,159.1487579345703]},{"page":96,"text":"diagram over j from 0 to 2p and over 2# from 0 to p in the polar coordinates with","rect":[53.815101623535159,180.00601196289063,385.11711970380318,170.85232543945313]},{"page":96,"text":"the vertical polar axis x. The angle g¼(p/2) \u0005 2y will be a polar angle and anglej","rect":[53.8131217956543,191.965576171875,385.18977615054396,182.7222442626953]},{"page":96,"text":"an azimuthal angle in the zy plane. Then, with a volume element dO¼singdgdj, the","rect":[53.81513214111328,203.92510986328126,385.17688787652818,194.81126403808595]},{"page":96,"text":"integral ÐÐ sin3gdgd’ ¼ 8p=3 and the overall scattering cross-section of an electron","rect":[53.81616973876953,216.80697631835938,385.11089411786568,205.00613403320313]},{"page":96,"text":"irradiated by a linearly polarized light is given by the Thomson formula:","rect":[53.81484603881836,227.84478759765626,345.6713090665061,218.87039184570313]},{"page":96,"text":"se ¼ 83p\u0004mee2c2\u00052","rect":[182.7955322265625,268.5750427246094,255.72488472542129,242.07540893554688]},{"page":96,"text":"(5.7)","rect":[366.09765625,260.8880920410156,385.1699155410923,252.29220581054688]},{"page":96,"text":"Since we are mostly interested in scattering unpolarised X-ray radiation we","rect":[65.76622772216797,292.06439208984377,385.1112750835594,283.12982177734377]},{"page":96,"text":"should average Eq. 5.7 over all directions of vector E perpendicular to the direction","rect":[53.81417465209961,304.0239562988281,385.09331599286568,295.0296325683594]},{"page":96,"text":"of the wavevector of the incident wave k0, i.e. around the z-axis. Then we find the","rect":[53.814205169677737,315.5655517578125,385.1752399273094,307.0290222167969]},{"page":96,"text":"differential cross-section of one-electron scattering in unpolarized light:","rect":[53.814491271972659,327.98297119140627,345.61421067783427,318.96875]},{"page":96,"text":"dse ¼ 12\u0002e2=mec2\u00032ð1 þ cos22yÞdO","rect":[148.2988739013672,356.3807067871094,290.65694951966136,342.17364501953127]},{"page":96,"text":"(5.8)","rect":[366.0979309082031,354.6949157714844,385.1701901992954,346.0990295410156]},{"page":96,"text":"As to the total cross-section of scattering by free electron irradiated by unpo-","rect":[65.76648712158203,379.95965576171877,385.12847266999855,370.9454345703125]},{"page":96,"text":"larizedlight, it is described by the same Thomson formula (5.7) that is easy to check","rect":[53.814476013183597,391.9192199707031,385.0955437760688,382.88507080078127]},{"page":96,"text":"by integrating (5.8) over 2y (from 0 to p) and over j (from 0 to 2p).","rect":[53.814476013183597,403.8388977050781,330.8899197151828,394.5955810546875]},{"page":96,"text":"5.2.2 Scattering by Two Material Points","rect":[53.812843322753909,453.94915771484377,260.0020359012858,443.3113708496094]},{"page":96,"text":"Let the plane wave with wavevector k0 is incident onto two scattering points fixed","rect":[53.812843322753909,481.4916076660156,385.1440056901313,472.55706787109377]},{"page":96,"text":"at O and O0, see Fig. 5.4. The center of the reference polar coordinate system is at","rect":[53.81315231323242,493.4512634277344,385.17878587314677,483.7915344238281]},{"page":96,"text":"point O and point O0 is characterized by radius-vector r. Both points are sources of","rect":[53.81304931640625,505.4107971191406,385.15035377351418,495.7511901855469]},{"page":96,"text":"secondary spherical waves propagating in all directions (Huygens’ principle). The","rect":[53.814483642578128,517.370361328125,385.1454242534813,508.43585205078127]},{"page":96,"text":"mechanism of scattering is not important because now we consider a very general","rect":[53.814483642578128,529.329833984375,385.181288314553,520.3953247070313]},{"page":96,"text":"geometry of wave scattering, not its amplitude. Consider a wave with wavevector k","rect":[53.814483642578128,541.2894287109375,385.18716782877996,532.3549194335938]},{"page":96,"text":"scattered by two points at angle 2y with respect to k0 and introduce the wavevector","rect":[53.814483642578128,553.2489624023438,385.15460600005346,544.0056762695313]},{"page":96,"text":"of scattering (or diffraction) as a difference between the two vectors","rect":[53.81379318237305,565.1920776367188,329.56930936919408,556.1778564453125]},{"page":96,"text":"q = k \u0005 k0","rect":[197.0688018798828,589.0016479492188,241.4503028162416,580.25634765625]},{"page":96,"text":"(5.9)","rect":[366.09765625,588.67333984375,385.1699155410923,580.0774536132813]},{"page":97,"text":"80","rect":[53.812843322753909,42.55789566040039,62.2739118972294,36.73301696777344]},{"page":97,"text":"5 Structure Analysis and X-Ray Diffraction","rect":[235.81301879882813,44.276573181152347,385.1466421523563,36.63142013549805]},{"page":97,"text":"It is equal to the momentum taken by a fixed material point. In our case, modulus","rect":[65.76496887207031,68.2883529663086,385.17868436919408,59.35380554199219]},{"page":97,"text":"|k| ¼ |k0| (i.e. l¼l0). This corresponds to elastic scattering because the points do","rect":[53.812950134277347,80.24800872802735,385.16960993817818,70.9947280883789]},{"page":97,"text":"not take energy from the photons and the light frequency remains unchanged.","rect":[53.81282424926758,92.20760345458985,385.1397976448703,83.27305603027344]},{"page":97,"text":"Hence, as seen from the figure the scattering wavevector amplitude is","rect":[53.81282424926758,104.11031341552735,334.82822050200658,95.17576599121094]},{"page":97,"text":"4p siny","rect":[235.8704071044922,125.57144165039063,266.28151789716255,118.3700180053711]},{"page":97,"text":"q ¼ 2k0 siny ¼ l0","rect":[171.0674285888672,140.6525115966797,255.3283087976869,125.1148910522461]},{"page":97,"text":"(5.10)","rect":[361.0561828613281,133.97972106933595,385.10555396882668,125.38382720947266]},{"page":97,"text":"and the scattering angle between incident and scattered waves is 2y.","rect":[53.81450653076172,164.19189453125,328.8224758675266,154.92864990234376]},{"page":97,"text":"This is very general equation that will be used further on. From the same figure","rect":[65.76750946044922,176.15139770507813,385.11753118707505,167.2168426513672]},{"page":97,"text":"we can extract another useful relationship between the q-vector and the wave path","rect":[53.81548309326172,188.11093139648438,385.16826716474068,179.17637634277345]},{"page":97,"text":"difference D accumulated along the distance between the particles. It is just a","rect":[53.815513610839847,200.07049560546876,385.1593402691063,190.8072509765625]},{"page":97,"text":"difference of two scalar products:","rect":[53.815513610839847,212.030029296875,189.38786180088114,203.09547424316407]},{"page":97,"text":"D ¼ kr \u0005 k0r = ðk \u0005 k0Þr = qr","rect":[154.58697509765626,236.28890991210938,284.3899616069969,226.33839416503907]},{"page":97,"text":"(5.11)","rect":[361.0553283691406,235.5518341064453,385.10469947663918,226.9559326171875]},{"page":97,"text":"5.2.3 Scattering by a Stack of Planes (Bragg Diffraction)","rect":[53.812843322753909,298.02032470703127,344.9467816426574,287.3825378417969]},{"page":97,"text":"Let an electromagnetic wave is incident on the system of two parallel planes at an","rect":[53.812843322753909,325.5626525878906,385.14870539716255,316.62811279296877]},{"page":97,"text":"angle y with respect to the planes. Then, as seen in Fig. 5.5, the scattering vector is","rect":[53.812843322753909,337.522216796875,385.1875344668503,328.2789001464844]},{"page":97,"text":"again described by Eq. 5.9. Now, let us introduce a new vector, a wavevector of the","rect":[53.81385040283203,349.48175048828127,385.1756061382469,340.4874267578125]},{"page":97,"text":"structure with period d: q0 ¼ 2p/d. Then, at a certain “resonance” angle y0 the","rect":[53.81385040283203,361.44158935546877,385.17554510309068,352.1982727050781]},{"page":97,"text":"wavevectors of scattering and structure coincide:","rect":[53.813838958740237,373.401123046875,251.21479661533426,364.466552734375]},{"page":97,"text":"ð4p=lÞsiny0 ¼ 2p=d or 2d siny0 ¼ l","rect":[142.29283142089845,397.6020812988281,296.70782822917058,387.65155029296877]},{"page":97,"text":"(5.12)","rect":[361.05584716796877,396.86541748046877,385.1052182754673,388.26953125]},{"page":97,"text":"The same condition can easily be found by comparison of the wave","rect":[65.76618194580078,421.1829528808594,362.9560931987938,412.2484130859375]},{"page":97,"text":"difference 2dsin# with wavelength l.","rect":[53.814144134521487,433.1424865722656,204.88323636557346,423.8891906738281]},{"page":97,"text":"For a stack of layers we will have m multiple reflections and equation","rect":[65.76715850830078,445.1020202636719,347.5350274186469,436.16748046875]},{"page":97,"text":"path","rect":[367.966064453125,421.1829528808594,385.16692439130318,412.2484130859375]},{"page":97,"text":"ml ¼ 2dsiny0; m ¼ 1;2;3:::","rect":[156.22804260253907,468.8026123046875,282.7363733021631,459.767822265625]},{"page":97,"text":"(5.13)","rect":[361.05712890625,468.6233215332031,385.10650001374855,460.0274353027344]},{"page":97,"text":"Fig. 5.5 Bragg scattering (or","rect":[53.812843322753909,530.1536254882813,155.04785245520763,522.423828125]},{"page":97,"text":"reflection) of an","rect":[53.812843322753909,539.7232666015625,108.50096649317666,532.467529296875]},{"page":97,"text":"electromagnetic wave bya","rect":[53.812843322753909,550.037841796875,145.45131823801519,542.4434814453125]},{"page":97,"text":"stack of parallel planes in","rect":[53.812843322753909,560.0137939453125,141.6497549453251,552.41943359375]},{"page":97,"text":"vacuum (d is period of the","rect":[53.812843322753909,569.9329833984375,144.3555998542261,562.3217163085938]},{"page":97,"text":"stack structure, q is vector of","rect":[53.812843322753909,579.8497314453125,153.53247159583263,572.3146362304688]},{"page":97,"text":"scattering)","rect":[53.812843322753909,589.8849487304688,89.64547818762948,582.2905883789063]},{"page":97,"text":"k0","rect":[273.73138427734377,510.28692626953127,281.14277654241388,502.6052551269531]},{"page":97,"text":"θ","rect":[298.178466796875,524.6236572265625,302.34384207922968,518.9595947265625]},{"page":97,"text":"dsinθ","rect":[277.46063232421877,571.9541625976563,295.7611089249328,566.0820922851563]},{"page":97,"text":"k","rect":[346.12408447265627,503.3873596191406,350.5373035817921,497.7312927246094]},{"page":97,"text":"θq","rect":[341.6429443359375,525.6088256835938,359.9569537333658,516.8866577148438]},{"page":97,"text":"k0","rect":[349.9158020019531,570.3088989257813,357.3271637494451,562.6272583007813]},{"page":97,"text":"d","rect":[380.633056640625,548.6901245117188,384.6305377177408,542.9860229492188]},{"page":98,"text":"5.2 X-Ray Scattering","rect":[53.812843322753909,44.275962829589847,126.13638824610635,36.63080978393555]},{"page":98,"text":"81","rect":[376.7464904785156,42.55728530883789,385.2075552382938,36.73240661621094]},{"page":98,"text":"called","rect":[53.812843322753909,67.0,77.82239619306097,59.35380554199219]},{"page":98,"text":"Bragg","rect":[82.87117004394531,68.2883529663086,107.21917048505316,59.553016662597659]},{"page":98,"text":"(sometimes","rect":[112.38340759277344,67.88993072509766,158.48234953032688,59.35380554199219]},{"page":98,"text":"Bragg-Wulf)","rect":[163.58985900878907,68.2883529663086,215.1755612930454,59.35380554199219]},{"page":98,"text":"formula","rect":[220.3468017578125,67.0,251.95142400934066,59.35380554199219]},{"page":98,"text":"for","rect":[257.10870361328127,67.0,268.7153256973423,59.35380554199219]},{"page":98,"text":"the","rect":[273.81884765625,67.0,286.04261053277818,59.35380554199219]},{"page":98,"text":"diffraction","rect":[291.152099609375,67.0,333.41472712567818,59.35380554199219]},{"page":98,"text":"(resonance)","rect":[338.50628662109377,67.88993072509766,385.10295997468605,59.41356658935547]},{"page":98,"text":"angles of X-ray scattering from the stack of planes. For example, it could be crystal","rect":[53.812843322753909,80.24788665771485,385.1218400723655,71.31333923339844]},{"page":98,"text":"planes (h k l) or smectic layers with interlayer distance d.","rect":[53.812843322753909,92.20748138427735,286.5631985237766,83.25301361083985]},{"page":98,"text":"This interlayer distance can be found as d¼l/2sin#0 from the X-ray experiment","rect":[65.7648696899414,104.11067962646485,385.15351731845927,94.8569107055664]},{"page":98,"text":"measuring the angle of the first-order diffraction spot (m ¼ 1) or from higher order","rect":[53.81370162963867,116.0702133178711,385.1485532364048,107.13566589355469]},{"page":98,"text":"reflections. It is convenient to plot the diffracted beam intensity as a function of q;","rect":[53.814674377441409,128.02981567382813,385.15254075595927,119.09526062011719]},{"page":98,"text":"then different diffraction orders are located at equidistant positions, as shown in","rect":[53.81470489501953,139.98934936523438,385.1436699967719,131.05479431152345]},{"page":98,"text":"Fig. 5.6:","rect":[53.81470489501953,151.94888305664063,87.65304429599832,142.95455932617188]},{"page":98,"text":"q ¼ ð4p=lÞ sin# ¼ mq0 ¼ 2pm=d","rect":[149.60108947753907,176.25416564941407,289.1102227311469,166.25608825683595]},{"page":98,"text":"Note that Eqs. 5.9–5.11 tell us nothing about the amplitude of waves and the","rect":[65.76663970947266,199.73098754882813,385.1753619976219,190.73666381835938]},{"page":98,"text":"intensity of scattering because we used only the momentum conservation law.","rect":[53.814598083496097,211.6905517578125,368.70415921713598,202.75599670410157]},{"page":98,"text":"5.2.4 Amplitude of Scattering for a System of Material Points","rect":[53.812843322753909,261.8579406738281,367.1227329227702,251.22015380859376]},{"page":98,"text":"Generally, the amplitude of a wave scattered by material point O and measured at","rect":[53.812843322753909,289.4002685546875,385.1785722500999,280.44580078125]},{"page":98,"text":"any distant point P (R) corresponds to the Huygens principle:","rect":[53.812843322753909,301.35980224609377,302.67304856845927,292.42523193359377]},{"page":98,"text":"1","rect":[207.6045379638672,322.2874755859375,212.58164301923285,315.55419921875]},{"page":98,"text":"Fp ¼ RfO expikR","rect":[183.6993865966797,335.8905029296875,255.2935408917781,322.255615234375]},{"page":98,"text":"(5.14)","rect":[361.0550537109375,330.8323059082031,385.10442481843605,322.2364196777344]},{"page":98,"text":"and is determined by a scattering efficiency fO of the point O (depending on its","rect":[53.81339645385742,359.4577941894531,385.1525002871628,350.4833068847656]},{"page":98,"text":"electron mass), a distance R between the scattering center and point P and a","rect":[53.81368637084961,371.4173583984375,385.1604694194969,362.4827880859375]},{"page":98,"text":"wavevector k of a scattered wave (through multiplier expikR). Below we shall","rect":[53.81368637084961,383.37689208984377,385.11970384189677,374.44232177734377]},{"page":98,"text":"disregard term (1/R) (it may be taken into account if necessary) but always operate","rect":[53.8137321472168,395.2796630859375,385.17942083551255,386.3450927734375]},{"page":98,"text":"with vector of scattering q ¼ k \u0005 k0 having in mind that k0 has fixed direction","rect":[53.8137321472168,407.23968505859377,385.09343806317818,398.3046875]},{"page":98,"text":"along the selected coordinate axis. It is vector q that is responsible for all the","rect":[53.814292907714847,419.19921875,385.17404974176255,410.2646484375]},{"page":98,"text":"interfering scattered beams propagating in direction to point P as was shown for two","rect":[53.81432342529297,431.1587829589844,385.1522149186469,422.2242431640625]},{"page":98,"text":"scattering points, see Eq. 5.11.","rect":[53.81432342529297,443.1183166503906,177.18381925131565,434.1239929199219]},{"page":98,"text":"Consider now N scattering points having different scattering efficiency fj and","rect":[65.7673568725586,455.99481201171877,385.14626399091255,446.10345458984377]},{"page":98,"text":"located at different distances rj from one of the scattering points O selected as","rect":[53.814414978027347,467.9544677734375,385.13901151763158,458.0835266113281]},{"page":98,"text":"I","rect":[266.61029052734377,503.11151123046877,268.8324177424305,497.3687438964844]},{"page":98,"text":"Fig. 5.6 Illustration of the","rect":[53.812843322753909,551.5789794921875,146.1121153571558,543.8491821289063]},{"page":98,"text":"Bragg diffraction with the","rect":[53.812843322753909,561.4304809570313,142.9967589118433,553.8361206054688]},{"page":98,"text":"qualitative angular diffraction","rect":[53.812843322753909,571.4064331054688,155.30591339259073,563.8120727539063]},{"page":98,"text":"spectrum of the smectic A","rect":[53.812843322753909,581.3823852539063,143.9486138973853,573.7880249023438]},{"page":98,"text":"phase","rect":[53.812843322753909,591.3016357421875,73.09562060617924,583.707275390625]},{"page":98,"text":"0","rect":[265.0995788574219,586.1282348632813,269.5438332875953,580.3054809570313]},{"page":98,"text":"2nd","rect":[325.4751281738281,564.2230834960938,336.58621206286446,556.916259765625]},{"page":99,"text":"82","rect":[53.81315231323242,42.55673599243164,62.27422088770791,36.73185729980469]},{"page":99,"text":"a","rect":[84.45791625976563,68.23336791992188,90.01317865699947,62.64461135864258]},{"page":99,"text":"k","rect":[126.053955078125,111.4937744140625,130.49820950829844,106.19090270996094]},{"page":99,"text":"to  P","rect":[160.17181396484376,98.96798706054688,174.83146146142344,93.56913757324219]},{"page":99,"text":"b","rect":[179.31446838378907,68.23336791992188,185.41926213326728,60.92499542236328]},{"page":99,"text":"Z","rect":[189.75265502929688,74.84271240234375,195.08416169283229,69.53984069824219]},{"page":99,"text":"15","rect":[187.08990478515626,87.04708862304688,195.08316427458409,81.56025695800781]},{"page":99,"text":"10","rect":[187.08990478515626,115.05868530273438,195.08316427458409,109.57185363769531]},{"page":99,"text":"5 Structure Analysis and X-Ray Diffraction","rect":[235.81332397460938,44.275413513183597,385.14694732813759,36.6302604675293]},{"page":99,"text":"rj","rect":[106.09237670898438,155.51063537597657,111.63650764957613,148.4627227783203]},{"page":99,"text":"O","rect":[108.26836395263672,163.31185913085938,114.03950010117129,157.77703857421876]},{"page":99,"text":"5","rect":[191.0865478515625,141.16989135742188,195.08317953337315,135.7870330810547]},{"page":99,"text":"sin qÔl (Å–1)","rect":[303.6614685058594,189.2255401611328,347.02631968836718,180.03550720214845]},{"page":99,"text":"Fig. 5.7 (a) Geometry ofscattering by several objects with vectors rj between them and the beams","rect":[53.812843322753909,210.62709045410157,385.15283663749019,202.2338104248047]},{"page":99,"text":"scattered in direction to point P; (b) angular dependencies of scattering intensity by different","rect":[53.81313705444336,219.87185668945313,385.13512901511259,212.27749633789063]},{"page":99,"text":"atoms. The plot shows strong scattering in forward and back directions and the weak scattering in","rect":[53.81315231323242,229.79107666015626,385.1697134414188,222.19671630859376]},{"page":99,"text":"the direction perpendicular to the incident beam","rect":[53.81315231323242,239.76702880859376,218.1905846440314,232.17266845703126]},{"page":99,"text":"a reference, Fig. 5.7a. All these points contribute to scattering in q direction defined","rect":[53.812843322753909,270.63897705078127,385.11184016278755,261.6446533203125]},{"page":99,"text":"by Eq. 5.10. Then the amplitude of the field of N scattering points “measured” at","rect":[53.8138313293457,282.6383361816406,385.181532455178,273.60418701171877]},{"page":99,"text":"point P is superposition of all N- amplitudes:","rect":[53.81385040283203,294.50128173828127,235.56379563877176,285.56671142578127]},{"page":99,"text":"N","rect":[211.51304626464845,316.46246337890627,216.15368591959317,311.791015625]},{"page":99,"text":"FpðqÞ ¼ Xfj expiðqrjÞ","rect":[170.276123046875,332.3114929199219,268.70514310942846,318.3768005371094]},{"page":99,"text":"j¼1","rect":[208.5677032470703,340.0400390625,219.35893318733535,333.8416442871094]},{"page":99,"text":"(5.15)","rect":[361.05633544921877,329.528564453125,385.1057065567173,320.93267822265627]},{"page":99,"text":"Here q is the wavevector of scattering defined by Eq. 5.9 for two material points.","rect":[65.7667007446289,366.5994873046875,385.1824001839328,357.60516357421877]},{"page":99,"text":"Now we make a generalization, i.e. consider a body with continuous density of","rect":[53.81468963623047,378.5022277832031,385.2570634610374,369.52783203125]},{"page":99,"text":"scattering points r(r) (that is density of electrons, atoms and molecules). Then the","rect":[53.81468963623047,390.5016174316406,385.15952337457505,381.5272216796875]},{"page":99,"text":"scattering amplitude is an integral over the scattering volume in the three-dimensional","rect":[53.81468963623047,402.42132568359377,385.235243392678,393.48675537109377]},{"page":99,"text":"r-space shown by dash line in Fig. 5.7a:","rect":[53.81468963623047,414.380859375,212.10156114170145,405.38653564453127]},{"page":99,"text":"FðqÞ ¼ ð rðrÞexpiðqrÞdV","rect":[166.0275115966797,452.7834777832031,272.4506911603328,430.6612548828125]},{"page":99,"text":"V","rect":[198.59793090820313,459.12713623046877,202.87623833783688,454.3999328613281]},{"page":99,"text":"(5.16)","rect":[361.0570983886719,445.9510498046875,385.1064694961704,437.35516357421877]},{"page":99,"text":"Thus, the amplitude of scattering in point P is just a Fourier integral of the","rect":[65.7674331665039,485.6296081542969,385.1761859722313,476.695068359375]},{"page":99,"text":"electron density function (generally complex). The variation of the position of point","rect":[53.815406799316409,497.58917236328127,385.14924485752177,488.65460205078127]},{"page":99,"text":"P means variation of scattering vector q, therefore suffix P at FP(q) is skipped. At","rect":[53.815406799316409,509.5487060546875,385.16108567783427,500.61395263671877]},{"page":99,"text":"each q we collect total amplitude of scattering from all the body with density r(r)","rect":[53.81332015991211,521.4512939453125,385.16011939851418,512.5167846679688]},{"page":99,"text":"usually situated far from point P. In the Cartesian system:","rect":[53.81426239013672,533.4107666015625,286.9847245938499,524.4762573242188]},{"page":99,"text":"FðqÞ ¼ ððð rðx;y;zÞexpiðqxx þ qyy þ qzzÞdxdydz","rect":[119.01236724853516,571.8134155273438,319.8957864199753,549.6912231445313]},{"page":99,"text":"V","rect":[155.15103149414063,578.6669311523438,159.42933892377438,573.939697265625]},{"page":99,"text":"(5.17)","rect":[361.05645751953127,564.9808349609375,385.1058286270298,556.3849487304688]},{"page":100,"text":"5.2 X-Ray Scattering","rect":[53.812843322753909,44.274620056152347,126.13638824610635,36.62946701049805]},{"page":100,"text":"83","rect":[376.7464904785156,42.55594253540039,385.2075552382938,36.73106384277344]},{"page":100,"text":"5.2.5 Scattering Amplitude for an Atom","rect":[53.812843322753909,80.50897216796875,258.63314782832267,69.95486450195313]},{"page":100,"text":"An atom has a spherical symmetry, therefore ra(r)¼ ra(r). However, the incident","rect":[53.812843322753909,108.13497161865235,385.1193681485374,99.20042419433594]},{"page":100,"text":"beam propagating along the x-axis breaks the overall spherical symmetry of","rect":[53.81338119506836,120.0945053100586,385.15117774812355,111.15995788574219]},{"page":100,"text":"scattering. In the spherical reference system, with radius r, polar angle # (0\u0005p),","rect":[53.81338119506836,132.05410766601563,385.15520902182348,122.90042114257813]},{"page":100,"text":"azimuthal angle c (0 \u0005 2p), a volume element is dV ¼ r2 sin# d#dcdr and D¼qr","rect":[53.813411712646487,143.95681762695313,385.1626361675438,132.9062957763672]},{"page":100,"text":"\u0001 qrcos#. The integral (5.16) is triple integral and, at first, we integrate with respect","rect":[53.81482696533203,155.91677856445313,385.15669114658427,146.76309204101563]},{"page":100,"text":"to #:","rect":[53.81482696533203,165.85435485839845,72.98166520664285,158.72265625]},{"page":100,"text":"p","rect":[89.32941436767578,186.24571228027345,93.17571029790025,183.09422302246095]},{"page":100,"text":"ð eiqrcos# sin#d# ¼ \u0005iq1r eiqrcos#jp0¼ q2r \u0004eiqr \u00052ie\u0005iqr\u0005 ¼ 2sqinrqr","rect":[88.36788940429688,211.15760803222657,350.32333768950658,186.7531280517578]},{"page":100,"text":"0","rect":[88.9893798828125,216.7181854248047,92.47334358772597,211.92123413085938]},{"page":100,"text":"Next integrating with respect to j results in 2p. Now we should integrate (5.16)","rect":[65.76631927490235,241.22091674804688,385.16011939851418,232.22659301757813]},{"page":100,"text":"with respect to r and find the angle (or q-) dependence of the field intensity scattered","rect":[53.81426239013672,253.18048095703126,385.0734795670844,244.2459259033203]},{"page":100,"text":"by an atom","rect":[53.814292907714847,265.1400146484375,98.7933878767546,256.2054443359375]},{"page":100,"text":"1","rect":[194.7460479736328,283.5308837890625,201.71397538345975,280.3863525390625]},{"page":100,"text":"FðqÞ ¼ ð 4pr2raðrÞsiqnrqrdr","rect":[162.4595184326172,307.8816833496094,276.23163236724096,285.45367431640627]},{"page":100,"text":"0","rect":[195.9926300048828,313.98248291015627,199.47659370979629,309.1855163574219]},{"page":100,"text":"(5.18)","rect":[361.056396484375,300.7340087890625,385.10576759187355,292.13812255859377]},{"page":100,"text":"We see that the scattering amplitude depends only on the modulus of q and is","rect":[65.76676177978516,338.4852294921875,385.1854287539597,329.5506591796875]},{"page":100,"text":"spherically symmetric in the q-space. Since ra(r) is unknown there is no universal","rect":[53.814735412597659,350.44476318359377,385.108534408303,341.51019287109377]},{"page":100,"text":"formula for each atom but we can analyze two asymptotic cases:","rect":[53.81351852416992,362.40509033203127,315.2290483243186,353.47052001953127]},{"page":100,"text":"1","rect":[230.88560485839845,380.795166015625,237.85353226822537,377.650634765625]},{"page":100,"text":"1","rect":[301.0124206542969,380.795166015625,307.9803480641238,377.650634765625]},{"page":100,"text":"forq ! 0; sinqr=qr ! 1and Fð0Þ ¼ ð 4pr2raðrÞdr ¼ ð raðrÞdV ¼ Z","rect":[77.37811279296875,404.8877868652344,361.2839390690143,382.7652893066406]},{"page":100,"text":"o","rect":[232.1314697265625,410.906494140625,235.61543343147597,407.7131652832031]},{"page":100,"text":"o","rect":[302.25830078125,410.906494140625,305.74226448616346,407.7131652832031]},{"page":100,"text":"and","rect":[53.81529998779297,433.43450927734377,68.21904078534613,426.53192138671877]},{"page":100,"text":"forq ! 1; sinqr=qr ! 0; FðqÞ ! 0:","rect":[142.068359375,459.7142028808594,296.90711915177249,449.7168884277344]},{"page":100,"text":"Indeed, according to (5.10), for a finite l0, the case of q!0 means y!0 (forward","rect":[65.77033996582031,483.2478332519531,385.1029900651313,473.9945373535156]},{"page":100,"text":"scattering) the scattering amplitude is proportional to the number of electrons Z in","rect":[53.81391525268555,495.2081604003906,385.1437920670844,486.27362060546877]},{"page":100,"text":"the atom. It means strong forward scattering as in case of a single electron. The","rect":[53.81391525268555,507.167724609375,385.1448444194969,498.233154296875]},{"page":100,"text":"intensity of scattering will be proportional to Z2. However, for directions strongly","rect":[53.81391525268555,519.127685546875,385.17730036786568,508.0766296386719]},{"page":100,"text":"perpendicular to the primary beam the scattering is absent. This is a result of","rect":[53.813533782958987,531.0872192382813,385.1484616836704,522.1527099609375]},{"page":100,"text":"interference of different scattered waves from individual electronic oscillators.","rect":[53.813533782958987,540.9580688476563,385.10662503744848,534.0554809570313]},{"page":100,"text":"The calculated angular dependencies of scattering intensity for different atoms","rect":[53.813533782958987,554.9495239257813,385.1116067324753,546.0150146484375]},{"page":100,"text":"are shown in Fig. 5.7b (in electron charge units) [2]. Since siny/l¼q/4p, see","rect":[53.813533782958987,566.9090576171875,385.11655462457505,557.6558227539063]},{"page":100,"text":"Eq. 5.10, the abscissa is, in fact, the vector of scattering.","rect":[53.814537048339847,578.8685913085938,281.67535062338598,569.8743286132813]},{"page":101,"text":"84","rect":[53.81366729736328,42.55740737915039,62.27473587183877,36.73252868652344]},{"page":101,"text":"5 Structure Analysis and X-Ray Diffraction","rect":[235.8138427734375,44.276084899902347,385.1474661269657,36.63093185424805]},{"page":101,"text":"5.3 Diffraction on a Periodic Structure","rect":[53.812843322753909,68.09864807128906,260.9480417582532,59.086421966552737]},{"page":101,"text":"5.3.1 Reciprocal Lattice","rect":[53.812843322753909,99.9498291015625,180.3704660746595,89.39572143554688]},{"page":101,"text":"Now we consider periodic crystalline structures. The simplest case is one-dimensional","rect":[53.812843322753909,127.5765609741211,385.1476884610374,118.64201354980469]},{"page":101,"text":"structure realized, for instance, in the smectic A phase, see Fig. 5.8a: the density is","rect":[53.812843322753909,139.53610229492188,385.24423612700658,130.54177856445313]},{"page":101,"text":"periodic along x with period a, and wavevector q ¼ qx ¼ h(2p/a), h is an integer.","rect":[53.812862396240237,151.43917846679688,385.1803860237766,142.4846954345703]},{"page":101,"text":"Then the density function can be written as","rect":[53.81367111206055,163.39871215820313,223.64641203032688,154.4641571044922]},{"page":101,"text":"1","rect":[211.003173828125,176.8576202392578,217.97110123795194,173.71311950683595]},{"page":101,"text":"rðxÞ ¼ Xdðx \u0005 haÞ","rect":[176.44955444335938,192.70640563964845,262.53073515044408,178.77169799804688]},{"page":101,"text":"\u00051","rect":[208.3407440185547,198.7367401123047,220.67398209732694,195.5922393798828]},{"page":101,"text":"and in accordance with (5.16) the scattering amplitude is given by","rect":[53.81382369995117,218.26541137695313,321.04851618817818,209.27108764648438]},{"page":101,"text":"1","rect":[165.91368103027345,231.72462463378907,172.88160844010037,228.5801239013672]},{"page":101,"text":"1","rect":[280.7333984375,231.72462463378907,287.7013258473269,228.5801239013672]},{"page":101,"text":"FðqxÞ ¼ X a1ð dðx \u0005 haÞeiqxxdx ¼ a1Xeiqxha:","rect":[123.14788055419922,251.67996215820313,315.87867676895999,229.5577392578125]},{"page":101,"text":"h¼\u00051","rect":[158.77642822265626,254.05625915527345,179.93036027115506,249.1616973876953]},{"page":101,"text":"\u00051","rect":[278.0709533691406,253.60374450683595,290.4041761891238,250.45924377441407]},{"page":101,"text":"As exp(iqxha) ¼ 1 only for qx ¼ 2p/a (otherwise it equals 0) the same equation","rect":[65.7662124633789,273.5865783691406,385.1760186295844,264.6318359375]},{"page":101,"text":"may be rewritten as","rect":[53.813350677490237,285.48907470703127,133.29773344145969,276.55450439453127]},{"page":101,"text":"11","rect":[219.7265167236328,303.52618408203127,238.64666886490506,295.8035888671875]},{"page":101,"text":"FhðqxÞ ¼ FðqxÞ ¼ Xdðqx \u0005","rect":[144.44544982910157,314.79644775390627,271.34790828916939,300.86175537109377]},{"page":101,"text":"a","rect":[219.7265167236328,317.2188415527344,224.70362177899848,312.68682861328127]},{"page":101,"text":"\u00051","rect":[229.0163116455078,320.8272399902344,241.34954972428006,317.6827087402344]},{"page":101,"text":"hÞ","rect":[285.71844482421877,312.807373046875,294.5358926211472,302.8568420410156]},{"page":101,"text":"(5.19)","rect":[361.0569152832031,312.0702819824219,385.10628639070168,303.4743957519531]},{"page":101,"text":"Therefore, F(q) is a set of the d-like peaks on the q-scale separated by distances","rect":[65.76725006103516,340.41180419921877,385.10129179106908,331.1784362792969]},{"page":101,"text":"2p/a! These peaks form a one-dimensional reciprocal lattice with basic vector 2p/a,","rect":[53.81623077392578,352.3146057128906,385.1869167854953,343.3601379394531]},{"page":101,"text":"shown in Fig. 5.8b.","rect":[53.81824493408203,364.274169921875,131.70098538901096,355.27984619140627]},{"page":101,"text":"In the three-dimensional-lattice, there are three basic vectors a, b, and c,","rect":[65.770263671875,375.8751220703125,385.16204495932348,367.29913330078127]},{"page":101,"text":"Fig. 5.9a, and we can introduce a concept of the reciprocal three-dimensional","rect":[53.81822967529297,388.1932373046875,385.13618333408427,379.19891357421877]},{"page":101,"text":"lattice. It is a lattice in the wavevector space having the dimension of inverse length","rect":[53.81824493408203,400.15277099609377,385.12627497724068,391.21820068359377]},{"page":101,"text":"for each coordinate in the inverse space. Such a lattice may be built by translations","rect":[53.81824493408203,412.1123352050781,385.17303861724096,403.17779541015627]},{"page":101,"text":"of the elementary cell shown in Fig. 5.9b. The basic vectors of the reciprocal lattice","rect":[53.81824493408203,424.0718688964844,385.16101873590318,415.0775451660156]},{"page":101,"text":"are a*, b*, c* and the vector of the reciprocal lattice is given by","rect":[53.81824493408203,436.0314025878906,314.65337458661568,427.0370788574219]},{"page":101,"text":"H ¼ Hhkl ¼ ha\u0006 þ kb\u0006 þ lc\u0006","rect":[162.18075561523438,459.4105529785156,276.3436743982728,450.424560546875]},{"page":101,"text":"(5.20)","rect":[361.0561828613281,459.4977722167969,385.10555396882668,450.9018859863281]},{"page":101,"text":"a","rect":[107.3991928100586,488.9748229980469,112.95445520729244,483.3860778808594]},{"page":101,"text":"r(x)","rect":[113.18431091308594,513.79833984375,129.9765511154044,506.3039245605469]},{"page":101,"text":"b","rect":[242.5095977783203,488.9748229980469,248.61439152779853,481.66644287109377]},{"page":101,"text":"a","rect":[258.7877197265625,516.0597534179688,262.78435140837316,512.3165893554688]},{"page":101,"text":"1D lattice","rect":[277.2931823730469,494.9129333496094,311.1705992847254,489.052734375]},{"page":101,"text":"x","rect":[315.84442138671877,517.8601684570313,320.2567027634377,514.1170043945313]},{"page":101,"text":"0","rect":[125.18138122558594,555.1990966796875,129.17801290739659,549.5283203125]},{"page":101,"text":"a","rect":[188.17550659179688,546.4905395507813,192.17213827360752,542.7473754882813]},{"page":101,"text":"Fig. 5.8 Periodic","rect":[53.812843322753909,584.0003662109375,116.01185748362065,576.0504150390625]},{"page":101,"text":"dimensional direct","rect":[53.81281280517578,592.1814575195313,117.08047203269068,586.314208984375]},{"page":101,"text":"density distribution (density wave) in one-dimensional","rect":[119.4648208618164,583.9326171875,311.5057650014407,576.3382568359375]},{"page":101,"text":"and reciprocal lattices with periods a and 2p/a (b)","rect":[119.46479797363281,593.9085693359375,291.19830411536386,586.314208984375]},{"page":101,"text":"crystal","rect":[315.00439453125,583.9326171875,337.6293306752688,576.3382568359375]},{"page":101,"text":"(a)","rect":[341.1178283691406,583.593994140625,350.95794767005136,576.3890991210938]},{"page":101,"text":"and","rect":[354.4278259277344,582.2055053710938,366.67099517970009,576.3382568359375]},{"page":101,"text":"one-","rect":[370.118896484375,582.0,385.1795968399732,578.0]},{"page":102,"text":"5.3 Diffraction on a Periodic Structure","rect":[53.81356430053711,42.55667495727539,185.6141905524683,36.63019943237305]},{"page":102,"text":"Fig. 5.9 A crystal lattice cell","rect":[53.812843322753909,67.58130645751953,154.5258150502688,59.546695709228519]},{"page":102,"text":"built on the a, b, c vector basis","rect":[53.812843322753909,76.2703628540039,155.34990389823236,69.89517211914063]},{"page":102,"text":"(a) and a cell of the reciprocal","rect":[53.81283950805664,87.4087142944336,155.42013267722192,79.81436157226563]},{"page":102,"text":"latticebased onvectors a*, b*,","rect":[53.81283950805664,96.1655044555664,155.33975479199848,89.79031372070313]},{"page":102,"text":"c* (b)","rect":[53.812843322753909,107.02202606201172,74.24801725012948,99.69859313964844]},{"page":102,"text":"a","rect":[260.05792236328127,68.23306274414063,265.6131847605151,62.64430618286133]},{"page":102,"text":"c","rect":[270.3119812011719,84.72928619384766,273.813030554438,80.82611846923828]},{"page":102,"text":"b","rect":[278.2860412597656,115.31716918945313,282.28267294157629,109.72636413574219]},{"page":102,"text":"a","rect":[293.4060974121094,132.15994262695313,297.40272909392,128.2647705078125]},{"page":102,"text":"b","rect":[318.4386291503906,68.23306274414063,324.5434228998688,60.92469024658203]},{"page":102,"text":"c*","rect":[322.5728454589844,114.5651626586914,329.0790472476861,109.25032043457031]},{"page":102,"text":"b*","rect":[362.7187805175781,94.43954467773438,369.70493225745175,88.84873962402344]},{"page":102,"text":"*","rect":[360.70062255859377,120.3927001953125,363.69809631995175,118.13117980957031]},{"page":102,"text":"a","rect":[356.7039794921875,123.43801879882813,360.70061117399816,119.54285430908203]},{"page":102,"text":"85","rect":[376.7472229003906,42.55667495727539,385.2082876601688,36.63019943237305]},{"page":102,"text":"k","rect":[178.0843048095703,185.789306640625,182.5295038031804,180.04530334472657]},{"page":102,"text":"H","rect":[193.39120483398438,189.96533203125,199.16356755590253,184.22132873535157]},{"page":102,"text":"2π/λ","rect":[195.0117950439453,204.6844940185547,210.4580513967509,198.64450073242188]},{"page":102,"text":"k0","rect":[196.0481719970703,220.62606811523438,203.8268482036408,212.7748565673828]},{"page":102,"text":"c*","rect":[243.224609375,229.28131103515626,250.77985536655988,224.0]},{"page":102,"text":"X-ray","rect":[169.8788299560547,252.2164764404297,189.8742303656804,244.7684783935547]},{"page":102,"text":"a*","rect":[260.9790344238281,253.52206420898438,268.53426515659899,248.0]},{"page":102,"text":"electrons","rect":[232.15878295898438,275.02606201171877,267.7043847846257,269.1780700683594]},{"page":102,"text":"Fig. 5.10 Projection of the Ewald sphere on the a*,c* plane in reciprocal lattice for crystal","rect":[53.812843322753909,297.9322509765625,384.58910850730009,290.2024230957031]},{"page":102,"text":"irradiated by X-rays (solid semicircle) and electrons (dash line). Radius of the sphere is 2p/l.","rect":[53.812843322753909,307.84051513671877,384.640658112311,299.9752197265625]},{"page":102,"text":"Lattice vector H connects two points of the reciprocal lattice. When vector of scattering (k \u0005 k0)","rect":[53.814517974853519,317.81646728515627,384.6058663712232,310.22210693359377]},{"page":102,"text":"coincides with H, a strong diffraction is observed at a particular angle defined by Eq. 5.10","rect":[53.81356430053711,327.7354431152344,363.0614065566532,320.09027099609377]},{"page":102,"text":"where h, k, l are integers. Vector H is a fundamental characteristic of a three-","rect":[53.812843322753909,350.7288818359375,385.16265235749855,341.7744140625]},{"page":102,"text":"dimensional crystal. In the simplest case of a rectangular cell, the reciprocal lattice","rect":[53.812843322753909,362.6884460449219,385.15271795465318,353.75390625]},{"page":102,"text":"has periods 2p/a, 2p/b and 2p/c. For crystals of other symmetry, a* ¼ 2p(b \u0003 c)/","rect":[53.812843322753909,374.6479797363281,385.208387923928,365.5440979003906]},{"page":102,"text":"(a\u0007b \u0003 c), b* ¼ 2p(c \u0003 a)/(a\u0007b \u0003 c), and c* ¼ 2p(a \u0003 b)/(a\u0007b \u0003 c) where we see","rect":[53.8138313293457,386.2091064453125,385.11975897027818,377.503662109375]},{"page":102,"text":"in denominator the mixed product of the three vectors corresponding to the volume","rect":[53.8177604675293,398.56707763671877,385.11878240777818,389.63250732421877]},{"page":102,"text":"of elementary cell.","rect":[53.8177604675293,410.526611328125,129.25674100424534,401.592041015625]},{"page":102,"text":"When the crystal is irradiated by an X-ray beam, its lattice scatters the radiation","rect":[65.76978302001953,422.48614501953127,385.0949639420844,413.55157470703127]},{"page":102,"text":"selectively. A strong diffraction is observed when the wavevector of scattering for a","rect":[53.8177604675293,434.4457092285156,385.16159856988755,425.51116943359377]},{"page":102,"text":"particular angle (i.e. q) coincides with the vector of reciprocal lattice, as shown in","rect":[53.8177604675293,446.3484802246094,385.14666071942818,437.2445983886719]},{"page":102,"text":"the Ewald sphere, Fig. 5.10. The condition","rect":[53.81874465942383,458.3080139160156,226.0465020280219,449.3136901855469]},{"page":102,"text":"q ¼ k \u0005 k0 ¼ H","rect":[186.08433532714845,480.1732177734375,252.87821148515304,471.3199768066406]},{"page":102,"text":"means","rect":[53.813899993896487,501.0,79.29668058501437,495.0]},{"page":102,"text":"the","rect":[84.57141876220703,501.0,96.7951892681297,493.2449951171875]},{"page":102,"text":"conservation","rect":[101.96142578125,501.0,153.1041649918891,493.2449951171875]},{"page":102,"text":"of","rect":[158.2664337158203,501.0,166.55828224031104,493.2449951171875]},{"page":102,"text":"linear","rect":[171.74742126464845,501.0,194.53260169831885,493.2449951171875]},{"page":102,"text":"momentum","rect":[199.78643798828126,501.0,245.3916544783171,494.2609558105469]},{"page":102,"text":"of","rect":[250.59671020507813,501.0,258.88855872956887,493.2449951171875]},{"page":102,"text":"electromagnetic","rect":[264.0776672363281,502.1795349121094,328.1599506206688,493.2449951171875]},{"page":102,"text":"according to (5.16) the amplitude of scattering is given by","rect":[53.813899993896487,514.082275390625,288.76015559247505,505.0879821777344]},{"page":102,"text":"wave.","rect":[333.3540344238281,501.0,356.86590238119848,495.0]},{"page":102,"text":"Then,","rect":[362.1296691894531,501.0,385.1338161995578,493.2449951171875]},{"page":102,"text":"Fhkl ¼ V1c ð rðrÞexpirHhkldV","rect":[158.72032165527345,548.5172729492188,280.0401992169734,526.3950805664063]},{"page":102,"text":"Vc","rect":[200.18399047851563,555.88720703125,206.71382540799065,550.1337890625]},{"page":102,"text":"(5.21)","rect":[361.0555419921875,541.6849975585938,385.10491309968605,533.089111328125]},{"page":102,"text":"where integrating is taken over the volume Vc of a single crystallographic cell in the","rect":[53.81388473510742,578.4157104492188,385.17334783746568,569.4806518554688]},{"page":102,"text":"direct space.","rect":[53.81364059448242,590.375244140625,104.12022824789767,581.4407348632813]},{"page":103,"text":"86","rect":[53.812843322753909,42.55594253540039,62.2739118972294,36.68026351928711]},{"page":103,"text":"5 Structure Analysis and X-Ray Diffraction","rect":[235.81301879882813,44.274620056152347,385.1466421523563,36.62946701049805]},{"page":103,"text":"Hence, for the elastic scattering, the ends of the scattering vector of the scattered","rect":[65.76496887207031,68.2883529663086,385.07122126630318,59.35380554199219]},{"page":103,"text":"beam k must coincide with the points of the reciprocal lattice determined by the","rect":[53.812950134277347,80.24788665771485,385.17173040582505,71.31333923339844]},{"page":103,"text":"three-dimensional Ewald sphere [1] of radius k ¼ 2p/l0. The center of the sphere is","rect":[53.812950134277347,92.20772552490235,385.18719877349096,82.9542007446289]},{"page":103,"text":"defined by the direction of k0 (horizontal in the figure) and one of the points. The","rect":[53.81357955932617,104.11067962646485,385.1460346050438,95.17594909667969]},{"page":103,"text":"X-ray wavelengths are close to the periods of crystal lattices and the sphere","rect":[53.813175201416019,116.0702133178711,385.1590045757469,107.13566589355469]},{"page":103,"text":"curvature is large. For electrons, the wavelength is much shorter, the sphere radius","rect":[53.813175201416019,128.02981567382813,385.1162148867722,119.09526062011719]},{"page":103,"text":"of the corresponding Ewald sphere is longer and the sphere surface in the figure is","rect":[53.813175201416019,139.98934936523438,385.18588651763158,131.05479431152345]},{"page":103,"text":"very flat.","rect":[53.813175201416019,151.94888305664063,89.37857480551486,143.0143280029297]},{"page":103,"text":"5.3.2 Intensity of Scattering","rect":[53.812843322753909,182.9015655517578,200.0236333837468,172.2637939453125]},{"page":103,"text":"Consider a three-dimensional crystal. For the scattering amplitude of a discrete","rect":[53.812843322753909,210.44384765625,385.0958942241844,201.50929260253907]},{"page":103,"text":"system of j atoms in an elementary cell we can write a formula similar to (5.15):","rect":[53.812843322753909,222.40341186523438,379.5166154629905,213.40908813476563]},{"page":103,"text":"N","rect":[199.5609130859375,238.35630798339845,204.20155274088223,233.6848602294922]},{"page":103,"text":"Fhkl ¼ Xfj expiðHhklrjÞ;","rect":[166.48057556152345,254.20530700683595,270.8462365834131,240.27059936523438]},{"page":103,"text":"j¼1","rect":[196.61557006835938,261.9903869628906,207.4632422693666,255.7919921875]},{"page":103,"text":"(5.22)","rect":[361.0562744140625,251.47914123535157,385.10564552156105,242.88323974609376]},{"page":103,"text":"Now the summation is performed over all atoms in one cell and fj is scattering","rect":[65.7666244506836,283.4584045410156,385.16606989911568,273.5107116699219]},{"page":103,"text":"efficiency of a particular atom. The vector Hhkl determines the angular positions of","rect":[53.814292907714847,294.4448547363281,385.14995704499855,285.51025390625]},{"page":103,"text":"the","rect":[53.814083099365237,305.0,66.0378536052879,297.4698486328125]},{"page":103,"text":"diffraction","rect":[71.14735412597656,305.0,113.40993586591253,297.4698486328125]},{"page":103,"text":"spots,","rect":[118.44477844238281,306.4043884277344,141.41909451987034,298.4858093261719]},{"page":103,"text":"the","rect":[146.48379516601563,305.0,158.70757330133285,297.4698486328125]},{"page":103,"text":"coefficient","rect":[163.8170623779297,305.0,206.09556443759989,297.4698486328125]},{"page":103,"text":"fj","rect":[211.1174774169922,307.33428955078127,216.04661453487243,297.42999267578127]},{"page":103,"text":"determines","rect":[221.08607482910157,305.0,265.0110207949753,297.469970703125]},{"page":103,"text":"their","rect":[270.08367919921877,305.0,288.4193357071079,297.469970703125]},{"page":103,"text":"form,","rect":[293.5348205566406,305.0,315.4042019417453,297.469970703125]},{"page":103,"text":"i.e.","rect":[320.4977722167969,305.0,332.7215542366672,297.469970703125]},{"page":103,"text":"the","rect":[337.8310546875,305.0,350.05481756402818,297.469970703125]},{"page":103,"text":"angular","rect":[355.164306640625,306.404541015625,385.1065610489048,297.469970703125]},{"page":103,"text":"distribution of the scattering intensity within the spot. But how to estimate the","rect":[53.813533782958987,318.36407470703127,385.1722797222313,309.42950439453127]},{"page":103,"text":"scattered field intensity related to the energy dW/dO scattered at a certain angle # in","rect":[53.813533782958987,330.3236389160156,385.14336482099068,321.1699523925781]},{"page":103,"text":"a unit solid angle?","rect":[53.81350326538086,342.2831726074219,127.77826727105939,333.3486328125]},{"page":103,"text":"The magnitude of the energy flux Q ¼ dW/dO scattered by an object is deter-","rect":[65.76551055908203,354.24273681640627,385.21904884187355,345.12890625]},{"page":103,"text":"mined by the number of electrons in the object, their spatial configuration and the","rect":[53.81350326538086,366.2022399902344,385.2290729351219,357.2677001953125]},{"page":103,"text":"differential cross-section of scattering by one electron, given by Eq. 5.8. The latter is","rect":[53.81350326538086,378.1050109863281,385.2439004336472,369.1106872558594]},{"page":103,"text":"normalized to the energy of the primary X-ray beam and is independent of the","rect":[53.812496185302737,390.0645751953125,385.22800481988755,381.1300048828125]},{"page":103,"text":"distance between an object and a detector. From the measurements of the flux we","rect":[53.812496185302737,402.02410888671877,385.1692584819969,393.08953857421877]},{"page":103,"text":"can find the scattering efficiency of an atom fj, molecule or any object. The spatial","rect":[53.812496185302737,414.90118408203127,385.14973313877177,405.00927734375]},{"page":103,"text":"configuration of electrons determines the scattering amplitude (electric field strength)","rect":[53.81391525268555,425.98406982421877,385.1587155899204,416.9698486328125]},{"page":103,"text":"at the detector and the flux of the energy is proportional to the squared modulus of the","rect":[53.81289291381836,437.9037780761719,385.22742498590318,428.96923828125]},{"page":103,"text":"complex amplitude that is |F(q)|2 ¼ F(q) F*(q). Therefore, for incident flux of","rect":[53.81289291381836,449.8633117675781,385.2104428848423,438.8127136230469]},{"page":103,"text":"unpolarized beam Q0 ¼ 1, on account of (5.8), the scattered flux is given by","rect":[53.81487274169922,461.82330322265627,355.77150813153755,452.8089599609375]},{"page":103,"text":"QðqÞ ¼ \u0004mec22\u00052 1 þ c2os22#jFðqÞj2","rect":[146.993896484375,498.5291442871094,291.46797249397596,472.0294494628906]},{"page":103,"text":"The differential intensity calculated in that way is related to a point in the","rect":[65.76496887207031,518.0506591796875,385.11499822809068,509.11614990234377]},{"page":103,"text":"diffraction pattern corresponding to wavevector q. Usually, all multipliers are","rect":[53.812950134277347,530.01025390625,385.1069721050438,521.0757446289063]},{"page":103,"text":"excluded, although they can be taken into account when necessary (for example","rect":[53.812923431396487,541.9697875976563,385.17270696832505,533.0352783203125]},{"page":103,"text":"cos22#), and the scattering intensity I(q) is expressed in relative, “electron units”","rect":[53.812923431396487,553.9296264648438,385.1114887066063,542.878662109375]},{"page":103,"text":"[electron2] as follows:","rect":[53.81349563598633,565.28173828125,143.040907455178,554.8383178710938]},{"page":103,"text":"IðqÞ ¼ jFðqÞj2 ¼ FðqÞF\u0006ðqÞ","rect":[162.17489624023438,592.0744018554688,276.80597318755346,580.1197509765625]},{"page":103,"text":"(5.23)","rect":[361.0564270019531,591.3372802734375,385.10579810945168,582.7413940429688]},{"page":104,"text":"5.3 Diffraction on a Periodic Structure","rect":[53.812843322753909,42.55594253540039,185.6134581305933,36.62946701049805]},{"page":104,"text":"87","rect":[376.7464904785156,42.55594253540039,385.2075552382938,36.73106384277344]},{"page":104,"text":"The intensity can be found from the X-ray diffraction experiment and the result","rect":[65.76496887207031,68.2883529663086,385.14692552158427,59.35380554199219]},{"page":104,"text":"compared with calculated diffraction pattern that is angular spectrum of the scat-","rect":[53.812950134277347,80.24788665771485,385.10799537507668,71.31333923339844]},{"page":104,"text":"tered X-ray intensity. To this effect, we should make a Fourier transform F(q) of the","rect":[53.812950134277347,92.20748138427735,385.1746295757469,83.27293395996094]},{"page":104,"text":"density function r(r) i.e. find the scattering amplitude and then take square of it,","rect":[53.812923431396487,104.11019134521485,385.1676907112766,95.17564392089844]},{"page":104,"text":"I(q) ¼ |F2(q)|. This works well for solid crystals, but is not always convenient for","rect":[53.81292724609375,116.07039642333985,385.1783689102329,105.01939392089844]},{"page":104,"text":"liquids, liquid crystals and other soft matter materials in which the thermal fluctua-","rect":[53.81258010864258,128.02999877929688,385.1136411270298,119.09544372558594]},{"page":104,"text":"tions play a very substantial role. In such cases, the so-called density autocorrela-","rect":[53.81258010864258,139.98953247070313,385.17131934968605,131.03504943847657]},{"page":104,"text":"tion function appears to be more convenient. However, before to proceed along that","rect":[53.81258010864258,151.94906616210938,385.1355424649436,142.97467041015626]},{"page":104,"text":"way, we should separate two sources of scattering.","rect":[53.81258010864258,163.90863037109376,258.1018337776828,154.9740753173828]},{"page":104,"text":"5.3.3 Form Factor and Structure Factor","rect":[53.812843322753909,203.9803466796875,262.38271217081708,195.44622802734376]},{"page":104,"text":"These are key functions in the X-ray analysis. Let us take Eq. 5.15 for the scattering","rect":[53.812843322753909,233.6263427734375,385.16460505536568,224.63201904296876]},{"page":104,"text":"amplitude of N scattering objects, e.g. by molecules forming a molecular crystal,","rect":[53.81282424926758,245.58590698242188,385.1168484261203,236.65135192871095]},{"page":104,"text":"and write the scattering intensity","rect":[53.812828063964847,257.58526611328127,186.18790472223129,248.6108856201172]},{"page":104,"text":"N","rect":[201.54344177246095,274.4619140625,206.18408142740567,269.79046630859377]},{"page":104,"text":"N","rect":[217.5173797607422,274.4619140625,222.1580194156869,269.79046630859377]},{"page":104,"text":"IðqÞ ¼ FðqÞF\u0006ðqÞ ¼ XXfjðqÞfkðqÞexpiqðrj \u0005 rkÞ","rect":[110.9669189453125,290.31097412109377,328.0126077090378,276.3760681152344]},{"page":104,"text":"j","rect":[203.01646423339845,298.03948974609377,204.9744518355598,291.9526672363281]},{"page":104,"text":"k","rect":[218.3667755126953,296.7795715332031,221.48143906488796,291.89892578125]},{"page":104,"text":"Here rj and rk are the same vectors corresponding to the distances shown in","rect":[65.76598358154297,320.4842834472656,385.1408623795844,310.61993408203127]},{"page":104,"text":"Fig. 7a and sign minus at rk comes from the complex conjugation. Both summations","rect":[53.81290817260742,331.5140686035156,385.12448515044408,322.57952880859377]},{"page":104,"text":"are made from 1 to N. The same equation may be presented in another form:","rect":[53.81350326538086,343.4736022949219,363.8015275723655,334.5390625]},{"page":104,"text":"N","rect":[97.88279724121094,360.4468078613281,102.52343689615567,355.7753601074219]},{"page":104,"text":"N N\u00051","rect":[225.22117614746095,360.4468078613281,250.5704505213197,355.6637878417969]},{"page":104,"text":"IðqÞ ¼ XfjðqÞfkðqÞexpiqðrj \u0005 rkÞ þ XXfjðqÞfkðqÞexpiqðrj \u0005 rkÞ (5.24)","rect":[63.15950012207031,376.3525085449219,385.16231666413918,362.41741943359377]},{"page":104,"text":"j¼k","rect":[95.05033874511719,384.13800048828127,105.52871079340358,377.8838195800781]},{"page":104,"text":"j6¼","rect":[224.03175354003907,384.3646240234375,231.3794337442386,377.8943176269531]},{"page":104,"text":"k","rect":[242.04519653320313,382.76446533203127,245.15986008539577,377.8838195800781]},{"page":104,"text":"In the first N terms j ¼ k, q(rj \u0005 rk) ¼ 0 and, this sum corresponds to the","rect":[65.76656341552735,406.8657531738281,385.17392767145005,396.981689453125]},{"page":104,"text":"intensity coming from the individual atoms or molecules without interference or","rect":[53.81417465209961,417.8387756347656,385.14910255281105,408.90423583984377]},{"page":104,"text":"diffraction. Such scattering and corresponding terms exist even in the gas phase","rect":[53.81417465209961,429.7983093261719,385.09729803277818,420.86376953125]},{"page":104,"text":"(so-called, “gas component”). Thus fjðqÞfkðqÞ ¼ Ff2ormðqÞ is a smooth decaying","rect":[53.81417465209961,443.87908935546877,385.10060969403755,431.2759704589844]},{"page":104,"text":"function of q like the square of the atom scattering amplitude shown in Fig. 5.7b.","rect":[53.81351852416992,453.7178039550781,385.1831936409641,444.7234802246094]},{"page":104,"text":"The second term includes N \u0005 1 times more terms than the first one and has very","rect":[53.81452178955078,465.6773681640625,385.11953059247505,456.7427978515625]},{"page":104,"text":"sharp maxima at q(rj \u0005 rk) ¼ 2p due to periodicity of the crystal lattice. For","rect":[53.81452178955078,478.5671081542969,385.14995704499855,468.70233154296877]},{"page":104,"text":"identical objects we may also extract fjfk ¼ F2form(q) from the second sum symbols:","rect":[53.814090728759769,490.51336669921877,385.2068315274436,478.5459289550781]},{"page":104,"text":"N ðN\u00051Þ","rect":[235.4739532470703,509.8233947753906,265.91299873247086,502.8580322265625]},{"page":104,"text":"IðqÞ ¼ NFf2ormðqÞ þ NFf2ormðqÞX X expiqðrj \u0005 rkÞ:","rect":[107.68244171142578,524.9129638671875,331.28660523575686,510.9779357910156]},{"page":104,"text":"j¼6","rect":[234.22808837890626,532.9249267578125,241.5757685831058,526.45458984375]},{"page":104,"text":"k","rect":[254.67723083496095,531.3246459960938,257.79189438715357,526.4440307617188]},{"page":104,"text":"The normalized intensity is given by","rect":[65.76702117919922,555.459716796875,213.70549861005316,546.5252075195313]},{"page":104,"text":"NFIf2oðrqmÞðqÞ ¼ 1 þ X jN6¼ ðX N\u0005k 1Þ expiqðrj \u0005 rkÞ ¼ SðqÞ","rect":[120.71174621582031,597.8245239257813,319.91354765044408,567.7575073242188]},{"page":104,"text":"(5.25)","rect":[361.114013671875,587.0296020507813,385.16338477937355,578.4337158203125]},{"page":105,"text":"88","rect":[53.812843322753909,42.55594253540039,62.2739118972294,36.73106384277344]},{"page":105,"text":"5 Structure Analysis and X-Ray Diffraction","rect":[235.81301879882813,44.274620056152347,385.1466421523563,36.62946701049805]},{"page":105,"text":"where S(q) is a structure factor determined by relative positions of the objects","rect":[53.812843322753909,68.2883529663086,385.15964140044408,59.35380554199219]},{"page":105,"text":"(atoms or molecules) in a medium of given symmetry and the character of their","rect":[53.813846588134769,80.24788665771485,385.10991798249855,71.31333923339844]},{"page":105,"text":"positional correlations. The normalized scattering intensity is given in “electron”","rect":[53.813846588134769,92.20748138427735,385.05616033746568,83.27293395996094]},{"page":105,"text":"units.","rect":[53.813846588134769,102.04837036132813,75.6832470467258,95.17564392089844]},{"page":105,"text":"From (5.25) we may conclude that the total intensity of scattering of a crystal isa","rect":[65.76586151123047,116.0697250366211,385.1587299175438,107.0754165649414]},{"page":105,"text":"product of a sharp structure factor and a smooth form-factor that is a series of sharp","rect":[53.813846588134769,128.02932739257813,385.15374079755318,119.09477233886719]},{"page":105,"text":"peaks with a smoothly decaying envelope fjfk. The structure factor can be found","rect":[53.813846588134769,140.9060516357422,385.12868586591255,131.01446533203126]},{"page":105,"text":"from the experimental angular dependence of the scattering intensity. But what is","rect":[53.81376266479492,151.94912719726563,385.18442167388158,143.0145721435547]},{"page":105,"text":"the relation between structure factor S(q) and density function?","rect":[53.81376266479492,163.90869140625,309.0148395366844,154.97413635253907]},{"page":105,"text":"Theoretically they are related by the Fourier transform","rect":[65.76580047607422,175.86822509765626,286.0066904890593,166.9336700439453]},{"page":105,"text":"SðqÞ ¼ ð GðrÞexpðiqrÞdV","rect":[166.593994140625,212.28671264648438,271.8856582013484,190.16448974609376]},{"page":105,"text":"(5.26)","rect":[361.0584411621094,205.4538116455078,385.1078122696079,196.85791015625]},{"page":105,"text":"of a new function, the so-called density correlation function G(r). According to","rect":[53.81678009033203,235.77947998046876,385.14666071942818,226.80508422851563]},{"page":105,"text":"(5.26) a diffraction structure factor S(q) related to intensity pattern may be calcu-","rect":[53.81678009033203,247.73904418945313,385.0988706192173,238.74472045898438]},{"page":105,"text":"lated form the known G(r) function by direct Fourier transform (this is a direct","rect":[53.815773010253909,259.69854736328127,385.184462142678,250.73411560058595]},{"page":105,"text":"problem of the X-ray analysis). On the contrary, the density correlation function","rect":[53.815773010253909,271.6581115722656,385.15960017255318,262.6837158203125]},{"page":105,"text":"G(r) may, in principle, be calculated from the measured function S(q) by the","rect":[53.815757751464847,283.6176452636719,385.1794818706688,274.6531982421875]},{"page":105,"text":"inverse Fourier transform (an inverse problem). Below we shall use these proce-","rect":[53.817771911621097,295.57720947265627,385.12774024812355,286.62274169921877]},{"page":105,"text":"dures, but, at first, let us consider the Fourier transforms and related operations","rect":[53.817787170410159,307.5367431640625,385.1029397402878,298.6021728515625]},{"page":105,"text":"more carefully.","rect":[53.817787170410159,319.4394836425781,115.27409024740939,310.50494384765627]},{"page":105,"text":"5.4 Fourier Transforms and Diffraction","rect":[53.812843322753909,361.648681640625,266.08563202714086,352.8635559082031]},{"page":105,"text":"5.4.1 Principle","rect":[53.812843322753909,393.4436950683594,134.91723426313608,382.88958740234377]},{"page":105,"text":"We know several important examples of the Foutier transform in physics. For","rect":[53.812843322753909,421.0697937011719,385.1477292617954,412.13525390625]},{"page":105,"text":"instance, the time evolution of the electric signal f(t) may be related to the","rect":[53.812843322753909,433.0293273925781,385.1715778179344,424.054931640625]},{"page":105,"text":"frequency specrum F(o) of the same signal by a Fourier transform. In the diffrac-","rect":[53.81187438964844,444.9888610839844,385.0939878067173,436.0543212890625]},{"page":105,"text":"tion study we relate spatial periodicity of a density r(r) to the spectrum of the","rect":[53.81186294555664,456.8916320800781,385.1725543804344,447.95709228515627]},{"page":105,"text":"wavevectors (or angular spectrum) F (q) of the same structure. The direct Fourier","rect":[53.81282424926758,468.8511657714844,385.1098569473423,459.9166259765625]},{"page":105,"text":"transform of density function is given by operation:","rect":[53.8138313293457,480.8106994628906,262.1086869961936,471.87615966796877]},{"page":105,"text":"FðqÞ ¼ ð rðrÞexpðiqrÞdV \b =½r\u0004","rect":[151.18690490722657,517.2291870117188,287.7838586537256,495.10699462890627]},{"page":105,"text":"V","rect":[183.81346130371095,523.5731811523438,188.09176873334469,518.845947265625]},{"page":105,"text":"The inverse Fourier transform of scattering amplitude is given by:","rect":[65.76717376708985,548.0914916992188,332.56794602939677,539.156982421875]},{"page":105,"text":"rðrÞ ¼ ð21pÞ3 ð FðqÞexpð\u0005iqrÞdq \b =\u00051½r\u0004","rect":[130.34213256835938,586.6897583007813,308.6854699818506,562.3873901367188]},{"page":105,"text":"q","rect":[187.3255157470703,591.8667602539063,190.80947945198379,587.2022705078125]},{"page":105,"text":"(5.27)","rect":[361.05682373046877,510.3970947265625,385.1061948379673,501.80120849609377]},{"page":105,"text":"(5.28)","rect":[361.05657958984377,577.62060546875,385.1059506973423,569.0247192382813]},{"page":106,"text":"5.4 Fourier Transforms and Diffraction","rect":[53.812843322753909,42.55594253540039,187.79556793360636,36.62946701049805]},{"page":106,"text":"89","rect":[376.7473449707031,42.62367248535156,385.2084097304813,36.73106384277344]},{"page":106,"text":"Note that factor (2p)3 correspond to three-dimensional (3D) case. For 2D and 1D","rect":[65.76496887207031,68.2883529663086,385.1936621661457,57.23747253417969]},{"page":106,"text":"cases we would have (2p)2 and (2p), respectively. However, very often the factor","rect":[53.812984466552737,80.24800872802735,385.0920346817173,69.19712829589844]},{"page":106,"text":"(2p)D is skipped at all. The direct and inverse Fourier operators applied consecu-","rect":[53.813899993896487,92.20772552490235,385.0932859024204,81.24967956542969]},{"page":106,"text":"tively restore the initial density function.","rect":[53.81315994262695,104.1104965209961,218.29751248862034,95.17594909667969]},{"page":106,"text":"=\u00051½=½rðrÞ\u0004\u0004 ¼ rðrÞ","rect":[177.97900390625,127.68840789794922,261.0036660586472,116.35739135742188]},{"page":106,"text":"We meet this case in technics. For instance, in an optical microscope, lenses","rect":[65.7677230834961,149.22897338867188,385.1118203555222,140.29441833496095]},{"page":106,"text":"fulfil the direct and inverse Fourier transforms: the light is focused by a condenser","rect":[53.815711975097659,161.18850708007813,385.16948829499855,152.2539520263672]},{"page":106,"text":"onto the object, then diffracted, then collected by an objective, and finally the image","rect":[53.815711975097659,173.14804077148438,385.10776556207505,164.21348571777345]},{"page":106,"text":"is taken by a video camera and seen on a screen. The form of the object is seen as an","rect":[53.815711975097659,185.10757446289063,385.1525811295844,176.1730194091797]},{"page":106,"text":"intensity pattern that is a flat distribution of the optical density, because the phases","rect":[53.815711975097659,197.067138671875,385.13959135161596,188.0529022216797]},{"page":106,"text":"of the waves forming the image are lost. A holographic technique, which always","rect":[53.815711975097659,209.02664184570313,385.0848733340378,200.0920867919922]},{"page":106,"text":"uses an interference of scattered rays with a reference beam having a known phase,","rect":[53.815711975097659,220.9862060546875,385.1834377815891,212.05165100097657]},{"page":106,"text":"allows the restoration of a volume image of the object.","rect":[53.815711975097659,232.94573974609376,274.87286038901098,224.0111846923828]},{"page":106,"text":"Unfortunately, some important information is also lost in the X-ray diffraction","rect":[65.7677230834961,244.8485107421875,385.09487238935005,235.91395568847657]},{"page":106,"text":"experiment:","rect":[53.815711975097659,256.8080749511719,101.69048936435769,247.87351989746095]},{"page":106,"text":"1.","rect":[53.815711975097659,273.0,61.2813686897922,265.9009704589844]},{"page":106,"text":"2.","rect":[53.815711975097659,285.0,61.2813686897922,277.8604736328125]},{"page":106,"text":"3.","rect":[53.815696716308597,309.0,61.28135343100314,301.72283935546877]},{"page":106,"text":"4.","rect":[53.814964294433597,333.0,61.28062100912814,325.64324951171877]},{"page":106,"text":"The phases of scattered rays are not recorded","rect":[66.27738952636719,274.7757568359375,248.54793635419379,265.8411865234375]},{"page":106,"text":"As density r is real quantity, F(q) ¼ F(\u0005q), the scattering pattern is always","rect":[66.27738952636719,286.7352600097656,385.0848428164597,277.80072021484377]},{"page":106,"text":"centrosymmetric (Friedel theorem)","rect":[66.27737426757813,298.6380615234375,206.97217689363135,289.7034912109375]},{"page":106,"text":"A possible range of vectors of scattering q ¼ (4p/l)sin# is limited by qmax ¼4","rect":[66.27737426757813,310.5976257324219,385.1796807389594,301.3443298339844]},{"page":106,"text":"p/l","rect":[66.2766342163086,321.4727783203125,80.0980488712604,313.30517578125]},{"page":106,"text":"An absence of lenses for the X-ray range restricts X-ray applications in compar-","rect":[66.27664184570313,334.5180358886719,385.15487037507668,325.58349609375]},{"page":106,"text":"ison with optics","rect":[66.2766342163086,346.4775695800781,130.0980569277878,337.54302978515627]},{"page":106,"text":"Therefore, it is very difficult to solve the inverse problem mentioned above, that","rect":[65.76696014404297,364.38848876953127,385.139784408303,355.43402099609377]},{"page":106,"text":"is to find r(r) from the data on scattering intensity I(q), and one usually tries","rect":[53.813961029052737,376.3480224609375,385.10104765044408,367.4134521484375]},{"page":106,"text":"different r(r) or G(r) model functions with subsequent calculations of S(q) and","rect":[53.813961029052737,388.3075866699219,385.14785090497505,379.3431396484375]},{"page":106,"text":"then I(q) forcomparison with experiment. Below we consider few examples of such","rect":[53.81597137451172,400.26708984375,385.1269463639594,391.33251953125]},{"page":106,"text":"direct problem solutions.","rect":[53.815940856933597,412.1170959472656,154.56348844076877,403.2721862792969]},{"page":106,"text":"5.4.2 Example: Form Factor of a Parallelepiped","rect":[53.812843322753909,454.3189697265625,302.63746395015309,443.7648620605469]},{"page":106,"text":"Consider diffraction","rect":[53.812843322753909,479.92303466796877,137.42324152997504,473.01043701171877]},{"page":106,"text":"A, B, C, Fig. 5.11a.","rect":[53.812843322753909,493.9045715332031,131.52537198569065,484.9102478027344]},{"page":106,"text":"by","rect":[142.801513671875,481.94500732421877,152.75571528485785,473.01043701171877]},{"page":106,"text":"a","rect":[158.03939819335938,479.8831787109375,162.48893011285629,475.2416076660156]},{"page":106,"text":"single","rect":[167.83932495117188,481.94500732421877,191.73938787652816,473.01043701171877]},{"page":106,"text":"transparent","rect":[197.06787109375,481.94500732421877,241.494520736428,474.02642822265627]},{"page":106,"text":"parallelepiped","rect":[246.85784912109376,481.94500732421877,303.61977473310005,473.01043701171877]},{"page":106,"text":"with","rect":[308.99603271484377,479.8831787109375,326.71453181317818,473.01043701171877]},{"page":106,"text":"edge","rect":[332.04998779296877,481.94500732421877,350.9032672710594,473.01043701171877]},{"page":106,"text":"lengths","rect":[356.29345703125,481.94500732421877,385.10092558013158,473.01043701171877]},{"page":106,"text":"\u0005 A=2","rect":[108.75809478759766,516.169189453125,134.75151911786566,506.23858642578127]},{"page":106,"text":"x","rect":[147.95777893066407,513.7318725585938,152.40731085016098,509.1998291015625]},{"page":106,"text":"A=2; \u0005B=2","rect":[165.63148498535157,516.169189453125,209.86501399091254,506.23858642578127]},{"page":106,"text":"y","rect":[223.1280059814453,515.7637939453125,227.57753790094223,509.130126953125]},{"page":106,"text":"B=2; \u0005C=2","rect":[240.8017120361328,516.169189453125,285.54485407880318,506.23858642578127]},{"page":106,"text":"z","rect":[298.8078308105469,513.6422119140625,302.64020170317846,509.3193664550781]},{"page":106,"text":"C=2","rect":[315.8573913574219,516.169189453125,332.44704524091255,506.23858642578127]},{"page":106,"text":"Assume density r¼const within the parallelepiped and r¼0 outside of its","rect":[65.77079010009766,537.7186889648438,385.1575967227097,528.7841796875]},{"page":106,"text":"volume. According to Eq. 5.17, the scattering amplitude is","rect":[53.818763732910159,549.67822265625,290.08591093169408,540.6839599609375]},{"page":106,"text":"A=2 B=2 C=2","rect":[147.33401489257813,568.9345092773438,199.92912361702285,561.9830932617188]},{"page":106,"text":"FðqÞ ¼ ð","rect":[111.59697723388672,592.1048583984375,154.8281252367903,569.982666015625]},{"page":106,"text":"ð","rect":[170.5585479736328,592.1048583984375,175.3266146166731,569.982666015625]},{"page":106,"text":"ð rexpiðqxx þ qyy þ qzzÞdxdydz","rect":[191.23402404785157,592.1048583984375,328.10947050200658,569.982666015625]},{"page":106,"text":"\u0005A=2 \u0005B=2 \u0005C=2","rect":[144.16151428222657,600.6759643554688,202.08151314339004,593.7245483398438]},{"page":107,"text":"90","rect":[53.8120002746582,42.62507629394531,62.27306884913369,36.73246765136719]},{"page":107,"text":"5 Structure Analysis and X-Ray Diffraction","rect":[235.81216430664063,44.276023864746097,385.14581817774697,36.6308708190918]},{"page":107,"text":"The integral over \u0005A/2 < x < A/2:","rect":[65.76496887207031,68.2883529663086,209.56550462314676,59.35380554199219]},{"page":107,"text":"A=2","rect":[75.11149597167969,90.54840850830078,86.35563728889784,83.59701538085938]},{"page":107,"text":"r ð expiqxxdx ¼ iqrx expðiqxxÞjA\u0005=A2=2 ¼ ArsiðnAðAqxq=x2=Þ2Þ ¼ ArDxðA;qxÞ (5.29)","rect":[64.74568176269531,114.42516326904297,385.1055844864048,90.87125396728516]},{"page":107,"text":"\u0005A=2","rect":[71.88255310058594,122.28986358642578,88.50803444465956,115.33847045898438]},{"page":107,"text":"The plot of scattered field amplitude Dx is shown in the upper part of Fig. 5.11b.","rect":[65.76656341552735,146.79135131835938,385.1831020882297,137.52786254882813]},{"page":107,"text":"It is the so-called sine-integral function. The scattering intensity is shown in the","rect":[53.814430236816409,158.75091552734376,385.17417181207505,149.8163604736328]},{"page":107,"text":"lower part of the figure. Integrating over the y and z co-ordinates we obtain the","rect":[53.814430236816409,170.71044921875,385.1731647319969,161.77589416503907]},{"page":107,"text":"three-dimensional","rect":[53.813411712646487,181.0,126.09391648838113,173.6786651611328]},{"page":107,"text":"r2V2(DxDyDz)2.","rect":[53.81418991088867,195.55970764160157,116.57190366293674,183.52232360839845]},{"page":107,"text":"scattering","rect":[131.75587463378907,182.61322021484376,170.70668879560004,173.6786651611328]},{"page":107,"text":"amplitude","rect":[176.3925323486328,182.61322021484376,216.4711841411766,173.6786651611328]},{"page":107,"text":"F(q)¼","rect":[222.1052703857422,182.54348754882813,248.06584957334906,173.73843383789063]},{"page":107,"text":"rVDxDyDz","rect":[253.71286010742188,183.6000518798828,296.1359407570378,173.3499755859375]},{"page":107,"text":"and","rect":[301.805419921875,181.0,316.20916071942818,173.67906188964845]},{"page":107,"text":"intensity","rect":[321.857177734375,182.61361694335938,356.3674248795844,173.67906188964845]},{"page":107,"text":"I(q)¼","rect":[362.0194396972656,182.54388427734376,385.14804867002877,173.73883056640626]},{"page":107,"text":"Note that, for infinitely thick parallelepiped (A!1), there is no diffraction, only","rect":[65.76561737060547,206.53280639648438,385.1285332780219,197.59825134277345]},{"page":107,"text":"directly transmitted beam is left and the integral becomes d-function. Generally, the","rect":[53.81455612182617,218.49234008789063,385.17530096246568,209.25897216796876]},{"page":107,"text":"larger parallelepiped dimensions the narrower is the central peak. We shall come","rect":[53.814537048339847,230.451904296875,385.14048040582505,221.51734924316407]},{"page":107,"text":"back to this point when discussing the diffraction on thin layers of a smectic","rect":[53.814537048339847,242.41143798828126,385.08872259332505,233.4768829345703]},{"page":107,"text":"A liquid crystal.","rect":[53.814537048339847,254.3709716796875,119.05941434408908,245.43641662597657]},{"page":107,"text":"Consider two interesting particular cases shown in Fig. 5.12:","rect":[65.76655578613281,266.33050537109377,310.66401536533427,257.336181640625]},{"page":107,"text":"1. In the top left sketch, the parallelepiped is degenerated into the infinitely thin","rect":[53.814537048339847,284.2414245605469,385.1563347916938,275.306884765625]},{"page":107,"text":"plane with dimensions A!1, B!1, C!d(z). All its density is concentrated in","rect":[66.2762222290039,296.2009582519531,385.14641657880318,286.96759033203127]},{"page":107,"text":"Fig. 5.11 Geometry of the","rect":[52.96326446533203,377.85223388671877,146.16872546457769,370.1224060058594]},{"page":107,"text":"parallelepiped discussed (a)","rect":[52.96326446533203,387.7604675292969,148.22561734778575,380.1661071777344]},{"page":107,"text":"and the patterns (b) of the","rect":[52.96326446533203,397.7364501953125,142.08963916086675,390.14208984375]},{"page":107,"text":"diffraction amplitude (above)","rect":[52.963260650634769,407.6556396484375,153.49347013098888,400.061279296875]},{"page":107,"text":"and intensity (below)","rect":[52.96326446533203,417.6316223144531,125.11418241126229,410.02032470703127]},{"page":107,"text":"a","rect":[204.94216918945313,325.28076171875,210.49743158668697,319.6920166015625]},{"page":107,"text":"A","rect":[226.75048828125,350.5193176269531,232.5205427056,345.1294860839844]},{"page":107,"text":"C","rect":[224.006103515625,380.8479919433594,229.3366108577877,375.33819580078127]},{"page":107,"text":"B","rect":[245.7716827392578,343.0910949707031,251.1021900814205,337.7972106933594]},{"page":107,"text":"b","rect":[296.1562805175781,325.28076171875,302.2610742670563,317.9723815917969]},{"page":107,"text":"4π","rect":[337.4888000488281,408.388427734375,344.1132171567679,404.04534912109377]},{"page":107,"text":"Α","rect":[338.6368103027344,414.50921630859377,342.9651624141353,410.4480895996094]},{"page":107,"text":"Fig. 5.12 Fourier transforms","rect":[53.812843322753909,562.3483276367188,154.41496737479486,554.6185302734375]},{"page":107,"text":"(lower drawings) of a plane","rect":[53.812843322753909,572.2565307617188,148.51674792551519,564.6621704101563]},{"page":107,"text":"into a line (a) and a line intoa","rect":[53.812843322753909,581.8370971679688,155.36429736399175,574.5813598632813]},{"page":107,"text":"plane (b)","rect":[53.8120002746582,592.1516723632813,84.89543241126229,584.5573120117188]},{"page":107,"text":"a","rect":[204.715576171875,455.9278564453125,210.26944934542767,450.3404846191406]},{"page":107,"text":"x","rect":[219.93896484375,508.0936279296875,224.38205267846437,503.83966064453127]},{"page":107,"text":"qx","rect":[213.8504638671875,576.0017700195313,222.06533439119355,569.6373291015625]},{"page":107,"text":"z","rect":[232.9070281982422,475.6890563964844,236.90261078341698,471.4350891113281]},{"page":107,"text":"qz","rect":[226.98397827148438,532.2426147460938,234.86305339266085,525.8779907226563]},{"page":107,"text":"x","rect":[327.0498962402344,509.7908935546875,331.49298407494879,505.53692626953127]},{"page":107,"text":"z","rect":[335.4637756347656,476.31884765625,339.45935821994046,472.06488037109377]},{"page":107,"text":"qz","rect":[338.74896240234377,543.5498046875,346.6284495108249,537.1859130859375]},{"page":107,"text":"qx","rect":[343.9585266113281,586.660400390625,352.17370231111547,580.2959594726563]},{"page":107,"text":"qy","rect":[376.94854736328127,568.0261840820313,385.16369254549047,560.38427734375]},{"page":108,"text":"5.4 Fourier Transforms and Diffraction","rect":[53.81282424926758,42.55807876586914,187.7955526748173,36.6316032409668]},{"page":108,"text":"91","rect":[376.7473449707031,42.62580871582031,385.2084097304813,36.73320007324219]},{"page":108,"text":"2.","rect":[53.813785552978519,127.0,61.27944226767306,119.15465545654297]},{"page":108,"text":"plane x,y that symbolically can be written as r110 Then, scattering field","rect":[66.27484893798828,68.2883529663086,385.11882868817818,59.35380554199219]},{"page":108,"text":"amplitude F(q) ¼ F001 is degenerated into a line along the z-axis, as shown","rect":[66.27552032470703,80.24800872802735,385.17852107099068,71.31333923339844]},{"page":108,"text":"in the bottom left sketch. The square of the field amplitude corresponds to the","rect":[66.27545928955078,92.20760345458985,385.1725543804344,83.27305603027344]},{"page":108,"text":"form-factor of an infinite square (or roughly speaking, to a very large square-like","rect":[66.27545928955078,104.11031341552735,385.14270818902818,95.17576599121094]},{"page":108,"text":"molecule).","rect":[66.27545928955078,115.67142486572266,108.83668180014377,107.13529968261719]},{"page":108,"text":"Density r001 is concentrated along the z line, C!1, A!d(x), B!d(y). Then","rect":[66.27545928955078,128.02999877929688,385.1409844498969,118.796630859375]},{"page":108,"text":"F(q) ¼ F110, that is the scattering amplitude is degenerated into the qx, qy-","rect":[66.2767562866211,140.91932678222657,385.15975318757668,131.05503845214845]},{"page":108,"text":"plane with Dz !d(qz), see right sketches. The intensity pattern corresponds to","rect":[66.27660369873047,151.94937133789063,385.1425713639594,142.68588256835938]},{"page":108,"text":"the form-factor of an infinite rod (or, roughly speaking, to a very long rod-like","rect":[66.27533721923828,163.908935546875,385.12168157770005,154.97438049316407]},{"page":108,"text":"molecule).","rect":[66.27533721923828,175.4700469970703,108.83655972983127,166.9339141845703]},{"page":108,"text":"5.4.3 Convolution of Two Functions","rect":[53.812843322753909,231.90350341796876,242.44405250284835,221.34939575195313]},{"page":108,"text":"The structure of a molecular or a liquid crystal is a result of convolution of two","rect":[53.812843322753909,259.529541015625,385.14470759442818,250.55514526367188]},{"page":108,"text":"density functions, the density of a group of atoms in a molecule and periodic density","rect":[53.812843322753909,271.48907470703127,385.19744196942818,262.5146789550781]},{"page":108,"text":"function of a lattice. Let us look at the convolution procedure. By definition, the","rect":[53.812843322753909,283.4486083984375,385.17261541559068,274.5140380859375]},{"page":108,"text":"convolution of two functions f1(x) and f2(x) is given by the expression","rect":[53.812843322753909,295.3517761230469,336.2692803483344,286.3769836425781]},{"page":108,"text":"1","rect":[223.86163330078126,313.7984313964844,230.82956071060819,310.6539001464844]},{"page":108,"text":"QðxÞ ¼ f1ðxÞ\u0006f2ðxÞ ¼ ð f1ðx0Þf2ðx \u0005 x0Þdx0","rect":[133.00332641601563,337.83526611328127,305.72597541041889,315.7130432128906]},{"page":108,"text":"\u00051","rect":[220.63270568847657,343.8393249511719,232.9659437672488,340.6947937011719]},{"page":108,"text":"(5.30)","rect":[361.0561828613281,331.0023498535156,385.10555396882668,322.4064636230469]},{"page":108,"text":"Here, the asterisk means the convolution operation. Such a convolution gives","rect":[53.81450653076172,368.4129333496094,385.15631498442846,359.4783935546875]},{"page":108,"text":"distribution of one function over a law given by the other. For example, on the","rect":[53.81450653076172,380.3724670410156,385.1732562847313,371.43792724609377]},{"page":108,"text":"top of Fig. 5.13 there are two functions of the same variable x, function f1(x) and","rect":[53.81450653076172,392.33203125,385.1459588151313,383.33770751953127]},{"page":108,"text":"function f2(x)¼d(x \u0005 a) located at different positions","rect":[53.81405258178711,404.2355651855469,275.86996854888158,395.002197265625]},{"page":108,"text":"convolution and using R\u000511 dðxÞdx ¼ 1, we shall get","rect":[53.81440353393555,417.5243225097656,265.31684739658427,405.7076110839844]},{"page":108,"text":"on","rect":[280.16619873046877,403.0,290.12041560224068,397.0]},{"page":108,"text":"the","rect":[294.32708740234377,403.0,306.5508502788719,395.301025390625]},{"page":108,"text":"x-axis.","rect":[310.8689880371094,402.1836853027344,337.2426418831516,395.301025390625]},{"page":108,"text":"After","rect":[341.4572448730469,402.1737365722656,362.48052345124855,395.301025390625]},{"page":108,"text":"their","rect":[366.77777099609377,402.1737365722656,385.1134275039829,395.301025390625]},{"page":108,"text":"1","rect":[82.6452407836914,434.6419372558594,89.61316819351834,431.4974060058594]},{"page":108,"text":"1","rect":[234.45431518554688,434.6419372558594,241.4222425953738,431.4974060058594]},{"page":108,"text":"ð f1ðx0Þdðx \u0005 a \u0005 x0Þdx0 ¼ f1ðx \u0005 aÞ ð dðx \u0005 a \u0005 x0Þdx0 ¼f1ðx \u0005 aÞ;","rect":[83.21450805664063,458.6787109375,359.7790063588037,436.5555419921875]},{"page":108,"text":"\u00051","rect":[79.41630554199219,464.6828918457031,91.7495359913699,461.5383605957031]},{"page":108,"text":"\u00051","rect":[231.2825164794922,464.6828918457031,243.61575455826444,461.5383605957031]},{"page":108,"text":"Fig. 5.13 Convolution","rect":[53.812843322753909,540.6395263671875,133.29698700098917,532.70654296875]},{"page":108,"text":"operation: function f1(x)","rect":[53.812843322753909,550.5477294921875,136.27356046302013,542.9195556640625]},{"page":108,"text":"convoluted with the other","rect":[53.81282424926758,558.73974609375,141.54734891516856,552.8724975585938]},{"page":108,"text":"function d(x\u0005a) occupies","rect":[53.81282424926758,570.4428100585938,140.92375643729486,562.594482421875]},{"page":108,"text":"position of the second","rect":[53.81282424926758,580.4187622070313,129.21280426173136,572.8244018554688]},{"page":108,"text":"function on axisx","rect":[53.81282424926758,588.6506958007813,115.14626452707768,582.8004150390625]},{"page":108,"text":"the","rect":[143.3690185546875,569.0,153.7592100837183,562.8484497070313]},{"page":108,"text":"a","rect":[265.4872741699219,510.061767578125,271.04353637325496,504.47198486328127]},{"page":108,"text":"b","rect":[265.21270751953127,554.408935546875,271.3185999767911,547.0992431640625]},{"page":108,"text":"f1(x)","rect":[275.4373474121094,532.8001708984375,293.5052047737107,525.06494140625]},{"page":108,"text":"f1(x–a)","rect":[309.9295959472656,569.346923828125,333.76706794094329,560.861083984375]},{"page":108,"text":"a","rect":[336.554931640625,540.9636840820313,340.5530622703726,537.2190551757813]},{"page":108,"text":"a","rect":[337.64288330078127,586.87939453125,341.64101393052888,583.134765625]},{"page":108,"text":"δ(x–a)","rect":[346.48260498046877,526.388427734375,368.3039636928964,518.9248657226563]},{"page":109,"text":"92","rect":[53.81319046020508,42.62525939941406,62.27425903468057,36.73265075683594]},{"page":109,"text":"5 Structure Analysis and X-Ray Diffraction","rect":[235.8133544921875,44.276206970214847,385.1470083632938,36.63105392456055]},{"page":109,"text":"and function f1(x) is translated into function f1(x \u0005 a) keeping the same form, as","rect":[53.812843322753909,68.2883529663086,385.14120878325658,59.31396484375]},{"page":109,"text":"seen in the lower plot. If the second function consists of two delta functions, f2(x)¼d","rect":[53.814292907714847,80.24788665771485,385.17873469403755,71.0146484375]},{"page":109,"text":"(x \u0005 a)+ d(x \u0005 b), we find our function f1(x) at both new positions, a and b, i. e.","rect":[53.81399154663086,92.20772552490235,385.1589016487766,82.9742431640625]},{"page":109,"text":"f1(x) will be doubled. An arbitrary smooth function f2(x \u0005 xn) can be represented as","rect":[53.815101623535159,104.11067962646485,385.14011015044408,95.1361083984375]},{"page":109,"text":"a sum of n columns of different height or, more strictly, an infinite sum of delta-","rect":[53.81417465209961,116.0702133178711,385.08736549226418,107.13566589355469]},{"page":109,"text":"functions f2(x \u0005 xn)¼Sand(x \u0005 xn) (n¼1). Then f1(x) will be distributed over the","rect":[53.81417465209961,128.01007080078126,385.17493475152818,118.796630859375]},{"page":109,"text":"whole sets of the columns, that is over the law given by the f2(x \u0005 xn) function.","rect":[53.81319808959961,139.98953247070313,375.8215298226047,131.01513671875]},{"page":109,"text":"Going back to solid or liquid crystals we can say that the convolution procedure","rect":[65.7657699584961,151.94912719726563,385.07996404840318,143.0145721435547]},{"page":109,"text":"distributes molecular density over the sites of the crystal lattice. On the left side of","rect":[53.81374740600586,163.90869140625,385.1486753067173,154.95420837402345]},{"page":109,"text":"Fig. 5.14, the two functions, the electron density of a molecule rmol(r) and discrete","rect":[53.81374740600586,175.86822509765626,385.09897649957505,166.8739013671875]},{"page":109,"text":"points of the lattice density rlattice(r) ¼Sd(ri \u0005 rj) are shown separately (before","rect":[53.813899993896487,188.75804138183595,385.0947650737938,178.59490966796876]},{"page":109,"text":"convolution). On the right side we see the result of their convolution. Note that the","rect":[53.813655853271487,199.73104858398438,385.17041814996568,190.7765655517578]},{"page":109,"text":"convolution operation f1(x)* f2(x) is dramatically different from the multiplication","rect":[53.813655853271487,211.69094848632813,385.12734309247505,202.71621704101563]},{"page":109,"text":"operation f1(x)f2(x). An example is illustrated by Fig. 5.15, in which function f2(x) is","rect":[53.81338882446289,223.65057373046876,385.18817533599096,214.65625]},{"page":109,"text":"the same rlattice(r) function as in the previous picture and f1(x) is the so called box-","rect":[53.813533782958987,235.61026000976563,385.1361020645298,226.6358642578125]},{"page":109,"text":"function. The latter is equal to 1 within its contour and 0 outside. The multiplication","rect":[53.81418991088867,247.56979370117188,385.12517634442818,238.63523864746095]},{"page":109,"text":"selects only few d-functions from the whole lattice. On the contrary, the convolu-","rect":[53.81418991088867,259.52935791015627,385.1002439102329,250.29598999023438]},{"page":109,"text":"tion translates rmol into new functional space, namely the space of rlattice.","rect":[53.81417465209961,271.4891662597656,352.89556546713598,262.534423828125]},{"page":109,"text":"For the future discussion of the liquid crystal structure we need two important","rect":[65.7663803100586,283.4486999511719,385.17219407627177,274.51416015625]},{"page":109,"text":"theorems. The first of them, the theorem of convolution is formulated as follows: a","rect":[53.814353942871097,295.33154296875,385.1611713237938,286.3770751953125]},{"page":109,"text":"Fourier transform of convolution of two functions f1(x) and f2(x) is a product of their","rect":[53.814353942871097,307.31146240234377,385.11348853913918,298.3366394042969]},{"page":109,"text":"Fourier transforms F1(q)\u0007F2(q):","rect":[53.814430236816409,319.1615295410156,179.9566026456077,310.33642578125]},{"page":109,"text":"=ðf1\u0006f2Þ ¼ =ðf1Þ \u0007 =ðf2Þ ¼ F1ðqÞ \u0007 F2ðqÞ","rect":[138.49830627441407,343.529052734375,300.48373808013158,333.57830810546877]},{"page":109,"text":"(5.31a)","rect":[356.6384582519531,342.7919616699219,385.1373532852329,334.1960754394531]},{"page":109,"text":"Fig. 5.14 Convolution","rect":[53.812843322753909,403.3586120605469,133.29698700098917,395.6287841796875]},{"page":109,"text":"operation rmol*rlattice that","rect":[53.812843322753909,413.2668762207031,142.54179100241724,405.62158203125]},{"page":109,"text":"distribute molecular density","rect":[53.8130989074707,423.18597412109377,149.16344207911417,415.59161376953127]},{"page":109,"text":"rmol over the sites of a crystal","rect":[53.8130989074707,433.1617126464844,155.3453493520266,425.5673522949219]},{"page":109,"text":"lattice (filled symbols","rect":[53.8125114440918,443.1376647949219,127.8874862956933,435.5433044433594]},{"page":109,"text":"represent molecules, O is the","rect":[53.8125114440918,453.1136169433594,153.07863757395269,445.5023193359375]},{"page":109,"text":"reference point, r and ri are","rect":[53.8125114440918,463.0328674316406,148.87911364817144,455.4385070800781]},{"page":109,"text":"radius-vectors of a molecule","rect":[53.81306838989258,471.28125,150.9749083259058,465.4140319824219]},{"page":109,"text":"and lattice points, T is vector","rect":[53.81306838989258,482.9843444824219,154.11059659583263,475.3899841308594]},{"page":109,"text":"of translations)","rect":[53.81307601928711,492.62164306640627,105.13200467688729,485.3659362792969]},{"page":109,"text":"r","rect":[220.87904357910157,450.44647216796877,223.9556038251207,446.6483154296875]},{"page":109,"text":"O•","rect":[219.38670349121095,480.20538330078127,225.96335045867526,473.9893798828125]},{"page":109,"text":"ρmol","rect":[228.09127807617188,399.9255065917969,241.722287028656,394.1677551269531]},{"page":109,"text":"ri","rect":[234.741943359375,454.8337097167969,239.44297489486696,449.0359191894531]},{"page":109,"text":"ρlattice","rect":[261.8854064941406,401.40631103515627,281.7631161165121,395.6406555175781]},{"page":109,"text":"j","rect":[264.036865234375,424.5038757324219,266.2030396059164,417.0414733886719]},{"page":109,"text":"••••","rect":[255.2371826171875,428.168212890625,288.40360494438189,425.8481750488281]},{"page":109,"text":"T","rect":[276.3681640625,440.51885986328127,281.6996707260354,435.0240173339844]},{"page":109,"text":"• • •i •","rect":[253.23687744140626,448.368408203125,286.40397115531939,442.6096496582031]},{"page":109,"text":"••••","rect":[251.2019805908203,463.42266845703127,284.40314718071,461.1026306152344]},{"page":109,"text":"••••","rect":[249.236328125,478.10052490234377,282.33317037406939,475.7804870605469]},{"page":109,"text":"convolution","rect":[296.5877990722656,402.6737365722656,334.56380087126379,396.94696044921877]},{"page":109,"text":"→","rect":[311.63165283203127,413.75726318359377,319.518834553644,409.66326904296877]},{"page":109,"text":"ρmol* ρlattice","rect":[345.3987731933594,404.1043701171875,385.19713711260587,397.8219299316406]},{"page":109,"text":"Fig. 5.15 A multiplication","rect":[53.812843322753909,567.3362426757813,147.09276336817667,559.6064453125]},{"page":109,"text":"operation with a box-function","rect":[53.812843322753909,577.2444458007813,155.32031768946573,569.6500854492188]},{"page":109,"text":"f1 and function f2¼rlattice(r)","rect":[53.812843322753909,587.1467895507813,148.84877103430919,579.5352783203125]},{"page":110,"text":"5.4 Fourier Transforms and Diffraction","rect":[53.812843322753909,42.55594253540039,187.79556793360636,36.62946701049805]},{"page":110,"text":"93","rect":[376.7473449707031,42.62367248535156,385.2084097304813,36.73106384277344]},{"page":110,"text":"The second one called the theorem of multiplication is an inverse of the first: the","rect":[65.76496887207031,68.2883529663086,385.17176092340318,59.35380554199219]},{"page":110,"text":"Fourier transform of the product of two functions f1(x) and f2(x) is convolution of","rect":[53.812950134277347,80.24788665771485,385.15215431062355,71.27349853515625]},{"page":110,"text":"Fourier transforms of each of them F1(q)*F2(q):","rect":[53.814292907714847,92.09815979003906,248.4964280850608,83.27305603027344]},{"page":110,"text":"=ðf1 \u0007 f2Þ ¼ =ðf1Þ\u0006=ðf2Þ ¼ F1ðqÞ\u0006F2ðqÞ","rect":[140.08413696289063,126.89507293701172,298.89755643950658,116.40440368652344]},{"page":110,"text":"(5.31b)","rect":[356.0715637207031,126.15787506103516,385.15975318757668,117.5619888305664]},{"page":110,"text":"5.4.4 Self-Convolution","rect":[53.812843322753909,188.54266357421876,174.09455841874243,177.98855590820313]},{"page":110,"text":"Let us make the inverse Fourier transform of the scattering intensity (5.23) and use","rect":[53.812843322753909,216.225341796875,385.1327289409813,207.23101806640626]},{"page":110,"text":"the properties of the Fourier integral:","rect":[53.812843322753909,228.12811279296876,203.34696824619364,219.1935577392578]},{"page":110,"text":"=\u00051fFðqÞ \u0007 F\u0006ðqÞg ¼ rðrÞ\u0006rð\u0005rÞ ¼ ð rðuÞrðr þ uÞdu ¼ PðrÞ","rect":[81.00276947021485,269.53485107421877,333.90622343169408,247.41262817382813]},{"page":110,"text":"(5.32)","rect":[361.0583801269531,262.70196533203127,385.10775123445168,254.1060791015625]},{"page":110,"text":"As a result, we obtain the convolution of the density function r(r) with the same","rect":[65.76871490478516,298.0726013183594,385.13462103082505,289.1380615234375]},{"page":110,"text":"function inverted with respect of the origin of the reference frame r(\u0005r). Note that","rect":[53.81769561767578,310.03216552734377,385.1435991055686,301.09759521484377]},{"page":110,"text":"the minus sign appears due to different signs in the exponents for two complex","rect":[53.81768035888672,321.9349365234375,385.1695183854438,313.0003662109375]},{"page":110,"text":"conjugates in (5.28). The P(r) function is known as density autocorrelation function","rect":[53.81768035888672,333.89447021484377,385.1744317155219,324.900146484375]},{"page":110,"text":"or the Paterson function when used in structural analysis. Thus, we may write the","rect":[53.81768035888672,345.85400390625,385.1745075054344,336.91943359375]},{"page":110,"text":"inverse and direct Fourier transforms as follows:","rect":[53.81768035888672,355.7915954589844,249.76335008213114,348.8790283203125]},{"page":110,"text":"PðrÞ ¼ ð IðqÞe\u0005iqrdq","rect":[176.85073852539063,399.2202453613281,262.15375108073308,377.0980224609375]},{"page":110,"text":"(5.33)","rect":[361.0557861328125,392.3876953125,385.10515724031105,383.79180908203127]},{"page":110,"text":"and","rect":[53.81412887573242,425.66961669921877,68.21787348798284,418.76702880859377]},{"page":110,"text":"IðqÞ ¼ ð PðrÞeiqrdV","rect":[179.1127471923828,469.1082458496094,259.3647536603328,446.98602294921877]},{"page":110,"text":"(5.34)","rect":[361.05596923828127,462.27508544921877,385.1053403457798,453.67919921875]},{"page":110,"text":"It means that the scattering (or diffraction) intensity and the autocorrelation","rect":[65.76631927490235,497.645751953125,385.1263360123969,488.711181640625]},{"page":110,"text":"function are reciprocal Fourier transforms similar to the reciprocal transforms of","rect":[53.814292907714847,509.60528564453127,385.1501706680454,500.67071533203127]},{"page":110,"text":"scattering amplitude F(q) and density r(r). It should be noted that in statistical","rect":[53.814292907714847,521.508056640625,385.0874467618186,512.5735473632813]},{"page":110,"text":"physics one widely uses the density correlation function G(r) mentioned earlier","rect":[53.814292907714847,533.4675903320313,385.0993894180454,524.4932250976563]},{"page":110,"text":"(5.26) that is related to the structure factor S(q) exactly as the Paterson function is","rect":[53.814292907714847,545.4271240234375,385.1889687930222,536.432861328125]},{"page":110,"text":"related to intensity of scattering I(q). Below we prefer to use G(r).","rect":[53.814308166503909,557.3866577148438,322.7890286019016,548.4222412109375]},{"page":110,"text":"Resuming this section, remember that there are two approaches to calculate the","rect":[65.76531219482422,569.34619140625,385.1701129741844,560.4116821289063]},{"page":110,"text":"scattering intensity I(q):","rect":[53.813289642333987,581.3057250976563,151.09379441806864,572.2018432617188]},{"page":111,"text":"94","rect":[53.812843322753909,42.62367248535156,62.2739118972294,36.73106384277344]},{"page":111,"text":"1.","rect":[53.812843322753909,67.0,61.27850003744845,59.41356658935547]},{"page":111,"text":"2.","rect":[53.81312942504883,103.0,61.278786139743377,95.23552703857422]},{"page":111,"text":"5 Structure Analysis and X-Ray Diffraction","rect":[235.81301879882813,44.274620056152347,385.1466421523563,36.62946701049805]},{"page":111,"text":"To make a Fourier transform of density r(r), in order to find the scattering field","rect":[66.27452087402344,68.2883529663086,385.11782160810005,59.35380554199219]},{"page":111,"text":"amplitude FðqÞ ¼ Ð rðrÞexpiðqrÞdV and then to make a product I(q) ¼","rect":[66.2745132446289,81.17020416259766,385.14698055479439,70.11407470703125]},{"page":111,"text":"F(q) F*(q).","rect":[66.2748031616211,92.13787841796875,113.08945889975314,83.33281707763672]},{"page":111,"text":"To make a Fourier transform directly of density correlation function G(r), and","rect":[66.2748031616211,104.11031341552735,385.1449822526313,95.14588165283203]},{"page":111,"text":"obtain intensity structure factor S(q) that, according to Eq. 5.25 is normalized","rect":[66.2748031616211,116.0698471069336,385.1469658952094,107.0755386352539]},{"page":111,"text":"intensity I(q):.","rect":[66.27379608154297,128.02944946289063,124.29489560629611,118.92556762695313]},{"page":111,"text":"IðqÞ","rect":[248.50225830078126,172.1829833984375,265.30694975005346,162.2324676513672]},{"page":111,"text":"SðqÞ ¼ =½GðrÞ\u0004 ¼ NF2","rect":[167.21446228027345,183.57577514648438,259.97322151741346,169.0337371826172]},{"page":111,"text":"form","rect":[256.4892578125,187.85223388671876,269.61683296424016,181.58412170410157]},{"page":111,"text":"(5.35)","rect":[361.0561828613281,178.24757385253907,385.10555396882668,169.65167236328126]},{"page":111,"text":"Further on we shall follow the first approach for discussion of crystals and the","rect":[65.76653289794922,213.33432006835938,385.17334783746568,204.39976501464845]},{"page":111,"text":"second one for discussion of liquids and liquid crystals.","rect":[53.81450653076172,225.29388427734376,277.8390163948703,216.3593292236328]},{"page":111,"text":"5.5 X-Ray Diffraction by Crystals","rect":[53.812843322753909,276.0231018066406,235.32601051066085,264.66815185546877]},{"page":111,"text":"We begin this section with an example of the X-ray diffraction on the nematic,","rect":[53.812843322753909,303.0036315917969,385.0959133675266,294.069091796875]},{"page":111,"text":"smectic A and crystalline smectic B phases. In Fig. 5.16 there is a series of X-ray","rect":[53.812843322753909,314.9631652832031,385.1068963151313,305.9688415527344]},{"page":111,"text":"photos of the same mesogenic compound at different temperatures. In this experi-","rect":[53.812843322753909,326.9227294921875,385.17656837312355,317.9881591796875]},{"page":111,"text":"ment, the material flow induced by the electric current aligns molecular axes in the","rect":[53.812843322753909,338.88226318359377,385.17066229059068,329.94769287109377]},{"page":111,"text":"nematic phase parallel to the field direction, which is horizontal, but in the SmA","rect":[53.812843322753909,350.8418273925781,385.1497168536457,341.90728759765627]},{"page":111,"text":"phase parallel to the smectic layers. Correspondingly diffraction patterns of the","rect":[53.812843322753909,362.74456787109377,385.17160833551255,353.80999755859377]},{"page":111,"text":"nematic and smectic phase considerably differ from each other. In the crystalline","rect":[53.812843322753909,374.7041320800781,385.17063177301255,365.76959228515627]},{"page":111,"text":"SmBcr phase the picture shows the six-fold rotation axis perpendicular to the figure","rect":[53.812843322753909,386.6645202636719,385.11597479059068,377.72998046875]},{"page":111,"text":"plane. Below we shall discuss such pictures in detail, but let us begin with solid","rect":[53.812931060791019,398.62408447265627,385.1358269791938,389.68951416015627]},{"page":111,"text":"crystals.","rect":[53.812931060791019,410.5836181640625,86.75140805746799,401.6490478515625]},{"page":111,"text":"Fig. 5.16","rect":[53.812843322753909,583.830322265625,86.29320282130166,576.1005249023438]},{"page":111,"text":"smectic A","rect":[53.812835693359378,592.0,88.26715637785404,586.0874633789063]},{"page":111,"text":"X-ray diffractograms of p-anisalamino-cinnamic","rect":[92.27517700195313,583.7625732421875,262.4484190680933,576.168212890625]},{"page":111,"text":"and crystalline smectic Bcr","rect":[90.63202667236328,593.6818237304688,182.02237347214638,586.0874633789063]},{"page":111,"text":"acid","rect":[266.2889099121094,582.0354614257813,280.4611868300907,576.168212890625]},{"page":111,"text":"in","rect":[284.2458190917969,582.0,290.8539175429813,576.168212890625]},{"page":111,"text":"different","rect":[294.6114807128906,582.0354614257813,323.84447960105009,576.168212890625]},{"page":111,"text":"phases,","rect":[327.6358947753906,583.7625732421875,352.2914759834047,576.168212890625]},{"page":111,"text":"nematic,","rect":[356.0718688964844,582.0100708007813,385.12719986035787,576.168212890625]},{"page":112,"text":"5.5 X-Ray Diffraction by Crystals","rect":[53.812835693359378,44.276390075683597,170.3674667644433,36.6312370300293]},{"page":112,"text":"5.5.1 Density Function and Structure Factor for Crystals","rect":[53.812843322753909,69.93675231933594,345.4162204715983,59.298980712890628]},{"page":112,"text":"95","rect":[376.7464904785156,42.62544250488281,385.2075552382938,36.6312370300293]},{"page":112,"text":"5.5.1.1 Density Function","rect":[53.812843322753909,97.87738800048828,164.31354111979557,88.46473693847656]},{"page":112,"text":"In crystals this function has three-dimensional periodicity. For simplicity, here we","rect":[53.812843322753909,121.3980941772461,385.16663397027818,112.46354675292969]},{"page":112,"text":"only consider the one-dimensional projection of the three-dimensional crystal. In","rect":[53.812843322753909,133.35763549804688,385.14873591474068,124.42308044433594]},{"page":112,"text":"this case, the density function with period a is very simple","rect":[53.812843322753909,145.31716918945313,289.7704242534813,136.3826141357422]},{"page":112,"text":"rðxÞ ¼ r0 þ Xrm cosmq0x;","rect":[130.51004028320313,173.5486297607422,249.88758790177247,159.61392211914063]},{"page":112,"text":"m","rect":[187.60877990722657,179.5773468017578,192.6117517874823,176.34219360351563]},{"page":112,"text":"q0 ¼ 2p=a;","rect":[261.47430419921877,171.49278259277345,306.8168176381006,161.5621795654297]},{"page":112,"text":"As shown in Fig. 5.17a it consists of density maxima with a constant amplitude. The","rect":[65.76764678955078,204.09542846679688,385.27588689996568,195.10110473632813]},{"page":112,"text":"width of the peaks is governed by the thermal fluctuations of atoms, D \u0001 (kT/b)1/2","rect":[53.816612243652347,216.05496215820313,385.18145399419168,204.96450805664063]},{"page":112,"text":"(b is a compressibility modulus). At room temperature, such fluctuations may be of the","rect":[53.812843322753909,228.01495361328126,385.26917303277818,218.77162170410157]},{"page":112,"text":"order of 10% of the interatomic distances. At zero temperature the maxima would have","rect":[53.8138427734375,239.9744873046875,385.26624334527818,231.03993225097657]},{"page":112,"text":"the size of atoms or molecules comprising a crystal.","rect":[53.8138427734375,251.93405151367188,256.59507413412816,242.99949645996095]},{"page":112,"text":"5.5.1.2 The Structure Factor","rect":[53.8138427734375,292.02093505859377,182.74973333551254,284.7398376464844]},{"page":112,"text":"According to (5.27) the amplitude of scattering F(q) for our one-dimensional","rect":[53.8138427734375,317.6263732910156,385.182509017678,308.6320495605469]},{"page":112,"text":"crystal is given by Fourier transform of density function r(x). Since we have only","rect":[53.814815521240237,329.5859069824219,385.12978449872505,320.6513671875]},{"page":112,"text":"the sum of cosine functions there are only discrete harmonics at wavevectors","rect":[53.81483459472656,341.54547119140627,385.15280546294408,332.61090087890627]},{"page":112,"text":"q ¼ mq0 ¼ 2pm/a. The structure factor (5.25) is proportional to scattered light","rect":[53.81483459472656,353.5062561035156,385.121718002053,344.5119323730469]},{"page":112,"text":"intensity F(q)F*(q) and also consists of harmonics represented by d-functions","rect":[53.813785552978519,365.4657897949219,385.10187162505346,356.232421875]},{"page":112,"text":"situated at the same wavevector values q ¼ 2pm/a and having amplitude r2m:","rect":[53.81576919555664,377.42535400390627,368.61152513095927,366.3745422363281]},{"page":112,"text":"SðqÞ ¼ Xr2mdðq \u0005 mq0Þ","rect":[167.6141357421875,405.65655517578127,271.36749662505346,391.72186279296877]},{"page":112,"text":"m","rect":[203.12950134277345,411.68536376953127,208.13247322302918,408.4502258300781]},{"page":112,"text":"(5.36)","rect":[361.0559387207031,402.8736572265625,385.10530982820168,394.27777099609377]},{"page":112,"text":"Fig. 5.17 Three-dimensional crystal considered along one direction: density function with equi-","rect":[53.812843322753909,554.1295776367188,385.17880338294199,546.3997802734375]},{"page":112,"text":"distant maxima blurred by thermal fluctuations (a) and the angular spectrum of the structure factor","rect":[53.812843322753909,564.0377807617188,385.16952603919199,556.4434204101563]},{"page":112,"text":"(b). The height of the d-type maxima is given by squared amplitudes of the density harmonics and","rect":[53.812828063964847,573.9569702148438,385.19574493555947,566.108642578125]},{"page":112,"text":"is additionally modulated by both the molecular form-factor (MMF) and Debye-Waller factor","rect":[53.812835693359378,583.9329223632813,385.1694344864576,576.3385620117188]},{"page":112,"text":"(DWF)","rect":[53.812835693359378,593.5702514648438,78.20610135657479,586.3653564453125]},{"page":113,"text":"96","rect":[53.812774658203128,42.62471008300781,62.273843232678618,36.68130111694336]},{"page":113,"text":"5 Structure Analysis and X-Ray Diffraction","rect":[235.8129425048828,44.275657653808597,385.14658111720009,36.6305046081543]},{"page":113,"text":"The correspondent X-ray picture consists of a set of narrow discrete equidistant","rect":[65.76496887207031,68.2883529663086,385.132948470803,59.35380554199219]},{"page":113,"text":"spots at q ¼ mq0 along the direction of periodicity. The angular spectrum of the","rect":[53.812950134277347,80.24800872802735,385.1717914409813,71.31346130371094]},{"page":113,"text":"structure factor is shown schematically in Fig. 5.17b. The amplitudes of harmonics","rect":[53.81302261352539,92.20760345458985,385.1010781680222,83.21329498291016]},{"page":113,"text":"depend mostly on the shape of the density curve and determine a number and","rect":[53.81400680541992,104.11031341552735,385.14492121747505,95.17576599121094]},{"page":113,"text":"the height of the d-type maxima. The peak amplitudes are weakly modulated","rect":[53.81400680541992,116.0698471069336,385.1130303483344,106.83648681640625]},{"page":113,"text":"by the molecular form factor (MFF) and additionally by thermal fluctuations","rect":[53.81399917602539,128.02944946289063,385.09811796294408,119.09489440917969]},{"page":113,"text":"through","rect":[53.81399917602539,139.98898315429688,84.81140986493597,131.05442810058595]},{"page":113,"text":"the","rect":[91.53945922851563,137.92715454101563,103.7632297344383,131.05442810058595]},{"page":113,"text":"factor","rect":[110.51516723632813,137.92715454101563,133.81798683015479,131.05442810058595]},{"page":113,"text":"of","rect":[140.59381103515626,137.92715454101563,148.88567481843604,131.05442810058595]},{"page":113,"text":"Debye-Waller","rect":[155.60476684570313,139.98898315429688,212.27108131257666,131.05442810058595]},{"page":113,"text":"(DWF):","rect":[218.99017333984376,139.59056091308595,250.48530442783426,131.11419677734376]},{"page":113,"text":"for","rect":[257.28204345703127,137.92715454101563,268.8886655410923,131.05442810058595]},{"page":113,"text":"the","rect":[275.5779113769531,137.92715454101563,287.8016742534813,131.05442810058595]},{"page":113,"text":"one-dimensional","rect":[294.5536193847656,138.0,361.174208236428,131.05442810058595]},{"page":113,"text":"case","rect":[367.9082336425781,137.92715454101563,385.0891803569969,133.2855682373047]},{"page":113,"text":"I / expð\u0005<u2>q2=3Þ [1]. Here < u2 > is the mean square amplitude of the ther-","rect":[53.81399917602539,153.08154296875,385.1195005020298,141.6918182373047]},{"page":113,"text":"mal oscillations of atoms proportional to temperature. Due to the exponential","rect":[53.81351852416992,164.70245361328126,385.10261399814677,155.7678985595703]},{"page":113,"text":"factor, higher harmonics are much more sensitive to temperature and strongly","rect":[53.81351852416992,176.6619873046875,385.1762627702094,167.72743225097657]},{"page":113,"text":"decrease Bragg diffraction intensity (but not the peak sharpness) with increasing","rect":[53.81351852416992,188.62155151367188,385.07973567060005,179.68699645996095]},{"page":113,"text":"temperature.","rect":[53.81351852416992,200.52432250976563,104.1589321663547,192.60574340820313]},{"page":113,"text":"5.5.2 A Crystal of a Finite Size","rect":[53.812843322753909,250.6918487548828,215.427961191847,240.0540771484375]},{"page":113,"text":"This case is important for thin crystalline films. At first, let us look at the simplest","rect":[53.812843322753909,278.2341613769531,385.1606584317405,269.29962158203127]},{"page":113,"text":"infinite one-dimensional model of the crystal structure, Fig. 5.18, having only zero","rect":[53.812843322753909,290.1936950683594,385.15566340497505,281.1993713378906]},{"page":113,"text":"and first harmonic of density,","rect":[53.812862396240237,302.15325927734377,172.37945218588596,293.21868896484377]},{"page":113,"text":"rðxÞ ¼ r0 þ r1 cosð2px=aÞ:","rect":[163.36392211914063,326.411376953125,275.60633790177249,316.4604797363281]},{"page":113,"text":"The direct Fourier transform of this function is two delta functions with ampli-","rect":[65.76860809326172,349.9918212890625,385.1246579727329,341.0572509765625]},{"page":113,"text":"tude r0 and r1 located at q ¼ 0 and q ¼ 2p/a. It is shown in the Inset to Fig. 5.18.","rect":[53.81658172607422,361.8949890136719,385.1838039925266,352.9006652832031]},{"page":113,"text":"Disregarding the zero Fourier harmonic the corresponding intensity of scattering","rect":[53.81511306762695,373.85455322265627,385.16497126630318,364.91998291015627]},{"page":113,"text":"for the infinite one-dimensional crystal is:","rect":[53.81511306762695,385.8140869140625,222.8107286221702,376.8795166015625]},{"page":113,"text":"IðqÞ ¼ FðqÞF\u0006ðqÞ ¼ r21dðq \u0005 2p=aÞ:","rect":[146.0896453857422,411.61676025390627,292.8823999134912,400.0447998046875]},{"page":113,"text":"A finite one-dimensional crystal is an analogue of a wave packet confined","rect":[65.76712799072266,435.1835021972656,385.1052178483344,426.24896240234377]},{"page":113,"text":"between \u0005 A/2 and A/2 points, shown in Fig. 5.18. Its scattering amplitude","rect":[53.815101623535159,447.14306640625,357.06018865777818,438.14874267578127]},{"page":113,"text":"A=2","rect":[172.6544952392578,469.40289306640627,183.95507881721816,462.4515075683594]},{"page":113,"text":"FðqÞ ¼ ð r1 \u0007 cosð2px=aÞ \u0007 expðiqxÞdx","rect":[136.91485595703126,492.57379150390627,302.2855914898094,470.4515686035156]},{"page":113,"text":"\u0005A=2","rect":[169.48199462890626,501.14434814453127,186.10746834358535,494.1929626464844]},{"page":113,"text":"(5.37)","rect":[361.0592346191406,485.7409973144531,385.10860572663918,477.1451110839844]},{"page":113,"text":"Fig. 5.18 Sine (or cosine)","rect":[53.812843322753909,542.5667114257813,145.0756539688795,534.6167602539063]},{"page":113,"text":"form density function for an","rect":[53.812843322753909,552.4749145507813,150.47549957423136,544.8805541992188]},{"page":113,"text":"infinite sample showing the","rect":[53.812843322753909,562.3941650390625,148.0953917243433,554.7998046875]},{"page":113,"text":"density wave components","rect":[53.812843322753909,572.3701171875,141.9010361003808,564.7757568359375]},{"page":113,"text":"r0 and r1 (main plot) and its","rect":[53.812843322753909,582.2614135742188,152.00433810233393,574.7343139648438]},{"page":113,"text":"angular spectrum (inset)","rect":[53.81277084350586,592.321533203125,136.83953946692638,584.7271728515625]},{"page":113,"text":"2ρ1","rect":[265.32598876953127,559.7892456054688,275.83191927276467,553.3646240234375]},{"page":113,"text":"ρ","rect":[279.03314208984377,522.7861938476563,283.41990771179067,517.3812255859375]},{"page":113,"text":"2","rect":[278.9780578613281,534.3547973632813,283.42075674257128,528.7339477539063]},{"page":113,"text":"1","rect":[280.1550598144531,558.0606079101563,284.59775869569628,552.4397583007813]},{"page":113,"text":"0","rect":[282.95892333984377,588.8217163085938,287.4016222210869,583.0569458007813]},{"page":113,"text":"–A/2","rect":[295.1116638183594,577.8798217773438,311.54804556093066,571.8432006835938]},{"page":113,"text":"0","rect":[322.7730712890625,577.8798217773438,327.21577017030566,572.1150512695313]},{"page":114,"text":"5.6 Structure of the Isotropic and Nematic Phase","rect":[53.812843322753909,44.274620056152347,220.47225329660894,36.62946701049805]},{"page":114,"text":"97","rect":[376.74566650390627,42.62367248535156,385.20673126368447,36.73106384277344]},{"page":114,"text":"Within these limits, direct integrating is difficult. However, the scattering ampli-","rect":[65.76496887207031,68.2883529663086,385.1240171035923,59.35380554199219]},{"page":114,"text":"tude may be found using the convolution theorem (Eq. 5.31a). The integral may be","rect":[53.812950134277347,80.24788665771485,385.1527789898094,71.25357818603516]},{"page":114,"text":"presented as a convolution f1(x)*f2 (x) where f1¼r1 (like in case of parallelepiped)","rect":[53.81294250488281,92.20772552490235,385.0897458633579,83.23309326171875]},{"page":114,"text":"and f2 ¼ cos(2px/a). Applying the convolution theorem we obtain the scattering","rect":[53.81362533569336,104.11067962646485,385.16326228192818,95.1361083984375]},{"page":114,"text":"amplitude from the two amplitudes found earlier, see Eqs. 5.29 and 5.36 for m ¼ 1:","rect":[53.81446075439453,116.0702133178711,385.2089982754905,107.0759048461914]},{"page":114,"text":"FðqÞ ¼ =½f1ðxÞ\u0006f2ðxÞ\u0004 ¼ F1ðqÞF2ðqÞ ¼ r1AsiðnAðAq=q2=Þ2Þ \u0007 dðq \u0005 q0Þ","rect":[88.4809799194336,153.93121337890626,350.50067533599096,130.3777313232422]},{"page":114,"text":"We have again found the scattering field amplitude in the form of sine integral.","rect":[65.76644134521485,177.51242065429688,385.1772121956516,168.57786560058595]},{"page":114,"text":"The correspondent intensity spectrum is similar to that for the parallelepiped, see","rect":[53.814414978027347,189.47198486328126,385.1124957866844,180.5374298095703]},{"page":114,"text":"Fig. 5.11b,","rect":[53.814414978027347,201.4315185546875,97.31430478598361,192.43719482421876]},{"page":114,"text":"IðqÞ ¼ FðqÞF\u0006ðqÞ ¼ A2r12 si½nA2ð½Aqð\u0005q \u0005qoÞq=02Þ=\u000422\u0004","rect":[130.3991241455078,241.95709228515626,306.92979371231936,215.66207885742188]},{"page":114,"text":"(5.38)","rect":[361.0561828613281,232.94456481933595,385.10555396882668,224.34866333007813]},{"page":114,"text":"However, there is a shift of the entire parallelepiped diffraction spectrum by q0","rect":[65.76653289794922,265.5372619628906,385.18130140630105,256.60272216796877]},{"page":114,"text":"on the wavevector scale; the curve for a parallelepiped without density modulation","rect":[53.812843322753909,277.497314453125,385.1476983170844,268.562744140625]},{"page":114,"text":"is centered at q ¼ 0 whereas the curve for the modulated structure is centered at","rect":[53.812843322753909,289.3472900390625,385.18061692783427,280.52227783203127]},{"page":114,"text":"q ¼ q0. Such a shifted angular spectrum of diffraction intensity is very similar to","rect":[53.813846588134769,301.4166259765625,385.13811579755318,292.4820556640625]},{"page":114,"text":"that observed on the freely suspended films of smectic A liquid crystals. It allows","rect":[53.81319046020508,313.37615966796877,385.1291848574753,304.44158935546877]},{"page":114,"text":"the determination of both the smectic layer period and the film thickness.","rect":[53.81319046020508,325.3357238769531,349.4543118050266,316.40118408203127]},{"page":114,"text":"5.6 Structure of the Isotropic and Nematic Phase","rect":[53.812843322753909,375.4462890625,313.235090098097,364.6531066894531]},{"page":114,"text":"5.6.1 Isotropic Liquid","rect":[53.812843322753909,405.23321533203127,169.4419683446843,394.6791076660156]},{"page":114,"text":"This is the other extreme case with respect to crystals. The density correlation","rect":[53.812843322753909,432.8594665527344,385.1248406510688,423.9249267578125]},{"page":114,"text":"function G(r) is spherically symmetric decaying function. It is very instructive to","rect":[53.812843322753909,444.8190002441406,385.14077082685005,435.85455322265627]},{"page":114,"text":"find, at first, the structure factor (5.26) for any function of the spherical symmetry.","rect":[53.81283950805664,456.778564453125,385.1686062386203,447.78424072265627]},{"page":114,"text":"We should use spherical frame with volume element dV ¼ r2sin#djd#dr:","rect":[53.812835693359378,468.73809814453127,354.09139381257668,457.6873779296875]},{"page":114,"text":"2p","rect":[175.2600860595703,488.7511901855469,182.5617805493651,483.9891052246094]},{"page":114,"text":"p","rect":[198.82452392578126,488.7511901855469,202.67081985600573,485.5997009277344]},{"page":114,"text":"1","rect":[236.0970001220703,488.7721252441406,243.06492753189725,485.6275939941406]},{"page":114,"text":"SðqÞ ¼ ð dfð sin#dy ð GðrÞeiqrr2dr","rect":[144.10633850097657,512.8088989257813,294.5843545352097,490.68634033203127]},{"page":114,"text":"0","rect":[176.67596435546876,519.167236328125,180.1599280603822,514.3703002929688]},{"page":114,"text":"0","rect":[198.4844970703125,519.167236328125,201.96846077522597,514.3703002929688]},{"page":114,"text":"0","rect":[237.34288024902345,519.167236328125,240.8268439539369,514.3703002929688]},{"page":114,"text":"Since qr ¼ qrcos#¼t we substitute sin#d# by –dt/qr and get","rect":[53.813899993896487,543.6704711914063,300.3776994473655,534.516845703125]},{"page":114,"text":"1","rect":[153.2251739501953,562.7405395507813,160.19310136002225,559.5960693359375]},{"page":114,"text":"qr","rect":[194.859375,563.5831298828125,201.02599006547869,558.9186401367188]},{"page":114,"text":"1","rect":[260.05792236328127,562.7405395507813,267.0258497731082,559.5960693359375]},{"page":114,"text":"SðqÞ ¼ 2p ð GðrÞdr ð eitqr2r dt ¼ 4p ð GðrÞr2 sinqrqrdr","rect":[109.89292907714844,587.0916748046875,328.74164213286596,563.1156005859375]},{"page":114,"text":"0","rect":[154.4145965576172,593.1356811523438,157.89856026253066,588.3387451171875]},{"page":114,"text":"\u0005qr","rect":[191.63043212890626,594.1343994140625,203.17837959184588,589.4699096679688]},{"page":114,"text":"0","rect":[261.3038024902344,593.1356811523438,264.7877661951478,588.3387451171875]},{"page":115,"text":"98","rect":[53.8120231628418,42.62513732910156,62.27309173731729,36.73252868652344]},{"page":115,"text":"5 Structure Analysis and X-Ray Diffraction","rect":[235.81219482421876,44.276084899902347,385.14581817774697,36.63093185424805]},{"page":115,"text":"Now, we","rect":[65.76496887207031,67.0,103.53124273492658,59.553016662597659]},{"page":115,"text":"unstructured","rect":[53.812950134277347,79.0,103.76218501142034,71.31333923339844]},{"page":115,"text":"may use","rect":[107.96484375,68.2883529663086,142.80558050348129,61.0]},{"page":115,"text":"liquid in","rect":[108.02159118652344,80.24788665771485,143.43868342450629,71.31333923339844]},{"page":115,"text":"the","rect":[147.2192840576172,67.0,159.4430621929344,59.35380554199219]},{"page":115,"text":"the","rect":[147.67318725585938,79.0,159.8969653911766,71.31333923339844]},{"page":115,"text":"simplest density correlation function for an isotropic","rect":[163.87266540527345,68.2883529663086,385.09705389215318,59.35380554199219]},{"page":115,"text":"form GðrÞ ¼ r2r\u00051e\u0005kr where k ¼ x\u00051 is an inverse","rect":[164.21310424804688,80.58654022216797,385.17112005426255,69.19712829589844]},{"page":115,"text":"value of the correlation length x comparable with molecular size [5]. Then, using","rect":[53.81438446044922,92.20760345458985,385.16713801435005,82.9543228149414]},{"page":115,"text":"Euler formula and integrating over r-coordinate we find the structure factor of a","rect":[53.81536102294922,104.11031341552735,385.1621478862938,95.17576599121094]},{"page":115,"text":"liquid without any short-range structure:","rect":[53.81536102294922,116.0698471069336,216.63139207431864,107.13529968261719]},{"page":115,"text":"1","rect":[130.3404541015625,134.51710510253907,137.30838151138944,131.3726043701172]},{"page":115,"text":"SðqÞ ¼ 2piqr2 ð e\u0005kr\u0002eiqr \u0005 e\u0005iqr\u0003dr ¼ 2piqr2 \u0004k \u00051 iq \u0005 k þ1iqÞ\u0005","rect":[75.90673065185547,159.40769958496095,334.4951445613481,134.83529663085938]},{"page":115,"text":"0","rect":[131.58631896972657,164.9121856689453,135.07028267464004,160.115234375]},{"page":115,"text":"4pr2","rect":[112.1007080078125,182.1937713623047,131.61528084358535,171.79119873046876]},{"page":115,"text":"¼","rect":[96.35458374023438,185.55194091796876,104.01932552549748,183.22119140625]},{"page":115,"text":"q2 þ k2","rect":[106.83275604248047,195.7674102783203,136.88333199104629,186.13137817382813]},{"page":115,"text":"(5.39a)","rect":[356.6379089355469,188.62013244628907,385.13680396882668,180.02423095703126]},{"page":115,"text":"The structure factor and intensity of scattering (5.25) have a spherically sym-","rect":[65.76592254638672,219.28646850585938,385.1457456192173,210.29214477539063]},{"page":115,"text":"metric Lorentzian form centered at the zero wavevector qc ¼ 0. The full width on","rect":[53.813899993896487,231.1364288330078,385.17107478192818,222.3112335205078]},{"page":115,"text":"the half a maximum (FWHM) is equal to 2k¼2/x.","rect":[53.81433868408203,243.1485595703125,256.2600521614719,233.895263671875]},{"page":115,"text":"In real liquids there is a short-range positional order because each particular","rect":[65.76732635498047,255.10809326171876,385.1482175430454,246.1735382080078]},{"page":115,"text":"molecule has nearest neighbors forming few so-called coordination spheres. There-","rect":[53.81529998779297,267.0676574707031,385.1223081192173,258.13311767578127]},{"page":115,"text":"fore, each selected molecule “feels” its nearest neighbors and the G-function","rect":[53.81529998779297,279.0271911621094,385.13320246747505,270.062744140625]},{"page":115,"text":"oscillates. For simplicity, we can take only the first harmonic of density oscillation","rect":[53.815284729003909,290.98675537109377,385.08644953778755,282.05218505859377]},{"page":115,"text":"and write the density correlation function as follows:","rect":[53.815284729003909,302.9462890625,267.2086625821311,294.01171875]},{"page":115,"text":"GðrÞ ffi r02 þ r12r\u00051 expð\u0005r=xÞcos2pr=a","rect":[137.64964294433595,328.8330993652344,301.3460625748969,317.1774597167969]},{"page":115,"text":"(5.39b)","rect":[356.07232666015627,327.71484375,385.1605161270298,319.11895751953127]},{"page":115,"text":"This equation shows that positional correlations described by the cosine multi-","rect":[65.76676177978516,352.373046875,385.11675391999855,343.4185791015625]},{"page":115,"text":"plier exponentially decay at a distance x, as shown in Fig. 5.19a. The scatteringfield","rect":[53.815757751464847,364.3325500488281,385.1187676530219,355.0792541503906]},{"page":115,"text":"intensity of a liquid can be found from (5.39b) and (5.35) with the help of the","rect":[53.815757751464847,376.2921142578125,385.17749822809068,367.29779052734377]},{"page":115,"text":"convolution theorem given by Eq. 5.31:","rect":[53.815757751464847,388.25164794921877,214.41799790927957,379.25732421875]},{"page":115,"text":"r","rect":[147.9004669189453,406.62860107421877,150.58311897172869,403.4841003417969]},{"page":115,"text":"r","rect":[221.3126678466797,406.62860107421877,223.99531989946306,403.4841003417969]},{"page":115,"text":"IðqÞ ¼ =½GðrÞ\u0004 ffi r02 ð expðiqrÞdr þ r21 ð r\u00051 expð\u0005r=xÞcosð2pr=aÞexpðiqrÞdr","rect":[60.95293045043945,430.79132080078127,377.68585600005346,408.6690979003906]},{"page":115,"text":"0","rect":[147.10751342773438,437.14990234375,150.59147713264785,432.3529357910156]},{"page":115,"text":"0","rect":[220.51971435546876,437.14990234375,224.0036780603822,432.3529357910156]},{"page":115,"text":"¼ r20dð0Þ þ r12½=ðr\u00051e\u0005r=xÞ\u0006=ðcos2pr=aÞ\u0004","rect":[80.04402160644531,454.32421875,251.6246789662256,442.1031494140625]},{"page":115,"text":"a","rect":[102.1312255859375,494.5863342285156,107.68648798317135,488.9975891113281]},{"page":115,"text":"G(r)","rect":[109.74101257324219,505.25067138671877,124.3926679549687,497.7002868652344]},{"page":115,"text":"exp(–r/ x)","rect":[149.46034240722657,515.9891967773438,184.49481028895307,507.7509460449219]},{"page":115,"text":"b","rect":[213.71905517578126,494.5863342285156,219.82384892525946,487.2779541015625]},{"page":115,"text":"S(q)","rect":[274.4222106933594,505.567626953125,289.961103623914,498.0172424316406]},{"page":115,"text":"Dq = 2/ x","rect":[292.9146423339844,524.3394165039063,324.6318714652184,516.1011352539063]},{"page":115,"text":"Fig. 5.19","rect":[53.812843322753909,584.0003662109375,85.38701385645791,575.9657592773438]},{"page":115,"text":"factor (b)","rect":[53.81201934814453,593.5699462890625,86.31183713538339,586.314208984375]},{"page":115,"text":"0","rect":[114.15489959716797,562.244140625,118.5991540273414,556.42138671875]},{"page":115,"text":"a","rect":[138.7301788330078,562.2035522460938,143.17443326318125,557.7324829101563]},{"page":115,"text":"2a","rect":[161.23681640625,562.2035522460938,170.12533047997813,556.4207763671875]},{"page":115,"text":"r","rect":[196.47752380371095,562.0995483398438,199.58690325215964,557.7324829101563]},{"page":115,"text":"–q","rect":[223.68177795410157,561.9481811523438,235.23204370521166,555.9334716796875]},{"page":115,"text":"– q0","rect":[247.7662811279297,562.407470703125,262.64910879138008,555.9334716796875]},{"page":115,"text":"0","rect":[277.3229675292969,562.244140625,281.7672219594703,556.42138671875]},{"page":115,"text":"q0","rect":[299.9044494628906,562.407470703125,308.1215514183332,555.932861328125]},{"page":115,"text":"q","rect":[329.7372131347656,561.9475708007813,334.6210970499382,555.932861328125]},{"page":115,"text":"Isotropic phase. Pair density correlation function (a) and the corresponding structure","rect":[91.36814880371094,583.9326171875,385.1771483161402,576.3382568359375]},{"page":116,"text":"5.6 Structure of the Isotropic and Nematic Phase","rect":[53.81200408935547,44.276145935058597,220.47139880442144,36.6309928894043]},{"page":116,"text":"99","rect":[376.7448425292969,42.62519836425781,385.20590728907509,36.73258972167969]},{"page":116,"text":"The first term for q ¼ 0 is not interesting (r0 can be found by other techniques,","rect":[65.76496887207031,68.2883529663086,385.1272854378391,59.35380554199219]},{"page":116,"text":"e.g. by dilatometry). The product term with r12 is a result of the convolution","rect":[53.81332015991211,80.24788665771485,385.1346062760688,69.19731140136719]},{"page":116,"text":"theorem and we already have the two Fourier transforms mentioned, namely, the","rect":[53.81370162963867,92.20760345458985,385.1724323101219,83.27305603027344]},{"page":116,"text":"structure factor of unstructured liquid, that is Lorentzian (5.39a) and the structure","rect":[53.81370162963867,104.11031341552735,385.0878375835594,95.11600494384766]},{"page":116,"text":"factor of a crystal that is delta-functions, Eq. 5.36:","rect":[53.81270980834961,116.0698471069336,257.07668931552959,107.0755386352539]},{"page":116,"text":"IðqÞ / SðqÞ / q24þprx21\u00052 \u0006 dðq \u0005 2p=aÞ ¼ jq \u0005 q40pjr2 21þ x\u00052","rect":[103.37771606445313,156.18894958496095,333.4986731775697,130.30081176757813]},{"page":116,"text":"(5.40)","rect":[361.0561828613281,147.1864471435547,385.10555396882668,138.59054565429688]},{"page":116,"text":"Thus, the structure factor of the liquid with a short-range periodicity is the two","rect":[65.76653289794922,179.66555786132813,385.15248957685005,170.7310028076172]},{"page":116,"text":"Lorentzians centered at q ¼ qo ¼2p/a and q ¼ –q0, Fig. 5.19b. Their positions are","rect":[53.81450653076172,191.62588500976563,385.16489446832505,182.63156127929688]},{"page":116,"text":"a measure of the molecular size a and their widths are a measure of the characteris-","rect":[53.814144134521487,201.553466796875,385.05144630281105,194.65086364746095]},{"page":116,"text":"tic distance x for the short range molecular correlations. The total intensity of","rect":[53.81411361694336,215.54498291015626,385.14800391999855,206.29168701171876]},{"page":116,"text":"scattering for positive q is shown in Fig. 5.20a. Note that the curve for the total","rect":[53.81411361694336,227.5045166015625,385.1838517911155,218.51019287109376]},{"page":116,"text":"intensity is slightly asymmetric because this function is a product of the form factor","rect":[53.814144134521487,239.46408081054688,385.0932553848423,230.52952575683595]},{"page":116,"text":"and the structure factor according to Eq. 5.25.","rect":[53.814144134521487,251.423583984375,239.20931668783909,242.42926025390626]},{"page":116,"text":"An experimental X-ray pattern for a liquid looks like that shown in Fig. 5.20b.","rect":[65.76618194580078,263.3831481933594,385.1838345101047,254.38882446289063]},{"page":116,"text":"The spot centered at q ¼ 0 is very strong and usually screened deliberately off.","rect":[53.815162658691409,275.285888671875,385.1769070198703,266.351318359375]},{"page":116,"text":"What is of importance is a diffused ring located at scattering vector q0 (or scattering","rect":[53.81517791748047,287.24639892578127,385.16594782880318,278.3109130859375]},{"page":116,"text":"angle #0) given by equation","rect":[53.81417465209961,299.2060546875,166.24506465009223,290.05224609375]},{"page":116,"text":"2p 4psin#0","rect":[159.90933227539063,322.1445007324219,218.39563057503066,313.5749816894531]},{"page":116,"text":"l","rect":[279.3172912597656,320.6070861816406,284.8120152408893,313.4753723144531]},{"page":116,"text":"q0 ¼","rect":[137.704345703125,329.3644104003906,157.09470394346625,322.7008361816406]},{"page":116,"text":"¼","rect":[173.16534423828126,326.00665283203127,180.83008602354438,323.6759033203125]},{"page":116,"text":"a","rect":[162.685546875,334.2798156738281,167.66265193036566,329.747802734375]},{"page":116,"text":"l","rect":[198.48463439941407,334.2104797363281,203.97935838053776,327.0787658691406]},{"page":116,"text":"i.e:; a ¼ 2sin#0 :","rect":[223.8618927001953,335.747802734375,301.26426636856936,320.5386962890625]},{"page":116,"text":"Therefore, the average molecular size a can be found from the angle #0.","rect":[65.76558685302735,359.2873229980469,357.3704495003391,350.1336364746094]},{"page":116,"text":"5.6.2 Nematic Phase","rect":[53.812843322753909,404.406494140625,164.2016886088392,396.0397033691406]},{"page":116,"text":"The density correlation function for nematics has the same liquid-like form, but","rect":[53.812843322753909,434.2197570800781,385.1318193204124,425.28521728515627]},{"page":116,"text":"anisotropic, namely, cylindrically symmetric. Along the two principal directions,","rect":[53.812843322753909,446.1793212890625,385.1646999886203,437.2447509765625]},{"page":116,"text":"Fig. 5.20 Isotropicphase.Angular dependence ofstructurefactor(SF), molecular form factor(FF)","rect":[53.812843322753909,581.7898559570313,385.1982735977857,574.06005859375]},{"page":116,"text":"and total intensity of scattering (a) and a typical pattern of scattering observed in experiment (b)","rect":[53.812843322753909,591.6980590820313,378.65695279700449,584.1036987304688]},{"page":117,"text":"100","rect":[53.812843322753909,42.55740737915039,66.50444931178018,36.73252868652344]},{"page":117,"text":"5 Structure Analysis and X-Ray Diffraction","rect":[235.8138427734375,44.276084899902347,385.1474661269657,36.63093185424805]},{"page":117,"text":"parallel and perpendicular to the director n ¼ nz, the correlation lengths x|| and","rect":[53.812843322753909,68.2883529663086,385.14626399091255,59.035072326660159]},{"page":117,"text":"x⊥are different for the x and y directions:","rect":[53.814414978027347,80.22808837890625,220.52094895908426,70.9946060180664]},{"page":117,"text":"G?ðr?Þ / r?\u00051 expð\u0005r?=x?Þ cos q?r?","rect":[142.12258911132813,111.03826904296875,296.3514796426761,99.46633911132813]},{"page":117,"text":"in the z-direction:","rect":[53.812843322753909,137.57093811035157,125.00834519931863,130.6583709716797]},{"page":117,"text":"GjjðzÞ / z\u00051 expð\u0005z=xjjÞ cos qjjz","rect":[153.50823974609376,171.2618865966797,285.5287209902878,158.75454711914063]},{"page":117,"text":"In the simplest approximation, functions G⊥(r⊥) and G|| (z) determine the","rect":[65.76563262939453,199.7310791015625,385.11887396051255,190.7666473388672]},{"page":117,"text":"structure factor. The form-factor contributes to diffraction pattern negligibly for","rect":[53.81487274169922,211.69082641601563,385.1816342910923,202.7562713623047]},{"page":117,"text":"small rod-like molecules comprising typical thermotropic nematics but may be","rect":[53.81487274169922,223.65036010742188,385.1507953472313,214.71580505371095]},{"page":117,"text":"important for biological materials.","rect":[53.81487274169922,235.55313110351563,192.13857694174534,226.6185760498047]},{"page":117,"text":"An X-ray pattern for a typical nematic with rod-like molecules taken with the","rect":[65.76689910888672,247.51266479492188,385.17368353082505,238.57810974121095]},{"page":117,"text":"help of a photographic film is presented in Fig. 5.21a [6]. The molecules are","rect":[53.81487274169922,259.47222900390627,385.1099323101219,250.4779052734375]},{"page":117,"text":"oriented vertically. Along the vertical, two spots are seen at small angles; they","rect":[53.815879821777347,271.4317626953125,385.1090020280219,262.4971923828125]},{"page":117,"text":"correspond to small wavevector q|| ¼ 2p/a||. From this angle of diffraction one can","rect":[53.815879821777347,283.39215087890627,385.12529841474068,274.45672607421877]},{"page":117,"text":"find a|| (length of a rod-like molecule). Along the equator (horizontal line) the spots","rect":[53.813350677490237,295.3517761230469,385.1510049258347,286.4171142578125]},{"page":117,"text":"are separated by larger distance, q⊥¼2p/a⊥> q||. The q⊥ position gives us diameter","rect":[53.813114166259769,307.31146240234377,385.0750669082798,298.37677001953127]},{"page":117,"text":"of a molecule. Usually a||/a⊥ \u0001 4–5. Thus two molecular dimensions and two","rect":[53.813899993896487,319.27099609375,385.1525811295844,310.2767639160156]},{"page":117,"text":"correlation lengths can be found [7].","rect":[53.81373977661133,331.1738586425781,200.83346219565159,322.23931884765627]},{"page":117,"text":"The equatorial spots are extended in the vertical direction and have the form","rect":[65.76676177978516,343.1333923339844,385.10904644609055,334.1988525390625]},{"page":117,"text":"of arcs: the intensity decreases with increasing #-angle as shown in Fig. 5.21b.","rect":[53.81474304199219,355.09295654296877,385.1834377815891,345.93927001953127]},{"page":117,"text":"This is a result of a non-ideal orientational order: the higher the order parameter","rect":[53.81475830078125,367.052490234375,385.10479102937355,358.117919921875]},{"page":117,"text":"S, the shorter the arcs. From the diffractogram one can find the distribution of","rect":[53.81475830078125,379.0120544433594,385.0908444961704,370.0775146484375]},{"page":117,"text":"intensity and calculate S [7]. In some cases, even the orientational distribution","rect":[53.81475830078125,390.9715881347656,385.1705254655219,382.03704833984377]},{"page":117,"text":"function for molecules f(#) can be calculated from experimental data as sche-","rect":[53.814781188964847,402.9311218261719,385.11882911531105,393.7774353027344]},{"page":117,"text":"matically shown in Fig. 5.22. Generally, the shape of the function is determined","rect":[53.813785552978519,414.8906555175781,385.1806267838813,405.8963317871094]},{"page":117,"text":"by different Legendre polynomials P2(cos#), P4(cos#), etc. (see Section 3.3)","rect":[53.813777923583987,426.7934265136719,385.1583493789829,417.6408996582031]},{"page":117,"text":"and, in principle, different order parameters P2, P4 etc. can be found from","rect":[53.8125114440918,438.7541198730469,385.14432476640305,429.819580078125]},{"page":117,"text":"experiment.","rect":[53.81345748901367,450.7137756347656,103.36849637533908,441.77923583984377]},{"page":117,"text":"Fig. 5.21 Nematic phase. Typical photo [6] of a diffraction pattern for a nematic liquid crystal","rect":[53.812843322753909,584.0003662109375,385.15429405417509,576.2705688476563]},{"page":117,"text":"with the director aligned vertically (a) and the scheme explaining this pattern (b)","rect":[53.812843322753909,593.9085693359375,331.41546720130136,586.314208984375]},{"page":118,"text":"5.7 Diffraction by Smectic Phases","rect":[53.812843322753909,44.274986267089847,170.4275253581933,36.62983322143555]},{"page":118,"text":"Fig. 5.22 Qualitative picture","rect":[53.812843322753909,67.58130645751953,154.6747374274683,59.85148620605469]},{"page":118,"text":"of the orientational","rect":[53.812843322753909,75.73698425292969,118.99100975241724,69.89517211914063]},{"page":118,"text":"distribution function","rect":[53.812843322753909,85.68157196044922,123.61074585108682,79.81436157226563]},{"page":118,"text":"calculated from the angular","rect":[53.812843322753909,97.3846664428711,147.80433744055919,89.79031372070313]},{"page":118,"text":"dependence of the diffracted","rect":[53.812843322753909,107.36067962646485,151.5162100234501,99.76632690429688]},{"page":118,"text":"intensity along the arc","rect":[53.812843322753909,117.33663177490235,129.84655902170659,109.74227905273438]},{"page":118,"text":"depicted in Fig. 5.22b","rect":[53.812843322753909,127.25582122802735,128.81599945216105,119.61066436767578]},{"page":118,"text":"1","rect":[259.2818603515625,74.522705078125,263.72611478173595,68.84391784667969]},{"page":118,"text":"0.5","rect":[252.6162872314453,108.90892028808594,263.7269082387672,103.08616638183594]},{"page":118,"text":"0","rect":[261.4808349609375,153.5018768310547,265.92508939111095,147.6791229248047]},{"page":118,"text":"101","rect":[372.4981689453125,42.55630874633789,385.18979400782509,36.73143005371094]},{"page":118,"text":"5.7 Diffraction by Smectic Phases","rect":[53.812843322753909,211.69041442871095,234.22582130167647,200.3354949951172]},{"page":118,"text":"5.7.1 SmecticA","rect":[53.812843322753909,238.70436096191407,139.71317388101796,230.43319702148438]},{"page":118,"text":"Smectic A is a one-dimensional crystal and, at the same time, a two-dimensional","rect":[53.812843322753909,268.5418395996094,385.17158372470927,259.6072998046875]},{"page":118,"text":"liquid. What kind of a diffraction pattern should it have? A naive expectation for a","rect":[53.812843322753909,280.5013732910156,385.15769231988755,271.56683349609377]},{"page":118,"text":"thick (or infinite) sample of the smectic A phase is as follows. If we have a one","rect":[53.812843322753909,292.4609069824219,385.14377630426255,283.5263671875]},{"page":118,"text":"dimensional density wave in the z-direction r ¼ r1 cosð2pz=aÞ þ ::: and neglect","rect":[53.812843322753909,304.7593994140625,385.1589799649436,294.8088684082031]},{"page":118,"text":"higher order terms, the intensity along the z-axis ought to be a single Bragg peak in","rect":[53.815162658691409,316.3802795410156,385.14309016278755,307.44573974609377]},{"page":118,"text":"the form of the delta function located at q ¼ 2p/a as shown in the Inset to Fig. 5.18.","rect":[53.815162658691409,328.33984375,385.18487210776098,319.34552001953127]},{"page":118,"text":"Note that an additional peak related to the r0 term is always situated at q ¼ 0. For","rect":[53.816200256347659,340.2995910644531,385.1522153457798,331.36480712890627]},{"page":118,"text":"the directions x and y perpendicular to the director there should be no difference","rect":[53.81438446044922,352.2591552734375,385.0656207866844,343.3245849609375]},{"page":118,"text":"between the density correlation functions for smectic A and nematic phases.","rect":[53.81438446044922,364.16192626953127,385.1611599495578,355.22735595703127]},{"page":118,"text":"Indeed, the naive expectation is correct for the x and y directions; we do have in","rect":[53.81438446044922,376.1214599609375,385.1442803483344,367.1868896484375]},{"page":118,"text":"smectic A liquid like correlations G(x,y)/exp(\u0005r⊥/x⊥) and the Lorentzian struc-","rect":[53.81439971923828,388.08099365234377,385.09957252351418,378.82818603515627]},{"page":118,"text":"ture factor, as in Fig. 5.19.","rect":[53.8134880065918,400.0410461425781,161.0382504036594,391.0467224121094]},{"page":118,"text":"However, in experiment [8], instead of the delta-function form of the intensity","rect":[65.76549530029297,412.0005798339844,385.1772088151313,403.0660400390625]},{"page":118,"text":"peaks","rect":[53.814476013183597,423.9601135253906,76.50012601958469,415.02557373046877]},{"page":118,"text":"along","rect":[82.59010314941406,423.9601135253906,104.76809016278753,415.02557373046877]},{"page":118,"text":"the","rect":[110.8560791015625,421.8982849121094,123.0798572368797,415.02557373046877]},{"page":118,"text":"z-direction","rect":[129.09518432617188,421.92816162109377,171.89032832196723,415.02557373046877]},{"page":118,"text":"dðqjj \u0005 q0Þ,","rect":[177.92356872558595,425.3619079589844,223.46175809408909,414.3482360839844]},{"page":118,"text":"quasi-Bragg","rect":[229.47010803222657,424.0003967285156,279.3018731217719,415.2451477050781]},{"page":118,"text":"singularities","rect":[285.3779296875,424.0003967285156,335.3152200137253,415.006103515625]},{"page":118,"text":"have","rect":[341.3434753417969,421.89874267578127,360.19675481988755,415.0260009765625]},{"page":118,"text":"been","rect":[366.2668151855469,421.89874267578127,385.1201104264594,415.0260009765625]},{"page":118,"text":"observed with the tails described by a power-law as shown in Fig. 5.23a,","rect":[53.814144134521487,435.9201354980469,347.7968411019016,426.9258117675781]},{"page":118,"text":"IðqÞ / ðqjj \u0005 q0Þ\u00052þ\u0002","rect":[176.56350708007813,463.16802978515627,261.7659895991085,450.1506042480469]},{"page":118,"text":"(5.41a)","rect":[356.6379089355469,461.3681945800781,385.13680396882668,452.7723083496094]},{"page":118,"text":"with small Z~0.1, depending on temperature. Such a structure factor may be","rect":[53.813899993896487,486.6498718261719,385.1527789898094,477.71533203125]},{"page":118,"text":"understood if the density correlation function is not a constant but obeys a power","rect":[53.813899993896487,498.60943603515627,385.1258786758579,489.67486572265627]},{"page":118,"text":"law of the type [8, 9]:","rect":[53.813899993896487,510.5690002441406,141.89170701572489,501.63446044921877]},{"page":118,"text":"GðzÞ / z\u0005\u0002","rect":[196.89967346191407,534.7699584960938,241.43028098094443,524.8194580078125]},{"page":118,"text":"(5.41b)","rect":[356.0715026855469,534.0330200195313,385.1596921524204,525.4371337890625]},{"page":118,"text":"Thus, instead of the true long-range order we have a quasi-long-range order with","rect":[65.76595306396485,558.3504638671875,385.11696711591255,549.4159545898438]},{"page":118,"text":"the density correlation function (5.41b) qualitatively shown in Fig. 5.23b.","rect":[53.81393051147461,570.31005859375,350.5738186409641,561.3157958984375]},{"page":118,"text":"But what is a physical sense of parameter Z? The answer is given by a theorem","rect":[65.76696014404297,582.2695922851563,385.14777325273118,573.3350830078125]},{"page":118,"text":"related to a more general question, whether true one- or two- dimensional crystals","rect":[53.81493377685547,594.2291259765625,385.1030312930222,585.2946166992188]},{"page":119,"text":"102","rect":[53.812828063964847,42.55618667602539,66.50443405299112,36.73130798339844]},{"page":119,"text":"a","rect":[108.47547149658203,68.23306274414063,114.03073389381588,62.64430618286133]},{"page":119,"text":"I(q)","rect":[117.68740844726563,78.43079376220703,130.11693400721479,70.9043960571289]},{"page":119,"text":"b","rect":[207.855712890625,68.23306274414063,213.9605066401032,60.92469024658203]},{"page":119,"text":"G(r)","rect":[217.5050048828125,78.43061065673828,232.156660264539,70.88021850585938]},{"page":119,"text":"5","rect":[235.8138427734375,42.55618667602539,240.04437774805948,36.62971115112305]},{"page":119,"text":"Structure Analysis","rect":[242.44140625,44.274864196777347,305.5068710613183,36.68050765991211]},{"page":119,"text":"r –h","rect":[268.0535888671875,92.67559814453125,280.3327573112758,86.43738555908203]},{"page":119,"text":"and","rect":[307.92333984375,43.0,320.1665396132938,36.68050765991211]},{"page":119,"text":"X-Ray","rect":[322.5381774902344,44.274864196777347,345.07846588282509,36.84983444213867]},{"page":119,"text":"Diffraction","rect":[347.4619445800781,43.0,385.1474661269657,36.68050765991211]},{"page":119,"text":"2p/a","rect":[144.73912048339845,141.72508239746095,160.23806241357188,135.50241088867188]},{"page":119,"text":"q","rect":[184.66136169433595,141.14544677734376,189.54524560950854,135.1307373046875]},{"page":119,"text":"0","rect":[220.51034545898438,142.28599548339845,224.9545998891578,136.46324157714845]},{"page":119,"text":"a","rect":[250.0510711669922,142.24600219726563,254.49532559716563,137.77496337890626]},{"page":119,"text":"2a","rect":[275.7629699707031,142.24600219726563,284.6514993032203,136.46324157714845]},{"page":119,"text":"3a","rect":[303.9048767089844,142.28599548339845,312.79337552392345,136.46324157714845]},{"page":119,"text":"r","rect":[326.58843994140627,139.49725341796876,329.69781938985497,135.13018798828126]},{"page":119,"text":"Fig. 5.23 Diffraction intensity (a) and density correlation function (b) for the smectic A phase","rect":[53.812843322753909,163.825439453125,385.1533751227808,155.89242553710938]},{"page":119,"text":"with a positional, quasi-long range, molecular order along the symmetry axis","rect":[53.812828063964847,173.73370361328126,317.54953463553707,166.13934326171876]},{"page":119,"text":"exist in Nature at all, for example, stable one-dimensional smectic A or two-","rect":[53.812843322753909,198.59759521484376,385.12383399812355,189.6630401611328]},{"page":119,"text":"dimensional discotic liquid crystals. Now we encounter a new type of order,","rect":[53.812843322753909,210.55715942382813,385.17657132651098,201.6226043701172]},{"page":119,"text":"known earlier only theoretically.","rect":[53.812843322753909,222.45989990234376,185.3477749397922,213.5253448486328]},{"page":119,"text":"5.7.2 Landau-Peierls Instability","rect":[53.812843322753909,266.6758117675781,219.34235938520639,256.03802490234377]},{"page":119,"text":"We know that, at a finite temperature, the position of atoms or molecules in a crystal","rect":[53.812843322753909,294.2181396484375,385.1218095547874,285.2835693359375]},{"page":119,"text":"(or liquid crystal) fluctuate that is density r(r) is a fluctuating value. With increas-","rect":[53.812843322753909,306.1777038574219,385.1048520645298,297.0738220214844]},{"page":119,"text":"ing size of a crystalline sample or a distance with respect to a reference point, the","rect":[53.812843322753909,318.0804443359375,385.1716693706688,309.1458740234375]},{"page":119,"text":"mean square displacement of atoms due to thermal fluctuations is growing. The","rect":[53.812843322753909,330.0400085449219,385.1447223491844,321.10546875]},{"page":119,"text":"question to be answered is whether the crystalline structure is stable for the infinite","rect":[53.812843322753909,341.9995422363281,385.18152654840318,333.06500244140627]},{"page":119,"text":"sample. Landau and Peierls [5] have found that the answer depends on dimension-","rect":[53.812843322753909,353.9591064453125,385.16265235749855,344.96478271484377]},{"page":119,"text":"ality of crystals.","rect":[53.81385040283203,365.91864013671877,118.52915616537814,356.98406982421877]},{"page":119,"text":"5.7.2.1 Displacement and Free Energy","rect":[53.81385040283203,402.1956787109375,223.74016657880316,392.78302001953127]},{"page":119,"text":"Let u(x, y, z) is a vector of displacement of a small piece of a three-dimensional","rect":[53.81385040283203,425.6595764160156,385.127821517678,416.72503662109377]},{"page":119,"text":"crystal at its position x, y, z. A characteristic linear size of the piece is L. Our task is","rect":[53.81385040283203,437.6191101074219,385.1884805117722,428.6646423339844]},{"page":119,"text":"to find an expression for the mean square value < u2(r) > of the displacement","rect":[53.81385040283203,449.5802917480469,385.175184798928,438.5292663574219]},{"page":119,"text":"[10]. We begin with the Fourier transform of u(x, y, z). Now each harmonic of","rect":[53.81444549560547,461.53985595703127,385.1512693008579,452.60528564453127]},{"page":119,"text":"displacement has its amplitude u [cm] and wavevector q [cm\u00051]:","rect":[53.81342697143555,473.4993896484375,316.3007646329124,462.44854736328127]},{"page":119,"text":"uðrÞ ¼ Xuq expðiqrÞ","rect":[172.8814697265625,499.7464294433594,266.09909452544408,485.8117370605469]},{"page":119,"text":"q","rect":[208.9639892578125,507.1002197265625,212.81028518803698,502.3590393066406]},{"page":119,"text":"(5.42)","rect":[361.0552978515625,496.9637756347656,385.10466895906105,488.3678894042969]},{"page":119,"text":"Here, the components of wavevector q acquire both positive and negative values","rect":[65.7656478881836,529.6699829101563,385.25595487700658,520.7354736328125]},{"page":119,"text":"u\u0005q ¼ u\u0006q in the range of L\u00051 < |q| < a\u00051 where a is a lattice constant. We are","rect":[53.81264877319336,543.7158813476563,385.1605609722313,530.5789794921875]},{"page":119,"text":"interested in the additional free energy term dF originated from the displacement:","rect":[53.81379318237305,555.2896728515625,374.9206987149436,546.0563354492188]},{"page":119,"text":"dF ¼ 21C3 ð \u0006@u@ðrrÞ\u00072d3r","rect":[169.08755493164063,595.33984375,269.6114541445847,569.5205078125]},{"page":119,"text":"V","rect":[212.4193572998047,600.8292236328125,216.69766472943844,596.1019897460938]},{"page":119,"text":"(5.43)","rect":[361.0558166503906,587.653076171875,385.10518775788918,579.0571899414063]},{"page":120,"text":"5.7 Diffraction by Smectic Phases","rect":[53.812843322753909,44.274620056152347,170.4275253581933,36.62946701049805]},{"page":120,"text":"103","rect":[372.4981689453125,42.55594253540039,385.18979400782509,36.73106384277344]},{"page":120,"text":"Here, for simplicity, we use a scalar displacement u and a single elasticity","rect":[65.76496887207031,68.2883529663086,385.1766289811469,59.35380554199219]},{"page":120,"text":"coefficient C3 for a three-dimensional crystal without anisotropy. C3 has an order","rect":[53.812923431396487,80.24800872802735,385.14861427156105,71.31333923339844]},{"page":120,"text":"of magnitude 1010–1011 erg/cm3 (or 109–1010 J/m3). Note that dF cannot depend on","rect":[53.81374740600586,92.20772552490235,385.1717461686469,81.15672302246094]},{"page":120,"text":"displacement explicitly since any vector u ¼ const corresponds to a shift of the","rect":[53.814022064208987,104.1104965209961,385.1737750835594,95.17594909667969]},{"page":120,"text":"whole crystal and dF ¼ 0. Linear terms ∂u/∂x, etc. do not contribute to dF because","rect":[53.814022064208987,116.0699691772461,385.10804022027818,106.12940979003906]},{"page":120,"text":"dF has minimum at u ¼ 0. Therefore, for small displacements, like in the Hooke","rect":[53.81501007080078,128.02957153320313,385.1608356304344,118.79620361328125]},{"page":120,"text":"law, only terms quadratic with respect to the first derivatives are important. Now,","rect":[53.816017150878909,139.98910522460938,385.1379360726047,131.05455017089845]},{"page":120,"text":"using (5.42) we find the Fourier expansion of derivatives","rect":[53.816017150878909,151.94863891601563,283.35116972075658,142.95431518554688]},{"page":120,"text":"@@ur ¼ XðiqÞeiqruq","rect":[181.03793334960938,187.08584594726563,259.14602432377918,166.15191650390626]},{"page":120,"text":"q","rect":[210.15347290039063,191.9549102783203,213.9997688306151,187.21372985839845]},{"page":120,"text":"and insert them into Eq. 5.43 to obtain the expansion of the free energy","rect":[53.812843322753909,216.5087890625,342.40825739911568,207.51446533203126]},{"page":120,"text":"dF ¼ 21XXC3uquq0ð\u00051Þqq0 ð eiðqþq0Þrd3r:","rect":[128.7530975341797,252.92691040039063,310.21412598770999,230.8046875]},{"page":120,"text":"q","rect":[163.47792053222657,256.23095703125,167.32421646245104,251.48976135253907]},{"page":120,"text":"0","rect":[182.3973846435547,252.97450256347657,183.74120300850343,250.41465759277345]},{"page":120,"text":"q","rect":[178.54551696777345,256.68438720703127,182.3918128979979,251.9431915283203]},{"page":120,"text":"V","rect":[256.82916259765627,259.2705078125,261.10747002729,254.5432891845703]},{"page":120,"text":"Note that uquq0 ¼ uquq\u0006 ¼ juqj2 and the integral Ð eiðqþq0Þrd3r ¼ Vdqþq0. Forq","rect":[65.7656021118164,284.7112731933594,385.18753403971746,272.1162109375]},{"page":120,"text":"¼ \u0005q0 the Kronecker symbol dii ¼1 and the integral eqVuals the crystal volume V; for","rect":[53.814842224121097,298.6958923339844,385.18111549226418,286.2846374511719]},{"page":120,"text":"q 6¼-q0 the symbol dij ¼0 and the integral vanishes. Hence,","rect":[53.8133659362793,311.572021484375,291.8126492073703,300.9959411621094]},{"page":120,"text":"dF ¼ 12VC3 Xq2juqj2 ¼ XdFq:","rect":[150.84608459472657,338.8865051269531,288.12257325333499,324.8860168457031]},{"page":120,"text":"q","rect":[202.3364715576172,346.2967224121094,206.18276748784167,341.5555419921875]},{"page":120,"text":"q","rect":[259.2082214355469,346.2967224121094,263.05451736577137,341.5555419921875]},{"page":120,"text":"From this equation and the equipartition theorem hdFqi ¼ kBT=2 we find the","rect":[65.76642608642578,371.6331481933594,385.1747516460594,361.24871826171877]},{"page":120,"text":"Fourier component of the mean square displacement (in q-space):","rect":[53.814022064208987,382.7533874511719,319.4789262540061,373.81884765625]},{"page":120,"text":"hjuqj2i ¼ VkCB3Tq2","rect":[185.9670867919922,419.4298400878906,250.85339424690566,396.9815368652344]},{"page":120,"text":"(5.44)","rect":[361.0561828613281,412.3393249511719,385.10555396882668,403.7434387207031]},{"page":120,"text":"For the mean square displacement in the r-space","rect":[65.76653289794922,442.9488830566406,262.0247272319969,434.01434326171877]},{"page":120,"text":"hu2ðrÞi ¼ Xq hjuqj2i ¼ ð2VpÞ3 ð hjuqj2id3q","rect":[134.13803100585938,482.6707763671875,304.85659113935005,457.2445068359375]},{"page":120,"text":"Here the summation is substituted by integration over the volume in the q-space","rect":[65.76641082763672,507.2246398925781,385.11338079645005,498.29010009765627]},{"page":120,"text":"and (2p)3 is a factor that relates the volumes in q- and r-spaces in three dimensions.","rect":[53.814369201660159,519.1842041015625,385.1665310433078,508.1333312988281]},{"page":120,"text":"More generally, (2p)D is a factor for any space of D dimension: in the one","rect":[53.81375503540039,531.1439208984375,385.14405096246568,520.1859741210938]},{"page":120,"text":"dimensional space (D ¼ 1), the reciprocal lattice vector length is q ¼ (2p)/a, for","rect":[53.81315994262695,543.046630859375,385.18087135163918,534.1121215820313]},{"page":120,"text":"D ¼ 2 the reciprocal lattice area is qxqy ¼ (2p)2/ab, etc.","rect":[53.81417465209961,555.9796752929688,281.57879300619848,543.9556274414063]},{"page":120,"text":"Finally, using Eq. 5.44, we can write the value of the mean square displacement","rect":[65.7655258178711,566.966064453125,385.17133958408427,557.9718017578125]},{"page":120,"text":"in the three dimensional r-space (d3q ¼ 4pq2dq is a volume of a spherical layer in","rect":[53.81351852416992,578.9259033203125,385.14275446942818,567.8749389648438]},{"page":120,"text":"the q-space):","rect":[53.81393051147461,590.8854370117188,105.5519166714866,581.950927734375]},{"page":121,"text":"104","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.73106384277344]},{"page":121,"text":"5 Structure Analysis and X-Ray Diffraction","rect":[235.8138427734375,44.274620056152347,385.1474661269657,36.62946701049805]},{"page":121,"text":"qmax","rect":[149.713134765625,65.13195037841797,161.82561917807346,60.46746826171875]},{"page":121,"text":"hu2ðrÞi ¼ ð2kpBÞT3C3 ð dq32q ¼ 2kpB2TC3 qj22pp==La ¼ pkBCT3 \u0004a1 \u0005 L1\u0005 ¼ const","rect":[71.31629180908203,90.56261444091797,343.5139299161155,64.60775756835938]},{"page":121,"text":"qmin","rect":[149.59991455078126,95.68297576904297,160.88608975912815,91.01849365234375]},{"page":121,"text":"(5.45)","rect":[361.0562438964844,81.4936294555664,385.1056150039829,72.89774322509766]},{"page":121,"text":"For a space of dimensionality D, we obtain a more general expression:","rect":[65.76659393310547,120.20796966552735,351.14692552158427,111.27342224121094]},{"page":121,"text":"hu2ðrÞi","rect":[164.3849334716797,157.1624755859375,193.1785012637253,145.88809204101563]},{"page":121,"text":"¼","rect":[195.9925537109375,156.0,203.65729549620063,149.0]},{"page":121,"text":"kBT","rect":[216.72433471679688,149.4110107421875,231.4522740299518,141.12440490722657]},{"page":121,"text":"ð2pÞDCD","rect":[206.4154510498047,165.43789672851563,241.7797205374823,153.5948028564453]},{"page":121,"text":"qmax","rect":[245.83999633789063,140.12059020996095,257.9523891976047,135.45611572265626]},{"page":121,"text":"ð dqd2q","rect":[249.2952880859375,163.5723114013672,274.60805598310005,139.5406951904297]},{"page":121,"text":"qmin","rect":[245.78334045410157,170.67164611816407,257.01287503744848,166.00717163085938]},{"page":121,"text":"(5.46)","rect":[361.0561828613281,156.42539978027345,385.10555396882668,147.82949829101563]},{"page":121,"text":"Now, elastic coefficients CD have different dimensions, particularly [erg\u0007cm\u00052]","rect":[65.76653289794922,195.1400146484375,385.15975318757668,184.08912658691407]},{"page":121,"text":"for D ¼ 2 and [erg\u0007cm\u00051] for D ¼ 1.","rect":[53.81393051147461,207.09957885742188,204.8173031380344,196.0487823486328]},{"page":121,"text":"5.7.2.2 Stability of Crystallographic Lattices of Different Dimensionality","rect":[53.81293869018555,249.328125,369.7258233170844,239.8656768798828]},{"page":121,"text":"Let us come back to the three dimensional crystal and Eq. 5.45. When crystal size L","rect":[53.81293869018555,272.8487854003906,385.1856419498737,263.8544616699219]},{"page":121,"text":"increases to infinity (i.e. approaches the so-called thermodynamic limit), then","rect":[53.81293869018555,284.808349609375,366.1313103776313,275.873779296875]},{"page":121,"text":"hu2ðrÞi ! kBT ¼ const","rect":[171.6339569091797,315.0752258300781,267.32697923252177,299.037109375]},{"page":121,"text":"pCa","rect":[215.93141174316407,319.692138671875,233.0327538590766,312.65008544921877]},{"page":121,"text":"and the mean square value of displacement remains independent of the crystal size.","rect":[53.81415939331055,343.1899108886719,385.1330837776828,334.25537109375]},{"page":121,"text":"From this equation, with kBT \u0001 4 \u0003 10\u000514 erg and a \u0001 10\u00057 cm (molecular crystal)","rect":[53.81415939331055,355.1500549316406,385.09688697663918,344.0989990234375]},{"page":121,"text":"we have small displacements of the other of u \u0001 10\u00059 cm (0.1 A˚). The crystalline","rect":[53.81379318237305,367.10968017578127,385.1742633648094,355.51068115234377]},{"page":121,"text":"order does not blurred, i.e., remains true long-range order.","rect":[53.813533782958987,379.1090393066406,291.1517605354953,370.11474609375]},{"page":121,"text":"A two-dimensional crystal (D ¼ 2) is nothing more than a single atomic or","rect":[65.76555633544922,391.0287780761719,385.15142188874855,382.0743103027344]},{"page":121,"text":"molecular monolayer. The latter may be prepared from graphene or in the form of a","rect":[53.814537048339847,402.9883117675781,385.1583942241844,394.05377197265627]},{"page":121,"text":"Langmuir film floating on water. For such a monolayer, d 2q ¼ 2pqdq is an area of","rect":[53.814537048339847,414.9478454589844,385.1523679336704,403.8972473144531]},{"page":121,"text":"a ring with circumference 2pq and width dq and Eq. 5.46 takes the form:","rect":[53.814537048339847,426.9078369140625,348.908094955178,417.91351318359377]},{"page":121,"text":"qmax","rect":[181.0945587158203,446.763916015625,193.2069515755344,442.09942626953127]},{"page":121,"text":"hu2ðrÞi ¼ 2kpBCT2 ð dqq ¼ 2kpBCT2 lnj22pp==aL ¼ 2kpBCT2 lnLa","rect":[114.36801147460938,470.2722473144531,322.9341771061237,447.8243103027344]},{"page":121,"text":"qmin","rect":[181.03793334960938,477.31488037109377,192.26754422690159,472.650390625]},{"page":121,"text":"(5.47)","rect":[361.05682373046877,463.1246032714844,385.1061948379673,454.5287170410156]},{"page":121,"text":"For L!1, this integral diverges with its area logarithmically, that is very","rect":[65.76717376708985,501.8389892578125,385.1171807389594,492.9044189453125]},{"page":121,"text":"slowly. Such a film has quasi-long-range order.","rect":[53.815147399902347,513.7985229492188,244.72100492026096,504.86395263671877]},{"page":121,"text":"A one-dimensional crystal (D ¼ 1) is a single chain of atoms or molecules","rect":[65.76717376708985,525.758056640625,385.07736600981908,516.8035888671875]},{"page":121,"text":"without any interaction with its surrounding. Eq. 5.46 reads:","rect":[53.815147399902347,537.717529296875,296.880018783303,528.7232666015625]},{"page":121,"text":"qmax","rect":[176.1664276123047,557.631591796875,188.2789272835422,552.9671020507813]},{"page":121,"text":"hu2ðrÞi ¼ 2kpBCT1 ð dqq2 ¼ 2kpBCT1 ð\u0005q\u00051Þj22pp==La ¼ 4kpB2TC1 L","rect":[109.44024658203125,581.0831909179688,329.56167954752996,558.63525390625]},{"page":121,"text":"qmin","rect":[176.10977172851563,588.1825561523438,187.3393978645969,583.51806640625]},{"page":121,"text":"(5.48)","rect":[361.0558166503906,573.936279296875,385.10518775788918,565.3403930664063]},{"page":122,"text":"5.7 Diffraction by Smectic Phases","rect":[53.812843322753909,44.274620056152347,170.4275253581933,36.62946701049805]},{"page":122,"text":"105","rect":[372.4981689453125,42.55594253540039,385.18979400782509,36.62946701049805]},{"page":122,"text":"For L!1 the mean square displacement grows linearly with the chain length","rect":[65.76496887207031,68.2883529663086,385.12197199872505,59.35380554199219]},{"page":122,"text":"and only the short-range order may exist.","rect":[53.81295394897461,80.24788665771485,220.2523159554172,71.31333923339844]},{"page":122,"text":"Recall that the smectic A phase is a three-dimensional phase which is simulta-","rect":[65.76496887207031,92.20748138427735,385.15679298249855,83.27293395996094]},{"page":122,"text":"neously one-dimensional crystal in the direction along the layer normal and two-","rect":[53.81295394897461,104.11019134521485,385.2613462051548,95.17564392089844]},{"page":122,"text":"dimensional liquid in the layer plane. So, the Eq. 5.46 cannot be applied to this","rect":[53.81295394897461,116.0697250366211,385.24435819731908,107.0754165649414]},{"page":122,"text":"strongly anisotropic system. We shall consider this problem in detail after discussion","rect":[53.81296157836914,128.02932739257813,385.2822198014594,119.09477233886719]},{"page":122,"text":"of anisotropic elastic properties of SmA in Chapter 8. Now we only mention that, in","rect":[53.81296157836914,139.98886108398438,385.2563408952094,131.05430603027345]},{"page":122,"text":"comparison to the linearly divergent order of a one-dimensional chain given by","rect":[53.813968658447269,151.94839477539063,385.2265557389594,143.0138397216797]},{"page":122,"text":"Eq. 5.48, the two- dimensional (x,y) liquid structure of smectic layers strongly","rect":[53.813968658447269,163.907958984375,385.2563408952094,154.91363525390626]},{"page":122,"text":"stabilizes the fluctuations along the smectic layer normal (z). As a result, the diver-","rect":[53.8139762878418,175.86749267578126,385.24639259187355,166.9329376220703]},{"page":122,"text":"gence of fluctuations with a distance follows the logarithmic law with increasing size","rect":[53.8149528503418,187.8270263671875,385.2513812847313,178.89247131347657]},{"page":122,"text":"ofthe SmAinthez-directionL,andthe order becomequasi-long-range.Itisthe order","rect":[53.8149528503418,199.72976684570313,385.1588071426548,190.7952117919922]},{"page":122,"text":"that results in the power law seen in Fig. 5.23b. Correspondingly, the power index Z","rect":[53.8149528503418,211.6893310546875,385.1896235626533,202.69500732421876]},{"page":122,"text":"in Eq. 5.41 can be expressed in terms of SmA interlayer distance l or q0 ¼2p/l and","rect":[53.81499099731445,223.64886474609376,385.17888728192818,214.654541015625]},{"page":122,"text":"elastic moduli for layers compressibility B and director distortion K [11].","rect":[53.81319808959961,235.61013793945313,341.3962673714328,226.6755828857422]},{"page":122,"text":"\u0002 \u0001 q02kBT=8pðKBÞ1=2","rect":[174.7508544921875,262.7433166503906,263.71174690803846,249.8984375]},{"page":122,"text":"(5.49)","rect":[361.0561828613281,261.6817626953125,385.10555396882668,253.08587646484376]},{"page":122,"text":"The estimates result in the values of Z of the order of 0.1–0.5.","rect":[65.76653289794922,286.2502746582031,317.2188381722141,277.3456115722656]},{"page":122,"text":"5.7.3 “Bond” Orientational Order in a Single Smectic Layer","rect":[53.812843322753909,336.449951171875,361.66026954386396,325.8121643066406]},{"page":122,"text":"and Hexatic Phase","rect":[89.66943359375,348.0510559082031,183.46064551801889,339.7559814453125]},{"page":122,"text":"Imagine that we have a film only one molecule thick (a smectic monolayer with or","rect":[53.812843322753909,377.9356689453125,385.14779029695168,369.0010986328125]},{"page":122,"text":"without tilt of molecules). In such a single layer, the nematic orientational order is","rect":[53.812843322753909,389.8952331542969,385.1855508242722,380.960693359375]},{"page":122,"text":"not discussed although some orientational order (or disorder) of long molecular","rect":[53.812843322753909,401.8547668457031,385.15166602937355,392.92022705078127]},{"page":122,"text":"axes may be important. We are interested in a new type of quasi-long-range order","rect":[53.812843322753909,413.8143005371094,385.1437009414829,404.8797607421875]},{"page":122,"text":"not forbidden by the Landau-Peierls theorem for the two-dimensional systems,","rect":[53.812843322753909,425.7738342285156,385.1576809456516,416.83929443359377]},{"page":122,"text":"Eq. 5.47. A monolayer can be liquid-like i.e. its translational order is absent and","rect":[53.812843322753909,437.7333984375,385.1437920670844,428.73907470703127]},{"page":122,"text":"molecules may be situated chaotically at any place in the layer plane. Liquid-like","rect":[53.81283950805664,449.63616943359377,385.1705707378563,440.70159912109377]},{"page":122,"text":"order means that not only the distances between molecules are not fixed but also no","rect":[53.81283950805664,461.5957336425781,385.16668025067818,452.66119384765627]},{"page":122,"text":"correlation exists in their angular positions. However, at a reduced temperature the","rect":[53.81283950805664,473.5552673339844,385.17160833551255,464.6207275390625]},{"page":122,"text":"full translational symmetry within the layer plane is broken only partially and the","rect":[53.81283950805664,485.5148010253906,385.17160833551255,476.58026123046877]},{"page":122,"text":"true positional order is not installed. The new order describes the positions of","rect":[53.81283950805664,497.4743347167969,385.14775977937355,488.539794921875]},{"page":122,"text":"molecular gravity centers along the connecting lines or “bonds” (not to be confused","rect":[53.81283950805664,509.47369384765627,385.1118096452094,500.4794006347656]},{"page":122,"text":"with chemical bonds). However, the distances between the molecules are not fixed.","rect":[53.812862396240237,520.9949951171875,385.1387905647922,512.4588623046875]},{"page":122,"text":"For example, in Fig. 5.24 all the molecules in the neighbour areas 1 and 2 sit in the","rect":[53.812862396240237,533.3529052734375,385.1715778179344,524.358642578125]},{"page":122,"text":"points of hexagonal lattice but, at some distance from the reference area 1, the","rect":[53.812862396240237,545.2557373046875,385.1726764507469,536.3212280273438]},{"page":122,"text":"positions of molecules in area 3 do not coincide with the lattice cross-points.","rect":[53.812862396240237,557.2152709960938,385.13775296713598,548.28076171875]},{"page":122,"text":"Nevertheless lattice vectors a and b still look along the grid lines and keep the","rect":[53.812862396240237,569.1748046875,385.17160833551255,560.2402954101563]},{"page":122,"text":"hexagonal order.","rect":[53.81187438964844,581.1343383789063,121.29843564535861,572.1998291015625]},{"page":123,"text":"106","rect":[53.81356430053711,42.55636978149414,66.50517028956338,36.68069076538086]},{"page":123,"text":"Fig. 5.24 Schematic picture","rect":[53.812843322753909,67.58130645751953,152.29548022531987,59.85148620605469]},{"page":123,"text":"of molecular ordering ina","rect":[53.812843322753909,77.4895248413086,143.5264219977808,69.89517211914063]},{"page":123,"text":"single smectic layer with","rect":[53.812843322753909,87.4087142944336,139.28320831446573,79.81436157226563]},{"page":123,"text":"liquid-like short range","rect":[53.812843322753909,97.3846664428711,129.75348040842534,89.79031372070313]},{"page":123,"text":"positional order and quasi-","rect":[53.812843322753909,107.36067962646485,145.258393226692,99.76632690429688]},{"page":123,"text":"long range hexatic order","rect":[53.812843322753909,117.33663177490235,137.3430642471998,109.74227905273438]},{"page":123,"text":"Table 5.1 In-plane order parameters","rect":[53.812843322753909,182.46246337890626,181.6257675456933,174.8003692626953]},{"page":123,"text":"Order/layer type","rect":[53.812843322753909,194.5355224609375,109.95458361887455,186.92422485351563]},{"page":123,"text":"Positional order","rect":[53.812843322753909,205.0,108.00091642005136,199.07098388671876]},{"page":123,"text":"Positional correlations","rect":[53.812843322753909,215.0,130.18585665702143,209.04693603515626]},{"page":123,"text":"“Bond” orientation order","rect":[53.812843322753909,225.0,139.21211331946544,219.02249145507813]},{"page":123,"text":"“Bond” correlations <cc>","rect":[53.812843322753909,236.45989990234376,148.92119756697253,228.67080688476563]},{"page":123,"text":"and","rect":[184.04055786132813,181.0,196.28372711329386,174.86810302734376]},{"page":123,"text":"5","rect":[235.8145751953125,42.55636978149414,240.04511016993448,36.6298942565918]},{"page":123,"text":"Structure","rect":[242.442138671875,43.0,273.5281004645777,36.73149108886719]},{"page":123,"text":"Analysis and","rect":[275.9192199707031,44.275047302246097,320.1672720351688,36.68069076538086]},{"page":123,"text":"a","rect":[302.5402526855469,70.57178497314453,306.5358352707217,66.6776351928711]},{"page":123,"text":"X-Ray","rect":[322.5389099121094,44.275047302246097,345.07919830470009,36.85001754760742]},{"page":123,"text":"b","rect":[331.15997314453127,91.00189971923828,335.15555572970609,85.41255950927735]},{"page":123,"text":"Diffraction","rect":[347.4626770019531,43.0,385.1481985488407,36.68069076538086]},{"page":123,"text":"correlations","rect":[198.65536499023438,181.0,238.71851046561518,174.86810302734376]},{"page":123,"text":"b","rect":[288.3894958496094,137.90692138671876,292.3850784347842,132.3175811767578]},{"page":123,"text":"2","rect":[278.431640625,155.70709228515626,282.9278512969321,149.4264373779297]},{"page":123,"text":"for two dimensional","rect":[241.1392364501953,181.0,309.9269075795657,174.86810302734376]},{"page":123,"text":"Hexatic layer","rect":[241.93121337890626,194.5355224609375,287.75384610755136,186.941162109375]},{"page":123,"text":"Liquid like","rect":[241.93121337890626,206.66534423828126,279.7361387946558,199.07098388671876]},{"page":123,"text":"exp(\u0005r/x)","rect":[241.9311981201172,216.64129638671876,276.24337857825449,208.7760009765625]},{"page":123,"text":"Quasi-long range","rect":[241.93121337890626,226.61685180664063,300.7652830817652,219.02249145507813]},{"page":123,"text":"r\u0005\u0002","rect":[241.9320526123047,234.7158203125,253.19150971598655,228.48585510253907]},{"page":123,"text":"single layers","rect":[312.342529296875,182.46246337890626,355.24780734061519,174.86810302734376]},{"page":123,"text":"Crystal layer","rect":[326.33294677734377,194.5355224609375,370.17221158606699,186.941162109375]},{"page":123,"text":"Quasi-long range","rect":[326.33209228515627,206.66534423828126,385.1660704352808,199.07098388671876]},{"page":123,"text":"r\u0005\u0002","rect":[326.33203125,214.82101440429688,337.5925869864944,208.5908966064453]},{"page":123,"text":"Long range","rect":[326.33209228515627,226.61685180664063,365.3968138434839,219.1918182373047]},{"page":123,"text":"Const","rect":[326.332763671875,234.81723022460938,346.0639925405032,228.9923553466797]},{"page":123,"text":"A two-dimensional phase with a bond orientation order is called hexatic phase","rect":[65.76496887207031,265.8777770996094,385.11792791559068,256.9233093261719]},{"page":123,"text":"[12, 13]. It has six-fold symmetry D6h and a new, two-component order parameter","rect":[53.81394958496094,277.83746337890627,385.14153419343605,268.90277099609377]},{"page":123,"text":"C ¼ C0 expi6jðrÞ","rect":[180.52774047851563,302.0386657714844,258.4528695498581,292.088134765625]},{"page":123,"text":"(5.50)","rect":[361.055908203125,301.30157470703127,385.10527931062355,292.7056884765625]},{"page":123,"text":"where j(r) is the angle a local “bond” vector forms with a reference system. It isa","rect":[53.814231872558597,325.6191101074219,385.15808904840318,316.6845703125]},{"page":123,"text":"phase with a new order parameter and C0 is its amplitude. The mean square","rect":[53.814231872558597,337.5789794921875,385.1608051128563,328.5345153808594]},{"page":123,"text":"displacements < (dj)2 > logarithmically decay with a distance from a reference","rect":[53.81400680541992,349.53863525390627,385.0806354351219,338.48760986328127]},{"page":123,"text":"point following Eq. 5.47 although with a special, “bond” elastic modulus Kbond. The","rect":[53.813472747802737,361.4981689453125,385.14768255426255,352.50384521484377]},{"page":123,"text":"density correlation function follows the power law decay GCðrÞ / r\u0005\u0002C due to","rect":[53.81380844116211,373.79656982421877,385.14409724286568,363.8460388183594]},{"page":123,"text":"fluctuations in the “bond” angle. The temperature dependent amplitude C0 of the","rect":[53.814205169677737,385.36065673828127,385.17554510309068,376.3165283203125]},{"page":123,"text":"two-component order parameter C takes the values between 0 and 1. Note the","rect":[53.813838958740237,397.3204650878906,385.17456854059068,388.2763366699219]},{"page":123,"text":"analogy with the two-component order parameter of the smectic C phase, although","rect":[53.813838958740237,409.2799987792969,385.10793391278755,400.345458984375]},{"page":123,"text":"the symmetry of the two phases is different.","rect":[53.813838958740237,421.2395324707031,231.3223080208469,412.30499267578127]},{"page":123,"text":"Depending on a material, single smectic monolayers can exist in two different","rect":[65.76586151123047,433.2389221191406,385.1745744473655,424.24462890625]},{"page":123,"text":"modifications, liquid-like and hexatic like. Properties of these monolayers are","rect":[53.81289291381836,445.15863037109377,385.10398138238755,436.22406005859377]},{"page":123,"text":"shown in Table 5.1. Upon melting, a two-dimensional hexatic layer undergoes","rect":[53.81289291381836,457.1181640625,385.10397733794408,448.12384033203127]},{"page":123,"text":"the transition into the liquid-like layer. It is spectacular that hexatic layers like","rect":[53.81289291381836,469.0777282714844,385.1358112163719,460.1431884765625]},{"page":123,"text":"liquid layers do not support the in-plane shear [14]. The layer can be sheared by as","rect":[53.81289291381836,480.9804992675781,385.13876737700658,472.04595947265627]},{"page":123,"text":"small force (stress) as is wished.","rect":[53.811885833740237,492.5415954589844,184.4579281380344,484.0054931640625]},{"page":123,"text":"5.7.4 Three-Dimensional Smectic Phases","rect":[53.812843322753909,528.7047729492188,265.8909519169108,520.170654296875]},{"page":123,"text":"5.7.4.1 Uniaxial Orthogonal","rect":[53.812843322753909,558.430419921875,179.40710313388895,549.3265380859375]},{"page":123,"text":"In three dimensions the situation is different, because there are interactions between","rect":[53.812843322753909,580.3374633789063,385.15166560224068,573.3352661132813]},{"page":123,"text":"the layers that may stabilize more ordered phases. Now the in-plane ordering and","rect":[53.812843322753909,594.2293090820313,385.1447686295844,585.2748413085938]},{"page":124,"text":"5.7 Diffraction by Smectic Phases","rect":[53.8134880065918,44.276084899902347,170.42816622733393,36.63093185424805]},{"page":124,"text":"107","rect":[372.4988098144531,42.55740737915039,385.1904348769657,36.73252868652344]},{"page":124,"text":"the ordering along the layer normal in the three-dimensional, uniaxial, orthogonal","rect":[53.812843322753909,68.2883529663086,385.16267259189677,59.35380554199219]},{"page":124,"text":"(without tilt) smectic phase should be discussed separately. The in-plane structural","rect":[53.812843322753909,80.24788665771485,385.1426835782249,71.29341888427735]},{"page":124,"text":"characteristics of the smectic A phase and smectic Bhex hexatic phase are presented","rect":[53.812843322753909,92.20772552490235,385.1412285905219,83.27293395996094]},{"page":124,"text":"in Table 5.2. Note that, in the three-dimensional hexatic phase,the quasi-long-range","rect":[53.81432342529297,104.1104965209961,385.0924762554344,95.1161880493164]},{"page":124,"text":"hexatic order inherent to a single monolayer is substituted by the true long-range","rect":[53.81432342529297,116.0699691772461,385.1432575054344,107.13542175292969]},{"page":124,"text":"hexatic order with constant correlation function GCðr?Þ. As to the out-of-plane","rect":[53.81432342529297,128.36865234375,385.1408160991844,118.41812896728516]},{"page":124,"text":"positional order in the hexatic phase, it is quasi-long-range with power law correla-","rect":[53.81391525268555,139.98953247070313,385.0980161270298,131.0549774169922]},{"page":124,"text":"tions <rzrz > of the z\u0005\u0002 type [14]. The same table is illustrated by Fig. 5.25. It is","rect":[53.81391525268555,151.94937133789063,385.1873818789597,142.4585418701172]},{"page":124,"text":"seen how a continuous, blurred diffraction ring typical of the smectic A phase is","rect":[53.81272506713867,163.908935546875,385.1854287539597,154.97438049316407]},{"page":124,"text":"substituted by a six-spot diffraction pattern for the hexatic Bhex and then by a six-","rect":[53.81272506713867,175.86868286132813,385.15316139070168,166.9339141845703]},{"page":124,"text":"point pattern for smectic Bcr (crystalline) phase. An example of the experimental","rect":[53.814353942871097,187.82827758789063,385.179091048928,178.89366149902345]},{"page":124,"text":"X-ray diffraction pattern for a thick layers of the smectic A and Bcr phases was","rect":[53.81337356567383,199.73126220703126,385.1522561465378,190.79649353027345]},{"page":124,"text":"illustrated by Fig. 5.16.","rect":[53.814414978027347,211.69082641601563,148.0111050179172,202.69650268554688]},{"page":124,"text":"On the other hand, the experiments with very thin free-suspended films of","rect":[65.76644134521485,223.65036010742188,385.1502927383579,214.6958770751953]},{"page":124,"text":"smectics show that the crystalline order in certain substances with weak interlayer","rect":[53.814414978027347,235.60992431640626,385.12542091218605,226.6753692626953]},{"page":124,"text":"interactions may exist only in the surface layers [11]. In thick films the smectic","rect":[53.814414978027347,247.5694580078125,385.1810993023094,238.63490295410157]},{"page":124,"text":"layers are mostly liquid. However, within the same thin film one may observe the","rect":[53.813411712646487,259.52899169921877,385.1721881694969,250.5944366455078]},{"page":124,"text":"layer-by-layer crystallization. For example, the entire sequence of phase transitions","rect":[53.813411712646487,271.488525390625,385.0985452090378,262.553955078125]},{"page":124,"text":"SmA-SmBhex-SmBcr is shifted downward as one advances into the bulk from the","rect":[53.813411712646487,283.0041809082031,385.17212713434068,274.51422119140627]},{"page":124,"text":"Table 5.2 Order parameters and density","rect":[53.812843322753909,307.44366455078127,196.91406768946573,299.8323669433594]},{"page":124,"text":"Bhex and crystalline Bcr","rect":[53.812843322753909,317.41949462890627,134.04391888230263,309.82513427734377]},{"page":124,"text":"Order/uniaxial phase","rect":[53.812843322753909,329.4925537109375,124.98314044260502,321.8812561035156]},{"page":124,"text":"In-plane positional order","rect":[53.812843322753909,341.6222229003906,138.2492379532545,334.0278625488281]},{"page":124,"text":"In-plane positional correlations","rect":[53.812843322753909,351.5981750488281,160.43417819022455,344.0038146972656]},{"page":124,"text":"In-plane “bond” orientation order","rect":[53.812843322753909,361.51739501953127,168.4408883438795,353.92303466796877]},{"page":124,"text":"In-plane “bond” order correlations","rect":[53.812843322753909,371.49334716796877,171.70599825858393,363.89898681640627]},{"page":124,"text":"Interlayer positional order","rect":[53.812843322753909,381.46929931640627,143.00775235755138,373.87493896484377]},{"page":124,"text":"Interlayer positional correlations","rect":[53.812843322753909,391.4452819824219,165.1926620769433,383.8509216308594]},{"page":124,"text":"correlations","rect":[199.95751953125,306.0,240.02068026541986,299.84930419921877]},{"page":124,"text":"for","rect":[243.06497192382813,306.0,252.9305886612623,299.84930419921877]},{"page":124,"text":"three-dimensional smectic","rect":[255.97994995117188,306.0,346.2959837653589,299.84930419921877]},{"page":124,"text":"Hexatic-Bhex","rect":[260.6251220703125,329.1343688964844,304.69992578341705,321.898193359375]},{"page":124,"text":"Liquid like","rect":[260.6242370605469,341.6222229003906,298.3724608161402,334.0278625488281]},{"page":124,"text":"exp(\u0005r/x)","rect":[260.6242370605469,351.5981750488281,294.87970059973886,343.7328796386719]},{"page":124,"text":"Long range","rect":[260.6242370605469,361.51739501953127,299.6889891364527,354.09234619140627]},{"page":124,"text":"const","rect":[260.6242370605469,370.0,278.5024691030032,364.7625427246094]},{"page":124,"text":"Quasi-long range","rect":[260.624267578125,381.46929931640627,319.4583372809839,373.87493896484377]},{"page":124,"text":"\u0005\u0002","rect":[263.8531188964844,387.0401306152344,271.88418244547878,383.3378601074219]},{"page":124,"text":"z","rect":[260.62432861328127,389.62432861328127,263.88184054374019,385.9499206542969]},{"page":124,"text":"A,","rect":[349.3876647949219,306.0,357.5779750068422,299.90008544921877]},{"page":124,"text":"hexatic","rect":[360.6036682128906,306.0,385.1661009528589,299.84930419921877]},{"page":124,"text":"Fig. 5.25 Comparison of in-plane the diffraction patterns for the smectic A (a), smectic Bhex (b)","rect":[53.812843322753909,574.0244750976563,385.1722421036451,566.294677734375]},{"page":124,"text":"and smectic Bcr (c) phase. Below are qualitative dependencies of scattering intensity on the","rect":[53.81307601928711,583.9326171875,385.1558165290308,576.3382568359375]},{"page":124,"text":"diffraction angle for the three phases","rect":[53.8134880065918,593.9085693359375,179.78780062674799,586.314208984375]},{"page":125,"text":"108","rect":[53.81370162963867,42.55740737915039,66.50530761866495,36.73252868652344]},{"page":125,"text":"5 Structure Analysis and X-Ray Diffraction","rect":[235.81471252441407,44.276084899902347,385.1483206191532,36.63093185424805]},{"page":125,"text":"first (surface) layer to the second, third, etc. layers of the same ��lm. It means that, in","rect":[53.812843322753909,68.2883529663086,385.1984490495063,59.31396484375]},{"page":125,"text":"the bulk, the liquid-like phase is more stable that the crystalline one and the surface","rect":[53.812862396240237,80.24788665771485,385.08408392145005,71.31333923339844]},{"page":125,"text":"forces are very important.","rect":[53.812862396240237,92.20748138427735,157.9816860726047,83.27293395996094]},{"page":125,"text":"5.7.4.2 Biaxial Orthogonal","rect":[53.812862396240237,131.96368408203126,173.285292220803,122.85980224609375]},{"page":125,"text":"In biaxial orthogonal smectics the symmetry is further reduced. For example, the","rect":[53.812862396240237,155.80313110351563,385.1726459331688,146.8685760498047]},{"page":125,"text":"group theory predicts phase transitions from smectic A (symmetry D1h) into","rect":[53.812862396240237,167.76266479492188,385.1577996354438,158.82810974121095]},{"page":125,"text":"smectic Ab with symmetry C2h, that is a biaxial SmA phase with a hindered rotation","rect":[53.814022064208987,179.72293090820313,385.1176690202094,170.78831481933595]},{"page":125,"text":"of molecules about their longitudinal axes. It is also possible a transition from SmB","rect":[53.81368637084961,191.68246459960938,385.1445796173408,182.74790954589845]},{"page":125,"text":"into an exotic SmBq phase with symmetry D3h due to specific distribution of","rect":[53.81368637084961,204.4989471435547,385.1506589492954,194.7074737548828]},{"page":125,"text":"positive and negative electric charges alternating along the perimeter of the hexag-","rect":[53.81380844116211,215.60177612304688,385.14666114656105,206.66722106933595]},{"page":125,"text":"onal elementary cell [15]. Such a phase has not been reported yet.","rect":[53.81380844116211,227.50454711914063,320.1426662972141,218.51022338867188]},{"page":125,"text":"On cooling, the smectic Bhex phase (symmetry D6h) can transit into smectic E","rect":[65.76583099365235,239.46450805664063,385.17963403142655,230.50962829589845]},{"page":125,"text":"(SmE) with herringbone packing and point group symmetry C2h. It is shown in","rect":[53.81394577026367,251.42404174804688,385.1424492936469,242.48948669433595]},{"page":125,"text":"Fig. 5.26 together with a sketch of the characteristic X-ray diffractogram. In fact,","rect":[53.81356430053711,263.3836975097656,385.1235622933078,254.38937377929688]},{"page":125,"text":"SmE is true crystalline phase.","rect":[53.81456756591797,275.3432312011719,173.58462186362034,266.40869140625]},{"page":125,"text":"5.7.4.3 Biaxial Tilted","rect":[53.81456756591797,313.2537536621094,149.5362584049518,305.8232421875]},{"page":125,"text":"When molecular axes (director) are tilted by some angle with respect to the smectic","rect":[53.81456756591797,338.9388732910156,385.1823200054344,330.00433349609377]},{"page":125,"text":"normal we have the remarkable correspondence between the tilted and orthogonal","rect":[53.81456756591797,350.8984069824219,385.167372298928,341.9638671875]},{"page":125,"text":"phases: SmC \u0005 SmA (both have liquid layer structure); SmF \u0005 SmBhex (hexatic","rect":[53.81456756591797,362.85797119140627,385.1111224956688,353.92340087890627]},{"page":125,"text":"layer structure); SmH \u0005 SmBcryst (crystalline layer structure).","rect":[53.814083099365237,375.74798583984377,303.13177152182348,365.88360595703127]},{"page":125,"text":"As an example, consider the X-ray diffraction by the smectic C phase of p-di-","rect":[65.76565551757813,386.77783203125,385.14547096101418,377.84326171875]},{"page":125,"text":"heptyloxyazoxybenzene [16]. Since always there is a possibility to align the","rect":[53.81362533569336,398.73736572265627,385.1743854351219,389.80279541015627]},{"page":125,"text":"director by a magnetic field along a certain, well defined direction (e.g., vertical","rect":[53.8126335144043,410.6968994140625,385.1783281094749,401.7623291015625]},{"page":125,"text":"as in Fig. 5.27) we expect that the diffraction pattern from the layered structure will","rect":[53.8126335144043,422.6564636230469,385.14753587314677,413.6621398925781]},{"page":125,"text":"also be tilted through the same angle with respect to the vertical. However, as a rule,","rect":[53.812618255615237,434.5592346191406,385.1405300667453,425.62469482421877]},{"page":125,"text":"Fig. 5.26 Smectic E phase. The herringbone structure (a) and corresponding diffraction pattern","rect":[53.812843322753909,560.7045288085938,385.1635794082157,552.9747314453125]},{"page":125,"text":"(b) for two different directions of scattering, parallel (q||) and perpendicular (q⊥) to the director. As","rect":[53.812843322753909,570.6127319335938,385.17310030936519,563.0182495117188]},{"page":125,"text":"an example, the two Miller indices are shown only for q||. They mark Bragg reflections of the first","rect":[53.81307601928711,580.5885620117188,385.1694918080813,572.9940795898438]},{"page":125,"text":"and second orders from the horizontal crystallographic planes","rect":[53.81370162963867,590.5076904296875,265.12219698905269,582.913330078125]},{"page":126,"text":"References","rect":[53.813690185546878,42.52305221557617,91.48237307547845,36.68124008178711]},{"page":126,"text":"109","rect":[372.4990234375,42.62464904785156,385.19061798243447,36.73204040527344]},{"page":126,"text":"Fig. 5.27 Smectic C phase in the magnetic film along the vertical direction, H||n. Typical","rect":[53.812843322753909,197.89071655273438,385.16183189597197,190.16090393066407]},{"page":126,"text":"diffractogram (a) and its scheme (b) showing the four-point picture of reflections from the smectic","rect":[53.81281280517578,207.74224853515626,385.1559385993433,200.14788818359376]},{"page":126,"text":"layers and blurred nematic-like arcs corresponding to sketch (c) of the uniform director alignment","rect":[53.81365966796875,217.71817016601563,385.14071373190947,210.12380981445313]},{"page":126,"text":"with broken smectic layers [16]","rect":[53.813682556152347,227.69415283203126,162.2744607315748,220.09979248046876]},{"page":126,"text":"the layers are broken and acquire a tilt in two opposite directions as shown in sketch","rect":[53.812843322753909,254.144775390625,385.11977473310005,245.21022033691407]},{"page":126,"text":"Fig. 5.27c. Therefore, the diffraction pattern becomes symmetric (degenerate) with","rect":[53.812843322753909,266.04754638671877,385.1138848405219,257.05322265625]},{"page":126,"text":"respect to the vertical and, instead a pair of the first order spots, we see four spots on","rect":[53.812843322753909,278.007080078125,385.16765681317818,269.072509765625]},{"page":126,"text":"photo (Fig. 5.27a). It is a so-called four-point pattern. The molecular tilt angle # can","rect":[53.812843322753909,289.9666442871094,385.1277703385688,280.8129577636719]},{"page":126,"text":"be found as shown in Fig. 5.27b. The broad arcs at the equatorial (horizontal) line","rect":[53.81385040283203,301.9261779785156,385.1387409038719,292.9318542480469]},{"page":126,"text":"are due to orientational (nematic) order.","rect":[53.81484603881836,313.4873046875,214.92672391440159,304.951171875]},{"page":126,"text":"References","rect":[53.812843322753909,362.04541015625,109.59614448282879,353.2602844238281]},{"page":126,"text":"1.","rect":[58.06126022338867,388.0,64.40706131055318,381.7472229003906]},{"page":126,"text":"2.","rect":[58.06126022338867,398.0,64.40706131055318,391.66644287109377]},{"page":126,"text":"3.","rect":[58.06126022338867,408.0,64.40706131055318,401.64239501953127]},{"page":126,"text":"4.","rect":[58.0612678527832,438.0,64.4070689399477,431.5135192871094]},{"page":126,"text":"5.","rect":[58.0612678527832,467.2095642089844,64.4070689399477,461.2830810546875]},{"page":126,"text":"6.","rect":[58.0612678527832,487.1614685058594,64.4070689399477,481.2857971191406]},{"page":126,"text":"7.","rect":[58.06126022338867,507.0566711425781,64.40706131055318,501.3503112792969]},{"page":126,"text":"8.","rect":[58.06126022338867,527.0086059570313,64.40706131055318,521.1837158203125]},{"page":126,"text":"9.","rect":[58.061256408691409,556.9474487304688,64.40705749585591,551.0548706054688]},{"page":126,"text":"10.","rect":[53.81295394897461,577.0,64.3892924506899,570.9500732421875]},{"page":126,"text":"Kittel, C.: Introduction to Solid State Physics, 4th edn. Wiley, New York (1971)","rect":[68.59698486328125,389.2908020019531,344.1317452774732,381.6964416503906]},{"page":126,"text":"Vainstein, B.K.: Diffraction of X-Rays on Chain Molecules. Elsevier, Amsterdam (1966).","rect":[68.59698486328125,399.21002197265627,376.3252894599672,391.61566162109377]},{"page":126,"text":"De Santo, M., Barberi, R., Blinov, L.M.: Electric force microscopic observations of electric","rect":[68.59698486328125,409.18597412109377,385.1162352302027,401.59161376953127]},{"page":126,"text":"surface potentials. In: Rasing, Th, Musˇevicˇ, I. (eds.) Surface and Interfaces of Liquid Crystals,","rect":[68.59698486328125,419.16192626953127,385.1298548896547,411.23822021484377]},{"page":126,"text":"pp. 194–210. Springer, Berlin (2004)","rect":[68.59699249267578,429.0811767578125,195.4944314346998,421.48681640625]},{"page":126,"text":"Landau, L.D., Lifshits, E.M.: Theory of Field, 5th edn. Nauka, Moscow (1967), Ch. 9 (in","rect":[68.59699249267578,439.0570983886719,385.1536001601688,431.41192626953127]},{"page":126,"text":"Russian) [see also Landau, L.D., Lifshits, E.M.: Theory of Field, 4th edn. Butterworth-","rect":[68.59699249267578,449.0330505371094,385.11285489661386,441.4386901855469]},{"page":126,"text":"Heinemann, Oxford (1980)]","rect":[68.59699249267578,458.67034912109377,164.38137906653575,451.397705078125]},{"page":126,"text":"Landau, L.D., Lifshits, E.M.: Statistical Physics, vol. pt.1. Nauka, Moscow (1976). (in","rect":[68.59699249267578,468.9282531738281,385.1527456679813,461.3338928222656]},{"page":126,"text":"Russian) [Statistical Physics, 3 rd edn, Pergamon, Oxford, 1980]","rect":[68.59699249267578,478.9042053222656,289.9369210587232,471.29290771484377]},{"page":126,"text":"Osipov, M.A., Ostrovskii, B.I.: Study of the orientational order in liquid crystals by x-ray","rect":[68.59699249267578,488.8801574707031,385.16623443751259,481.26885986328127]},{"page":126,"text":"scattering. Cryst. Rev. 3, 113–156 (1992)","rect":[68.59699249267578,498.8561096191406,210.392670570442,490.9908142089844]},{"page":126,"text":"Vertogen, G., de Jeu, W.H.: Thermotropic Liquid Crystals, Fundamentals. Springer Verlag,","rect":[68.59698486328125,508.7753601074219,385.1610743720766,501.1809997558594]},{"page":126,"text":"Berlin (1988)","rect":[68.59698486328125,518.4126586914063,114.77496427161386,511.1569519042969]},{"page":126,"text":"Als-Nielsen, J., Litster, J.D., Birgenau, R.J., Kaplan, M., Safinia, C.R., Lindegaard-Andersen, A.,","rect":[68.59698486328125,528.7272338867188,385.286410065436,521.1328735351563]},{"page":126,"text":"Mathiesen, S.: Observation of algebraic decay of positional order in a smectic liquid crystal.","rect":[68.59698486328125,538.6464233398438,385.1729457099672,531.03515625]},{"page":126,"text":"Phys. Rev. B 22, 314–320 (1980)","rect":[68.59698486328125,548.6224365234375,183.146332679817,541.028076171875]},{"page":126,"text":"Caille´, A.: Remarques sur la diffusion des rayons X dans les smectiques. C. R. Acad. Sci.B","rect":[68.59698486328125,558.598388671875,385.1577776643215,550.0]},{"page":126,"text":"247, 891 (1972)","rect":[68.59782409667969,568.2357177734375,123.44248288733651,560.9884643554688]},{"page":126,"text":"de Gennes, P.G., Prost, J.: The Physics of Liquid Crystals, 2nd edn. Oxford Science Publica-","rect":[68.59698486328125,578.4935913085938,385.1906747208326,570.88232421875]},{"page":126,"text":"tions, Oxford (1995)","rect":[68.59697723388672,588.1309204101563,138.9626168595045,580.8244018554688]},{"page":127,"text":"110","rect":[53.81199264526367,42.55691909790039,66.50359863428995,36.73204040527344]},{"page":127,"text":"5 Structure Analysis and X-Ray Diffraction","rect":[235.81300354003907,44.275596618652347,385.1466116347782,36.63044357299805]},{"page":127,"text":"11.","rect":[53.812843322753909,65.12664794921875,64.38918182446919,59.40336608886719]},{"page":127,"text":"12.","rect":[53.81200408935547,85.07855224609375,64.38834259107076,79.35527038574219]},{"page":127,"text":"13.","rect":[53.812835693359378,105.07534790039063,64.38917419507466,99.25047302246094]},{"page":127,"text":"14.","rect":[53.812835693359378,115.0,64.38917419507466,109.22642517089844]},{"page":127,"text":"15.","rect":[53.812835693359378,125.02725219726563,64.38917419507466,119.10077667236328]},{"page":127,"text":"16.","rect":[53.812835693359378,145.0,64.38917419507466,139.04672241210938]},{"page":127,"text":"de","rect":[68.59687042236328,65.21977996826172,76.60950610422612,59.35256576538086]},{"page":127,"text":"Jeu,","rect":[80.94580078125,65.22824096679688,94.33121749951802,59.52189254760742]},{"page":127,"text":"W.H.,","rect":[98.67597961425781,65.22824096679688,119.1094615180727,59.52189254760742]},{"page":127,"text":"Ostrovskii,","rect":[123.4296875,65.22824096679688,160.77685806348286,59.33563232421875]},{"page":127,"text":"B.I.,","rect":[165.1207733154297,65.12664794921875,179.91919204785786,59.52189254760742]},{"page":127,"text":"Shalaginov,","rect":[184.21066284179688,66.9469223022461,224.46841690137348,59.35256576538086]},{"page":127,"text":"A.N.:","rect":[228.8470458984375,65.22824096679688,247.60522178855005,59.40336608886719]},{"page":127,"text":"Structure","rect":[251.90176391601563,65.22824096679688,282.9877257087183,59.40336608886719]},{"page":127,"text":"and","rect":[287.36126708984377,65.21977996826172,299.60443634180947,59.35256576538086]},{"page":127,"text":"fluctuations","rect":[303.9018249511719,65.19438171386719,343.44882662772457,59.35256576538086]},{"page":127,"text":"of","rect":[347.8020324707031,65.12664794921875,354.8500985489576,59.35256576538086]},{"page":127,"text":"smectic","rect":[359.1880798339844,65.19438171386719,385.15502307199957,59.35256576538086]},{"page":127,"text":"membranes. Rev. Mod. Phys. 75, 181–235 (2003)","rect":[68.59687042236328,76.9228744506836,239.22448819739513,69.27771759033203]},{"page":127,"text":"Halperin, B.I., Nelson, D.R.: Theory of two-dimensional melting. Phys. Rev. Lett. 19, 2456","rect":[68.59603118896485,86.8988265991211,385.1711782851688,78.93194580078125]},{"page":127,"text":"(1979)","rect":[68.59686279296875,96.53612518310547,91.1540765152662,89.33122253417969]},{"page":127,"text":"Pershan, P.S.: Structure of Liquid Crystal Phases. World Scientific, Singapore (1988)","rect":[68.59686279296875,106.79402923583985,360.4446114884107,99.19967651367188]},{"page":127,"text":"Pindak, R., Moncton, D.: Two-dimensional systems. Phys. Today (May), 57–66 (1982)","rect":[68.59686279296875,116.76998138427735,367.0730294571607,109.12482452392578]},{"page":127,"text":"Pikin, S.A.: Structural Transformations in Liquid Crystals. Gordon & Breach, New York","rect":[68.59686279296875,126.74593353271485,385.1499380507938,119.15158081054688]},{"page":127,"text":"(1981)","rect":[68.59686279296875,136.3264617919922,91.1540765152662,129.12156677246095]},{"page":127,"text":"Ostrovskii, B.I.: X-ray diffraction study of nematic, smectic A and C liquid crystals. Soviet","rect":[68.59686279296875,146.64108276367188,385.1600923940188,139.02978515625]},{"page":127,"text":"Sci. Rev. A 12(pt.2), 85–146 (1989)","rect":[68.59686279296875,156.6170654296875,192.435639320442,148.95497131347657]},{"page":128,"text":"Chapter6","rect":[53.812843322753909,72.10812377929688,114.14115996551633,59.571903228759769]},{"page":128,"text":"Phase Transitions","rect":[53.812843322753909,87.58094787597656,175.80796445758848,76.122314453125]},{"page":128,"text":"Liquid crystals manifest a number of transitions between different phases upon","rect":[53.812843322753909,211.74758911132813,385.15273371747505,202.8130340576172]},{"page":128,"text":"variation of temperature, pressure or a content of various compounds in a mixture.","rect":[53.812843322753909,223.65036010742188,385.11285062338598,214.71580505371095]},{"page":128,"text":"All the transitions are divided into two groups, namely, first and second order","rect":[53.812843322753909,235.60992431640626,385.1527036270298,226.6753692626953]},{"page":128,"text":"transitions both accompanied by interesting pre-transitional phenomena and usually","rect":[53.812843322753909,247.5694580078125,385.1009148698188,238.63490295410157]},{"page":128,"text":"described","rect":[53.812843322753909,258.0,92.12859431317816,250.59446716308595]},{"page":128,"text":"by","rect":[98.56199645996094,259.5290222167969,108.51620570233831,250.59446716308595]},{"page":128,"text":"the","rect":[114.93269348144531,258.0,127.15646398736799,250.59446716308595]},{"page":128,"text":"Landau","rect":[133.62570190429688,258.0,163.56795588544379,250.59446716308595]},{"page":128,"text":"(phenomenological)","rect":[170.10488891601563,259.5290222167969,250.70415626374854,250.59446716308595]},{"page":128,"text":"theory","rect":[257.1106872558594,259.5290222167969,282.60341731122505,250.59446716308595]},{"page":128,"text":"or","rect":[289.1144714355469,258.0,297.40633521882668,252.0]},{"page":128,"text":"molecular-statistical","rect":[303.84173583984377,258.0,385.1477494961936,250.59446716308595]},{"page":128,"text":"approach. In this chapter we are going to consider the most important phase transi-","rect":[53.812843322753909,271.4885559082031,385.11882911531105,262.55401611328127]},{"page":128,"text":"tions between isotropic, nematic, smectic A and C phases. The phase transitions in","rect":[53.812843322753909,283.44805908203127,385.1407403092719,274.51348876953127]},{"page":128,"text":"ferroelectric liquid crystals are discussed in Chapter 13.","rect":[53.812843322753909,295.4076232910156,276.5761379769016,286.47308349609377]},{"page":128,"text":"6.1 Landau Approach","rect":[53.812843322753909,345.5187683105469,174.75152558182837,334.7853698730469]},{"page":128,"text":"In Fig. 6.1 we have an example of the experimental phase diagram for homologues","rect":[53.812843322753909,373.0611877441406,385.1367532168503,364.12664794921877]},{"page":128,"text":"of 4-ethoxybenzene-40-amino-n-alkyl a-methyl cinnamates [1]. We see that, with","rect":[53.812843322753909,385.0207824707031,385.1170586686469,375.4178466796875]},{"page":128,"text":"increasing length of the alkyl chain, the temperature range of the nematic phase","rect":[53.81306076049805,396.9803161621094,385.09717596246568,388.0457763671875]},{"page":128,"text":"between the isotropic and smectic A phase becomes narrower. This range is limited","rect":[53.81306076049805,408.93988037109377,385.14992610028755,400.00531005859377]},{"page":128,"text":"by solid lines corresponding to the phase transitions between different phases. How","rect":[53.81306076049805,420.8993835449219,385.13802862122386,411.96484375]},{"page":128,"text":"to explain this diagram? We may begin with the molecular properties and intermo-","rect":[53.81306076049805,432.85894775390627,385.08919654695168,423.92437744140627]},{"page":128,"text":"lecular interaction and try to calculate the temperature range of stability of a","rect":[53.81306076049805,444.8185119628906,385.15992010309068,435.88397216796877]},{"page":128,"text":"particular phase, the values of the order parameters and thermodynamic functions","rect":[53.81306076049805,456.7780456542969,385.16888822661596,447.843505859375]},{"page":128,"text":"such as free energy and others. This approach will be discussed in the end of this","rect":[53.81306076049805,468.6808166503906,385.1439553652878,459.74627685546877]},{"page":128,"text":"chapter. Another approach is based on phenomenological description of the phase","rect":[53.81306076049805,480.6403503417969,385.09714544488755,471.705810546875]},{"page":128,"text":"transitions and called Landau theory of phase transitions. The key issue is the","rect":[53.81306076049805,492.5998840332031,385.11606634332505,483.66534423828127]},{"page":128,"text":"symmetry of the phases and corresponding order parameters related to a particular","rect":[53.81306076049805,504.5594482421875,385.13994727937355,495.6248779296875]},{"page":128,"text":"transition. Such an approach appeared to be very powerful and relatively simple.","rect":[53.81306076049805,516.5189208984375,379.6034206917453,507.58441162109377]},{"page":128,"text":"Imagine a series of transitions between phases of different symmetry, as shown","rect":[65.76508331298828,528.478515625,385.1509636979438,519.5440063476563]},{"page":128,"text":"in Fig. 6.2, for instance, with decreasing temperature. Our task is to select one of","rect":[53.81306076049805,540.43798828125,385.14791236726418,531.4835205078125]},{"page":128,"text":"these transitions, find the temperature behaviour of the order parameter and other","rect":[53.81306076049805,552.3975830078125,385.1359799942173,543.4630737304688]},{"page":128,"text":"thermodynamic functions close to the phase transition [2]. To this effect, we should","rect":[53.81306076049805,564.3003540039063,385.1787957291938,555.3658447265625]},{"page":128,"text":"make the following steps.","rect":[53.81406021118164,576.2598876953125,157.44236417319065,567.3253784179688]},{"page":128,"text":"L.M. Blinov, Structure and Properties of Liquid Crystals,","rect":[53.812843322753909,623.7623291015625,251.63688919141254,616.1510620117188]},{"page":128,"text":"DOI 10.1007/978-90-481-8829-1_6, # Springer ScienceþBusiness Media B.V. 2011","rect":[53.812843322753909,633.698486328125,347.38995880274697,625.4920043945313]},{"page":128,"text":"111","rect":[372.4981994628906,621.958984375,385.18979400782509,616.2357177734375]},{"page":129,"text":"112","rect":[53.8134880065918,42.455322265625,66.50509399561807,36.73204040527344]},{"page":129,"text":"Fig. 6.1 An example of the","rect":[53.812843322753909,67.58130645751953,150.35958239817144,59.85148620605469]},{"page":129,"text":"experimental phase diagram","rect":[53.812843322753909,77.4895248413086,150.12521599168765,69.89517211914063]},{"page":129,"text":"for a homological series of","rect":[53.812843322753909,87.4087142944336,145.77197354895763,79.81436157226563]},{"page":129,"text":"cinnamate derivatives. The","rect":[53.812843322753909,95.65752410888672,146.1544279792261,89.79031372070313]},{"page":129,"text":"scale of the abscissa meansa","rect":[53.812843322753909,105.60813903808594,153.38102099680425,99.76632690429688]},{"page":129,"text":"number of alkyl chains in the","rect":[53.812843322753909,117.33663177490235,154.32528064035894,109.74227905273438]},{"page":129,"text":"tail of a particular molecule.","rect":[53.812843322753909,127.25582122802735,150.93747207715473,119.66146850585938]},{"page":129,"text":"The lines show the location of","rect":[53.812843322753909,135.47923278808595,155.3456735001295,129.63742065429688]},{"page":129,"text":"phase transition temperatures","rect":[53.812843322753909,147.20773315429688,153.7930801677636,139.61337280273438]},{"page":129,"text":"T","rect":[269.07171630859377,66.38525390625,273.95560022376636,60.642478942871097]},{"page":129,"text":"120","rect":[265.2693176269531,104.10679626464844,278.60209134423595,98.28404235839844]},{"page":129,"text":"100","rect":[265.2693176269531,156.2205047607422,278.60209134423595,150.3977508544922]},{"page":129,"text":"80","rect":[269.71356201171877,207.28721618652345,278.6020608266578,201.46446228027345]},{"page":129,"text":"2","rect":[286.7472229003906,239.064453125,291.19147733056408,233.3856658935547]},{"page":129,"text":"TNI","rect":[293.8846435546875,160.53997802734376,305.90817255975255,151.45143127441407]},{"page":129,"text":"6 Phase Transitions","rect":[318.28961181640627,42.55691909790039,385.1428879070214,36.68124008178711]},{"page":129,"text":"EOBAAMC","rect":[335.20849609375,80.91606140136719,376.5016964512689,74.87734985351563]},{"page":129,"text":"Iso","rect":[316.6534118652344,120.48983764648438,329.647445612976,113.91223907470703]},{"page":129,"text":"N","rect":[316.8215637207031,182.0277099609375,323.31409193097309,175.5670928955078]},{"page":129,"text":"4","rect":[307.85504150390627,239.064453125,312.2992959340797,233.3856658935547]},{"page":129,"text":"6","rect":[328.3377685546875,239.2084197998047,332.78202298486095,233.3856658935547]},{"page":129,"text":"8","rect":[349.19537353515627,239.2084197998047,353.6396279653297,233.3856658935547]},{"page":129,"text":"10","rect":[367.5806884765625,239.2084197998047,376.4692178090797,233.3856658935547]},{"page":129,"text":"alkyl chain length","rect":[299.60198974609377,252.15765380859376,366.68146081946949,244.63125610351563]},{"page":129,"text":"High","rect":[116.49029541015625,294.5875549316406,133.64362567266006,287.125244140625]},{"page":129,"text":"symmetry","rect":[116.49029541015625,304.18536376953127,149.95767173329575,297.5948486328125]},{"page":129,"text":"phase,","rect":[116.49029541015625,313.75921630859377,138.05585672446584,306.3448791503906]},{"page":129,"text":"η=0","rect":[116.49029541015625,323.3970031738281,130.79805717030747,316.09466552734377]},{"page":129,"text":"PT1","rect":[158.7545928955078,295.6034851074219,173.39646873036606,290.020751953125]},{"page":129,"text":"Low","rect":[187.50909423828126,292.34747314453127,202.98385503597539,286.7487487792969]},{"page":129,"text":"symmetry","rect":[187.50909423828126,303.5449523925781,220.97647056142075,296.9544372558594]},{"page":129,"text":"phase,","rect":[187.50909423828126,313.1188049316406,209.07465555259084,305.7044677734375]},{"page":129,"text":"η1≠0,","rect":[187.50909423828126,323.0364074707031,206.19756509848927,315.4541015625]},{"page":129,"text":"η2=0","rect":[187.50955200195313,332.6349792480469,204.97381553212387,325.05267333984377]},{"page":129,"text":"PT2","rect":[231.05345153808595,295.8826904296875,245.71292075673325,290.2999572753906]},{"page":129,"text":"Even","rect":[257.8897399902344,292.3878479003906,275.13100665410539,286.78912353515627]},{"page":129,"text":"lower","rect":[257.8897399902344,301.9856872558594,276.79355439039946,296.1470031738281]},{"page":129,"text":"symmetry","rect":[257.8897399902344,313.18316650390627,291.35711631337389,306.5926513671875]},{"page":129,"text":"phase,","rect":[257.8897399902344,322.75701904296877,279.4553013045439,315.3426818847656]},{"page":129,"text":"η1≠0,","rect":[257.8897399902344,332.67498779296877,276.57776834555957,325.0926818847656]},{"page":129,"text":"η2≠0","rect":[257.8897399902344,342.27239990234377,274.4116207079051,334.690185546875]},{"page":129,"text":"Next","rect":[303.99151611328127,290.8691711425781,320.7291977410742,285.27044677734377]},{"page":129,"text":"PT","rect":[303.99151611328127,300.43023681640627,314.15083688762669,294.93548583984377]},{"page":129,"text":"Fig 6.2 Sequence of the phase transitions with decreasing temperature and lowering the phase","rect":[53.812843322753909,374.1679992675781,385.15514514231207,366.43817138671877]},{"page":129,"text":"symmetry. A new order parameter Z1, Z2, etc. is introduced for each new phase with lower","rect":[53.812843322753909,384.01953125,385.1540841446607,376.425048828125]},{"page":129,"text":"symmetry","rect":[53.8134880065918,393.995361328125,87.72545379786416,387.2645568847656]},{"page":129,"text":"1.","rect":[53.812843322753909,419.0,61.27850003744845,412.0249328613281]},{"page":129,"text":"2.","rect":[53.812843322753909,443.0,61.27850003744845,435.94403076171877]},{"page":129,"text":"3.","rect":[53.812843322753909,467.0,61.27850003744845,459.86309814453127]},{"page":129,"text":"4.","rect":[53.812843322753909,491.0,61.27850003744845,483.7254638671875]},{"page":129,"text":"5.","rect":[53.812843322753909,503.0,61.27850003744845,495.5654602050781]},{"page":129,"text":"From symmetry consideration we should choose a proper order parameter for","rect":[66.27452087402344,420.89971923828127,385.1785825332798,411.96514892578127]},{"page":129,"text":"the lower symmetry phase (on account of molecular distribution functions).","rect":[66.27452087402344,432.8592834472656,371.1302151253391,423.92474365234377]},{"page":129,"text":"Using smallness of the chosen parameter we expand the free energy density in","rect":[66.27452087402344,444.8188171386719,385.1387566666938,435.88427734375]},{"page":129,"text":"powers of this parameter, with only the first term temperature dependent.","rect":[66.27452087402344,456.7783508300781,361.4348415901828,447.84381103515627]},{"page":129,"text":"The thermodynamic behaviour of the order parameter in the low symmetry","rect":[66.27452087402344,468.7378845214844,385.17656794599068,459.8033447265625]},{"page":129,"text":"phase is found by a minimization procedure for free energy density.","rect":[66.27452087402344,480.64068603515627,340.54583402182348,471.70611572265627]},{"page":129,"text":"With the order parameter found the free energy may be written explicitly.","rect":[66.27452087402344,492.6002502441406,364.23492093588598,483.66571044921877]},{"page":129,"text":"Other thermodynamic functions are found from the temperature behaviour of the","rect":[66.27452087402344,504.5597839355469,385.16956365777818,495.6053161621094]},{"page":129,"text":"free energy.","rect":[66.27452087402344,516.519287109375,114.34638638998752,507.58477783203127]},{"page":129,"text":"As to free energy, below we shall use the Helmholtz free energy F ¼ U \u0002 TS","rect":[65.76486206054688,534.43017578125,385.1785821061469,525.4956665039063]},{"page":129,"text":"(U, S and T are total energy, entropy and temperature, respectively) that is more","rect":[53.81288146972656,546.3897705078125,385.1338275737938,537.4552612304688]},{"page":129,"text":"appropriate for discussion of the systems in terms of temperature and volumeV","rect":[53.81188201904297,558.3493041992188,385.17856591619218,549.414794921875]},{"page":129,"text":"(or density r) at constant pressure p. In a more general case, the thermodynamic","rect":[53.811893463134769,570.308837890625,385.1796039409813,561.3743286132813]},{"page":129,"text":"potential (or Gibbs free energy) F ¼ F þ pV appears to be more suitable for an","rect":[53.8119010925293,582.2683715820313,385.14971247724068,573.303955078125]},{"page":129,"text":"expansion, e.g. when varying pressure p.","rect":[53.811893463134769,594.2279052734375,221.53435941244846,585.2933959960938]},{"page":130,"text":"6.1 Landau Approach","rect":[53.812843322753909,44.274986267089847,127.9766592422001,36.68062973022461]},{"page":130,"text":"Fig. 6.3 Disorder–order","rect":[53.812843322753909,67.58130645751953,138.36431974036388,59.648292541503909]},{"page":130,"text":"transition in Cu–Au alloy.","rect":[53.812843322753909,77.4895248413086,143.16597244336567,69.89517211914063]},{"page":130,"text":"In the low temperature, low","rect":[53.812843322753909,87.4087142944336,148.85773254972905,79.81436157226563]},{"page":130,"text":"symmetry phase, the atom of","rect":[53.812843322753909,97.3846664428711,153.13565152747325,89.79031372070313]},{"page":130,"text":"gold is more often occupies","rect":[53.812843322753909,107.36067962646485,148.57175143241205,99.76632690429688]},{"page":130,"text":"the central position in the","rect":[53.812843322753909,117.33663177490235,141.18441149973394,109.74227905273438]},{"page":130,"text":"cubic lattice","rect":[53.812843322753909,125.50328063964844,95.45823046946049,119.66146850585938]},{"page":130,"text":"High-T phase ","rect":[220.28746032714845,67.94453430175781,268.9281025345388,60.639442443847659]},{"page":130,"text":"Cu","rect":[235.9888458251953,89.68891906738281,245.73616347908127,83.96006774902344]},{"page":130,"text":"Au","rect":[237.5369110107422,136.65272521972657,247.28422866462814,130.9638671875]},{"page":130,"text":"113","rect":[372.4981689453125,42.55630874633789,385.18979400782509,36.73143005371094]},{"page":130,"text":"Low-T phase","rect":[326.3477478027344,67.94453430175781,370.2466728478119,60.76746368408203]},{"page":130,"text":"Consider, as an example, a disorder-order transition in Cu-Au (8:1) alloy. In this","rect":[65.76496887207031,184.59738159179688,385.1428567324753,175.66282653808595]},{"page":130,"text":"case, we can choose the simplest (scalar) order parameter Z, which is a normalized","rect":[53.812950134277347,196.55691528320313,385.1478203873969,187.6223602294922]},{"page":130,"text":"difference between probabilities to find either gold or copper atoms in the center of","rect":[53.81296157836914,208.5164794921875,385.14788184968605,199.58192443847657]},{"page":130,"text":"the cubic cell, see Fig. 6.3. In the higher temperature (and higher symmetry) phase","rect":[53.81296157836914,220.41925048828126,385.0990375347313,211.4846954345703]},{"page":130,"text":"an atom of gold has equal probability to be at any lattice site included the central","rect":[53.81295394897461,232.3787841796875,385.1150041348655,223.44422912597657]},{"page":130,"text":"one (but not between the sites), and order parameter Z ¼ 0. In the ideally ordered,","rect":[53.81295394897461,244.33834838867188,385.1099819710422,235.40379333496095]},{"page":130,"text":"zero-temperature phase, an Au atom is always in the central position and Z ¼ 1.","rect":[53.81296157836914,256.2978515625,385.1737026741672,247.36329650878907]},{"page":130,"text":"Generally, in the low-temperature phase, the central position is more often popu-","rect":[53.81199264526367,268.2574157714844,385.12593971101418,259.3228759765625]},{"page":130,"text":"lated by a gold atom than by a particular copper atom and 0 < Z < 1.","rect":[53.81199264526367,280.2169494628906,336.5165676644016,271.28240966796877]},{"page":130,"text":"Neglecting the mass density change at the transition, the Landau expansion for","rect":[65.76498413085938,292.1764831542969,385.1797116836704,283.241943359375]},{"page":130,"text":"the free energy density is","rect":[53.81296920776367,304.13604736328127,155.5489846621628,295.20147705078127]},{"page":130,"text":"gðT;ZÞ ¼ gðT;0Þ þ lZ þ 21AðTÞZ2 þ 31BZ3 þ 41CZ4 þ \u0003 \u0003 \u0003","rect":[99.18524932861328,338.7874755859375,339.7837671009912,318.3316345214844]},{"page":130,"text":"(6.1)","rect":[366.0976867675781,333.55291748046877,385.1699460586704,325.01678466796877]},{"page":130,"text":"Here g(T, 0) is free energy of the high-temperature phase and the fractional form","rect":[65.7662582397461,362.33172607421877,385.13715313554368,353.35736083984377]},{"page":130,"text":"of (1/2), (1/3), etc. coefficients is adopted for convenience. In equilibrium, function","rect":[53.814231872558597,374.2514343261719,385.1670769791938,365.31689453125]},{"page":130,"text":"g(T, Z) must have a minimum value, therefore at any temperature:","rect":[53.814231872558597,386.2508239746094,322.43757493564677,377.27642822265627]},{"page":130,"text":"dg","rect":[175.88316345214845,410.9637451171875,185.83738032392035,401.9694519042969]},{"page":130,"text":"d2g","rect":[233.03817749023438,410.9637451171875,247.24842158368598,400.4415283203125]},{"page":130,"text":"¼ 0 and","rect":[189.5911102294922,415.6462097167969,224.95343867352973,408.733642578125]},{"page":130,"text":">0","rect":[251.44789123535157,415.906005859375,265.77101985028755,408.7942199707031]},{"page":130,"text":"dZ","rect":[174.92059326171876,424.4372863769531,186.81785842593457,415.5724792480469]},{"page":130,"text":"dZ2","rect":[232.18870544433595,424.4372863769531,247.62458870491347,414.7818603515625]},{"page":130,"text":"Thus, derivative of (6.1) is","rect":[65.76500701904297,447.4816589355469,173.5054055606003,438.945556640625]},{"page":130,"text":"l þ AðTÞZ þ BZ2 þ CZ3 þ \u0003 \u0003 \u0003 ¼0","rect":[146.87986755371095,473.44171142578127,292.11171809247505,462.11029052734377]},{"page":130,"text":"From here we conclude that coefficient l in the expansion must be zero;","rect":[65.76590728759766,497.0218811035156,385.1208635098655,487.7685852050781]},{"page":130,"text":"otherwise, the non-zero derivative of free energy dg(T, Z ¼ 0)/dZ ¼ l would be","rect":[53.8128776550293,509.021240234375,385.1516803569969,499.7281188964844]},{"page":130,"text":"present also in the high symmetry phase. However, the presence of such an","rect":[53.81187057495117,520.8841552734375,385.1497429948188,511.94964599609377]},{"page":130,"text":"additional constant term in the high symmetry phase would smooth its own energy","rect":[53.81187057495117,532.8436889648438,385.1368340592719,523.9091796875]},{"page":130,"text":"minimum what is senseless.","rect":[53.81187057495117,542.741455078125,166.26458402182346,535.8687133789063]},{"page":130,"text":"As to the “leading” coefficient A(T) in Eq. 6.1, in the high symmetry phase, it","rect":[65.7638931274414,556.7627563476563,385.1716447598655,547.8282470703125]},{"page":130,"text":"must be positive to provide a minimum of free energy at Z ¼ 0. On the other hand, it","rect":[53.81187057495117,568.7222900390625,385.17060716220927,559.767822265625]},{"page":130,"text":"must be negative to provide a minimum of the free energy density in the low","rect":[53.81283187866211,580.681884765625,385.14877080872386,571.7473754882813]},{"page":130,"text":"symmetry phase at a finite value of order parameter Z ¼6 0, Fig. 6.4a. Thus, in","rect":[53.81283187866211,592.6414184570313,385.1426934342719,583.378173828125]},{"page":131,"text":"114","rect":[53.812843322753909,42.4547119140625,66.50444931178018,36.73143005371094]},{"page":131,"text":"Fig. 6.4 The forms of the","rect":[53.812843322753909,67.58130645751953,144.186379883523,59.85148620605469]},{"page":131,"text":"free energy density in the high","rect":[53.812843322753909,77.4895248413086,155.3219961562626,69.89517211914063]},{"page":131,"text":"symmetry (A > 0) and low","rect":[53.812843322753909,87.4087142944336,147.0453698788306,79.81436157226563]},{"page":131,"text":"symmetry (A < 0) phases","rect":[53.812843322753909,97.3846664428711,142.28772433280268,89.79031372070313]},{"page":131,"text":"(a) and the temperature","rect":[53.812843322753909,107.36067962646485,134.05170581125737,99.76632690429688]},{"page":131,"text":"dependence ofthe first term in","rect":[53.812843322753909,117.33663177490235,155.35838836817667,109.74227905273438]},{"page":131,"text":"the Landau expansion (b)","rect":[53.812843322753909,127.25582122802735,141.03123563391856,119.66146850585938]},{"page":131,"text":"a","rect":[214.3452606201172,68.23126220703125,219.89924507462909,62.64379119873047]},{"page":131,"text":"h","rect":[287.6809997558594,145.58535766601563,292.50346798520868,139.53546142578126]},{"page":131,"text":"6 Phase Transitions","rect":[318.2889709472656,42.55630874633789,385.1422470378808,36.68062973022461]},{"page":131,"text":"general case, when not only temperature is varied but pressure p, composition X,","rect":[53.812843322753909,175.585205078125,385.1824917366672,166.65065002441407]},{"page":131,"text":"etc., the coefficient A(T, p, X..) should change sign at the phase transition. There-","rect":[53.8138313293457,187.54473876953126,385.12481056062355,178.6101837158203]},{"page":131,"text":"fore, for a transition at temperature T ¼ Tc, we can make an expansion of coeffi-","rect":[53.81582260131836,199.5045166015625,385.1291440567173,190.5697479248047]},{"page":131,"text":"cient A in a Taylor series over temperature (for p, X ffi const close to Tc) and write","rect":[53.81321334838867,211.46405029296876,385.13001287652818,202.5294952392578]},{"page":131,"text":"dA\u0001","rect":[259.2082214355469,233.59130859375,273.516412882218,225.55020141601563]},{"page":131,"text":"A ¼ aðT \u0002 TcÞ with a ¼ dT\u0001\u0001\u0001Tc>0","rect":[144.04901123046876,251.05776977539063,294.9432000260688,231.44482421875]},{"page":131,"text":"as sketched in Fig. 6.4b.","rect":[53.81339645385742,274.6063232421875,151.80563016440159,265.6717529296875]},{"page":131,"text":"Now the excess of the free energy density acquired by the low symmetry phase","rect":[65.76541900634766,286.5658874511719,385.09848821832505,277.63134765625]},{"page":131,"text":"at the transition is","rect":[53.81339645385742,296.4635925292969,126.15065397368625,289.59088134765627]},{"page":131,"text":"Dg ¼ gðT;ZÞ \u0002 gðT;0Þ ¼ 12aðT \u0002 TcÞZ2 \u0002 31BZ3 þ 41CZ4 þ \u0003 \u0003 \u0003","rect":[79.30313110351563,333.1760559082031,340.57679688614749,312.72021484375]},{"page":131,"text":"(6.2)","rect":[366.0973815917969,327.941650390625,385.16964088288918,319.405517578125]},{"page":131,"text":"Here the minus sign at the B-term is taken for convenience. The thermodynamic","rect":[65.76595306396485,356.7373962402344,385.1815875835594,347.8028564453125]},{"page":131,"text":"stability condition reads:","rect":[53.813961029052737,368.6401672363281,153.39687974521707,359.70562744140627]},{"page":131,"text":"dDg ¼ aðT \u0002 TcÞZ \u0002 BZ2 þ CZ3 þ \u0003 \u0003 \u0003 ¼0","rect":[131.416748046875,399.2462463378906,309.2184990983344,382.8993225097656]},{"page":131,"text":"dZ","rect":[133.90928649902345,405.6759338378906,145.8065669220283,396.8111267089844]},{"page":131,"text":"(6.3)","rect":[366.09783935546877,398.5091552734375,385.17012916413918,389.9730224609375]},{"page":131,"text":"This equation has three roots: Z ¼ 0 for the high-symmetry phase and","rect":[65.7663803100586,429.11871337890627,349.0064629655219,420.18414306640627]},{"page":131,"text":"B \u0005 ½B2 \u0002 4aCðT \u0002 TcÞ\u00061=2","rect":[174.41041564941407,456.7206115722656,282.2913062342103,443.4634704589844]},{"page":131,"text":"Z¼","rect":[154.52908325195313,463.0370788574219,171.59656551573188,456.26397705078127]},{"page":131,"text":"2C","rect":[222.7854461669922,468.0829162597656,234.40001174624707,461.0408630371094]},{"page":131,"text":"(6.4)","rect":[366.0971984863281,462.7281188964844,385.1694577774204,454.1920166015625]},{"page":131,"text":"for the low symmetry phase. Therefore, the correct temperature dependence of","rect":[53.8137321472168,491.5806579589844,385.14766822663918,482.6461181640625]},{"page":131,"text":"the order parameter is found using only symmetry arguments! Note that coefficients","rect":[53.8137321472168,503.54022216796877,385.1765481387253,494.60565185546877]},{"page":131,"text":"a, B and C are independent of temperature, although their physical sense (molecular","rect":[53.8137321472168,515.499755859375,385.1077817520298,506.5552673339844]},{"page":131,"text":"nature) is unknown. They may only be found experimentally or using some","rect":[53.81269454956055,527.4592895507813,385.14356268121568,518.5247802734375]},{"page":131,"text":"microscopic molecular models.","rect":[53.81269454956055,539.4188232421875,179.7882046272922,530.4843139648438]},{"page":131,"text":"Consider a particular case of B ¼ 0. Then in the low symmetry phase,","rect":[65.76471710205078,551.3783569335938,349.72051663901098,542.44384765625]},{"page":131,"text":"Z ¼ \u0005\u0003aðTcC\u0002 TÞ\u00041=2","rect":[175.0888214111328,591.9957275390625,263.4284827478822,565.6672973632813]},{"page":131,"text":"(6.5)","rect":[366.09765625,584.3088989257813,385.1699155410923,575.7130126953125]},{"page":132,"text":"6.2 Isotropic Liquid–Nematic Transition","rect":[53.812843322753909,44.274620056152347,192.2367910781376,36.68026351928711]},{"page":132,"text":"115","rect":[372.4981994628906,42.55594253540039,385.1898245254032,36.62946701049805]},{"page":132,"text":"Thus, if the cubic term is absent in the expansion, the system becomes insensitive to","rect":[65.76496887207031,68.2883529663086,385.2534112077094,59.35380554199219]},{"page":132,"text":"the sign of the order parameter. Moreover, the parameter must change continuously at","rect":[53.812950134277347,80.24788665771485,385.23444993564677,71.31333923339844]},{"page":132,"text":"the phase transitionfromzerotoa finitevalue.Sucha Z ¼ \u00051symmetrycorresponds","rect":[53.812950134277347,92.20748138427735,385.16672147856908,83.27293395996094]},{"page":132,"text":"tosecond order transition (a caseoftheN-SmA orSmA-SmC transitions).Atsecond","rect":[53.81294250488281,103.7117691040039,385.2822198014594,95.15572357177735]},{"page":132,"text":"order transitions the symmetry changes abruptly but thermodynamic functions change","rect":[53.81392288208008,116.0697250366211,385.26828802301255,107.13517761230469]},{"page":132,"text":"continuously (only their temperature derivatives may change stepwise).","rect":[53.81392288208008,128.02932739257813,333.8040432503391,119.09477233886719]},{"page":132,"text":"When B 6¼ 0 the two non-zero roots are different, there is no longer Z ¼ \u00051","rect":[65.76593780517578,139.98886108398438,385.17864314130318,130.72561645507813]},{"page":132,"text":"symmetry; the order parameter and other thermodynamic functions change discon-","rect":[53.81389236450195,151.94839477539063,385.1109555801548,143.0138397216797]},{"page":132,"text":"tinuously. This situation corresponds to first order transition (a case of the Iso-N","rect":[53.81389236450195,163.907958984375,385.1148962970051,154.93356323242188]},{"page":132,"text":"transition). There is, however, a possibility to discuss the first order transition even","rect":[53.81288528442383,175.86749267578126,385.1179131608344,166.9329376220703]},{"page":132,"text":"for B ¼ 0 when the order parameter is symmetric: to this effect we should put","rect":[53.81288528442383,187.8270263671875,385.1318498379905,178.89247131347657]},{"page":132,"text":"C < 0,ignore the fifth order term (D ¼ 0) and add a sixth order term. Then we have","rect":[53.81288528442383,199.72976684570313,385.11890447809068,190.78524780273438]},{"page":132,"text":"the Landau expansion of the following type:","rect":[53.813899993896487,211.6893310546875,232.75578172275614,202.75477600097657]},{"page":132,"text":"Dg ¼ 12aðT \u0002 TcÞZ2 \u0002 41CZ4 þ 14EZ6 þ \u0003 \u0003 \u0003","rect":[130.3408660888672,246.2215576171875,308.64186036270999,225.88525390625]},{"page":132,"text":"(6.6)","rect":[366.0975646972656,241.16322326660157,385.1698239883579,232.62709045410157]},{"page":132,"text":"This biquadratic equationalso describes discontinuity of thermodynamic proper-","rect":[65.76612091064453,276.7599792480469,385.1858151992954,267.825439453125]},{"page":132,"text":"ties at temperature Tc. We shall discuss such a case later.","rect":[53.8140983581543,288.71954345703127,284.0351223519016,279.78497314453127]},{"page":132,"text":"6.2 Isotropic Liquid–Nematic Transition","rect":[53.812843322753909,331.9153747558594,269.9892880330002,321.1819763183594]},{"page":132,"text":"6.2.1 Landau-De Gennes Equation","rect":[53.812843322753909,361.70233154296877,235.64849823807837,351.1482238769531]},{"page":132,"text":"What is known from experiments on this transition?","rect":[53.812843322753909,389.32855224609377,263.7610553569969,380.39398193359377]},{"page":132,"text":"1.","rect":[53.812843322753909,406.0,61.27850003744845,398.36468505859377]},{"page":132,"text":"2.","rect":[53.81456756591797,442.0,61.280224280612518,434.2435302734375]},{"page":132,"text":"3.","rect":[53.814720153808597,477.0,61.28037686850314,470.1225280761719]},{"page":132,"text":"There is only a small jump of density at transition temperature TNI, about 0.3%.","rect":[66.27452087402344,407.2394714355469,385.1285366585422,398.304931640625]},{"page":132,"text":"Therefore, the density can approximately be considered constant at both sides of","rect":[66.27623748779297,419.19921875,385.1484616836704,410.2646484375]},{"page":132,"text":"the transition; the pressure is also considered constant.","rect":[66.27623748779297,431.1587829589844,286.1100124886203,422.2242431640625]},{"page":132,"text":"The order parameter is not symmetric, its magnitudes Smax ¼ þ1, Smin ¼ \u00021/2.","rect":[66.2762451171875,443.1183166503906,385.1325954964328,434.18377685546877]},{"page":132,"text":"This asymmetry generates the cubic term, coefficient B must be finite, and the","rect":[66.27635955810547,455.0782165527344,385.17542303277818,446.1436767578125]},{"page":132,"text":"first order transition is expected.","rect":[66.2763900756836,467.03778076171877,196.54314847494846,458.10321044921877]},{"page":132,"text":"The tensor form of the order parameter should be taken into account, in the","rect":[66.27639770507813,478.997314453125,385.17246282770005,470.062744140625]},{"page":132,"text":"simplest case, the uniaxial one [3, 4]:","rect":[66.2763900756836,490.9568786621094,217.00305039951395,482.0223388671875]},{"page":132,"text":"Qab ¼ Sðnab \u0002 1=3dabÞ","rect":[173.788818359375,518.012451171875,265.19308866606908,505.15155029296877]},{"page":132,"text":"(6.7)","rect":[366.096923828125,516.6320190429688,385.1691831192173,508.09588623046877]},{"page":132,"text":"With that order parameter, the Landau-de Gennes expansion of free energy","rect":[65.7654800415039,542.5361938476563,385.1672295670844,533.6016845703125]},{"page":132,"text":"reads:","rect":[53.81345748901367,552.4737548828125,77.63387925693582,545.5612182617188]},{"page":132,"text":"gN ¼ gIso þ 12AQabQba \u0002 13BQabQbgQga þ 41CðQabQbaÞ2 A ¼ aðT \u0002 Tc\u0007Þ (6.8)","rect":[59.30818557739258,589.1465454101563,385.1694577774204,568.6907348632813]},{"page":133,"text":"116","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.68026351928711]},{"page":133,"text":"6 Phase Transitions","rect":[318.2889709472656,42.55594253540039,385.1422470378808,36.68026351928711]},{"page":133,"text":"Here Tc* is virtual second order transition temperature. In real substances it is","rect":[65.76496887207031,68.2883529663086,385.1834756289597,59.294044494628909]},{"page":133,"text":"slightly below TNI. Coefficients B and C are independent of T. Now we choose a","rect":[53.8127555847168,80.24800872802735,385.15918768121568,71.30349731445313]},{"page":133,"text":"proper coordinate system wherein matrices Qab in Eq. 6.8 become diagonal. Then","rect":[53.813350677490237,93.07765197753906,385.13893977216255,83.19337463378906]},{"page":133,"text":"we contract indices (reduce tensor valence) by multiplying the diagonal elements","rect":[53.81306076049805,104.1104965209961,385.1777688418503,95.17594909667969]},{"page":133,"text":"and writing the traces QabQba ¼ (2/3)S2 and QabQbgQga ¼ (2/9)S3. Then, we have","rect":[53.81306076049805,116.99702453613281,385.1196979351219,105.01939392089844]},{"page":133,"text":"equations for the excess of the normalized free energy density (in units [erg/cm3] or","rect":[53.81374740600586,128.02999877929688,385.1522153457798,116.97911071777344]},{"page":133,"text":"[J/m3] in the SI system) [5, 6]:","rect":[53.81438446044922,139.98959350585938,177.2372728116233,128.93870544433595]},{"page":133,"text":"Dg ¼ gN \u0002 gIso ¼ 1aðT \u0002 Tc\u0007ÞS2 \u0002 2 BS3 þ 1CS4","rect":[115.83969116210938,170.19912719726563,322.67942879280408,154.18389892578126]},{"page":133,"text":"(6.9)","rect":[366.09765625,169.46205139160157,385.1699155410923,160.92591857910157]},{"page":133,"text":"and stability equation:","rect":[53.81417465209961,195.31048583984376,143.4389024502952,186.3759307861328]},{"page":133,"text":"ddDSg ¼ aðT \u0002 Tc\u0007ÞS \u0002 13BS2 þ 32CS3 ¼0","rect":[137.59103393554688,230.49649047851563,303.0441826920844,209.51275634765626]},{"page":133,"text":"(6.10)","rect":[361.05633544921877,225.1792449951172,385.1057065567173,216.6431121826172]},{"page":133,"text":"6.2.2 Temperature Dependence of the Nematic","rect":[53.812843322753909,287.1105651855469,293.966535410597,276.55645751953127]},{"page":133,"text":"Order Parameter","rect":[89.66943359375,298.8431701660156,174.05512122843428,290.5002746582031]},{"page":133,"text":"Equation 6.10 has three solutions: S ¼ 0 for the isotropic phase and","rect":[53.812843322753909,328.7367858886719,328.38579646161568,319.80224609375]},{"page":133,"text":"S\u0005 ¼ 4BC(1 \u0005 \u00031 \u0002 24aCðBT2\u0002 Tc\u0007Þ\u00041=2)","rect":[134.7584686279297,372.7791748046875,304.23699965333779,342.91766357421877]},{"page":133,"text":"(6.11)","rect":[361.0556335449219,362.1204528808594,385.1050046524204,353.5843505859375]},{"page":133,"text":"for the nematic phase. First of all, we should establish which sign is correct in","rect":[53.813961029052737,391.369384765625,385.14287653974068,382.434814453125]},{"page":133,"text":"solutions (6.11). It is a bit tricky. We define a certain temperature Tc (not necessar-","rect":[53.813961029052737,403.3289489746094,385.10530982820168,394.3944091796875]},{"page":133,"text":"ily equal to Tc*), at which free energy of the nematic and the isotropic phases are","rect":[53.81426239013672,415.2317199707031,385.16452825738755,406.296875]},{"page":133,"text":"equal, i.e. Dg in Eq. 6.9 is zero. Assuming T ¼ Tc, we multiplied Eq. 6.10 by S/3","rect":[53.81272506713867,427.2310791015625,385.14333430341255,417.9280090332031]},{"page":133,"text":"and subtract it from Eq. 6.9. Then we get","rect":[53.81344223022461,439.1509094238281,220.28068406650614,430.21636962890627]},{"page":133,"text":"217BSc3 ¼ 91CSc4 or Sc3ðCSc \u0002 B=3Þ ¼0","rect":[142.57571411132813,473.880859375,298.05983820966255,453.3453369140625]},{"page":133,"text":"(6.12)","rect":[361.05706787109377,468.6233215332031,385.1064389785923,460.08721923828127]},{"page":133,"text":"As seen from Eq. 6.11, only negative sign in front of brackets can give us Sc ¼0","rect":[65.7663803100586,492.4876403808594,385.1796807389594,483.5531005859375]},{"page":133,"text":"at Tc ¼ Tc* that is at the same characteristic temperature. Another solution of","rect":[53.81493377685547,504.4472961425781,385.1470883926548,495.51275634765627]},{"page":133,"text":"Eq. 6.12, namely, Sc ¼ B/3C, if substituted into Eq. 6.11, results in branch S+","rect":[53.81324005126953,516.4069213867188,367.40438629630736,507.4624328613281]},{"page":133,"text":"13 ¼ \u00031 \u0002 24aCðBT2\u0002 Tc\u0007Þ\u00041=2","rect":[161.72189331054688,557.0237426757813,278.4394690760072,530.6952514648438]},{"page":133,"text":"with a new characteristic temperature Tc ¼ Tc* þ B2/27aC. This solution showsa","rect":[53.812843322753909,580.512939453125,385.16138494684068,569.4619750976563]},{"page":133,"text":"positive jump of S ¼ B/3C at a temperature Tc that is higher than the “second order","rect":[53.814552307128909,592.512451171875,385.14827857820168,583.5181274414063]},{"page":134,"text":"6.2 Isotropic Liquid–Nematic Transition","rect":[53.813236236572269,44.275901794433597,192.23718780665323,36.68154525756836]},{"page":134,"text":"117","rect":[372.49859619140627,42.55722427368164,385.1902212539188,36.73234558105469]},{"page":134,"text":"transition” temperature Tc*. Hence S+ from (6.11) is a more stable solution than S\u0002.","rect":[53.812843322753909,68.2883529663086,385.1832241585422,59.294044494628909]},{"page":134,"text":"Therefore, we take sign (þ) in solution (6.11) of stability equation:","rect":[53.814537048339847,80.24788665771485,324.98615892002177,71.31333923339844]},{"page":134,"text":"S ¼ 4BC(1 þ \u00031 \u0002 24aCðBT2\u0002 Tc\u0007Þ\u00041=2)","rect":[137.7056427001953,124.29035949707031,301.2914735302909,94.42884826660156]},{"page":134,"text":"(6.13)","rect":[361.0555725097656,113.63162994384766,385.10494361726418,105.09550476074219]},{"page":134,"text":"Finally, from (6.13) we find one more critical temperature: Tc+ ¼ Tc* þ B2/","rect":[65.76592254638672,147.86880493164063,385.151991439553,136.81736755371095]},{"page":134,"text":"24aC that is even higher than Tc and there is no other real solutions of the stability","rect":[53.814144134521487,159.8280029296875,385.11895075849068,150.88323974609376]},{"page":134,"text":"equation. Totally, we have now three characteristic temperatures:","rect":[53.812984466552737,171.73077392578126,317.48713548252177,162.7962188720703]},{"page":134,"text":"Tc\u0007 ðvirtualsecondordertransitionÞ<Tc ¼ Tc\u0007 þ B2\u000527aCðDg = 0Þ<Tcþ ¼","rect":[64.85917663574219,198.24119567871095,374.15894344541939,185.96136474609376]},{"page":134,"text":"Tc\u0007 þ B2\u000524aCðjumpof SÞ:","rect":[56.58975601196289,213.6017303466797,167.8645776478662,201.66908264160157]},{"page":134,"text":"Within the range of Tc* < Tc+ a hysteresis in the order parameter should be","rect":[65.7652359008789,233.73983764648438,385.15149725152818,223.77793884277345]},{"page":134,"text":"observed upon the heating and cooling scans. Such hysteresis is often observed","rect":[53.81368637084961,245.69927978515626,385.11077204755318,236.7647247314453]},{"page":134,"text":"under polarization microscope in the form of two-phase textures. If a sample is","rect":[53.81368637084961,257.6588439941406,385.1864053164597,248.7242889404297]},{"page":134,"text":"placed between crossed polarizers, dark spots of the isotropic phase sharply con-","rect":[53.81368637084961,269.6183776855469,385.1157163223423,260.683837890625]},{"page":134,"text":"trasts with bright nematic background (like in Fig. 1.3c) or vice versa. The temper-","rect":[53.81368637084961,281.5779113769531,385.1057370742954,272.64337158203127]},{"page":134,"text":"ature behaviour of function S(T) is shown in Fig. 6.5a. In the figure:","rect":[53.81368637084961,293.4806823730469,328.32490403720927,284.4863586425781]},{"page":134,"text":"Sþ ¼ B=4CðatTcþÞ; Sc ¼ B=3CðatT ¼ TcÞ; S\u0007 ¼ B=2CðatT ¼ Tc\u0007Þ (6.14)","rect":[62.9904899597168,318.23956298828127,385.1053403457798,307.793212890625]},{"page":134,"text":"Hence, the universal ratio Sc/S* ¼ 2/3 is valid for any expansion up to the fourth","rect":[65.76630401611328,341.7734069824219,385.1290215592719,332.7790832519531]},{"page":134,"text":"order term and Tc* can be found by plotting S vs temperature. By the way, in","rect":[53.8140754699707,353.7330322265625,385.14199152997505,344.79840087890627]},{"page":134,"text":"experiment, the phase transition point TNI is associated with temperature Tc.","rect":[53.813114166259769,365.635986328125,360.4293179085422,356.70123291015627]},{"page":134,"text":"a","rect":[75.3380126953125,423.95977783203127,80.89160852760482,418.3726806640625]},{"page":134,"text":"S","rect":[84.48084259033203,447.00811767578127,89.8106830825127,440.9712829589844]},{"page":134,"text":"B/2C","rect":[79.64722442626953,459.4239807128906,97.41069763800782,453.38714599609377]},{"page":134,"text":"B/3C","rect":[79.64722442626953,485.01214599609377,97.41069763800782,478.9753112792969]},{"page":134,"text":"B/4C","rect":[79.64722442626953,495.4290466308594,97.41069763800782,489.3922119140625]},{"page":134,"text":"*","rect":[112.63397216796875,515.3030395507813,115.7423798912945,513.0082397460938]},{"page":134,"text":"T","rect":[106.14866638183594,520.674560546875,111.03102401410338,514.93359375]},{"page":134,"text":"c","rect":[111.03092193603516,522.7583618164063,114.02889516783698,519.446533203125]},{"page":134,"text":"T","rect":[163.1508026123047,520.6588134765625,168.03316024457213,514.9178466796875]},{"page":134,"text":"c","rect":[168.03289794921876,522.7423095703125,171.0308711810206,519.4304809570313]},{"page":134,"text":"TNI","rect":[186.18408203125,483.3817138671875,197.0590244196115,475.64166259765627]},{"page":134,"text":"T+ T","rect":[179.3848114013672,520.6997680664063,203.57281783734556,512.8574829101563]},{"page":134,"text":"c","rect":[184.26669311523438,522.7403564453125,187.2646663470362,519.4285278320313]},{"page":134,"text":"b","rect":[211.31369018554688,423.95928955078127,217.41665251204078,416.6531066894531]},{"page":134,"text":"S","rect":[217.35958862304688,470.0687255859375,222.68942911522755,464.0318908691406]},{"page":134,"text":"0.6","rect":[226.66366577148438,438.6412048339844,237.77082269103014,432.8762512207031]},{"page":134,"text":"0.4","rect":[226.66366577148438,462.8197021484375,237.77082269103014,457.05474853515627]},{"page":134,"text":"T","rect":[348.39044189453127,448.23712158203127,353.2743869541914,442.4942626953125]},{"page":134,"text":"c","rect":[354.7660827636719,452.23626708984377,357.7640559954737,448.9244384765625]},{"page":134,"text":"T+","rect":[333.41619873046877,491.38250732421877,344.6251946141508,484.5693664550781]},{"page":134,"text":"c","rect":[338.2989501953125,493.4665832519531,341.29692342711436,490.1547546386719]},{"page":134,"text":"35","rect":[345.5002136230469,521.2686157226563,354.38595025450669,515.503662109375]},{"page":134,"text":"Fig. 6.5 (a) Temperature dependence of the order parameter in the Landau-de Gennes model; (B)","rect":[53.812843322753909,554.1295776367188,385.1711739884107,546.3997802734375]},{"page":134,"text":"and (C) are coefficients of the expansion. TNI \b Tc is experimental value of the isotropic–nematic","rect":[53.81200408935547,564.0377807617188,385.14537951731207,556.3672485351563]},{"page":134,"text":"phase transition temperature corresponding to equality of free energy densities for the two phases.","rect":[53.813228607177737,574.0136108398438,385.1477992255922,566.4192504882813]},{"page":134,"text":"(b) Experimental dependence of the order parameter for 5CB and the characteristic temperature","rect":[53.813228607177737,583.9895629882813,385.1605467536402,576.3444213867188]},{"page":134,"text":"points Tc*, Tc and Tc+ defined in accordance with the model of panel (a)","rect":[53.813228607177737,593.9088134765625,302.2158517227857,585.4974365234375]},{"page":135,"text":"118","rect":[53.8111457824707,42.55734634399414,66.50275177149698,36.73246765136719]},{"page":135,"text":"6.2.3 Free Energy","rect":[53.812843322753909,69.93675231933594,152.09181006880014,59.370697021484378]},{"page":135,"text":"6 Phase Transitions","rect":[318.28729248046877,42.55734634399414,385.1405685710839,36.68166732788086]},{"page":135,"text":"It is instructive to look at the free energy density dependence on the nematic order","rect":[53.812843322753909,97.47896575927735,385.14470802156105,88.54441833496094]},{"page":135,"text":"parameter, Eq. 6.9, on both sides of the phase transition. To this effect we need","rect":[53.812843322753909,109.43856048583985,385.11586848310005,100.50401306152344]},{"page":135,"text":"Landau expansion coefficients a, B and C. We mayfindthem,atleast,approximately","rect":[53.81184387207031,121.3980941772461,385.17949763349068,112.45358276367188]},{"page":135,"text":"from the experimental dependence of order parameter on temperature S(T). Let us take","rect":[53.81086730957031,133.35763549804688,385.14466131402818,124.42308044433594]},{"page":135,"text":"as an example a nematic compound p-pentyl-cyano-biphenyl (5CB). Function S(T)","rect":[53.810848236083987,145.31716918945313,385.1566403946079,136.32284545898438]},{"page":135,"text":"has been measured earlier by both the optical and nuclear magnetic resonance","rect":[53.81086730957031,157.27670288085938,385.13294256402818,148.34214782714845]},{"page":135,"text":"techniques [7]. The Landau coefficients may be found from this curve as follows.","rect":[53.81086730957031,169.23623657226563,385.18157620932348,160.3016815185547]},{"page":135,"text":"Using the values of the order parameter S*, Sc and S+ expressed in terms of coefficients","rect":[53.81086730957031,181.13992309570313,385.1617776309128,171.17784118652345]},{"page":135,"text":"B and C, we can mark the corresponding temperatures T*, Tc and T+ on the experi-","rect":[53.81399154663086,193.09957885742188,385.1601804336704,183.1375274658203]},{"page":135,"text":"mentalplot, as shown in Fig. 6.5b. Thehighest temperature point of the nematic phase","rect":[53.81438446044922,205.05911254882813,385.16611517145005,196.06478881835938]},{"page":135,"text":"is T+ ¼ 35.3 C where, according to the model, S+ ¼ B/4C. In the experimental plot","rect":[53.81438446044922,217.01885986328126,385.1258378750999,207.05674743652345]},{"page":135,"text":"S ¼ 0.3 and, assuming B ¼ 1, we find C ¼ 0.838. Then, using the difference","rect":[53.81388473510742,228.97842407226563,385.16761053277818,220.03390502929688]},{"page":135,"text":"between the two characteristic temperatures from the experimental plot (b),","rect":[53.816856384277347,240.93795776367188,348.7630886604953,232.00340270996095]},{"page":135,"text":"Tc \u0002 T\u0007 ¼ B2=27aC ¼ 33:7oC \u0002 22:7oC ¼ 11oC,","rect":[119.80829620361328,266.43310546875,319.1936306526828,255.1685791015625]},{"page":135,"text":"we find a ¼ 0.004. Now we have all necessary data to plot the normalized free","rect":[53.814353942871097,290.02349853515627,385.1362994976219,281.08892822265627]},{"page":135,"text":"energy Dg(S). If necessary, the absolute values of free energy may be found by","rect":[53.81432342529297,302.02288818359377,385.17009821942818,292.7198181152344]},{"page":135,"text":"multiplying the dimensionless Landau coefficient B by the free energy density of","rect":[53.814292907714847,313.9425964355469,385.15117774812355,305.008056640625]},{"page":135,"text":"the isotropic phase giso ¼ (rNINAv/M) kB TNI. (where kB is Boltzmann constant, NAv","rect":[53.814292907714847,325.94195556640627,385.1776582863347,316.9482116699219]},{"page":135,"text":"Avogadro number, M molecular weight, rNI the density at the transition tempera-","rect":[53.812843322753909,337.8624267578125,385.0828794082798,328.9278564453125]},{"page":135,"text":"ture). In our example, M ¼ 249, rNI \b 1 g/cm3, TNI ¼ Tc \b 307 K and giso \b1","rect":[53.813716888427737,349.8052062988281,374.8135613541938,338.71435546875]},{"page":135,"text":"108 erg/cm3 or 1 107 J/m3 in the SI system.","rect":[53.81417465209961,361.7250671386719,238.9813198616672,350.67401123046877]},{"page":135,"text":"The result of Dg(S) calculation is shown in Fig. 6.6. Considering a cooling","rect":[65.76510620117188,373.72442626953127,385.1160515885688,364.4213562011719]},{"page":135,"text":"process from the stable isotropic phase we shall better understand the physical","rect":[53.8130989074707,385.6441345214844,385.15595872470927,376.7095947265625]},{"page":135,"text":"sense of the three critical temperatures. For T > Tc+ (dot curve 3 in the figure) the","rect":[53.8130989074707,397.6039733886719,385.1734088726219,387.6421813964844]},{"page":135,"text":"absolute minimum is situated at S ¼ 0 and this corresponds to the stable isotropic","rect":[53.81370162963867,409.5635070800781,385.17346990777818,400.62896728515627]},{"page":135,"text":"phase. As the temperature approaches Tc from above, in the range of Tc < T < Tc+,","rect":[53.81370162963867,421.5232849121094,385.1832241585422,411.6079406738281]},{"page":135,"text":"a second minimum appears above the abscissa axis, which corresponds to the","rect":[53.814537048339847,433.4828186035156,385.1653522319969,424.54827880859377]},{"page":135,"text":"Fig. 6.6 Normalized free","rect":[53.812843322753909,480.7279052734375,141.07692858957769,472.9980773925781]},{"page":135,"text":"energy of 5CB as a function of","rect":[53.812843322753909,490.57940673828127,155.40237516028575,482.9342346191406]},{"page":135,"text":"order parameter S in the","rect":[53.812843322753909,500.5553894042969,133.838479492898,492.9610290527344]},{"page":135,"text":"vicinityoftheN–I transition at","rect":[53.81199645996094,510.5313415527344,155.41930870261255,502.9369812011719]},{"page":135,"text":"different temperatures: Tc*","rect":[53.81199645996094,520.5072631835938,143.2944540420048,512.8786010742188]},{"page":135,"text":"(curve 1), Tc (curve 2) and Tc+","rect":[53.81271743774414,530.1073608398438,154.92212209813887,521.975341796875]},{"page":135,"text":"(curve 3). The curves are","rect":[53.812843322753909,540.0631103515625,137.18314502024175,532.807373046875]},{"page":135,"text":"calculated with dimensionless","rect":[53.812843322753909,548.6505737304688,153.53671725272455,542.7833251953125]},{"page":135,"text":"Landau expansion coefficients","rect":[53.812843322753909,560.2969360351563,154.8963974285058,552.7025756835938]},{"page":135,"text":"a ¼ 0.004, B ¼ 1 and","rect":[53.812843322753909,568.5542602539063,128.50970977930948,562.6785278320313]},{"page":135,"text":"C ¼ 0.838 obtained from the","rect":[53.81284713745117,578.6317749023438,151.45357653879644,572.6460571289063]},{"page":135,"text":"experimental curve S(T)","rect":[53.8120002746582,590.2247924804688,134.17610257727794,582.6304321289063]},{"page":135,"text":"Δg","rect":[191.68722534179688,510.25860595703127,200.99800439857104,503.6310119628906]},{"page":135,"text":"0.004","rect":[210.54393005371095,462.1593017578125,230.1481201056704,456.3958435058594]},{"page":135,"text":"0.002","rect":[210.54393005371095,485.4169616699219,230.1481201056704,479.65350341796877]},{"page":135,"text":"0.000","rect":[210.54393005371095,508.72418212890627,230.1481201056704,502.9607238769531]},{"page":135,"text":"–0.4","rect":[227.1302947998047,575.3369140625,242.6203185919985,569.573486328125]},{"page":135,"text":"–0.2","rect":[247.85609436035157,575.3369140625,263.28221434395166,569.573486328125]},{"page":135,"text":"0.0","rect":[270.711669921875,575.3369140625,281.81590575020166,569.573486328125]},{"page":135,"text":"0.2 0.4","rect":[291.4334716796875,575.3369140625,323.2043945197329,569.573486328125]},{"page":135,"text":"S","rect":[304.6354675292969,586.7305908203125,309.96864004653539,580.68994140625]},{"page":135,"text":"0.6","rect":[332.8211669921875,575.3369140625,343.86949461738916,569.573486328125]},{"page":135,"text":"0.8","rect":[353.48785400390627,575.3369140625,364.5920898322329,569.573486328125]},{"page":135,"text":"1.0","rect":[374.2088317871094,575.3369140625,385.25636595527979,569.573486328125]},{"page":136,"text":"6.2 Isotropic Liquid–Nematic Transition","rect":[53.812843322753909,44.274986267089847,192.2367910781376,36.68062973022461]},{"page":136,"text":"Stable","rect":[102.60941314697266,72.91129302978516,123.75932156141394,67.0566177368164]},{"page":136,"text":"Nematic","rect":[102.60941314697266,82.49319458007813,131.6549667030155,76.63052368164063]},{"page":136,"text":"phase","rect":[102.60941314697266,93.66661834716797,122.00881802137488,86.2522964477539]},{"page":136,"text":"Overheated","rect":[225.33164978027345,72.90335845947266,264.55410235723039,67.05667877197266]},{"page":136,"text":"Nematic","rect":[225.33164978027345,82.49319458007813,254.37721096571083,76.63052368164063]},{"page":136,"text":"(metastable)","rect":[225.33164978027345,93.66661834716797,267.2078354363867,86.2522964477539]},{"page":136,"text":"Stable","rect":[298.08221435546877,72.9104995727539,319.23211514051556,67.05582427978516]},{"page":136,"text":"Isotropic","rect":[298.08221435546877,84.06804656982422,328.92462002332806,76.62973022460938]},{"page":136,"text":"Phase","rect":[298.08221435546877,92.09017944335938,317.89727627332806,86.25150299072266]},{"page":136,"text":"119","rect":[372.4981994628906,42.62403869628906,385.1898245254032,36.73143005371094]},{"page":136,"text":"T*","rect":[135.31985473632813,128.182861328125,146.71888694325669,122.46015167236328]},{"page":136,"text":"c","rect":[140.6512451171875,130.2604217529297,143.2770321321371,127.33303833007813]},{"page":136,"text":"T+","rect":[276.6002502441406,127.22320556640625,290.304594697781,120.52615356445313]},{"page":136,"text":"c","rect":[281.9317932128906,129.30076599121095,284.5575802278402,126.37338256835938]},{"page":136,"text":"Fig. 6.7 Sequence of phase states observed during the","rect":[53.812843322753909,151.80926513671876,254.64394519114019,144.07945251464845]},{"page":136,"text":"manifests a temperature hysteresis near the N–I transition","rect":[53.812843322753909,161.71749877929688,250.9921621718876,154.12313842773438]},{"page":136,"text":"up-and-down","rect":[258.9244079589844,151.74154663085938,304.03037017970009,144.14718627929688]},{"page":136,"text":"temperature","rect":[308.2626037597656,151.74154663085938,348.8503660895777,145.01075744628907]},{"page":136,"text":"scan","rect":[353.1257629394531,150.0,368.1780142226688,146.0]},{"page":136,"text":"that","rect":[372.3848571777344,150.0,385.15258506980009,144.14718627929688]},{"page":136,"text":"overheated (metastable) nematic phase with at S > 0.At T ¼ Tc the twominima have","rect":[53.812843322753909,185.67434692382813,385.16077459527818,176.7397918701172]},{"page":136,"text":"the same zero free energy density (solid line 2), but between them there is a barrier","rect":[53.813961029052737,197.6339111328125,385.1969235977329,188.69935607910157]},{"page":136,"text":"shown by the arrow. The right minimum for S ¼6 0 corresponds to the stable nematic","rect":[53.813961029052737,209.59344482421876,385.1726459331688,200.3302001953125]},{"page":136,"text":"stateandtheleftonewithS ¼ 0representstheovercooled(metastable)isotropicstate.","rect":[53.813961029052737,221.55300903320313,385.1497768929172,212.6184539794922]},{"page":136,"text":"Between Tc and Tc* the two minima coexist. Finally, for T < Tc* the left minimum","rect":[53.813961029052737,233.512939453125,385.1436533796843,224.51861572265626]},{"page":136,"text":"disappears, the metastable isotropic phase becomes unstable and the nematic state","rect":[53.81379318237305,245.41571044921876,385.16049993707505,236.4811553955078]},{"page":136,"text":"becomes absolutely stable (dash curve 1 with a deep minimum). Fig. 6.7illustrates the","rect":[53.81379318237305,257.3752746582031,385.15760076715318,248.4407196044922]},{"page":136,"text":"sequence of the intermediate phases in the proximity of the NI transition.","rect":[53.81379318237305,269.3348083496094,340.24810452963598,260.4002685546875]},{"page":136,"text":"6.2.4 Physical Properties in the Vicinity of the N–Iso","rect":[53.812843322753909,308.50628662109377,324.56016292476246,297.8684997558594]},{"page":136,"text":"Transition","rect":[89.66943359375,320.346435546875,141.4033657917893,311.884033203125]},{"page":136,"text":"Physical properties of substance close to N–I phase transition may be related to the","rect":[53.812843322753909,349.9920349121094,385.17063177301255,341.0574951171875]},{"page":136,"text":"parameters of Landau expansion [8]. For example we can calculate an entropy","rect":[53.812843322753909,361.9515686035156,385.1227959733344,353.01702880859377]},{"page":136,"text":"density change at the transition temperature Tc from Eq. 6.9 and Sc ¼ B/3C:","rect":[53.81285095214844,373.9114074707031,361.81465013095927,364.9668884277344]},{"page":136,"text":"DS ¼ \u0002@ðgN@\u0002TgIsoÞ\u0001\u0001\u0001\u0001Tc ¼ \u000231aSc2 ¼ \u00022a7BC22","rect":[127.4527359008789,414.1285400390625,309.36787483772596,388.141845703125]},{"page":136,"text":"Correspondingly, the latent heat of the N–Iso transition is","rect":[65.76496887207031,431.66912841796877,298.2933694277878,422.73455810546877]},{"page":136,"text":"aB2","rect":[242.7245330810547,454.330078125,257.3109290369447,445.8996276855469]},{"page":136,"text":"DH ¼ DSTc ¼ \u000227C2 Tc","rect":[167.61245727539063,468.0829162597656,270.8762602318801,453.9198303222656]},{"page":136,"text":"As we have seen above, Landau expansion coefficients a, B and C can be found","rect":[65.76496887207031,486.6495056152344,385.1288994889594,477.7049865722656]},{"page":136,"text":"from the measurements of order parameter S(T) and DH by different techniques,","rect":[53.81493377685547,498.6090393066406,385.1288723519016,489.3457946777344]},{"page":136,"text":"such as microscopy (for Tc), differential scanning calorimetry (for Tc, and DH),","rect":[53.81493377685547,510.5688171386719,385.15581937338598,501.3055725097656]},{"page":136,"text":"refractometry or NMR (for Sc).","rect":[53.814022064208987,522.5283203125,179.53400846030002,513.5938110351563]},{"page":136,"text":"Two other calculated temperature dependencies are shown in Fig. 6.8 with the","rect":[65.76534271240235,534.43115234375,385.17408025934068,525.4966430664063]},{"page":136,"text":"characteristic temperatures discussed above. The excess of the specific heat in the","rect":[53.8133430480957,546.3907470703125,385.17212713434068,537.4562377929688]},{"page":136,"text":"nematic phase Cp ¼ aS\u0006@2S\u0005@T2\u0007P is shown in Fig. 6.8a. The temperature depen-","rect":[53.8133430480957,560.0068969726563,385.1403745254673,547.7049560546875]},{"page":136,"text":"dence obtained from Eq. 6.13 follows a law Cp / (Tc+\u0002T)\u00021/2 with a step at Tc in","rect":[53.81345748901367,571.1671142578125,385.1442498307563,559.2196044921875]},{"page":136,"text":"agreement with experiment [8]. With decreasing temperature Cp achieves a plateau","rect":[53.81438446044922,583.1267700195313,385.1790703873969,573.325439453125]},{"page":136,"text":"equal to a2/2C. Another important characteristic is structural susceptibility (not to","rect":[53.814353942871097,594.2296142578125,385.1404656510688,583.1787719726563]},{"page":137,"text":"120","rect":[53.81321716308594,42.55600357055664,66.50482315211221,36.73112487792969]},{"page":137,"text":"Fig. 6.8 Temperature","rect":[53.812843322753909,67.58130645751953,130.0724653937769,59.6313591003418]},{"page":137,"text":"dependencies of heat capacity","rect":[53.812843322753909,77.4895248413086,155.28644317774698,69.89517211914063]},{"page":137,"text":"(a), and inverse value of","rect":[53.812843322753909,87.07006072998047,137.50128263098888,79.81436157226563]},{"page":137,"text":"nematic-like structural","rect":[53.812843322753909,95.63212585449219,130.78320030417505,89.79031372070313]},{"page":137,"text":"susceptibility w\u00021 (b) in the","rect":[53.812843322753909,107.36067962646485,148.66097399973394,97.92735290527344]},{"page":137,"text":"vicinity of N–I phase","rect":[53.81321716308594,117.3363265991211,126.39904162668705,109.74197387695313]},{"page":137,"text":"transition","rect":[53.81321716308594,125.50297546386719,85.87221283106729,119.66116333007813]},{"page":137,"text":"a","rect":[242.12603759765626,68.2304458618164,247.67929978575169,62.643707275390628]},{"page":137,"text":"C","rect":[251.1691131591797,104.31963348388672,256.93808517122548,98.28318786621094]},{"page":137,"text":"p","rect":[256.937744140625,107.40959930419922,260.2697129922677,102.94818878173828]},{"page":137,"text":"b","rect":[241.30804443359376,166.4362030029297,247.41064011151875,159.13046264648438]},{"page":137,"text":"–1","rect":[252.17050170898438,173.50198364257813,261.0556860097632,167.88128662109376]},{"page":137,"text":"z","rect":[247.03298950195313,180.38352966308595,250.980180878616,172.5161590576172]},{"page":137,"text":"6","rect":[318.28936767578127,42.55600357055664,322.5199026504032,36.68032455444336]},{"page":137,"text":"Phase Transitions","rect":[324.9168701171875,42.52213668823242,385.1426437663964,36.68032455444336]},{"page":137,"text":"a2/2C","rect":[337.34881591796877,137.77423095703126,358.3555503811864,130.41639709472657]},{"page":137,"text":"T","rect":[370.758544921875,147.40972900390626,375.6405974140633,141.6691131591797]},{"page":137,"text":"T*","rect":[274.7297668457031,237.95462036132813,282.6537151614578,232.21400451660157]},{"page":137,"text":"c","rect":[279.6117248535156,240.03717041015626,282.6080997201008,236.72708129882813]},{"page":137,"text":"Tc T+","rect":[317.51043701171877,239.81524658203126,339.5592158571724,231.99220275878907]},{"page":137,"text":"T","rect":[370.758544921875,224.35964965820313,375.6405974140633,218.61903381347657]},{"page":137,"text":"be confused with the electric wE or magnetic susceptibility wM) determining the","rect":[53.812843322753909,269.78875732421877,385.17429388238755,260.85418701171877]},{"page":137,"text":"development of order parameter fluctuations in the isotropic phase (short-range","rect":[53.813533782958987,281.748291015625,385.14243353082505,272.813720703125]},{"page":137,"text":"order [3]) near the transition to the nematic phase z\u00021 ¼ @2Dg\u0005@S2. This is a","rect":[53.813533782958987,294.99591064453127,385.16065252496568,282.48876953125]},{"page":137,"text":"steepness of the free energy (dot curve 3 in Fig. 6.6) close to its minimum.","rect":[53.813838958740237,305.6675720214844,385.1815456917453,296.7330322265625]},{"page":137,"text":"Above Tc the order parameter is small, S ! 0, and the terms with S3 and S4 in","rect":[53.81283187866211,317.62738037109377,385.1442498307563,306.5762939453125]},{"page":137,"text":"Eq. 6.9 may be disregarded. Then the inverse susceptibility z\u00021 ¼ 2=3aðT \u0002 TcÞ","rect":[53.81438446044922,330.569091796875,385.1676064883347,317.7080993652344]},{"page":137,"text":"follows the Curie law z / 1/T, see Fig. 6.8b.","rect":[53.81380844116211,341.48974609375,234.11045499350315,332.2364501953125]},{"page":137,"text":"This susceptibility can be studied in the isotropic phase by electro-optical","rect":[65.76583099365235,353.44927978515627,385.160536361428,344.51470947265627]},{"page":137,"text":"or magneto-optical techniques. Indeed, anisotropy of the electric and magnetic sus-","rect":[53.81380844116211,365.4088439941406,385.16362891999855,356.47430419921877]},{"page":137,"text":"ceptibilities is proportional to the order parameter S, see Eq. 3.15. For example, nematic","rect":[53.81380844116211,377.3683776855469,385.2203754253563,368.3740539550781]},{"page":137,"text":"liquid crystal acquires an additional free energy \u0002 (1/2)waH2 ¼ \u0002(1/2)wamaxSH2 in","rect":[53.81380844116211,389.32794189453127,385.17983332685005,378.2776184082031]},{"page":137,"text":"the magnetic field parallel to the director (wmax is anisotropy of magnetic susceptibility","rect":[53.814144134521487,401.2881774902344,385.1778496842719,392.353515625]},{"page":137,"text":"for the ideal nematic). Then, on account of Eq. 6.9, the energy in the isotropic phase","rect":[53.8140983581543,413.2477111816406,385.1648639507469,404.31317138671877]},{"page":137,"text":"with a short-range nematic order is given by","rect":[53.81411361694336,425.207275390625,225.67354670575629,416.272705078125]},{"page":137,"text":"DgðisoÞ ¼ \u000212wamaxH2S þ 31aðT \u0002 Tc\u0007ÞS2","rect":[136.5724334716797,456.8536071777344,301.8906714685853,436.39776611328127]},{"page":137,"text":"After minimization we obtain the contribution of the magnetic field to the order","rect":[65.76496887207031,477.4105529785156,385.1448911270298,468.47601318359377]},{"page":137,"text":"parameter","rect":[53.812950134277347,489.3701171875,93.88561378327978,481.4515380859375]},{"page":137,"text":"SH ¼ 4a3ðwTam\u0002axHTc2\u0007Þ","rect":[182.22723388671876,525.0209350585938,255.05363859282688,500.5965881347656]},{"page":137,"text":"that may be related to a change of the refraction index parallel to the field. The","rect":[53.813350677490237,545.5972900390625,385.1432575054344,536.6627807617188]},{"page":137,"text":"appearance of the birefringence dn(H) in the isotropic liquid induced by the","rect":[53.813350677490237,557.5568237304688,385.11634100152818,548.323486328125]},{"page":137,"text":"magnetic effect is called Cotton-Mouton effect. Its electric field analogue is Kerr","rect":[53.8133659362793,569.516357421875,385.1124204239048,560.5818481445313]},{"page":137,"text":"effect. In both cases, dn follows the Curie law and the Landau coefficient a may be","rect":[53.8133659362793,581.4191284179688,385.15317571832505,572.185791015625]},{"page":137,"text":"found from these experiments. Note, however, that the numerical coefficient (3/4)","rect":[53.81338119506836,593.378662109375,385.13628516999855,584.4441528320313]},{"page":138,"text":"6.3 Nematic–Smectic A Transition","rect":[53.812843322753909,42.55594253540039,172.92102569727823,36.68026351928711]},{"page":138,"text":"121","rect":[372.4981994628906,42.454345703125,385.1898245254032,36.73106384277344]},{"page":138,"text":"depends on arbitrariness of numerical coefficients in the original Landau expansion","rect":[53.812843322753909,68.2883529663086,385.09697810224068,59.35380554199219]},{"page":138,"text":"(6.8). We should also underline that here we did not discuss any effects of fluctua-","rect":[53.812843322753909,80.24788665771485,385.18648658601418,71.31333923339844]},{"page":138,"text":"tions of the order parameter in space dS(r); the S-value was considered dependent","rect":[53.812843322753909,92.20748138427735,385.1745439297874,82.97412109375]},{"page":138,"text":"only on temperature and magnetic field.","rect":[53.81385040283203,104.11019134521485,214.78237576987034,95.17564392089844]},{"page":138,"text":"In the frame of Landau theory we can also consider the influence of the external","rect":[65.765869140625,116.0697250366211,385.18061692783427,107.13517761230469]},{"page":138,"text":"magnetic or electric field on the N–I phase transition temperature. Imagine that we","rect":[53.81385040283203,128.02932739257813,385.1706928081688,119.09477233886719]},{"page":138,"text":"apply the electric field E along the director of a nematic and increase temperature.","rect":[53.81385040283203,139.98886108398438,385.2602200081516,131.05430603027345]},{"page":138,"text":"In the case of positive dielectric anisotropy ea, even a weak field changes the","rect":[53.81385040283203,151.94937133789063,385.23044622613755,143.0138397216797]},{"page":138,"text":"symmetry of both phases to conical (C1v), and, strictly speaking, the phase transition","rect":[53.8139762878418,163.908935546875,385.21343318036568,154.97438049316407]},{"page":138,"text":"vanishes. However, in the continuous temperature dependence of the order parame-","rect":[53.8138542175293,175.86846923828126,385.18758521882668,166.9339141845703]},{"page":138,"text":"ter, a characteristic inflection point appears that may be considered as an apparent","rect":[53.8138542175293,187.8280029296875,385.202528548928,178.89344787597657]},{"page":138,"text":"N–I phase transition temperature Tc. The latter may be changed with an applied field.","rect":[53.8138542175293,199.73126220703126,385.2271999886203,190.7962188720703]},{"page":138,"text":"As the equation for the enthalpy is given by ∂Dg/∂T¼\u0002DH/Tc we may write the","rect":[65.76573944091797,211.7306671142578,385.17502630426255,201.75025939941407]},{"page":138,"text":"discontinuity of the free energy density as dDg¼\u0002DHdT/Tc. When E || n there is a","rect":[53.814292907714847,223.69032287597657,385.1614154644188,214.38723754882813]},{"page":138,"text":"difference between quadratic-in-field energy terms in the nematic and isotropic","rect":[53.814598083496097,235.61013793945313,385.1743549175438,226.6755828857422]},{"page":138,"text":"phases DgE ¼ (1/8p)(e|| \u0002eiso)E2. From comparison of the two contributions, the","rect":[53.814598083496097,247.60951232910157,385.17234075738755,236.5188751220703]},{"page":138,"text":"field induced shift dTE of the transition temperature Tc is given by [9]:","rect":[53.813594818115237,259.529541015625,337.9962602383811,250.29605102539063]},{"page":138,"text":"DTE ¼ ðejj \u0002 eisoÞE2","rect":[177.8101348876953,290.9429626464844,259.0103003748353,273.7599182128906]},{"page":138,"text":"8p \u0003 DH","rect":[217.85728454589845,296.3905944824219,249.30563116028635,289.1493225097656]},{"page":138,"text":"For nematics with high positive dielectric anisotropy the difference e||\u0002eiso \b 10","rect":[65.7658920288086,319.8943176269531,385.1712578873969,310.95977783203127]},{"page":138,"text":"is substantial and, for a typical value of DH \b 2.5 107 erg/cm3, the shift DTE \b","rect":[53.81350326538086,331.8542175292969,385.14807918760689,320.8032531738281]},{"page":138,"text":"1K is expected for a field strength of 500 statV/cm (or 1.5 107 V/m) in agreement","rect":[53.814231872558597,343.8138732910156,385.17500169345927,332.85589599609377]},{"page":138,"text":"with experiment [9].","rect":[53.81426239013672,355.7734069824219,136.37147183920627,346.8388671875]},{"page":138,"text":"Concluding this section I would like to underline the significance of coefficient B","rect":[65.7662582397461,367.73297119140627,385.1809462872859,358.79840087890627]},{"page":138,"text":"in the Landau expansion:","rect":[53.81426239013672,379.6925048828125,155.56919725009989,370.7579345703125]},{"page":138,"text":"1.","rect":[53.81426239013672,395.4619140625,61.279919104831268,388.7286376953125]},{"page":138,"text":"2.","rect":[53.81302261352539,444.0,61.27867932821994,436.56768798828127]},{"page":138,"text":"3.","rect":[53.81301498413086,468.0,61.278671698825409,460.4300231933594]},{"page":138,"text":"For B ¼ 0, the free energy is symmetric with respect to \u0005Z and we have a","rect":[66.27593994140625,397.6034240722656,385.1620258159813,388.66888427734377]},{"page":138,"text":"second order transition. For small B ¼6 0 the transition is called weak first order","rect":[66.27690887451172,409.54302978515627,385.1470883926548,400.2997131347656]},{"page":138,"text":"transition","rect":[66.27690887451172,420.0,104.08698359540472,412.58795166015627]},{"page":138,"text":"becomes close to Tc.","rect":[66.27632904052735,433.0847473144531,149.36945767905002,424.54827880859377]},{"page":138,"text":"The biaxiality of molecules influences the value of B and, in turn, a variation of","rect":[66.27469635009766,445.4424743652344,385.14986549226418,436.5079345703125]},{"page":138,"text":"B may provide, at least, theoretically the biaxiality of the nematic phase.","rect":[66.27468872070313,457.3452453613281,359.4610867073703,448.41070556640627]},{"page":138,"text":"Flexibility of mesogenic molecules also strongly influences B.","rect":[66.27468872070313,469.3048095703125,317.2651028206516,460.3702392578125]},{"page":138,"text":"6.3 Nematic–Smectic A Transition","rect":[53.812843322753909,511.1522216796875,238.21072357987524,501.8770446777344]},{"page":138,"text":"6.3.1 Order Parameter","rect":[53.812843322753909,540.4969482421875,174.05512122843428,532.1301879882813]},{"page":138,"text":"As both the nematic and smectic A phases have quantitatively similar orientational","rect":[53.812843322753909,570.310302734375,385.1238237149436,561.3757934570313]},{"page":138,"text":"order, we may fix the free energy of the nematic phase and assume the orientational","rect":[53.812843322753909,582.2698974609375,385.1268144375999,573.3353881835938]},{"page":138,"text":"order parameter to be equal in both phases, SA ¼ SN. Then we introduce a new","rect":[53.812843322753909,594.2294311523438,385.16271734192699,585.294921875]},{"page":139,"text":"122","rect":[53.81303024291992,42.454345703125,66.5046362319462,36.73106384277344]},{"page":139,"text":"Fig. 6.9 Below: A schematic","rect":[53.812843322753909,67.58130645751953,155.08678576731206,59.546695709228519]},{"page":139,"text":"picture of molecular packing","rect":[53.81283950805664,77.4895248413086,152.50275177149698,69.89517211914063]},{"page":139,"text":"in the vertically oriented","rect":[53.81283950805664,87.4087142944336,137.9463400283329,79.81436157226563]},{"page":139,"text":"smectic layers. Above:","rect":[53.81283950805664,97.3846664428711,130.56489280905786,89.79031372070313]},{"page":139,"text":"Average density r modulated","rect":[53.811988830566409,107.36067962646485,154.69078582911417,99.76632690429688]},{"page":139,"text":"with amplitude r1 and period","rect":[53.81283950805664,117.33663177490235,154.1621603408329,109.74191284179688]},{"page":139,"text":"l of the density wave","rect":[53.81303024291992,127.25545501708985,125.49520251535893,119.6441650390625]},{"page":139,"text":"r(z)","rect":[192.63975524902345,70.06765747070313,204.01670789572109,62.65302276611328]},{"page":139,"text":"r","rect":[196.66464233398438,88.81193542480469,201.05184695247957,82.68677520751953]},{"page":139,"text":"0","rect":[201.05206298828126,91.96768951416016,204.38442079339525,87.60171508789063]},{"page":139,"text":"–l/2","rect":[208.7644805908203,122.01429748535156,221.64640580429887,115.97708892822266]},{"page":139,"text":"0","rect":[241.03001403808595,122.01429748535156,245.47315751328325,116.24896240234375]},{"page":139,"text":"6 Phase Transitions","rect":[318.2891540527344,42.55594253540039,385.14243014334957,36.68026351928711]},{"page":139,"text":"r","rect":[378.54345703125,68.21537780761719,382.93066164974518,62.09021759033203]},{"page":139,"text":"core","rect":[225.35617065429688,157.74131774902345,242.2337165953145,153.27137756347657]},{"page":139,"text":"tails","rect":[261.5310363769531,157.74131774902345,277.5215431334004,151.8400421142578]},{"page":139,"text":"order parameter because, in the smectic A phase, a new symmetry element appears,","rect":[53.812843322753909,187.54486083984376,385.09698148276098,178.6103057861328]},{"page":139,"text":"namely, one-dimensional positional order. Recall that, in the SmA phase, local","rect":[53.812843322753909,199.50442504882813,385.11189134189677,190.5698699951172]},{"page":139,"text":"density is modulated, Fig. 6.9,","rect":[53.812843322753909,211.46395874023438,175.9918179085422,202.52940368652345]},{"page":139,"text":"drðzÞ ¼ rðzÞ \u0002 r0 ¼ Xrm cos\b2plmz þ jm","rect":[124.7883529663086,246.5828399658203,306.4118463919745,222.69761657714845]},{"page":139,"text":"m","rect":[216.5544891357422,247.7077178955078,221.55746101599793,244.47256469726563]},{"page":139,"text":"Here l is interlayer distance and rm is the infinite set of possible complex order","rect":[53.81394577026367,269.2220153808594,385.1466001114048,260.2675476074219]},{"page":139,"text":"parameters (amplitudes and phases of density harmonics with m ¼ 1,2,3 ...) In","rect":[53.8137321472168,281.1247253417969,385.15056696942818,272.190185546875]},{"page":139,"text":"fact, usually the modulation is not deep and, in the simplest approach, we can leave","rect":[53.81272506713867,293.0842590332031,385.1237262554344,284.14971923828127]},{"page":139,"text":"only the first strongest Fourier harmonic with m ¼ 1 and the role of highest","rect":[53.81272506713867,305.0438232421875,385.10578782627177,296.1092529296875]},{"page":139,"text":"harmonics will be discussed later. Then,","rect":[53.8137321472168,315.0,216.0494350960422,308.06878662109377]},{"page":139,"text":"drðzÞ ¼ r1 cos\b2pl z þ j1","rect":[163.81771850585938,352.1225280761719,267.39378426155408,328.2373046875]},{"page":139,"text":"(6.15)","rect":[361.05615234375,344.4360046386719,385.10552345124855,335.8401184082031]},{"page":139,"text":"This density wave is usually considered as a complex order parameter r1 ¼","rect":[65.7665023803711,372.6082458496094,385.14807918760689,363.6737060546875]},{"page":139,"text":"exp (ip) of the smectic A phase in the Landau expansion or free energy at the","rect":[53.814231872558597,384.5672607421875,385.1720355816063,375.6326904296875]},{"page":139,"text":"SmA–N phase transition. Typically, when there is no distortion, one assumes","rect":[53.814231872558597,396.52679443359377,385.0824319277878,387.59222412109377]},{"page":139,"text":"j1 ¼ 0 at z ¼ 0 and operates only with the wave amplitude r1 as the real part of","rect":[53.814231872558597,408.48663330078127,385.1516049942173,399.55206298828127]},{"page":139,"text":"the order parameter.","rect":[53.81374740600586,420.4461975097656,135.27901883627659,411.51165771484377]},{"page":139,"text":"6.3.2 Free Energy Expansion","rect":[53.812843322753909,462.6214599609375,208.47952240800024,452.0553894042969]},{"page":139,"text":"Due to symmetry \u0005r1 the free energy density is expanded over even powers of r1","rect":[53.812843322753909,490.16375732421877,385.18130140630105,481.22918701171877]},{"page":139,"text":"that is without B-term:","rect":[53.812843322753909,500.1014099121094,145.08996446201395,493.1888427734375]},{"page":139,"text":"gSmA ¼ gN þ 21Ar21 þ 41Cr41 þ \u0003\u0003\u0003 with A ¼ aðT \u0002 TNAÞ;a >0","rect":[80.15267944335938,536.6539916992188,334.70937434247505,516.3176879882813]},{"page":139,"text":"(6.16)","rect":[361.0571594238281,531.595703125,385.1065610489048,523.0595703125]},{"page":139,"text":"Assuming gN ¼ const, after minimization of (6.16) with respect to r1 we have","rect":[65.76651763916016,560.2609252929688,383.08164251520005,551.2865600585938]},{"page":139,"text":"@@grSm1A ¼ r1½aðT \u0002 TNAÞ þ Cr12\u0006 ¼0","rect":[148.18370056152345,597.472900390625,292.45082942060005,574.423828125]},{"page":139,"text":"(6.17)","rect":[361.05523681640627,590.0903930664063,385.1046079239048,581.5542602539063]},{"page":140,"text":"6.3 Nematic–Smectic A Transition","rect":[53.8129997253418,42.55679702758789,172.9211935439579,36.68111801147461]},{"page":140,"text":"123","rect":[372.49835205078127,42.55679702758789,385.1899771132938,36.73191833496094]},{"page":140,"text":"Solution r1 ¼ 0 corresponds to the positionally symmetric (not modulated)","rect":[65.76496887207031,68.2883529663086,385.0771421035923,59.35380554199219]},{"page":140,"text":"nematic phase. The other two solutions correspond to the positionally ordered","rect":[53.813926696777347,80.24788665771485,385.15081111005318,71.31333923339844]},{"page":140,"text":"SmA phase with a continuous growth of the order parameter:","rect":[53.813926696777347,92.20748138427735,301.03376634189677,83.27293395996094]},{"page":140,"text":"r1 ¼ \u0005\u0003aðTNAC\u0002 TÞ\u00041=2","rect":[170.72911071777345,132.82432556152345,267.7334449060853,106.49600219726563]},{"page":140,"text":"(6.18)","rect":[361.0561828613281,125.13761138916016,385.10555396882668,116.60148620605469]},{"page":140,"text":"The temperature dependence of r1 may be found from the intensity of the X-ray","rect":[65.76653289794922,168.27346801757813,385.1076592545844,159.3388214111328]},{"page":140,"text":"diffraction at the Bragg angle determined by period l of the smectic layers. The","rect":[53.81362533569336,180.23291015625,385.14649236871568,171.27842712402345]},{"page":140,"text":"experimental data on r1(TNA \u0002 T) for cholesteryl nonanoate [10] (solid bold curve)","rect":[53.8126335144043,192.19268798828126,385.12175880281105,183.25791931152345]},{"page":140,"text":"are","rect":[53.81375503540039,203.0,66.02756918145978,197.0]},{"page":140,"text":"compared","rect":[72.16633605957031,204.1522216796875,111.61088648847113,195.21766662597657]},{"page":140,"text":"with","rect":[117.76557922363281,203.0,135.48407832196723,195.21766662597657]},{"page":140,"text":"the","rect":[141.66961669921876,203.0,153.89339483453598,195.21766662597657]},{"page":140,"text":"corresponding","rect":[160.02218627929688,204.1522216796875,217.11057368329535,195.21766662597657]},{"page":140,"text":"theoretical","rect":[223.2941436767578,203.0,265.660292220803,195.21766662597657]},{"page":140,"text":"dependence","rect":[271.7821350097656,204.1522216796875,318.99597967340318,195.21766662597657]},{"page":140,"text":"(dash","rect":[325.1407165527344,203.75379943847657,346.7712029557563,195.21766662597657]},{"page":140,"text":"line)","rect":[352.8960266113281,203.75379943847657,371.2316831192173,195.21766662597657]},{"page":140,"text":"in","rect":[377.3664855957031,203.0,385.1407097916938,195.21766662597657]},{"page":140,"text":"Fig. 6.10a. Note that the helical structure of the cholesteric is disregarded because","rect":[53.81375503540039,216.11178588867188,385.10483587457505,207.17723083496095]},{"page":140,"text":"locally, on the scale of the size l, the nematic and cholesteric phases are indistin-","rect":[53.81375503540039,228.01455688476563,385.1715024551548,219.06007385253907]},{"page":140,"text":"guishable. From this plot we can find r1 ¼ 0.31 at TNA \u0002 T ¼10 C.","rect":[53.81377029418945,239.97409057617188,329.1632046272922,231.03953552246095]},{"page":140,"text":"Therefore, with TNA ¼ 348 K and found ratio a1/C1 ¼ 0.01 we can plot the free","rect":[65.76583099365235,251.93423461914063,385.1378558941063,242.98971557617188]},{"page":140,"text":"energy in arbitrary units. To this effect, let us write Eq. 6.16 in the dimensionless","rect":[53.8139762878418,263.8937683105469,385.0970803652878,254.95921325683595]},{"page":140,"text":"form:","rect":[53.81399154663086,273.8313293457031,75.99197251865457,266.91876220703127]},{"page":140,"text":"gS1=m2AaT\u0002NAgN ¼ ðTNA \u0002 TÞr12 þ 0:143r14 þ \u0003\u0003\u0003","rect":[133.45602416992188,310.757568359375,305.51267945450686,290.08221435546877]},{"page":140,"text":"a","rect":[55.455562591552737,341.7413635253906,61.01082498878658,336.1526184082031]},{"page":140,"text":"0.6","rect":[75.57317352294922,372.4554138183594,86.67853269638865,366.69140625]},{"page":140,"text":"b","rect":[219.39060974121095,341.49139404296877,225.49540349068915,334.1830139160156]},{"page":140,"text":"0.015","rect":[239.7429656982422,348.67431640625,259.7341183495312,342.90985107421877]},{"page":140,"text":"0.010","rect":[239.64390563964845,391.344970703125,259.6350582909374,385.58050537109377]},{"page":140,"text":"3","rect":[354.23223876953127,356.12274169921877,358.6747709942802,350.3581848144531]},{"page":140,"text":"2","rect":[372.3475646972656,385.5362548828125,376.7900969220146,379.9156188964844]},{"page":140,"text":"0.4","rect":[75.47090911865235,419.9027404785156,86.57626829209177,414.13873291015627]},{"page":140,"text":"0.2","rect":[75.63948822021485,467.3468933105469,86.74484739365427,461.5828857421875]},{"page":140,"text":"T","rect":[168.9912872314453,428.2681884765625,173.87279058790257,422.5281982421875]},{"page":140,"text":"TNA","rect":[185.95298767089845,431.134765625,199.1617311939406,423.3951416015625]},{"page":140,"text":"0.005","rect":[239.7429656982422,434.06597900390627,259.7341183495312,428.301513671875]},{"page":140,"text":"0.000","rect":[239.67825317382813,476.73895263671877,259.6694058251171,470.9744873046875]},{"page":140,"text":"1","rect":[373.88958740234377,457.7641906738281,378.3321196270927,452.1435546875]},{"page":140,"text":"0.0","rect":[75.50446319580078,514.7446899414063,86.6098223692402,508.98065185546877]},{"page":140,"text":"30","rect":[87.26215362548828,528.2257080078125,96.1464434629902,522.461669921875]},{"page":140,"text":"40","rect":[109.44730377197266,528.2257080078125,118.33159360947458,522.461669921875]},{"page":140,"text":"50","rect":[131.63246154785157,528.2257080078125,140.51674375595895,522.461669921875]},{"page":140,"text":"60","rect":[153.8184051513672,528.2257080078125,162.70268735947458,522.461669921875]},{"page":140,"text":"T (oC)","rect":[136.23768615722657,541.9747314453125,158.1010419784062,534.0443725585938]},{"page":140,"text":"70","rect":[176.0035400390625,528.2257080078125,184.88782224716989,522.461669921875]},{"page":140,"text":"80","rect":[198.19110107421876,528.2257080078125,207.07538328232614,522.461669921875]},{"page":140,"text":"–0.005","rect":[235.3340606689453,519.4096069335938,259.7676876854687,513.6451416015625]},{"page":140,"text":"–0.6–0.4–0.2 0.0","rect":[265.63140869140627,528.4746704101563,329.69974029777338,522.710205078125]},{"page":140,"text":"0.2","rect":[335.5061340332031,528.4746704101563,346.61230743644526,522.710205078125]},{"page":140,"text":"r1 (arb. u.)","rect":[304.494384765625,541.8267822265625,343.75007457117968,533.9566650390625]},{"page":140,"text":"0.4","rect":[352.3731384277344,528.4746704101563,363.4793423485546,522.710205078125]},{"page":140,"text":"0.6","rect":[369.2873229980469,528.4746704101563,380.393496401289,522.710205078125]},{"page":140,"text":"Fig. 6.10 (a) Theoretical (dash curve) and experimental (solid curve) dependence of the smectic","rect":[53.812843322753909,564.7855834960938,385.1525511481714,557.0557861328125]},{"page":140,"text":"A order parameter r1 on temperature; TNA ¼ 75 C for cholesteryl nonanoate [10]. (b) Free energy","rect":[53.812843322753909,574.6937866210938,385.1500296035282,567.048583984375]},{"page":140,"text":"of a smectic A as a function of order parameter r1 for different temperatures: 10 C below the","rect":[53.81368637084961,584.6129150390625,385.1543516852808,577.0184326171875]},{"page":140,"text":"transition (curve 1), T ¼ TNA (curve 2) and 10 C above TNA (curve 3)","rect":[53.813838958740237,594.2501220703125,292.8229989395826,586.9943237304688]},{"page":141,"text":"124","rect":[53.812843322753909,42.454345703125,66.50444931178018,36.73106384277344]},{"page":141,"text":"6 Phase Transitions","rect":[318.2889709472656,42.55594253540039,385.1422470378808,36.68026351928711]},{"page":141,"text":"The dependencies of the dimensionless free energy on the order parameter at","rect":[65.76496887207031,68.2883529663086,385.1787248379905,59.35380554199219]},{"page":141,"text":"T < TNA (curve 1), T ¼ TNA (curve 2) and T > TNA (curve 3) are presented in","rect":[53.812950134277347,80.24800872802735,385.1434258561469,71.31346130371094]},{"page":141,"text":"Fig. 6.10b. The energy is symmetric about r1 ¼ 0. For T \u000B Tc the higher symmetry","rect":[53.81357955932617,92.20772552490235,385.17363825849068,83.27305603027344]},{"page":141,"text":"nematic state is stable; curve 3 at finite r1 in the nematic manifests short-range","rect":[53.81388473510742,104.11067962646485,385.0801471538719,95.17594909667969]},{"page":141,"text":"smectic order effect. For T < Tc, in the smectic A state, the two minima in curve1","rect":[53.81400680541992,115.68238830566406,385.1768731217719,107.13566589355469]},{"page":141,"text":"situated exactly at r1¼ \u0005 0.31 reflect the symmetry of the energy with respect to","rect":[53.8131217956543,128.02999877929688,385.1409844498969,119.09544372558594]},{"page":141,"text":"the phase (0 or p) of the density wave.","rect":[53.81306076049805,139.98953247070313,209.53374906088596,131.0549774169922]},{"page":141,"text":"We can also discuss the structural susceptibility in the more symmetric","rect":[65.76509094238281,151.94906616210938,385.17981756402818,143.01451110839845]},{"page":141,"text":"(nematic) phase near the SmA-N transition.","rect":[53.81306838989258,163.90863037109376,229.62434049155002,154.9740753173828]},{"page":141,"text":"wN\u0002A1 ¼ @@2rg12A ¼ aðT \u0002 TNAÞ","rect":[166.4798126220703,206.03543090820313,272.50030912505346,181.1434326171875]},{"page":141,"text":"(6.19)","rect":[361.05596923828127,197.9724578857422,385.1053403457798,189.4363250732422]},{"page":141,"text":"It is a special layer formation susceptibility: close to the phase transition the","rect":[65.76630401611328,232.54925537109376,385.13916814996568,223.57485961914063]},{"page":141,"text":"nematic is very sensitive to the spatially periodic molecular field, which induce","rect":[53.81426239013672,244.5087890625,385.15183294488755,235.57423400878907]},{"page":141,"text":"the density wave with period l. In order to study this phenomenon one is","rect":[53.81426239013672,256.4683532714844,385.13327421294408,247.5138702392578]},{"page":141,"text":"tempted to use an external spatially periodic force with the same period, but,","rect":[53.81426239013672,268.4278869628906,385.2397121956516,259.4534912109375]},{"page":141,"text":"at present, it is technically impossible. Therefore, we cannot find the Landau","rect":[53.81327438354492,280.3874206542969,385.1692742448188,271.452880859375]},{"page":141,"text":"coefficient a above TNA using some analogy with the Kerr or Cotton-Mouton","rect":[53.81327438354492,292.2910461425781,385.1226739030219,283.35565185546877]},{"page":141,"text":"effects.","rect":[53.81374740600586,302.1887512207031,84.22684140707736,295.3160400390625]},{"page":141,"text":"However, there is a great deal of studies of pre-transitional effects by the","rect":[65.76576232910156,316.21014404296877,385.18933904840318,307.27557373046877]},{"page":141,"text":"calorimetric and X-ray scattering techniques showing that, in the vicinity of","rect":[53.81373977661133,328.169677734375,385.12871681062355,319.235107421875]},{"page":141,"text":"second order N–A transition, a strong fluctuations of the smectic order occur. It","rect":[53.81373977661133,340.12921142578127,385.2558427579124,331.19464111328127]},{"page":141,"text":"means that the order parameter changes in time and space r1¼r1(r, t). For","rect":[53.81373977661133,352.0887451171875,385.15911231843605,343.1541748046875]},{"page":141,"text":"example,","rect":[53.813289642333987,364.0487976074219,90.96937222983127,355.1142578125]},{"page":141,"text":"a","rect":[96.63729858398438,363.0,101.08683050348127,357.0]},{"page":141,"text":"character","rect":[106.77665710449219,363.0,144.28612647859229,355.1142578125]},{"page":141,"text":"of","rect":[149.9779510498047,363.0,158.33451209870948,355.1142578125]},{"page":141,"text":"the","rect":[164.02633666992188,363.0,176.4929889507469,355.1142578125]},{"page":141,"text":"functional","rect":[182.1828155517578,363.0,223.698866439553,355.1142578125]},{"page":141,"text":"dependence","rect":[229.3419189453125,364.0487976074219,277.48740423395005,355.1142578125]},{"page":141,"text":"of","rect":[283.17724609375,363.0,291.5338071426548,355.1142578125]},{"page":141,"text":"heat","rect":[297.2256164550781,363.0,314.2393632657249,355.1142578125]},{"page":141,"text":"capacity","rect":[319.8824157714844,364.0487976074219,354.02236262372505,355.1142578125]},{"page":141,"text":"at","rect":[359.69427490234377,363.0,367.0623613125999,356.1302185058594]},{"page":141,"text":"the","rect":[372.7054138183594,363.0,385.1720355816063,355.1142578125]},{"page":141,"text":"nematic–smectic A transition may be very different, varying from a simple","rect":[53.813289642333987,376.00836181640627,385.2517474956688,367.07379150390627]},{"page":141,"text":"step to the divergent cusp-like maximum [11]. The experiment shows that","rect":[53.813289642333987,387.9111328125,385.1272416836936,378.9765625]},{"page":141,"text":"N–A transition may be second or first order. This depends on the width of the","rect":[53.813289642333987,399.87066650390627,385.1154254741844,390.93609619140627]},{"page":141,"text":"temperature range of the intermediate nematic phase between smectic A and","rect":[53.813289642333987,411.8302307128906,385.22686091474068,402.89569091796877]},{"page":141,"text":"isotropic phases: the narrower the range the closer the N–A transition to the first","rect":[53.813289642333987,423.7897644042969,385.2129045254905,414.855224609375]},{"page":141,"text":"order.","rect":[53.813289642333987,433.71734619140627,77.86265988852267,426.81475830078127]},{"page":141,"text":"6.3.3 Weak First Order Transition","rect":[53.812843322753909,488.7447204589844,232.37557953690649,480.2106018066406]},{"page":141,"text":"In reality, the N–A transition is, as a rule, weak first order transition. There are, at","rect":[53.812843322753909,518.3906860351563,385.1796098477561,509.4561767578125]},{"page":141,"text":"least, two ways to understand this in framework of Landau approach. We still use","rect":[53.812843322753909,530.3502197265625,385.13083685113755,521.4157104492188]},{"page":141,"text":"the same smectic order parameter r1 but include additional factors, either (a)","rect":[53.812843322753909,542.309814453125,385.1622251114048,533.3753051757813]},{"page":141,"text":"higher harmonics of the density wave, or (b) consider the influence of the","rect":[53.813411712646487,554.26953125,385.11743963434068,545.3350219726563]},{"page":141,"text":"positional order on the orientational order of SmA, the so-called interaction of","rect":[53.813411712646487,566.2290649414063,385.14730201570168,557.2945556640625]},{"page":141,"text":"order parameters.","rect":[53.813411712646487,578.1885986328125,124.79090543295627,569.2540893554688]},{"page":142,"text":"6.3 Nematic–Smectic A Transition","rect":[53.812843322753909,42.55594253540039,172.92102569727823,36.68026351928711]},{"page":142,"text":"125","rect":[372.4981994628906,42.55594253540039,385.1898245254032,36.62946701049805]},{"page":142,"text":"6.3.3.1 Role of Higher Order Fourier Components","rect":[53.812843322753909,68.36803436279297,276.01373685942846,59.035072326660159]},{"page":142,"text":"Wekeepequality ofnematicorientationalorderparametersinbothphasesSN ¼ SA,","rect":[53.812843322753909,92.20748138427735,385.1832241585422,83.27293395996094]},{"page":142,"text":"and take only the amplitude of the second harmonic r2 of the density wave as an","rect":[53.814537048339847,104.11067962646485,385.15288630536568,95.17594909667969]},{"page":142,"text":"additional SmA order parameter","rect":[53.8140983581543,116.0702133178711,182.97693763581885,107.13566589355469]},{"page":142,"text":"drðzÞ ¼ r1 cosð2pz=lÞ þ r2 cosð4pz=lÞ","rect":[144.89910888671876,143.18539428710938,294.1386758242722,130.26470947265626]},{"page":142,"text":"Therefore we have two order parameters (for the same transition) and the free","rect":[65.7662124633789,166.6861572265625,385.1371234722313,157.75160217285157]},{"page":142,"text":"energy density is:","rect":[53.81418991088867,178.64572143554688,124.92707688877175,169.71116638183595]},{"page":142,"text":"gSmA �� gN þ 21A1r12 þ 12A2r22 \u0002 Br21r2 þ 14C1r14 þ 14C2r24 þ C12r21r22","rect":[67.97504425048828,213.17657470703126,346.4137427576478,192.84027099609376]},{"page":142,"text":"with A1 ¼ a1ðT \u0002 T1Þ; A2 ¼ a2ðT \u0002 T2Þ;","rect":[68.37067413330078,228.126953125,238.3885034291162,218.1764373779297]},{"page":142,"text":"(6.20)","rect":[361.0562438964844,227.38987731933595,385.1056150039829,218.85374450683595]},{"page":142,"text":"For a typical situation r1 > r2, it is sufficient to take only one cross-term with","rect":[65.76659393310547,252.67108154296876,385.11663142255318,243.7360076904297]},{"page":142,"text":"coefficient B. Coefficients a1 and a2 are assumed positive and, in addition, we","rect":[53.813655853271487,264.6307373046875,385.16968572809068,255.6960906982422]},{"page":142,"text":"assume T1 > T2 because on cooling, the first Fourier harmonic appears at higher","rect":[53.81393051147461,276.5903625488281,385.1302426895298,267.65582275390627]},{"page":142,"text":"temperature, and afterwards, at a lower temperature, the single harmonic law is","rect":[53.813236236572269,288.5499267578125,385.18393339263158,279.6153564453125]},{"page":142,"text":"violated and r2 appears. The minimization of (6.20) with respect to r2 results in","rect":[53.813236236572269,300.509765625,377.61061945966255,291.57489013671877]},{"page":142,"text":"@@grSm2A ¼ A2r2 þ C2r32 \u0002 Br12 þ 2C12r21r2 ¼0","rect":[127.73484802246094,337.7613830566406,312.9005059342719,314.7123107910156]},{"page":142,"text":"(6.21)","rect":[361.0559997558594,330.3221130371094,385.1053708633579,321.7860107421875]},{"page":142,"text":"Due to smallness of r2, the second and fourth terms are small and we can find r2:","rect":[65.76636505126953,361.2286071777344,385.151991439553,352.3935546875]},{"page":142,"text":"B2","rect":[222.21890258789063,384.80645751953127,240.9970862635072,375.5685119628906]},{"page":142,"text":"r2 \b A2 r1","rect":[197.5220947265625,397.2196350097656,240.9970862635072,384.1431884765625]},{"page":142,"text":"(6.22)","rect":[361.0561828613281,390.6872253417969,385.10555396882668,382.151123046875]},{"page":142,"text":"Then, substituting r2 into expression (6.20) for free energy and omitting the","rect":[65.76653289794922,420.7864685058594,385.1727985210594,411.85125732421877]},{"page":142,"text":"terms with r18, we obtain a biquadratic equation of the (6.6) type:","rect":[53.81308364868164,432.7461242675781,320.4030900723655,421.6953430175781]},{"page":142,"text":"gSmA ¼ gN þ 12A1r21 \u0002 41\b2AB22 \u0002 C1 r14 þ 61D1r16","rect":[119.35094451904297,471.4924011230469,319.1674048670228,446.9764709472656]},{"page":142,"text":"(6.23)","rect":[361.0561828613281,463.8054504394531,385.10555396882668,455.26934814453127]},{"page":142,"text":"where D1 ¼ 12C12B2/A22. Like in the case of Eq. 6.6, the condition for first order","rect":[53.81450653076172,495.0382995605469,385.14742408601418,483.9872741699219]},{"page":142,"text":"transition is","rect":[53.812557220458987,504.9360046386719,101.0562325869675,498.06329345703127]},{"page":142,"text":"2B2\u0005A2 \u0002 C1 >0;","rect":[184.15298461914063,537.0510864257813,253.17239319962403,521.2283935546875]},{"page":142,"text":"otherwise, the transition is second order. Therefore, an appearance of small terms","rect":[53.81342697143555,560.6181640625,385.17816557036596,551.6836547851563]},{"page":142,"text":"with r2 in the free energy (6.20) results in a weak first order N–SmA transition.","rect":[53.81342697143555,572.52099609375,375.6919827034641,563.5863647460938]},{"page":143,"text":"126","rect":[53.812843322753909,42.55783462524414,66.50444931178018,36.68215560913086]},{"page":143,"text":"6 Phase Transitions","rect":[318.2889709472656,42.55783462524414,385.1422470378808,36.68215560913086]},{"page":143,"text":"6.3.3.2 Interaction of Two Order Parameters","rect":[53.812843322753909,66.76439666748047,253.24945463286594,59.035072326660159]},{"page":143,"text":"Experiment shows that:","rect":[53.812843322753909,92.20748138427735,148.7809739834983,83.27293395996094]},{"page":143,"text":"1.","rect":[53.812843322753909,109.0,61.27850003744845,101.24358367919922]},{"page":143,"text":"2.","rect":[53.81183624267578,168.0,61.27749295737033,160.98452758789063]},{"page":143,"text":"The narrower a range of the nematic phase in a homological series of different","rect":[66.27452087402344,110.11837005615235,385.17256028720927,101.18382263183594]},{"page":143,"text":"compounds (as an example see Fig. 6.11a) the stronger are first order features of","rect":[66.27452087402344,122.0779037475586,385.1496823867954,113.14335632324219]},{"page":143,"text":"the N–SmA transition. In some sense, the SmA phase “feels” the proximity of","rect":[66.27352905273438,134.03750610351563,385.14678321687355,125.10295104980469]},{"page":143,"text":"the isotropic phase. In other words, we may say that, in the isotropic phase, there","rect":[66.27352905273438,145.99703979492188,385.11792791559068,137.06248474121095]},{"page":143,"text":"are traces of both nematic and smectic A short-range order.","rect":[66.27352905273438,157.9396209716797,307.70891995932348,148.94529724121095]},{"page":143,"text":"Appearance of the positional order in the SmA phase is accompanied by an","rect":[66.27351379394531,169.85931396484376,385.2064141373969,160.9247589111328]},{"page":143,"text":"increaseintheorientationalorder, DS ¼ SA \u0002 SN. Thereasonis densermolecular","rect":[66.27351379394531,181.27870178222657,385.19158302156105,172.55560302734376]},{"page":143,"text":"packing within the smectic layers that is more favorable for higher S, Fig. 6.11b.","rect":[66.27562713623047,193.7796630859375,385.18365140463598,184.84510803222657]},{"page":143,"text":"On account of DS, the free energy density of smectic A may be written as","rect":[65.7669906616211,211.69058227539063,385.14191068755346,202.42733764648438]},{"page":143,"text":"follows:","rect":[53.815956115722659,221.6281280517578,86.45580918857644,214.71556091308595]},{"page":143,"text":"gSmA ¼ gN þ 12A1r12 þ 21A2ðDSÞ2 \u0002 Br12ðDSÞ þ 41C1r14","rect":[110.29116821289063,258.18115234375,328.2306372888978,237.8448486328125]},{"page":143,"text":"(6.24)","rect":[361.0561828613281,253.1228485107422,385.10555396882668,244.5867156982422]},{"page":143,"text":"Now we make minimization with respect to DS and obtain","rect":[65.76653289794922,288.7196350097656,302.0456475358344,279.4563903808594]},{"page":143,"text":"@@gDSmSA ¼ A2DS \u0002 Br12 ¼ 0 or DS ¼ AB2 r12","rect":[137.36453247070313,325.35955810546877,302.7970130213197,302.97943115234377]},{"page":143,"text":"(6.25)","rect":[361.0561828613281,318.5892028808594,385.10555396882668,309.9933166503906]},{"page":143,"text":"Hence, we arrive at exactly the same form the Eq. 6.23 has.","rect":[65.76653289794922,343.92669677734377,307.0735134651828,334.99212646484377]},{"page":143,"text":"Therefore we again obtain the first order transition for 2B2\u0005A2 \u0002 C1 >0 and","rect":[65.76555633544922,357.17425537109377,385.1790398698188,345.2416076660156]},{"page":143,"text":"second order for 2B2\u0005A2 \u0002 C1 <0 and a tricritical point for 2B2\u0005A2 \u0002 C1 ¼ 0.","rect":[53.81332015991211,369.1338806152344,385.17596097494848,357.20123291015627]},{"page":143,"text":"The tricritical point (TCP) is located in the continuous phase transition line","rect":[53.81426239013672,379.7493896484375,385.0991596050438,370.8148193359375]},{"page":143,"text":"separating the nematic and smectic A phases [12], see a phase diagram schema-","rect":[53.81426239013672,391.70892333984377,385.11528907624855,382.77435302734377]},{"page":143,"text":"tically shown in Fig. 6.12. Such a point should not be confused with the triple","rect":[53.81426239013672,403.6684875488281,385.15015447809068,394.7140197753906]},{"page":143,"text":"point common for the isotropic, nematic and SmA phases. In Fig. 6.12, for","rect":[53.81426239013672,415.6280212402344,385.1590818008579,406.6934814453125]},{"page":143,"text":"homologues with alkyl chains shorter than lcr, the N–SmA transition is second","rect":[53.81426239013672,427.5875549316406,385.1830681901313,418.6330871582031]},{"page":143,"text":"order and shown by the dashed curve. With increasing chain length the nematic","rect":[53.81338119506836,439.5476379394531,385.1711810894188,430.61309814453127]},{"page":143,"text":"temperature range becomes narrower (like in Fig. 6.1) and, at TCP, the N–SmA","rect":[53.81338119506836,451.4504089355469,385.1422705645832,442.515869140625]},{"page":143,"text":"transition becomes first order (solid curve).","rect":[53.81338119506836,463.0115051269531,231.74288602377659,454.47540283203127]},{"page":143,"text":"Fig. 6.11 Sequence of","rect":[53.812843322753909,493.9911804199219,132.85701078040294,486.2613525390625]},{"page":143,"text":"phases on the temperature","rect":[53.812843322753909,503.8994140625,142.94515368723394,496.3050537109375]},{"page":143,"text":"scale (a) and qualitative","rect":[53.812843322753909,513.818603515625,135.96137378000737,506.2242736816406]},{"page":143,"text":"dependence of orientational","rect":[53.812843322753909,523.7945556640625,148.6732759877688,516.2001953125]},{"page":143,"text":"order parameter S in the","rect":[53.812843322753909,533.7705078125,136.425881836648,526.1761474609375]},{"page":143,"text":"nematic and smectic A phases","rect":[53.812843322753909,543.6897583007813,155.31689913749018,536.0953979492188]},{"page":143,"text":"(b). With increasing","rect":[53.812843322753909,553.6657104492188,122.83740753321573,546.0713500976563]},{"page":143,"text":"increment DS the N–A","rect":[53.812843322753909,562.0415649414063,130.2763728887772,555.7679443359375]},{"page":143,"text":"transition acquires more","rect":[53.812843322753909,573.6176147460938,136.39289233469487,566.0232543945313]},{"page":143,"text":"features of first order","rect":[53.812843322753909,581.8097534179688,125.84362119300057,575.9425048828125]},{"page":143,"text":"transition","rect":[53.812843322753909,591.7603149414063,85.87183136134073,585.91845703125]},{"page":143,"text":"b","rect":[265.21270751953127,544.1485595703125,271.3148761323439,536.8433227539063]},{"page":143,"text":"S","rect":[278.32672119140627,549.9608764648438,283.1573676367824,544.2308349609375]},{"page":143,"text":"SmA","rect":[292.7646789550781,571.0624389648438,309.9278688806561,565.2044067382813]},{"page":144,"text":"6.3 Nematic–Smectic A Transition","rect":[53.812843322753909,42.55673599243164,172.92102569727823,36.68105697631836]},{"page":144,"text":"Fig. 6.12 Phase diagram","rect":[53.812843322753909,67.58130645751953,140.8358025395392,59.85148620605469]},{"page":144,"text":"“isotropic liquid–nematic","rect":[53.812843322753909,77.4895248413086,140.6640410163355,69.89517211914063]},{"page":144,"text":"–smectic A” with a tricritical","rect":[53.812843322753909,85.65617370605469,153.48678306784692,79.81436157226563]},{"page":144,"text":"point TCP at temperature Tcr","rect":[53.812843322753909,97.3846664428711,153.01018430830926,89.79031372070313]},{"page":144,"text":"and length lcr of alkyl chains","rect":[53.812843322753909,107.3603744506836,152.56685336112299,99.74908447265625]},{"page":144,"text":"Fig. 6.13 Phase diagram","rect":[53.812843322753909,204.57901000976563,136.79480217332827,196.64599609375]},{"page":144,"text":"“isotropic phase (Iso)–nematic","rect":[53.812843322753909,214.43051147460938,148.69698474192144,206.83615112304688]},{"page":144,"text":"(N)–smectic A(SmA)–reentrant","rect":[53.812843322753909,224.0677947998047,154.58756736960474,216.81210327148438]},{"page":144,"text":"nematic (Re-N)” of p-octyloxy-","rect":[53.812843322753909,234.38241577148438,152.466401039192,226.78805541992188]},{"page":144,"text":"cyanobiphenyl inthe","rect":[53.812843322753909,244.35836791992188,119.10352465891362,236.76400756835938]},{"page":144,"text":"pressure–temperature","rect":[53.812843322753909,254.27761840820313,120.4793028571558,247.5468292236328]},{"page":144,"text":"coordinates [14]","rect":[53.812843322753909,263.7370910644531,104.32543271643807,256.6591796875]},{"page":144,"text":"2.0","rect":[221.4222869873047,226.04100036621095,232.5355660117074,220.2728729248047]},{"page":144,"text":"1.5","rect":[221.4222869873047,265.3283996582031,232.5355660117074,259.56024169921877]},{"page":144,"text":"1.0","rect":[221.4222869873047,304.603759765625,232.5355660117074,298.8356018066406]},{"page":144,"text":"0.5","rect":[221.4222869873047,343.787109375,232.5355660117074,338.0189514160156]},{"page":144,"text":"0","rect":[228.0902557373047,382.4888610839844,232.5355660117074,376.720703125]},{"page":144,"text":"50","rect":[232.0398712158203,389.05462646484377,240.93049399022304,383.2864685058594]},{"page":144,"text":"T","rect":[248.16024780273438,66.3836669921875,253.04266657951954,60.64261245727539]},{"page":144,"text":"N","rect":[274.3710021972656,89.83648681640625,280.14040703660916,84.0954360961914]},{"page":144,"text":"icr","rect":[299.8038635253906,166.70257568359376,307.4252839514148,160.598388671875]},{"page":144,"text":"SmA","rect":[322.7485656738281,144.55328369140626,340.06477052202856,138.51638793945313]},{"page":144,"text":"127","rect":[372.4981994628906,42.55673599243164,385.1898245254032,36.73185729980469]},{"page":144,"text":"6.3.4 Re-entrant Phases","rect":[53.812843322753909,456.836181640625,180.63924293253585,448.4693908691406]},{"page":144,"text":"As a rule, increasing pressure or decreasing temperature promotes a denser, more","rect":[53.812843322753909,486.6495056152344,385.13376653863755,477.7149658203125]},{"page":144,"text":"ordered phase. There are, however, cases when, upon increasing pressure or","rect":[53.812843322753909,498.6090393066406,385.1487058242954,489.67449951171877]},{"page":144,"text":"decreasing temperature, the smectic A phase is substituted by the nematic phase","rect":[53.812843322753909,510.568603515625,385.09790838434068,501.634033203125]},{"page":144,"text":"[13]. Therefore, a more symmetric phase reappears or “re-enters” into consider-","rect":[53.812843322753909,522.4713134765625,385.1507505020298,513.5368041992188]},{"page":144,"text":"ation. An example [14] is shown in Fig. 6.13. Following the horizontal line at","rect":[53.81184768676758,534.430908203125,385.179579330178,525.4963989257813]},{"page":144,"text":"constant pressure of about 2 kbar (1bar ¼ 106 dyn/cm2 or 105 Pa) from the right to","rect":[53.81086730957031,546.3910522460938,385.1431817155219,535.2604370117188]},{"page":144,"text":"the left we begin from the nematic phase then, with decreasing temperature, cross","rect":[53.81427764892578,558.3505859375,385.14218534575658,549.4160766601563]},{"page":144,"text":"the SmA phase and enter again the nematic phase. Similar sequence is observed on","rect":[53.81427764892578,570.3101196289063,385.17006770185005,561.3756103515625]},{"page":144,"text":"the down-up way along the vertical line at constant temperature. Such an abnormal","rect":[53.81427764892578,582.2696533203125,385.1372209317405,573.3351440429688]},{"page":144,"text":"behavior can be explained with a molecular model, Fig. 6.14. In fact, reentrant","rect":[53.81427764892578,594.229248046875,385.17597825595927,585.2947387695313]},{"page":145,"text":"128","rect":[53.81200408935547,42.55636978149414,66.50361007838174,36.73149108886719]},{"page":145,"text":"6 Phase Transitions","rect":[318.28814697265627,42.55636978149414,385.1414230632714,36.68069076538086]},{"page":145,"text":"Fig. 6.14 Packing of molecular dimers in the nematic (a), smectic A (b) and reentrant nematic (c)","rect":[53.812843322753909,161.44488525390626,385.1711739884107,153.71507263183595]},{"page":145,"text":"phases. The middle part of dimers formed by rigid biphenyl cores is broader than their end parts","rect":[53.81200408935547,171.35311889648438,385.16867526053707,163.75875854492188]},{"page":145,"text":"formed by molecular tails and the length of the dimers depends on the pressure and temperature","rect":[53.81200408935547,181.3291015625,382.6666502692652,173.7347412109375]},{"page":145,"text":"phases are observed in liquid crystals with strongly asymmetric polar molecules","rect":[53.812843322753909,206.19290161132813,385.1606484805222,197.2583465576172]},{"page":145,"text":"such","rect":[53.812843322753909,216.09063720703126,72.04895869306097,209.21791076660157]},{"page":145,"text":"as","rect":[77.20722961425781,216.09063720703126,85.48913206206515,211.4490509033203]},{"page":145,"text":"p-octyloxy-p0-cyanobiphenyl","rect":[90.6324691772461,218.1524658203125,206.83183152744364,208.4928436279297]},{"page":145,"text":"(8OCB).","rect":[212.02194213867188,217.75404357910157,246.6029018929172,209.19798278808595]},{"page":145,"text":"They","rect":[251.73031616210938,218.1524658203125,272.31560603192818,209.21791076660157]},{"page":145,"text":"form","rect":[277.447021484375,217.0,296.8278879988249,209.21791076660157]},{"page":145,"text":"pairs","rect":[301.97418212890627,218.1524658203125,321.4098245059128,209.21791076660157]},{"page":145,"text":"of","rect":[326.5581359863281,217.0,334.8499692520298,209.21791076660157]},{"page":145,"text":"antiparallel","rect":[339.9833679199219,218.1524658203125,385.14265306064677,209.21791076660157]},{"page":145,"text":"dipoles (or molecular dimers as earlier shown in Fig. 3.9) whose length may depend","rect":[53.813716888427737,230.11199951171876,385.10280695966255,221.1774444580078]},{"page":145,"text":"on temperature and pressure. Such dimers are building elements of mesophases.","rect":[53.813716888427737,242.0147705078125,385.1346706917453,233.08021545410157]},{"page":145,"text":"Then a subtle change inthe dimer geometry determines the packing structureshown","rect":[53.813716888427737,253.97430419921876,385.1764763932563,245.0397491455078]},{"page":145,"text":"in Fig. 6.14 that explains the re-entrance phenomena.","rect":[53.813716888427737,265.9338684082031,268.7191128304172,256.99932861328127]},{"page":145,"text":"6.4 Smectic A–Smectic C Transition","rect":[53.812843322753909,314.3564147949219,247.5565182576096,305.35614013671877]},{"page":145,"text":"6.4.1 Landau Expansion","rect":[53.812843322753909,345.888427734375,184.46163910721899,335.3343200683594]},{"page":145,"text":"In the smectic C phase a new feature appears, a uniform molecular tilt that is","rect":[53.812843322753909,373.5146179199219,385.1875039492722,364.580078125]},{"page":145,"text":"characterized by the two-component order parameter W exp(ij) [15]. For simplicity,","rect":[53.812843322753909,385.47418212890627,385.15667386557348,376.2408142089844]},{"page":145,"text":"we can fix the azimuth angle j and operate with a real order parameter W:","rect":[53.81185531616211,397.4337158203125,352.5217729336936,388.2003479003906]},{"page":145,"text":"gSmC ¼ gSmA þ 12A#2 þ 41C#4 þ 16D#6","rect":[141.61097717285157,432.08404541015627,296.9058997400697,411.6282043457031]},{"page":145,"text":"(6.26)","rect":[361.0561828613281,426.8496398925781,385.10555396882668,418.31353759765627]},{"page":145,"text":"The odd terms are absent due to the \u0005W symmetry. After minimization with","rect":[65.76653289794922,455.6454162597656,385.1175469498969,446.41204833984377]},{"page":145,"text":"respect to W we get","rect":[53.81450653076172,467.5481872558594,130.49973161289285,458.3148193359375]},{"page":145,"text":"# ¼ \u0005\bCA 1=2 ¼ \u0003aðT C\u0002 TcÞ\u00041=2","rect":[152.8290252685547,508.22149658203127,285.69001839241346,481.8930969238281]},{"page":145,"text":"that is the same temperature dependence of the order parameter W as presented in","rect":[53.812843322753909,531.7107543945313,385.14272395185005,522.4774169921875]},{"page":145,"text":"Fig. 6.10a. The inverse “soft-mode” susceptibility for the uniform tilt","rect":[53.81385040283203,543.6702880859375,333.723280502053,534.7357788085938]},{"page":145,"text":"zC\u00021 ¼ @@2#g2C ¼ aðT \u0002 TCAÞ;","rect":[164.44094848632813,581.0179443359375,272.8854516713037,557.90087890625]},{"page":145,"text":"(6.27)","rect":[361.0558776855469,574.7298583984375,385.1052487930454,566.1937255859375]},{"page":146,"text":"6.4 Smectic A–Smectic C Transition","rect":[53.812843322753909,42.55630874633789,179.54858917384073,36.68062973022461]},{"page":146,"text":"Fig. 6.15 Phase diagram","rect":[53.812843322753909,67.58130645751953,140.8358025395392,59.85148620605469]},{"page":146,"text":"“nematic- smectic A- smectic","rect":[53.812843322753909,75.73698425292969,155.28815600656987,69.89517211914063]},{"page":146,"text":"C” with the triple NAC point","rect":[53.812843322753909,87.4087142944336,153.7913942739016,79.81436157226563]},{"page":146,"text":"at temperature T* and mixture","rect":[53.812843322753909,97.3846664428711,155.34822985910894,89.7564468383789]},{"page":146,"text":"concentration M*","rect":[53.812843322753909,105.60813903808594,113.74937194971963,99.73246002197266]},{"page":146,"text":"T","rect":[268.4466552734375,66.38238525390625,273.32858605898385,60.64190673828125]},{"page":146,"text":"SmA","rect":[283.65887451171877,94.79399871826172,300.9733564572589,88.75769805908203]},{"page":146,"text":"129","rect":[372.4981994628906,42.62403869628906,385.1898245254032,36.73143005371094]},{"page":146,"text":"0","rect":[276.17864990234377,159.8834686279297,280.62112701652338,154.11900329589845]},{"page":146,"text":"as expected, has the same linear temperature dependence as in nematics, see","rect":[53.812843322753909,211.74758911132813,385.11289251520005,202.8130340576172]},{"page":146,"text":"Eq. 6.10 and SmA. This is a typical picture for second order transition.","rect":[53.812843322753909,223.65036010742188,340.5150417854953,214.71580505371095]},{"page":146,"text":"However, if you take into account the “interaction” of SmA and SmC order","rect":[65.76486206054688,235.60992431640626,385.14376197663918,226.6753692626953]},{"page":146,"text":"parameters, a cross term r12#2 would result in the appearance of the NAC triple","rect":[53.812843322753909,248.20550537109376,385.13208807184068,236.4639892578125]},{"page":146,"text":"point in the phase diagram [8, 16], see Fig. 6.15. In this case, the phase transition","rect":[53.813167572021487,259.5294189453125,385.1569756608344,250.53509521484376]},{"page":146,"text":"lines might correspond to either second or first order transitions; it depends on","rect":[53.813167572021487,271.48895263671877,385.16896906903755,262.55438232421877]},{"page":146,"text":"parameters of the Landau expansion. In experiment, such a phase diagram may be","rect":[53.813167572021487,283.448486328125,385.1510394878563,274.513916015625]},{"page":146,"text":"observed when a content of binary mixtures is varied.","rect":[53.813167572021487,295.4080505371094,270.7423061897922,286.4735107421875]},{"page":146,"text":"6.4.2 Influence of External Fields","rect":[53.812843322753909,345.4350891113281,230.48107032511397,334.8809814453125]},{"page":146,"text":"In the framework of the Landau theory one can analyze the influence of a magnetic","rect":[53.812843322753909,373.0611877441406,385.17954290582505,364.12664794921877]},{"page":146,"text":"or an electric field on the phase symmetry and order parameters [17]. Here we","rect":[53.812843322753909,385.020751953125,385.1705707378563,376.086181640625]},{"page":146,"text":"consider the magnetic field influence on the temperature of the smectic A–C","rect":[53.812843322753909,396.98028564453127,385.1288325470283,388.04571533203127]},{"page":146,"text":"transition [11]. Let the magnetic field is applied along the smectic layer normal.","rect":[53.812843322753909,408.9398498535156,385.1775784065891,400.00531005859377]},{"page":146,"text":"Then, it is sufficient to add the field term to the expansion (6.26).","rect":[53.812843322753909,420.8993835449219,317.5775112679172,411.96484375]},{"page":146,"text":"gSmC ¼ gSmA þ 12aðT \u0002 TCAÞ#2 \u0002 21waðHnÞ2","rect":[130.90719604492188,455.430419921875,307.55513069710096,435.0941162109375]},{"page":146,"text":"We assumed small tilt angles and disregard the term with coefficients C and D.","rect":[65.76496887207031,478.9976501464844,385.1816372444797,470.0531311035156]},{"page":146,"text":"For small W the scalar product is Hn(1\u0002W2/2) and after minimization we obtain","rect":[53.81294250488281,490.9571838378906,372.60418025067818,479.9062194824219]},{"page":146,"text":"aðT \u0002 TCAÞ# þ waH2# ¼0","rect":[165.46104431152345,516.462158203125,273.5324334245063,505.1877746582031]},{"page":146,"text":"Finely, the temperature shift of the A–C phase transition point by the magnetic","rect":[65.76618194580078,540.0426025390625,385.1819232769188,531.1080932617188]},{"page":146,"text":"field is given by:","rect":[53.814144134521487,552.0021362304688,121.56747300693582,543.067626953125]},{"page":146,"text":"T \u0002 ðTCA \u0002 1waH2Þ ¼0","rect":[170.44564819335938,582.1552734375,268.54741755536568,566.19677734375]},{"page":146,"text":"a","rect":[219.443359375,586.6227416992188,224.42046443036566,582.0906982421875]},{"page":146,"text":"(6.28)","rect":[361.0558776855469,581.418212890625,385.1052487930454,572.882080078125]},{"page":147,"text":"130","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.73106384277344]},{"page":147,"text":"Therefore, for positive wa (or ea in case","rect":[65.76496887207031,68.2883529663086,236.62374151422348,59.35380554199219]},{"page":147,"text":"temperature reduced with increasing field.","rect":[53.81368637084961,80.24788665771485,223.67532010580784,71.31333923339844]},{"page":147,"text":"of","rect":[241.25045776367188,66.22653198242188,249.54232154695166,59.35380554199219]},{"page":147,"text":"the","rect":[254.05258178710938,66.22653198242188,266.27634466363755,59.35380554199219]},{"page":147,"text":"electric","rect":[270.8194580078125,66.22653198242188,300.4093402691063,59.35380554199219]},{"page":147,"text":"6 Phase Transitions","rect":[318.2889709472656,42.55594253540039,385.1422470378808,36.68026351928711]},{"page":147,"text":"field) the transition","rect":[304.975341796875,67.88993072509766,385.1555108170844,59.35380554199219]},{"page":147,"text":"6.5 Dynamics of Order Parameter","rect":[53.812843322753909,125.98899841308594,237.69825538130014,114.63407897949219]},{"page":147,"text":"6.5.1 Landau-Khalatnikov Approach","rect":[53.812843322753909,155.2142333984375,244.37075775956274,144.66012573242188]},{"page":147,"text":"Going back to the beginning of this section, let us recall the conditions of the","rect":[53.812843322753909,182.84036254882813,385.17261541559068,173.9058074951172]},{"page":147,"text":"thermodynamic equilibrium giving the free energy density minimum: dg=d\u0002 ¼0","rect":[53.812843322753909,195.1286163330078,385.17852107099068,185.19801330566407]},{"page":147,"text":"and d2\u0002=d\u00022 >0.What would happen if these conditions are not fulfilled? For","rect":[53.81379318237305,207.08836364746095,385.15020118562355,196.2772216796875]},{"page":147,"text":"example, due to disturbance by an external field, the order parameter of the system","rect":[53.81235885620117,218.66244506835938,385.1691660749968,209.72789001464845]},{"page":147,"text":"may become different from the equilibrium value. Then, after switching the field","rect":[53.81235885620117,230.62197875976563,385.1143426041938,221.6874237060547]},{"page":147,"text":"off, the order parameter will relax to its equilibrium value. The problem is how","rect":[53.81235885620117,242.58154296875,385.12035894348949,233.64698791503907]},{"page":147,"text":"to find its relaxation time? Below we shall only consider the second order transition","rect":[53.81235885620117,254.54107666015626,385.14821711591255,245.6065216064453]},{"page":147,"text":"and only a weak deviation from the equilibrium, i.e., small values of derivative","rect":[53.81235885620117,266.5006408691406,385.1701129741844,257.56610107421877]},{"page":147,"text":"dg/dZ.","rect":[53.81235885620117,278.5,80.7146191170383,269.5057067871094]},{"page":147,"text":"We neglect fluctuations of the order parameter, and assume that there is a simple","rect":[65.7643814086914,290.4197082519531,385.1313861675438,281.48516845703127]},{"page":147,"text":"linear relationship between a torque dg/dZ and a relaxation rate dZ/dt. Physically it","rect":[53.8123664855957,302.4190673828125,385.1700883633811,293.4247741699219]},{"page":147,"text":"means that the steeper potential well g(Z) (larger dg/dZ), the faster is relaxation","rect":[53.81134033203125,314.32183837890627,385.1680840592719,305.3275451660156]},{"page":147,"text":"(larger dZ/dt) of the induced order parameter. Hence the Landau–Khalatnikov","rect":[53.81134033203125,326.2415466308594,385.09543646051255,317.2870788574219]},{"page":147,"text":"equation reads","rect":[53.81135940551758,338.091552734375,112.70245756255344,329.26654052734377]},{"page":147,"text":"d\u0002 ¼ \u0002G@g","rect":[194.34950256347657,366.3078308105469,244.6423272233344,352.4618225097656]},{"page":147,"text":"dt","rect":[195.6525115966797,373.3658752441406,203.42675645420145,366.3736267089844]},{"page":147,"text":"(6.29)","rect":[361.0548400878906,368.0708312988281,385.10421119538918,359.53472900390627]},{"page":147,"text":"where the rate","rect":[53.81318283081055,397.0,116.80242956353986,389.8583984375]},{"page":147,"text":"temperature.","rect":[53.813167572021487,410.75250244140627,104.15858121420627,402.83392333984377]},{"page":147,"text":"controlling","rect":[122.52311706542969,398.79296875,166.43711176923285,389.8583984375]},{"page":147,"text":"coefficient","rect":[172.14385986328126,397.0,214.38554246494364,389.8583984375]},{"page":147,"text":"G","rect":[220.06640625,396.63153076171877,226.69591018374707,389.8185729980469]},{"page":147,"text":"is","rect":[232.41461181640626,397.0,239.04411710845188,389.8583984375]},{"page":147,"text":"considered","rect":[244.70606994628907,397.0,288.0098809342719,389.8583984375]},{"page":147,"text":"to","rect":[293.70367431640627,397.0,301.4778985123969,390.8743896484375]},{"page":147,"text":"be","rect":[307.12890625,397.0,316.55554998590318,389.8583984375]},{"page":147,"text":"independent","rect":[322.25335693359377,398.79296875,371.2161088711936,389.8583984375]},{"page":147,"text":"of","rect":[376.859130859375,397.0,385.1509946426548,389.8583984375]},{"page":147,"text":"6.5.2 Relaxation Rate","rect":[53.812843322753909,453.82940673828127,169.5626047465345,445.2952880859375]},{"page":147,"text":"The relaxation rate","rect":[53.812843322753909,482.0,132.8313678569969,474.54083251953127]},{"page":147,"text":"without B-term:","rect":[53.81185531616211,493.4129333496094,117.74078233310769,486.5003662109375]},{"page":147,"text":"can","rect":[137.07981872558595,482.0,150.95598689130316,476.0]},{"page":147,"text":"be","rect":[155.14968872070313,482.0,164.57633245660629,474.54083251953127]},{"page":147,"text":"found","rect":[168.80088806152345,482.0,192.02407160810004,474.54083251953127]},{"page":147,"text":"from","rect":[196.21676635742188,482.0,215.59763287187179,474.54083251953127]},{"page":147,"text":"the","rect":[219.83712768554688,482.0,232.06089056207504,474.54083251953127]},{"page":147,"text":"equilibrium","rect":[236.26455688476563,483.4753723144531,283.0025706160124,474.54083251953127]},{"page":147,"text":"Landau","rect":[287.1883239746094,482.0,317.1305779557563,474.54083251953127]},{"page":147,"text":"expansion","rect":[321.4019470214844,483.4753723144531,361.8160332780219,474.54083251953127]},{"page":147,"text":"(6.1)","rect":[366.0943603515625,483.0769348144531,385.15767799226418,474.54083251953127]},{"page":147,"text":"g ¼ g0 ¼ 12aðT \u0002 TcÞZ2 þ 41CZ4 þ \u0003\u0003\u0003","rect":[140.08201599121095,529.9656372070313,297.24244630020999,509.62933349609377]},{"page":147,"text":"Its solution for T < Tc has been found above. The equilibrium value of the order","rect":[65.7662582397461,553.5328369140625,385.14653907624855,544.5983276367188]},{"page":147,"text":"parameter is:","rect":[53.81367111206055,565.4356079101563,106.12007005283425,556.5010986328125]},{"page":147,"text":"Z\u0002 ¼ ½aðTc \u0002 TÞ=C\u00061=2","rect":[175.59945678710938,592.2444458007813,262.9185034998353,579.78076171875]},{"page":147,"text":"(6.30)","rect":[361.0561828613281,591.5073852539063,385.10555396882668,582.9712524414063]},{"page":148,"text":"6.5 Dynamics of Order Parameter","rect":[53.812843322753909,44.274620056152347,169.461274086067,36.62946701049805]},{"page":148,"text":"131","rect":[372.4981689453125,42.55594253540039,385.18979400782509,36.73106384277344]},{"page":148,"text":"Eq.(6.29) reads:","rect":[65.76496887207031,68.2883529663086,130.08708818027567,59.35380554199219]},{"page":148,"text":"ddZt ¼ \u0002G½aðT \u0002 TcÞZ þ CZ3\u0006","rect":[159.90933227539063,103.11160278320313,280.75868165177249,82.51631927490235]},{"page":148,"text":"(6.31)","rect":[361.0563049316406,97.8174819946289,385.10567603913918,89.28135681152344]},{"page":148,"text":"It is convenient to introduce a difference of the order parameters for the distorted","rect":[65.76665496826172,126.5564193725586,385.1734551530219,117.62187194824219]},{"page":148,"text":"and equilibrium medium dZ ¼ Z \u0002 Z\u0002:","rect":[53.81462860107422,138.51596069335938,209.16004883927247,129.2825927734375]},{"page":148,"text":"Then, for the high symmetry (high T) phase, Z\u0002 ¼ 0 and the linearized","rect":[65.76665496826172,150.5153350830078,385.1494988541938,141.52101135253907]},{"page":148,"text":"Landau–Khalatnikov equation for dZ is given by:","rect":[53.81464385986328,162.43505859375,263.0820146329124,153.20169067382813]},{"page":148,"text":"ddZt \b \u0002GaðT \u0002 TcÞdZ ¼ \u0002dtZ","rect":[156.85040283203126,196.578369140625,282.08726760562208,175.70419311523438]},{"page":148,"text":"(6.32)","rect":[361.0559997558594,191.28334045410157,385.1053708633579,182.74720764160157]},{"page":148,"text":"with a characteristic relaxation time","rect":[53.81433868408203,216.99630737304688,198.62918890191879,210.1235809326172]},{"page":148,"text":"1","rect":[201.71340942382813,239.0234375,206.69051447919379,232.2901611328125]},{"page":148,"text":"ths ¼ GaðT \u0002 TcÞ ¼ aðT \u0002 TcÞ","rect":[156.68115234375,255.10699462890626,280.6572610293503,241.16543579101563]},{"page":148,"text":"(6.33)","rect":[361.0564270019531,247.51112365722657,385.10579810945168,238.97499084472657]},{"page":148,"text":"Coefficient g ¼ G\u00021 is a kind of friction coefficient (viscosity for liquid crystals)","rect":[65.76677703857422,277.6673889160156,385.25090919343605,266.6163330078125]},{"page":148,"text":"controlling the relaxation process. In the Gauss system [g] ¼ s.erg/cm3 ¼ g.cm/s or","rect":[53.81350326538086,289.6269836425781,385.2659848770298,278.5760498046875]},{"page":148,"text":"Poise. In the SI system [g] ¼ s\u0003J/m3 ¼ s\u0003N/m2 or Pa\u0003s (1 Pa\u0003s ¼ 10 P).","rect":[53.814659118652347,301.5297546386719,336.8939175179172,290.47900390625]},{"page":148,"text":"For the low-symmetry phase we make linearization of the right part of Eq. 6.31","rect":[65.7661361694336,313.4895324707031,385.1598748307563,304.5350646972656]},{"page":148,"text":"\u0002 GZ½ðT \u0002 TcÞa þ CZ2\u0006 ¼ \u0002GZ½ðT \u0002 TcÞa þ CðZ\u00022 þ 2Z\u0002dZ þ \u0003\u0003\u0003Þ\u0006","rect":[83.89275360107422,339.0513000488281,357.2909387318506,327.72003173828127]},{"page":148,"text":"In the brackets, only term 2\u0002\u0002d\u0002C includes increment dZ. Thus we keep it and","rect":[65.76497650146485,362.6317443847656,385.14385310224068,353.39837646484377]},{"page":148,"text":"then ignore higher order term with (dZ)2:","rect":[53.81296920776367,374.5345153808594,220.9375901944358,363.4838562011719]},{"page":148,"text":"ddZ=dt ¼ \u0002GZ2CZ\u0002dZ ¼ \u00022GCðZ\u0002 þ dZÞZ\u0002dZ \b \u00022GCZ\u00022dZ","rect":[95.73054504394531,402.1033935546875,343.2720210235908,388.8240966796875]},{"page":148,"text":"Finally, using Eq. 6.30 for the equilibrium order parameter, we exclude C and","rect":[65.76656341552735,425.6040954589844,385.14739314130318,416.6595764160156]},{"page":148,"text":"obtain for the low-symmetry phase:","rect":[53.81554412841797,437.4869079589844,195.9586778409202,428.5523681640625]},{"page":148,"text":"dddtZ ¼ 2GaðT \u0002 TcÞdZ ¼ \u0002dtZ","rect":[155.71754455566407,472.6705627441406,283.22020217593458,451.79638671875]},{"page":148,"text":"(6.34)","rect":[361.05615234375,467.37554931640627,385.10552345124855,458.83941650390627]},{"page":148,"text":"with a relaxation time","rect":[53.814491271972659,494.1094665527344,142.32534063531723,487.23675537109377]},{"page":148,"text":"1","rect":[198.48463439941407,517.0994262695313,203.46173945477973,510.36614990234377]},{"page":148,"text":"tls ¼ 2GaðTc \u0002 TÞ ¼ 2aðTc \u0002 TÞ","rect":[152.432861328125,533.1829833984375,284.84943021880346,519.2417602539063]},{"page":148,"text":"(6.35)","rect":[361.0558776855469,525.587158203125,385.1052487930454,516.9912719726563]},{"page":148,"text":"The results (6.33) and (6.35) are of principal importance. We already know that","rect":[65.76622772216797,556.7066650390625,385.138075423928,547.71240234375]},{"page":148,"text":"at the second order transition the structural susceptibility diverges (Curie law). Now","rect":[53.81321334838867,568.6661987304688,385.1391882891926,559.731689453125]},{"page":148,"text":"we see that relaxation times diverge as well, i.e., on approaching the transition from","rect":[53.81321334838867,580.6655883789063,385.1361155378874,571.6712646484375]},{"page":148,"text":"any side, the relaxation of the order parameter becomes slower and slower and, at","rect":[53.81220626831055,592.5852661132813,385.1789689786155,583.6507568359375]},{"page":149,"text":"132","rect":[53.812843322753909,42.55630874633789,66.50444931178018,36.73143005371094]},{"page":149,"text":"Fig. 6.16 Softening of the","rect":[53.812843322753909,67.58130645751953,146.05544421457769,59.85148620605469]},{"page":149,"text":"N–I phase transition:","rect":[53.812843322753909,77.4895248413086,125.40872673239771,69.89517211914063]},{"page":149,"text":"relaxation time of the","rect":[53.812843322753909,85.65617370605469,127.53245684885502,79.81436157226563]},{"page":149,"text":"orientational order parameter","rect":[53.812843322753909,97.3846664428711,153.44447415930919,89.79031372070313]},{"page":149,"text":"as a function of temperature","rect":[53.812843322753909,107.36067962646485,150.08205554270269,99.76632690429688]},{"page":149,"text":"in p-pentyl-cyanobiphenyl","rect":[53.812843322753909,117.33663177490235,143.8208589955813,109.74227905273438]},{"page":149,"text":"(5CB)","rect":[53.812843322753909,126.91716766357422,74.94859403235604,119.61066436767578]},{"page":149,"text":"0.6","rect":[213.21530151367188,66.52958679199219,224.3259377797828,60.7628173828125]},{"page":149,"text":"0.5","rect":[213.21530151367188,87.42832946777344,224.3259377797828,81.66156005859375]},{"page":149,"text":"0.4","rect":[213.21530151367188,108.32548522949219,224.3259377797828,102.5587158203125]},{"page":149,"text":"0.3","rect":[213.21530151367188,129.22422790527345,224.3259377797828,123.45745849609375]},{"page":149,"text":"0.2","rect":[213.21530151367188,150.12217712402345,224.3259377797828,144.35540771484376]},{"page":149,"text":"0.1","rect":[213.21530151367188,171.02012634277345,224.3259377797828,165.25335693359376]},{"page":149,"text":"24","rect":[234.72756958007813,189.25302124023438,243.61608365380625,183.63021850585938]},{"page":149,"text":"26","rect":[252.0569610595703,189.39698791503907,260.94547513329845,183.63021850585938]},{"page":149,"text":"28","rect":[269.05224609375,189.39698791503907,277.94074490868908,183.63021850585938]},{"page":149,"text":"30","rect":[286.2145690917969,189.39698791503907,295.10306790673595,183.63021850585938]},{"page":149,"text":"6 Phase Transitions","rect":[318.2889709472656,42.55630874633789,385.1422470378808,36.68062973022461]},{"page":149,"text":"32 34 36 38 40","rect":[303.37689208984377,189.39698791503907,380.91471341454845,183.63021850585938]},{"page":149,"text":"T, oC","rect":[298.07177734375,201.46202087402345,316.5042294102533,195.0913848876953]},{"page":149,"text":"the transition, in the linear approximation considered above, the relaxation times","rect":[53.812843322753909,238.21743774414063,385.1397744570847,229.2828826904297]},{"page":149,"text":"are infinite (softening of the transition).","rect":[53.812843322753909,250.12020874023438,213.09316678549534,241.18565368652345]},{"page":149,"text":"Had we taken higher order terms of","rect":[65.76486206054688,262.07977294921877,218.04735694734229,253.1452178955078]},{"page":149,"text":"dZ","rect":[222.2181854248047,261.9901123046875,233.824801174958,252.84640502929688]},{"page":149,"text":"into","rect":[238.07923889160157,261.0,253.62770930341254,253.1452178955078]},{"page":149,"text":"account,","rect":[257.8482971191406,261.0,291.4139065315891,254.16119384765626]},{"page":149,"text":"the","rect":[295.6862487792969,261.0,307.91001165582505,253.1452178955078]},{"page":149,"text":"divergence","rect":[312.1127014160156,262.07977294921877,356.0067828960594,253.1452178955078]},{"page":149,"text":"would","rect":[360.26019287109377,261.0,385.13579646161568,253.1452178955078]},{"page":149,"text":"disappear. The Curie-type order parameter relaxation has been studied on a typical","rect":[53.812862396240237,274.039306640625,385.1228471524436,265.104736328125]},{"page":149,"text":"nematic (5CB), see Fig. 6.16. The measurements have been made using a pulse","rect":[53.812862396240237,285.9988708496094,385.1437457866844,277.0045471191406]},{"page":149,"text":"pyroelectric technique [18]. As the nematic–isotropic transition in 5CB is weak first","rect":[53.812862396240237,297.9584045410156,385.15766770908427,288.9640808105469]},{"page":149,"text":"order, it clearly demonstrates some features of the softening: the relaxation time of","rect":[53.81186294555664,309.91796875,385.1457456192173,300.9833984375]},{"page":149,"text":"the orientational order parameter on the nematic side of the NI transition increases","rect":[53.81186294555664,321.8774719238281,385.08502592192846,312.94293212890627]},{"page":149,"text":"five times.","rect":[53.81186294555664,331.7751770019531,95.76188321615939,324.9024658203125]},{"page":149,"text":"There are also other reasons that truncate the order parameter divergence such","rect":[65.76388549804688,345.7397766113281,385.24208918622505,336.80523681640627]},{"page":149,"text":"as spatial inhomogeneities or external fields. For example, to describe a spatial","rect":[53.81186294555664,357.6993408203125,385.126844955178,348.7647705078125]},{"page":149,"text":"inhomogeneous system, a term quadratic in the gradient of the order parameter","rect":[53.81187438964844,369.65887451171877,385.12078224031105,360.72430419921877]},{"page":149,"text":"G(rZ)2 must be added to the density of free energy and all the Landau","rect":[53.81187438964844,381.6199951171875,385.1726922135688,370.5689392089844]},{"page":149,"text":"expansion should be integrated over the system volume:","rect":[53.81277084350586,393.57952880859377,288.6276078946311,384.64495849609377]},{"page":149,"text":"F ¼ ð ½g0 þ 12aðT \u0002 TcÞZ2 þ 14CZ4 þ GðrZÞ2 þ \u0003\u0003\u0003\u0006dV","rect":[105.98278045654297,429.9980163574219,332.4935683575984,407.87579345703127]},{"page":149,"text":"(6.36)","rect":[361.0561828613281,423.1653137207031,385.10555396882668,414.62921142578127]},{"page":149,"text":"Then in the relaxation equation an additional term appears. E.g., in the low-","rect":[65.76653289794922,453.5477600097656,385.1235288223423,444.61322021484377]},{"page":149,"text":"symmetry phase, at T < Tc:","rect":[53.81450653076172,465.4505310058594,166.161757064553,456.5159912109375]},{"page":149,"text":"@@dtZ ¼ \u0002\bdtZ \u0002 2GGDðdZÞ","rect":[159.68272399902345,503.57269287109377,273.6905862246628,479.6874694824219]},{"page":149,"text":"where D is Laplace operator. Then, the inhomogeneous distribution of dZ(r) can be","rect":[53.81403732299805,527.0614624023438,385.1538776226219,517.7982177734375]},{"page":149,"text":"expanded in the Fourier series of spatial harmonics and, for each Fourier compo-","rect":[53.81403732299805,539.02099609375,385.12301002351418,530.0864868164063]},{"page":149,"text":"nents with number m, we have different Landau–Khalatnikov equations","rect":[53.81403732299805,550.9805297851563,342.9271584902878,542.0460205078125]},{"page":149,"text":"ddZm","rect":[188.4018096923828,574.7559814453125,210.3416101859198,565.2147216796875]},{"page":149,"text":"dZm","rect":[233.43466186523438,574.7559814453125,250.04995369177918,565.2147216796875]},{"page":149,"text":"¼\u0002","rect":[213.60862731933595,578.0,231.7531664678803,575.0]},{"page":149,"text":"dt","rect":[195.7657012939453,586.0892333984375,203.53994615146707,579.0969848632813]},{"page":149,"text":"tm","rect":[237.05999755859376,587.5658569335938,246.4246179984198,581.3082275390625]},{"page":150,"text":"6.6 Molecular Statistic Approach to Phase Transitions","rect":[53.812843322753909,44.274620056152347,238.82855685233393,36.68026351928711]},{"page":150,"text":"with different relaxation times","rect":[53.812843322753909,66.25641632080078,176.2585793887253,59.35380554199219]},{"page":150,"text":"1 ¼ 1 þ GGm2","rect":[190.2144012451172,96.77690124511719,252.38278267464004,82.48265838623047]},{"page":150,"text":"tm","rect":[187.72186279296876,104.41859436035156,197.08666633826355,98.21717071533203]},{"page":150,"text":"t","rect":[211.17274475097657,102.88863372802735,215.52273456936616,98.21717071533203]},{"page":150,"text":"Therefore, at T ¼ Tc, the relaxation time becomes","rect":[65.76653289794922,127.57511901855469,275.64548124419408,118.98173522949219]},{"page":150,"text":"situation in the helical phases as well.","rect":[53.81293869018555,139.87625122070313,206.49360318686252,130.9416961669922]},{"page":150,"text":"finite.","rect":[279.7117919921875,126.0,302.7159695198703,118.98216247558594]},{"page":150,"text":"We","rect":[306.7882080078125,126.0,320.6643756694969,119.1813735961914]},{"page":150,"text":"meet","rect":[324.74462890625,126.0,344.2151017911155,119.99813079833985]},{"page":150,"text":"133","rect":[372.4990539550781,42.55594253540039,385.19064850001259,36.73106384277344]},{"page":150,"text":"(6.37)","rect":[361.0561828613281,97.70413970947266,385.10555396882668,89.16801452636719]},{"page":150,"text":"the same","rect":[348.3082275390625,126.0,385.1278766460594,118.98216247558594]},{"page":150,"text":"6.6 Molecular Statistic Approach to Phase Transitions","rect":[53.812843322753909,177.17677307128907,340.52355079386396,166.43142700195313]},{"page":150,"text":"The problem is to derive the equation of state and thermodynamic functions of a","rect":[53.812843322753909,204.71908569335938,385.15869939996568,195.78453063964845]},{"page":150,"text":"particular liquid crystal phase from properties of constituting molecules (a form, a","rect":[53.812843322753909,216.67864990234376,385.15869939996568,207.7440948486328]},{"page":150,"text":"polarizability, chirality, etc.). The problem we are going to discuss is one of the","rect":[53.812843322753909,228.5814208984375,385.17258489801255,219.64686584472657]},{"page":150,"text":"most difficult in physics of liquid crystals and the aim of this chapter isvery modest:","rect":[53.812843322753909,240.54095458984376,385.2024064786155,231.6063995361328]},{"page":150,"text":"just to introduce the reader to the basic ideas of the theory with the help of","rect":[53.812843322753909,252.50051879882813,385.1487058242954,243.5659637451172]},{"page":150,"text":"comprehensive works of the others [2, 5, 19]. To consider the problem quantita-","rect":[53.812843322753909,264.4600524902344,385.0919431289829,255.46572875976563]},{"page":150,"text":"tively we need special methods of the statistical physics. In this context, the most","rect":[53.812843322753909,276.41961669921877,385.13276536533427,267.48504638671877]},{"page":150,"text":"useful function is free energy F, which is based microscopically on the so-called","rect":[53.812843322753909,288.3791198730469,385.0889824967719,279.444580078125]},{"page":150,"text":"partition function, see below. For the partition function, we need that energy","rect":[53.812843322753909,300.3386535644531,385.16762629560005,291.40411376953127]},{"page":150,"text":"spectrum of a molecular system, which is relevant to the problem under consider-","rect":[53.812843322753909,312.2982177734375,385.1507505020298,303.3636474609375]},{"page":150,"text":"ation. The energy spectrum is related to the entropy of the system and we would like","rect":[53.812843322753909,324.20098876953127,385.1377643413719,315.26641845703127]},{"page":150,"text":"to recall the microscopic sense of the entropy.","rect":[53.812843322753909,336.1605224609375,239.76742215658909,327.2259521484375]},{"page":150,"text":"6.6.1 Entropy, Partition Function and Free Energy","rect":[53.812843322753909,375.4462890625,316.5904672953626,364.8085021972656]},{"page":150,"text":"6.6.1.1 Entropy","rect":[53.812843322753909,403.38702392578127,126.61694422772894,393.974365234375]},{"page":150,"text":"We consider a small but macroscopic part of a larger molecular system. Even under","rect":[53.812843322753909,426.90771484375,385.2224362930454,417.97314453125]},{"page":150,"text":"equilibrium conditions such a subsystem can be found in any of a tremendous","rect":[53.812843322753909,438.81048583984377,385.22043241606908,429.87591552734377]},{"page":150,"text":"number n of statistical configurations or quantum states [2]. Any change ina","rect":[53.812843322753909,450.7700500488281,385.15967596246568,441.83551025390627]},{"page":150,"text":"position, velocity or internal motion of a particular molecule will bring our macro-","rect":[53.812843322753909,462.7295837402344,385.1537107071079,453.7950439453125]},{"page":150,"text":"scopic subsystem in the new state with energy En. The set of corresponding energy","rect":[53.812843322753909,474.6898498535156,385.25420466474068,465.75457763671877]},{"page":150,"text":"levels isextremelydenseaspicturedschematicallyinFig.6.17andwe mayconsider","rect":[53.8138542175293,486.6494140625,385.2044614395298,477.71484375]},{"page":150,"text":"a continuous distribution function w(E) ¼ w(En) of the probability for the subsys-","rect":[53.81386947631836,498.6091613769531,385.18828712312355,489.6744384765625]},{"page":150,"text":"tem to be in a state with energy E ¼ En. A number of the levels below a particular","rect":[53.8136100769043,510.6085205078125,385.2340024551548,501.6142272949219]},{"page":150,"text":"energy E is G(E). Now we would like to relate the entropy to this energy spectrum.","rect":[53.813533782958987,522.5283203125,385.23797269369848,513.5938110351563]},{"page":150,"text":"By definition, the dimensionless entropy is given by","rect":[65.76654815673828,534.4310913085938,276.01641169599068,525.49658203125]},{"page":150,"text":"s ¼ lnDGðEÞ","rect":[192.70761108398438,558.6888427734375,246.3319436465378,548.7383422851563]},{"page":150,"text":"(6.38)","rect":[361.0572204589844,557.9517822265625,385.1065915664829,549.4156494140625]},{"page":150,"text":"where DG is the so-called statistic weight of a macroscopic state of our molecular","rect":[53.81554412841797,582.3091430664063,385.1553891739048,573.294921875]},{"page":150,"text":"subsystem related to the formidable number of the microscopic quantum states.","rect":[53.81554412841797,594.2288208007813,385.15935941244848,585.2943115234375]},{"page":151,"text":"134","rect":[53.81303024291992,42.55655288696289,66.5046362319462,36.73167419433594]},{"page":151,"text":"E","rect":[105.13368225097656,66.38671875,110.46518891451196,60.643943786621097]},{"page":151,"text":"ΔE","rect":[100.24180603027344,121.250244140625,110.46518891451196,115.5074691772461]},{"page":151,"text":"E","rect":[153.6759796142578,66.38592529296875,159.00748627779323,60.643150329589847]},{"page":151,"text":"<E>","rect":[189.72409057617188,94.3406982421875,204.39172913443296,88.5979232788086]},{"page":151,"text":"Real","rect":[236.4678955078125,137.11001586914063,252.9020374257083,131.25526428222657]},{"page":151,"text":"6 Phase Transitions","rect":[318.2891540527344,42.55655288696289,385.14243014334957,36.68087387084961]},{"page":151,"text":"0","rect":[106.77879333496094,172.9655303955078,111.22304776513437,167.19876098632813]},{"page":151,"text":"States","rect":[120.19949340820313,177.6317596435547,142.86040121306065,171.59304809570313]},{"page":151,"text":"w(E)","rect":[184.18235778808595,186.79937744140626,200.6085187850468,179.2889862060547]},{"page":151,"text":"W(E)","rect":[251.2386474609375,186.79937744140626,269.4393140731328,179.2889862060547]},{"page":151,"text":"W(E)","rect":[323.8453369140625,186.79946899414063,342.04450816492968,179.2889862060547]},{"page":151,"text":"Fig. 6.17 Energy levels of a macroscopic quantum subsystem, probability w(E) for the subsystem","rect":[53.812843322753909,206.84619140625,385.1847252690314,199.1163787841797]},{"page":151,"text":"to be in a state En and probability W(E) for the subsystem to be within the energy interval between","rect":[53.811988830566409,216.6976318359375,385.1815237441532,209.103271484375]},{"page":151,"text":"E and E þ dE, more or less realistic and approximated by a rectangular","rect":[53.81303024291992,226.67361450195313,299.85418790442636,219.06231689453126]},{"page":151,"text":"The probability for a molecular subsystem to have energy in the interval betweenE","rect":[53.812843322753909,250.63064575195313,385.17951196111405,241.6960906982422]},{"page":151,"text":"and E þ dE reads:","rect":[53.812843322753909,261.26544189453127,129.23689134189676,253.6356964111328]},{"page":151,"text":"dGðEÞ","rect":[213.49559020996095,283.8441467285156,239.25011076079563,273.89361572265627]},{"page":151,"text":"WðEÞ ¼","rect":[177.6392364501953,290.5890808105469,210.68125179502875,280.6385498046875]},{"page":151,"text":"dE","rect":[220.80296325683595,295.14630126953127,231.89196008123123,288.154052734375]},{"page":151,"text":"(6.39)","rect":[361.0558166503906,289.8512878417969,385.10518775788918,281.315185546875]},{"page":151,"text":"where dG/dE is density of possible states on the energy scale. Since the energy of","rect":[53.81415939331055,315.6429748535156,385.15007911531105,306.6685791015625]},{"page":151,"text":"our subsystem under fixed experimental conditions fluctuates only negligibly about","rect":[53.81415939331055,327.6025085449219,385.1559892422874,318.66796875]},{"page":151,"text":"the average value <E>, the density of states and probability w(E) have extremely","rect":[53.81415939331055,339.5052795410156,385.1758660416938,330.57073974609377]},{"page":151,"text":"sharp maxima at E ¼ <E>, close to d-function shown as “realistic” function in","rect":[53.815162658691409,351.4648132324219,385.1440362077094,342.2314453125]},{"page":151,"text":"Fig. 6.17. Therefore, the normalization condition may be written as","rect":[53.816139221191409,363.42437744140627,325.72238554106908,354.48980712890627]},{"page":151,"text":"ð WðEÞdE \b WðhEiÞDE ¼1","rect":[162.7440643310547,397.8587341308594,277.9521111588813,375.73651123046877]},{"page":151,"text":"(6.40)","rect":[361.05780029296877,391.02581787109377,385.1072019180454,382.48968505859377]},{"page":151,"text":"On the other hand, the number of the quantum states DG in the DE interval is:","rect":[65.76818084716797,419.3674011230469,385.15595872470927,410.1041564941406]},{"page":151,"text":"dGðhEiÞ","rect":[80.32280731201172,431.66766357421877,113.76099027739719,421.7171325683594]},{"page":151,"text":"dE","rect":[236.49359130859376,429.3667907714844,247.58258813298904,422.3745422363281]},{"page":151,"text":"DG ¼ dE DE; that means DE ¼ DGdGðhEiÞ: Now the condition (6.40) on","rect":[53.816139221191409,445.27069091796877,385.17067805341255,428.8656921386719]},{"page":151,"text":"account of (6.39) results in:","rect":[53.812923431396487,456.4931335449219,165.21247727939676,447.95703125]},{"page":151,"text":"WðhEiÞ \u0003 DE ¼ dGðhEiÞwðhEiÞ \u0003 DG dE ¼1","rect":[119.29470825195313,485.91046142578127,319.69768611005318,469.2156677246094]},{"page":151,"text":"dE","rect":[193.89596557617188,490.4683532714844,204.98496240056716,483.4761047363281]},{"page":151,"text":"dGðhEiÞ","rect":[268.0453186035156,492.76922607421877,301.48349393950658,482.8186950683594]},{"page":151,"text":"Hence wðhEiÞDG ¼ 1 and the dimensionless entropy is found to be related to the","rect":[65.76580047607422,514.6473999023438,385.1725238628563,504.6968688964844]},{"page":151,"text":"distribution function w(En)","rect":[53.81379318237305,525.94775390625,162.0344937881626,517.3338012695313]},{"page":151,"text":"s ¼ lnDG ¼ \u0002lnwðhEiÞ ¼ \u0002hlnwðEnÞi","rect":[138.27081298828126,548.5435180664063,300.6902200137253,538.5925903320313]},{"page":151,"text":"(6.41)","rect":[361.0565490722656,547.8064575195313,385.10592017976418,539.2703247070313]},{"page":151,"text":"The transformation of average values in (6.41) follows from the statistical","rect":[65.76688385009766,570.1398315429688,385.1098771817405,561.205322265625]},{"page":151,"text":"independence of the events described by a distribution function (for w12 ¼ w1w2,","rect":[53.81482696533203,582.0426025390625,385.1832241585422,573.1080932617188]},{"page":151,"text":"lnw12 ¼ lnw1 þ lnw2, etc.).","rect":[53.814537048339847,593.6046142578125,166.01899381186252,585.068359375]},{"page":152,"text":"6.6 Molecular Statistic Approach to Phase Transitions","rect":[53.812843322753909,44.274620056152347,238.82855685233393,36.68026351928711]},{"page":152,"text":"135","rect":[372.4990539550781,42.55594253540039,385.19064850001259,36.62946701049805]},{"page":152,"text":"To have the entropy in the Boltzmann form we write","rect":[65.76496887207031,68.2883529663086,279.25891912652818,59.35380554199219]},{"page":152,"text":"S ¼ kBs ¼ \u0002kBhlnwðEnÞi","rect":[167.951904296875,92.54637908935547,271.0080911074753,82.59586334228516]},{"page":152,"text":"(6.42)","rect":[361.0558776855469,91.8093032836914,385.1052487930454,83.27317810058594]},{"page":152,"text":"Dimension of entropy is [erg/K] or [J/K]. To have an idea of the entropy value,","rect":[65.76624298095703,116.0699691772461,385.1372036507297,107.13542175292969]},{"page":152,"text":"let us take a tremendous number of points in the phase space, e.g. 10100. Then,","rect":[53.814205169677737,128.02957153320313,385.1371731331516,116.97911071777344]},{"page":152,"text":"s ¼ 230 and S ¼ 3\u000310\u000214 erg/K are extremely small values with respect to experi-","rect":[53.814231872558597,139.98959350585938,385.1787046035923,128.93870544433595]},{"page":152,"text":"mentally measured quantities.","rect":[53.8129768371582,151.94912719726563,174.25494046713596,143.0145721435547]},{"page":152,"text":"Up to now we considered number DG of the quantum states or “cells” in a","rect":[65.7649917602539,163.90869140625,385.1587604351219,154.64544677734376]},{"page":152,"text":"multidimensional","rect":[53.8129768371582,174.0,123.92645127842019,166.9336700439453]},{"page":152,"text":"configuration","rect":[128.9792022705078,175.86822509765626,182.24118891767035,166.9336700439453]},{"page":152,"text":"or","rect":[187.3795623779297,174.0,195.6714109024204,168.0]},{"page":152,"text":"phase","rect":[200.74806213378907,175.86822509765626,223.4337085552391,166.9336700439453]},{"page":152,"text":"space,","rect":[228.56011962890626,175.86822509765626,253.3102993538547,168.0]},{"page":152,"text":"formed","rect":[258.3551025390625,174.0,287.1625908952094,166.9336700439453]},{"page":152,"text":"by","rect":[292.3417663574219,175.86822509765626,302.2959832291938,166.9336700439453]},{"page":152,"text":"coordinates","rect":[307.35272216796877,174.0,353.4267922793503,166.9336700439453]},{"page":152,"text":"qi","rect":[358.5631103515625,175.75865173339845,365.64639175655216,169.0950927734375]},{"page":152,"text":"and","rect":[370.7425231933594,173.83673095703126,385.14626399091255,166.9341278076172]},{"page":152,"text":"momenta pi (i ¼ 1,2...l) where l is the number of degrees of freedom. Each cell","rect":[53.814414978027347,187.82827758789063,385.120253158303,178.87379455566407]},{"page":152,"text":"had volume of hl (h ¼ 2ph\u0002 is Planck constant). In general, these states include","rect":[53.81325912475586,199.73126220703126,385.0993121929344,188.62710571289063]},{"page":152,"text":"all possible degrees of freedom, such as translational and rotational motion of all","rect":[53.814205169677737,211.69082641601563,385.14607102939677,202.7562713623047]},{"page":152,"text":"molecules, their internal (atomic) motion, interactions with other molecules, etc.","rect":[53.814205169677737,223.25193786621095,385.14910550619848,214.71580505371095]},{"page":152,"text":"Now, in the classical limit, instead of DG we introduce a volume in the phase space","rect":[53.814205169677737,235.60992431640626,385.1400836773094,226.3466796875]},{"page":152,"text":"DpDq, in which a subsystem evolves in time. Additionally, to have the absolute","rect":[53.814205169677737,247.5694580078125,385.17395818902818,238.6050262451172]},{"page":152,"text":"value of the entropy, we introduce the volume of the elementary cell in the phase","rect":[53.815208435058597,259.52899169921877,385.0983356304344,250.5944366455078]},{"page":152,"text":"space ð2p\u0002hÞl and write the dimensionless entropy in the form","rect":[53.815208435058597,271.8271789550781,300.62165020585618,259.7607421875]},{"page":152,"text":"DpDq","rect":[220.29302978515626,288.89398193359377,243.56697932294379,279.74029541015627]},{"page":152,"text":"s ¼ lnð2ph\u0002Þl","rect":[192.5935516357422,304.47613525390627,244.22930413048167,286.8143310546875]},{"page":152,"text":"(6.43)","rect":[361.0561828613281,295.35003662109377,385.10555396882668,286.81390380859377]},{"page":152,"text":"6.6.1.2 Partition Function and Free Energy","rect":[53.81450653076172,350.3906555175781,245.43605891278754,340.9779968261719]},{"page":152,"text":"In the quantum-mechanical case, a probability w(En) for a subsystem","rect":[53.81450653076172,373.8547668457031,349.1742930281218,364.91998291015627]},{"page":152,"text":"energy En in a quantum state is given by the Gibbs distribution:","rect":[53.81394577026367,385.8144226074219,311.01277024814677,376.8798828125]},{"page":152,"text":"to","rect":[353.80499267578127,372.0,361.5792168717719,365.9361877441406]},{"page":152,"text":"have","rect":[366.26666259765627,372.0,385.1199420757469,364.92022705078127]},{"page":152,"text":"wn ¼ Z\u00021 exp \u0002En.kBT\u000B","rect":[166.30941772460938,418.01751708984377,272.7015831679293,400.1082763671875]},{"page":152,"text":"(6.44)","rect":[361.0561828613281,413.3028869628906,385.10555396882668,404.76678466796877]},{"page":152,"text":"where Z is a constant to be found from normalization procedure","rect":[53.81450653076172,441.5318908691406,311.61365545465318,432.59735107421877]},{"page":152,"text":"Xwn ¼ Z\u00021 Xexpð\u0002En.kBTÞ ¼ 1;","rect":[140.31161499023438,473.7348327636719,298.6592248646631,455.8258972167969]},{"page":152,"text":"n","rect":[145.69131469726563,477.83184814453127,149.1752784021791,474.5967102050781]},{"page":152,"text":"n","rect":[202.39312744140626,477.83184814453127,205.8770911463197,474.5967102050781]},{"page":152,"text":"and called a partition function that includes all degrees of freedom of the subsystem:","rect":[53.81452178955078,502.3498840332031,385.24690110752177,493.37548828125]},{"page":152,"text":"Z ¼ Xexpð\u0002En.kBTÞ","rect":[171.23936462402345,533.5330810546875,267.7420998965378,515.6241455078125]},{"page":152,"text":"n","rect":[195.65240478515626,537.6300659179688,199.1363684900697,534.3948974609375]},{"page":152,"text":"Using Eqs. 6.42 and 6.44 we write entropy in the form","rect":[65.76619720458985,561.128173828125,286.4052500593718,552.1936645507813]},{"page":152,"text":"(6.45)","rect":[361.0558776855469,528.8184204101563,385.1052487930454,520.2225341796875]},{"page":152,"text":"S ¼ \u0002kBhlnwðEnÞi ¼ \u0002kBDlnZ\u00021 exp \u0002En.kBT\u000BE ¼ \u0002kB lnZ\u00021 þ hETi;","rect":[70.69351959228516,596.5089111328125,366.6902307240381,575.4454956054688]},{"page":153,"text":"136","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.68026351928711]},{"page":153,"text":"6 Phase Transitions","rect":[318.2889709472656,42.55594253540039,385.1422470378808,36.68026351928711]},{"page":153,"text":"equivalent to","rect":[53.812843322753909,68.2883529663086,106.10927668622503,59.35380554199219]},{"page":153,"text":"hEi \u0002 TS ¼ kBT lnZ\u00021:","rect":[172.5406951904297,93.78318786621094,266.4274438588037,82.51895141601563]},{"page":153,"text":"(6.46)","rect":[361.05615234375,93.05631256103516,385.10552345124855,84.52018737792969]},{"page":153,"text":"As the macroscopic definition of the free energy is F ¼ hEi \u0002 TS; we relate the","rect":[65.76648712158203,117.70256042480469,385.17322576715318,107.77196502685547]},{"page":153,"text":"free energy to partition function:","rect":[53.81246566772461,129.33340454101563,185.60522325107645,120.39884948730469]},{"page":153,"text":"F ¼ kBT lnZ\u00021 ¼ \u0002kBT lnZ","rect":[162.51443481445313,153.88885498046876,276.20148057292058,143.56411743164063]},{"page":153,"text":"(6.47)","rect":[361.0551452636719,154.1581268310547,385.1045163711704,145.6219940185547]},{"page":153,"text":"Finally, on account of (6.45) the free energy acquires a desired microscopic","rect":[65.76549530029297,178.41885375976563,385.14136541559068,169.42453002929688]},{"page":153,"text":"form:","rect":[53.8134880065918,190.35848999023438,75.41709764072488,181.40402221679688]},{"page":153,"text":"F ¼ \u0002kBT lnXexpð\u0002En=kBTÞ","rect":[155.54849243164063,218.731689453125,283.43283475981908,204.61923217773438]},{"page":153,"text":"n","rect":[214.3452606201172,224.80870056152345,217.82922432503066,221.57354736328126]},{"page":153,"text":"(6.48)","rect":[361.0558166503906,215.94029235839845,385.10518775788918,207.40415954589845]},{"page":153,"text":"This formula is a base for calculation of all thermodynamic functions of any","rect":[65.76615142822266,249.27005004882813,385.14202204755318,240.3354949951172]},{"page":153,"text":"system if the energy spectrum of the latter is known. We shall illustrate this","rect":[53.81412887573242,261.2296142578125,385.14401640044408,252.29505920410157]},{"page":153,"text":"approach considering two simple systems, the ideal gas and a liquid, both consisted","rect":[53.81412887573242,273.18914794921877,385.1549920182563,264.25457763671877]},{"page":153,"text":"of spherical particles.","rect":[53.81412887573242,285.1487121582031,140.28237576987034,276.21417236328127]},{"page":153,"text":"In the classical case, instead of discrete distribution wn (En) we have a continu-","rect":[65.76615142822266,296.7305908203125,385.1658872207798,288.1737060546875]},{"page":153,"text":"ous probability function r (p,q) that is probability to have a subsystem with given","rect":[53.814144134521487,309.06842041015627,385.16692439130318,300.13385009765627]},{"page":153,"text":"momentum p and co-ordinate q in the configuration space:","rect":[53.815147399902347,320.97119140625,290.0554338223655,312.03662109375]},{"page":153,"text":"rðp;qÞ ¼ Aexpð\u0002Eðp;qÞ=kBTÞ","rect":[156.9656524658203,350.2878112792969,282.0167275820847,335.2784118652344]},{"page":153,"text":"and, in the expression for the free energy, instead of the partition function we have a","rect":[53.81350326538086,373.7980651855469,385.15930975152818,364.863525390625]},{"page":153,"text":"configuration integral","rect":[53.81350326538086,385.75762939453127,140.53259141513895,376.82305908203127]},{"page":153,"text":"=","rect":[200.52389526367188,406.997314453125,204.00785896858535,400.0459289550781]},{"page":153,"text":"F ¼ \u0002kBT lnð exp \u0002Eðp;qÞ=kBT\u000BdG","rect":[141.83958435058595,424.7269287109375,297.1621577423408,402.6047058105469]},{"page":153,"text":"where dG ¼ dpi dqi\u0005ð2p\u0002hÞl and prime (/) means integrating over physically","rect":[53.81357955932617,449.56396484375,385.14824763349068,436.5481262207031]},{"page":153,"text":"different states. To avoid (/) we may integrate over all states of N particles","rect":[53.81338119506836,460.1795349121094,385.16696561919408,449.0887756347656]},{"page":153,"text":"(molecules) but afterward to divide the result by the number of permutations","rect":[53.81319808959961,472.1390686035156,385.14508451567846,463.20452880859377]},{"page":153,"text":"N!: Ð= :::dG ¼ ð1=N!ÞÐ :::dG","rect":[53.81319808959961,485.0212707519531,172.59901687320019,471.4637756347656]},{"page":153,"text":"6.6.2 Equations of State for Gas and Liquid","rect":[53.812843322753909,529.1943359375,279.6719182958562,518.6402587890625]},{"page":153,"text":"6.6.2.1 Ideal Gas of Spherical Particles","rect":[53.812843322753909,556.7506103515625,226.70154203520969,547.7562866210938]},{"page":153,"text":"To illustrate the technique, consider an ideal gas of N spherical, point-like, non-","rect":[53.812843322753909,580.7393798828125,385.13375221101418,571.8048706054688]},{"page":153,"text":"interacting particles or molecules without internaldegrees of freedom. The partition","rect":[53.812862396240237,592.698974609375,385.1696709733344,583.7644653320313]},{"page":154,"text":"6.6 Molecular Statistic Approach to Phase Transitions","rect":[53.812843322753909,44.274620056152347,238.82855685233393,36.68026351928711]},{"page":154,"text":"137","rect":[372.4990539550781,42.55594253540039,385.19064850001259,36.73106384277344]},{"page":154,"text":"function","rect":[53.812843322753909,67.0,87.16143122724066,59.35380554199219]},{"page":154,"text":"freedom","rect":[53.812843322753909,79.0,87.06986187577803,71.31333923339844]},{"page":154,"text":"takes into account only the translational motion","rect":[91.36808776855469,68.2883529663086,291.7682732682563,59.35380554199219]},{"page":154,"text":"(no rotation assumed for spherical particles):","rect":[89.95159912109375,80.24788665771485,270.6225419766624,71.31333923339844]},{"page":154,"text":"with","rect":[295.96795654296877,67.0,313.68645564130318,59.35380554199219]},{"page":154,"text":"three","rect":[317.88909912109377,67.0,337.87717474176255,59.35380554199219]},{"page":154,"text":"N","rect":[322.5941162109375,97.1643295288086,327.2347558658822,92.49287414550781]},{"page":154,"text":"ZIG ¼ Xn expð\u0002En=kBTÞ ¼ N1! Xk expð\u0002ek=kBTÞ!","rect":[110.96792602539063,124.17768859863281,322.55179546952805,94.31617736816406]},{"page":154,"text":"degrees of","rect":[342.1325988769531,68.2883529663086,385.1477292617954,59.35380554199219]},{"page":154,"text":"(6.49)","rect":[361.0561828613281,113.51810455322266,385.10555396882668,104.98197937011719]},{"page":154,"text":"Here, k states of energy ek belong to an individual particle and the summation","rect":[65.76653289794922,145.77114868164063,385.1562432389594,136.81666564941407]},{"page":154,"text":"should be made over all these states. Since particles do not interact, the statistic sum","rect":[53.81344223022461,157.73062133789063,385.1712412703093,148.7960662841797]},{"page":154,"text":"for N independent particles is a product of N sums calculated for each particle. As","rect":[53.81344223022461,169.63339233398438,385.17514433013158,160.69883728027345]},{"page":154,"text":"explained above, to exclude identical sum corresponding to the same state of the","rect":[53.81345748901367,181.59292602539063,385.1721881694969,172.6583709716797]},{"page":154,"text":"gas, the number of permutation N! is introduced in the denominator.","rect":[53.81345748901367,193.55245971679688,329.5162014534641,184.61790466308595]},{"page":154,"text":"The translational motion of a particle is classic and the kinetic energy of one","rect":[65.76547241210938,205.51202392578126,385.1433795757469,196.5774688720703]},{"page":154,"text":"molecule is ekðpx;py;pzÞ ¼ ðp2x þ p2y þ pz2Þ=2m: Therefore, the summation may be","rect":[53.81345748901367,219.5931396484375,385.15296209527818,206.989990234375]},{"page":154,"text":"substituted by integration over the phase space (Vis the physical volume ofthe gas):","rect":[53.815162658691409,229.43182373046876,385.2077775723655,220.4972686767578]},{"page":154,"text":"Xk exp \u0002ek=kBT\u000B ¼ Xk ð2p1\u0002hÞ3","rect":[55.39891052246094,265.20819091796877,179.82034370979629,241.7524871826172]},{"page":154,"text":"\u0002px2 þ p2y þ pz2.2mkBT\u000B dpxdpydpzdV","rect":[239.8359832763672,260.5897216796875,384.7323036593992,243.43569946289063]},{"page":154,"text":"¼ V\bm2pkBh\u0002T2","rect":[129.2080535888672,301.80560302734377,173.39936812329513,279.11505126953127]},{"page":154,"text":"Note that, in the triple integral, each integral with respect to pi with limits","rect":[65.76496887207031,323.29541015625,385.1671487246628,314.36083984375]},{"page":154,"text":"1","rect":[260.05792236328127,325.1913146972656,267.0258497731082,322.0467834472656]},{"page":154,"text":"ffiffiffiffiffiffiffi","rect":[347.29180908203127,327.0,361.78513634756339,326.0]},{"page":154,"text":"(\u00021,1) has the known Gaussian form: R expð\u0002ax2Þdx ¼ pp=a and","rect":[53.81438446044922,338.2181701660156,385.14666071942818,326.2534484863281]},{"page":154,"text":"lðTÞ ¼ ð2pÞ1=2h\u0002=ðmkBTÞ1=2","rect":[53.81481170654297,352.0779113769531,165.9421546228822,339.6239929199219]},{"page":154,"text":"\u00021","rect":[255.92315673828126,344.17974853515627,268.2563795582644,341.03521728515627]},{"page":154,"text":"Now the partition function for N molecules is found:","rect":[65.76496887207031,363.6521301269531,278.54218919345927,354.71759033203127]},{"page":154,"text":"ZIG ¼ lN\u00023!N ð dr1dr2:::drN ¼ l\u00023NN!VN","rect":[142.40542602539063,400.3538818359375,294.1540368717416,375.8987731933594]},{"page":154,"text":"(6.50)","rect":[361.055908203125,393.52099609375,385.10527931062355,384.92510986328127]},{"page":154,"text":"In (6.50) integrating is made over N-dimensional coordinate space of volume VN.","rect":[65.7662582397461,421.862548828125,385.1832241585422,410.91864013671877]},{"page":154,"text":"Then, the free energy (6.47) reads:","rect":[53.814537048339847,433.8228454589844,193.8135667569358,424.8883056640625]},{"page":154,"text":"F ¼ kBTð3N lnlðTÞ \u0002 N lnV þ lnN!Þ ¼ \u0002kBTNðlnV \u0002 lnN \u0002 3lnlðTÞ þ 1Þ","rect":[63.27406311035156,458.0809326171875,375.70782865630346,448.1304016113281]},{"page":154,"text":"Here we used the Stirling","rect":[65.76656341552735,481.6613464355469,191.34688654950629,472.726806640625]},{"page":154,"text":"e ¼ 2.718....)","rect":[53.817543029785159,493.1656799316406,112.30149970857275,484.6893310546875]},{"page":154,"text":"Now we can find pressure","rect":[65.77053833007813,505.523681640625,170.4828571148094,496.589111328125]},{"page":154,"text":"formula","rect":[199.84280395507813,480.0,231.53800237848129,472.726806640625]},{"page":154,"text":"for","rect":[240.059814453125,480.0,251.66643653718604,472.726806640625]},{"page":154,"text":"large","rect":[260.1683044433594,481.6613464355469,280.15638006402818,472.726806640625]},{"page":154,"text":"p ¼ \u0002@F=@V ¼ NkVBT","rect":[175.2648468017578,538.3636474609375,261.58746690104558,517.7683715820313]},{"page":154,"text":"and obtain the equation of state for the ideal gas with","rect":[53.81426239013672,559.8802490234375,280.3810967545844,550.9457397460938]},{"page":154,"text":"number density r¼N/V:","rect":[53.81426239013672,571.7830810546875,150.41587693759989,562.8485717773438]},{"page":154,"text":"N ðlnN! ffi N lnðN=eÞ;","rect":[288.720947265625,482.0,383.5159143177881,472.0494689941406]},{"page":154,"text":"particle concentration or","rect":[284.3578186035156,559.8802490234375,385.15023170320168,550.9457397460938]},{"page":154,"text":"pIGV ¼ NkBT or pIG ¼ rkBT","rect":[151.3575439453125,595.6038208007813,287.13443345377996,586.7489624023438]},{"page":154,"text":"(6.51)","rect":[361.056396484375,595.3050537109375,385.10576759187355,586.7091674804688]},{"page":155,"text":"138","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.73106384277344]},{"page":155,"text":"6 Phase Transitions","rect":[318.2889709472656,42.55594253540039,385.1422470378808,36.68026351928711]},{"page":155,"text":"6.6.2.2 Equation of State for a Dense Gas or a Liquid","rect":[53.812843322753909,68.2186279296875,288.47940415690496,59.22431945800781]},{"page":155,"text":"If we would like to discuss a non-ideal dense gas of interacting hard spheres of a","rect":[53.812843322753909,92.20748138427735,385.15869939996568,83.27293395996094]},{"page":155,"text":"finite size, we should introduce a concept of excluded volume to take into account","rect":[53.812843322753909,104.11019134521485,385.20246751377177,95.17564392089844]},{"page":155,"text":"the repulsion of molecules at short intermolecular distances and write the energy of","rect":[53.812843322753909,116.0697250366211,385.14681373445168,107.13517761230469]},{"page":155,"text":"attraction between molecules at large distances. Then the partition function of type","rect":[53.812843322753909,128.02932739257813,385.1626666851219,119.09477233886719]},{"page":155,"text":"(6.48) will include two additional contributions and becomes quite cumbersome.","rect":[53.812843322753909,139.98886108398438,385.16766019369848,131.05430603027345]},{"page":155,"text":"Nevertheless it allows the discussion of the Van der Waals equation of state","rect":[53.812843322753909,151.94839477539063,360.5150836773094,143.0138397216797]},{"page":155,"text":"p ¼ ½NkBT=ðV \u0002 bÞ\u0006 \u0002 a=V2","rect":[162.1743621826172,175.47036743164063,276.2870337732728,164.19598388671876]},{"page":155,"text":"(6.52)","rect":[361.0561828613281,174.73329162597657,385.10555396882668,166.13739013671876]},{"page":155,"text":"on the microscopic level and find the physical sense of parameters a (for attraction)","rect":[53.81450653076172,197.06668090820313,385.1662534317173,188.1321258544922]},{"page":155,"text":"and b (for repulsion) introduced by Van der Waals phenomenologically. For a","rect":[53.81450653076172,209.02621459960938,385.1603473491844,200.09165954589845]},{"page":155,"text":"simple liquid consisting of hard spherical molecules the equation of state may be","rect":[53.815513610839847,220.98577880859376,385.15137518121568,212.0512237548828]},{"page":155,"text":"written in terms of number density r:","rect":[53.815513610839847,232.9453125,204.96621568271707,224.01075744628907]},{"page":155,"text":"p ¼ phs \u0002 21 J0r2","rect":[186.1388397216797,257.38763427734377,252.38278267464004,245.1361083984375]},{"page":155,"text":"(6.53)","rect":[361.0561828613281,255.73020935058595,385.10555396882668,247.13430786132813]},{"page":155,"text":"Here, the first and second terms describe correspondingly positive pressure due","rect":[65.76653289794922,278.97100830078127,385.14344061090318,270.03643798828127]},{"page":155,"text":"to molecular repulsion and negative pressure due to molecular attractive forces. Our","rect":[53.81450653076172,290.873779296875,385.15935645906105,281.9193115234375]},{"page":155,"text":"task is to understand the microscopic sense of parameters phs (index means hard","rect":[53.81450653076172,302.83331298828127,385.1175164323188,293.89874267578127]},{"page":155,"text":"spheres) and J0. Therefore we need a proper partition function.","rect":[53.81356430053711,314.7932434082031,307.5941433479953,305.85870361328127]},{"page":155,"text":"Let u(rij) is repulsive and –W(rij) attractive parts of the intermolecular potential","rect":[65.76558685302735,327.66937255859377,385.1601396329124,317.818359375]},{"page":155,"text":"for molecules i and j; then the partition function for N spherically symmetric","rect":[53.81333541870117,338.7124328613281,385.10837591363755,329.77789306640627]},{"page":155,"text":"particles of mass m and radius r0 reads [5]:","rect":[53.81333541870117,350.6719970703125,227.48181016513895,341.6778869628906]},{"page":155,"text":"N","rect":[316.9296875,369.57232666015627,321.5703271549447,364.90087890625]},{"page":155,"text":"Z ¼ lN\u00023!N ð dr1:::drNeiNj(exp\"1=2bXWðrklÞ#)","rect":[116.63369750976563,396.585693359375,316.86810316896279,366.7242126464844]},{"page":155,"text":"k¼6 l","rect":[263.17340087890627,396.7774963378906,273.74164849083328,390.30718994140627]},{"page":155,"text":"(6.54a)","rect":[356.6379089355469,385.9827575683594,385.13680396882668,377.3868713378906]},{"page":155,"text":"Here b ¼ 1=kBT; l is given above when discussing the ideal gas,","rect":[65.76592254638672,421.6253356933594,385.1286892464328,411.69476318359377]},{"page":155,"text":"eij ¼ exp½\u0002buðrijÞ\u0006: This function can be written shortly as a product of the part","rect":[53.813724517822269,434.073486328125,385.1440263516624,423.6443176269531]},{"page":155,"text":"Z0 including solely repulsive interactions and the thermal average of the term","rect":[53.813106536865237,445.6126403808594,385.10904644609055,436.6781005859375]},{"page":155,"text":"describing attraction between different particles k and l:","rect":[53.81296157836914,457.5721740722656,279.22407395908427,448.6177062988281]},{"page":155,"text":"Z ¼ Z0*exp\"1=2bXWðrklÞ#+","rect":[153.9602813720703,501.6809387207031,285.03372206544716,471.8094787597656]},{"page":155,"text":"k¼6 l","rect":[231.2821807861328,501.8070373535156,241.90688591759105,495.33673095703127]},{"page":155,"text":"(6.54b)","rect":[356.0699157714844,491.0122375488281,385.1581052383579,482.4163513183594]},{"page":155,"text":"First we estimate only the repulsive part Z0 and then the attractive one.","rect":[65.76436614990235,526.3261108398438,352.3438381722141,517.3916015625]},{"page":155,"text":"In Eq. 6.54a, averaging over ensemble (reference system) depends on the type of","rect":[65.76610565185547,538.28564453125,385.15093360749855,529.2913818359375]},{"page":155,"text":"the ensemble. In the simplest case, we take as a reference the system of hard spheres","rect":[53.81309127807617,550.2451782226563,385.1091348086472,541.3106689453125]},{"page":155,"text":"with repulsive potential of the type of a hard wall, shown in Fig. 6.18a, namely,","rect":[53.81309127807617,562.2047119140625,376.4569363167453,553.2702026367188]},{"page":155,"text":"Forrij \f 2r0 : u\u0006rij\u0007 ¼ 1andeij ¼0","rect":[140.13897705078126,588.3186645507813,298.85256281903755,576.385986328125]},{"page":156,"text":"6.6 Molecular Statistic Approach to Phase Transitions","rect":[53.813175201416019,44.274681091308597,238.82887728690424,36.68032455444336]},{"page":156,"text":"Fig. 6.18 Hard core","rect":[53.812843322753909,67.58130645751953,123.37215564035893,59.6313591003418]},{"page":156,"text":"intermolecular potential (a) and","rect":[53.812843322753909,77.4895248413086,155.3592276015751,69.89517211914063]},{"page":156,"text":"illustration of the excluded","rect":[53.81285095214844,85.68157196044922,141.09976715235636,79.81436157226563]},{"page":156,"text":"volume for two spherical","rect":[53.81285095214844,97.3846664428711,135.4824800893313,89.79031372070313]},{"page":156,"text":"particles of radius r0 (b)","rect":[53.81285095214844,107.36067962646485,132.08229154212169,99.76632690429688]},{"page":156,"text":"r","rect":[270.230224609375,125.05267333984375,272.8933453819338,120.74738311767578]},{"page":156,"text":"b","rect":[297.03515625,68.23553466796875,303.14153720871,60.92526626586914]},{"page":156,"text":"139","rect":[372.4993896484375,42.62373352050781,385.19098419337197,36.73112487792969]},{"page":156,"text":"and","rect":[53.812843322753909,162.55722045898438,68.21658412030706,155.6546173095703]},{"page":156,"text":"forrij \u000B 2r0: u\u0006rij\u0007 ¼ 0andeij ¼1","rect":[144.55841064453126,190.7027130126953,294.4341973405219,178.7700653076172]},{"page":156,"text":"Next, since the hard spheres (or molecules) have their own volume Vm ¼ (4/3)pr3","rect":[65.7660903930664,214.24188232421876,385.18130140630105,203.19056701660157]},{"page":156,"text":"and touch each other, the free volume for their translational motion is reduced. As","rect":[53.812843322753909,224.16928100585938,385.22638334380346,217.2666778564453]},{"page":156,"text":"seen in Fig. 6.18b the presence of sphere 1 reduces the available volume for sphere","rect":[53.812843322753909,238.16079711914063,385.24524725152818,229.2262420654297]},{"page":156,"text":"2 by 8Vm. This value is common for two spheres; therefore, the excluded volume per","rect":[53.812843322753909,250.12039184570313,385.15740333406105,241.18577575683595]},{"page":156,"text":"one sphere is 4Vm. It means that the volume of the whole system is diminished down","rect":[53.81352615356445,262.0802001953125,385.24541560224068,253.14540100097657]},{"page":156,"text":"tothe value ofV \u0002 4NVm. Thenthe partition function(6.50) for the ideal gas ofpoint","rect":[53.81300735473633,273.9831237792969,385.207167220803,264.9888000488281]},{"page":156,"text":"spheres, which corresponds to the same translational degrees of freedom may be","rect":[53.814605712890628,285.9426574707031,385.26502264215318,277.00811767578127]},{"page":156,"text":"corrected for the excluded volume or a packing fraction Z¼rVm:","rect":[53.814605712890628,297.9021911621094,311.1733537442405,288.9676513671875]},{"page":156,"text":"l\u00023NVNð1 \u0002 4ZÞN","rect":[193.16000366210938,324.0307922363281,266.11466919596037,312.187744140625]},{"page":156,"text":"Z0 ¼","rect":[170.44570922851563,329.9096984863281,190.345375330185,321.77862548828127]},{"page":156,"text":"N!","rect":[225.16453552246095,335.2536315917969,234.87586915177247,328.0223083496094]},{"page":156,"text":"(6.55)","rect":[361.0554504394531,330.0950927734375,385.10482154695168,321.49920654296877]},{"page":156,"text":"Therefore, the part of free energy related to hard sphere repulsion is expressed in","rect":[65.76578521728516,358.8340759277344,385.1416863541938,349.8995361328125]},{"page":156,"text":"terms of density of particles r ¼N/V that on account of Stirling formula is given by","rect":[53.81376266479492,370.7936096191406,385.1705254655219,361.85906982421877]},{"page":156,"text":"Fhs ¼ \u0002kBT lnZ0 ¼ NkBT½3lnl þ lnr \u0002 1 \u0002 lnð1 \u0002 4ZÞ\u0006","rect":[103.37776947021485,394.99517822265627,335.56907593888186,385.0446472167969]},{"page":156,"text":"From here the repulsive part of pressure is found","rect":[65.7631607055664,418.5556640625,264.4153679948188,409.6011962890625]},{"page":156,"text":"phs ¼ \u0002 @V ¼ V \u0002 4NVm ¼ Vð1 \u0002 4ZÞ","rect":[133.9634552001953,456.0400695800781,298.2174416934128,442.1789245605469]},{"page":156,"text":"\b rkBTð1 þ 4Z þ 16Z2 þ 64Z3 þ \u0003\u0003\u0003Þ","rect":[148.4115753173828,473.2713623046875,304.9842568789597,461.9403076171875]},{"page":156,"text":"(6.56)","rect":[361.0563659667969,472.5342712402344,385.1057370742954,463.9383850097656]},{"page":156,"text":"The approximation is correct for small density r, i.e. small packing fraction Z.","rect":[65.7667007446289,497.8160095214844,385.1833767464328,488.8814697265625]},{"page":156,"text":"Note that for Z¼0, we obtain the equation of state (6.51) for the ideal gas p¼rkBT.","rect":[53.81468963623047,509.7755432128906,385.1833767464328,500.7812194824219]},{"page":156,"text":"Now we try to estimate the attractive term (a thermal average) in the partition","rect":[65.76671600341797,521.7348022460938,385.1715020280219,512.80029296875]},{"page":156,"text":"function (6.54a). Due to enormous mathematical difficulties, we get rid of the","rect":[53.814659118652347,533.6375732421875,385.11670721246568,524.643310546875]},{"page":156,"text":"summation:","rect":[53.813655853271487,543.5751342773438,101.04836900302957,536.66259765625]},{"page":156,"text":"*exp\"1=2bXWðrklÞ#+ \b exph1=2bNrJ0i","rect":[130.62327575683595,589.7062377929688,308.3335276072129,559.8347778320313]},{"page":156,"text":"k¼6 l","rect":[179.45187377929688,589.8323974609375,190.02013665001292,583.362060546875]},{"page":156,"text":"(6.57)","rect":[361.0559997558594,579.0380249023438,385.1053708633579,570.442138671875]},{"page":157,"text":"140","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.73106384277344]},{"page":157,"text":"6 Phase Transitions","rect":[318.2889709472656,42.55594253540039,385.1422470378808,36.68026351928711]},{"page":157,"text":"In fact, instead of summation we have averaged the potential W(rkl) with the","rect":[65.76496887207031,68.2883529663086,385.1747211284813,59.35380554199219]},{"page":157,"text":"hard-sphere radial density distribution function rhs familiar to us from the discus-","rect":[53.813961029052737,80.24788665771485,385.1004880508579,71.31333923339844]},{"page":157,"text":"sion of density correlation function of isotropic liquids in Chapter 5. As a result, we","rect":[53.81342697143555,92.20760345458985,385.17017400934068,83.21329498291016]},{"page":157,"text":"obtain a new constant J0 ¼ Ð WðrÞrhsðrÞdr: From (6.57) the contribution to free","rect":[53.81342697143555,105.0898208618164,385.13721502496568,94.03369140625]},{"page":157,"text":"energy due to attraction is found:","rect":[53.81433868408203,116.0702133178711,187.79902513095926,107.13566589355469]},{"page":157,"text":"Fattr ¼ \u0002kBT\b2k1BT NrJ0 ¼ \u000221V N2J0","rect":[136.51593017578126,154.19309997558595,301.9473120935853,130.30787658691407]},{"page":157,"text":"The pressure due to attractive forces is given by","rect":[65.76496887207031,177.682373046875,259.75660029462349,168.74781799316407]},{"page":157,"text":"pattr ¼ \u0002@Fattr=@V ¼ 1=2N2J0ð\u00021\u0005V2Þ ¼ \u00021=2r2J0","rect":[118.72833251953125,205.4969024658203,319.79048225960096,191.94168090820313]},{"page":157,"text":"Finally, we obtain the equation of state for interacting hard spheres (attraction","rect":[65.76496887207031,229.07508850097657,385.16970149091255,220.06085205078126]},{"page":157,"text":"and repulsion):","rect":[53.81294250488281,240.99478149414063,114.3444963223655,232.0602264404297]},{"page":157,"text":"p ¼ rkBT \u0002 1J0r2","rect":[177.69508361816407,268.83892822265627,260.82286140999158,253.18235778808595]},{"page":157,"text":"1 \u0002 4Z2","rect":[195.87876892089845,275.65020751953127,241.6971825700141,266.8650817871094]},{"page":157,"text":"(6.58)","rect":[361.0561828613281,268.54010009765627,385.10555396882668,259.9442138671875]},{"page":157,"text":"This equation corresponds to both the equation of state for liquids (6.53) and the","rect":[65.76653289794922,297.16552734375,385.17624700738755,288.17120361328127]},{"page":157,"text":"Van der Waals equation(6.52). However, the phenomenological parameters aandb","rect":[53.81450653076172,309.12506103515627,385.18023005536568,300.1307373046875]},{"page":157,"text":"in (6.52) acquired physical sense. Parameter b ¼ 4NrVm is related to particular","rect":[53.815513610839847,321.0848693847656,385.1480649551548,312.09027099609377]},{"page":157,"text":"molecular volume and density of spheres (molecules), parameter a ¼ N2J0/2 points","rect":[53.814231872558597,333.0444641113281,385.1305276309128,321.9367980957031]},{"page":157,"text":"to the properly averaged potential describing molecular attraction.","rect":[53.814598083496097,344.9472351074219,320.6690640022922,336.0126953125]},{"page":157,"text":"Thus, we have seen how intermolecular interactions can be taken into account","rect":[65.7666244506836,355.0,385.1067033536155,347.97222900390627]},{"page":157,"text":"for description of non-ideal gases and even liquids. Now we are much closer to","rect":[53.814598083496097,368.8663330078125,385.14153376630318,359.9317626953125]},{"page":157,"text":"liquid crystals.","rect":[53.814598083496097,380.82586669921877,112.92269559408908,371.89129638671877]},{"page":157,"text":"6.7 Nematic–Isotropic Transition (Molecular Approach)","rect":[53.812843322753909,429.2363586425781,350.54544497273556,418.3953857421875]},{"page":157,"text":"6.7.1 Interaction Potential and Partition Function","rect":[53.812843322753909,457.0032958984375,311.23431366776586,448.46917724609377]},{"page":157,"text":"Consider the simplest case, namely, the nematic phase consisting of uniaxial rod-","rect":[53.812843322753909,486.6495056152344,385.1706479629673,477.7149658203125]},{"page":157,"text":"like molecules. Generally, the intermolecular interaction again consists of the","rect":[53.812843322753909,498.6090393066406,385.11389959527818,489.67449951171877]},{"page":157,"text":"repulsive and attractive parts but both of them become anisotropic. The potential","rect":[53.812843322753909,510.568603515625,385.1576371915061,501.634033203125]},{"page":157,"text":"of pair molecular interaction can be written in the following general form:","rect":[53.812843322753909,522.528076171875,356.211805892678,513.5736083984375]},{"page":157,"text":"W12 ¼ W12ðr12;y1;f1;y2;f2Þ","rect":[158.9452362060547,546.729736328125,280.0341836367722,536.7792358398438]},{"page":157,"text":"(6.59)","rect":[361.0555114746094,545.99267578125,385.1048825821079,537.3967895507813]},{"page":157,"text":"Note that Euler angle C is not considered due to the rod-like form of a molecule;","rect":[65.76583099365235,570.3101196289063,385.1446977383811,561.2660522460938]},{"page":157,"text":"the other angles are shown in Fig. 6.19a. Vector r12 connects the gravity centers of","rect":[53.81382369995117,582.2699584960938,385.14998756257668,573.3351440429688]},{"page":157,"text":"rods. If the particular form of W12 is known, it can be used for calculation of the","rect":[53.814144134521487,594.2294921875,385.17383611871568,585.2949829101563]},{"page":158,"text":"6.7 Nematic–Isotropic Transition (Molecular Approach)","rect":[53.812843322753909,44.274986267089847,245.8969430313795,36.68062973022461]},{"page":158,"text":"Fig. 6.19 Geometry of","rect":[53.812843322753909,67.58130645751953,131.674164711067,59.546695709228519]},{"page":158,"text":"interaction between two rod-like","rect":[53.812843322753909,75.76238250732422,155.36092517160894,69.89517211914063]},{"page":158,"text":"molecules (a) and geometry of a","rect":[53.812843322753909,87.4087142944336,155.3634581305933,79.81436157226563]},{"page":158,"text":"spherocylinder (b)","rect":[53.8120002746582,97.3846664428711,112.59528440100839,89.79031372070313]},{"page":158,"text":"1","rect":[238.95028686523438,144.240234375,242.9483172587816,138.6551971435547]},{"page":158,"text":"r12","rect":[266.0426940917969,113.94189453125,275.13023654321958,108.28466033935547]},{"page":158,"text":"2","rect":[285.6344909667969,143.1600341796875,289.63252136034409,137.5749969482422]},{"page":158,"text":"a1","rect":[316.97808837890627,82.41259765625,323.9742612014227,76.75530242919922]},{"page":158,"text":"b","rect":[349.55548095703127,68.23651885986328,355.66198434384889,60.92573928833008]},{"page":158,"text":"141","rect":[372.49822998046877,42.4547119140625,385.1898550429813,36.73143005371094]},{"page":158,"text":"partition function for N molecules with numbers k and l. The partition function may","rect":[53.812843322753909,198.20089721679688,385.16558161786568,189.2464141845703]},{"page":158,"text":"be written in analogy to expression (6.54a) for simple liquids:","rect":[53.81183624267578,210.16043090820313,303.61775071689677,201.16610717773438]},{"page":158,"text":"Z ¼ ZHR*exp\"1=2bXWðrklakalÞ#+","rect":[142.63026428222657,254.21249389648438,296.36370863771279,224.3410186767578]},{"page":158,"text":"k¼6 l","rect":[225.78761291503907,254.39529418945313,236.41230278770824,247.92498779296876]},{"page":158,"text":"(6.60)","rect":[361.05609130859377,243.54380798339845,385.1054624160923,235.00767517089845]},{"page":158,"text":"Again this function includes the repulsive multiplier ZHR and the attractive part.","rect":[65.76644134521485,278.9144592285156,385.1435513069797,269.979736328125]},{"page":158,"text":"The repulsive part may be considered as a reference for calculation of the thermal","rect":[53.81368637084961,290.8170471191406,385.1804643399436,281.88250732421877]},{"page":158,"text":"average necessary for the attractive part. To find ZHR, we may operate with an","rect":[53.81368637084961,302.7769775390625,385.1529473405219,293.842041015625]},{"page":158,"text":"excluded volume, although even for hard rods (suffix HR) it is very difficult to","rect":[53.81411361694336,314.7365417480469,385.1440056901313,305.802001953125]},{"page":158,"text":"calculate it. The total procedure is enormously complicated because, even for ZHR","rect":[53.81411361694336,326.6960754394531,385.1616112714689,317.76153564453127]},{"page":158,"text":"known, it requires multiple averaging over (a) all orientations of molecule 1, (b) all","rect":[53.812843322753909,338.6559753417969,385.14472825595927,329.721435546875]},{"page":158,"text":"orientations of molecule 2, and (c) all distances r12.","rect":[53.812843322753909,350.2170715332031,261.5268520882297,341.68096923828127]},{"page":158,"text":"Below we shall consider two extreme cases, long hard rods without attraction","rect":[65.7653579711914,362.5751647949219,385.09548274091255,353.640625]},{"page":158,"text":"and rod-like molecules without repulsion. The first approach (by Onsager) may be","rect":[53.81333541870117,374.5346984863281,385.1492389507469,365.5802307128906]},{"page":158,"text":"applied to very long molecules like tobacco mosaic viruses and calls for hard","rect":[53.81333541870117,386.4374694824219,385.11733332685005,377.5029296875]},{"page":158,"text":"mathematics [20]. We shall discuss it very schematically in the next section. The","rect":[53.81333541870117,398.3970031738281,385.1442035503563,389.46246337890627]},{"page":158,"text":"second one (by Maier and Saupe [21]) appeared to be simpler but very powerful and","rect":[53.81333541870117,410.3565673828125,385.14421931317818,401.4219970703125]},{"page":158,"text":"can be applied to many typical nematic liquid crystals. We shall consider it in the","rect":[53.81333541870117,422.31610107421877,385.1720660991844,413.38153076171877]},{"page":158,"text":"subsequent Section 6.7.3.","rect":[53.81333541870117,434.275634765625,156.2232632210422,425.341064453125]},{"page":158,"text":"6.7.2 Onsager’s Results","rect":[53.812843322753909,479.4285583496094,179.11266212198897,468.87445068359377]},{"page":158,"text":"Consider a medium consisting of elongated, cylindrically symmetric hard-core","rect":[53.812843322753909,507.0546569824219,385.1547321148094,498.1201171875]},{"page":158,"text":"molecules","rect":[53.812843322753909,516.8956298828125,94.40312589751437,510.02288818359377]},{"page":158,"text":"in","rect":[101.79014587402344,516.81591796875,109.56438532880316,510.02288818359377]},{"page":158,"text":"the","rect":[116.9713134765625,516.8956298828125,129.1950839824852,510.02288818359377]},{"page":158,"text":"form","rect":[136.62689208984376,516.8956298828125,156.0077433455046,510.02288818359377]},{"page":158,"text":"of","rect":[163.41964721679688,516.8956298828125,171.71151100007666,510.02288818359377]},{"page":158,"text":"spherocylinders.","rect":[179.05374145507813,518.9573974609375,244.82418485190159,510.02288818359377]},{"page":158,"text":"For","rect":[252.23809814453126,516.8956298828125,266.0246823867954,510.2220764160156]},{"page":158,"text":"spherocylinders","rect":[273.42364501953127,518.9573974609375,336.6299172793503,510.02288818359377]},{"page":158,"text":"shown","rect":[344.05975341796877,516.8956298828125,369.9705437760688,510.02288818359377]},{"page":158,"text":"in","rect":[377.36651611328127,516.81591796875,385.1407403092719,510.02288818359377]},{"page":158,"text":"Fig. 6.19b we may introduce parameter","rect":[53.812843322753909,530.9169311523438,212.7148679336704,521.982421875]},{"page":158,"text":"x ¼ ðL þ DÞ=D;","rect":[187.8343505859375,558.7026977539063,249.4887231556787,545.2241821289063]},{"page":158,"text":"(6.61)","rect":[361.0534973144531,557.4417114257813,385.10286842195168,548.9055786132813]},{"page":158,"text":"that reduces to x ¼ 2D for spherical particles discussed above. A rod has kinetic","rect":[53.81185531616211,582.2691650390625,385.1248248882469,573.3346557617188]},{"page":158,"text":"energy of translations p2/m and rotation about its short axes py2/2I and pf2/2I","rect":[53.81285095214844,595.1063232421875,385.1596616348423,583.1787719726563]},{"page":159,"text":"142","rect":[53.812843322753909,42.454345703125,66.50444931178018,36.73106384277344]},{"page":159,"text":"6 Phase Transitions","rect":[318.2889709472656,42.55594253540039,385.1422470378808,36.68026351928711]},{"page":159,"text":"(angles j and y are in Fig. 6.19a, I is moment of inertia). Therefore, a rod has five","rect":[53.812843322753909,68.2883529663086,385.14771307184068,59.04502868652344]},{"page":159,"text":"degrees of freedom. In addition rods k and l interact with each other. The interaction","rect":[53.81086730957031,80.24788665771485,385.08501521161568,71.29341888427735]},{"page":159,"text":"potential is ukl ¼ 1 if they overlap each other and ukl ¼ 0 without overlapping.","rect":[53.809898376464847,92.20772552490235,385.1318630745578,83.27293395996094]},{"page":159,"text":"The Hamiltonian of the system consisting of N rods is given by","rect":[53.81393051147461,104.1104965209961,309.88994685224068,95.17594909667969]},{"page":159,"text":"N","rect":[197.57826232910157,124.08773040771485,202.21890198404629,119.41627502441406]},{"page":159,"text":"H ¼ XTk þ 2 X ukl","rect":[172.2013397216797,142.26885986328126,266.2634075240364,126.00146484375]},{"page":159,"text":"k¼1","rect":[194.00929260253907,146.40538024902345,206.1035315760072,141.52476501464845]},{"page":159,"text":"(6.62)","rect":[361.0561828613281,137.1538543701172,385.10555396882668,128.6177215576172]},{"page":159,"text":"and the partition function can be written as","rect":[53.81450653076172,172.12710571289063,227.38507475005344,163.1925506591797]},{"page":159,"text":"ZHR","rect":[108.70304870605469,201.44403076171876,123.47792290814935,193.39569091796876]},{"page":159,"text":"¼","rect":[126.77186584472656,198.6456298828125,134.43660762998969,196.31488037109376]},{"page":159,"text":"1","rect":[155.66090393066407,193.1685791015625,160.63800898602973,186.435302734375]},{"page":159,"text":"N!ð2p\u0002hÞ5N","rect":[137.19485473632813,210.89627075195313,178.25788819010098,198.8576202392578]},{"page":159,"text":"ð","rect":[182.3973846435547,208.54583740234376,187.165451286595,186.42361450195313]},{"page":159,"text":"N","rect":[324.9166564941406,194.31532287597657,329.55729614908537,189.6438751220703]},{"page":159,"text":"drdpidpyidpfidyidfi expð\u0002bHÞ\u000E","rect":[193.38682556152345,203.3997344970703,324.8786895783944,191.4670867919922]},{"page":159,"text":"(6.63)","rect":[361.0561828613281,201.7134552001953,385.10555396882668,193.1773223876953]},{"page":159,"text":"Onsager used a low-density expansion, that is small packing factor Z¼rVm.","rect":[65.76653289794922,234.47647094726563,385.1832241585422,225.52198791503907]},{"page":159,"text":"After a cumbersome calculation procedure he has found the excluded volume","rect":[53.814537048339847,246.43612670898438,385.11557806207505,237.50157165527345]},{"page":159,"text":"VexclðaiajÞ that depends on orientation of the rods. Then, using (6.63) and several","rect":[53.814537048339847,259.2132873535156,385.18244798252177,248.7840118408203]},{"page":159,"text":"approximations concerning averaging, the free energy and the equation of state for","rect":[53.81275177001953,270.80865478515627,385.17550025788918,261.87408447265627]},{"page":159,"text":"hard spherocylinders have been found.","rect":[53.81275177001953,282.7681579589844,209.52447171713596,273.8336181640625]},{"page":159,"text":"At that stage a uniaxial, orientational order parameter S is introduced in terms of","rect":[65.7647705078125,294.72772216796877,385.1496518692173,285.79315185546877]},{"page":159,"text":"the mean square projections ax,y,z of molecular vector a:","rect":[53.81277084350586,307.6044921875,280.1322465665061,297.6959228515625]},{"page":159,"text":"ha2xi ¼ hay2i ¼ ð1=3Þð1 \u0002 SÞ;haz2i ¼ ð1=3Þð1 þ 2SÞ","rect":[117.59689331054688,333.9752502441406,321.38556303130346,320.8616943359375]},{"page":159,"text":"The order parameter depends on the packing fraction and temperature. With","rect":[65.76641082763672,357.4735412597656,385.13039485028755,348.53900146484377]},{"page":159,"text":"increasing density or decreasing temperature the isotropic phase is substituted by","rect":[53.81438446044922,369.43310546875,385.1691826920844,360.49853515625]},{"page":159,"text":"the nematic phase. The equation for S is found in terms of Z and g.","rect":[53.81438446044922,381.39263916015627,325.6499294808078,372.45806884765627]},{"page":159,"text":"Sð2S3 \u0002 S \u0002 1 þ 3=8ZgÞ","rect":[170.04969787597657,406.9449462890625,268.9312783633347,395.6236267089844]},{"page":159,"text":"(6.64)","rect":[361.0555114746094,406.1610412597656,385.1048825821079,397.62493896484377]},{"page":159,"text":"where factor g includes the molecular anisotropy x:","rect":[53.8138542175293,430.4786071777344,261.2695756680686,421.5440673828125]},{"page":159,"text":"2ðx \u0002 1Þ2","rect":[208.3408966064453,456.6639099121094,246.0952002771791,444.7092590332031]},{"page":159,"text":"g ¼ pð3x \u0002 1Þ","rect":[189.9884490966797,470.2673034667969,247.35170377837376,456.4236145019531]},{"page":159,"text":"Equation 6.64 has three solutions, S ¼ 0 for the isotropic phase and","rect":[65.76799774169922,493.8477478027344,340.00249568036568,484.9132080078125]},{"page":159,"text":"S\u0005 ¼ 41 \u0005 34\b1 \u0002 3Z1g 1=2","rect":[166.25575256347657,534.464599609375,272.20848152717908,508.1362609863281]},{"page":159,"text":"(6.65)","rect":[361.0561828613281,526.7778930664063,385.10555396882668,518.1820068359375]},{"page":159,"text":"for the nematic phase. The solution with sign (þ) is stable. A critical condi-","rect":[53.81450653076172,558.0108642578125,385.1583188614048,549.0763549804688]},{"page":159,"text":"tion 3Zg¼1 corresponds to the transition from the isotropic to the nematic phase.","rect":[53.81450653076172,569.9703979492188,385.18120999838598,561.035888671875]},{"page":159,"text":"It","rect":[53.81548309326172,579.8682250976563,59.92736680576394,573.1947021484375]},{"page":159,"text":"is","rect":[64.91741180419922,580.0,71.54691709624484,572.9954833984375]},{"page":159,"text":"solely","rect":[76.58573913574219,581.9299926757813,100.41611567548284,572.9954833984375]},{"page":159,"text":"determined","rect":[105.474853515625,580.0,150.53157893231879,572.9954833984375]},{"page":159,"text":"by","rect":[155.54849243164063,581.9299926757813,165.56740656903754,572.9954833984375]},{"page":159,"text":"molecular","rect":[170.5594482421875,580.0,210.6321042617954,572.9954833984375]},{"page":159,"text":"parameter","rect":[215.64804077148438,581.9299926757813,255.72075782624854,574.0]},{"page":159,"text":"g","rect":[260.7396240234375,581.8403930664063,265.71672907880318,575.1170654296875]},{"page":159,"text":"(or","rect":[270.7655029296875,581.5316162109375,282.37212501374855,573.0552368164063]},{"page":159,"text":"x)","rect":[287.3621826171875,581.5316162109375,295.1523374160923,573.0552368164063]},{"page":159,"text":"and","rect":[300.1642761230469,580.0,314.56801692060005,572.9954833984375]},{"page":159,"text":"packing","rect":[319.5928955078125,581.9299926757813,351.2990655045844,572.9954833984375]},{"page":159,"text":"density","rect":[356.299072265625,581.9299926757813,385.1065606217719,572.9954833984375]},{"page":159,"text":"dependent on temperature. The qualitative temperature dependencies are shown","rect":[53.815513610839847,593.832763671875,385.1783074479438,584.8982543945313]},{"page":160,"text":"6.7 Nematic–Isotropic Transition","rect":[53.8123893737793,44.275352478027347,167.19920868067667,36.68099594116211]},{"page":160,"text":"Fig. 6.20 Onsager model:","rect":[53.812843322753909,67.58130645751953,141.76313499655786,59.85148620605469]},{"page":160,"text":"order parameter dependence on","rect":[53.812843322753909,77.4895248413086,154.2880148330204,69.89517211914063]},{"page":160,"text":"molecular packing factor Z for","rect":[53.812843322753909,87.4087142944336,152.4164742813795,79.81436157226563]},{"page":160,"text":"two values of spherocylinder","rect":[53.812843322753909,97.3846664428711,145.8879098282545,89.79031372070313]},{"page":160,"text":"anisotropy ratio x ¼ 4 (dash","rect":[53.812843322753909,107.36067962646485,145.16194671778605,99.7493896484375]},{"page":160,"text":"curve)andx ¼ 11(solidcurve).","rect":[53.81200408935547,116.99797821044922,155.33891555860004,109.725341796875]},{"page":160,"text":"Sc ¼ 0.25 is the amplitude of","rect":[53.8111572265625,127.25533294677735,148.08734220130138,119.61017608642578]},{"page":160,"text":"the order parameter jump at the","rect":[53.8123893737793,137.23129272460938,154.4010100104761,129.63693237304688]},{"page":160,"text":"phase transition","rect":[53.8123893737793,147.20724487304688,103.73608917384073,139.61288452148438]},{"page":160,"text":"(Molecular","rect":[169.5953826904297,43.93669891357422,207.28100675208263,36.68099594116211]},{"page":160,"text":"Approach)","rect":[209.7000274658203,44.275352478027347,245.89650052649669,36.68099594116211]},{"page":160,"text":"143","rect":[372.4977722167969,42.55667495727539,385.18939727930947,36.73179626464844]},{"page":160,"text":"in Fig. 6.20. For very elongated molecules (x ¼ 11) the phase transition appears at","rect":[53.812843322753909,200.41143798828126,385.181532455178,191.4768829345703]},{"page":160,"text":"much smaller packing density (compare with the curve for x ¼ 4). There is no","rect":[53.81385040283203,212.37100219726563,385.17058650067818,203.4364471435547]},{"page":160,"text":"solution for short spherocylinders anisotropy x < 3.08. The order parameter","rect":[53.8138313293457,224.33053588867188,385.15462623445168,215.39598083496095]},{"page":160,"text":"changes discontinuously with a jump (Sc ¼ 0.25) at the isotropic-nematic transition","rect":[53.813812255859378,236.29037475585938,385.15923396161568,227.29605102539063]},{"page":160,"text":"point (first order transition).","rect":[53.814430236816409,248.24990844726563,166.25123258139377,239.3153533935547]},{"page":160,"text":"6.7.3 Mean Field Approach for the Nematic Phase","rect":[53.812843322753909,285.6935729980469,313.7451303813001,275.13946533203127]},{"page":160,"text":"6.7.3.1 Interaction Potential and Partition Function","rect":[53.812843322753909,311.7956237792969,280.94903916667058,304.0662841796875]},{"page":160,"text":"The potential of pair molecular interaction (6.59) is too difficult to deal with.","rect":[53.812843322753909,337.2386779785156,385.1656155159641,328.2443542480469]},{"page":160,"text":"Much simpler is to use symmetry of the phase of interest (nematic) and construct","rect":[53.81280517578125,349.1982116699219,385.2421098477561,340.263671875]},{"page":160,"text":"the form of potential energy of a molecule as if it interacts not with other","rect":[53.81280517578125,361.15777587890627,385.25240455476418,352.22320556640627]},{"page":160,"text":"surrounding molecules but with an average molecular field. This is an essence","rect":[53.81280517578125,373.1172790527344,385.20630682184068,364.1827392578125]},{"page":160,"text":"of the mean field approximation [1]. The single molecule potential represents the","rect":[53.81280517578125,385.0200500488281,385.17163885309068,376.045654296875]},{"page":160,"text":"mean field of all intermolecular forces acting on a given molecule. In this case, we","rect":[53.81181716918945,396.9795837402344,385.2063373394188,388.0450439453125]},{"page":160,"text":"neglect the intermolecular short-range order. Such theories have appeared to","rect":[53.81181716918945,408.93914794921877,385.2112969498969,400.00457763671877]},{"page":160,"text":"be very powerful in the physics of solid state, for instance in magnetism, ferro-","rect":[53.81181716918945,420.898681640625,385.1784604629673,411.964111328125]},{"page":160,"text":"electricity and superconductivity.","rect":[53.81181716918945,432.8582458496094,191.25159879233127,423.9237060546875]},{"page":160,"text":"Consider the nematic phase. It has cylindrical symmetry and the orientational","rect":[65.76383972167969,444.8177795410156,385.1247697598655,435.88323974609377]},{"page":160,"text":"order parameter <P2> ¼ 1=2\u000F3cos2# \u0002 1\u0010 with angle W between a molecular long","rect":[53.81181716918945,458.07659912109377,385.12908259442818,444.89990234375]},{"page":160,"text":"axis and the symmetry axis (the director n). The tasks of the molecular theory is to","rect":[53.81412887573242,468.7383117675781,385.19976130536568,459.80377197265627]},{"page":160,"text":"use the symmetry arguments and properties of molecules and (a) to find the","rect":[53.81415939331055,480.6410827636719,385.16989935113755,471.70654296875]},{"page":160,"text":"temperature dependence of <P2> (T), (b) to calculate thermodynamic and other","rect":[53.81415939331055,492.6010437011719,385.1401303848423,483.66607666015627]},{"page":160,"text":"properties in terms of <P2>, (c) to discuss the phase transition from finite <P2>","rect":[53.814205169677737,504.5606994628906,385.14807918760689,495.62603759765627]},{"page":160,"text":"to zero (N–Iso transition), and (d) to discuss the role of the higher order parameters","rect":[53.814231872558597,516.5202026367188,385.1242715273972,507.585693359375]},{"page":160,"text":"<P4>, <P6> etc.","rect":[53.814231872558597,528.0618896484375,127.07881589438205,519.7643432617188]},{"page":160,"text":"The key problem is a form of the interaction potential. The two-pair potential","rect":[65.76585388183594,540.439453125,385.15974290439677,531.5049438476563]},{"page":160,"text":"(6.59) is too complicated and we would like to substitute it by a single molecule","rect":[53.8138313293457,552.3990478515625,385.17856634332505,543.40478515625]},{"page":160,"text":"potential:","rect":[53.8138313293457,564.3585815429688,91.72643144199441,555.424072265625]},{"page":160,"text":"1. At first, the pair potential W12 is expanded into two series of spherical harmonics","rect":[53.8138313293457,582.2699584960938,385.09833158599096,573.3350219726563]},{"page":160,"text":"Y1 and Y2. Then the dependence of W12(r12) on the intermolecular distance","rect":[66.27490997314453,594.2296142578125,385.1598590679344,585.2951049804688]},{"page":161,"text":"144","rect":[53.813961029052737,42.4559326171875,66.505567018079,36.73265075683594]},{"page":161,"text":"6 Phase Transitions","rect":[318.29010009765627,42.55752944946289,385.1433761882714,36.68185043334961]},{"page":161,"text":"2.","rect":[53.813961029052737,127.0,61.27961774374728,119.15465545654297]},{"page":161,"text":"becomes separated from the dependence of W12 on molecular orientation. Then","rect":[66.27484893798828,68.2883529663086,385.1394890885688,59.35380554199219]},{"page":161,"text":"only the difference between two angles f1\u0002f2 is considered essential, but not","rect":[66.27526092529297,80.24788665771485,385.135878158303,70.9846420288086]},{"page":161,"text":"each of the two angles. In addition, due to head-to-tail symmetry, only even","rect":[66.27562713623047,92.20760345458985,385.1199883561469,83.27305603027344]},{"page":161,"text":"terms are left in the expansion. Fortunately, the coefficients of the expansion","rect":[66.27562713623047,104.11031341552735,385.09511652997505,95.17576599121094]},{"page":161,"text":"decrease rapidly with the number of a harmonic.","rect":[66.27562713623047,116.0698471069336,262.2929653694797,107.13529968261719]},{"page":161,"text":"Then, a new, polar coordinate frame was introduced based on the director n as a","rect":[66.275634765625,128.02944946289063,385.16175115777818,119.09489440917969]},{"page":161,"text":"polar axis. To obtain the single molecular potential W1 as a function of the first","rect":[66.27661895751953,139.98898315429688,385.1625200040061,131.05442810058595]},{"page":161,"text":"molecule orientation with respect to n, one has to take three successive averages","rect":[66.2754135131836,151.94912719726563,385.0788918887253,143.0145721435547]},{"page":161,"text":"of W12: (a) over all orientations of intermolecular vector r, (b) over all orienta-","rect":[66.2754135131836,163.51051330566407,385.1083921035923,154.97413635253907]},{"page":161,"text":"tions of molecule 2, and (c) over all intermolecular separations |r|.","rect":[66.27503204345703,175.86846923828126,332.7691616585422,166.9339141845703]},{"page":161,"text":"Finally, the single-molecule potential has been found in the form of expansion","rect":[65.76537322998047,193.77938842773438,385.09548274091255,184.84483337402345]},{"page":161,"text":"over Legendre polynomials:","rect":[53.8133544921875,205.73892211914063,167.21769578525614,196.8043670654297]},{"page":161,"text":"W1ðcos#Þ ¼ vhP2iP2ðcos#Þ þ mhP4iP4ðcos#Þ þ 6th þ \u0003\u0003\u0003 terms","rect":[76.98078155517578,231.35763549804688,337.8514748965378,219.97084045410157]},{"page":161,"text":"(6.66)","rect":[361.05572509765627,230.6205596923828,385.1050962051548,222.0844268798828]},{"page":161,"text":"In the simplest case, we use only the first term:","rect":[65.76607513427735,254.88128662109376,256.096571517678,245.9467315673828]},{"page":161,"text":"W1ðcos#Þ ¼ \u0002vhP2iP2ðcos#Þ:","rect":[156.90782165527345,279.13970947265627,282.1185754506006,269.1891784667969]},{"page":161,"text":"(6.67)","rect":[361.0564880371094,278.4026184082031,385.1058591446079,269.86651611328127]},{"page":161,"text":"Its form is shown in Fig. 6.21. P2(cosW) is a universal function varying from \u00021/2 to","rect":[65.7668228149414,302.7203369140625,385.1795281510688,293.4869689941406]},{"page":161,"text":"þ1 and v <P2> is a number determining the depth of the potential well. Note that","rect":[53.813838958740237,314.67987060546877,385.1318498379905,305.74530029296877]},{"page":161,"text":"parameter v depends on properties of a molecule (shape, electronic structure, etc.).","rect":[53.81292724609375,326.6394348144531,365.8146633675266,317.70489501953127]},{"page":161,"text":"The even function fðcos#Þ given by the Gibbs distribution describes the proba-","rect":[65.76590728759766,338.9376220703125,385.13982520906105,328.9870910644531]},{"page":161,"text":"bility for a molecule to be at an angle W with respect to the director n:","rect":[53.81389236450195,350.5017395019531,336.5503068692405,341.26837158203127]},{"page":161,"text":"fðcos#Þ ¼ Z\u00021 exp\b\u0002WðkcBoTs#Þ","rect":[150.3368682861328,388.6253967285156,281.39328397856908,364.74017333984377]},{"page":161,"text":"Fig. 6.21 The dimensionless","rect":[53.812843322753909,522.2183227539063,152.3470810222558,514.488525390625]},{"page":161,"text":"form of the nematic mean field","rect":[53.812843322753909,530.3994750976563,155.36176056055948,524.5322265625]},{"page":161,"text":"potential as a function of","rect":[53.812843322753909,542.1025390625,135.59414762122325,534.5081787109375]},{"page":161,"text":"molecular angle W at three","rect":[53.812843322753909,552.0784912109375,140.01337573313237,544.2301635742188]},{"page":161,"text":"different values of <P2> ¼ 0","rect":[53.812843322753909,561.5938720703125,153.71634430079386,554.4033203125]},{"page":161,"text":"(horizontal line), 0.5 (dot","rect":[53.8122673034668,571.6344604492188,136.94819359030786,564.3279418945313]},{"page":161,"text":"curve) and 1 (solid curve)","rect":[53.813106536865237,581.6104125976563,138.76648038489513,574.3377685546875]},{"page":162,"text":"6.7 Nematic–Isotropic Transition (Molecular Approach)","rect":[53.812843322753909,44.274620056152347,245.8969430313795,36.68026351928711]},{"page":162,"text":"145","rect":[372.49822998046877,42.55594253540039,385.1898550429813,36.62946701049805]},{"page":162,"text":"The probability to find the molecule at any angle W within 0 and p equals unity.","rect":[65.76496887207031,68.2883529663086,385.14483304526098,59.05499267578125]},{"page":162,"text":"From here we find the single-molecule partition function","rect":[53.811954498291019,80.28772735595703,282.74086848310005,71.27349853515625]},{"page":162,"text":"1","rect":[175.0334930419922,100.1553955078125,178.51745674690566,95.44210815429688]},{"page":162,"text":"Z ¼ ð exp\b\u0002WðkcBoTs#Þ","rect":[154.86610412597657,125.11613464355469,249.16222013579563,101.23091125488281]},{"page":162,"text":"0","rect":[174.5234375,130.67674255371095,178.00740120491347,125.87979125976563]},{"page":162,"text":"d cos#","rect":[256.4330749511719,115.86587524414063,284.15755254143138,108.67440795898438]},{"page":162,"text":"(6.68)","rect":[361.0558166503906,117.4296646118164,385.10518775788918,108.89353942871094]},{"page":162,"text":"The configuration integral includes only one degree of freedom (W-orientation).","rect":[65.76619720458985,155.18093872070313,385.1619839241672,145.94757080078126]},{"page":162,"text":"Other degrees are ignored because we are only interested in an excess of free energy","rect":[53.81418991088867,167.14044189453126,385.1709832291938,158.1859588623047]},{"page":162,"text":"of the nematic phase with respect to the isotropic phase. At this stage Eq. 6.67 is not","rect":[53.81418991088867,179.0999755859375,385.1360917813499,170.16542053222657]},{"page":162,"text":"yet helpful for the calculation of thermodynamic parameters since it includes","rect":[53.81418991088867,191.05953979492188,385.16302885161596,182.12498474121095]},{"page":162,"text":"unknown value of <P2> in the mean-field potential W. However, using the theo-","rect":[53.81418991088867,203.01864624023438,385.1393064102329,194.0641632080078]},{"page":162,"text":"rem of average, <P2> may be written in the form valid for any weight function f","rect":[53.81342697143555,215.01805114746095,385.1514726407249,206.00381469726563]},{"page":162,"text":"(cosW):","rect":[53.81364822387695,226.48268127441407,82.07365281650613,217.64773559570313]},{"page":162,"text":"1","rect":[183.19039916992188,246.8453369140625,186.67436287483535,242.13204956054688]},{"page":162,"text":"hP2i ¼ ð P2ðcos#Þfðcos#Þdcos#","rect":[151.01651000976563,270.951904296875,287.95403081291576,248.82968139648438]},{"page":162,"text":"0","rect":[182.6803436279297,277.3101806640625,186.16430733284316,272.5132141113281]},{"page":162,"text":"(6.69)","rect":[361.0567626953125,264.1190185546875,385.10613380281105,255.58290100097657]},{"page":162,"text":"Now we combine (6.67), (6.68) and (6.69) and obtain the self-consistent equa-","rect":[65.76709747314453,308.8415832519531,385.0931638321079,299.90704345703127]},{"page":162,"text":"tion for the determination of the orientational order parameter <P2> as a function","rect":[53.816078186035159,320.8011169433594,385.1718377213813,311.8665771484375]},{"page":162,"text":"of kBT/v:","rect":[53.81405258178711,332.26336669921877,89.52105577060769,323.8065490722656]},{"page":162,"text":"hP2i","rect":[122.97615814208985,379.3980407714844,140.7436868106003,369.4674377441406]},{"page":162,"text":"¼","rect":[143.53883361816407,378.0,151.20357540342719,371.0]},{"page":162,"text":"Ð01P2ðcos#Þexp\u0003vhP2iPkB2Tðcos#Þ\u0004d cos#","rect":[154.0181884765625,372.0800476074219,314.3485010277595,347.95513916015627]},{"page":162,"text":"Ð01exp\u0003vhP2iPkB2Tðcos#Þ\u0004d cos#","rect":[174.12652587890626,400.8739013671875,294.23946172111888,376.7489929199219]},{"page":162,"text":"(6.70)","rect":[361.05511474609377,378.6712951660156,385.1044858535923,370.13519287109377]},{"page":162,"text":"The equation is complicated but we can vary kBT/v and, for each given value,","rect":[65.76546478271485,419.4266052246094,385.13689847494848,410.4721374511719]},{"page":162,"text":"calculate numerically <P2> by integrating over W. The result is universal and","rect":[53.81301498413086,431.38568115234377,385.14421931317818,422.1523132324219]},{"page":162,"text":"shown in Fig. 6.22.","rect":[53.81332015991211,443.3452453613281,131.69605680014377,434.41070556640627]},{"page":162,"text":"Now we can summarize some preliminary, but important conclusions:","rect":[65.76533508300781,455.24798583984377,348.8103166348655,446.31341552734377]},{"page":162,"text":"1.","rect":[53.81332015991211,472.0,61.278976874606659,464.34088134765627]},{"page":162,"text":"2.","rect":[53.813289642333987,506.8966064453125,61.27894635702853,500.163330078125]},{"page":162,"text":"3.","rect":[53.81376266479492,531.0,61.27941937948947,524.082763671875]},{"page":162,"text":"The two branches correspond to two ordered phases, one is nematic with positive","rect":[66.27499389648438,473.2156677246094,385.1561664409813,464.2811279296875]},{"page":162,"text":"<P2> > 0, and the other phase (not observed yet) with negative <P2>. The","rect":[66.27499389648438,485.1758117675781,385.2048724956688,476.24127197265627]},{"page":162,"text":"positive branch corresponds to the results of the Landau approach, see Fig. 6.5.","rect":[66.27599334716797,497.1353454589844,384.2761501839328,488.1410217285156]},{"page":162,"text":"At Tc the order parameter <P2> increases by a jump from 0 to 0.429, therefore,","rect":[66.27496337890625,509.0384216308594,385.0888943245578,500.1038818359375]},{"page":162,"text":"the N–Iso transition is first order transition.","rect":[66.27543640136719,518.9759521484375,240.87228055502659,512.0634155273438]},{"page":162,"text":"The N–Iso transition takes place at kBTc/v ¼ 0.222. From this value and a","rect":[66.27543640136719,532.95751953125,385.15992010309068,524.0030517578125]},{"page":162,"text":"typical","rect":[66.27579498291016,544.917236328125,93.52046830722878,535.9827270507813]},{"page":162,"text":"experimental","rect":[100.60189819335938,544.917236328125,152.90530259677957,535.9827270507813]},{"page":162,"text":"transition","rect":[160.02256774902345,543.0,197.73906794599066,535.9827270507813]},{"page":162,"text":"temperature","rect":[204.82846069335938,544.917236328125,252.68534124566879,536.9986572265625]},{"page":162,"text":"Tc ¼ 400K","rect":[259.7746887207031,544.5192260742188,305.10094976380199,536.0426025390625]},{"page":162,"text":"we","rect":[312.11468505859377,542.8555908203125,323.7113422222313,538.2139892578125]},{"page":162,"text":"can","rect":[330.80767822265627,542.8555908203125,344.68386164716255,538.2139892578125]},{"page":162,"text":"estimate","rect":[351.70953369140627,542.8555908203125,385.1357806987938,535.9828491210938]},{"page":162,"text":"the height of the potential barrier v ¼ kBTc=0:222 ¼ ð1:38 \u0003 10\u000216 \u0003 400Þ=0:222","rect":[66.2755355834961,557.2156982421875,385.1646660905219,546.3528442382813]},{"page":162,"text":"¼ 2:5 \u0003 10\u000213erg ¼ 0:15eV: However, the molecular nature of parameter v is not","rect":[66.27654266357422,569.6301879882813,385.2073808438499,559.14794921875]},{"page":162,"text":"clear yet and we cannot calculate the partition function and free energy. It should","rect":[66.27549743652344,581.5897216796875,385.17461482099068,572.6552124023438]},{"page":162,"text":"be discussed using specific molecular models.","rect":[66.27549743652344,593.4924926757813,251.47555203940159,584.5579833984375]},{"page":163,"text":"146","rect":[53.8111457824707,42.55716323852539,66.50275177149698,36.68148422241211]},{"page":163,"text":"Fig. 6.22 Mean field model","rect":[53.812843322753909,67.58130645751953,151.2293825551516,59.85148620605469]},{"page":163,"text":"of the nematic phase:","rect":[53.812843322753909,77.4895248413086,127.0197115346438,69.89517211914063]},{"page":163,"text":"temperature dependence of","rect":[53.812843322753909,87.4087142944336,146.33886808020763,79.81436157226563]},{"page":163,"text":"the order parameter. The two","rect":[53.812843322753909,97.3846664428711,153.61624664454386,89.79031372070313]},{"page":163,"text":"branches correspond to stable","rect":[53.812843322753909,107.36067962646485,155.13247058176519,99.76632690429688]},{"page":163,"text":"nematic phase with positive","rect":[53.812843322753909,117.33663177490235,149.16063830637456,109.74227905273438]},{"page":163,"text":"order parameter (solid line),","rect":[53.812843322753909,127.25582122802735,149.88745376660786,119.64453125]},{"page":163,"text":"and unstable phase with","rect":[53.812843322753909,137.23178100585938,135.82852691798136,129.63742065429688]},{"page":163,"text":"negative order parameter","rect":[53.812843322753909,147.20773315429688,139.22564786536388,139.61337280273438]},{"page":163,"text":"(dash line). Order parameter","rect":[53.812843322753909,157.1837158203125,151.1210641251295,149.57241821289063]},{"page":163,"text":"discontinuity at S ¼ 0.429","rect":[53.81199645996094,167.10293579101563,144.50450653223917,159.50857543945313]},{"page":163,"text":"indicates the first order N–I","rect":[53.8111457824707,175.36019897460938,148.04969876868419,169.48452758789063]},{"page":163,"text":"transition","rect":[53.8111457824707,185.3023223876953,85.87013763575479,179.46051025390626]},{"page":163,"text":"1.0","rect":[212.43865966796876,76.18036651611328,223.55582830629226,70.41020965576172]},{"page":163,"text":"0.5","rect":[212.43865966796876,109.45435333251953,223.55582830629226,103.68419647216797]},{"page":163,"text":"0.0","rect":[212.43865966796876,141.89210510253907,223.55582830629226,136.1219482421875]},{"page":163,"text":"–0.5","rect":[207.99180603027345,174.1922149658203,223.55582830629226,168.42205810546876]},{"page":163,"text":"0.0","rect":[221.6251220703125,187.2331085205078,232.742290708636,181.46295166015626]},{"page":163,"text":"6 Phase Transitions","rect":[318.28729248046877,42.55716323852539,385.1405685710839,36.68148422241211]},{"page":163,"text":"6.7.3.2 Maier–Saupe Theory","rect":[53.812843322753909,241.62002563476563,182.1843194108344,231.96832275390626]},{"page":163,"text":"The results of the simplest mean field approach are very impressing. However,","rect":[53.812843322753909,265.0839538574219,385.1537136604953,256.1494140625]},{"page":163,"text":"some experimental observations, e.g., different temperature dependencies of the","rect":[53.812843322753909,277.0434875488281,385.1726459331688,268.10894775390627]},{"page":163,"text":"order parameter for different substances, a discontinuity of density at the N–Iso","rect":[53.812843322753909,289.0030517578125,385.12981501630318,280.0684814453125]},{"page":163,"text":"transition have not been explained. The main disadvantages of the simplest theory","rect":[53.812843322753909,300.96258544921877,385.1308220963813,292.02801513671877]},{"page":163,"text":"are (a) lack of the density (or volume) dependence of <P2> showing a jump at the","rect":[53.812843322753909,312.9228515625,385.1756976909813,303.98760986328127]},{"page":163,"text":"transition; (b) an oversimplified form of the potential well; and (c) pure phenome-","rect":[53.8139762878418,324.88238525390627,385.09111915437355,315.94781494140627]},{"page":163,"text":"nological nature of the depth of the potential well v.","rect":[53.8139762878418,336.8419189453125,263.9065212776828,327.9073486328125]},{"page":163,"text":"If we come back to Eq. 6.66, then with additional terms (fourth, sixth, etc.) we","rect":[65.76599884033203,348.8014831542969,385.1687702007469,339.866943359375]},{"page":163,"text":"can find a new partition function and calculate more precisely the thermodynamic","rect":[53.8139762878418,360.7042541503906,385.18369329645005,351.76971435546877]},{"page":163,"text":"parameters (free energy, entropy, etc.). Indeed, the results of such calculations fit","rect":[53.8139762878418,372.6637878417969,385.1478105313499,363.729248046875]},{"page":163,"text":"much better the experimental data on <P2> and <P4> for different materials","rect":[53.8139762878418,384.62335205078127,385.08490385161596,375.68878173828127]},{"page":163,"text":"[22]. But what about the nature of parameters v and m? For the first and most","rect":[53.81379318237305,396.5836181640625,385.1376786954124,387.6490478515625]},{"page":163,"text":"important of them, the answer is given by the Maier–Saupe theory [21].","rect":[53.81376266479492,408.5431823730469,343.8605618050266,399.608642578125]},{"page":163,"text":"All physics of the intermolecular interactions is included in potential W1. Its","rect":[65.7647933959961,420.5027160644531,385.1613809023972,411.56817626953127]},{"page":163,"text":"general dependence on density (or volume V) is not known. However, a dependence","rect":[53.814598083496097,432.4625549316406,385.07776678277818,423.52801513671877]},{"page":163,"text":"of the form W1 / \u0002A\u0005V2 is consistent with the typical r-6-dependence of intermo-","rect":[53.814598083496097,445.710205078125,385.0914548477329,432.5832824707031]},{"page":163,"text":"lecular attractive energy (for instance, in the Lennard-Jones potential r12\u00026). The","rect":[53.813350677490237,456.3250732421875,385.1483539409813,445.2344970703125]},{"page":163,"text":"V\u00022 form is a result of averaging over three-dimensional volume. This law is valid","rect":[53.81450653076172,468.284912109375,385.1285332780219,457.2339782714844]},{"page":163,"text":"for London dispersion forces related to “induced dipole – induced dipole” interac-","rect":[53.81257629394531,480.2444763183594,385.08672462312355,471.3099365234375]},{"page":163,"text":"tions. Since elongated molecules have anisotropic (tensorial) polarizabilities, the","rect":[53.81257629394531,492.2040100097656,385.1703571148094,483.26947021484377]},{"page":163,"text":"idea of Maier and Saupe was to describe their interaction in terms of anisotropic","rect":[53.81257629394531,504.1635437011719,385.17334783746568,495.22900390625]},{"page":163,"text":"dispersion forces. On account of the molecular volume term, Vm ¼ V/N, the mean","rect":[53.81258010864258,516.1031494140625,385.14235774091255,507.148681640625]},{"page":163,"text":"field potential (6.67) takes the form:","rect":[53.814476013183597,528.0830078125,199.76212937900614,519.1484985351563]},{"page":163,"text":"W1 ¼ \u0002A\u0005V2hP2iP2ðcos#Þ","rect":[164.61181640625,555.4442749023438,274.36969389067846,542.3173828125]},{"page":163,"text":"m","rect":[212.4757537841797,557.921630859375,217.47872566443543,554.6864624023438]},{"page":163,"text":"(6.71)","rect":[361.0559387207031,553.7579956054688,385.10530982820168,545.2218627929688]},{"page":163,"text":"This form allows for separation of effects related to packing of molecules (Vm),","rect":[65.76630401611328,581.420166015625,385.1559109261203,572.4856567382813]},{"page":163,"text":"their ordering (<P2>) and molecular spectral properties (A). The coefficient A was","rect":[53.81411361694336,593.3795166015625,385.15192045317846,584.4448852539063]},{"page":164,"text":"References","rect":[53.81215286254883,42.52561569213867,91.48083956717767,36.68380355834961]},{"page":164,"text":"147","rect":[372.49749755859377,42.55948257446289,385.1890921035282,36.73460388183594]},{"page":164,"text":"considered as a constant to be found from experiment. It can also be estimated from","rect":[53.812843322753909,68.2883529663086,385.1337962019499,59.35380554199219]},{"page":164,"text":"the theory of dispersion forces. The classical formula for the interaction energy","rect":[53.812843322753909,80.24788665771485,385.16765681317818,71.31333923339844]},{"page":164,"text":"between the oscillating dipole of one atom with another, neutral atom (or a","rect":[53.812843322753909,92.20748138427735,385.15866888238755,83.27293395996094]},{"page":164,"text":"spherical molecule):","rect":[53.812843322753909,104.11019134521485,135.59663255283426,95.17564392089844]},{"page":164,"text":"wðrÞ ¼ \u00022h\u0002ro6a20 ¼ \u0002rC6","rect":[169.70872497558595,140.285400390625,267.11055061897596,118.34115600585938]},{"page":164,"text":"(6.72)","rect":[361.0561828613281,135.2268524169922,385.10555396882668,126.69071960449219]},{"page":164,"text":"Here o is frequency of oscillating electron in the first atom, a0 is polarizability of","rect":[65.76653289794922,163.85232543945313,385.15096412507668,154.91770935058595]},{"page":164,"text":"the second atom, r is distance between the two. This result is in qualitative","rect":[53.81411361694336,175.81185913085938,385.0972369976219,166.87730407714845]},{"page":164,"text":"agreement with a quantum–mechanical theory developed by London. For calcula-","rect":[53.81313705444336,187.77142333984376,385.0882505020298,178.8368682861328]},{"page":164,"text":"tion of parameter A in (6.71) Maier and Saupe used this basic formula, but in","rect":[53.81313705444336,199.73095703125,385.14202204755318,190.79640197753907]},{"page":164,"text":"addition they took the anisotropy of molecular polarizability Da into account. It is","rect":[53.81313705444336,211.69052124023438,385.18683256255346,202.42727661132813]},{"page":164,"text":"Da that determines the stability of the nematic phase.","rect":[53.813106536865237,223.59329223632813,269.4541897347141,214.33004760742188]},{"page":164,"text":"The Maier–Saupe theory is very successful in explanation of density jump at TNI.","rect":[65.76514434814453,235.55282592773438,385.1832241585422,226.61827087402345]},{"page":164,"text":"It can also explain some correlation between the thermal stability of the nematic","rect":[53.814537048339847,247.5130615234375,385.0996784038719,238.57850646972657]},{"page":164,"text":"phase and the anisotropy of molecular polarizability Da. Up to now it is very","rect":[53.814537048339847,259.4726257324219,385.11953059247505,250.20938110351563]},{"page":164,"text":"popular among chemists although there are some substances (e.g., cyclohexyl-","rect":[53.81554412841797,271.4321594238281,385.1404965957798,262.49761962890627]},{"page":164,"text":"cyclohexanes), which have a very stable nematic phase but Da \b 0. Its main","rect":[53.81554412841797,283.3917236328125,385.1344842057563,274.12847900390627]},{"page":164,"text":"drawback is a neglect of short-range (steric) effects taken into account, for instance,","rect":[53.81554412841797,295.3512268066406,385.1554226448703,286.41668701171877]},{"page":164,"text":"by hard-rod Onsager-type models. On the contrary, the hard-rod models do not take","rect":[53.81554412841797,307.310791015625,385.1514362163719,298.3563232421875]},{"page":164,"text":"long-range interaction into account. The two approaches taken together result in","rect":[53.81554412841797,319.21356201171877,385.14446345380318,310.27899169921877]},{"page":164,"text":"more realistic predictions. However, in general, due to complexity of the problem,","rect":[53.81554412841797,331.173095703125,385.1693996956516,322.238525390625]},{"page":164,"text":"all such models present only semi-quantitative picture [23].","rect":[53.81554412841797,343.13262939453127,293.7886929085422,334.19805908203127]},{"page":164,"text":"References","rect":[53.812843322753909,387.3817138671875,109.59614448282879,378.5965881347656]},{"page":164,"text":"1.","rect":[58.06126022338867,413.0,64.40706131055318,407.08349609375]},{"page":164,"text":"2.","rect":[58.06126022338867,443.0,64.40706131055318,436.9546203613281]},{"page":164,"text":"3.","rect":[58.06126022338867,463.0,64.40706131055318,456.8498229980469]},{"page":164,"text":"4.","rect":[58.0612678527832,483.0,64.4070689399477,476.8017272949219]},{"page":164,"text":"5.","rect":[58.0612678527832,493.0,64.4070689399477,486.619384765625]},{"page":164,"text":"6.","rect":[58.0612678527832,513.0,64.4070689399477,506.6220397949219]},{"page":164,"text":"7.","rect":[58.0612678527832,543.0,64.4070689399477,536.66259765625]},{"page":164,"text":"8.","rect":[58.0612678527832,563.0,64.4070689399477,556.4959716796875]},{"page":164,"text":"9.","rect":[58.06130599975586,582.28369140625,64.40710708692036,576.39111328125]},{"page":164,"text":"Wojtowicz, P.: Introduction to molecular theory of nematic liquid crystals. In: Priestley, E.B.,","rect":[68.59698486328125,414.6270751953125,385.199984284186,407.03271484375]},{"page":164,"text":"Wojtowicz, P., Sheng, P. (eds.) Introduction to Liquid Crystals, Chapter 3, pp. 31–43. Plenum","rect":[68.59698486328125,424.5462951660156,385.2001061284064,416.9519348144531]},{"page":164,"text":"Press, New York (1975)","rect":[68.59698486328125,434.18359375,151.53831571204356,426.8770751953125]},{"page":164,"text":"Landau, L.D., Lifshitz, E.M.: Statistical Physics, 5th edn. Nauka, Moscow (2001).","rect":[68.59698486328125,444.4981994628906,385.161165924811,436.85302734375]},{"page":164,"text":"(in Russian) [see also “Statistical Physics”, v. 1, 3rd ed., Pergamon, Oxford, 1980]","rect":[68.59698486328125,454.4741516113281,351.5106515274732,446.86285400390627]},{"page":164,"text":"de Gennes, P.G.: Short-range order effects in the isotropic phase of nematics and cholesterics.","rect":[68.59698486328125,464.3934020996094,385.1576869209047,456.7990417480469]},{"page":164,"text":"Mol. Cryst. Liq. Cryst. 12, 193–214 (1971)","rect":[68.59698486328125,474.3693542480469,216.62339871985606,466.7072448730469]},{"page":164,"text":"De Gennes, P.G., Prost, J.: The Physics of Liquid Crystals, 2nd edn. Clarendon, Oxford (1993)","rect":[68.59699249267578,484.3453063964844,385.2001046524732,476.7340087890625]},{"page":164,"text":"Vertogen, G., de Jeu, V.H.: Thermotropic Liquid Crystals. Fundamentals. Springer-Verlag,","rect":[68.59699249267578,494.2645568847656,385.1315028388735,486.6701965332031]},{"page":164,"text":"Berlin (1987)","rect":[68.59699249267578,503.9018249511719,114.77497953040292,496.6461181640625]},{"page":164,"text":"Priestley, E.B., Sheng, P.: Landau-de Gennes theory of liquid crystal phase transitions. In:","rect":[68.59699249267578,514.2163696289063,385.1534700795657,506.6220397949219]},{"page":164,"text":"Priestley, E.B., Wojtowicz, P., Sheng, P. (eds.) Introduction to Liquid Crystals, Chapter 10,","rect":[68.59699249267578,524.1923828125,385.170443268561,516.5980224609375]},{"page":164,"text":"pp. 143–201. Plenum Press, New York (1975)","rect":[68.59699249267578,534.111572265625,226.876557289192,526.4664306640625]},{"page":164,"text":"Khoo, I.C., Wu, S.-T.: Optics and Nonlinear Optics of Liquid Crystals. Word Scientific,","rect":[68.59699249267578,544.0875854492188,385.13830825879537,536.476318359375]},{"page":164,"text":"Singapore (1993). Chapt. 1","rect":[68.59699249267578,554.0634765625,161.4200186172001,546.4691162109375]},{"page":164,"text":"Anisimov, M.A.: Critical phenomena in liquid crystals. Mol. Cryst. Liq. Cryst. (special","rect":[68.59699249267578,564.0394897460938,385.1222505971438,556.3012084960938]},{"page":164,"text":"topics) 162a, 1–96 (1988)","rect":[68.5970230102539,573.8994140625,158.221772132942,566.2203979492188]},{"page":164,"text":"Helfrich, W.: Effect of electric fields on the temperature of phase transitions of liquid crystals.","rect":[68.59703063964844,583.9346313476563,385.146761627936,576.3402709960938]},{"page":164,"text":"Phys. Rev. Lett. 24, 201–203 (1970)","rect":[68.59703063964844,593.9105834960938,193.3986520157545,586.3162231445313]},{"page":165,"text":"148","rect":[53.80946350097656,42.56039810180664,66.50106949000284,36.73551940917969]},{"page":165,"text":"6 Phase Transitions","rect":[318.28558349609377,42.56039810180664,385.1388595867089,36.68471908569336]},{"page":165,"text":"10.","rect":[53.812843322753909,65.22824096679688,64.38918182446919,59.40336608886719]},{"page":165,"text":"11.","rect":[53.81200408935547,85.07855224609375,64.38834259107076,79.35527038574219]},{"page":165,"text":"12.","rect":[53.81200408935547,105.0,64.38834259107076,99.25047302246094]},{"page":165,"text":"13.","rect":[53.81200408935547,125.02725219726563,64.38834259107076,119.20237731933594]},{"page":165,"text":"14.","rect":[53.81200408935547,155.0,64.38834259107076,149.07350158691407]},{"page":165,"text":"15.","rect":[53.81199645996094,185.0,64.38833496167622,178.8430633544922]},{"page":165,"text":"16.","rect":[53.81199645996094,205.0,64.38833496167622,198.84576416015626]},{"page":165,"text":"17.","rect":[53.81199645996094,235.0,64.38833496167622,228.76768493652345]},{"page":165,"text":"18.","rect":[53.8111457824707,255.0,64.38748428418599,248.66285705566407]},{"page":165,"text":"19.","rect":[53.810306549072269,275.0,64.38664505078755,268.6147766113281]},{"page":165,"text":"20.","rect":[53.810306549072269,305.0,64.38664505078755,298.4859619140625]},{"page":165,"text":"21.","rect":[53.810306549072269,324.10443115234377,64.38664505078755,318.3811340332031]},{"page":165,"text":"22.","rect":[53.80946350097656,354.03228759765627,64.38580200269185,348.3089904785156]},{"page":165,"text":"23.","rect":[53.80946350097656,384.0050354003906,64.38580200269185,378.1801452636719]},{"page":165,"text":"McMillan, W.L.: X-ray scattering from liquid crystals: 1. Cholesteryl nonanoate and myristate.","rect":[68.59687042236328,66.9469223022461,385.2464320380922,59.35256576538086]},{"page":165,"text":"Phys. Rev. A 6, 936–947 (1972)","rect":[68.59687042236328,76.9228744506836,177.43499845130138,69.26925659179688]},{"page":165,"text":"Thoen, J.: Thermal methods. In: Demus, D., Goodby, J., Gray, G.W., Spiess, H.-W., Vill, V.","rect":[68.59603118896485,86.8988265991211,385.1626918037172,79.30447387695313]},{"page":165,"text":"(eds.) Physical Properties of Liquid Crystals, pp. 208–232. Wiley-VCH, Weinheim (1999)","rect":[68.59603118896485,96.8747787475586,377.88958829505136,89.28042602539063]},{"page":165,"text":"Pikin, S.A.: Structural Transformations in Liquid Crystals. Gordon & Breach, New York","rect":[68.59603118896485,106.79402923583985,385.14911407618447,99.19967651367188]},{"page":165,"text":"(1981)","rect":[68.59603118896485,116.43132781982422,91.15324491126229,109.22642517089844]},{"page":165,"text":"Cladis, P.E.: Re-entrant phase transitions in liquid crystals. In: Demus, D., Goodby, J., Gray,","rect":[68.59603118896485,126.74593353271485,385.1737696845766,119.15158081054688]},{"page":165,"text":"G.W., Spiess, H.-W., Vill, V. (eds.) Physical Properties of Liquid Crystals, pp. 289–303.","rect":[68.59603118896485,136.66513061523438,385.192415924811,129.07077026367188]},{"page":165,"text":"Wiley-VCH, Weinheim (1999)","rect":[68.59603118896485,146.64108276367188,174.8746804581373,139.04672241210938]},{"page":165,"text":"Kasting, G.B., Lushington, K.T., Garland, C.W.: Critical heat capacity near the nematic-","rect":[68.59603118896485,156.6170654296875,385.1483163223951,149.022705078125]},{"page":165,"text":"smectic A transition in octyloxycyanobiphenyl in the range 1-2000 bar. Phys. Rev. B 22,","rect":[68.59603118896485,166.59304809570313,385.204134674811,158.99868774414063]},{"page":165,"text":"321–331 (1980)","rect":[68.59602355957031,176.17356872558595,123.1580819473951,168.9686737060547]},{"page":165,"text":"de Gennes, P.G.: Sur la transition smectique A – smectique C. C. R. Acad. Sci. 274, 758–760","rect":[68.59602355957031,186.48822021484376,385.1737112441532,178.8430633544922]},{"page":165,"text":"(1972)","rect":[68.59602355957031,196.12550354003907,91.15323728186776,188.9206085205078]},{"page":165,"text":"Barois, Ph: Phase transition theories. In: Demus, D., Goodby, J., Gray, G.W., Spiess, H.-W.,","rect":[68.59602355957031,206.44012451171876,385.155062409186,198.84576416015626]},{"page":165,"text":"Vill, V. (eds.) Physical Properties of Liquid Crystals, pp. 179–207. Wiley-VCH, Weinheim","rect":[68.59602355957031,216.35934448242188,385.126284106922,208.76498413085938]},{"page":165,"text":"(1999)","rect":[68.59602355957031,225.9966278076172,91.15323728186776,218.79173278808595]},{"page":165,"text":"Dunmur, D.A., Palffy-Muhoray, P.: Effect of electric and magnetic fields on orientational","rect":[68.59602355957031,236.31124877929688,385.1660738393313,228.71688842773438]},{"page":165,"text":"disorder in liquid crystals. J. Phys. Chem. 92, 1406–1419 (1988)","rect":[68.59602355957031,246.23046875,289.69394010169199,238.26358032226563]},{"page":165,"text":"Blinov, L.M., Ozaki, M., Yoshino, K.: Flexoelectric polarization in nematic liquid crystals","rect":[68.59517669677735,256.2064208984375,385.1254013347558,248.59512329101563]},{"page":165,"text":"measured by a field on-off pyroelectric technique. JETP Lett. 69, 236–242 (1999)","rect":[68.59516906738281,266.1824035644531,348.43405240637949,258.21551513671877]},{"page":165,"text":"Osipov, M.A.: Molecular theories of liquid crystals. In: Demus, D., Goodby, J., Gray, G.W.,","rect":[68.5943374633789,276.1583557128906,385.1686122138735,268.54705810546877]},{"page":165,"text":"Spiess, H.-W., Vill, V. (eds.) Physical Properties of Liquid Crystals, pp. 40–71. Wiley-VCH,","rect":[68.5943374633789,286.07763671875,385.1228968818422,278.4832763671875]},{"page":165,"text":"Weinheim (1999)","rect":[68.5943374633789,295.7149353027344,128.9903039322584,288.459228515625]},{"page":165,"text":"Onsager, L.: The effects of shape on the interaction of colloidal particles. Ann. N.Y. Acad.","rect":[68.5943374633789,306.029541015625,385.1737391669985,298.4182434082031]},{"page":165,"text":"Sci. 51, 627–659 (1949)","rect":[68.5943374633789,315.6668701171875,151.4222878800123,308.3434143066406]},{"page":165,"text":"Maier, W., Saupe, A.: Eine einfache molekular-statistische Theorie der nematischen kristal-","rect":[68.5943374633789,325.9247131347656,385.14743131262949,318.3303527832031]},{"page":165,"text":"linflu€ssigen Phase. Teil I, Z. Naturforschg. 14a, 882–889 (1959). Teil II, Z. Naturforschg. 15a,","rect":[68.5943374633789,335.90069580078127,385.1542689521547,328.0]},{"page":165,"text":"287-292 (1960)","rect":[68.59349060058594,345.5379943847656,121.73916715247323,338.28228759765627]},{"page":165,"text":"Wojtowicz, P.: Generalized mean field theory of nematic liquid crystals. In: Priestley, E.B.,","rect":[68.59349060058594,355.8525695800781,385.1965052802797,348.2582092285156]},{"page":165,"text":"Wojtowicz, P., Sheng, P. (eds.) Introduction to Liquid Crystals, Chapter 4, pp. 45–81. Plenum","rect":[68.59349060058594,365.7718200683594,385.196596606922,358.12664794921877]},{"page":165,"text":"Press, New York (1975)","rect":[68.59349060058594,375.40911865234377,151.53482144934825,368.10260009765627]},{"page":165,"text":"Averyanov, E.M.: Steric Effects of Substituents and Mesomorphism. Siberian Department of","rect":[68.59349060058594,385.7237243652344,385.20748990637949,378.1293640136719]},{"page":165,"text":"Russian Academy of Science Publishers, Novosibirsk (2004) (in Russian)","rect":[68.59349060058594,395.6429443359375,320.6843270645826,388.048583984375]},{"page":166,"text":"Part II","rect":[338.284912109375,70.58036041259766,385.21503843219787,59.488277435302737]},{"page":166,"text":"Physical Properties","rect":[236.49359130859376,94.86486053466797,385.10479621428967,77.9219741821289]},{"page":167,"text":"Chapter7","rect":[53.812843322753909,72.10812377929688,114.14115996551633,59.571903228759769]},{"page":167,"text":"Magnetic, Electric and Transport Properties","rect":[53.812843322753909,90.6727066040039,358.9654961958697,76.122314453125]},{"page":167,"text":"Some properties of liquid crystals depend mainly on properties of individual","rect":[53.812843322753909,211.74758911132813,385.16462571689677,202.8130340576172]},{"page":167,"text":"molecules and approximately obey the additivity law. Thus, molecular properties","rect":[53.812843322753909,223.65036010742188,385.15265287505346,214.71580505371095]},{"page":167,"text":"can","rect":[53.812843322753909,234.0,67.68901148847113,228.0]},{"page":167,"text":"be","rect":[72.90203094482422,234.0,82.32866705133283,226.6753692626953]},{"page":167,"text":"translated,","rect":[87.51679992675781,234.0,128.94324155356174,226.6753692626953]},{"page":167,"text":"of","rect":[134.1343536376953,234.0,142.42620216218604,226.6753692626953]},{"page":167,"text":"course","rect":[147.55960083007813,234.0,173.64759100152816,228.0]},{"page":167,"text":"with","rect":[178.8277587890625,234.0,196.5462578873969,226.6753692626953]},{"page":167,"text":"some","rect":[201.71249389648438,234.0,222.81543768121566,228.0]},{"page":167,"text":"precautions,","rect":[227.99560546875,235.60992431640626,276.5931057503391,226.6753692626953]},{"page":167,"text":"onto","rect":[281.7513732910156,234.0,299.4798211198188,227.69134521484376]},{"page":167,"text":"the","rect":[304.6928405761719,234.0,316.91660345270005,226.6753692626953]},{"page":167,"text":"properties","rect":[322.08282470703127,235.60992431640626,362.04501737700658,226.6753692626953]},{"page":167,"text":"of","rect":[367.22821044921877,234.0,375.5200437149204,226.6753692626953]},{"page":167,"text":"a","rect":[380.70916748046877,234.0,385.15869939996568,228.0]},{"page":167,"text":"mesophase on account of the symmetry of the latter. Quantitatively, the relevant","rect":[53.812843322753909,247.5694580078125,385.1079545743186,238.63490295410157]},{"page":167,"text":"phenomenological characteristics such as magnetic and dielectric susceptibilities,","rect":[53.812843322753909,259.5290222167969,385.09893460776098,250.59446716308595]},{"page":167,"text":"electric and thermal conductivity, diffusion coefficients, etc. can be calculated by","rect":[53.812843322753909,271.4885559082031,385.16668025067818,262.55401611328127]},{"page":167,"text":"averaging molecular parameters with the corresponding single-particle distribution","rect":[53.812843322753909,283.4878845214844,385.17754450849068,274.49359130859377]},{"page":167,"text":"function. Other properties of liquid crystals such as elasticity or viscosity dramati-","rect":[53.81282424926758,295.4076232910156,385.17559181062355,286.4531555175781]},{"page":167,"text":"cally depend on intermolecular interactions and the corresponding many-particle","rect":[53.81282424926758,307.3671569824219,385.14267767145005,298.4126892089844]},{"page":167,"text":"distribution functions have to be taken into account. Here we shall start with a","rect":[53.81282424926758,317.23797607421877,385.15869939996568,310.33538818359377]},{"page":167,"text":"discussion of the properties of the first sort. Moreover, we shall limit ourselves","rect":[53.81282424926758,331.2294921875,385.1546975527878,322.294921875]},{"page":167,"text":"mostly to the phases of highest symmetry (uniaxial nematics and smectic A) whose","rect":[53.81282424926758,343.1889953613281,385.13278997613755,334.25445556640627]},{"page":167,"text":"properties","rect":[53.81282424926758,355.1485595703125,93.77499784575656,346.2139892578125]},{"page":167,"text":"are","rect":[99.63802337646485,354.0,111.85184515191877,348.0]},{"page":167,"text":"represented","rect":[117.70790100097656,355.1485595703125,163.79688349774848,346.2139892578125]},{"page":167,"text":"by","rect":[169.64996337890626,355.1485595703125,179.6041649918891,346.2139892578125]},{"page":167,"text":"second-rank","rect":[185.45425415039063,354.0,234.23586360028754,346.2139892578125]},{"page":167,"text":"tensors,","rect":[240.11680603027345,354.0,270.8713650276828,347.22998046875]},{"page":167,"text":"discussed","rect":[276.7095031738281,354.0,314.93666926435005,346.2139892578125]},{"page":167,"text":"in","rect":[320.7220458984375,354.0,328.4963006120063,346.2139892578125]},{"page":167,"text":"Section","rect":[334.3752136230469,353.1265869140625,364.3174676041938,346.2139892578125]},{"page":167,"text":"2.5.","rect":[370.23126220703127,353.1265869140625,385.1815151741672,346.15423583984377]},{"page":167,"text":"Throughout this chapter, the director field is considered to be non-distorted, n(r) ¼","rect":[53.812843322753909,367.10809326171877,385.14771297666939,358.17352294921877]},{"page":167,"text":"constant.","rect":[53.81386947631836,377.00579833984377,89.66595120932345,371.1490478515625]},{"page":167,"text":"7.1 Magnetic Phenomena","rect":[53.812843322753909,429.4156188964844,191.67286434077807,418.5029602050781]},{"page":167,"text":"7.1.1 Magnetic Anisotropy","rect":[53.812843322753909,459.1069641113281,193.29142554731576,448.5408935546875]},{"page":167,"text":"Inthe Gauss system, magneticinduction B ¼ mH where H is magnetic field strength.","rect":[53.812843322753909,486.6495056152344,385.1845364144016,477.7149658203125]},{"page":167,"text":"The magnetic permeability m ¼ 1 þ 4pw where w is dimensionless magnetic sus-","rect":[53.814815521240237,498.6090393066406,385.2562192520298,489.67449951171877]},{"page":167,"text":"ceptibility.","rect":[53.81582260131836,510.568603515625,96.00375028158908,501.634033203125]},{"page":167,"text":"Except","rect":[101.05650329589844,510.568603515625,128.48632676670145,501.833251953125]},{"page":167,"text":"ferromagnetic","rect":[133.5699462890625,510.568603515625,188.6247028179344,501.634033203125]},{"page":167,"text":"materials,","rect":[193.72622680664063,509.0,232.06983609701877,501.634033203125]},{"page":167,"text":"<w> ¼ \u00025 \u0003 10\u00027, therefore m \u0004 1.","rect":[53.8157844543457,522.3822021484375,204.0813412239719,511.5137634277344]},{"page":167,"text":"4pw","rect":[237.2281036376953,510.47894287109377,253.0931431705768,501.6938171386719]},{"page":167,"text":"<<","rect":[256.43475341796877,508.80560302734377,271.7642290899506,503.0484313964844]},{"page":167,"text":"1.","rect":[275.1277770996094,509.0,282.59344144369848,501.6938171386719]},{"page":167,"text":"e.g.,","rect":[287.64617919921877,510.568603515625,304.1831326058078,503.0]},{"page":167,"text":"for","rect":[309.3414001464844,509.0,320.7558835586704,501.634033203125]},{"page":167,"text":"p-azoxyanisole","rect":[325.8813171386719,510.568603515625,385.2402728862938,501.634033203125]},{"page":167,"text":"By definition, a magnetic moment of substance per unit volume is magnetization","rect":[65.76557922363281,534.431396484375,385.12456599286568,525.4968872070313]},{"page":167,"text":"M ¼ wH. For an anisotropic material, the magnetization vector components are","rect":[53.81356430053711,546.3909301757813,385.1633991069969,537.4564208984375]},{"page":167,"text":"Ma ¼ wabHb and the contribution to the free energy density of the mesophase from","rect":[53.81455993652344,559.9693603515625,385.1356272566374,549.4161987304688]},{"page":167,"text":"the magnetic field is given by [1]:","rect":[53.812713623046878,571.50048828125,190.60449082920145,562.5659790039063]},{"page":167,"text":"L.M. Blinov, Structure and Properties of Liquid Crystals,","rect":[53.812843322753909,623.7623291015625,251.63688919141254,616.1510620117188]},{"page":167,"text":"DOI 10.1007/978-90-481-8829-1_7, # Springer Science+Business Media B.V. 2011","rect":[53.812843322753909,633.698486328125,345.57761139063759,625.4920043945313]},{"page":167,"text":"151","rect":[372.4981994628906,622.0606079101563,385.18979400782509,616.1340942382813]},{"page":168,"text":"152","rect":[53.813690185546878,42.55777359008789,66.50529617457315,36.63129806518555]},{"page":168,"text":"7 Magnetic, Electric and Transport Properties","rect":[228.79029846191407,44.276451110839847,385.1491134929589,36.68209457397461]},{"page":168,"text":"Ha","rect":[153.33847045898438,67.20203399658203,160.86837430502659,61.54512405395508]},{"page":168,"text":"Hb","rect":[214.28858947753907,67.22942352294922,222.07740959798373,60.46828079223633]},{"page":168,"text":"gmagn ¼ \u0002 ð HadMa ¼ \u0002 ð HawabdHb ¼ \u000212wabHaHb","rect":[109.55176544189453,90.42349243164063,328.9327827466307,68.30126953125]},{"page":168,"text":"0","rect":[155.09449768066407,96.78154754638672,158.57846138557754,91.98458862304688]},{"page":168,"text":"0","rect":[216.15792846679688,96.78154754638672,219.64189217171035,91.98458862304688]},{"page":168,"text":"For uniaxial phases wab ¼ w?dab þ wananb, therefore","rect":[65.76496887207031,122.96034240722656,278.37204778863755,112.1082763671875]},{"page":168,"text":"gmagn ¼ \u000212hw?H2 þ waðnaHaÞ2i","rect":[152.14968872070313,155.87228393554688,286.8646555369004,135.53594970703126]},{"page":168,"text":"(7.1)","rect":[366.0971374511719,150.7577362060547,385.16939674226418,142.2813720703125]},{"page":168,"text":"or in the vector form","rect":[53.81367111206055,177.37811279296876,137.7684855330046,170.50538635253907]},{"page":168,"text":"gmagn ¼ \u000212hw?H2 þ waðHnÞ2i","rect":[155.54769897460938,213.97006225585938,283.465638202916,193.63375854492188]},{"page":168,"text":"(7.2)","rect":[366.0965576171875,208.8556365966797,385.1688169082798,200.3792724609375]},{"page":168,"text":"Here Hn ¼ Hcosa where a is the angle between the director and magnetic field.","rect":[65.76508331298828,237.53781127929688,385.1141018440891,228.60325622558595]},{"page":168,"text":"The second term determines an orientation of the director in the field: for wa > 0 the","rect":[53.81306076049805,249.40773010253907,385.1755756206688,240.5624237060547]},{"page":168,"text":"director n tends to align parallel to the field; for wa < 0 it tends to be perpendicular","rect":[53.814842224121097,261.4000244140625,385.1377194961704,252.4651641845703]},{"page":168,"text":"to H. A sign of wa is determined by competition of diamagnetic and paramagnetic","rect":[53.81379318237305,273.3595886230469,385.1780170269188,264.425048828125]},{"page":168,"text":"terms.","rect":[53.81325912475586,283.2572937011719,78.46983762289767,277.4005432128906]},{"page":168,"text":"7.1.2 Diamagnetism","rect":[53.812843322753909,335.40264892578127,161.423263184768,324.9202575683594]},{"page":168,"text":"Diamagnetism is caused by an additional electric current induced by a magnetic","rect":[53.812843322753909,363.0286560058594,385.17758978082505,354.0941162109375]},{"page":168,"text":"field in a molecule. The diamagnetic contribution to wab is negative, independent","rect":[53.812843322753909,375.8582763671875,385.1260820157249,366.05364990234377]},{"page":168,"text":"of permanent magnetic moments of molecules and is present in all molecular","rect":[53.81411361694336,386.94781494140627,385.1510251602329,378.01324462890627]},{"page":168,"text":"materials [2].","rect":[53.81411361694336,398.24298095703127,107.48222775961642,389.916015625]},{"page":168,"text":"7.1.2.1 Single Electron","rect":[53.8140983581543,440.81707763671877,156.65004319499088,431.71319580078127]},{"page":168,"text":"Consider a classical model of a current i caused by a rotating electron in the absence","rect":[53.8140983581543,464.5997009277344,385.1061176128563,455.6651611328125]},{"page":168,"text":"of a magnetic field, see Fig. 7.1. When an external magnetic field is applied, an","rect":[53.8140983581543,476.5592346191406,385.14995661786568,467.62469482421877]},{"page":168,"text":"additional, namely, induced current appears due to the Lorentz force acting on a","rect":[53.813106536865237,488.518798828125,385.15891302301255,479.584228515625]},{"page":168,"text":"moving electron. The induced current component di tries to screen the external field","rect":[53.813106536865237,500.47833251953127,385.11806574872505,491.2449645996094]},{"page":168,"text":"Fig. 7.1 Diamagnetism of","rect":[53.812843322753909,531.4005737304688,145.6027535782545,523.6707763671875]},{"page":168,"text":"a single electron: electron","rect":[53.812843322753909,541.3087768554688,142.1430410781376,533.7144165039063]},{"page":168,"text":"rotation creates current i in","rect":[53.812843322753909,549.5576171875,146.4082540908329,543.6903686523438]},{"page":168,"text":"the magnetic field absence","rect":[53.812843322753909,561.2607421875,144.44188830637456,553.6663818359375]},{"page":168,"text":"and the additional current","rect":[53.812843322753909,569.4528198242188,141.6336794301516,563.5855712890625]},{"page":168,"text":"di is induced by the magnetic","rect":[53.812843322753909,581.1559448242188,155.2805266120386,573.3330078125]},{"page":168,"text":"field H","rect":[53.813690185546878,589.4047241210938,77.86850975389469,583.5374755859375]},{"page":168,"text":"i","rect":[300.7238464355469,573.6197509765625,302.89002080708829,568.2769165039063]},{"page":168,"text":"ρ","rect":[316.7183532714844,539.1630859375,321.10665485811247,533.7562866210938]},{"page":168,"text":"H","rect":[379.4136962890625,542.5676879882813,385.18483243759706,537.2728271484375]},{"page":168,"text":"δH","rect":[362.0459289550781,573.580322265625,371.76652799423769,567.91748046875]},{"page":169,"text":"7.1 Magnetic Phenomena","rect":[53.812843322753909,44.274620056152347,140.7714934089136,36.68026351928711]},{"page":169,"text":"153","rect":[372.49737548828127,42.55594253540039,385.1889700332157,36.62946701049805]},{"page":169,"text":"H ¼ Hz (the Lenz law). In fact, di comes about due to precession of electronic","rect":[53.812843322753909,68.2883529663086,385.1392291851219,59.05499267578125]},{"page":169,"text":"orbits with angular frequency according to the Larmor theorem:","rect":[53.813316345214847,80.24788665771485,311.8513322598655,71.31333923339844]},{"page":169,"text":"eH","rect":[224.82461547851563,103.2552490234375,236.42127752259104,96.47216796875]},{"page":169,"text":"oL ¼","rect":[196.44522094726563,111.3214111328125,217.87488582823188,105.12620544433594]},{"page":169,"text":"2mec","rect":[220.63290405273438,118.32622528076172,240.8864673443016,109.91582489013672]},{"page":169,"text":"(7.3)","rect":[366.09747314453127,111.5905532836914,385.16973243562355,103.11418914794922]},{"page":169,"text":"where e=2mec ¼ g [cm1/2g\u00021/2] is gyromagnetic ratio, e and me are the charge and","rect":[53.814022064208987,144.17222595214845,385.14614192060005,132.75311279296876]},{"page":169,"text":"mass of an electron, c is light velocity. Such a precession of an electron is","rect":[53.814292907714847,155.80355834960938,385.1312295352097,146.86900329589845]},{"page":169,"text":"equivalent to a diamagnetic current:","rect":[53.814292907714847,167.76309204101563,199.1845842129905,158.8285369873047]},{"page":169,"text":"dI ¼ \u0002eoL ¼ \u0002 e2H","rect":[169.1995086669922,199.27203369140626,260.9174323077473,183.97756958007813]},{"page":169,"text":"2p","rect":[205.1121368408203,205.99147033691407,215.5919224674518,199.1884765625]},{"page":169,"text":"4pmec2","rect":[238.419921875,207.5418701171875,267.62031624397596,198.31781005859376]},{"page":169,"text":"Generally, the magnetic moment of a frame with current is pm ¼ dI \u0003 s where","rect":[65.76496887207031,233.11624145507813,385.1118549175438,223.88287353515626]},{"page":169,"text":"s is vector of the current loop area (s ¼ p <r2>). Therefore, the induced moment","rect":[53.81481170654297,245.01901245117188,385.17341477939677,233.9681854248047]},{"page":169,"text":"and the susceptibility of single electron moving along the contour perpendicular to","rect":[53.813655853271487,256.9787292480469,385.13958064130318,248.04417419433595]},{"page":169,"text":"H are given by","rect":[53.813655853271487,268.93829345703127,114.40790644696722,260.00372314453127]},{"page":169,"text":"pme ¼ \u00024em2eHc2 \u0002r2\u0003;\u0002r2\u0003 ¼ \u0002x2\u0003 þ \u0002y2\u0003; gemagn ¼ pHem ¼ \u00024mee2c2 \u0002r2\u0003","rect":[79.53034973144531,308.71722412109377,359.48339943338478,285.1528015136719]},{"page":169,"text":"Here, there is no component of the current along z and <r2> is mean square of","rect":[65.7665023803711,334.2914733886719,385.15108619538918,323.24053955078127]},{"page":169,"text":"the distance between the electron and the field axis z. For a circular electron orbit of","rect":[53.814205169677737,344.2190856933594,385.15203224031105,337.31646728515627]},{"page":169,"text":"radius r, we have <r2> ¼ <r2>. In a more general case of electron orbits tilted","rect":[53.81519317626953,358.2107849121094,385.1243828873969,347.1597595214844]},{"page":169,"text":"with respect to H, the mean square distance between the electron and the nucleus is","rect":[53.81339645385742,370.17034912109377,385.18707670317846,361.23577880859377]},{"page":169,"text":"2","rect":[65.3685073852539,375.5108337402344,69.10126142827369,371.0224609375]},{"page":169,"text":"2","rect":[103.26417541503906,375.5108337402344,106.99692945805885,371.0224609375]},{"page":169,"text":"2","rect":[139.12045288085938,375.5108337402344,142.85320692387917,371.0224609375]},{"page":169,"text":"2","rect":[174.29708862304688,375.5108337402344,178.02984266606667,371.0224609375]},{"page":169,"text":"<r > ¼ <x > + <y > + <z >.","rect":[53.81339645385742,382.1101379394531,188.22787137533909,374.6097106933594]},{"page":169,"text":"7.1.2.2 Molecules","rect":[53.81318283081055,422.21697998046877,133.94755949126438,414.9458312988281]},{"page":169,"text":"For spherically symmetric molecules with electron orbit radius r, <x2> ¼ <y2>","rect":[53.81318283081055,447.80322265625,385.14807918760689,436.7721252441406]},{"page":169,"text":"¼ <z2> and <r2> ¼ (2/3) <r2>. Then the magnetic susceptibility of a spherical","rect":[53.814231872558597,459.7828063964844,385.1798539883811,448.7317810058594]},{"page":169,"text":"molecule having Z electrons is","rect":[53.81315231323242,471.74237060546877,177.24437345610813,462.80780029296877]},{"page":169,"text":"gmagn ¼ \u00026mZee2c2 \u0002r2\u0003","rect":[176.0528564453125,511.52105712890627,262.96078590799416,487.9567565917969]},{"page":169,"text":"(7.4)","rect":[366.0973205566406,504.7861022949219,385.1695798477329,496.30975341796877]},{"page":169,"text":"For cylindrical molecules with length L and diameter D, gmagn is a tensor with","rect":[65.7658920288086,538.025146484375,385.1193474870063,528.1414184570313]},{"page":169,"text":"principal components","rect":[53.81432342529297,549.054931640625,140.48364652739719,540.1204223632813]},{"page":169,"text":"gmjjagn ¼ \u00022mZee2c2 \u0002D2\u0003 and gm?agn ¼ \u00024mZee2c2 \u0002L2 þ D2\u0003","rect":[108.19319152832031,586.8499145507813,330.76501564432228,563.28564453125]},{"page":169,"text":"(7.5)","rect":[366.09747314453127,580.1140747070313,385.16973243562355,571.5181884765625]},{"page":170,"text":"154","rect":[53.812843322753909,42.55801773071289,66.50444931178018,36.63154220581055]},{"page":170,"text":"7 Magnetic, Electric and Transport Properties","rect":[228.78945922851563,44.276695251464847,385.1482590007714,36.68233871459961]},{"page":170,"text":"Hence, the diamagnetic anisotropy of a uniaxial liquid crystal phase with","rect":[65.76496887207031,68.2883529663086,385.1149834733344,59.35380554199219]},{"page":170,"text":"orientational order S and nv molecules per unit volume can be found:","rect":[53.812950134277347,80.24800872802735,332.4124589688499,71.31333923339844]},{"page":170,"text":"wadia ¼ \u00022nmvZeec22 S\u0004\u0002D2\u0003 \u0002 21\u0002L2 þ D2\u0003\u0005","rect":[142.40550231933595,118.04290008544922,296.6137970002694,94.47854614257813]},{"page":170,"text":"(7.6)","rect":[366.0971374511719,111.30802154541016,385.16939674226418,102.77189636230469]},{"page":170,"text":"According to (7.6), anisotropy wadia may be either positive or negative depending","rect":[65.76570892333985,142.275146484375,385.1109551530219,131.03855895996095]},{"page":170,"text":"on the molecular geometry. A very important structural unit of liquid crystal","rect":[53.813961029052737,153.53616333007813,385.1428666836936,144.6016082763672]},{"page":170,"text":"molecules, a benzene ring has negative diamagnetic susceptibility with the maxi-","rect":[53.813961029052737,165.4957275390625,385.17766700593605,156.56117248535157]},{"page":170,"text":"mum absolute value along its normal due to the maximum di current along the ring","rect":[53.813961029052737,177.45529174804688,385.1647576432563,168.221923828125]},{"page":170,"text":"perimeter. For this reason, elongated molecules containing two or three benzene","rect":[53.8139762878418,189.41482543945313,385.1508258648094,180.4802703857422]},{"page":170,"text":"rings have negative susceptibility with minimum absolute value along their longi-","rect":[53.8139762878418,201.41419982910157,385.12093482820168,192.4198760986328]},{"page":170,"text":"tudinal axes that is along the director. For such molecules <L2 þ D2>/2 exceeds","rect":[53.81296920776367,213.27713012695313,385.10702909575658,202.2269744873047]},{"page":170,"text":"<D2> in (7.6) and calamitic uniaxial phases formed by several benzene fragments","rect":[53.81399154663086,225.237548828125,385.17673124419408,214.18663024902345]},{"page":170,"text":"have positive diamagnetic anisotropy wdaia. along longitudinal axes. Typically, in","rect":[53.81296157836914,237.89566040039063,385.14397517255318,226.6591033935547]},{"page":170,"text":"nematic and SmA phases shown in Fig. 7.2, the diamagnetic susceptibilities are","rect":[53.81412887573242,249.15682983398438,385.1638873882469,240.22227478027345]},{"page":170,"text":"almost independent of temperature.","rect":[53.8140983581543,261.1163635253906,196.4967464974094,252.1818084716797]},{"page":170,"text":"In some rare cases, e.g., when a calamitic phase consists of solely aliphatic","rect":[65.76612091064453,273.075927734375,385.1808551616844,264.141357421875]},{"page":170,"text":"compounds or cyclohexane derivatives, its anisotropy wadia is very small and can","rect":[53.8140983581543,285.7342529296875,385.1285332780219,274.4976806640625]},{"page":170,"text":"even vanish. As to the discotic mesophases, they have, as a rule, negative diamag-","rect":[53.813594818115237,296.9954528808594,385.15548072663918,288.0609130859375]},{"page":170,"text":"netic anisotropy wadia ¼ wdjjia \u0002 wd?ia < 0 due to a considerably larger value of the","rect":[53.813594818115237,311.77386474609377,385.17346990777818,298.4169921875]},{"page":170,"text":"susceptibility component perpendicular to the director (for discotics w?dia would be","rect":[53.8137321472168,321.4941101074219,385.15317571832505,310.3767395019531]},{"page":170,"text":"closer to zero line than wdjjia in the plot similar to Fig. 7.2).","rect":[53.814353942871097,335.6929626464844,288.2355618050266,322.27960205078127]},{"page":170,"text":"Fig. 7.2 Qualitative","rect":[53.812843322753909,521.0845947265625,124.43738696360112,513.3547973632813]},{"page":170,"text":"temperature dependences of","rect":[53.812843322753909,530.9360961914063,149.62429898841075,523.3417358398438]},{"page":170,"text":"the principal components","rect":[53.812843322753909,540.9120483398438,140.14535983084955,533.3176879882813]},{"page":170,"text":"of diamagnetic (negative)","rect":[53.812843322753909,550.8880004882813,141.65736478430919,543.2936401367188]},{"page":170,"text":"and paramagnetic (positive)","rect":[53.812843322753909,560.8071899414063,149.1318826065748,553.2128295898438]},{"page":170,"text":"susceptibility for calamitic","rect":[53.812843322753909,570.7831420898438,144.99018237375737,563.1887817382813]},{"page":170,"text":"compounds in the isotropic,","rect":[53.812843322753909,580.7590942382813,148.76889297559223,573.1647338867188]},{"page":170,"text":"nematic and SmA phases","rect":[53.812843322753909,590.7350463867188,140.0793808269433,583.1406860351563]},{"page":170,"text":"0","rect":[207.8665008544922,526.8447265625,212.31253313057304,521.0756225585938]},{"page":170,"text":"cpara","rect":[274.94091796875,450.4416198730469,292.13100234606767,442.2522888183594]},{"page":170,"text":"||","rect":[280.0268859863281,453.523681640625,283.14425764496868,448.9946594238281]},{"page":170,"text":"cpara","rect":[256.1170349121094,511.3541564941406,273.30714980700517,503.1649169921875]},{"page":170,"text":"⊥","rect":[260.0906677246094,513.8548583984375,264.0353431945565,509.8836975097656]},{"page":170,"text":"dia","rect":[258.6919860839844,580.3173828125,268.02586754291687,575.1924438476563]},{"page":170,"text":"c⊥","rect":[253.50253295898438,587.6348876953125,261.3136329894784,579.1788940429688]},{"page":170,"text":"<c>dia","rect":[332.60772705078127,555.5665283203125,354.3373622093489,546.2969970703125]},{"page":171,"text":"7.1 Magnetic Phenomena","rect":[53.812843322753909,44.274620056152347,140.7714934089136,36.68026351928711]},{"page":171,"text":"7.1.3 Paramagnetism and Ferromagnetism","rect":[53.812843322753909,69.85308837890625,275.13522302363517,59.298980712890628]},{"page":171,"text":"155","rect":[372.49737548828127,42.55594253540039,385.1889700332157,36.62946701049805]},{"page":171,"text":"7.1.3.1 Paramagnetism","rect":[53.812843322753909,97.55864715576172,158.2096121491923,88.2256851196289]},{"page":171,"text":"The susceptibility of paramagnetic substances is mostly determined by permanent","rect":[53.812843322753909,121.3980941772461,385.0989824063499,112.46354675292969]},{"page":171,"text":"magnetic moments pm, which are aligned by the magnetic field. The field induced","rect":[53.812843322753909,133.35800170898438,385.1253594498969,124.42308044433594]},{"page":171,"text":"magnetization is determined by the total projection of nv molecular magnetic","rect":[53.81342697143555,145.31759643554688,385.09458196832505,136.3829803466797]},{"page":171,"text":"moments in a unit volume onto the field axis","rect":[53.8134880065918,155.24520874023438,234.4784280215378,148.3426055908203]},{"page":171,"text":"Mpara ¼ nvpmhcosyi","rect":[178.5457000732422,183.93495178222657,260.4340859805222,173.57850646972657]},{"page":171,"text":"where y is an angle between individual dipole pm and field H. In the isotropic phase","rect":[53.81356430053711,209.4237060546875,385.1005328960594,200.17994689941407]},{"page":171,"text":"such a distribution is given by the Langevin formula [3]:","rect":[53.81344223022461,221.38323974609376,282.8229814297874,212.4486846923828]},{"page":171,"text":"Ð expð\u0002pmH=kBTÞcosydO","rect":[153.11184692382813,248.71914672851563,263.46667242005199,237.66302490234376]},{"page":171,"text":"hcosyi ¼O","rect":[111.59064483642578,262.5787658691406,157.43510078650574,250.2967529296875]},{"page":171,"text":"Ð expð\u0002pmH=kBTÞdO","rect":[163.9315185546875,270.0312805175781,252.6473975177082,258.97515869140627]},{"page":171,"text":"O","rect":[163.25131225585938,276.48248291015627,168.2542841361151,271.60882568359377]},{"page":171,"text":"¼ cthðpmH=kBTÞ \u0002 ðkBT=pmHÞ","rect":[141.95347595214845,294.4434814453125,268.59222807036596,284.49285888671877]},{"page":171,"text":"(7.7)","rect":[366.0976867675781,293.706298828125,385.1699460586704,285.2299499511719]},{"page":171,"text":"Herex ¼ pmH/kBTandL(x) ¼ cth(x)\u0002(1/x)istheLangevinfunction.Forx << 1,","rect":[65.76622772216797,320.97149658203127,385.2892117073703,312.01702880859377]},{"page":171,"text":"cth(x) \u0004 1/x + x/3 -... and <cosy> \u0004 x/3. Therefore, for a weak magnetic field,","rect":[53.81499481201172,332.9310607910156,385.1169704964328,323.687744140625]},{"page":171,"text":"pmH << kBT, the magnetization is:","rect":[53.817962646484378,344.89080810546877,197.6010118252952,335.93634033203127]},{"page":171,"text":"Mpara ¼ n3vkpB2mTH","rect":[187.77818298339845,384.52301025390627,249.2485938067707,361.1050720214844]},{"page":171,"text":"(7.8)","rect":[366.0968933105469,377.93359375,385.16915260163918,369.4572448730469]},{"page":171,"text":"What is a nature of pm in a molecular system? The molecular paramagnetism is","rect":[65.76543426513672,410.07373046875,385.1855508242722,401.1385498046875]},{"page":171,"text":"mostly originated from the unpaired electron spins. The magnetic moment for a free","rect":[53.81283950805664,422.0332946777344,385.1367572612938,413.05889892578127]},{"page":171,"text":"electron spin is","rect":[53.81284713745117,433.8832702636719,115.44433988677219,425.0383605957031]},{"page":171,"text":"ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi","rect":[233.26470947265626,450.0,265.69651330068839,448.0]},{"page":171,"text":"pem ¼ \u0002g mBpsðs þ 1Þ","rect":[173.2763214111328,460.99761962890627,265.7034951602097,448.6840515136719]},{"page":171,"text":"where s is spin quantum number, mB ¼ e\u0002h=2mec is Bohr magneton for electron and","rect":[53.81418991088867,486.97833251953127,385.1437920670844,472.537109375]},{"page":171,"text":"g ¼ 2.0023 is Lande factor. According to (7.8), the spin magnetization and suscep-","rect":[53.81393051147461,496.4949951171875,385.1099790176548,487.5206298828125]},{"page":171,"text":"tibility follow the Curie law, w/T\u00021:","rect":[53.81393051147461,508.4147033691406,203.03776414463114,497.3073425292969]},{"page":171,"text":"Ms ¼ nv g2m2B3skðBsTþ 1ÞH and ws ¼ nv g2mB23skðBsTþ 1Þ","rect":[118.50162506103516,546.0633544921875,318.8368264590378,522.5887451171875]},{"page":171,"text":"(7.9)","rect":[366.097412109375,539.4740600585938,385.1696714004673,530.9976806640625]},{"page":171,"text":"Typical temperature dependencies of paramagnetic susceptibility wpara are pic-","rect":[65.7659683227539,569.573486328125,385.1475766739048,559.9966430664063]},{"page":171,"text":"tured in the same Fig. 7.2 in comparison with wdia. The order of magnitude of both","rect":[53.813716888427737,581.5331420898438,385.12896052411568,570.4290771484375]},{"page":171,"text":"wdia and wpara is 10\u00027. As far as the nature of the paramagnetic anisotropy is","rect":[53.81403732299805,593.5326538085938,385.18854154692846,582.3887329101563]},{"page":172,"text":"156","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.62946701049805]},{"page":172,"text":"7 Magnetic, Electric and Transport Properties","rect":[228.78945922851563,44.274620056152347,385.1482590007714,36.68026351928711]},{"page":172,"text":"concerned, it should be noted that in molecules the unpaired spins are almost free to","rect":[53.812843322753909,68.2883529663086,385.1975030045844,59.35380554199219]},{"page":172,"text":"rotate. Therefore, their alignment, e.g. by an external magnetic field, needs not be","rect":[53.812843322753909,80.24788665771485,385.1487201519188,71.31333923339844]},{"page":172,"text":"accompanied by alignment of molecular skeletons. In reality, however, there is","rect":[53.812843322753909,92.20748138427735,385.18552030669408,83.27293395996094]},{"page":172,"text":"some coupling between spins and molecular axes. The g-factor becomes a tensor","rect":[53.812843322753909,104.15003204345703,385.11980567781105,95.17564392089844]},{"page":172,"text":"due to interaction of the unpaired electron spins with the angular momentum of","rect":[53.8138313293457,116.0697250366211,385.1477292617954,107.13517761230469]},{"page":172,"text":"molecular orbitals (the so-called spin-orbit interaction). This is a reason for the","rect":[53.8138313293457,128.02932739257813,385.1716693706688,119.09477233886719]},{"page":172,"text":"anisotropy of paramagnetic susceptibility of liquid crystals. For a uniaxial phase,","rect":[53.8138313293457,139.98886108398438,385.1824917366672,131.0343780517578]},{"page":172,"text":"the paramagnetic anisotropy is given by","rect":[53.8138313293457,151.94839477539063,215.06606379560004,143.0138397216797]},{"page":172,"text":"wpaara ¼ nvmB2ksBðsTþ 1ÞSðg2? \u0002 g2jjÞ","rect":[154.58529663085938,192.60183715820313,284.39581693755346,169.183837890625]},{"page":172,"text":"(7.10)","rect":[361.0561828613281,186.01280212402345,385.10555396882668,177.53643798828126]},{"page":172,"text":"fwrohmerethSe igs2thvealourei.enTthaetiolantatel rodrdeeterrmpairnaems eatesriganndofðgp?2ar\u0002amga2jjgÞniestiacnaisnoitsrootproypcyowmpaianrag.","rect":[53.814308166503909,231.774169921875,385.1832241585422,208.69039916992188]},{"page":172,"text":"Like diamagnetic anisotropy wadia, wpaaramay be either positive or negative depending","rect":[53.814537048339847,243.73382568359376,385.1093377213813,232.4972686767578]},{"page":172,"text":"on orientation of the g-tensor with respect to the director. For instance, wapara < 0 for","rect":[53.81332015991211,255.6934814453125,385.1817868789829,246.06031799316407]},{"page":172,"text":"elongated calamitic complexes of copper II with d9 electron configuration. Differ-","rect":[53.81505584716797,266.9546203613281,385.1623776992954,255.9037322998047]},{"page":172,"text":"ent compounds of this sort can be oriented either perpendicular or parallel to the","rect":[53.81356430053711,278.9141540527344,385.17136419488755,269.9796142578125]},{"page":172,"text":"magnetic field depending on competition with the positive diamagnetic contribu-","rect":[53.81356430053711,290.87371826171877,385.11565528718605,281.93914794921877]},{"page":172,"text":"tion. On the other hand, vanadyl (VO) d1 complexes manifest both wpaara > 0 and","rect":[53.81356430053711,303.4755859375,385.1458672623969,291.7259826660156]},{"page":172,"text":"wadia > 0 and are always oriented along the magnetic field.","rect":[53.814022064208987,316.2854309082031,287.5132107308078,305.1053466796875]},{"page":172,"text":"7.1.3.2 Ferromagnetism","rect":[53.81324005126953,363.7884216308594,162.18672396559857,354.4554443359375]},{"page":172,"text":"The ferromagnetism of organic compounds has been observed only recently. These","rect":[53.81324005126953,387.6278381347656,385.09934271051255,378.69329833984377]},{"page":172,"text":"are compounds containing Fe, Ni, Co atoms and the ferromagnetic state is found at","rect":[53.81324005126953,399.58734130859377,385.18000657627177,390.65277099609377]},{"page":172,"text":"very low temperatures (few K), at which a liquid crystal state is not observed yet.","rect":[53.81324005126953,411.5469055175781,385.1670193245578,402.61236572265627]},{"page":172,"text":"However, ferromagnetic materials can be prepared from colloidal suspensions of","rect":[53.81324005126953,423.4496765136719,385.14812599031105,414.51513671875]},{"page":172,"text":"small solid ferromagnetic particles, even nanoparticles (e.g., magnetite Fe2O3 or","rect":[53.81324005126953,435.4092102050781,385.1521237930454,426.4555358886719]},{"page":172,"text":"ferrite Fe3O4) in liquids. Such solutions are called ferrofluids. Since these particles","rect":[53.81432342529297,447.36968994140627,385.1616250430222,438.3554382324219]},{"page":172,"text":"have permanent magnetic moments pm, under a magnetic field they can be oriented.","rect":[53.813838958740237,459.3293151855469,385.1368679573703,450.3946533203125]},{"page":172,"text":"In ferrofluids they form chains, which are arranged in ordered patterns.","rect":[53.81399154663086,471.2888488769531,340.4604763558078,462.35430908203127]},{"page":172,"text":"The same particles can be introduced into liquid crystals, e.g. into nematics [4].","rect":[65.7660140991211,483.2484130859375,385.1557888558078,474.3138427734375]},{"page":172,"text":"If the guest particles are elongated they may be aligned by a liquid crystal (host)","rect":[53.81400680541992,495.20794677734377,385.11702857820168,486.27337646484377]},{"page":172,"text":"even in the absence of the magnetic field, e.g. by a surface treatment (without","rect":[53.81400680541992,507.1675109863281,385.0951371915061,498.23297119140627]},{"page":172,"text":"macroscopic magnetization). The external magnetic field will orient the magnetic","rect":[53.81400680541992,519.0702514648438,385.1797260112938,510.1357421875]},{"page":172,"text":"moments of particles, which, in turn, orient the liquid crystal matrix. Such nematic","rect":[53.81400680541992,531.02978515625,385.10108221246568,522.0952758789063]},{"page":172,"text":"suspensions of particles show very interesting magneto-optical properties (a guest-","rect":[53.81400680541992,542.9893188476563,385.1209958633579,534.0548095703125]},{"page":172,"text":"host effect in ferrofluids).","rect":[53.81400680541992,554.5504760742188,156.79131742026096,546.0143432617188]},{"page":173,"text":"7.2 Dielectric Properties","rect":[53.812843322753909,44.274620056152347,137.0410049724511,36.68026351928711]},{"page":173,"text":"7.2 Dielectric Properties","rect":[53.812843322753909,70.10667419433594,186.7288120243327,59.37328338623047]},{"page":173,"text":"157","rect":[372.4981994628906,42.55594253540039,385.1898245254032,36.62946701049805]},{"page":173,"text":"7.2.1 Permittivity of Isotropic Liquids","rect":[53.812843322753909,100.03349304199219,248.1884067508952,89.39572143554688]},{"page":173,"text":"Liquid crystals are anisotropic fluids and the discussion of their dielectric properties","rect":[53.812843322753909,127.5765609741211,385.1546975527878,118.64201354980469]},{"page":173,"text":"is based on the fundamental ideas obtained for isotropic liquids. We recall the","rect":[53.812843322753909,139.53610229492188,385.17359197809068,130.60154724121095]},{"page":173,"text":"relevant results.","rect":[53.812843322753909,149.37704467773438,117.42124600912814,142.5043182373047]},{"page":173,"text":"7.2.1.1 Dielectric Spectrum","rect":[53.812843322753909,187.58843994140626,176.70652987380169,178.63397216796876]},{"page":173,"text":"The Maxwell equations for the electromagnetic field in conductive materials such","rect":[53.812843322753909,211.52047729492188,385.1219109635688,202.58592224121095]},{"page":173,"text":"as organic liquids read:","rect":[53.812843322753909,223.48001098632813,147.72184617588114,214.5454559326172]},{"page":173,"text":"curlE ¼ \u0002m \u0005 @H and curlH ¼ þe \u0005 @E þ 4psE","rect":[113.12002563476563,252.42247009277345,325.8252436798408,237.73974609375]},{"page":173,"text":"c @t","rect":[158.4363555908203,258.6736755371094,181.56791551181864,251.3427734375]},{"page":173,"text":"(7.11)","rect":[361.0561828613281,253.3488006591797,385.10555396882668,244.8724365234375]},{"page":173,"text":"Here E and H are vectors of electric and magnetic field strength, e(o) and m(o)","rect":[65.76654815673828,282.2013244628906,385.1583493789829,273.26678466796877]},{"page":173,"text":"are frequency dependent dielectric and magnetic permittivities, s is permanent","rect":[53.812496185302737,294.160888671875,385.09455735752177,285.226318359375]},{"page":173,"text":"conductivity. In the second equation, the two terms describe the displacement and","rect":[53.81246566772461,306.12042236328127,385.1433953385688,297.18585205078127]},{"page":173,"text":"Ohmic current, respectively.","rect":[53.81246566772461,318.0799255371094,168.35755582358127,309.1254577636719]},{"page":173,"text":"In the limit of o ¼ 2pf ! 1 no dynamic process in medium can follow the","rect":[65.7645034790039,330.03948974609377,385.17322576715318,321.0650939941406]},{"page":173,"text":"field; the electric polarization P ¼ wEE vanishes (i.e. dielectric susceptibility","rect":[53.813472747802737,341.9434509277344,385.12258235028755,330.9853820800781]},{"page":173,"text":"wE ! 0) and the displacement vector D ¼ (1 + 4pwE)E coincides with E, that is","rect":[53.8136100769043,353.90301513671877,385.1866494570847,342.9450378417969]},{"page":173,"text":"e ¼ 1 þ 4pwE ! 1. With decreasing frequency, fast electronic processes have","rect":[53.81296920776367,365.8626708984375,385.11887396051255,354.9046630859375]},{"page":173,"text":"enough time to follow the field and, at optical frequencies, e ¼ n2 (n is refraction","rect":[53.8128662109375,377.82220458984377,385.1578301530219,366.7713928222656]},{"page":173,"text":"index)","rect":[53.814022064208987,389.383544921875,79.30675636140478,380.847412109375]},{"page":173,"text":"shows","rect":[84.45905303955078,388.0,109.22513212310031,380.847412109375]},{"page":173,"text":"peculiarities","rect":[114.36747741699219,389.781982421875,163.36308683501438,380.847412109375]},{"page":173,"text":"related","rect":[168.46263122558595,388.0,195.69733515790473,380.847412109375]},{"page":173,"text":"to","rect":[200.8635711669922,388.0,208.6378106217719,381.8634033203125]},{"page":173,"text":"electronic","rect":[213.721435546875,388.0,253.26953924371566,380.847412109375]},{"page":173,"text":"absorption","rect":[258.3571472167969,389.781982421875,300.5212029557563,380.847412109375]},{"page":173,"text":"bands","rect":[305.65557861328127,388.0,328.8688088809128,380.847412109375]},{"page":173,"text":"(normal","rect":[333.9783020019531,389.383544921875,365.6635881192405,380.847412109375]},{"page":173,"text":"and","rect":[370.740234375,388.0,385.14397517255318,380.847412109375]},{"page":173,"text":"abnormal dispersion). With further decreasing frequency other processes such as","rect":[53.814022064208987,401.74151611328127,385.13995756255346,392.80694580078127]},{"page":173,"text":"molecular rotations and vibrations begin to contribute to the electric polarization","rect":[53.814022064208987,413.7010498046875,385.0872429948188,404.7664794921875]},{"page":173,"text":"and e ¼ n2 again increases, see Fig. 7.3.","rect":[53.814022064208987,425.6609191894531,216.77633328940159,414.6099548339844]},{"page":173,"text":"On the other hand, since for the sine-form field ∂E/∂t / oE the role of","rect":[65.76499938964844,436.4779968261719,385.1487973770298,427.6231384277344]},{"page":173,"text":"permanent conductivity s decreases with increasing frequency, in the high fre-","rect":[53.81195831298828,449.5232238769531,385.1308835586704,440.58868408203127]},{"page":173,"text":"quency limit o >> 1/tM ¼ 4ps/e a material can be considered as non-conductive.","rect":[53.81095504760742,461.4827575683594,385.1685452034641,452.5482177734375]},{"page":173,"text":"The time tM ¼ e/4ps is called Maxwell dielectric relaxation time. Later we shall","rect":[53.81376266479492,472.9451599121094,385.12055833408427,464.50823974609377]},{"page":173,"text":"meet it again under another name “space charge relaxation time”.","rect":[53.81359100341797,485.4024658203125,318.51702542807348,476.4678955078125]},{"page":173,"text":"7.2.1.2 Local Field, Clausius-Mossotti and Onsager Equations","rect":[53.81359100341797,521.6446533203125,324.73736967192846,512.540771484375]},{"page":173,"text":"The vectors of electric displacement D and polarization P are also coupled by the","rect":[53.81359100341797,545.4840087890625,385.17432439996568,536.5494995117188]},{"page":173,"text":"additional Maxwell equation:","rect":[53.81357955932617,557.443603515625,172.33537156650614,548.5090942382813]},{"page":173,"text":"e ¼ D ¼ E þ 4pP ¼ 1 þ 4pP=E and wE ¼ P ¼ e \u00021","rect":[104.22766876220703,587.756103515625,334.76552668622505,571.7017822265625]},{"page":173,"text":"(7.12)","rect":[361.0566101074219,586.9722290039063,385.1059812149204,578.495849609375]},{"page":174,"text":"158","rect":[53.812843322753909,42.55649185180664,66.50444931178018,36.6300163269043]},{"page":174,"text":"7 Magnetic, Electric and Transport Properties","rect":[228.78945922851563,44.275169372558597,385.1482590007714,36.68081283569336]},{"page":174,"text":"Fig. 7.3 Qualitative frequency","rect":[53.812843322753909,226.79794311523438,160.73199981837198,218.86492919921876]},{"page":174,"text":"Fig. 7.4 Lorentz model for","rect":[53.812843322753909,266.5878601074219,148.9296578751295,258.8580322265625]},{"page":174,"text":"the local field. Polarization of","rect":[53.812843322753909,274.76898193359377,155.34483426673106,268.9017639160156]},{"page":174,"text":"an ellipsoidal form dielectric","rect":[53.812843322753909,286.4720764160156,152.5670714118433,278.8777160644531]},{"page":174,"text":"sample and appearance of","rect":[53.812843322753909,296.447998046875,142.54406827552013,288.8536376953125]},{"page":174,"text":"depolarizing field E1 (a),","rect":[53.812843322753909,306.36724853515627,139.29466507031879,298.77288818359377]},{"page":174,"text":"Lorentz cavity field E2 and","rect":[53.81246566772461,316.3426818847656,146.6620840468876,308.7481384277344]},{"page":174,"text":"the field of individual","rect":[53.81283187866211,324.5913391113281,127.90896324362818,318.72412109375]},{"page":174,"text":"molecules within the cavity","rect":[53.81283187866211,336.29443359375,148.23160308985636,328.7000732421875]},{"page":174,"text":"E3 (b)","rect":[53.81283187866211,345.8777770996094,75.32320493323495,338.62774658203127]},{"page":174,"text":"spectrum","rect":[163.13833618164063,226.730224609375,194.22430656785952,219.9994354248047]},{"page":174,"text":"of","rect":[196.6154022216797,225.0,203.66346829993419,219.1358642578125]},{"page":174,"text":"the","rect":[206.0748748779297,225.0,216.46508166574956,219.1358642578125]},{"page":174,"text":"dielectric","rect":[218.8197784423828,225.00306701660157,250.50650927805425,219.1358642578125]},{"page":174,"text":"permittivity","rect":[252.9204559326172,226.730224609375,293.08514923243447,219.1358642578125]},{"page":174,"text":"How can we relate these macroscopic quantities to the microscopic parameters","rect":[65.76496887207031,438.2479248046875,385.0841103945847,429.42291259765627]},{"page":174,"text":"of molecules such as polarizability or a dipole moment? With some precautions, the","rect":[53.81296157836914,450.3170471191406,385.17273748590318,441.38250732421877]},{"page":174,"text":"polarization of an isotropic liquid may be found as a sum of the field induced","rect":[53.81296157836914,462.2765808105469,385.12294856122505,453.342041015625]},{"page":174,"text":"molecular dipole moments whose number coincides with the amount of dipolar","rect":[53.81296157836914,474.1793518066406,385.09407935945168,465.24481201171877]},{"page":174,"text":"molecules in the unit volume nv ¼ rNA/M (r is mass density, NA is Avogadro","rect":[53.81296157836914,486.1394348144531,385.10552302411568,477.204345703125]},{"page":174,"text":"number, M is molecular mass):","rect":[53.81350326538086,497.7005310058594,179.0543809659202,489.1644287109375]},{"page":174,"text":"P ¼ Xpe ¼ nvgEloc","rect":[175.9954376220703,526.3298950195313,262.49689621820826,512.3952026367188]},{"page":174,"text":"nv","rect":[199.39096069335938,533.35498046875,205.12776583767815,529.1802368164063]},{"page":174,"text":"(7.13)","rect":[361.0561828613281,523.6038208007813,385.10555396882668,515.12744140625]},{"page":174,"text":"Here g is average molecular polarizability (generally, gij is a tensor), Eloc is a","rect":[65.76653289794922,558.8270263671875,385.1612018413719,548.9622192382813]},{"page":174,"text":"local electric field acting on each molecule and pe is a field induced electric dipole","rect":[53.814414978027347,569.8002319335938,385.12473333551255,560.8655395507813]},{"page":174,"text":"in a molecule. So, if we find Eloc we could calculate P and then the value of e, using","rect":[53.813777923583987,581.7598876953125,385.16533747724068,572.8252563476563]},{"page":174,"text":"Eq. 7.12, and known macroscopic field E in the sample, see Fig. 7.4a.","rect":[53.81354904174805,593.7194213867188,336.2980923226047,584.784912109375]},{"page":175,"text":"7.2 Dielectric Properties","rect":[53.812843322753909,44.274620056152347,137.0410049724511,36.68026351928711]},{"page":175,"text":"159","rect":[372.4981994628906,42.62367248535156,385.1898245254032,36.62946701049805]},{"page":175,"text":"The local field can be found, e.g., from some models, particularly the Lorentz","rect":[65.76496887207031,68.2883529663086,385.12790311919408,59.35380554199219]},{"page":175,"text":"model [3]. For its discussion we select a single molecule and surround it by a","rect":[53.81294250488281,80.24788665771485,385.1597369976219,71.29341888427735]},{"page":175,"text":"fictitious spherical cavity shown in Fig. 7.4b. Then Eloc is a sum of four fields:","rect":[53.81393814086914,92.20748138427735,371.4673906094749,83.27293395996094]},{"page":175,"text":"Eloc ¼ E0 þ E1 þ E2 þ E3","rect":[165.5745086669922,115.54296112060547,272.8883216150697,107.1453857421875]},{"page":175,"text":"(7.14)","rect":[361.0561828613281,115.6719741821289,385.10555396882668,107.19561004638672]},{"page":175,"text":"Here, E0 is an external field created by charges located outside of the sample;","rect":[53.81450653076172,157.90078735351563,367.64619309970927,148.9662322998047]},{"page":175,"text":"E1¼\u0002kP is a depolarizing field from the charges formed at the external surfaces","rect":[53.812828063964847,169.8604736328125,385.1654397402878,160.90599060058595]},{"page":175,"text":"(upper sketch);","rect":[65.76461029052735,181.82000732421876,126.27924211093972,172.8854522705078]},{"page":175,"text":"E2 is the Lorentz field coming from the charges at the inner surfaces of the cavity;","rect":[53.812591552734378,193.77975463867188,385.125135970803,184.84519958496095]},{"page":175,"text":"and","rect":[65.76512908935547,203.70733642578126,80.16886988690863,196.8047332763672]},{"page":175,"text":"E3 is the field from all molecules inside the cavity except that one we have selected.","rect":[53.813106536865237,217.69906616210938,385.09417386557348,208.76451110839845]},{"page":175,"text":"The depolarization field is opposite to the external field and factor k is generally","rect":[65.7650146484375,235.6099853515625,385.10210505536568,226.65550231933595]},{"page":175,"text":"a tensor dependent on the shape of the sample. For samples in the form of the","rect":[53.812007904052737,247.56951904296876,385.17075384332505,238.6349639892578]},{"page":175,"text":"ellipsoid, oriented with one of its axes along the field, depolarizing factors become","rect":[53.812007904052737,259.5290832519531,385.10511053277818,250.5945281982422]},{"page":175,"text":"scalars ki dependent on the ratios of ellipsoid axes. For instance, for a spherical","rect":[53.812007904052737,271.4891662597656,385.1782060391624,262.5341491699219]},{"page":175,"text":"sample k ¼ 4p/3, for a thin plate with the field perpendicular to its surface, k ¼ 4p.","rect":[53.81346130371094,283.4486999511719,385.18413968588598,274.4942321777344]},{"page":175,"text":"From Fig. 7.4a follows that the macroscopic field in the sample E ¼ E0þ","rect":[65.76746368408203,295.3514709472656,385.14807918760689,286.41693115234377]},{"page":175,"text":"E1 ¼ E0\u0002kP. When polarization P is very high, the macroscopic field is consider-","rect":[53.814231872558597,307.31146240234377,385.15520606843605,298.35699462890627]},{"page":175,"text":"ably reduced. The Lorentz field E2 is parallel to the external field and, for a","rect":[53.81340408325195,319.2710876464844,385.1599506206688,310.33642578125]},{"page":175,"text":"spherical cavity, is equal exactly to þ4pP/3. Therefore, when both the sample","rect":[53.81318283081055,331.23065185546877,385.1738971538719,322.29608154296877]},{"page":175,"text":"and the cavity are spherical,","rect":[53.81417465209961,343.190185546875,167.03533597494846,334.255615234375]},{"page":175,"text":"Eloc ¼ E0 \u0002 4 pP=3 þ 4 pP=3 þ E3 ¼ E0 þ E3:","rect":[121.67501831054688,367.4951477050781,317.35142457169436,357.5645751953125]},{"page":175,"text":"Due to high symmetry, for all isotropic liquids (and all cubic crystals), field E3","rect":[65.76573944091797,390.97198486328127,385.18130140630105,382.03741455078127]},{"page":175,"text":"acting on the selected molecule from its neighbors is exactly compensated. Thus,","rect":[53.812843322753909,402.93194580078127,385.1397366097141,393.99737548828127]},{"page":175,"text":"for a spherical isotropic sample the local field is equal to the external field:","rect":[53.812843322753909,414.8914794921875,358.45942552158427,405.9170837402344]},{"page":175,"text":"Eloc ¼ E0","rect":[199.73062133789063,438.22674560546877,238.7879493006166,429.885986328125]},{"page":175,"text":"For an isotropic sample of an arbitrary form, the depolarization field E1 is form-","rect":[65.76496887207031,462.7301940917969,385.1112302383579,453.75579833984377]},{"page":175,"text":"dependent and Eloc should be written as","rect":[53.814205169677737,474.6897277832031,214.6404153262253,465.75518798828127]},{"page":175,"text":"Eloc ¼ E0 þ E1 þ 4 pP=3 ¼ E þ 4 pP=3","rect":[136.4013214111328,498.9378967285156,302.59169856122505,489.00732421875]},{"page":175,"text":"(7.15)","rect":[361.0567626953125,498.1540222167969,385.10613380281105,489.5581359863281]},{"page":175,"text":"It should be noted that, if a sample is connected directly to the electric voltage","rect":[65.76708221435547,522.4714965820313,385.10019720270005,513.5369873046875]},{"page":175,"text":"source (fixed potential difference across electrodes), there is no depolarization","rect":[53.81505584716797,534.4310302734375,385.1698845963813,525.4965209960938]},{"page":175,"text":"charges on the external surfaces of the sample. In this case, E1 ¼ 0 and the local","rect":[53.81505584716797,546.390625,385.1145463711936,537.4561157226563]},{"page":175,"text":"field acting on a molecule in the cavity is given by","rect":[53.814537048339847,558.3505859375,258.56865778974068,549.4160766601563]},{"page":175,"text":"Eloc ¼ E0 þ 4 pP=3 ¼ ð2E0 þ DÞ=3 ¼ E0ð e þ 2Þ=3:","rect":[111.30906677246094,582.5989379882813,327.6615442006006,572.6015625]},{"page":175,"text":"(7.16)","rect":[361.05694580078127,581.8150024414063,385.1063169082798,573.2788696289063]},{"page":176,"text":"160","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.68026351928711]},{"page":176,"text":"7 Magnetic, Electric and Transport Properties","rect":[228.78945922851563,44.274620056152347,385.1482590007714,36.68026351928711]},{"page":176,"text":"With the local field found, we obtain a relation between the macroscopic field in","rect":[65.76496887207031,68.2883529663086,385.14092341474068,59.35380554199219]},{"page":176,"text":"the sample E, polarization P and the local field acting on a particular molecule, see","rect":[53.812950134277347,80.24788665771485,385.1149371929344,71.31333923339844]},{"page":176,"text":"Eqs. 7.13 and 7.15:","rect":[53.81393814086914,92.20748138427735,131.77830369541239,83.21317291259766]},{"page":176,"text":"4p","rect":[256.1494140625,110.20088958740235,266.62918443034246,103.39788055419922]},{"page":176,"text":"P ¼ nvgEloc ¼ nvgðE þ 3 PÞ:","rect":[158.0404815673828,123.85365295410156,280.92942750138186,109.4050064086914]},{"page":176,"text":"(7.17)","rect":[361.0568542480469,118.61844635009766,385.1062253555454,110.14208221435547]},{"page":176,"text":"Solving the latter for P we find the microscopic value of electric susceptibility:","rect":[65.76720428466797,144.41012573242188,385.10725267002177,135.47557067871095]},{"page":176,"text":"w ¼ E ¼ 1 \u0002 43pnvg","rect":[177.41363525390626,179.8753204345703,259.93694392255318,164.73643493652345]},{"page":176,"text":"(7.18)","rect":[361.0558166503906,170.9924774169922,385.10518775788918,162.51611328125]},{"page":176,"text":"This equation has a singularity at g ! 3=4pnv: for large enough molecular","rect":[65.76615142822266,200.79722595214845,385.15483985749855,190.8666229248047]},{"page":176,"text":"polarizability","rect":[53.814022064208987,212.37088012695313,107.33582392743597,203.4363250732422]},{"page":176,"text":"the","rect":[114.25399780273438,211.0,126.47777593805158,203.4363250732422]},{"page":176,"text":"macroscopic","rect":[133.39993286132813,212.37088012695313,183.93845403863754,203.4363250732422]},{"page":176,"text":"susceptibility","rect":[190.89344787597657,212.37088012695313,244.32366267255316,203.4363250732422]},{"page":176,"text":"and,","rect":[251.21995544433595,211.0,268.11224027182348,203.4363250732422]},{"page":176,"text":"consequently,","rect":[275.0672607421875,212.37088012695313,330.2792934944797,203.4363250732422]},{"page":176,"text":"polarization","rect":[337.2631530761719,212.37088012695313,385.09914485028755,203.4363250732422]},{"page":176,"text":"become infinite. This phenomenon is called polarization catastrophe. In a more","rect":[53.814022064208987,224.33041381835938,385.1379779644188,215.39585876464845]},{"page":176,"text":"subtle approach, the polarization remains finite and exists even in the absence of the","rect":[53.814022064208987,236.28997802734376,385.1717914409813,227.3554229736328]},{"page":176,"text":"external","rect":[53.814022064208987,246.18768310546876,86.12240464756082,239.31495666503907]},{"page":176,"text":"field","rect":[91.76544189453125,246.21755981445313,109.48394099286566,239.31495666503907]},{"page":176,"text":"(spontaneous","rect":[115.10308837890625,248.1399383544922,168.2744943057175,239.37472534179688]},{"page":176,"text":"polarization","rect":[173.9573516845703,248.1399383544922,222.74892512372504,239.29502868652345]},{"page":176,"text":"Ps).","rect":[228.39393615722657,247.8516082763672,243.2031216194797,239.32492065429688]},{"page":176,"text":"The","rect":[248.84217834472657,247.0,264.3806842632469,239.3154754638672]},{"page":176,"text":"spontaneous","rect":[270.0267333984375,248.25003051757813,319.32794584380346,240.33145141601563]},{"page":176,"text":"polarization","rect":[325.0287170410156,248.25003051757813,372.8994988541938,239.3154754638672]},{"page":176,"text":"is","rect":[378.500732421875,247.0,385.13022245513158,239.3154754638672]},{"page":176,"text":"responsible for pyro- and ferroelectricity in solid and liquid crystals, however it is","rect":[53.81332015991211,260.2095947265625,385.18698515044408,251.27503967285157]},{"page":176,"text":"not observed in the isotropic liquid (see Chapters 4 and 13).","rect":[53.81332015991211,272.16912841796877,295.75335355307348,263.23455810546877]},{"page":176,"text":"Finally,combining(7.18)withdefinitione ¼ 1 þ 4pwEwearriveattheClausius-","rect":[65.76432037353516,284.1286926269531,385.15950904695168,273.1709899902344]},{"page":176,"text":"Mossotti equation (NAv is Avogadro number):","rect":[53.81374740600586,296.0885925292969,234.67600877353739,287.154052734375]},{"page":176,"text":"e \u0002 1 4pnv","rect":[164.89402770996095,315.5486145019531,218.19628281488796,307.27886962890627]},{"page":176,"text":"4prNAv","rect":[238.5327606201172,316.0540466308594,268.583749245552,307.27886962890627]},{"page":176,"text":"(7.19)","rect":[361.0558776855469,322.49981689453127,385.1052487930454,314.0234680175781]},{"page":176,"text":"This equation relates the macroscopic value of dielectric permittivity e to","rect":[65.76622772216797,348.9150085449219,385.14406672528755,339.98046875]},{"page":176,"text":"microscopic parameters of medium. For instance, from measurements of capaci-","rect":[53.81418991088867,360.87457275390627,385.1789487442173,351.94000244140627]},{"page":176,"text":"tance of a liquid by a dielectric bridge one finds dielectric constant and then","rect":[53.81418991088867,372.8341064453125,385.10927668622505,363.8995361328125]},{"page":176,"text":"calculates molecular polarizability g.","rect":[53.81418991088867,384.73687744140627,202.84365506674534,375.80230712890627]},{"page":176,"text":"Molecular polarizability includes electronic gel and orientational gor parts. The","rect":[65.76720428466797,396.6970520019531,385.1477741069969,387.7618408203125]},{"page":176,"text":"first of them is frequency and temperature independent and, at optical frequencies,","rect":[53.81393051147461,408.6566162109375,385.1557583382297,399.7220458984375]},{"page":176,"text":"the Lorenz-Lorentz formula is valid:","rect":[53.81393051147461,420.5962219238281,199.6411271817405,411.6417541503906]},{"page":176,"text":"n2 \u0002 1 4prNA","rect":[180.8679656982422,442.2255554199219,247.05948113568844,431.8427429199219]},{"page":176,"text":"n2 þ 2 ¼ 3M gel","rect":[180.86740112304688,454.60382080078127,259.29609185020828,442.3135681152344]},{"page":176,"text":"(7.20)","rect":[361.0561828613281,448.7283630371094,385.10555396882668,440.25201416015627]},{"page":176,"text":"The second, dipolar part of polarizability is related to the orientational suscepti-","rect":[65.76653289794922,475.0867614746094,385.0956967910923,466.1522216796875]},{"page":176,"text":"bility of permanent dipole moments pe and can be found from the Langevin","rect":[53.81450653076172,487.04632568359377,385.17852107099068,478.11175537109377]},{"page":176,"text":"equation (7.7) as in the case of paramagnetism (only pm is substituted by pe):","rect":[53.81382369995117,499.0059509277344,365.0722795254905,490.0712890625]},{"page":176,"text":"Pdip ¼ n3vkpB2eTE","rect":[192.1407470703125,534.6704711914063,245.16226455386426,511.2525939941406]},{"page":176,"text":"(7.21)","rect":[361.05548095703127,528.0811157226563,385.1048520645298,519.604736328125]},{"page":176,"text":"We see that the dipolar susceptibility obeys the Curie law:","rect":[65.76581573486328,556.2532958984375,301.6040178067405,547.3187866210938]},{"page":176,"text":"2","rect":[231.7352752685547,573.2135009765625,235.21923897346816,568.500244140625]},{"page":176,"text":"pe","rect":[226.75059509277345,579.7081909179688,234.84993882074734,572.2623901367188]},{"page":176,"text":"gor ¼ 3kBT","rect":[195.99276733398438,591.9180908203125,240.85536546061588,578.9144287109375]},{"page":176,"text":"(7.22)","rect":[361.0554504394531,585.3287353515625,385.10482154695168,576.8523559570313]},{"page":177,"text":"7.2 Dielectric Properties","rect":[53.812843322753909,44.274620056152347,137.0410049724511,36.68026351928711]},{"page":177,"text":"161","rect":[372.4981994628906,42.55594253540039,385.1898245254032,36.68026351928711]},{"page":177,"text":"Using the last three equations one can find molecular parameters gel and pe from","rect":[65.76496887207031,68.2883529663086,385.13858746171555,59.35380554199219]},{"page":177,"text":"independent measurements of density, refraction index and temperature depen-","rect":[53.81368637084961,80.24788665771485,385.0819333633579,71.31333923339844]},{"page":177,"text":"dence of dielectric permittivity at a low frequency.","rect":[53.81368637084961,92.20748138427735,258.9670071175266,83.27293395996094]},{"page":177,"text":"The Clausius-Mossotti equation is based on the simplest (Lorentz) form of the","rect":[65.76573944091797,104.11019134521485,385.1725543804344,95.17564392089844]},{"page":177,"text":"local field. In reality, the induced dipole in the selected molecule also creates an","rect":[53.81368637084961,116.0697250366211,385.14959040692818,107.13517761230469]},{"page":177,"text":"additional, reaction field that modifies the cavity field. On account of these factors","rect":[53.81368637084961,128.02932739257813,385.12961210356908,119.054931640625]},{"page":177,"text":"Onsager has obtained the following equation for dielectric permittivity","rect":[53.81368637084961,139.98886108398438,339.1103448014594,131.0343780517578]},{"page":177,"text":"e \u0002 1 ¼ 4pMrNA Fh\u0007gel þ F3kpB2eT\b","rect":[149.65682983398438,178.7929229736328,289.2928740535356,154.2200927734375]},{"page":177,"text":"(7.23)","rect":[361.0557556152344,171.1063995361328,385.1051267227329,162.63003540039063]},{"page":177,"text":"where the cavity h and reaction field F factors are:","rect":[53.8140983581543,202.28268432617188,257.96198899814677,193.3282012939453]},{"page":177,"text":"h ¼ 3e=2e þ 1; F ¼ ð2e3þð21eÞðþnn22þÞ 2Þ","rect":[149.7696990966797,234.80799865722657,289.2104836856003,216.58937072753907]},{"page":177,"text":"The Onsager equation agrees quite well with experimental data on liquids and","rect":[65.76496887207031,258.3958740234375,385.14287653974068,249.44139099121095]},{"page":177,"text":"liquid crystals and will be generalized for calculations of the tensor eij in nematic","rect":[53.812950134277347,271.2852478027344,385.10340154840318,261.36407470703127]},{"page":177,"text":"liquid crystals.","rect":[53.81432342529297,282.2584228515625,112.92242093588595,273.3238525390625]},{"page":177,"text":"7.2.2 Static Dielectric Anisotropy of Nematics and Smectics","rect":[53.812843322753909,320.1258239746094,357.29555396769208,309.488037109375]},{"page":177,"text":"7.2.2.1 Maier-Meier Theory","rect":[53.812843322753909,348.0665283203125,180.2024315934516,338.65386962890627]},{"page":177,"text":"In experiment on nematic liquid crystals, both positive and negative anisotropy ea is","rect":[53.812843322753909,371.58721923828127,385.1891213809128,362.65264892578127]},{"page":177,"text":"observed, the sign depending on chemical structure. The magnitude of ea is often","rect":[53.81450653076172,383.5469665527344,385.1407097916938,374.6124267578125]},{"page":177,"text":"proportional to orientational order parameter S. In the isotropic phase the anisotropy","rect":[53.813838958740237,395.50665283203127,385.1486748795844,386.57208251953127]},{"page":177,"text":"disappears. Typical temperature dependencies of e|| and e⊥ are shown in Fig. 7.5.","rect":[53.813838958740237,407.46636962890627,385.1824917366672,398.4720458984375]},{"page":177,"text":"These observations can be accounted for by the Maier-Meier theory [5]. The latter","rect":[53.81382369995117,419.4259338378906,385.1178220352329,410.4316101074219]},{"page":177,"text":"is based on the following seven assumptions:","rect":[53.814796447753909,431.3287048339844,235.89225633213114,422.3941650390625]},{"page":177,"text":"1.","rect":[53.814796447753909,448.0,61.28045316244845,440.4216003417969]},{"page":177,"text":"2.","rect":[53.81260299682617,473.0,61.27825971152072,465.8154296875]},{"page":177,"text":"3.","rect":[53.8138542175293,497.0,61.27951093222384,489.73419189453127]},{"page":177,"text":"4.","rect":[53.813838958740237,521.0,61.27949567343478,513.6532592773438]},{"page":177,"text":"5.","rect":[53.813838958740237,557.0,61.27949567343478,549.3555908203125]},{"page":177,"text":"6.","rect":[53.81444549560547,569.0,61.280102210300018,561.375732421875]},{"page":177,"text":"7.","rect":[53.814979553222659,593.0,61.2806362679172,585.4942016601563]},{"page":177,"text":"the molecules are spherical with radius a, but their polarizability is tensorial,","rect":[66.27647399902344,449.29638671875,385.1785549690891,440.36181640625]},{"page":177,"text":"ga ¼ gjj \u0002 g?>0","rect":[66.27749633789063,463.16802978515627,133.39153376630316,452.3817138671875]},{"page":177,"text":"the point molecular dipole pe makes an angle b with the axis of maximum","rect":[66.27427673339844,474.6902160644531,385.0949473249968,465.446533203125]},{"page":177,"text":"molecular polarizability","rect":[66.27552795410156,486.6494140625,162.67607966474066,477.71484375]},{"page":177,"text":"a nematic liquid crystal has a center of symmetry and characterized by orienta-","rect":[66.27552795410156,498.6089782714844,385.10588966218605,489.6744384765625]},{"page":177,"text":"tional quadrupolar order parameter S","rect":[66.27552795410156,510.5685119628906,214.7346123795844,501.63397216796877]},{"page":177,"text":"the analysis is performed within the framework of Onsager’s theory of polar","rect":[66.2755126953125,522.5280151367188,385.13778053132668,513.5735473632813]},{"page":177,"text":"liquids, and the mean dielectric susceptibility <e> was taken for the calculation","rect":[66.27552032470703,534.4307861328125,385.1198357682563,525.4962768554688]},{"page":177,"text":"of the Onsager factors h and F, introduced above","rect":[66.27552795410156,546.3903198242188,264.2876666851219,537.4358520507813]},{"page":177,"text":"the dielectric anisotropy is assumed to be small, ea ¼ |e||-e⊥| << <e>","rect":[66.2755126953125,558.349853515625,351.44434383604439,549.4153442382813]},{"page":177,"text":"when calculating the reaction field the tensor nature of electronic polarizability","rect":[66.276123046875,570.3102416992188,385.17620173505318,561.375732421875]},{"page":177,"text":"gij was neglected and the average value <g> ¼ ðgjj þ 2g?Þ=3 was used","rect":[66.27613067626953,584.1815185546875,356.23781672528755,572.6581420898438]},{"page":177,"text":"the interaction between molecules is disregarded","rect":[66.27665710449219,594.2294921875,262.5936211686469,585.2949829101563]},{"page":178,"text":"162","rect":[53.81199264526367,42.55716323852539,66.50359863428995,36.68148422241211]},{"page":178,"text":"Fig. 7.5 Typical temperature","rect":[53.812843322753909,67.58130645751953,155.2940764167261,59.85148620605469]},{"page":178,"text":"behavior of principal","rect":[53.812843322753909,77.4895248413086,125.5398760243899,69.89517211914063]},{"page":178,"text":"dielectric permittivities for","rect":[53.812843322753909,87.4087142944336,145.530854164192,79.81436157226563]},{"page":178,"text":"two nematic liquid crystals,","rect":[53.812843322753909,97.3846664428711,148.42538711621723,89.79031372070313]},{"page":178,"text":"one with positive (solid lines)","rect":[53.812843322753909,107.36067962646485,155.3625954971998,99.7493896484375]},{"page":178,"text":"and the other with negative","rect":[53.812843322753909,117.33663177490235,147.2382902839136,109.74227905273438]},{"page":178,"text":"(dash lines) dielectric","rect":[53.812843322753909,126.91716766357422,127.8691954352808,119.64453125]},{"page":178,"text":"anisotropy","rect":[53.81199264526367,137.23178100585938,89.57693237452432,129.63742065429688]},{"page":178,"text":"7 Magnetic, Electric and Transport Properties","rect":[228.78860473632813,44.275840759277347,385.1474045085839,36.68148422241211]},{"page":178,"text":"30","rect":[256.3664245605469,66.52958679199219,265.25492337548595,60.7628173828125]},{"page":178,"text":"ε||","rect":[326.8648376464844,91.93077850341797,333.21055566665259,84.49336242675781]},{"page":178,"text":"With these restrictions, Maier and Meier have calculated two principal compo-","rect":[65.76496887207031,280.954833984375,385.12102638093605,272.020263671875]},{"page":178,"text":"nents of dielectric permittivity and dielectric anisotropy:","rect":[53.812950134277347,292.85760498046877,281.52351243564677,283.92303466796877]},{"page":178,"text":"ejj","rect":[82.9847640991211,323.7899475097656,91.10945929713765,315.5555114746094]},{"page":178,"text":"e?","rect":[81.3441162109375,356.2393493652344,91.06945907382846,350.07366943359377]},{"page":178,"text":"\u00021","rect":[93.86099243164063,320.24700927734377,108.69476405194769,313.51373291015627]},{"page":178,"text":"\u00021","rect":[93.8043441772461,354.7656555175781,108.63810816815863,348.0323791503906]},{"page":178,"text":"ea","rect":[99.69683074951172,390.8206481933594,107.54116127571425,384.5923156738281]},{"page":178,"text":"¼","rect":[111.47795867919922,318.9222717285156,119.14270046446232,316.5915222167969]},{"page":178,"text":"¼","rect":[111.42131042480469,353.44091796875,119.08605221006779,351.11016845703127]},{"page":178,"text":"¼","rect":[110.79791259765625,387.9594421386719,118.46265438291936,385.6286926269531]},{"page":178,"text":"4prNA","rect":[121.95702362060547,315.4871826171875,148.89341973432125,306.7120056152344]},{"page":178,"text":"Fh","rect":[151.01560974121095,320.42584228515627,162.1046074967719,313.43359375]},{"page":178,"text":"M","rect":[131.416748046875,327.1053466796875,139.70860506911419,320.53143310546877]},{"page":178,"text":"4prNA","rect":[121.90036010742188,350.0058288574219,148.83677910932125,341.23065185546877]},{"page":178,"text":"Fh","rect":[150.95892333984376,354.944580078125,162.04792109540473,347.95233154296877]},{"page":178,"text":"M","rect":[131.36007690429688,361.56732177734377,139.65193392653607,354.993408203125]},{"page":178,"text":"ejj \u0002 e? ¼","rect":[121.2777099609375,392.77044677734377,162.70258358213813,384.5928039550781]},{"page":178,"text":"hgi þ","rect":[169.65187072753907,322.71673583984377,192.15833309385688,312.7861633300781]},{"page":178,"text":"hgi \u0002","rect":[169.59521484375,357.2354736328125,192.10166195127875,347.3049011230469]},{"page":178,"text":"4prNA","rect":[165.46055603027345,384.52447509765627,192.39686822064938,375.7492980957031]},{"page":178,"text":"M","rect":[174.92027282714845,396.0859375,183.21212984938763,389.51202392578127]},{"page":178,"text":"2=3gaS þ","rect":[194.406005859375,323.37017822265627,229.88361385557563,310.50921630859377]},{"page":178,"text":"1=3gaS þ","rect":[194.34933471679688,357.888916015625,229.82697323057563,345.0279541015625]},{"page":178,"text":"Fh\u000Bga \u0002","rect":[194.57586669921876,398.7137756347656,229.71378353331,374.82855224609377]},{"page":178,"text":"2","rect":[250.5415496826172,309.8177185058594,254.02551338753066,305.10443115234377]},{"page":178,"text":"F pe","rect":[232.1312713623047,320.24700927734377,253.65621323480984,308.9232177734375]},{"page":178,"text":"3kBT","rect":[239.9488983154297,328.57904052734377,259.66151780436589,320.2923889160156]},{"page":178,"text":"2","rect":[250.48489379882813,344.3363342285156,253.9688575037416,339.623046875]},{"page":178,"text":"F pe","rect":[232.07464599609376,354.7656555175781,253.59955735102077,343.4418640136719]},{"page":178,"text":"3kBT","rect":[239.8922119140625,363.0976867675781,259.6048619205768,354.7543640136719]},{"page":178,"text":"2","rect":[250.37158203125,378.85498046875,253.85554573616347,374.1416931152344]},{"page":178,"text":"F pe","rect":[231.96145629882813,389.2841796875,253.48624558344265,377.96051025390627]},{"page":178,"text":"2kBT","rect":[239.77891540527345,397.55963134765627,259.4916569645221,389.2729797363281]},{"page":178,"text":"½1 \u0002 1ð1 \u0002 3cos2bÞS\u0006","rect":[261.8135986328125,322.7271728515625,348.5071557728662,306.7120056152344]},{"page":178,"text":"2","rect":[281.63970947265627,327.1050720214844,286.6168145280219,320.3717956542969]},{"page":178,"text":"½1 þ ð1 \u0002 3cos2bÞS\u0006","rect":[261.7569580078125,357.2458190917969,341.8230432240381,345.9146423339844]},{"page":178,"text":"½ð1 \u0002 3cos2bÞ\fS","rect":[261.70050048828127,398.7137756347656,327.11830988935005,374.82855224609377]},{"page":178,"text":"(7.24)","rect":[361.05621337890627,391.0272521972656,385.1055844864048,382.5509033203125]},{"page":178,"text":"The equations have the Onsager form. In the isotropic phase, S ¼ 0, e|| ¼ e⊥ ¼","rect":[65.76656341552735,420.2194519042969,385.14807918760689,411.2649841308594]},{"page":178,"text":"eiso and equation (7.24) reduces to (7.23). The theory results in the following","rect":[53.814231872558597,432.1792297363281,385.1693047623969,423.24468994140627]},{"page":178,"text":"conclusions:","rect":[53.81255340576172,442.1167907714844,103.75979478427957,435.2042236328125]},{"page":178,"text":"1.","rect":[53.81255340576172,461.0,61.278210120456268,453.1749267578125]},{"page":178,"text":"2.","rect":[53.814491271972659,508.0,61.2801479866672,500.9568786621094]},{"page":178,"text":"The average molar dielectric susceptibility of the nematic phase he \u0002 1iM=4pr","rect":[66.27423095703125,462.3783874511719,385.18719834635808,452.44781494140627]},{"page":178,"text":"is independent of parameter S and equal to the molar susceptibility of the","rect":[66.27621459960938,474.0092468261719,385.11853826715318,465.07470703125]},{"page":178,"text":"isotropic phase <e> N ¼ eiso. Thus, the theory cannot explain a discontinuity","rect":[66.27620697021485,485.9693298339844,385.1204766373969,477.03424072265627]},{"page":178,"text":"of <e> at the Iso-N transition shown in Fig. 7.6 by a dashed line.","rect":[66.27519989013672,497.92889404296877,333.36358304526098,488.99432373046877]},{"page":178,"text":"\u0007","rect":[231.7919464111328,500.6285400390625,234.40484393992655,498.8252868652344]},{"page":178,"text":"For a specific value of the angle (b \u0004 55 ) between the dipole moment and the","rect":[66.27616882324219,509.8319396972656,385.1732868023094,500.5883483886719]},{"page":178,"text":"axis of maximum polarizability of the molecule, given by 1 \u0002 3cos2b ¼ 0, the","rect":[66.2752456665039,521.79150390625,385.1752094097313,511.3092956542969]},{"page":178,"text":"contribution from the orientational polarization to ea becomes zero [6]. For","rect":[66.2761459350586,533.7510986328125,385.15252052156105,524.8165893554688]},{"page":178,"text":"somewhat larger value of the angle b determined by condition","rect":[66.27632904052735,545.7108154296875,317.5544365983344,536.467529296875]},{"page":178,"text":"2","rect":[194.68943786621095,562.6708374023438,198.1734015711244,557.9575805664063]},{"page":178,"text":"ga \u0002 F2kpB2T ½ð1 \u0002 3cos2bÞ ¼ 0;","rect":[155.209228515625,581.3755493164063,282.11900269669436,561.7764282226563]},{"page":179,"text":"7.2 Dielectric Properties","rect":[53.81281280517578,44.274925231933597,137.04097445487299,36.68056869506836]},{"page":179,"text":"Fig. 7.6 Discontinuity of the","rect":[53.812843322753909,67.58130645751953,155.0055479743433,59.85148620605469]},{"page":179,"text":"average dielectric","rect":[53.812843322753909,77.4895248413086,114.33234545969487,69.89517211914063]},{"page":179,"text":"permittivity at the nematic –","rect":[53.812843322753909,87.4087142944336,151.28096527247355,79.81436157226563]},{"page":179,"text":"isotropic phase transition fora","rect":[53.812843322753909,97.3846664428711,155.3643126227808,89.79031372070313]},{"page":179,"text":"nematic with negative","rect":[53.812843322753909,107.36067962646485,129.39474627756597,99.76632690429688]},{"page":179,"text":"dielectric anisotropy","rect":[53.812843322753909,117.33663177490235,123.6784414687626,109.74227905273438]},{"page":179,"text":"ω","rect":[265.042724609375,80.78731536865235,270.52843345221,76.99451446533203]},{"page":179,"text":"e⊥","rect":[285.3398742675781,86.8272705078125,292.7947120910409,80.55290222167969]},{"page":179,"text":"e||","rect":[288.4416809082031,127.43163299560547,294.3503872096213,119.86747741699219]},{"page":179,"text":"N","rect":[303.3935241699219,67.22943115234375,309.16711277710388,61.484214782714847]},{"page":179,"text":"Iso","rect":[342.93060302734377,67.80799865722656,354.48578233182556,61.958763122558597]},{"page":179,"text":"εiso","rect":[352.7607727050781,99.92191314697266,364.9342447234888,94.154541015625]},{"page":179,"text":"T","rect":[334.2326354980469,142.99029541015626,339.11859483182556,137.24508666992188]},{"page":179,"text":"c","rect":[339.1180725097656,145.06846618652345,342.45409655878327,141.7123260498047]},{"page":179,"text":"163","rect":[372.4981689453125,42.55624771118164,385.18979400782509,36.68056869506836]},{"page":179,"text":"β","rect":[134.9646453857422,204.62454223632813,139.352892032397,197.35418701171876]},{"page":179,"text":"P","rect":[150.69451904296876,180.5977783203125,156.02595895793918,174.85507202148438]},{"page":179,"text":"e","rect":[156.02590942382813,182.6753387451172,159.3591002464582,179.32205200195313]},{"page":179,"text":"Fig. 7.7 Location of a molecular dipole moment with respect to the longitudinal molecular axis of","rect":[53.812843322753909,241.13809204101563,385.21176236731699,233.4082794189453]},{"page":179,"text":"a molecule. Note that in the Maier-Meier theory the dipole moment forms angle b with the axis of","rect":[53.812843322753909,251.04635620117188,385.21090787512949,243.1895294189453]},{"page":179,"text":"maximum polarizability of a spherical molecule","rect":[53.81281280517578,261.0223083496094,218.21224353098394,253.42794799804688]},{"page":179,"text":"the dielectric anisotropy completely vanishes. This agrees with experiment: the","rect":[53.812843322753909,287.4164733886719,385.1695941753563,278.48193359375]},{"page":179,"text":"anisotropy changes sign with a change of the angle b the dipole forms with the long","rect":[53.812843322753909,299.3760070800781,385.1267937760688,290.1326904296875]},{"page":179,"text":"molecular axis, Fig. 7.7, which, indeed, is the axis of maximum polarizability for","rect":[53.812843322753909,311.3355712890625,385.17855201570168,302.4010009765625]},{"page":179,"text":"rod-like molecules. For nematics with molecules having large longitudinal dipole","rect":[53.81183624267578,323.3349304199219,385.12278021051255,314.34063720703127]},{"page":179,"text":"moment, the anisotropy is positive, ea > 0. For molecules with large transverse","rect":[53.81183624267578,335.2550964355469,385.10776556207505,326.32012939453127]},{"page":179,"text":"dipole moment ea < 0.","rect":[53.81370162963867,347.21466064453127,147.1529507210422,338.28009033203127]},{"page":179,"text":"3. The temperature dependence of average dielectric permittivity <e> enters the","rect":[53.81330490112305,359.11749267578127,385.1749957866844,350.18292236328127]},{"page":179,"text":"equations both explicitly (term kBT) and through S (the additional contribution","rect":[66.27593994140625,371.0773620605469,385.1359795670844,362.12255859375]},{"page":179,"text":"from h and F is weak) while ea is directly proportional to S. The latter corre-","rect":[66.2757339477539,383.0369873046875,385.10250221101418,374.08245849609377]},{"page":179,"text":"sponds to the uniaxial symmetry of the dielectric permittivity with a tensor form","rect":[66.27613067626953,394.99652099609377,385.13541363359055,386.06195068359377]},{"page":179,"text":"of Eq. 3.16.","rect":[66.27613067626953,406.9560852050781,113.72188230063205,398.02154541015627]},{"page":179,"text":"_","rect":[117.08551788330078,422.5836486816406,122.06262293866644,421.3385925292969]},{"page":179,"text":"e ¼ hei þ ea½nanb \u0002 ð1=3Þdab\u0006 ¼ e?dab þ eaðnanbÞ","rect":[117.42681884765625,431.9275207519531,321.8949014102097,421.5471496582031]},{"page":179,"text":"(7.25a)","rect":[356.6380920410156,430.7605895996094,385.1369870742954,422.1647033691406]},{"page":179,"text":"Note that","rect":[65.7660903930664,457.4208068847656,102.99185044834207,450.50823974609377]},{"page":179,"text":"hei ¼ ðejj þ 2e?Þ=3 ¼ e? þ ejj=3 \u0002 e?=3 ¼ e? þ ea=3","rect":[109.83538055419922,482.72296142578127,329.15761652997505,471.70965576171877]},{"page":179,"text":"Using Eq. 7.25a one can calculate the value of the dielectric permittivity e(W,j)","rect":[65.76624298095703,504.2206726074219,385.1600583633579,494.9873046875]},{"page":179,"text":"of a uniaxial phase at any angle with respect to the director. Let the director is","rect":[53.814231872558597,516.18017578125,385.1859475527878,507.24566650390627]},{"page":179,"text":"rigidly fixed by a strong magnetic field along the z-axis, n ¼ (0, 0, 1). Then the","rect":[53.814231872558597,528.1397705078125,385.17697942926255,519.2052612304688]},{"page":179,"text":"single term nznz ¼ 1 is finite and Eq. 7.25a has a familiar form:","rect":[53.81523895263672,540.0994873046875,310.7277971036155,531.105224609375]},{"page":179,"text":"_e ¼ @0e00? e00? e00? 1A þ 0@000 000 e00a A1 ¼ 0@e00? e00? e00jj 1A","rect":[97.59963989257813,589.6470336914063,341.3866916593899,554.2903442382813]},{"page":180,"text":"164","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.68026351928711]},{"page":180,"text":"7 Magnetic, Electric and Transport Properties","rect":[228.78945922851563,44.274620056152347,385.1482590007714,36.68026351928711]},{"page":180,"text":"Assume that a very weak electric field is applied at angles W and j, respectively,","rect":[65.76496887207031,68.2883529663086,385.0980190804172,59.05499267578125]},{"page":180,"text":"to the z and x axes: E ¼ (EcosWcosj, EcosWsinj, EcosW). Then we can calculate","rect":[53.812923431396487,80.16820526123047,385.15772283746568,71.0145263671875]},{"page":180,"text":"the components of the displacement vector:","rect":[53.81491470336914,92.20748138427735,229.5315843350608,83.27293395996094]},{"page":180,"text":"DDDxyx ¼ @0e00? e00? e00jj 1A \u0005 EEssEiinncWWoscsoiWnsjj ¼ EEee??EessjiijnncyyoscsoiWnsjj","rect":[104.39701080322266,139.7146759033203,334.60496780093458,104.41476440429688]},{"page":180,"text":"and find D2 ¼ Dx2 þ D2y þ D2z ¼ e2E2 ¼ E2ðe?2 sin2W þ ej2jcos2WÞ: Hence, the dielec-","rect":[53.81344223022461,164.00648498535157,385.11931739656105,150.30917358398438]},{"page":180,"text":"tric permittivity e(W,j) is found to be independent of the azymuthal angle j as","rect":[53.81433868408203,173.1478271484375,385.1432229434128,163.91445922851563]},{"page":180,"text":"expected for a uniaxial material:","rect":[53.81537628173828,185.10736083984376,184.66148240634989,176.1728057861328]},{"page":180,"text":"eðWÞ ¼ ðe?2 sin2W þ ej2jcos2WÞ1=2:","rect":[156.00331115722657,211.90223693847657,283.0245813099756,197.41204833984376]},{"page":180,"text":"(7.25b)","rect":[356.071533203125,209.19520568847657,385.15972266999855,200.59930419921876]},{"page":180,"text":"It is evident that the same formula (7.25b) is valid for any properties of uniaxial","rect":[65.76598358154297,233.45596313476563,385.12092454502177,224.46163940429688]},{"page":180,"text":"phases described by a tensor of the type (7.25a) such as magnetic susceptibility,","rect":[53.814964294433597,245.35873413085938,385.11895413901098,236.36441040039063]},{"page":180,"text":"thermal and electric conductivity, diffusion and others.","rect":[53.814964294433597,257.3182678222656,275.1856655647922,248.3837127685547]},{"page":180,"text":"The displacement can be written in the vector form as D ¼ e?E þ eaðnEÞn and","rect":[65.7669906616211,269.6173095703125,385.1468438248969,259.6667785644531]},{"page":180,"text":"the electric field contribution to the free energy density is given by:","rect":[53.814964294433597,281.23822021484377,326.395155502053,272.30364990234377]},{"page":180,"text":"gel ¼ \u0002E8Dp ¼ \u00028e?pE2 \u0002 8epa ðnEÞ2","rect":[149.54461669921876,316.0020751953125,288.91890022835096,295.49615478515627]},{"page":180,"text":"(7.26)","rect":[361.0561828613281,310.7673034667969,385.10555396882668,302.231201171875]},{"page":180,"text":"to be compared with the magnetic counterpart (7.2).","rect":[53.81450653076172,339.56304931640627,264.0494350960422,330.62847900390627]},{"page":180,"text":"7.2.2.2 SmA Phase and the Role of the Positional Order","rect":[53.81450653076172,377.21661376953127,298.5676654644188,370.0052490234375]},{"page":180,"text":"Generally speaking, the Maier-Meier theory [5] explains all essential static dielec-","rect":[53.81450653076172,402.9883728027344,385.11946998445168,393.9940490722656]},{"page":180,"text":"tric properties of the nematic phase [7, 8]. The transition from the nematic to the","rect":[53.81450653076172,414.94793701171877,385.17530096246568,406.01336669921877]},{"page":180,"text":"smectic A phase is accompanied by an increase in the orientational order S. When","rect":[53.815513610839847,426.9074401855469,385.11452570966255,417.972900390625]},{"page":180,"text":"molecules do not possess very large longitudinal dipoles, the set of Maier-Meier","rect":[53.815513610839847,438.8102111816406,385.15532813874855,429.87567138671877]},{"page":180,"text":"equations is still valid even in the SmA phase. Typically, the dielectric anisotropy","rect":[53.815513610839847,450.769775390625,385.1504754166938,441.835205078125]},{"page":180,"text":"increases proportionally to S, as shown in Fig. 7.8. In this case, a periodicity of the","rect":[53.815513610839847,462.72930908203127,385.1752399273094,453.79473876953127]},{"page":180,"text":"smectic A density is not important. However, in many compounds, on approaching","rect":[53.815513610839847,474.6888732910156,385.13552180341255,465.75433349609377]},{"page":180,"text":"the SmA phase, the dielectric anisotropy decreases despite increasing orientational","rect":[53.815513610839847,486.6484069824219,385.127577377053,477.7138671875]},{"page":180,"text":"order [9]. It can even change sign either in the nematic or in the smectic phase as","rect":[53.815513610839847,498.60797119140627,385.14047636138158,489.67340087890627]},{"page":180,"text":"shown in Fig. 7.9.","rect":[53.815513610839847,510.5675354003906,126.71318479086642,501.63299560546877]},{"page":180,"text":"This","rect":[65.76753997802735,521.0,83.48603452788547,513.592529296875]},{"page":180,"text":"effect","rect":[89.72831726074219,521.0,112.503553939553,513.592529296875]},{"page":180,"text":"originates","rect":[118.78665161132813,522.5270385742188,158.22522367583469,513.592529296875]},{"page":180,"text":"from","rect":[164.49838256835938,521.0,183.87923382402023,513.592529296875]},{"page":180,"text":"the","rect":[190.15835571289063,521.0,202.38211858941879,513.592529296875]},{"page":180,"text":"anisotropic","rect":[208.62442016601563,522.5270385742188,253.0789264507469,513.592529296875]},{"page":180,"text":"dipole-dipole","rect":[259.32122802734377,522.5270385742188,312.66883123590318,513.592529296875]},{"page":180,"text":"correlations","rect":[318.9111328125,521.0,366.1358987246628,513.592529296875]},{"page":180,"text":"not","rect":[372.3841857910156,521.0,385.1355119473655,514.6084594726563]},{"page":180,"text":"accounted for by the Maier-Meier theory operating with a single particle distribu-","rect":[53.815513610839847,534.4298095703125,385.09771095124855,525.4953002929688]},{"page":180,"text":"tion function. When, with decreasing temperature, the smectic density wave r(z)","rect":[53.815513610839847,546.3893432617188,385.16231666413918,537.454833984375]},{"page":180,"text":"develops (even at the short-range scale) the longitudinal dipole moments prefer to","rect":[53.816490173339847,558.348876953125,385.14345637372505,549.4143676757813]},{"page":180,"text":"form antiparallel pairs and the “apparent” molecular dipole moment becomes","rect":[53.816490173339847,570.3084716796875,385.0896340762253,561.3739624023438]},{"page":180,"text":"smaller. This would reduce positive ea. Theoretically, dipole-dipole correlations","rect":[53.816490173339847,582.2699584960938,385.10666288481908,573.33349609375]},{"page":180,"text":"may be taken into account by introducing the so-called Kirkwood factors.","rect":[53.813594818115237,594.2294921875,350.9686550667453,585.255126953125]},{"page":181,"text":"7.2 Dielectric Properties","rect":[53.81284713745117,44.275718688964847,137.0410049724511,36.68136215209961]},{"page":181,"text":"Fig. 7.8 Typical temperature","rect":[53.812843322753909,67.58130645751953,155.2940764167261,59.6313591003418]},{"page":181,"text":"behavior of the principal","rect":[53.812843322753909,77.4895248413086,138.28478721823755,69.89517211914063]},{"page":181,"text":"dielectric permittivities in the","rect":[53.812843322753909,87.4087142944336,155.06224963449956,79.81436157226563]},{"page":181,"text":"nematic and SmA phases","rect":[53.812843322753909,97.3846664428711,140.0793808269433,89.79031372070313]},{"page":181,"text":"SmA","rect":[228.13232421875,66.82969665527344,245.45372956392604,60.790985107421878]},{"page":181,"text":"ε||","rect":[286.7530212402344,99.88750457763672,292.66023218032447,92.9109878540039]},{"page":181,"text":"Iso","rect":[339.3865661621094,66.7977066040039,350.0495705922828,60.942955017089847]},{"page":181,"text":"165","rect":[372.4981994628906,42.55704116821289,385.1898245254032,36.63056564331055]},{"page":181,"text":"ε⊥","rect":[286.1698303222656,163.05044555664063,293.62341692502528,157.3640594482422]},{"page":181,"text":"Fig. 7.9 Anomalous","rect":[53.812843322753909,239.04095458984376,125.68963320731439,231.00634765625]},{"page":181,"text":"temperature behavior of the","rect":[53.812843322753909,248.94918823242188,148.71813342356206,241.35482788085938]},{"page":181,"text":"principal dielectric","rect":[53.812843322753909,258.9251403808594,118.1271300055933,251.33078002929688]},{"page":181,"text":"permittivities within the","rect":[53.812843322753909,268.9010925292969,136.0290617682886,261.3067321777344]},{"page":181,"text":"nematic and SmA phase in","rect":[53.812843322753909,278.8203430175781,145.78552765040323,271.2259826660156]},{"page":181,"text":"di-n-heptyl-azoxybenzene","rect":[53.812843322753909,288.7962951660156,142.4036192634058,281.2019348144531]},{"page":181,"text":"(Adapted from [7])","rect":[53.812843322753909,298.7722473144531,119.26514524085214,291.1778869628906]},{"page":181,"text":"3.5","rect":[211.9938507080078,235.15296936035157,223.10448697411875,229.38619995117188]},{"page":181,"text":"3.4","rect":[211.9938507080078,251.02638244628907,223.10448697411875,245.25961303710938]},{"page":181,"text":"3.3","rect":[211.9938507080078,269.6056213378906,223.10448697411875,263.8388366699219]},{"page":181,"text":"3.2","rect":[211.9938507080078,288.72637939453127,223.10448697411875,282.9595947265625]},{"page":181,"text":"30","rect":[221.31639099121095,295.2857666015625,230.20490506493906,289.51898193359377]},{"page":181,"text":"40","rect":[244.39134216308595,295.2857666015625,253.27985623681406,289.51898193359377]},{"page":181,"text":"^","rect":[268.0226135253906,235.83840942382813,273.28218081865347,230.5435333251953]},{"page":181,"text":"||","rect":[263.79498291015627,260.88348388671877,267.95147845501028,254.84475708007813]},{"page":181,"text":"50","rect":[268.0977783203125,295.2857666015625,276.98627713525158,289.51898193359377]},{"page":181,"text":"SN","rect":[289.6412048339844,221.9109649658203,300.7438534336908,215.87225341796876]},{"page":181,"text":"60","rect":[291.8041687011719,295.2857666015625,300.69266751611095,289.51898193359377]},{"page":181,"text":"70","rect":[316.10211181640627,295.2857666015625,324.9906106313453,289.51898193359377]},{"page":181,"text":"NI","rect":[328.2270812988281,221.4071044921875,336.2195332209461,215.66432189941407]},{"page":181,"text":"80","rect":[339.0347595214844,295.2857666015625,347.92325833642345,289.51898193359377]},{"page":181,"text":"T","rect":[376.3546142578125,193.40277099609376,381.2384981729851,187.6599884033203]},{"page":181,"text":"T°C","rect":[371.34912109375,286.4300537109375,385.20143400009706,280.3913269042969]},{"page":181,"text":"90","rect":[362.10888671875,295.2857666015625,370.9974160512672,289.51898193359377]},{"page":181,"text":"7.2.2.3 Smectic C Case","rect":[53.812843322753909,330.16778564453127,157.71291387750473,322.6575927734375]},{"page":181,"text":"The point group symmetry of the SmC phase (C2h) is different from that of the N","rect":[53.812843322753909,355.8299865722656,385.13952398255199,346.89544677734377]},{"page":181,"text":"and SmA phases (D1h). Now the tensor of dielectric permittivity is represented by a","rect":[53.81362533569336,367.7898254394531,385.1582721538719,358.855224609375]},{"page":181,"text":"biaxial ellipsoid with three different components e1, e2 and e3 as shown in Fig. 7.10.","rect":[53.81344985961914,379.7495422363281,385.1827358772922,370.8148193359375]},{"page":181,"text":"The component e3 is parallel to the director (e3 ¼ e||), e2 is parallel to the symmetry","rect":[53.814083099365237,391.7091369628906,385.17327204755318,382.7745361328125]},{"page":181,"text":"axis C2, and e1 is perpendicular to the both e3 and e2. The biaxiality, however, is","rect":[53.813533782958987,403.61212158203127,385.1879922305222,394.6773681640625]},{"page":181,"text":"weak e1 \u0004 e2.","rect":[53.81432342529297,415.13067626953127,110.7374233772922,406.63714599609377]},{"page":181,"text":"7.2.3 Dipole Dynamics of an Isotropic Liquid","rect":[53.812843322753909,456.44317626953127,287.1507085790593,445.8053894042969]},{"page":181,"text":"To set the stage for discussion of frequency dispersion of liquid crystal permittivity","rect":[53.812843322753909,483.9854431152344,385.17656794599068,475.0509033203125]},{"page":181,"text":"we turn back to the isotropic liquids. First we shall find a characteristic relaxation","rect":[53.812843322753909,495.94500732421877,385.16765681317818,487.01043701171877]},{"page":181,"text":"time for molecular dipoles and then discuss real and imaginary components of the","rect":[53.812843322753909,507.9045715332031,385.1695941753563,498.97003173828127]},{"page":181,"text":"permittivity [10].","rect":[53.812843322753909,519.8640747070313,123.05831571127658,510.9295654296875]},{"page":181,"text":"7.2.3.1 Dipole Relaxation","rect":[53.812843322753909,558.2806396484375,167.73880357096744,549.0971069335938]},{"page":181,"text":"An applied electric field reduces the symmetry of an isotropic liquid from Kh to","rect":[53.812843322753909,582.2694091796875,385.1442498307563,573.3348999023438]},{"page":181,"text":"C1v and creates anisotropy in the angular distribution function of dipoles; the","rect":[53.81438446044922,594.2296142578125,385.11609686090318,585.2951049804688]},{"page":182,"text":"166","rect":[53.812843322753909,42.55777359008789,66.50444931178018,36.68209457397461]},{"page":182,"text":"Fig. 7.10 Three principal","rect":[53.812843322753909,67.58130645751953,143.32674888815942,59.85148620605469]},{"page":182,"text":"permittivities of the biaxial","rect":[53.812843322753909,77.4895248413086,146.92690757956567,69.89517211914063]},{"page":182,"text":"SmC phase; e3 is parallel to","rect":[53.812843322753909,87.4087142944336,148.95743316798136,79.81430053710938]},{"page":182,"text":"the director, e2 is parallel","rect":[53.81270980834961,97.3844223022461,140.29040245261255,89.79006958007813]},{"page":182,"text":"to symmetry axis C2, and e1","rect":[53.81319808959961,107.36043548583985,149.94043481662016,99.76602172851563]},{"page":182,"text":"is perpendicular to the both","rect":[53.812843322753909,117.33626556396485,147.73242706446573,109.74191284179688]},{"page":182,"text":"e3 and e2","rect":[53.812843322753909,126.91971588134766,84.57192743136625,119.66098022460938]},{"page":182,"text":"7 Magnetic, Electric and Transport Properties","rect":[228.78945922851563,44.276451110839847,385.1482590007714,36.68209457397461]},{"page":182,"text":"e3","rect":[316.14886474609377,69.09949493408203,323.4277737510645,62.84474182128906]},{"page":182,"text":"e2","rect":[197.91819763183595,207.92404174804688,205.1972134483301,201.77745056152345]},{"page":182,"text":"e1","rect":[377.9104919433594,192.67645263671876,385.1894009483301,186.52976989746095]},{"page":182,"text":"Fig. 7.11 Angular","rect":[53.812843322753909,266.6445617675781,118.65933316809823,258.91473388671877]},{"page":182,"text":"distribution of molecular","rect":[53.812843322753909,274.7689514160156,138.72474759680919,268.9017333984375]},{"page":182,"text":"dipoles in the isotropic phase","rect":[53.812843322753909,286.4720153808594,153.8150725104761,278.8776550292969]},{"page":182,"text":"without external field (dash","rect":[53.812843322753909,296.10931396484377,148.10216278223917,288.836669921875]},{"page":182,"text":"curve sphere) and with the","rect":[53.812843322753909,306.330810546875,145.2626280524683,298.8126525878906]},{"page":182,"text":"electric field applied (solid","rect":[53.812843322753909,316.3431701660156,145.4546865859501,308.73187255859377]},{"page":182,"text":"line)","rect":[53.812843322753909,325.98046875,69.43197721106698,318.70782470703127]},{"page":182,"text":"f(0)","rect":[266.88995361328127,296.448974609375,279.6710035262578,289.1385498046875]},{"page":182,"text":"f(E)","rect":[359.0185241699219,272.9188232421875,373.1346265731328,265.6083984375]},{"page":182,"text":"distribution function becomes elongated inthe field direction, Fig. 7.11. Ifwe forget","rect":[53.812843322753909,359.1177062988281,385.181532455178,350.18316650390627]},{"page":182,"text":"for a while about the role of temperature then the angular motion of the dipoles in","rect":[53.812843322753909,371.0772399902344,385.19643488935005,362.1427001953125]},{"page":182,"text":"the electric field E can be described by equation of motion:","rect":[53.812843322753909,383.03680419921877,293.6038041836936,374.10223388671877]},{"page":182,"text":"d2W","rect":[171.91810607910157,405.78759765625,186.70259446452213,397.2674865722656]},{"page":182,"text":"dW","rect":[205.45196533203126,405.78759765625,216.27119798014713,398.5164489746094]},{"page":182,"text":"I dt2 þ x dt ¼ \u0002peEsinW","rect":[166.6497802734375,419.390625,272.35013169596746,405.2416076660156]},{"page":182,"text":"(7.27)","rect":[361.05609130859377,414.0964660644531,385.1054624160923,405.6201171875]},{"page":182,"text":"Here, W is the decreasing with time angle between the dipole and E and x is a","rect":[65.76644134521485,442.8354187011719,385.16318548395005,433.5821228027344]},{"page":182,"text":"friction coefficient for a change of W angle [g.cm2s\u00021]. Usually, due to high","rect":[53.81543731689453,454.7949523925781,385.12786189130318,443.74395751953127]},{"page":182,"text":"viscosity of a liquid, the inertial term may be neglected. Then","rect":[53.813899993896487,466.7544250488281,303.11516657880318,457.81988525390627]},{"page":182,"text":"dW","rect":[185.2296905517578,488.2582092285156,195.99217576335026,480.987060546875]},{"page":182,"text":"peE sin W","rect":[218.5936737060547,490.1108703613281,253.7705113346393,480.987060546875]},{"page":182,"text":"¼\u0002","rect":[198.7674102783203,494.0,216.91194942686469,491.0]},{"page":182,"text":"dt","rect":[186.70191955566407,501.9179992675781,194.4761644131858,494.9257507324219]},{"page":182,"text":"x","rect":[233.6610107421875,503.7905578613281,238.63811579755316,494.6269226074219]},{"page":182,"text":"Since the contribution of the considered dipole to the field-induced polarization","rect":[65.7653579711914,527.2325439453125,385.07952204755318,518.2980346679688]},{"page":182,"text":"is given by its projection on the E-axis, peE ¼ pecosW, the rate of the increase of this","rect":[53.81333541870117,539.1920776367188,385.14560331450658,528.2482299804688]},{"page":182,"text":"projection is","rect":[53.813777923583987,551.152099609375,103.83269132720187,542.2175903320313]},{"page":182,"text":"ddpteE ¼ \u0002pe sinWddWt ¼ p2e Esixn2W","rect":[155.3209991455078,589.7752685546875,283.6224095279987,565.3828125]},{"page":183,"text":"7.2 Dielectric Properties","rect":[53.812843322753909,44.274620056152347,137.0410049724511,36.68026351928711]},{"page":183,"text":"167","rect":[372.4981994628906,42.55594253540039,385.1898245254032,36.68026351928711]},{"page":183,"text":"From here the kinetic equation may be written for the electric field induced","rect":[65.76496887207031,68.2883529663086,385.1219414811469,59.35380554199219]},{"page":183,"text":"polarization in the system of nv dipoles in a unit volume with initial arbitrary","rect":[53.812950134277347,80.24800872802735,385.1762627702094,71.31333923339844]},{"page":183,"text":"orientation:","rect":[53.81350326538086,90.18562316894531,100.00301988193582,83.27305603027344]},{"page":183,"text":"dPE","rect":[167.159912109375,113.70208740234375,182.5406517899853,105.41548919677735]},{"page":183,"text":"nvpe2Ehsin2Wi","rect":[196.2754669189453,115.09557342529297,248.82292570220188,103.43405151367188]},{"page":183,"text":"¼","rect":[185.85305786132813,117.6483383178711,193.51779964659125,115.31758880615235]},{"page":183,"text":"dt","rect":[171.23828125,126.01113891601563,179.01252610752176,119.01888275146485]},{"page":183,"text":"PE","rect":[260.9075927734375,113.70208740234375,271.3601860917431,105.654541015625]},{"page":183,"text":"(7.28)","rect":[361.11285400390627,120.71659088134766,385.1622251114048,112.24022674560547]},{"page":183,"text":"The second term on the right of (7.28) i.e. the dipole disorienting factor describes","rect":[65.76647186279297,147.92459106445313,385.1134377871628,138.9900360107422]},{"page":183,"text":"the relaxation of dipoles due to a finite temperature. The multiplier <sin2W> may be","rect":[53.81444549560547,159.88467407226563,385.15427435113755,148.83360290527345]},{"page":183,"text":"considered as a numerical coefficient k \u0004 2/3, as if the distribution function is","rect":[53.81444549560547,170.75851440429688,385.1881448184128,162.8897247314453]},{"page":183,"text":"spherical even in the electric field. In fact, a more precise value was found by Debye","rect":[53.814476013183597,183.80377197265626,385.1294635601219,174.8692169189453]},{"page":183,"text":"by averaging the PE value over W with the field-induced dipole distribution function","rect":[53.814476013183597,195.76361083984376,385.19023982099068,186.53024291992188]},{"page":183,"text":"shown qualitatively in Fig. 7.11. Since the thermal motion of dipolar molecules","rect":[53.81356430053711,207.72314453125,385.1623879824753,198.78858947753907]},{"page":183,"text":"destroys the field induced polar order, we introduce a thermal relaxation time tD, as","rect":[53.81356430053711,219.68270874023438,385.14203275786596,210.74815368652345]},{"page":183,"text":"the first (linear) approximation of the relaxation rate. In order to find this time, we","rect":[53.814144134521487,231.58560180664063,385.16892278863755,222.6510467529297]},{"page":183,"text":"should exclude PE from the kinetic equation.","rect":[53.814144134521487,243.5455322265625,235.00745816733127,234.61058044433595]},{"page":183,"text":"In the steady-state regime, dPE/dt ¼ 0, and the value of the dipole polarization is","rect":[65.76496887207031,255.50518798828126,385.1875344668503,246.5506134033203]},{"page":183,"text":"PE ¼ tD 2nvp2eE","rect":[186.4766387939453,284.3680419921875,250.81894677556716,267.7518005371094]},{"page":183,"text":"3x","rect":[231.50877380371095,291.7475280761719,241.4709405045844,282.5838928222656]},{"page":183,"text":"(7.29)","rect":[361.1125793457031,284.63671875,385.16195045320168,276.1603698730469]},{"page":183,"text":"This value may be compared with that found from the Langevin formula, see","rect":[65.7662124633789,313.26214599609377,385.1132282085594,304.32757568359377]},{"page":183,"text":"Eq. 7.21. From the comparison, the relaxation time for molecular dipoles is found:","rect":[53.81418991088867,325.2217102050781,385.14704759189677,316.28717041015627]},{"page":183,"text":"x","rect":[227.71347045898438,346.614990234375,232.69057551435004,337.45135498046877]},{"page":183,"text":"tD¼","rect":[197.01246643066407,352.8384704589844,217.3084643194428,346.8747863769531]},{"page":183,"text":"2kBT","rect":[220.066650390625,359.6402282714844,239.83571214030338,351.3531799316406]},{"page":183,"text":"(7.30)","rect":[361.1128234863281,353.0508117675781,385.16219459382668,344.574462890625]},{"page":183,"text":"Now, if we assume that a dipolar molecule has a spherical form of volume (4/3)pa3","rect":[65.76642608642578,381.2230529785156,385.18130140630105,370.1722412109375]},{"page":183,"text":"and rotates in continuous medium with viscosity Z [units g.cm\u00021s\u00021 (Poise)], then","rect":[53.812843322753909,393.1828308105469,385.1669854264594,382.13189697265627]},{"page":183,"text":"the friction force may be written as x ¼ 8pZa3 and tD ¼ 4p\u0002a3=kBT. This model is","rect":[53.814231872558597,405.4711608886719,385.18939603911596,394.091552734375]},{"page":183,"text":"very simple, however, it predicts a correct magnitude and temperature dependence","rect":[53.813777923583987,417.1020202636719,385.0750202007469,408.16748046875]},{"page":183,"text":"of relaxation times for dipoles in an isotropic liquid.","rect":[53.813777923583987,429.0047912597656,264.88189359213598,420.07025146484377]},{"page":183,"text":"In the dispersion region o \u0004 t\u0002D1, molecular dipoles follow the electric field with","rect":[65.76580047607422,441.6004638671875,385.19927302411568,430.4823913574219]},{"page":183,"text":"some","rect":[53.813655853271487,451.0,74.84690130426252,445.0]},{"page":183,"text":"lag,","rect":[79.9832763671875,452.9242858886719,94.69559903647189,443.98974609375]},{"page":183,"text":"i.e.","rect":[99.80907440185547,451.0,112.03284879233127,443.98974609375]},{"page":183,"text":"the","rect":[117.14234161376953,451.0,129.36611211969223,443.98974609375]},{"page":183,"text":"orientational","rect":[134.47561645507813,451.0,185.11466844150614,443.98974609375]},{"page":183,"text":"component","rect":[190.27093505859376,452.9242858886719,234.62690599033426,445.0]},{"page":183,"text":"of","rect":[239.77821350097657,451.0,248.07006202546729,443.98974609375]},{"page":183,"text":"polarization","rect":[253.2034454345703,452.9242858886719,301.07427302411568,443.98974609375]},{"page":183,"text":"P","rect":[306.16680908203127,450.78277587890627,312.2786940900203,444.2088623046875]},{"page":183,"text":"has","rect":[317.3822326660156,450.8624572753906,330.6412393008347,443.98974609375]},{"page":183,"text":"some","rect":[335.73480224609377,450.8624572753906,356.76807439996568,446.22088623046877]},{"page":183,"text":"phase","rect":[361.9053955078125,452.8147277832031,385.1185993023094,443.9698181152344]},{"page":183,"text":"retardation with respect to field E. Therefore, the dielectric permittivity becomes","rect":[53.813655853271487,464.8838195800781,385.0887795840378,455.9293518066406]},{"page":183,"text":"complex functions of frequency","rect":[53.813655853271487,476.8433532714844,182.1612481217719,467.9088134765625]},{"page":183,"text":"e\b ¼ e0 þ ie00","rect":[193.55682373046876,497.90789794921877,244.90116156764547,489.1195068359375]},{"page":183,"text":"(7.31)","rect":[361.11285400390627,498.8341979980469,385.1622251114048,490.35784912109377]},{"page":183,"text":"The imaginary part describes dissipation of energy due to molecular friction. It is","rect":[65.76647186279297,521.1675415039063,385.18622221099096,512.2330322265625]},{"page":183,"text":"called dielectric losses and equivalent to appearance of non-Ohmic electric con-","rect":[53.81444549560547,533.1270751953125,385.11748634187355,524.172607421875]},{"page":183,"text":"ductivity. The frequency dependence of e* can be written in the form of the Debye","rect":[53.81444549560547,545.086669921875,385.1134723491844,536.0924072265625]},{"page":183,"text":"dispersion law [10]","rect":[53.81444549560547,556.8798828125,132.01270423738135,548.0349731445313]},{"page":183,"text":"e\b \u0002 eð1Þ ¼ eð0Þ \u0002 eð1Þ","rect":[165.23388671875,586.009521484375,272.0698281680222,569.3140869140625]},{"page":183,"text":"1 \u0002 iotD","rect":[228.33798217773438,591.8615112304688,264.21119331580266,583.6541748046875]},{"page":183,"text":"(7.32)","rect":[361.11285400390627,585.2724609375,385.1622251114048,576.7960815429688]},{"page":184,"text":"168","rect":[53.81287384033203,42.55740737915039,66.5044798293583,36.68172836303711]},{"page":184,"text":"7 Magnetic, Electric and Transport Properties","rect":[228.78948974609376,44.276084899902347,385.14828951834957,36.68172836303711]},{"page":184,"text":"where e(0) and e(1) correspond respectively to zero frequency and to the frequen-","rect":[53.812843322753909,68.2883529663086,385.08199439851418,59.35380554199219]},{"page":184,"text":"cies","rect":[53.81183624267578,79.0,69.34040464507297,71.31333923339844]},{"page":184,"text":"essentially","rect":[74.26077270507813,80.24788665771485,116.50644007733831,71.31333923339844]},{"page":184,"text":"exceeding","rect":[121.44572448730469,80.24788665771485,162.0519188981391,71.31333923339844]},{"page":184,"text":"the","rect":[166.9314727783203,79.0,179.15525091363754,71.31333923339844]},{"page":184,"text":"relaxation","rect":[184.03778076171876,79.0,224.11141291669379,71.31333923339844]},{"page":184,"text":"frequency","rect":[229.06961059570313,80.24788665771485,269.0527276139594,71.31333923339844]},{"page":184,"text":"region,","rect":[273.9322814941406,80.24788665771485,301.9135708382297,71.31333923339844]},{"page":184,"text":"o >>","rect":[306.8448791503906,78.4848861694336,332.1435015508881,72.72772979736328]},{"page":184,"text":"\u00021","rect":[339.8709411621094,74.47900390625,348.7362220056947,69.76571655273438]},{"page":184,"text":"tD","rect":[335.5069885253906,80.8837890625,344.87391304236516,73.6739730834961]},{"page":184,"text":"(in","rect":[354.1455383300781,79.8495864868164,365.23452082685005,71.31346130371094]},{"page":184,"text":"that","rect":[370.1190490722656,78.18618774414063,385.13996751377177,71.31346130371094]},{"page":184,"text":"range, e(1) ¼ n2, n is refraction index, as shown in Fig. 7.3).","rect":[53.814083099365237,92.20772552490235,304.3792385628391,81.15672302246094]},{"page":184,"text":"The two components of the permittivity are:","rect":[65.7648696899414,104.1104965209961,244.30857713291239,95.17594909667969]},{"page":184,"text":"e0 ¼ eð1Þ þ eð10Þþ\u0002oe2ðt1D2 Þ e00 ¼ ½eð0Þ \u0002 eð1Þ\u0006otD","rect":[114.42303466796875,142.38250732421876,324.08512520056828,118.41812896728516]},{"page":184,"text":"(7.33)","rect":[361.0561828613281,134.4332733154297,385.10555396882668,125.95690155029297]},{"page":184,"text":"and the corresponding spectra of e0 and e00 are illustrated by Fig.7.12a. The ratio of","rect":[53.81450653076172,165.949462890625,385.1520932754673,156.2898406982422]},{"page":184,"text":"the two components determines the phase angle, Fig.7.12b:","rect":[53.81325912475586,177.90902709960938,293.046766830178,168.97447204589845]},{"page":184,"text":"00","rect":[222.38888549804688,195.7526092529297,226.15157355495016,192.16883850097657]},{"page":184,"text":"e","rect":[218.02723693847657,199.71986389160157,222.37722675686616,194.93882751464845]},{"page":184,"text":"tanf ¼ e0 \u0002 eð1Þ ¼ otD","rect":[168.1234130859375,215.77047729492188,272.02824043494328,199.31008911132813]},{"page":184,"text":"(7.34)","rect":[361.0561828613281,208.17503356933595,385.10555396882668,199.69866943359376]},{"page":184,"text":"7.2.3.2 Debye and Cole-Cole Diagrams","rect":[53.81450653076172,269.5634460449219,225.85412992583469,259.9117431640625]},{"page":184,"text":"Very often a rotation of a complex molecule includes a motion of different","rect":[53.81450653076172,293.0841369628906,385.1722856290061,284.14959716796877]},{"page":184,"text":"molecular dipoles and the dielectric spectrum e00(o) is not as simple as shown in","rect":[53.81450653076172,305.0442199707031,385.1437615495063,295.3845520019531]},{"page":184,"text":"the picture. It becomes somewhat blurred and the correspondent time tD cannot be","rect":[53.812862396240237,317.0037536621094,385.1537250347313,308.0692138671875]},{"page":184,"text":"found with sufficient accuracy. In order to improve the analysis, a simple procedure","rect":[53.813899993896487,328.96337890625,385.0791400737938,320.02880859375]},{"page":184,"text":"is used based on the Debye Eq. 7.32.","rect":[53.813899993896487,340.9229431152344,202.84336514975315,331.9884033203125]},{"page":184,"text":"ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi","rect":[181.49253845214845,342.0,220.93509972646963,341.0]},{"page":184,"text":"Note that sinf ¼ tanf=p1 þ tan2f and","rect":[65.76689910888672,353.3868713378906,239.1712578873969,341.44427490234377]},{"page":184,"text":"ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi","rect":[298.9731750488281,342.0,338.40180504873526,341.0]},{"page":184,"text":"cosf ¼ 1=p1 þ tan2f.","rect":[244.19613647460938,353.3868713378906,341.45318265463598,341.44427490234377]},{"page":184,"text":"Then","rect":[346.8951416015625,350.8210144042969,367.4804619889594,343.94830322265627]},{"page":184,"text":"the","rect":[372.9512939453125,350.8210144042969,385.17505682184068,343.94830322265627]},{"page":184,"text":"equations (7.33) can be cast in the new form:","rect":[53.81432342529297,364.7856140136719,236.353651595803,355.5323181152344]},{"page":184,"text":"a","rect":[70.74977111816406,410.6422424316406,76.30503351539791,405.0534973144531]},{"page":184,"text":"ε′(0)","rect":[75.61965942382813,436.91143798828127,91.3103679745742,429.4005432128906]},{"page":184,"text":"ε′","rect":[170.6401824951172,444.2890625,176.56221258346415,438.7478942871094]},{"page":184,"text":"ε′(∞)","rect":[74.36392211914063,532.0249633789063,91.30910149508202,524.5140991210938]},{"page":184,"text":"ε′′","rect":[163.23233032226563,511.97125244140627,171.12913763229228,506.4300842285156]},{"page":184,"text":"0.01  ","rect":[87.57738494873047,559.6638793945313,107.57778044516583,553.896728515625]},{"page":184,"text":"0.1","rect":[132.4695587158203,559.6638793945313,143.58088437665979,553.896728515625]},{"page":184,"text":"1","rect":[178.16909790039063,559.5198974609375,182.6136297379879,553.896728515625]},{"page":184,"text":"10","rect":[217.46804809570313,559.6638793945313,226.35709897626917,553.896728515625]},{"page":184,"text":"Fig. 7.12 Frequency dependence of the real (solid","rect":[53.812843322753909,584.0003662109375,228.72261566309855,576.2705688476563]},{"page":184,"text":"dielectric permittivity of an isotropic liquid (a) and","rect":[53.812843322753909,593.9085693359375,229.0805410781376,586.314208984375]},{"page":184,"text":"ε(0)+ε(∞)","rect":[241.70579528808595,471.9344787597656,274.7822841733047,464.423583984375]},{"page":184,"text":"2","rect":[258.0166320800781,482.0670166015625,262.46116391767546,476.4438781738281]},{"page":184,"text":"b","rect":[308.2703857421875,410.9106750488281,314.3751794916657,403.602294921875]},{"page":184,"text":"100","rect":[258.2074279785156,559.6638793945313,271.54102841962858,553.896728515625]},{"page":184,"text":"ωτD","rect":[288.71551513671877,562.3931884765625,302.036922990153,556.714111328125]},{"page":184,"text":"line) and imaginary (dash curve) parts","rect":[231.22540283203126,583.9326171875,362.79333956717769,576.3213500976563]},{"page":184,"text":"the definition of the phase angle f (b)","rect":[231.45217895507813,593.9085693359375,362.06063932044199,586.0348510742188]},{"page":184,"text":"of","rect":[365.2478942871094,582.1123657226563,372.29596036536386,576.3382568359375]},{"page":184,"text":"the","rect":[374.7640380859375,582.1801147460938,385.1542601325464,576.3382568359375]},{"page":185,"text":"7.2 Dielectric Properties","rect":[53.812843322753909,44.276084899902347,137.0410049724511,36.68172836303711]},{"page":185,"text":"169","rect":[372.4981994628906,42.62513732910156,385.1898245254032,36.68172836303711]},{"page":185,"text":"eðe00Þ\u0002\u0002eðe1ð1ÞÞ ¼ cos2f ¼ 21 þ 21cos2f eð0Þ \u0002e00eð1Þ ¼ sinfcosf ¼ 12sin2f","rect":[64.63162231445313,83.08061981201172,374.3693110626533,59.4790153503418]},{"page":185,"text":"or","rect":[53.81386947631836,104.59957885742188,62.10572563020361,99.95800018310547]},{"page":185,"text":"e0 ¼ eð1Þ þ eð0Þ þ eð0Þ \u0002 eð1Þ cos2f e00 ¼ eð0Þ \u0002 eð1Þ sin2f","rect":[69.67491912841797,137.2352752685547,345.1986567657783,120.91202545166016]},{"page":185,"text":"(7.35)","rect":[361.0567321777344,136.92649841308595,385.1061032852329,128.33059692382813]},{"page":185,"text":"The new equations may be regarded as the parametric representation of the","rect":[65.7670669555664,165.5518798828125,385.15992010309068,156.61732482910157]},{"page":185,"text":"equation of a circle of radius R with a center at x0, y0: x ¼ x0 þ Rcos2f","rect":[53.815025329589847,177.51141357421876,385.1893489044502,168.24905395507813]},{"page":185,"text":"and y ¼ y0 þ Rsin2f with angle f related to frequency by Eq. 7.34. Consequently,","rect":[53.814720153808597,189.4720458984375,385.2906155159641,180.20880126953126]},{"page":185,"text":"plottingtheexperimentaldependenceofe00 againste0 atdifferentfrequenciesoshould","rect":[53.81542205810547,201.43170166015626,385.2571648698188,191.7720489501953]},{"page":185,"text":"giveusa semi-circle(e00 > 0)withits centerata point e0 ¼ 12½eð1Þ þ eð0Þ\u0006; e00 ¼0","rect":[53.81380844116211,214.59359741210938,383.53704157880318,202.85223388671876]},{"page":185,"text":"and a radius 12½eð1Þ \u0002 eð0Þ\u0006, as shown by the Debye diagram, Fig.7.13a. If, in the","rect":[53.81481170654297,226.55349731445313,385.16086614801255,214.86859130859376]},{"page":185,"text":"experiment, the points do lie on such a circle we can find the single dipole relaxation","rect":[53.81407165527344,237.25381469726563,385.2863701920844,228.3192596435547]},{"page":185,"text":"time from any particular point on the circle using Eq. 7.34.","rect":[53.81407165527344,249.21337890625,285.3181423714328,240.27882385253907]},{"page":185,"text":"In a number of cases, the experimental points also form a part of a circle with a","rect":[65.76609802246094,261.17291259765627,385.15796697809068,252.2383575439453]},{"page":185,"text":"center that, however, lies below the e00 axis, see Cole-Cole diagram in Fig. 7.13b.","rect":[53.81407928466797,273.1727294921875,385.1832241585422,263.47320556640627]},{"page":185,"text":"Then the frequency dependence of the dielectric permittivity can be described by","rect":[53.814537048339847,285.09246826171877,385.1683281998969,276.15789794921877]},{"page":185,"text":"the empirical equation","rect":[53.814537048339847,297.052001953125,144.05542841962348,288.117431640625]},{"page":185,"text":"e\bðoÞ \u0002 eð1Þ ¼ 1eþð0Þði\u0002oteDð1Þ1\u0002Þh ;","rect":[153.00027465820313,336.4442443847656,284.3273462025537,311.302978515625]},{"page":185,"text":"(7.36)","rect":[361.056396484375,327.31805419921877,385.10576759187355,318.78192138671877]},{"page":185,"text":"where the angle ph/2 defines the position of the center. The relaxation time can be","rect":[53.814720153808597,360.0242919921875,385.15161932184068,351.06982421875]},{"page":185,"text":"found from the relationship otD ¼ v=u\u000E1\u0002h after location of the circle center. For","rect":[53.814720153808597,373.27215576171877,385.1509946426548,359.348876953125]},{"page":185,"text":"h ¼ 0, v/u ¼ tanf and the Cole-Cole equation reduces to the Debye equation. The","rect":[53.81318283081055,383.94378662109377,385.1460651226219,374.6805419921875]},{"page":185,"text":"parameter h tends to increase with the number of degrees of freedom in the","rect":[53.81418991088867,395.9033508300781,385.1172565288719,386.9488830566406]},{"page":185,"text":"molecule (for example, through the rotation of the dipole moments of various","rect":[53.81418991088867,407.8061218261719,385.1092568789597,398.87158203125]},{"page":185,"text":"molecular groups) or with increasing temperature.","rect":[53.81418991088867,419.7656555175781,255.85781522299534,410.83111572265627]},{"page":185,"text":"In a mixture of different dipolar molecules with strongly different relaxation","rect":[65.76622772216797,431.7251892089844,385.16899958661568,422.7906494140625]},{"page":185,"text":"times, several maxima of e00 will be observed and several characteristic semi-circles","rect":[53.81418991088867,441.65350341796877,385.0985452090378,434.0257568359375]},{"page":185,"text":"can be drawn. As a rule, the relaxation times do not differ so much and the","rect":[53.81344985961914,453.613037109375,385.11646307184068,446.71044921875]},{"page":185,"text":"Fig. 7.13","rect":[53.812843322753909,584.0003662109375,85.78299469141885,576.0673828125]},{"page":185,"text":"relaxation","rect":[53.812843322753909,592.1560668945313,87.80096191553995,586.314208984375]},{"page":185,"text":"a","rect":[100.7723159790039,492.9991760253906,106.32757837623775,487.4104309082031]},{"page":185,"text":"e//","rect":[100.93473815917969,502.4675598144531,108.97624964715489,496.1968994140625]},{"page":185,"text":"e(∞)","rect":[111.94874572753906,561.4061279296875,127.67989177088667,553.895751953125]},{"page":185,"text":"The Debye","rect":[91.76497650146485,583.9326171875,130.2924589850855,576.3382568359375]},{"page":185,"text":"times","rect":[90.23605346679688,592.1560668945313,108.63889010428705,586.314208984375]},{"page":185,"text":"(a)","rect":[133.5127410888672,583.593994140625,143.410492836067,576.3890991210938]},{"page":185,"text":"and","rect":[146.65362548828126,582.2055053710938,158.89679474024698,576.3382568359375]},{"page":185,"text":"ω2","rect":[170.53729248046876,515.3843994140625,179.35464468005197,509.7049560546875]},{"page":185,"text":"ω1","rect":[187.59799194335938,527.7815551757813,196.41435232165353,522.102783203125]},{"page":185,"text":"2f","rect":[177.40628051757813,544.2760620117188,185.54272507572794,537.1255493164063]},{"page":185,"text":"e(0)","rect":[188.43858337402345,562.5962524414063,202.71394175867963,555.0858764648438]},{"page":185,"text":"e/","rect":[209.2737274169922,559.6016235351563,215.64836115838535,553.3309326171875]},{"page":185,"text":"b","rect":[225.2021942138672,492.9991760253906,231.3069879633454,485.6907958984375]},{"page":185,"text":"e//","rect":[226.12741088867188,502.5261535644531,234.17024226190098,496.2554931640625]},{"page":185,"text":"e(∞)","rect":[241.33949279785157,561.9762573242188,257.0704481063359,554.4658813476563]},{"page":185,"text":"Cole-Cole (b) diagrams for","rect":[162.117919921875,583.9326171875,257.9708566055982,576.3382568359375]},{"page":185,"text":"ω2","rect":[279.2343444824219,521.3361206054688,288.0509032249738,515.657470703125]},{"page":185,"text":"u","rect":[271.644775390625,537.5057983398438,276.08902982079845,533.210693359375]},{"page":185,"text":"ph/2","rect":[272.8197937011719,556.7528686523438,289.1180520375953,550.6661987304688]},{"page":185,"text":"ω1","rect":[301.2137145996094,528.6149291992188,310.03030385973946,522.9361572265625]},{"page":185,"text":"v","rect":[302.3045349121094,544.5307006835938,306.30116659392,540.3475952148438]},{"page":185,"text":"e(0) e/","rect":[309.71160888671877,559.5042114257813,338.18334956170568,551.9938354492188]},{"page":185,"text":"calculations of characteristic","rect":[261.1903076171875,582.1801147460938,360.6696104743433,576.3382568359375]},{"page":185,"text":"dipole","rect":[363.94573974609377,583.9326171875,385.1745848395777,576.3382568359375]},{"page":186,"text":"170","rect":[53.812843322753909,42.55679702758789,66.50444931178018,36.73191833496094]},{"page":186,"text":"7 Magnetic, Electric and Transport Properties","rect":[228.78945922851563,44.275474548339847,385.1482590007714,36.68111801147461]},{"page":186,"text":"corresponding maxima in the dielectric spectra are not resolved. In this case, the","rect":[53.812843322753909,68.2883529663086,385.16965521051255,59.35380554199219]},{"page":186,"text":"Debye and Cole-Cole diagrams are very useful for calculations of different tD.","rect":[53.812843322753909,80.24788665771485,371.8149685433078,71.31333923339844]},{"page":186,"text":"7.2.4 Frequency Dispersion of e|| and e⊥ in Nematics","rect":[53.812843322753909,130.3584442138672,327.1514499637858,119.72067260742188]},{"page":186,"text":"7.2.4.1 Relaxation Modes","rect":[53.812843322753909,156.04812622070313,168.20563139067844,148.8865509033203]},{"page":186,"text":"Basically the experimental observations of dielectric relaxation in nematics are","rect":[53.812843322753909,181.81988525390626,385.15964544488755,172.8853302001953]},{"page":186,"text":"consistent with Fig.7.14. There are three characteristic modes: the rotation of","rect":[53.812843322753909,193.77944946289063,385.1486753067173,184.8448944091797]},{"page":186,"text":"molecules about short molecular axes (the lowest frequency o1 \u0004 106 Hz); the","rect":[53.81285095214844,205.73898315429688,385.17490423395005,194.64862060546876]},{"page":186,"text":"precession of long molecular axes about the director n (the middle frequency","rect":[53.81417465209961,217.69894409179688,385.17888728192818,208.76438903808595]},{"page":186,"text":"o2 \u0004 108 Hz); and the fast rotation of molecules about long molecular axes (the","rect":[53.81417465209961,229.65866088867188,385.1452106304344,218.60768127441407]},{"page":186,"text":"highest frequency o3 \u0004 109 Hz) [6]. The corresponding dielectric spectra are","rect":[53.813316345214847,241.61834716796876,385.10669744684068,230.51072692871095]},{"page":186,"text":"shown in Fig.7.15. The most striking feature is strong retardation of the permittivity","rect":[53.81367874145508,253.52108764648438,385.1763848405219,244.52676391601563]},{"page":186,"text":"component parallel to the director, i.e. e||-relaxation, tjj ¼ o\u000211 ¼ jjjtiso (retardation","rect":[53.81367874145508,266.8823547363281,385.1789177995063,255.05532836914063]},{"page":186,"text":"factor j|| ¼ 10–100) and some acceleration of e⊥-relaxation t? ¼ o3\u00021 ¼ j?tiso","rect":[53.814231872558597,278.1600036621094,384.6449439295228,266.9582824707031]},{"page":186,"text":"(acceleration factor j⊥ \u0004 0.5) with respect to the e-relaxation in the isotropic","rect":[53.812843322753909,289.4002685546875,385.0986713237938,280.40594482421877]},{"page":186,"text":"n","rect":[164.44091796875,327.6671447753906,168.8861169623601,323.3631591796875]},{"page":186,"text":"ω2","rect":[212.9661102294922,329.0037536621094,221.78470342825018,323.32366943359377]},{"page":186,"text":"ω3","rect":[265.78399658203127,333.85736083984377,274.6021625346955,328.06927490234377]},{"page":186,"text":"Pt","rect":[196.52391052246095,364.260009765625,202.56332935469465,356.480712890625]},{"page":186,"text":"P","rect":[251.0885772705078,376.664794921875,256.42121707039618,370.9208068847656]},{"page":186,"text":"i","rect":[255.46182250976563,378.8982849121094,257.43458758990837,375.5082702636719]},{"page":186,"text":"ω1","rect":[238.30355834960938,404.41253662109377,247.1213181332174,398.73297119140627]},{"page":186,"text":"Fig. 7.14","rect":[53.812843322753909,426.0309753417969,86.51995605616495,418.3011474609375]},{"page":186,"text":"Three characteristic relaxation modes for rotation of molecules in nematic liquid","rect":[92.50108337402344,425.9632568359375,385.1289114394657,418.368896484375]},{"page":186,"text":"crystals: slow rotation about short molecular axes with frequency o1; the precession of long","rect":[53.812843322753909,435.9392395019531,385.1890005507938,428.34466552734377]},{"page":186,"text":"molecular axes about the director n with middle frequency o2; and fast rotation of molecules about","rect":[53.8137321472168,445.8582458496094,385.17541221823759,438.2638244628906]},{"page":186,"text":"long molecular axes with frequency o3","rect":[53.81370162963867,455.83416748046877,187.6661275656436,448.23980712890627]},{"page":186,"text":"Fig. 7.15 Spectra of","rect":[53.812843322753909,523.6919555664063,125.77679532630136,515.962158203125]},{"page":186,"text":"principal dielectric","rect":[53.812843322753909,533.6001586914063,118.1271300055933,526.0057983398438]},{"page":186,"text":"permittivities for nematic","rect":[53.812843322753909,543.5761108398438,140.74018237375737,535.9817504882813]},{"page":186,"text":"phase. Characteristic","rect":[53.812843322753909,553.5520629882813,124.78260943674565,545.9577026367188]},{"page":186,"text":"dispersion ranges correspond","rect":[53.812843322753909,563.4713134765625,153.10941070704386,555.876953125]},{"page":186,"text":"to relaxation modes with","rect":[53.812843322753909,571.7201538085938,138.7179922500126,565.8529052734375]},{"page":186,"text":"frequencies o1, o2 and o3","rect":[53.812843322753909,583.4232177734375,143.25627649142485,575.8281860351563]},{"page":186,"text":"illustrated by Fig. 7.14","rect":[53.812843322753909,593.3983154296875,131.60984558253214,585.803955078125]},{"page":186,"text":"'","rect":[208.2870330810547,480.5882263183594,209.81374638350636,478.5486755371094]},{"page":186,"text":"ε","rect":[204.7779998779297,484.4034118652344,208.28704249455945,480.6042175292969]},{"page":186,"text":"ε||(0)","rect":[195.1962890625,508.34100341796877,211.59026927332807,500.3706970214844]},{"page":186,"text":"ε^(0)","rect":[194.17965698242188,526.0011596679688,211.401045030164,518.0548706054688]},{"page":186,"text":"0","rect":[208.9388885498047,578.8579711914063,213.38314297997813,573.0911865234375]},{"page":186,"text":"ω1","rect":[264.60919189453127,513.5115966796875,273.42544546130196,507.8329162597656]},{"page":186,"text":"log ω","rect":[284.59295654296877,591.5128173828125,302.96149756392858,584.0183715820313]},{"page":186,"text":"ω2","rect":[322.7315979003906,527.1248779296875,331.5480650902082,521.4461669921875]},{"page":186,"text":"ω3","rect":[344.37225341796877,547.0322875976563,353.18972768786446,541.2454833984375]},{"page":187,"text":"7.2 Dielectric Properties","rect":[53.81205368041992,44.276390075683597,137.04021151541986,36.68203353881836]},{"page":187,"text":"171","rect":[372.4974060058594,42.55771255493164,385.18903106837197,36.73283386230469]},{"page":187,"text":"phase. Moreover, a growth of t||(T) with decreasing temperature (i.e., with increas-","rect":[53.812843322753909,68.2883529663086,385.1035092910923,59.35380554199219]},{"page":187,"text":"ing S) is much faster than that predicted by the Arrhenius law for viscosity.","rect":[53.81245040893555,80.24788665771485,358.0639309456516,71.31333923339844]},{"page":187,"text":"Evidently, the nematic order strongly influences the relaxation of e|| and e⊥.","rect":[65.76448059082031,92.20748138427735,385.1832241585422,83.27293395996094]},{"page":187,"text":"An individual rod-like molecule feels the nematic potential curve W(cosW), (see","rect":[53.814537048339847,104.1104965209961,385.1464008159813,94.87713623046875]},{"page":187,"text":"Fig. 6.21), whose form is an inverse of the molecular distribution function in","rect":[53.814537048339847,116.0699691772461,385.1414727311469,107.13542175292969]},{"page":187,"text":"Fig. 3.15. In fact, each molecule moves in the potential well of the depth about","rect":[53.814537048339847,128.02957153320313,385.0997148282249,119.0352554321289]},{"page":187,"text":"0.15 eV with a minimum centered at W \u0004 0 or p. This prevents deviation of the","rect":[53.814537048339847,139.98910522460938,385.17530096246568,130.7557373046875]},{"page":187,"text":"molecule through large angles from the director n. A primitive, but useful mechan-","rect":[53.81554412841797,151.94863891601563,385.08269630281105,143.0140838623047]},{"page":187,"text":"ical model for this situation is a rod-like molecule in a rubber tube, Fig. 7.16. ForW","rect":[53.81652069091797,163.908203125,385.1901890690143,154.67483520507813]},{"page":187,"text":"deviation from 0 to p/2 the molecule has to overcome a high barrier WN and its","rect":[53.81749725341797,175.86773681640626,385.1538125430222,166.9331817626953]},{"page":187,"text":"angular velocity and frequency decreases dramatically down to o1 (case a). The","rect":[53.814022064208987,187.82821655273438,385.14820135309068,178.89366149902345]},{"page":187,"text":"retardation factor depends on the height of the barrier. Theoretically this can","rect":[53.81432342529297,199.73104858398438,385.1253289323188,190.79649353027345]},{"page":187,"text":"approximately be written as","rect":[53.81432342529297,211.69061279296876,166.15256132231907,202.7560577392578]},{"page":187,"text":"kBT","rect":[203.8658905029297,234.26226806640626,218.59382981608463,225.91893005371095]},{"page":187,"text":"WN","rect":[239.26913452148438,234.3319854736328,252.23666321451504,226.15798950195313]},{"page":187,"text":"jjj ¼ WN expkBT","rect":[183.53163146972657,247.9353790283203,253.26060838053776,232.95919799804688]},{"page":187,"text":"This expression gives a correct order of magnitude for the retardation factor (for","rect":[65.76607513427735,271.48944091796877,385.1529477676548,262.55487060546877]},{"page":187,"text":"WN ¼ 0.16eV and T ¼ 400K, kBT/WN \u0004 0.21 and the retardation factor is about","rect":[53.81405258178711,283.0307312011719,385.1014848477561,274.49432373046877]},{"page":187,"text":"25). Note that the retardation is controlled not by a molecular dipole moment, but","rect":[53.81344223022461,295.4083557128906,385.134413314553,286.4140319824219]},{"page":187,"text":"rather by a molecular shape.","rect":[53.81344223022461,307.3111267089844,168.11861844565159,298.3765869140625]},{"page":187,"text":"For rotation of the same molecule about its long axis (frequency o3) there is no","rect":[65.76544952392578,319.2706604003906,385.17095271161568,310.33612060546877]},{"page":187,"text":"barrier (case b). To some extent, such rotation in the nematic phase is even easier","rect":[53.814231872558597,331.23065185546877,385.08342872468605,322.29608154296877]},{"page":187,"text":"than in the isotropic phase (friction is less). Therefore, instead of retardation we","rect":[53.814205169677737,343.190185546875,385.1680072612938,334.255615234375]},{"page":187,"text":"have acceleration, j⊥ < 1.","rect":[53.814205169677737,355.12982177734377,159.84151883627659,346.2152099609375]},{"page":187,"text":"When rigid molecules precess (case c) about the director at small W angles within","rect":[65.7662582397461,367.1095886230469,385.1331719498969,357.876220703125]},{"page":187,"text":"the flat potential minimum they are more or less free. Therefore, frequency o2 cor-","rect":[53.814239501953128,379.0691223144531,385.15596900788918,370.13458251953127]},{"page":187,"text":"respond to a quite fast molecular motion and the precession contributes to both e||","rect":[53.81411361694336,391.0289001464844,385.1829548488092,382.0943603515625]},{"page":187,"text":"and e⊥(case c). All the three dispersion regions are observed by dielectric spectros-","rect":[53.812843322753909,402.93194580078127,385.10299049226418,393.99737548828127]},{"page":187,"text":"copy techniques [11].","rect":[53.81388473510742,414.8914794921875,140.7320522835422,405.9569091796875]},{"page":187,"text":"Fig. 7.16 A mechanical model that helps to understand the process of retardation or acceleration","rect":[53.812843322753909,564.0487670898438,385.1110891738407,556.3189697265625]},{"page":187,"text":"of molecular rotation in the nematic potential: slow hindered rotation of molecule at the angles","rect":[53.812843322753909,573.9002075195313,385.1525009441308,566.3058471679688]},{"page":187,"text":"W \u0004 p/2 (a), fast accelerated rotation about long molecular axes at the angles W \u0004 0 or p (b) and","rect":[53.812843322753909,583.8761596679688,385.19318145899697,576.02783203125]},{"page":187,"text":"quite fast molecular precession within small W-angles (c)","rect":[53.812034606933597,593.8521118164063,248.20343106848888,586.0037841796875]},{"page":188,"text":"172","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.73106384277344]},{"page":188,"text":"7 Magnetic, Electric and Transport Properties","rect":[228.78945922851563,44.274620056152347,385.1482590007714,36.68026351928711]},{"page":188,"text":"7.2.4.2 Dual Frequency Addressing","rect":[53.812843322753909,68.68677520751953,210.8056573014594,59.27412033081055]},{"page":188,"text":"Experiments show that with increasing molecular length j||-factor increases dramat-","rect":[53.812843322753909,92.20748138427735,385.1082700332798,83.27293395996094]},{"page":188,"text":"ically. For terphenyl derivatives, frequency o1 can be shifted down to 1–10 kHz.","rect":[53.81421661376953,104.1104965209961,385.15826077963598,95.17594909667969]},{"page":188,"text":"The simple theory discussed above does not consider a molecular length but","rect":[53.81344223022461,116.0702133178711,385.1324296719749,107.13566589355469]},{"page":188,"text":"intuitively it is understandable in the framework of the same mechanical model.","rect":[53.81344223022461,128.02981567382813,385.12041898276098,119.09526062011719]},{"page":188,"text":"The low frequency dispersion of e|| is seen in Fig. 7.17 against the background of","rect":[53.81344223022461,139.98959350585938,385.1506284317173,131.05479431152345]},{"page":188,"text":"almost constant e⊥. Therefore there is an inversion point for the sign of dielectric","rect":[53.81374740600586,151.94937133789063,385.1786578960594,143.0145721435547]},{"page":188,"text":"anisotropy at a certain frequency finv; ea > 0 for f < finv and ea < 0 for f > finv.","rect":[53.812931060791019,163.908935546875,385.1832241585422,154.93453979492188]},{"page":188,"text":"This is very interesting for display applications because an external field of low","rect":[53.814537048339847,175.86846923828126,385.15047979309886,166.9339141845703]},{"page":188,"text":"frequency (say, at 1 kHz) aligns the director (that is the optical axis) along the field,","rect":[53.814537048339847,187.8280029296875,385.1136135628391,178.89344787597657]},{"page":188,"text":"while at an enhanced frequency (say, at 10 kHz) the director is aligned perpendicu-","rect":[53.814537048339847,199.73077392578126,385.1324704727329,190.7962188720703]},{"page":188,"text":"lar to the field. Changing frequency of the field one can switch the director very fast","rect":[53.814537048339847,211.69033813476563,385.1345353848655,202.7557830810547]},{"page":188,"text":"because, in this, so-called dual-frequency addressing regime, the director always","rect":[53.814537048339847,223.64987182617188,385.08273710356908,214.71531677246095]},{"page":188,"text":"suffers a torque eaE2 from a strong field and the switching rate is high t\u00021 / eaE2.","rect":[53.814537048339847,235.61026000976563,385.1832241585422,224.5591583251953]},{"page":188,"text":"Since the field is never switched off, the slowest process of the director free","rect":[53.814537048339847,247.56979370117188,385.15940130426255,238.63523864746095]},{"page":188,"text":"relaxation is excluded.","rect":[53.814537048339847,257.4974060058594,144.69646878744846,250.5948028564453]},{"page":188,"text":"7.3 Transport Properties","rect":[53.812843322753909,301.7045593261719,189.6493442508952,290.6842956542969]},{"page":188,"text":"7.3.1 Thermal Conductivity","rect":[53.812843322753909,331.5751953125,198.141095957472,320.9374084472656]},{"page":188,"text":"According to Fourier law, the scalar coefficient of thermal conductivity k relates","rect":[53.812843322753909,359.1177062988281,385.1058694277878,350.1632385253906]},{"page":188,"text":"!","rect":[369.94952392578127,360.6049499511719,376.9174513356082,358.1297607421875]},{"page":188,"text":"the thermal flux density Q [in erg/cm2s] to the gradient of temperature Q ¼ \u0002krT","rect":[53.812843322753909,371.2367248535156,384.6774632389362,360.0262756347656]},{"page":188,"text":"[units of k: erg/cm.s.K]. The corresponding thermal diffusion coefficient [in cm2/s]","rect":[53.81444549560547,383.03692626953127,385.16182838288918,371.986083984375]},{"page":188,"text":"includes density of substance r and heat capacitance Cp (at constant pressure)","rect":[53.814022064208987,395.9264221191406,368.25631080476418,386.052001953125]},{"page":188,"text":"Dth ¼ k\u000FrCp","rect":[194.74647521972657,422.9580078125,243.77275154671035,407.2408447265625]},{"page":188,"text":"and determines the time tT of the heat transfer over the distance LT called a thermal","rect":[53.812843322753909,443.9812316894531,385.18305833408427,435.54443359375]},{"page":188,"text":"diffusion length:","rect":[53.814353942871097,456.4385681152344,120.44485337802957,447.5040283203125]},{"page":188,"text":"tth ¼ L2T\u000F2Dth","rect":[192.36801147460938,484.63348388671877,246.1272132166322,468.6852722167969]},{"page":188,"text":"(7.37)","rect":[361.0561828613281,481.7166442871094,385.10555396882668,473.24029541015627]},{"page":188,"text":"Fig. 7.17 Spectra of","rect":[53.812843322753909,533.7244873046875,125.77679532630136,525.9946899414063]},{"page":188,"text":"principal dielectric","rect":[53.812843322753909,543.6326904296875,118.1271300055933,536.038330078125]},{"page":188,"text":"permittivity components","rect":[53.812843322753909,553.608642578125,137.3929794597558,546.0142822265625]},{"page":188,"text":"showing the inversion of the","rect":[53.812843322753909,563.5278930664063,151.3232969977808,555.9335327148438]},{"page":188,"text":"sign of dielectric anisotropy","rect":[53.812843322753909,573.5038452148438,149.56509918360636,565.9094848632813]},{"page":188,"text":"at a particularly low","rect":[53.812843322753909,583.4797973632813,122.80103088957279,575.8854370117188]},{"page":188,"text":"frequency finv","rect":[53.812843322753909,593.3990478515625,100.29401087205452,585.7708740234375]},{"page":188,"text":"ε > 0","rect":[233.41587829589845,526.8367309570313,254.28891030704657,521.0728149414063]},{"page":188,"text":"a","rect":[237.36373901367188,529.7745971679688,240.69526327136044,526.4649658203125]},{"page":188,"text":"ε < 0","rect":[323.8074645996094,552.99267578125,344.0817112103669,547.228759765625]},{"page":188,"text":"a ","rect":[327.75616455078127,555.9301147460938,332.7534391307974,552.6204833984375]},{"page":188,"text":"10 kHz","rect":[323.51275634765627,587.9400634765625,348.3753706277889,582.0562133789063]},{"page":188,"text":"ε^","rect":[353.22161865234377,533.5101928710938,361.11198417413109,527.2794189453125]},{"page":189,"text":"7.3 Transport Properties","rect":[53.81285095214844,44.276145935058597,136.41827853202143,36.68178939819336]},{"page":189,"text":"173","rect":[372.49822998046877,42.55746841430664,385.1898245254032,36.73258972167969]},{"page":189,"text":"This formula comes about from the general expression for the diffusion (ran-","rect":[65.76496887207031,68.2883529663086,385.1269467910923,59.35380554199219]},{"page":189,"text":"dom) process, like the Brownian motion, and relate average distance passed by a","rect":[53.812950134277347,80.24788665771485,385.1558307476219,71.31333923339844]},{"page":189,"text":"random-walking particle to time t hx2i ¼ 2Dt derived by Einstein and Smolu-","rect":[53.812950134277347,92.53642272949219,385.1237729629673,81.72531127929688]},{"page":189,"text":"chowski in the beginning of twentieth century. The temperature dependence of the","rect":[53.814857482910159,104.1104965209961,385.1736224956688,95.17594909667969]},{"page":189,"text":"thermal conductivity in the nematic and smectic A phase resembles that of the","rect":[53.814857482910159,116.0699691772461,385.1726764507469,107.13542175292969]},{"page":189,"text":"magnetic susceptibility [12]. A good example is p-octyl-p0-cyanobiphenyl (8CB),","rect":[53.814857482910159,128.02999877929688,385.12383695151098,118.37037658691406]},{"page":189,"text":"see Fig. 7.18. No such sharp anomalies in k at the phase transitions are observed as","rect":[53.81386947631836,139.98953247070313,385.13876737700658,131.03504943847657]},{"page":189,"text":"manifested, for instance, by the specific heat discussed in Section 6.2.4. The reason","rect":[53.812862396240237,151.94906616210938,385.12978449872505,143.01451110839845]},{"page":189,"text":"is that the thermal conductivity is mainly determined by a single-particle molecular","rect":[53.81290817260742,163.90863037109376,385.1507810196079,154.9740753173828]},{"page":189,"text":"distribution function whereas the specific heat dramatically depends on the long-","rect":[53.81290817260742,175.8681640625,385.1547788223423,166.93360900878907]},{"page":189,"text":"range fluctuations of the order parameter.","rect":[53.81290817260742,187.82766723632813,220.8107113411594,178.8931121826172]},{"page":189,"text":"In anisotropic phases the magnitude of the thermal flux depends on the direction","rect":[65.76493072509766,199.73043823242188,385.08904353192818,190.79588317871095]},{"page":189,"text":"of gradient rT:","rect":[53.81290817260742,211.69000244140626,117.05000932285378,202.7454833984375]},{"page":189,"text":"@T","rect":[232.0185546875,233.28106689453126,243.1782261783893,225.95016479492188]},{"page":189,"text":"Qi ¼ \u0002kij @xj:","rect":[190.44041442871095,249.63409423828126,248.5276330677881,232.9436492919922]},{"page":189,"text":"In the case of a uniaxial phase, the thermal conductivity tensor has a familiar","rect":[65.7653579711914,273.1895446777344,385.0994809707798,264.2550048828125]},{"page":189,"text":"form (7.25a): kij ¼ k?dij þ kaninj, where ka ¼ kjj \u0002 k?>0 for calamitic phases and","rect":[53.81332015991211,286.4939880371094,385.1450127702094,275.85919189453127]},{"page":189,"text":"ka < 0 for discotic ones.","rect":[53.81313705444336,296.67083740234377,153.2174038460422,288.0976257324219]},{"page":189,"text":"At present the coefficients k|| and k⊥ are measured by sophisticated techniques","rect":[65.76567840576172,309.0118408203125,385.1301614199753,300.05731201171877]},{"page":189,"text":"such as a.c. adiabatic calorimetry, photoacoustic and photopyroelectric methods.","rect":[53.81321334838867,320.97137451171877,385.1798977425266,312.03680419921877]},{"page":189,"text":"The latter is very sensitive and allows the measurements using small amounts of","rect":[53.81321334838867,332.9309387207031,385.14714942781105,323.99639892578127]},{"page":189,"text":"liquid crystals [13]. The idea is demonstrated by Fig. 7.19. The light beam (shown","rect":[53.81321334838867,344.8904724121094,385.12514582685005,335.9559326171875]},{"page":189,"text":"by arrows) of intensity I is modulated by a chopper according to the law of","rect":[53.81321334838867,356.85003662109377,385.1470883926548,347.91546630859377]},{"page":189,"text":"I ¼ Imcosot and absorbed by black paint on the bottom of a quartz block. The","rect":[53.81222152709961,368.81011962890627,385.14612615777818,359.87554931640627]},{"page":189,"text":"heat flux traverses the properly aligned liquid crystal layer (LC) and reachesa","rect":[53.8132209777832,380.712890625,385.15808904840318,371.7783203125]},{"page":189,"text":"crystalline pyroelectric detector. The latter generates an electric signal at frequency","rect":[53.8132209777832,392.67242431640627,385.1730584245063,383.73785400390627]},{"page":189,"text":"o. A lock-in amplifier (LA) analyzes the amplitude and phase of the signal. The","rect":[53.8132209777832,404.6319885253906,385.14316595270005,395.69744873046877]},{"page":189,"text":"measured amplitude provides the thermal energy reached the detector; the phase","rect":[53.8132209777832,416.5915222167969,385.09934271051255,407.656982421875]},{"page":189,"text":"Fig. 7.18 Anisotropy of","rect":[53.812843322753909,560.0244140625,138.40832608802013,552.074462890625]},{"page":189,"text":"thermal conductivity of 8CB","rect":[53.812843322753909,569.8758544921875,151.95872676100115,562.281494140625]},{"page":189,"text":"in the nematic and smectic A","rect":[53.812843322753909,578.1246948242188,153.9758355770728,572.2574462890625]},{"page":189,"text":"phases (Adapted from [12])","rect":[53.812843322753909,589.8277587890625,148.43721097571544,582.2333984375]},{"page":190,"text":"174","rect":[53.812843322753909,42.55630874633789,66.50444931178018,36.73143005371094]},{"page":190,"text":"Fig. 7.19 Scheme of the","rect":[53.812843322753909,67.58130645751953,139.9372343757105,59.546695709228519]},{"page":190,"text":"set-up for measuring thermal","rect":[53.812843322753909,77.4895248413086,152.86066917624536,69.89517211914063]},{"page":190,"text":"conductivity and specific","rect":[53.812843322753909,87.4087142944336,139.17318103098394,79.81436157226563]},{"page":190,"text":"heat in liquid crystals","rect":[53.812843322753909,97.3846664428711,127.44785006522455,89.79031372070313]},{"page":190,"text":"7 Magnetic,","rect":[228.78945922851563,44.274986267089847,269.52612564160787,36.68062973022461]},{"page":190,"text":"light","rect":[207.63870239257813,71.50753021240235,223.0736982284062,64.04512023925781]},{"page":190,"text":"absorber","rect":[207.63870239257813,79.5058364868164,237.60545204929108,73.66707611083985]},{"page":190,"text":"quartz","rect":[193.3865966796875,97.77967834472656,215.503948523188,91.21308135986328]},{"page":190,"text":"block","rect":[193.3865966796875,105.80191802978516,212.31624255835957,99.9631576538086]},{"page":190,"text":"LC","rect":[205.4510498046875,127.6542739868164,216.52971861430309,121.91949462890625]},{"page":190,"text":"Electric","rect":[271.95361328125,43.0,298.4452452399683,36.68062973022461]},{"page":190,"text":"and","rect":[300.8431091308594,43.0,313.08627838282509,36.68062973022461]},{"page":190,"text":"Transport","rect":[315.4579162597656,44.274986267089847,348.39684776511259,36.84995651245117]},{"page":190,"text":"Properties","rect":[350.8048400878906,44.274986267089847,385.1482590007714,36.68062973022461]},{"page":190,"text":"contains information on the time of the heat transfer tT related to thermal diffusion","rect":[53.812843322753909,192.0350341796875,385.1251153092719,183.59825134277345]},{"page":190,"text":"coefficient by Eq. 7.37.","rect":[53.81417465209961,204.49234008789063,148.0108608772922,195.5577850341797]},{"page":190,"text":"7.3.2 Diffusion","rect":[53.812843322753909,250.26812744140626,136.67315644120337,239.71401977539063]},{"page":190,"text":"The diffusion is a kinetic process of molecular transport due to a gradient of","rect":[53.812843322753909,277.89410400390627,385.1467221817173,268.95953369140627]},{"page":190,"text":"molecular concentration c.The coefficient of diffusion D relates the flux of particles","rect":[53.812843322753909,289.8536682128906,385.1636086856003,280.91912841796877]},{"page":190,"text":"to their concentration gradient (first Fick law):","rect":[53.81385040283203,301.8132019042969,241.30241258213114,292.8587341308594]},{"page":190,"text":"~","rect":[230.26220703125,317.9979553222656,237.30978778964778,316.0158386230469]},{"page":190,"text":"J ¼ \u0002Drc","rect":[196.614990234375,326.5211486816406,242.35855901910629,318.7718811035156]},{"page":190,"text":"Note that, in contrast to hydrodynamic processes, there is no mass transport","rect":[65.7648696899414,351.635498046875,385.1059709317405,342.700927734375]},{"page":190,"text":"during the diffusion process, the mass velocity v(x,y,z) ¼ 0 and the mass density is","rect":[53.81285095214844,363.5950622558594,385.1875039492722,354.6605224609375]},{"page":190,"text":"constant. For this reason, the diffusion in anisotropic media is described by a","rect":[53.813812255859378,375.5545959472656,385.1566547222313,366.62005615234377]},{"page":190,"text":"simplest second rank tensor Dij:","rect":[53.813812255859378,388.4313049316406,182.4189592129905,378.57958984375]},{"page":190,"text":"@c","rect":[150.95936584472657,409.0482482910156,161.1295856304344,401.71734619140627]},{"page":190,"text":"Ji ¼ \u0002Dij @xj Dij ¼ D?dij þ Daninj Da ¼ Djj \u0002 D?","rect":[108.64531707763672,425.40130615234377,329.828744046973,408.4923400878906]},{"page":190,"text":"Microscopically, the diffusion in the isotropic and the nematic phase is thermally","rect":[65.76496887207031,448.9568176269531,385.1508721452094,440.02227783203127]},{"page":190,"text":"activated. However, in this case, the Arrhenius-type process with activation energy","rect":[53.812950134277347,460.9163818359375,385.16774836591255,451.9818115234375]},{"page":190,"text":"DE is not related to the orientational potential Wcosy. In fact, this process is","rect":[53.812950134277347,472.8759460449219,385.1299172793503,463.6127014160156]},{"page":190,"text":"controlled by another potential barrier, namely, the barrier for translational jumps","rect":[53.81296157836914,484.7787170410156,385.1269875918503,475.84417724609377]},{"page":190,"text":"of a molecule from site to site:","rect":[53.81296157836914,494.7162780761719,177.8832383877952,487.8037109375]},{"page":190,"text":"Di / Ai exp\u0007\u0002kDBET\b","rect":[176.1661376953125,534.8617553710938,262.8395506892778,510.9765625]},{"page":190,"text":"Coefficient Ai, however, depends on molecular orientation function.","rect":[65.76607513427735,558.3507080078125,339.3960232308078,549.4160766601563]},{"page":190,"text":"Recall the Stokes law, related the force (F) acting on a sphere of radius R to","rect":[65.7657241821289,570.3102416992188,385.14458552411568,561.375732421875]},{"page":190,"text":"velocity of the sphere in a viscous liquid, v ¼ F/4pZR (Z is viscosity of the liquid).","rect":[53.814697265625,582.269775390625,385.1426052620578,573.3352661132813]},{"page":190,"text":"Roughly, by analogy with the sphere in viscous liquid, the diffusion coefficient is","rect":[53.81568145751953,594.2293090820313,385.18637479888158,585.2947998046875]},{"page":191,"text":"7.3 Transport Properties","rect":[53.81306076049805,44.275169372558597,136.4184921550683,36.68081283569336]},{"page":191,"text":"a","rect":[97.65629577636719,68.23348999023438,103.21155817360104,62.64473342895508]},{"page":191,"text":"n","rect":[124.447021484375,97.6859130859375,128.8580344410731,93.8878173828125]},{"page":191,"text":"dye","rect":[310.6985168457031,91.9854965209961,322.60505387588207,84.5492172241211]},{"page":191,"text":"glasses","rect":[318.49896240234377,110.3102035522461,341.3850849504088,102.8739242553711]},{"page":191,"text":"175","rect":[372.4984130859375,42.55649185180664,385.19003814845009,36.6300163269043]},{"page":191,"text":"spacer","rect":[311.4028015136719,154.6715850830078,332.96241649337756,149.1943359375]},{"page":191,"text":"Fig. 7.20","rect":[53.812843322753909,180.82974243164063,86.12312835841104,173.0999298095703]},{"page":191,"text":"Anisotropy of diffusion amplitudes Ai for a cylindrical molecule (a) and a simple","rect":[92.1051025390625,180.76202392578126,385.16112658762457,173.16766357421876]},{"page":191,"text":"technique for measurements of anisotropy of diffusion coefficients D||/D⊥ for a dye dissolved ina","rect":[53.812923431396487,190.73797607421876,385.17479846262457,183.1435546875]},{"page":191,"text":"nematic liquid crystal","rect":[53.81306076049805,200.65716552734376,127.98533348288599,193.06280517578126]},{"page":191,"text":"proportional to the corresponding molecular dimension. In the nematic phase a","rect":[53.812843322753909,228.52511596679688,385.15869939996568,219.59056091308595]},{"page":191,"text":"cylindrical molecule of length L and diameter D is typically aligns parallel to the","rect":[53.812843322753909,240.48468017578126,385.17456854059068,231.5501251220703]},{"page":191,"text":"director as in Fig. 7.20a. Then the diffusion is also easier (faster) along the director,","rect":[53.8138313293457,252.4442138671875,385.1039089729953,243.50965881347657]},{"page":191,"text":"A||/1/pD > A⊥/1/(LD)1/2. Consequently, the ratio D||/D⊥/L/D > 1 and anisot-","rect":[53.8138313293457,264.40411376953127,385.1733335098423,253.31317138671876]},{"page":191,"text":"ropy Da ¼ D||-D⊥ > 0. Evidently, the order parameter influences the anisotropy","rect":[53.8136100769043,276.3637390136719,385.1464776139594,267.42919921875]},{"page":191,"text":"of diffusion; roughly Da/S and for S ! 0, D|| ¼ D⊥ ¼ Diso. For nematics at","rect":[53.813602447509769,288.3232727050781,385.1255937344749,279.38873291015627]},{"page":191,"text":"room temperature, typical values of diffusion coefficient are Diso \u0004 10\u00026 cm2/s,","rect":[53.813655853271487,300.282958984375,385.1852993538547,289.19219970703127]},{"page":191,"text":"D||/D⊥ \u0004 1–2.","rect":[53.81462860107422,311.74493408203127,111.56441922690158,303.30810546875]},{"page":191,"text":"For smectics the situation is different, because an additional potential barrier Wtr","rect":[65.76509857177735,324.14544677734377,385.1742999588283,315.21087646484377]},{"page":191,"text":"(for translations) appears for molecules penetrating smectic layers. For instance, in","rect":[53.812843322753909,336.1053161621094,385.1387566666938,327.1707763671875]},{"page":191,"text":"a smectic A, the component parallel to the layers (D⊥) follows the same Arrhenius","rect":[53.812843322753909,348.0648498535156,385.1805459414597,339.13031005859377]},{"page":191,"text":"law with approximately same activation energy DE as for nematics, however, for","rect":[53.813838958740237,360.02447509765627,385.1815122207798,350.76123046875]},{"page":191,"text":"the D|| component, the activation energy is roughly DE þ Wtr and the diffusion","rect":[53.814842224121097,371.9842529296875,385.12453547528755,363.0198059082031]},{"page":191,"text":"anisotropy becomes negative:","rect":[53.813533782958987,383.94378662109377,173.328749252053,375.00921630859377]},{"page":191,"text":"DD?jj / DL exp\u0007\u0002kUBtTr \b<1;Djj \u0002 D?<0","rect":[142.00942993164063,422.0102844238281,298.62538996747505,398.12506103515627]},{"page":191,"text":"The diffusion coefficient can be measured by several techniques. One of them is","rect":[65.76558685302735,445.5561828613281,385.1843301211472,436.6017150878906]},{"page":191,"text":"very simple, see Fig. 7.20b. A small amount of a dye solution in a liquid crystal is","rect":[53.81356430053711,457.5157165527344,385.1861916934128,448.5811767578125]},{"page":191,"text":"introduced through a hole in a top glass of a sandwich cell filled with the same","rect":[53.81456756591797,469.4184875488281,385.12854803277818,460.48394775390627]},{"page":191,"text":"liquid crystal. The latter is oriented homogeneously, therefore, using a microscope,","rect":[53.81456756591797,481.3780212402344,385.18133206869848,472.4434814453125]},{"page":191,"text":"one can observe the diffusion of dye parallel and perpendicular to the director. After","rect":[53.81456756591797,493.33758544921877,385.1345456680454,484.40301513671877]},{"page":191,"text":"some time tD, a dye stain acquires an elliptic form and the ratio of ellipse axes","rect":[53.81456756591797,505.2975158691406,385.1057168398972,496.36260986328127]},{"page":191,"text":"provides the ratio of diffusion coefficients (l||/l⊥)2 ¼ D||/D⊥. The absolute value of,","rect":[53.812644958496097,517.2570190429688,385.14788480307348,506.20623779296877]},{"page":191,"text":"e.g., D|| can be found from the well known solution of the diffusion equation,","rect":[53.814022064208987,529.216796875,385.1704067757297,520.2822875976563]},{"page":191,"text":"tjDj ¼ lj2j=2Djj. In the same way, a small amount of a cholesteric liquid crystal can","rect":[53.8126220703125,546.8292236328125,385.12505427411568,531.5454711914063]},{"page":191,"text":"be introduced into a nematic and a spot is observed, in which the initially homoge-","rect":[53.81303405761719,558.3505249023438,385.11602149812355,549.416015625]},{"page":191,"text":"neous texture is substituted by an inhomogeneous (e.g., fingerprint) cholesteric","rect":[53.81303405761719,570.31005859375,385.10413397027818,561.3755493164063]},{"page":191,"text":"texture. The self-diffusion of liquid crystal molecules is studied using quasi-elastic","rect":[53.81303405761719,582.2696533203125,385.1210407085594,573.3351440429688]},{"page":191,"text":"neutron scattering or a spin-echo technique.","rect":[53.81303405761719,594.2291870117188,230.24941678549534,585.294677734375]},{"page":192,"text":"176","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.68026351928711]},{"page":192,"text":"7.3.3 Electric Conductivity","rect":[53.812843322753909,69.93675231933594,194.28999122114389,59.298980712890628]},{"page":192,"text":"7 Magnetic, Electric and Transport Properties","rect":[228.78945922851563,44.274620056152347,385.1482590007714,36.68026351928711]},{"page":192,"text":"7.3.3.1 Mobility of Ions","rect":[53.812843322753909,97.87738800048828,159.9177590762253,88.2256851196289]},{"page":192,"text":"Now we are dealing with the current term 4psE=c in the Maxwell equation","rect":[53.812843322753909,121.72679138183594,385.17653742841255,111.79619598388672]},{"page":192,"text":"(7.11) for curlH. The overwhelming majority of liquid crystalline phases (nematic,","rect":[53.81282424926758,133.35763549804688,385.17559476401098,124.42308044433594]},{"page":192,"text":"smectic A, C, B, etc.) may be considered as weak electrolytes. The charge carriers","rect":[53.81282043457031,145.31716918945313,385.1944924746628,136.36268615722657]},{"page":192,"text":"are ions, which move rather slowly in the electric field. There are some interesting","rect":[53.812843322753909,157.27670288085938,385.0900506120063,148.34214782714845]},{"page":192,"text":"publications concerning columnar discotic phases, in which high mobility of charge","rect":[53.812843322753909,169.23623657226563,385.12180364801255,160.3016815185547]},{"page":192,"text":"carriers has been reported. In principle, in such well-ordered columns of organic","rect":[53.812843322753909,181.13900756835938,385.17856634332505,172.20445251464845]},{"page":192,"text":"molecules, the electron or hole conductivity is possible. Electronic processes are","rect":[53.812843322753909,193.09857177734376,385.1606830425438,184.1640167236328]},{"page":192,"text":"faster than ionic ones and may be studied by a time-of-flight technique. However,","rect":[53.812843322753909,205.05810546875,385.1507229378391,196.12355041503907]},{"page":192,"text":"below we shall consider only ionic processes as the most important issue for major","rect":[53.812843322753909,217.01763916015626,385.09990821687355,208.0830841064453]},{"page":192,"text":"number of mesophases [14].","rect":[53.812843322753909,228.97720336914063,167.80747647787815,220.0426483154297]},{"page":192,"text":"The electric conductivity of a liquid is related to the drift velocity vE of ions with","rect":[65.76485443115235,240.93673706054688,385.11867610028755,232.00218200683595]},{"page":192,"text":"a charge qi moved by field E. The current density depends on concentration of ions","rect":[53.813716888427737,252.8978271484375,385.1717873965378,243.94334411621095]},{"page":192,"text":"nv in a unit volume of the liquid:","rect":[53.81304931640625,264.85748291015627,186.26914842197489,255.9229278564453]},{"page":192,"text":"j ¼ nqivE","rect":[200.63705444335938,288.69989013671877,237.82625054486813,280.00439453125]},{"page":192,"text":"(7.38)","rect":[361.0561828613281,288.3216552734375,385.10555396882668,279.8453063964844]},{"page":192,"text":"Units of j: [cm-3 \u0003 CGSQ \u0003 cm/s] ¼ [CGSI/cm2] or [A/m2] in the SI system.","rect":[65.76653289794922,312.639404296875,383.5964626839328,301.5884704589844]},{"page":192,"text":"In the linear regime, vE ¼ mE where coefficient m is called ion mobility. Hence,","rect":[65.76648712158203,324.5989685058594,385.1070828011203,315.6445617675781]},{"page":192,"text":"the conductivity is","rect":[53.812068939208987,336.53863525390627,127.28310026274875,327.6040954589844]},{"page":192,"text":"s ¼ j=E ¼ qimn","rect":[188.68365478515626,363.3811340332031,250.30739680341254,350.799072265625]},{"page":192,"text":"(7.39)","rect":[361.05596923828127,362.1201477050781,385.1053403457798,353.643798828125]},{"page":192,"text":"The mobility of ions and their diffusion coefficient are coupled by the Einstein","rect":[65.76631927490235,386.9476318359375,385.1750420670844,378.0130615234375]},{"page":192,"text":"relationship:","rect":[53.814292907714847,398.7976379394531,104.92916734287332,389.9527282714844]},{"page":192,"text":"m ¼ qi=kBTD","rect":[194.34982299804688,425.2765197753906,244.60363531067697,413.1058654785156]},{"page":192,"text":"(7.40)","rect":[361.0559387207031,422.4851379394531,385.10530982820168,414.0087890625]},{"page":192,"text":"and both of them depend on viscosity of a liquid. Note that the energy kBT at room","rect":[53.814292907714847,448.7867736816406,385.15897320390305,439.8323059082031]},{"page":192,"text":"temperature is about 0.025 eV and the factor kBT/q has dimension of voltage. For","rect":[53.81417465209961,460.7464294433594,385.1516355117954,451.7519226074219]},{"page":192,"text":"ions with charge qi ¼ e (charge of an electron) and typical diffusion coefficient of","rect":[53.81379318237305,472.7060852050781,385.1512387832798,463.77142333984377]},{"page":192,"text":"organic liquid D \u0004 5 \u0003 10\u00027 cm2/s, the ion mobility m \u0004 6 \u0003 10\u00023 cm2/statV.s (or","rect":[53.81337356567383,484.6657409667969,385.18169532624855,473.61468505859377]},{"page":192,"text":"2 \u0003 10\u00029 m2/V\u0005s in the SI system).","rect":[53.814022064208987,496.6253662109375,195.9603695442844,485.5743408203125]},{"page":192,"text":"In a liquid crystal, the anisotropy of diffusion results in an anisotropy of mobility","rect":[65.76526641845703,508.5849304199219,385.1640252213813,499.650390625]},{"page":192,"text":"and, consequently, conductivity. The corresponding tensor for a uniaxial phase has","rect":[53.8132438659668,520.54443359375,385.1301614199753,511.60992431640627]},{"page":192,"text":"a standard form:","rect":[53.8132438659668,530.4819946289063,119.89028794834207,523.5694580078125]},{"page":192,"text":"sij ¼ s?dij þ saninj","rect":[177.58290100097657,557.2409057617188,260.93961602013015,547.1336059570313]},{"page":192,"text":"Like in the case of diffusion, the anisotropy of conductivity sa ¼ sjj \u0002 s?can be","rect":[65.76496887207031,582.14111328125,385.1534808941063,571.7652587890625]},{"page":192,"text":"positive (e.g., in conventional nematics) or negative (in smectic A, discotic","rect":[53.81462860107422,592.6991577148438,385.10764349176255,583.7646484375]},{"page":193,"text":"7.3 Transport Properties","rect":[53.812843322753909,44.274620056152347,136.41827853202143,36.68026351928711]},{"page":193,"text":"177","rect":[372.4981994628906,42.55594253540039,385.1898245254032,36.73106384277344]},{"page":193,"text":"mesophases). In typical nematics at room temperature, sa \u0004 1.2–1.6 depending on","rect":[53.812843322753909,68.2883529663086,385.1722039323188,59.35380554199219]},{"page":193,"text":"the type of ions and their concentration. Generally, sa is proportional to the","rect":[53.814430236816409,80.24800872802735,385.11850774957505,71.31333923339844]},{"page":193,"text":"orientational order parameter, as for other transport phenomena under discussion.","rect":[53.813533782958987,92.20760345458985,382.1632046272922,83.27305603027344]},{"page":193,"text":"7.3.3.2 Ion Concentration","rect":[53.813533782958987,126.28636169433594,169.79999131510807,118.77616119384766]},{"page":193,"text":"Where do ions emerge from? Their sources can be different:","rect":[53.813533782958987,151.94851684570313,297.8121782071311,143.0139617919922]},{"page":193,"text":"1.","rect":[53.813533782958987,168.0,61.27919049765353,160.98468017578126]},{"page":193,"text":"2.","rect":[53.813533782958987,192.0,61.27919049765353,184.90377807617188]},{"page":193,"text":"3.","rect":[53.813533782958987,204.0,61.27919049765353,196.86328125]},{"page":193,"text":"a residual concentration of ionic impurities can remain after the synthesis of a","rect":[66.27520751953125,169.85946655273438,385.1564105816063,160.92491149902345]},{"page":193,"text":"substance","rect":[66.27521514892578,179.75717163085938,105.10758245660627,172.8844451904297]},{"page":193,"text":"a liquid crystal material can deliberately be doped with some compounds","rect":[66.27520751953125,193.778564453125,361.61862577544408,184.84400939941407]},{"page":193,"text":"ions can be created by an external electric field either in the bulk (due to the field","rect":[66.27520751953125,205.73806762695313,385.11757746747505,196.8035125732422]},{"page":193,"text":"ionization of neutral molecules) or at the electrodes. The latter is more probable:","rect":[66.27521514892578,217.6976318359375,385.1355424649436,208.76307678222657]},{"page":193,"text":"electrons or holes are injected from an electrode and almost immediately (within","rect":[66.27521514892578,229.65716552734376,385.1026238541938,220.7226104736328]},{"page":193,"text":"10\u00025 to 10\u00028 s) trapped by neutral molecules of a liquid crystal forming negative","rect":[66.27521514892578,241.61834716796876,385.1725238628563,230.43104553222657]},{"page":193,"text":"or positive ions. Before their recombination the ions participate in the electric","rect":[66.27546691894531,253.52108764648438,385.15369451715318,244.58653259277345]},{"page":193,"text":"current","rect":[66.27546691894531,263.4188232421875,94.55538041660378,257.56207275390627]},{"page":193,"text":"Consider a doping process. Let an organic salt AB of volume concentration c","rect":[65.76580810546875,283.4483337402344,385.16159856988755,274.5137939453125]},{"page":193,"text":"[cm\u00023] is introduced into an isotropic solvent. The salt will dissociate to yield","rect":[53.81382369995117,295.3517761230469,385.1291436295844,284.30072021484377]},{"page":193,"text":"anions A\u0002 and cations B+ with a subsequent recombination, according to the","rect":[53.81312942504883,307.31146240234377,385.11521185113755,297.34942626953127]},{"page":193,"text":"reaction AB,A\u0002 þ B+. Then the mass action law reads:","rect":[53.81320571899414,317.94635009765627,284.5856157071311,309.3089599609375]},{"page":193,"text":"KDcð1 \u0002 aÞ ¼ KRðacÞ2","rect":[173.39111328125,343.47235107421877,265.07127449592908,331.5177307128906]},{"page":193,"text":"Here the rate of the ion dissociation is on the left-hand side (a is the degree of","rect":[65.76496887207031,365.0692138671875,385.15081153718605,356.1346435546875]},{"page":193,"text":"ionization), and the rate of their bimolecular recombination is on the right-hand side","rect":[53.81296157836914,377.0287780761719,385.1527484722313,368.09423828125]},{"page":193,"text":"(ac ¼ nv+¼nv\u0002 is a volume concentration of ions), KD and KR are corresponding","rect":[53.81296157836914,388.9883728027344,385.11724177411568,379.026611328125]},{"page":193,"text":"dissociation and recombination constants. The temperature dependent ionization","rect":[53.814231872558597,400.94793701171877,385.17897883466255,392.01336669921877]},{"page":193,"text":"coefficient can be written as follows:","rect":[53.814231872558597,412.83074951171877,201.771620345803,403.87628173828127]},{"page":193,"text":"K ¼ KKDR ¼ 1a\u00022ca or c ¼ Kð1a\u00022 aÞ","rect":[157.92831420898438,439.6734619140625,281.0930219258302,425.1485595703125]},{"page":193,"text":"and the degree of ionization is given by","rect":[53.81450653076172,460.23577880859377,213.98970881513129,451.30120849609377]},{"page":193,"text":"(7.41)","rect":[361.0561828613281,436.65533447265627,385.10555396882668,428.1789855957031]},{"page":193,"text":"1.","rect":[53.81418991088867,532.0,61.27984662558322,524.8760375976563]},{"page":193,"text":"2.","rect":[53.814491271972659,568.0,61.2801479866672,560.7553100585938]},{"page":193,"text":"\u0002K þ Kð1 þ 4c=KÞ1=2","rect":[182.22743225097657,484.040771484375,272.8315589197572,471.5771789550781]},{"page":193,"text":"a¼","rect":[163.98870849609376,488.3751220703125,179.41376522276313,483.7434997558594]},{"page":193,"text":"2c","rect":[223.06858825683595,495.3133239746094,232.50319708062973,488.4306335449219]},{"page":193,"text":"(7.42)","rect":[361.0558776855469,490.0481872558594,385.1052487930454,481.57183837890627]},{"page":193,"text":"Consider three particular cases:","rect":[65.7662124633789,515.83984375,192.53803880283426,506.90533447265627]},{"page":193,"text":"For very small concentration of a dopant c ! 0 and the first term of the square","rect":[66.27586364746094,533.7507934570313,385.13013494684068,524.8162841796875]},{"page":193,"text":"root expansion [1 þ (2c/K)] results in a ! 1. The recombination is absent and","rect":[66.27588653564453,545.7103271484375,385.14507380536568,536.7758178710938]},{"page":193,"text":"the concentration of ions is nv+¼nv\u0002 ¼c","rect":[66.27689361572266,557.2325439453125,230.63411749078598,547.708740234375]},{"page":193,"text":"For higher concentration of salt, the situation depends on the ionization coef-","rect":[66.27616882324219,569.6300659179688,385.13045631257668,560.695556640625]},{"page":193,"text":"ficient. If K is large (strong electrolytes) then again 4c/K << 1, a ! 1 and","rect":[66.27617645263672,581.589599609375,385.1463555436469,572.6550903320313]},{"page":193,"text":"+","rect":[71.25965118408203,586.8837890625,75.45526672843627,583.53076171875]},{"page":193,"text":"n ¼n ¼c","rect":[66.27617645263672,591.5872802734375,119.32622564508283,586.885986328125]},{"page":194,"text":"178","rect":[53.8120002746582,42.55759048461914,66.50360626368448,36.73271179199219]},{"page":194,"text":"7 Magnetic, Electric and Transport Properties","rect":[228.78860473632813,44.276268005371097,385.14743502616207,36.68191146850586]},{"page":194,"text":"3. If, however, K is small (weak electrolyte), 4c/K >> 1 the numerator of Eq. 7.42","rect":[53.812843322753909,68.2883529663086,385.15960017255318,59.35380554199219]},{"page":194,"text":"is approximately equal to \u0002K þ (4cK)1/2, the degree of ionization a¼\u0002K/2 þ","rect":[66.27552032470703,80.24800872802735,385.1472552129975,69.15728759765625]},{"page":194,"text":"(K/c)1/2 and","rect":[66.27509307861328,91.8091812133789,113.58944026044378,81.11688232421875]},{"page":194,"text":"ffiffiffiffiffi","rect":[238.87330627441407,108.0,249.22568810537588,106.0]},{"page":194,"text":"nþv ¼ nv\u0002 ¼ ca ¼ pKc \u0002 K=2","rect":[159.85264587402345,118.86640167236328,279.14104548505318,106.88935089111328]},{"page":194,"text":"(7.43)","rect":[361.0572204589844,117.7691421508789,385.1065915664829,109.29277801513672]},{"page":194,"text":"Figure 7.21 illustrates a dependence of the ion concentration on concentration of","rect":[65.7676010131836,148.32192993164063,385.15142188874855,139.3873748779297]},{"page":194,"text":"a salt. It is well seen how the linear dependence becomes sub-linear. A typical value","rect":[53.815574645996097,160.28146362304688,385.1435016460594,151.34690856933595]},{"page":194,"text":"of K is on the order of 1019 cm\u00023.","rect":[53.815574645996097,170.21900939941407,191.06031461264377,161.1900177001953]},{"page":194,"text":"7.3.3.3 Current-Voltage Curve for Thin Cells","rect":[53.813655853271487,214.15069580078126,254.25963224517063,204.81771850585938]},{"page":194,"text":"When investigating electro-optical effects, usually we have to deal with the layer","rect":[53.813655853271487,237.9901123046875,385.1186460098423,229.05555725097657]},{"page":194,"text":"thickness of a liquid crystals in the range of 1–50 mm and with the electric field","rect":[53.813655853271487,249.94964599609376,385.1176385026313,240.955322265625]},{"page":194,"text":"strengths of 104 to 105 V/cm (\u000430–300 statV/cm in the Gauss system). In such","rect":[53.813655853271487,261.9101257324219,385.12481013349068,250.7227020263672]},{"page":194,"text":"instances the field induced drift of charge carriers to electrodes cannot be neglected.","rect":[53.81283950805664,273.8128967285156,385.11687894369848,264.87835693359377]},{"page":194,"text":"However, in many cases, we may still neglect electro-chemical processes at the","rect":[53.81283950805664,285.7724304199219,385.17063177301255,276.837890625]},{"page":194,"text":"electrodes.","rect":[53.81283950805664,295.7000427246094,96.88470120932345,288.79742431640627]},{"page":194,"text":"Assume the simplest ionization-recombination model with field independent KD","rect":[65.76485443115235,309.6915283203125,385.16602093858895,300.7569580078125]},{"page":194,"text":"and KR coefficients in a weak or intermediate electric field. The strong field limit","rect":[53.812843322753909,321.6517028808594,385.1122880704124,312.7171630859375]},{"page":194,"text":"will be discussed separately. In the case of a << 1, the volume ion concentration","rect":[53.81324768066406,333.6112365722656,385.1182793717719,324.67669677734377]},{"page":194,"text":"nv+ ¼ nv\u0002 ¼ nv (cm\u00023) is governed by equation","rect":[53.812278747558597,345.5709533691406,245.94428340009223,334.5199279785156]},{"page":194,"text":"ddntv ¼ KDc \u0002 KRn2v \u0002 Eðmþdþ m\u0002Þnv;","rect":[144.6151123046875,381.07476806640627,294.35353028458499,359.8216857910156]},{"page":194,"text":"(7.44)","rect":[361.0556945800781,375.8369140625,385.10506568757668,367.3605651855469]},{"page":194,"text":"where d is the gap between the electrodes, m+ and m\u0002 are mobilities of the positive","rect":[53.81403732299805,404.5758972167969,385.1262897319969,394.6135559082031]},{"page":194,"text":"and negative ions, and c is the concentration of a dopant. The third term on the","rect":[53.81427764892578,416.5352478027344,385.1730426616844,407.6007080078125]},{"page":194,"text":"right-hand side describes the process of ion drift to the electrodes. It has a typical","rect":[53.81427764892578,428.4947814941406,385.1272416836936,419.56024169921877]},{"page":194,"text":"form of –nv/t [cm\u00023s\u00021].","rect":[53.81427764892578,440.0166320800781,154.61019559408909,429.4037170410156]},{"page":194,"text":"n+, n–","rect":[241.98805236816407,472.1163330078125,259.97988931095287,464.21234130859377]},{"page":194,"text":"n+, n–=÷Kc","rect":[332.02386474609377,495.6187744140625,368.3902021707577,487.01287841796877]},{"page":194,"text":"Fig. 7.21 Qualitative","rect":[53.812843322753909,559.9110107421875,128.68570849680425,552.1812133789063]},{"page":194,"text":"dependence of either positive","rect":[53.812843322753909,569.8192138671875,154.31513354563237,562.224853515625]},{"page":194,"text":"or negative ion concentration","rect":[53.812843322753909,579.7384643554688,153.8565420547001,572.1441040039063]},{"page":194,"text":"on concentration c of a salt","rect":[53.812843322753909,588.0211791992188,146.83553795065942,582.1200561523438]},{"page":194,"text":"n+,n–= c","rect":[247.45750427246095,544.7333374023438,276.8100630106014,536.8285522460938]},{"page":194,"text":"4c/K=1","rect":[304.1485290527344,580.1480712890625,330.04173318302318,574.1123657226563]},{"page":195,"text":"7.3 Transport Properties","rect":[53.812843322753909,44.277061462402347,136.41827853202143,36.68270492553711]},{"page":195,"text":"179","rect":[372.4981994628906,42.62611389160156,385.1898245254032,36.73350524902344]},{"page":195,"text":"In the steady-state regime, the same equation can be written in the form","rect":[65.76496887207031,68.2883529663086,355.8505930769499,59.35380554199219]},{"page":195,"text":"dnv","rect":[169.36904907226563,90.86579132080078,182.45321030512234,82.51631927490235]},{"page":195,"text":"c","rect":[198.93785095214845,89.47869110107422,203.38738287164535,84.8072280883789]},{"page":195,"text":"nv 2nv","rect":[218.48036193847657,90.86579132080078,252.5232481469192,82.59600067138672]},{"page":195,"text":"¼\u0002\u0002","rect":[185.73980712890626,94.74922943115235,237.2475512335053,92.4184799194336]},{"page":195,"text":"¼0","rect":[255.8094940185547,96.19392395019531,271.26638117841255,89.34111785888672]},{"page":195,"text":"dt","rect":[172.31455993652345,103.11203002929688,180.0888047940452,96.1197738647461]},{"page":195,"text":"tD tR","rect":[196.2193603515625,104.40643310546875,226.83711480268063,98.4999008178711]},{"page":195,"text":"tT","rect":[241.7048797607422,104.40643310546875,249.91291580327136,98.50032806396485]},{"page":195,"text":"(7.45)","rect":[361.05633544921877,97.8174819946289,385.1057065567173,89.22159576416016]},{"page":195,"text":"where tD ¼ 1/KD, tR ¼ 1/KRnv, tT ¼ 2d/(m++m\u0002)E are characteristic times for","rect":[53.81368637084961,127.88370513916016,385.1802915176548,118.01117706298828]},{"page":195,"text":"ionization of molecules, recombination of ions and ion transit to electrodes. Con-","rect":[53.81456756591797,138.0,385.1445249160923,130.9983367919922]},{"page":195,"text":"sider again three particular cases.","rect":[53.81456756591797,151.89242553710938,188.1137661507297,142.95787048339845]},{"page":195,"text":"In the low field regime (region 1), tT is large, the third term may be neglected.","rect":[65.76659393310547,163.89183044433595,385.1184963753391,154.87759399414063]},{"page":195,"text":"Then from (7.45) we have the previous result n2v ¼ ðKD=KRÞc ¼ Kc and conductiv-","rect":[53.813533782958987,176.453857421875,385.1386655410923,165.32955932617188]},{"page":195,"text":"ity (7.39) is field independent:","rect":[53.814781188964847,187.71481323242188,175.81358201572489,178.78025817871095]},{"page":195,"text":"ffiffiffiffiffi","rect":[253.77037048339845,203.0,264.12275231436026,202.0]},{"page":195,"text":"s1 ¼ qðmþ þ m\u0002ÞpKc","rect":[174.12835693359376,214.07003784179688,264.8494037456688,202.453125]},{"page":195,"text":"(7.46)","rect":[361.0558776855469,213.3329620361328,385.1052487930454,204.7968292236328]},{"page":195,"text":"This corresponds to region 1 of the current-voltage curve in Fig. 7.22.","rect":[65.76622772216797,237.65048217773438,348.2507595589328,228.71592712402345]},{"page":195,"text":"For intermediate fields (region 2), the drift term is important but the recombina-","rect":[65.76619720458985,249.55325317382813,385.06646095124855,240.578857421875]},{"page":195,"text":"tion rate may be neglected, because the field rapidly removes the generated ions.","rect":[53.81418991088867,261.5128173828125,385.1679348519016,252.57826232910157]},{"page":195,"text":"Now the ion concentration is given by","rect":[53.81418991088867,273.47235107421877,208.26868525556098,264.53778076171877]},{"page":195,"text":"tT","rect":[191.0075225830078,295.9876708984375,199.21543655522448,290.08148193359377]},{"page":195,"text":"dKDc","rect":[235.8704071044922,295.9876708984375,257.4834674663719,287.700927734375]},{"page":195,"text":"nv ¼ 2tD c ¼ ðmþ þ m\u0002ÞE","rect":[166.31069946289063,310.59747314453127,271.04172509587968,296.7935791015625]},{"page":195,"text":"and the apparent conductivity is field dependent","rect":[53.8139762878418,334.17791748046877,247.28303392002176,325.24334716796877]},{"page":195,"text":"qdKDc","rect":[216.72433471679688,357.2510070800781,243.32212103082504,348.4060974121094]},{"page":195,"text":"s2¼","rect":[194.01007080078126,363.4944763183594,213.90969112608344,357.5212707519531]},{"page":195,"text":"E","rect":[226.97731018066407,368.82208251953127,233.0891951886531,362.2481689453125]},{"page":195,"text":"(7.47)","rect":[361.0555419921875,363.70635986328127,385.10491309968605,355.2300109863281]},{"page":195,"text":"j","rect":[271.13006591796877,430.3546447753906,272.90543554559567,422.9366149902344]},{"page":195,"text":"Fig. 7.22 Current-voltage","rect":[53.812843322753909,501.07635498046877,144.6348204352808,493.3465270996094]},{"page":195,"text":"curve for a thin layer ofa","rect":[53.812843322753909,510.9845886230469,141.59984729074956,503.3902282714844]},{"page":195,"text":"weak electrolyte between","rect":[53.812843322753909,520.9605102539063,140.6437277236454,513.3661499023438]},{"page":195,"text":"plane electrodes. Solid","rect":[53.812843322753909,530.8797607421875,131.46008819727823,523.285400390625]},{"page":195,"text":"curve corresponds to a direct","rect":[53.812843322753909,540.855712890625,152.8234224721438,533.2613525390625]},{"page":195,"text":"current (d.c.) in weak (1),","rect":[53.812843322753909,550.4930419921875,141.75467178418598,543.2373046875]},{"page":195,"text":"intermediate (2) and strong","rect":[53.812843322753909,560.8076171875,146.78394073634073,553.2132568359375]},{"page":195,"text":"(3) field regimes. Dash branch","rect":[53.812843322753909,570.7268676757813,155.33975738673136,563.1325073242188]},{"page":195,"text":"is a part of the same curve for","rect":[53.812843322753909,580.7028198242188,155.33129972083263,573.1084594726563]},{"page":195,"text":"an alternating (a.c.) field","rect":[53.812843322753909,590.6787719726563,137.86681121973917,583.0844116210938]},{"page":195,"text":"1","rect":[286.41143798828127,555.087646484375,290.8578592178333,549.4620971679688]},{"page":195,"text":"tgα = σ","rect":[295.6777648925781,576.4271240234375,320.6049110633536,569.3211669921875]},{"page":195,"text":"a.c.","rect":[324.2140197753906,500.8023986816406,337.1054432807917,496.38519287109377]},{"page":195,"text":"d.c.","rect":[362.9346923828125,525.066650390625,375.82611588821359,519.208984375]},{"page":196,"text":"180","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.73106384277344]},{"page":196,"text":"7 Magnetic, Electric and Transport Properties","rect":[228.78945922851563,44.274620056152347,385.1482590007714,36.68026351928711]},{"page":196,"text":"Therefore, with increasing field, the current density saturates at the value","rect":[65.76496887207031,68.2883529663086,360.5181354351219,59.35380554199219]},{"page":196,"text":"j ¼ s2E ¼ qdKDc","rect":[183.75619506835938,91.16754150390625,255.21758306695785,82.23299407958985]},{"page":196,"text":"This regime is designated as region 2 in the same figure. The order of magnitude","rect":[65.76607513427735,113.1224594116211,385.10315740777818,104.18791198730469]},{"page":196,"text":"of the ion transit time is tT ¼ d2/mU \u0004 2.5ms (d ¼ 10 mm, U ¼ 10 V, m ¼ 4.10\u00029","rect":[53.81405258178711,124.99295806884766,385.1809962305198,113.97502136230469]},{"page":196,"text":"m2/Vs).","rect":[53.812843322753909,136.58714294433595,84.94843717123752,125.93467712402344]},{"page":196,"text":"For the a.c. current at angular frequency o > 1/tT, the drift term may be","rect":[65.76468658447266,148.94522094726563,385.1535114116844,140.01060485839845]},{"page":196,"text":"neglected from the beginning. Then, according to Eq. 7.45, the ohmic regime","rect":[53.81368637084961,160.90472412109376,385.16147649957505,151.910400390625]},{"page":196,"text":"with constant KD and KR is valid not only for weak but for intermediate fields as","rect":[53.81367111206055,172.864501953125,385.1389504824753,163.9297332763672]},{"page":196,"text":"well. In Fig. 7.22, it is pictured by the dash line.","rect":[53.813053131103519,184.82403564453126,248.56417508627659,175.8894805908203]},{"page":196,"text":"In a strong field, coefficient KD becomes a function of E due to the field induced","rect":[65.76506805419922,196.8234405517578,385.12664118817818,187.8092041015625]},{"page":196,"text":"ionization of molecules. Then we cross again the recombination term in (7.45) and","rect":[53.81370162963867,208.74334716796876,385.14556208661568,199.7490234375]},{"page":196,"text":"get","rect":[53.81368637084961,220.70291137695313,66.03745896884988,212.78433227539063]},{"page":196,"text":"KDðEÞc \u0002 Eðmþdþ m\u0002Þ nv ¼ 0 or nvðEÞ ¼ ðmdþKþDðmE\u0002ÞcÞE","rect":[117.14237976074219,253.17259216308595,321.83110283979,234.9539337158203]},{"page":196,"text":"and the current acquires the form","rect":[53.812843322753909,276.7037658691406,187.78556011796554,267.76922607421877]},{"page":196,"text":"j ¼ qEðmþ þ m\u0002ÞnvðEÞ ¼ qcdKDðEÞ:","rect":[145.23728942871095,301.6418762207031,293.7313989369287,291.0158386230469]},{"page":196,"text":"It depends on the field only implicitly through dissociation constant KD.","rect":[65.76703643798828,325.2223205566406,356.0109829476047,316.28778076171877]},{"page":196,"text":"Further, the zero-field dissociation constant is described by the equation","rect":[65.7661361694336,337.18206787109377,356.1742791276313,328.24749755859377]},{"page":196,"text":"KDðE ¼ 0Þ ¼ KD0 exp\u0007\u0002kWBT\b","rect":[157.07809448242188,375.2485656738281,281.8722807918169,351.36334228515627]},{"page":196,"text":"where W is the electrostatic binding energy of the ion pair. Now, if we assume that","rect":[53.8133659362793,398.7941589355469,385.137343002053,389.859619140625]},{"page":196,"text":"the field reduces the energy barrier by DW \u0004 constE1/2 (exactly as in the Schottky","rect":[53.8133659362793,410.7538757324219,385.1174248795844,399.66314697265627]},{"page":196,"text":"model for the barrier at the metal-insulator contact), then KD would depend","rect":[53.81344223022461,422.7135314941406,385.1047295670844,413.77886962890627]},{"page":196,"text":"exponentially on the square root of the field strength. Then the current would also","rect":[53.813716888427737,434.6163024902344,385.1515740495063,425.6817626953125]},{"page":196,"text":"exponentially depend on pE and proportional to the cell thickness for a given field","rect":[53.813716888427737,446.9483642578125,385.11773005536568,436.9978332519531]},{"page":196,"text":"strength:","rect":[53.813716888427737,458.535400390625,88.83362443515847,449.600830078125]},{"page":196,"text":"j ¼ const \u0005 KD0 exp\u0007bkEB1T=2\b","rect":[164.66778564453126,497.906005859375,274.28191824298878,472.8241271972656]},{"page":196,"text":"(7.48)","rect":[361.0557556152344,490.2194519042969,385.1051267227329,481.74310302734377]},{"page":196,"text":"This corresponds to violation of the current saturation regime and region 3 in","rect":[65.76610565185547,521.395751953125,385.14003840497505,512.4612426757813]},{"page":196,"text":"Fig. 7.24.","rect":[53.81406784057617,533.3552856445313,92.32889981772189,524.4207763671875]},{"page":196,"text":"This simple picture qualitatively agrees with the experimental data. For instance,","rect":[65.76610565185547,545.3148193359375,385.15496488119848,536.3803100585938]},{"page":196,"text":"the linear current growth at low fields with subsequent saturation and further strong","rect":[53.81406784057617,557.2743530273438,385.13701716474068,548.33984375]},{"page":196,"text":"increase of the current is often observed in the direct current (d.c.) regime. On the","rect":[53.81406784057617,569.23388671875,385.17383611871568,560.2794189453125]},{"page":196,"text":"other hand, due to simplicity of the model (as the injection and space charge","rect":[53.81406784057617,581.1366577148438,385.15195501520005,572.2021484375]},{"page":196,"text":"phenomena are not taken into account) it is not easy to obtain precise quantitative","rect":[53.81406784057617,593.09619140625,385.17383611871568,584.1616821289063]},{"page":197,"text":"7.3 Transport Properties","rect":[53.812843322753909,44.274620056152347,136.41827853202143,36.68026351928711]},{"page":197,"text":"181","rect":[372.4981994628906,42.55594253540039,385.1898245254032,36.73106384277344]},{"page":197,"text":"data and determine relevant material parameters. A technical problem is to avoid","rect":[53.812843322753909,68.2883529663086,385.1775750260688,59.35380554199219]},{"page":197,"text":"uncontrollable impurities. High purity of a liquid crystal material with low conduc-","rect":[53.812843322753909,80.24788665771485,385.1536801895298,71.31333923339844]},{"page":197,"text":"tivity is always desirable, because, if necessary, a well controllable conductive","rect":[53.812843322753909,92.20748138427735,385.1367572612938,83.27293395996094]},{"page":197,"text":"dopants can be introduced on purpose.","rect":[53.812843322753909,104.11019134521485,208.9691128304172,95.17564392089844]},{"page":197,"text":"7.3.3.4 Frequency Dependence of Ionic Conductivity","rect":[53.812843322753909,146.39544677734376,284.6808709245063,136.74374389648438]},{"page":197,"text":"In conventional liquids and liquid crystals, the ionic conductivity has no dispersion","rect":[53.812843322753909,169.85934448242188,385.10393611005318,160.92478942871095]},{"page":197,"text":"up to microwave or even optical frequencies. It can be shown by consideration of","rect":[53.812843322753909,181.81887817382813,385.14669166413918,172.8843231201172]},{"page":197,"text":"the equation for the ion oscillation under an external electric field (force qEexp","rect":[53.812843322753909,193.77841186523438,385.14467707685005,184.84385681152345]},{"page":197,"text":"(iot)):","rect":[53.812843322753909,205.3395233154297,78.81582505771707,196.8631591796875]},{"page":197,"text":"dv","rect":[175.59999084472657,226.95962524414063,185.02663458062973,219.9673614501953]},{"page":197,"text":"mi ¼ qEexpð\u0002iotÞ \u0002 xv","rect":[164.32647705078126,236.00473022460938,274.6473773784813,226.05421447753907]},{"page":197,"text":"dt","rect":[176.4500732421875,240.56265258789063,184.22431809970926,233.5703887939453]},{"page":197,"text":"(7.49)","rect":[361.055908203125,235.2676544189453,385.10527931062355,226.79129028320313]},{"page":197,"text":"Here v is velocity of an ion of mass mi and \u0002xv is a friction force; in this case,","rect":[65.7662582397461,264.06341552734377,385.1429104378391,254.8106689453125]},{"page":197,"text":"there is no elastic restoring force familiar from the problem of a pendulum.","rect":[53.81403732299805,275.96673583984377,357.7718777229953,267.03216552734377]},{"page":197,"text":"Substituting to (7.49) a solution in the form of v ¼ v0 expð\u0002iotÞ we obtain","rect":[65.76604461669922,288.26519775390627,367.5647820573188,278.3146667480469]},{"page":197,"text":"v0ðx \u0002 iomiÞ ¼ qE","rect":[181.15126037597657,312.18450927734377,257.77902978337968,302.2339782714844]},{"page":197,"text":"From here, introducing inverse relaxation time of ions ti\u00021 ¼ x/mi we find the","rect":[65.76628875732422,335.76495361328127,385.1747516460594,324.7144775390625]},{"page":197,"text":"complex amplitude of ion velocity","rect":[53.81399154663086,347.7247009277344,192.86635676435004,338.7901611328125]},{"page":197,"text":"v0 ¼ miðti\u0002q1E\u0002 ioÞ ¼ miðoqt2i\u0002þ1Eti\u00022Þ þ iomiðo2qEþ t\u0002i 2Þ","rect":[108.47655487060547,386.8592224121094,328.8623086367722,361.955322265625]},{"page":197,"text":"(7.50)","rect":[361.05621337890627,378.78466796875,385.1055844864048,370.18878173828127]},{"page":197,"text":"and the complex conductivity","rect":[53.814537048339847,410.3573913574219,173.4044121842719,401.4228515625]},{"page":197,"text":"2","rect":[216.5544891357422,429.3008728027344,220.03845284065566,424.58758544921877]},{"page":197,"text":"s\b ¼ s0 þ ios00 ¼ qnvv ¼ mið1qþnvoti2t2i Þ þ iomið1q þnvoti2ti2Þ","rect":[86.15876770019531,449.54803466796877,327.04965604888158,428.2968444824219]},{"page":197,"text":"(7.51)","rect":[361.05621337890627,441.4736022949219,385.1055844864048,432.8777160644531]},{"page":197,"text":"Therefore,forfrequencieso <<titheionicconductivityisconstant,s ¼ q2nv/x.","rect":[65.76656341552735,473.04632568359377,385.18316312338598,461.9951477050781]},{"page":197,"text":"The friction coefficient can be estimated from the Stokes formula for the friction","rect":[65.7665023803711,482.9836730957031,385.0986260514594,476.07110595703127]},{"page":197,"text":"force","rect":[53.814476013183597,494.9033508300781,74.32015264703596,488.0306396484375]},{"page":197,"text":"Fx ¼ xv ¼ 6pr0\u0002v","rect":[182.56817626953126,521.3603515625,256.4075397319969,511.6214294433594]},{"page":197,"text":"(7.52)","rect":[361.0561828613281,520.4862670898438,385.10555396882668,511.8903503417969]},{"page":197,"text":"where r0 is radius of a spherical ion and Z is viscosity of the liquid: x ¼ 6pZr0 \u0004","rect":[53.81450653076172,544.8606567382813,381.86267879698189,535.607421875]},{"page":197,"text":"2 \u0003 10\u00027 g/s (we take r0 ¼ 5A˚ ¼ 5 \u0003 10\u00028 cm and viscosity Z ¼ 0.1P). Therefore,","rect":[53.814659118652347,556.8203125,385.1516995003391,545.2213745117188]},{"page":197,"text":"in the framework of the Stokes model, the ion conductivity and mobility at o << ti","rect":[53.813838958740237,568.7798461914063,385.18902481319278,559.8453369140625]},{"page":197,"text":"can be written as","rect":[53.812843322753909,578.6778564453125,122.19528593169406,571.8051147460938]},{"page":198,"text":"182","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.73106384277344]},{"page":198,"text":"7 Magnetic, Electric and Transport Properties","rect":[228.78945922851563,44.274620056152347,385.1482590007714,36.68026351928711]},{"page":198,"text":"2","rect":[162.28842163085938,64.1630859375,165.77238533577285,59.44980239868164]},{"page":198,"text":"s ¼ 6qprn0vZ; and mi ¼ s=qnv ¼ q=6pr0Z","rect":[133.73919677734376,83.44583892822266,305.20613357241896,63.159027099609378]},{"page":198,"text":"(7.53)","rect":[361.0562438964844,76.2787857055664,385.1056150039829,67.68289947509766]},{"page":198,"text":"Numerically, for q ¼ e ¼ 4.8 \u0003 10\u000210 CGSQ and x \u0004 2 \u0003 10\u00027 g/s we get","rect":[65.7665786743164,106.88826751708985,385.174940658303,95.83702087402344]},{"page":198,"text":"the order of magnitude for the mobility mi \u0004 2.4 \u0003 10\u00023 cm2/statV\u0005s (or 0.8 \u0003 10\u00029","rect":[53.814231872558597,118.84761810302735,385.1809962305198,107.79673767089844]},{"page":198,"text":"m2/V\u0005s) that is close to typical experimental data.","rect":[53.812843322753909,130.80740356445313,253.61459012533909,119.75645446777344]},{"page":198,"text":"Finally, we estimate the ion translational relaxation time for spherical molecules","rect":[65.76468658447266,142.76693725585938,385.1605264102097,133.83238220214845]},{"page":198,"text":"in a viscous liquid:","rect":[53.81266403198242,154.72647094726563,130.53076036045145,145.7919158935547]},{"page":198,"text":"ti ¼ mi=x ¼ mi=6pr0\u0002","rect":[175.71290588378907,181.6500244140625,263.0563294472726,169.04408264160157]},{"page":198,"text":"(7.54)","rect":[361.0555114746094,178.2475128173828,385.1048825821079,169.651611328125]},{"page":198,"text":"Typically a mass of an organic molecule (or ion) is mi ¼ Mmp \u0004 4 \u0003 10\u000222g","rect":[65.76586151123047,206.04566955566407,385.17937556317818,194.0648956298828]},{"page":198,"text":"(here M \u0004 200 is molecular mass, mp is proton mass) and, indeed, the estimated","rect":[53.81462860107422,218.0053253173828,385.1261834245063,208.1409149169922]},{"page":198,"text":"relaxation time is very short ti \u0004 2 \u0003 10\u000215s. Thus, the dispersion of the ionic","rect":[53.813228607177737,229.03524780273438,385.1317523784813,217.9046173095703]},{"page":198,"text":"conductivity can only occur in the range of optical frequencies where the physical","rect":[53.81380844116211,240.99478149414063,385.154676986428,232.0602264404297]},{"page":198,"text":"sense of the friction force is doubtful. By the way, the Stokes approximation seems","rect":[53.81380844116211,252.95431518554688,385.10788358794408,244.01976013183595]},{"page":198,"text":"to be quite good at lower frequencies.","rect":[53.81380844116211,264.8570861816406,206.3869595101047,255.9225311279297]},{"page":198,"text":"7.3.3.5 Conductivity due to Dielectric Losses","rect":[53.81380844116211,310.089599609375,250.33581937895969,300.4378967285156]},{"page":198,"text":"Let us go back to the complex dielectric permittivity e\b ¼ e0 þ ie00. Here e00","rect":[53.81380844116211,333.61029052734377,384.64467658229389,323.9516296386719]},{"page":198,"text":"describes dielectric losses, i.e. energy dissipation in the range of Debye relaxation","rect":[53.812843322753909,345.5709533691406,385.16762629560005,336.63641357421877]},{"page":198,"text":"00","rect":[264.5895080566406,348.45062255859377,268.35219611354389,344.8668518066406]},{"page":198,"text":"of dipoles. We can introduce a parameter sD ¼ oe =4p, which has dimension of","rect":[53.812843322753909,358.4628601074219,385.15157447663918,347.6637878417969]},{"page":198,"text":"electric conductivity ([s\u00021] in the Gauss system and [F/m\u0005s ¼ C/V\u0005m\u0005s ¼ A/V\u0005m ¼","rect":[53.81376266479492,369.4902648925781,385.14609554502877,358.4393310546875]},{"page":198,"text":"O\u00021m\u00021] in the SI system and, in fact, is indistinguishable from the a.c. ohmic","rect":[53.812259674072269,381.39324951171877,385.12046087457505,370.34228515625]},{"page":198,"text":"conductivity. The values of e00 and sD are essential in the frequency range of","rect":[53.81343460083008,393.3529052734375,385.14946876374855,383.7499084472656]},{"page":198,"text":"dispersion of the e0 component, as we have seen in Section 7.2.4.","rect":[53.813655853271487,405.31256103515627,316.64153714682348,395.6529235839844]},{"page":198,"text":"What has been said is true for both isotropic liquids and liquid crystals. How-","rect":[65.76460266113281,417.2721252441406,385.10967384187355,408.33758544921877]},{"page":198,"text":"ever, in liquid crystals, due to their specific anisotropy, the ratio","rect":[53.81258773803711,429.2316589355469,385.1086358170844,420.297119140625]},{"page":198,"text":"sjDj .s?D","rect":[53.81258773803711,449.4187316894531,85.97136794834018,431.5097961425781]},{"page":198,"text":"¼ ðe00jjojjÞ\u000Fðe00?o?Þ","rect":[89.27276611328125,446.447265625,171.6720315371628,434.4568786621094]},{"page":198,"text":"dramatically","rect":[176.7894744873047,445.0458679199219,226.92983332685004,436.111328125]},{"page":198,"text":"depends","rect":[232.01742553710938,445.0458679199219,264.73788847075658,436.111328125]},{"page":198,"text":"on","rect":[269.8553466796875,444.0,279.8095635514594,438.0]},{"page":198,"text":"frequency","rect":[284.8663330078125,445.0458679199219,324.84938899091255,436.111328125]},{"page":198,"text":"and,","rect":[329.95489501953127,444.0,346.8471951058078,436.111328125]},{"page":198,"text":"at","rect":[351.93280029296877,444.0,359.1794877774436,437.1272888183594]},{"page":198,"text":"least,","rect":[364.281005859375,444.0,385.0953030159641,436.111328125]},{"page":198,"text":"theoretically can vary from þ1 to \u00021. It is seen from a qualitative picture in","rect":[53.81318283081055,460.9158935546875,385.1440362077094,451.9813232421875]},{"page":198,"text":"Fig. 7.23. Due to strongly different dispersion frequency range for real components","rect":[53.81416702270508,472.8754577636719,385.13217558013158,463.94091796875]},{"page":198,"text":"e0|| and e0⊥, the band maxima for imaginary components e00|| and e00⊥ are well","rect":[53.81416702270508,484.835693359375,385.100966048928,475.17608642578127]},{"page":198,"text":"separated and the ratios e00||/e00⊥ (and sjDj=s?D) at the maxima are very large. In","rect":[53.81393051147461,497.3748779296875,385.15215388349068,484.3303527832031]},{"page":198,"text":"addition, at a certain frequency oinv, the anisotropy of both e00|| \u0002 e00⊥ and sjDj \u0002 s?D","rect":[53.8133659362793,509.33428955078127,384.66135513095738,496.2333068847656]},{"page":198,"text":"changes sign.","rect":[53.812843322753909,520.6580200195313,107.89107938315158,511.7235107421875]},{"page":198,"text":"The","rect":[65.76486206054688,531.0,81.30338323785627,523.6830444335938]},{"page":198,"text":"high","rect":[86.78018951416016,532.6175537109375,104.50863734296331,523.6830444335938]},{"page":198,"text":"anisotropy","rect":[110.00436401367188,532.6175537109375,152.0808037858344,523.6830444335938]},{"page":198,"text":"of","rect":[157.58547973632813,531.0,165.9420407852329,523.6830444335938]},{"page":198,"text":"sD","rect":[171.35116577148438,532.1199951171875,183.39598614854985,525.8743286132813]},{"page":198,"text":"can","rect":[188.8549346923828,531.0,202.73110285810004,525.0]},{"page":198,"text":"influence","rect":[208.17108154296876,531.0,244.81848943902816,523.6831665039063]},{"page":198,"text":"the","rect":[250.31521606445313,531.0,262.5389789409813,523.6831665039063]},{"page":198,"text":"electro-optical","rect":[267.9879150390625,532.61767578125,325.9482866055686,523.6831665039063]},{"page":198,"text":"behaviour","rect":[331.4290771484375,531.0,371.3255551895298,523.6831665039063]},{"page":198,"text":"of","rect":[376.85809326171877,531.0,385.14995704499855,523.6831665039063]},{"page":198,"text":"nematics even at low frequencies, especially for those materials, which have a","rect":[53.8131217956543,544.5772094726563,385.15695989801255,535.6427001953125]},{"page":198,"text":"low frequency inversion of dielectric anisotropy ea. An example is shown in","rect":[53.8131217956543,556.536865234375,385.1424187760688,547.6022338867188]},{"page":198,"text":"Fig. 7.24 related to a material with ea inversion at f ¼ 700 kHz. The electric","rect":[53.81357955932617,568.4963989257813,385.15439642145005,559.5221557617188]},{"page":198,"text":"conductivity begins to deviate from the ionic, low frequency plateau at 200 Hz","rect":[53.814552307128909,580.4560546875,385.17136419488755,571.5215454101563]},{"page":198,"text":"and 10 kHz for the longitudinal and transverse components, respectively. At f ¼ 10","rect":[53.814552307128909,592.4156494140625,385.1723260026313,583.4412841796875]},{"page":199,"text":"7.3 Transport Properties","rect":[53.813167572021487,44.275230407714847,136.41859896659174,36.68087387084961]},{"page":199,"text":"a","rect":[131.4814910888672,73.50039672851563,137.03675348610103,67.9116439819336]},{"page":199,"text":"e′||","rect":[185.59417724609376,81.28424835205078,193.07557783539836,72.76203155517578]},{"page":199,"text":"′","rect":[261.208251953125,92.37583923339844,263.18258800393945,90.09632110595703]},{"page":199,"text":"e","rect":[256.90069580078127,95.79110717773438,260.409738417411,91.36804962158203]},{"page":199,"text":"⊥","rect":[261.2079162597656,98.5103759765625,264.4951458180549,95.20108032226563]},{"page":199,"text":"183","rect":[372.49853515625,42.55655288696289,385.19016021876259,36.73167419433594]},{"page":199,"text":"b","rect":[133.82748413085938,151.48403930664063,139.93227788033759,144.1756591796875]},{"page":199,"text":"′′","rect":[200.280517578125,153.1212615966797,203.9094556797207,150.8417510986328]},{"page":199,"text":"e ||","rect":[196.4517364501953,159.36392211914063,205.1639720004374,152.1134796142578]},{"page":199,"text":"′′","rect":[276.356201171875,173.35813903808595,279.9851392734707,171.07862854003907]},{"page":199,"text":"e","rect":[271.887939453125,176.77340698242188,275.39698206975478,172.35035705566407]},{"page":199,"text":"⊥","rect":[278.01080322265627,179.49282836914063,281.2980327809455,176.18353271484376]},{"page":199,"text":"ω","rect":[322.8450012207031,148.86959838867188,328.3283798881473,145.0784149169922]},{"page":199,"text":"c","rect":[133.81349182128907,241.9508056640625,139.3687542185229,236.36204528808595]},{"page":199,"text":"d","rect":[134.16419982910157,343.2921447753906,140.26899357857978,335.9837646484375]},{"page":199,"text":"ω","rect":[180.9647979736328,289.8347473144531,186.448176641077,286.0435485839844]},{"page":199,"text":"||","rect":[186.44798278808595,294.0118713378906,188.8459621607932,288.55902099609377]},{"page":199,"text":"s ∼ ω","rect":[184.79464721679688,388.820068359375,203.48290564498326,384.5409851074219]},{"page":199,"text":"e","rect":[205.481201171875,388.820068359375,208.99024378850477,384.3970031738281]},{"page":199,"text":"ωinv","rect":[246.1087646484375,376.0428161621094,259.2536993472955,369.3643493652344]},{"page":199,"text":"ω⊥","rect":[264.7283630371094,278.80413818359377,274.1560402160409,272.1257019042969]},{"page":199,"text":"ω","rect":[318.3487854003906,227.85275268554688,323.8321640678348,224.0615692138672]},{"page":199,"text":"ω","rect":[321.34625244140627,295.3383483886719,326.82963110885046,291.5471496582031]},{"page":199,"text":"ω","rect":[311.51934814453127,375.4860534667969,317.00272681197546,371.6948547363281]},{"page":199,"text":"Fig. 7.23 Qualitative frequency spectra of two components of dielectric permittivity e0 (a) and","rect":[53.812843322753909,443.03515625,385.1944937148563,434.7228698730469]},{"page":199,"text":"e00 (b), and corresponding anisotropy of imaginary component of dielectric permittivity e00|| \u0002 e00⊥","rect":[53.81332015991211,452.9204406738281,385.1651324428814,444.69873046875]},{"page":199,"text":"(c) and real a.c. conductivity sjDj \u0002 sD? (d) in the Debye relaxation range. oinv is inversion","rect":[53.812843322753909,463.42279052734377,385.1700491347782,452.2956848144531]},{"page":199,"text":"frequency, at which anisotropy ea00 and sjDj \u0002 s?D changes sign","rect":[53.81338119506836,473.3419494628906,267.1302847304813,462.21484375]},{"page":199,"text":"kHz the ratio of the two conductivity components reaches 400. The law for the","rect":[53.812843322753909,497.8156433105469,385.1695941753563,488.881103515625]},{"page":199,"text":"conductivity growth at otD < 1 can be derived from the Debye formula (7.32) and","rect":[53.812843322753909,509.7752990722656,385.14544001630318,500.84063720703127]},{"page":199,"text":"allows tD to be determined.","rect":[53.8136100769043,521.2371826171875,165.53709073569065,512.80029296875]},{"page":199,"text":"oe00 ½eð0Þ \u0002 eð1Þ\u0006o2tD","rect":[178.54551696777345,546.729736328125,283.6404535697089,535.9654541015625]},{"page":199,"text":"sD ¼","rect":[153.16799926757813,552.5250244140625,175.78754451963813,546.4498291015625]},{"page":199,"text":"¼","rect":[196.6153564453125,549.7261962890625,204.28009823057563,547.3954467773438]},{"page":199,"text":"4p","rect":[180.9812774658203,557.9226684570313,191.4610630924518,551.1196899414063]},{"page":199,"text":"(7.55)","rect":[361.0561828613281,552.7940673828125,385.10555396882668,544.1981811523438]},{"page":199,"text":"Such a great growth of conductivity anisotropy dramatically","rect":[65.76653289794922,581.41943359375,322.77236262372505,572.4849243164063]},{"page":199,"text":"electro-optical behaviour of nematics in this frequency interval.","rect":[53.814491271972659,593.3789672851563,310.5754360726047,584.4444580078125]},{"page":199,"text":"influences","rect":[327.5772705078125,580.0,368.05707182036596,572.4849243164063]},{"page":199,"text":"the","rect":[372.892822265625,580.0,385.1166156597313,572.4849243164063]},{"page":200,"text":"184","rect":[53.81330108642578,42.55582046508789,66.50490707545205,36.73094177246094]},{"page":200,"text":"Fig. 7.24 Low frequency","rect":[53.812843322753909,67.58130645751953,142.66254181055948,59.85148620605469]},{"page":200,"text":"spectrum of the principal","rect":[53.812843322753909,77.4895248413086,139.70031456198755,69.89517211914063]},{"page":200,"text":"components of real a.c.","rect":[53.812843322753909,87.4087142944336,133.52456924512348,79.81436157226563]},{"page":200,"text":"conductivity s|| and s⊥ fora","rect":[53.812843322753909,97.3846664428711,152.87230822824956,89.79006958007813]},{"page":200,"text":"typical nematic liquid crystal","rect":[53.81264877319336,107.36043548583985,153.5322847768313,99.76608276367188]},{"page":200,"text":"having frequency dispersion","rect":[53.81264877319336,117.33638763427735,150.4050802627079,109.74203491210938]},{"page":200,"text":"of e|| in the 1 MHz range","rect":[53.81264877319336,127.25533294677735,138.7607512214136,119.66098022460938]},{"page":200,"text":"7 Magnetic, Electric and Transport Properties","rect":[228.7899169921875,44.274497985839847,385.1487167644433,36.68014144897461]},{"page":200,"text":"It is instructive to compare the ratio s||/s⊥ \u0004 1.5 for ionic processes witha","rect":[65.76496887207031,222.8570556640625,385.16065252496568,213.86273193359376]},{"page":200,"text":"much higher ratio sDjj=s?D \u0004 102 \u0002 103 for the conductivity due to dielectric losses.","rect":[53.81380844116211,235.45230102539063,385.1118435433078,222.35159301757813]},{"page":200,"text":"In the first case, the orientational nematic potential WN does not influence the","rect":[53.813838958740237,246.7762451171875,385.1172260112938,237.84169006347657]},{"page":200,"text":"translational motion of ions and s||/s⊥. In the second case, the losses ej00j come in","rect":[53.814231872558597,261.4974670410156,385.1436699967719,249.07627868652345]},{"page":200,"text":"from the field induced alignment of the longitudinal molecular axes against the","rect":[53.813777923583987,270.6387634277344,385.17356146051255,261.7042236328125]},{"page":200,"text":"potential barrier and dramatically depend on WN whereas losses e0?0 caused by","rect":[53.813777923583987,283.23443603515627,385.17131892255318,272.9955749511719]},{"page":200,"text":"molecular rotation about the longitudinal molecular axes are independent of WN.","rect":[53.814537048339847,294.55816650390627,385.1832241585422,285.62359619140627]},{"page":200,"text":"Therefore, ratios e00||/e00⊥ (and sjDj=sD?) are very large.","rect":[53.814537048339847,307.1536560058594,269.8206753304172,294.0528564453125]},{"page":200,"text":"7.3.3.6 Space Charge Relaxation","rect":[53.81330490112305,344.74322509765627,199.2891117985065,335.4102478027344]},{"page":200,"text":"Many phenomena in liquid crystals such as formation of double electric layers at","rect":[53.81330490112305,368.5826416015625,385.18000657627177,359.6480712890625]},{"page":200,"text":"interfaces, screening any electric polarization by ions, triggering electro-hydrody-","rect":[53.81330490112305,380.5421447753906,385.1611569961704,371.60760498046877]},{"page":200,"text":"namic processes, etc. are related to the so-called space charge. The latter can be","rect":[53.81330490112305,392.501708984375,385.1502155132469,383.567138671875]},{"page":200,"text":"imagined as a cloud of a non-compensated charge, say, positive þdq(r) at some","rect":[53.81330490112305,404.46124267578127,385.14817083551255,395.2278747558594]},{"page":200,"text":"place in medium with a coordinate r. The charge of the opposite sign, \u0002dq(r) is","rect":[53.81333541870117,416.364013671875,385.1889687930222,407.1306457519531]},{"page":200,"text":"situated in another place, see Fig. 7.25a. The total charge is zero but the electrical","rect":[53.814308166503909,428.3235778808594,385.1491838223655,419.3292541503906]},{"page":200,"text":"neutrality is disturbed locally.","rect":[53.81330490112305,440.2831115722656,174.28413053061252,431.34857177734377]},{"page":200,"text":"In strongly conductive materials, like metals, the local electric current in the","rect":[65.76531219482422,452.2426452636719,385.1710895366844,443.30810546875]},{"page":200,"text":"medium will immediately restore the charge neutrality everywhere. If the local","rect":[53.81330490112305,464.2021789550781,385.1113725430686,455.26763916015627]},{"page":200,"text":"neutrality is disturbed in weakly conductive materials, it takes some time to restore","rect":[53.81330490112305,476.1617126464844,385.1581500835594,467.2271728515625]},{"page":200,"text":"it. It is very easy to find this time. Consider a capacitor with a gap d, capacitanceC","rect":[53.81330490112305,488.1212463378906,385.18794509585646,479.1667785644531]},{"page":200,"text":"and charge q0 on the limiting plates of area A, Fig.7.25b. At first, the dielectric is","rect":[53.81332015991211,500.0828552246094,385.18698515044408,491.0885314941406]},{"page":200,"text":"assumed to be ideal, s ¼ 0. Upon connecting the external resistor R the charge will","rect":[53.81332015991211,511.9856262207031,385.14619309970927,503.05108642578127]},{"page":200,"text":"relax producing current I given by equation","rect":[53.8133430480957,523.9451293945313,229.47528925946723,515.0106201171875]},{"page":200,"text":"dq","rect":[193.4431610107422,547.0184326171875,203.3973778825141,538.1735229492188]},{"page":200,"text":"dU U","rect":[224.9379425048828,545.265380859375,257.7453131671223,538.1735229492188]},{"page":200,"text":"I¼","rect":[176.56369018554688,551.7876586914063,190.62897518370063,545.2137451171875]},{"page":200,"text":"¼C","rect":[206.18853759765626,552.0265502929688,223.24110671695019,544.9844970703125]},{"page":200,"text":"¼","rect":[240.1190948486328,550.4627075195313,247.78383663389594,548.1319580078125]},{"page":200,"text":":","rect":[259.7182312011719,551.7874755859375,262.40586793106936,550.7416381835938]},{"page":200,"text":"dt","rect":[194.5192108154297,558.768798828125,202.29345567295145,551.7765502929688]},{"page":200,"text":"dt","rect":[227.26026916503907,558.768798828125,235.0345140225608,551.7765502929688]},{"page":200,"text":"R","rect":[251.278076171875,558.5895385742188,257.38996117986405,552.015625]},{"page":200,"text":"Hence, q ¼ CU ¼ q0 expð\u0002t=RCÞ, that is the charge relaxes with time constant","rect":[65.76644134521485,582.608642578125,385.17317063877177,572.6581420898438]},{"page":200,"text":"RC.","rect":[53.812442779541019,592.3270874023438,69.10310025717502,585.2850341796875]},{"page":201,"text":"7.3 Transport Properties","rect":[53.813682556152347,44.275352478027347,136.4191025066308,36.68099594116211]},{"page":201,"text":"a","rect":[103.77393341064453,68.23287963867188,109.3397512511414,62.64412307739258]},{"page":201,"text":"+δq","rect":[120.58926391601563,124.20527648925781,133.8660774825919,116.97481536865235]},{"page":201,"text":"–δq","rect":[183.0414276123047,129.28346252441407,195.78268820036534,122.0530014038086]},{"page":201,"text":"b","rect":[235.51321411132813,68.23287963867188,241.62960745762954,60.92450714111328]},{"page":201,"text":"+q0","rect":[242.97183227539063,89.3298110961914,254.47412781409236,82.20530700683594]},{"page":201,"text":"185","rect":[372.4990539550781,42.55667495727539,385.1906790175907,36.63019943237305]},{"page":201,"text":"R","rect":[307.52313232421877,166.6336669921875,312.85463898775415,161.3387908935547]},{"page":201,"text":"Fig. 7.25 Formation of space charge in a weakly conductive material (a) and the capacitor-","rect":[53.812843322753909,188.25485229492188,385.1229256973951,180.52503967285157]},{"page":201,"text":"resistance circuit for the calculation of charge relaxation time (b)","rect":[53.813682556152347,198.1630859375,277.26376432044199,190.5687255859375]},{"page":201,"text":"The same charge would relax with time constant RC even without the external","rect":[65.76496887207031,222.68698120117188,385.10896165439677,213.74246215820313]},{"page":201,"text":"resistance if the material inside is conductive with the same resistance R. In that","rect":[53.81296157836914,232.61456298828126,385.13984544345927,225.7119598388672]},{"page":201,"text":"case the relaxation time can be expressed as follows (A and d are area and thickness","rect":[53.81294250488281,246.54928588867188,385.1368447695847,237.5948028564453]},{"page":201,"text":"of the sample):","rect":[53.81294250488281,258.50885009765627,114.37435777256082,249.5742950439453]},{"page":201,"text":"tM ¼ RC ¼ \u0007sdA\b \u0005 \u00074epAd\b ¼ 4pes","rect":[146.0874786376953,296.63201904296877,291.21556350405958,272.7467956542969]},{"page":201,"text":"(7.56)","rect":[361.0562744140625,288.9444885253906,385.10564552156105,280.3486022949219]},{"page":201,"text":"We can see that relaxation time tM is independent of the sample dimensions and","rect":[65.7666244506836,320.1213073730469,385.14324275067818,311.18621826171877]},{"page":201,"text":"includes only material parameters, namely, specific conductivity s and dielectric","rect":[53.81339645385742,332.08087158203127,385.1810993023094,323.14630126953127]},{"page":201,"text":"constant (real part e ¼ e0). This time is called space charge relaxation time. It is the","rect":[53.8133659362793,344.0804138183594,385.17444647027818,334.3809814453125]},{"page":201,"text":"same Maxwell dielectric relaxation time we met in Section 7.2.1. Note that time tM","rect":[53.8137321472168,355.50250244140627,385.16660600380126,347.06561279296877]},{"page":201,"text":"has no relation to the dispersion frequency of ionic conductivity (ti)\u00021, neither to","rect":[53.812843322753909,367.9598693847656,385.1441277604438,356.90887451171877]},{"page":201,"text":"Debye dipole relaxation time.","rect":[53.81426239013672,379.9194030761719,173.7017788460422,370.98486328125]},{"page":201,"text":"As the space charge relaxation in medium is caused by the counter motion of","rect":[65.76628875732422,391.8221740722656,385.1501401504673,382.88763427734377]},{"page":201,"text":"positive and negative charge carriers in a diffusion process, there should be a","rect":[53.81426239013672,403.78173828125,385.1590961284813,394.84716796875]},{"page":201,"text":"characteristic diffusion length related to this motion:","rect":[53.81426239013672,415.74127197265627,265.8479142911155,406.80670166015627]},{"page":201,"text":"ffiffiffiffiffiffiffiffi","rect":[252.80740356445313,431.0,268.8127852733446,430.0]},{"page":201,"text":"LD ¼ ð2DtÞ1=2 ¼ r2Dpes","rect":[170.21978759765626,454.3848876953125,268.78216904948308,430.4897155761719]},{"page":201,"text":"(7.57)","rect":[361.0566711425781,448.3319396972656,385.10604225007668,439.7360534667969]},{"page":201,"text":"Recalling the Einstein relationship D ¼ mkBT=qi","rect":[65.76703643798828,478.1926574707031,267.85041313927078,468.2620849609375]},{"page":201,"text":"charge of a particle (e.g. ion) we obtain","rect":[53.81450653076172,489.8235168457031,213.43425837567816,480.88897705078127]},{"page":201,"text":"ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi","rect":[221.31256103515626,505.0,251.8749648387743,504.0]},{"page":201,"text":"LD ¼ s2pkBqTi2env","rect":[187.1003875732422,534.4427490234375,251.39042038813015,504.5713195800781]},{"page":201,"text":"and","rect":[272.9164123535156,475.83203125,287.3201531510688,468.929443359375]},{"page":201,"text":"s ¼ qinvm","rect":[291.9488525390625,477.7743225097656,333.35687608073308,471.0908508300781]},{"page":201,"text":"where qi is","rect":[337.9447937011719,477.7544250488281,385.1891213809128,468.929443359375]},{"page":201,"text":"(7.58)","rect":[361.055419921875,524.5101928710938,385.10479102937355,515.914306640625]},{"page":201,"text":"This is a so-called Debye screening length playing a crucial role in any phenom-","rect":[65.7657699584961,557.9933471679688,385.1157468399204,548.9990234375]},{"page":201,"text":"ena related to the local disturbance of the electric neutrality. For instance, of great","rect":[53.8127555847168,569.9130249023438,385.2053666836936,560.978515625]},{"page":201,"text":"technical importance are metal-dielectric contacts or p-n semiconductor junctions","rect":[53.8127555847168,581.87255859375,385.11777128325658,572.9380493164063]},{"page":201,"text":"or contacts between liquid crystal and colloidal particles or metal electrodes. For","rect":[53.812801361083987,593.8320922851563,385.14669166413918,584.8975830078125]},{"page":202,"text":"186","rect":[53.812843322753909,42.55740737915039,66.50444931178018,36.68172836303711]},{"page":202,"text":"7 Magnetic, Electric and Transport Properties","rect":[228.78945922851563,44.276084899902347,385.1482590007714,36.68172836303711]},{"page":202,"text":"liquid crystals, due to anisotropy of e, the length LD is also different for two","rect":[53.812843322753909,68.2883529663086,385.15190974286568,59.35380554199219]},{"page":202,"text":"principal directions (|| and ⊥ to the director).","rect":[53.81406784057617,80.24788665771485,235.06829495932346,71.31333923339844]},{"page":202,"text":"7.3.3.7 Measurements of Anisotropy «a (v) and sa(v)","rect":[53.81403732299805,122.47632598876953,291.46819434968605,112.82462310791016]},{"page":202,"text":"A widely spread technique includes a dielectric bridge (DB) measuring the capaci-","rect":[53.813411712646487,145.99765014648438,385.1761716446079,137.06309509277345]},{"page":202,"text":"tance and conductance of a liquid crystal cell at various frequencies (typically","rect":[53.813411712646487,157.900390625,385.12441340497505,148.96583557128907]},{"page":202,"text":"1–106 Hz) and temperatures. A small thermostat with a liquid crystal cell is placed","rect":[53.813411712646487,169.8604736328125,385.1133965592719,158.76959228515626]},{"page":202,"text":"in the gap between two poles of a magnet. The thickness of the liquid crystal layer","rect":[53.813289642333987,181.82000732421876,385.11834083406105,172.8854522705078]},{"page":202,"text":"should be fairly large (~100 mm) to avoid effects of boundaries. The electrodes must","rect":[53.813289642333987,193.77957153320313,385.2068925625999,184.8450164794922]},{"page":202,"text":"have high conductivity (e.g., made of gold) and must not be covered by any aligning","rect":[53.813289642333987,205.73910522460938,385.1690301041938,196.80455017089845]},{"page":202,"text":"layers. The amplitude of the a.c. voltage across the electrodes must be small enough","rect":[53.813289642333987,217.69866943359376,385.1203545670844,208.7641143798828]},{"page":202,"text":"(usually about 0.1 V) to avoid any influence of the electric field on the liquid crystal","rect":[53.813289642333987,229.65817260742188,385.1193071133811,220.72361755371095]},{"page":202,"text":"alignment. The orientation of the liquid crystal director is fixed by the external","rect":[53.813289642333987,241.61773681640626,385.10835130283427,232.6831817626953]},{"page":202,"text":"magnetic field of high enough strength (H 2 kOe), therefore, one always has n||","rect":[53.813289642333987,253.52047729492188,385.14319985079848,244.5659942626953]},{"page":202,"text":"H. In Fig. 7.26 the magnetic field is directed vertically but the cell is rotated about","rect":[53.813316345214847,265.48004150390627,385.15815599033427,256.54547119140627]},{"page":202,"text":"the horizontal axis. In that way, the cell normal (i.e., the direction of E) is installed","rect":[53.81332015991211,277.4395751953125,385.1740349870063,268.5050048828125]},{"page":202,"text":"either parallel to H for measurements of e|| and s|| (as shown in the figure) or","rect":[53.81333541870117,289.4002685546875,385.1515134414829,280.464599609375]},{"page":202,"text":"perpendicular to it for measurements of e⊥ and s⊥.","rect":[53.81370162963867,301.35980224609377,261.5268520882297,292.42523193359377]},{"page":202,"text":"7.3.3.8 Characteristic Times Related to the Discussed Phenomena (Resume)","rect":[53.81333541870117,342.7118225097656,383.9417966446079,333.9167175292969]},{"page":202,"text":"In the present chapter we have met many characteristic times related to different","rect":[53.81333541870117,367.1090393066406,385.1711259610374,358.17449951171877]},{"page":202,"text":"physical processes. It will be useful to collect them altogether:","rect":[53.81333541870117,379.0685729980469,306.0690141446311,370.134033203125]},{"page":202,"text":"T°C","rect":[133.78863525390626,534.847412109375,148.03263304301403,529.1126098632813]},{"page":202,"text":"H","rect":[219.9126434326172,562.3463745117188,226.4910991808775,556.8515625]},{"page":202,"text":"Fig. 7.26 Scheme of the setup for measurements of the principal","rect":[53.812843322753909,584.0003662109375,293.26705650534697,576.2705688476563]},{"page":202,"text":"permittivity and electric conductivity using a dielectric bridge (DB)","rect":[53.812843322753909,593.9085693359375,285.73077481848886,586.314208984375]},{"page":202,"text":"components","rect":[297.3317565917969,583.9326171875,338.2918136882714,577.2018432617188]},{"page":202,"text":"of","rect":[342.364990234375,582.1123657226563,349.41305631262949,576.3382568359375]},{"page":202,"text":"dielectric","rect":[353.46759033203127,582.2055053710938,385.1542296149683,576.3382568359375]},{"page":203,"text":"References","rect":[53.80965042114258,42.52677536010742,91.47833712577142,36.68496322631836]},{"page":203,"text":"187","rect":[372.4950256347656,42.56064224243164,385.18662017970009,36.73576354980469]},{"page":203,"text":"t ¼ L2\u000F2D","rect":[53.812843322753909,86.72028350830078,89.230161443772,74.15701293945313]},{"page":203,"text":"tth ¼ L2\u000F2Dth ¼ rCpL2\u000F2k","rect":[53.812965393066409,107.63053894042969,142.6194395759058,94.10879516601563]},{"page":203,"text":"tL ¼ 12ðKRKDcÞ\u00021=2","rect":[53.81332015991211,125.435302734375,119.4230234175507,114.08126831054688]},{"page":203,"text":"tT ¼ L2\u000FmU","rect":[53.813411712646487,147.8673095703125,93.70544220549076,133.95535278320313]},{"page":203,"text":"tM ¼ e=4ps","rect":[53.813175201416019,158.4346160888672,93.944894668716,149.27734375]},{"page":203,"text":"ti ¼ mi=x","rect":[53.812339782714847,179.8358154296875,84.77955383448526,169.2139129638672]},{"page":203,"text":"tD ¼ 4pZa3\u000FkBT","rect":[53.81380844116211,202.52093505859376,110.37033208415063,189.049072265625]},{"page":203,"text":"tjDj ¼ jjjtD; tD? ¼ j?tD","rect":[53.81147766113281,220.82481384277345,129.94793309366725,209.07830810546876]},{"page":203,"text":"General expression for a time of diffusion of a particle along distance","rect":[148.8070831298828,82.4201889038086,385.1400694587183,74.82583618164063]},{"page":203,"text":"L (D is diffusion coefficient).","rect":[160.7592010498047,92.00072479248047,260.7758204658266,84.74502563476563]},{"page":203,"text":"Thermal diffusion time (r, Cp and k are density, specific heat and","rect":[148.80702209472657,103.12580871582031,373.24237579493447,94.70440673828125]},{"page":203,"text":"thermal conductivity).","rect":[160.7595672607422,112.2916488647461,236.55552932813129,104.69729614257813]},{"page":203,"text":"Langevin time for chemical relaxation (KD, KR are dissociation and","rect":[148.806884765625,122.21077728271485,380.21017975001259,114.61642456054688]},{"page":203,"text":"recombination rates for ions, c is solute concentration).","rect":[160.7596435546875,131.8480682373047,349.40088150098287,124.59237670898438]},{"page":203,"text":"Voltage induced transit time for charge carriers with mobility m.","rect":[148.8073272705078,142.16183471679688,368.9491297919985,134.56747436523438]},{"page":203,"text":"Maxwell space charge relaxation time, e and s are dielectric","rect":[148.8064727783203,157.466064453125,356.7719359137964,149.8717041015625]},{"page":203,"text":"permittivity (real part) and conductivity.","rect":[160.75857543945313,167.4420166015625,298.936312409186,159.84765625]},{"page":203,"text":"Relaxation (collision) time for motion of ions of mass mi and friction","rect":[148.80616760253907,177.02244567871095,385.1611380019657,169.76675415039063]},{"page":203,"text":"coefficient x.","rect":[160.7592010498047,187.26092529296876,205.30163833692036,179.47183227539063]},{"page":203,"text":"Debye relaxation time of dipoles in a liquid (a is molecular radius, Z is","rect":[148.8064422607422,197.31259155273438,385.15625460624019,189.71823120117188]},{"page":203,"text":"viscosity of medium).","rect":[160.75770568847657,207.28854370117188,235.44441482617817,199.69418334960938]},{"page":203,"text":"Debye relaxation times in a liquid crystal (j||, j⊥ are amplification","rect":[148.806884765625,217.20773315429688,372.4054006972782,209.61337280273438]},{"page":203,"text":"factors).","rect":[160.7593536376953,226.8450164794922,188.75703689649067,219.58932495117188]},{"page":203,"text":"References","rect":[53.812843322753909,255.2021026611328,109.59614448282879,246.41697692871095]},{"page":203,"text":"1.","rect":[58.06126022338867,281.0,64.40706131055318,274.90386962890627]},{"page":203,"text":"2.","rect":[58.06126022338867,291.0,64.40706131055318,284.87982177734377]},{"page":203,"text":"3.","rect":[58.06126022338867,321.0,64.40706131055318,314.7509765625]},{"page":203,"text":"4.","rect":[58.06126022338867,331.0,64.40706131055318,324.7269287109375]},{"page":203,"text":"5.","rect":[58.06041717529297,351.0,64.40621826245747,344.5205078125]},{"page":203,"text":"6.","rect":[58.05955123901367,391.0,64.40535232617818,384.4184265136719]},{"page":203,"text":"7.","rect":[58.05955123901367,420.16522216796877,64.40535232617818,414.4588623046875]},{"page":203,"text":"8.","rect":[58.05955123901367,450.0931091308594,64.40535232617818,444.2682189941406]},{"page":203,"text":"9.","rect":[58.05955123901367,470.0560607910156,64.40535232617818,464.1634216308594]},{"page":203,"text":"10.","rect":[53.810489654541019,489.9402160644531,64.3868281562563,484.1153259277344]},{"page":203,"text":"11.","rect":[53.810489654541019,500.0,64.3868281562563,494.0345764160156]},{"page":203,"text":"12.","rect":[53.810489654541019,520.0,64.3868281562563,513.9865112304688]},{"page":203,"text":"13.","rect":[53.810489654541019,539.7066040039063,64.3868281562563,533.8817138671875]},{"page":203,"text":"14.","rect":[53.80965042114258,570.0,64.38598892285786,563.7528686523438]},{"page":203,"text":"De Gennes, Prost, J.: The Physics of Liquid Crystals, 2nd edn. Clarendon, Oxford (1995)","rect":[68.59698486328125,282.44744873046877,374.1558846817701,274.8022766113281]},{"page":203,"text":"Dunmur, D., Toriyama, K.: Magnetic properties of liquid crystals. In: Demus, D., Goodby, J.,","rect":[68.59698486328125,292.42340087890627,385.19751236035787,284.82904052734377]},{"page":203,"text":"Gray, G.W., Spiess, H.-W., Vill, V. (eds.) Physical Properties of Liquid Crystals, pp. 102–112.","rect":[68.59698486328125,302.3993835449219,385.19592544629537,294.8050231933594]},{"page":203,"text":"Wiley-VCH, Weinheim (1999)","rect":[68.59698486328125,312.3185729980469,174.87564176184825,304.7242126464844]},{"page":203,"text":"Kittel, Ch: Introduction to Solid State Physics, 4th edn. Wiley, New-York (1971)","rect":[68.59698486328125,322.2945556640625,346.68109220130136,314.7001953125]},{"page":203,"text":"Brochard, F., de Gennes, P.G.: Theory of magnetic suspensions in liquid crystals. J. Phys.","rect":[68.59698486328125,332.2705078125,385.1686732490297,324.6761474609375]},{"page":203,"text":"(Paris) 31, 691–708 (1970)","rect":[68.59698486328125,341.9078063964844,161.05447476966075,334.38116455078127]},{"page":203,"text":"Maier, W., Meier, G.: Eine einfache Theorie der dielektrischen Eigenschaften homogen","rect":[68.59614562988281,352.1656799316406,385.17807525782509,344.5713195800781]},{"page":203,"text":"orientierter kristallinflu€ssiger Phasen des nematischen Typs, Z. Naturforschg. 16a, 262–267","rect":[68.59613800048828,362.1416320800781,385.1746878066532,354.0]},{"page":203,"text":"(1961); Hauptdielektrizit€atskonstanten der homogen geordneten kristallinflu€ssigen Phase des","rect":[68.59613800048828,372.11761474609377,385.18396456717769,364.0]},{"page":203,"text":"p-Azoxyanizols, ibid, 16a, 470–477 (1961)","rect":[68.59612274169922,382.0935363769531,215.773041664192,374.48223876953127]},{"page":203,"text":"Dunmur, D., Toriyama, K.: Dielectric properties. In: Demus, D., Goodby, J., Gray, G.W.,","rect":[68.59527587890625,392.0127868652344,385.1694972236391,384.4184265136719]},{"page":203,"text":"Spiess, H.-W., Vill, V. (eds.) Physical Properties of Liquid Crystals, pp. 129–150. Wiley-","rect":[68.59526824951172,401.9887390136719,385.16363614661386,394.34356689453127]},{"page":203,"text":"VCH, Weinheim (1999)","rect":[68.59526824951172,411.62603759765627,151.25315946204356,404.3703308105469]},{"page":203,"text":"de Jeu, W.H.: The dielectric permittivity of liquid crystals. In: Liebert, L. (ed.) Liquid","rect":[68.59527587890625,421.8839111328125,385.1704153457157,414.28955078125]},{"page":203,"text":"Crystals, pp.109–174 (series Solid State Physics, eds.: Ehrenreich, H., Seitz, F., Turnbull,","rect":[68.59526824951172,431.8598937988281,385.1568934638735,424.2655334472656]},{"page":203,"text":"D.), Academic, New York (1978)","rect":[68.59526824951172,441.4971923828125,183.7106637345045,434.2414855957031]},{"page":203,"text":"de Jeu, W.H.: Physical Properties of Liquid Crystalline Materials. Gordon & Breach, New","rect":[68.59527587890625,451.8117980957031,385.14414002043216,444.2174377441406]},{"page":203,"text":"York (1980)","rect":[68.59526824951172,461.3923645019531,110.92176908118417,454.13665771484377]},{"page":203,"text":"de Jeu, W.H., Lathouwers, Th W., Bordewijk, P.: Dielectric properties of di-n-heptyl azox-","rect":[68.59527587890625,471.7070007324219,385.1645516739576,464.1126403808594]},{"page":203,"text":"ybenzene in the nematic and in the smectic A phases. Phys. Rev. Lett. 32, 40–43 (1974)","rect":[68.59534454345703,481.6829528808594,370.6953133927076,473.8176574707031]},{"page":203,"text":"€","rect":[76.12825012207031,485.0,80.35878509669229,483.0]},{"page":203,"text":"Frolich, H.: Theory of dielectrics, 2nd ed., Chapt. 3, Clarendon Press, London (1978)","rect":[68.59452056884766,491.6589050292969,360.7823800430982,484.0645446777344]},{"page":203,"text":"Wrobel, S.: Dielectric relaxation spectroscopy. In: Haase, W., Wrobel, S. (eds.) Relaxation","rect":[68.59452056884766,501.5781555175781,385.20261139063759,493.9837951660156]},{"page":203,"text":"Phenomena, pp. 13–35. Springer-Verlag, Berlin (2003)","rect":[68.59451293945313,511.5541076660156,257.00985807044199,503.908935546875]},{"page":203,"text":"Thoen, J.: Thermal methods. In: Demus, D., Goodby, J., Gray, G.W., Spiess, H.-W., Vill, V.","rect":[68.59452056884766,521.530029296875,385.161165924811,513.9356689453125]},{"page":203,"text":"(eds.) Physical Properties of Liquid Crystals, pp. 208–232. Wiley-VCH, Weinheim (1999)","rect":[68.59451293945313,531.5060424804688,377.8880624161451,523.9116821289063]},{"page":203,"text":"Zammit, U., Martinelli, M., Pizzoferrato, R., Scudieri, F., Martelucci, M.: Thermal conduc-","rect":[68.59452056884766,539.7066040039063,385.18658536536386,533.8308715820313]},{"page":203,"text":"tivity, diffusivity, and heat capacity studies at the smdctic-A – nematic transition in alkylcya-","rect":[68.59451293945313,551.4011840820313,385.13156217200449,543.8068237304688]},{"page":203,"text":"nobiphenyl liquid crystals. Phys. Rev. A 41, 1153–1155 (1990)","rect":[68.59451293945313,561.3771362304688,285.21734708411386,553.715087890625]},{"page":203,"text":"Blinov, L.M.: Electro-Optical and Magneto-Optical Properties of Liquid Crystals. Wiley,","rect":[68.59368133544922,571.29638671875,385.13830825879537,563.6851196289063]},{"page":203,"text":"Chichester (1983)","rect":[68.59368133544922,580.9337158203125,129.83998197180919,573.677978515625]},{"page":204,"text":"Chapter8","rect":[53.812843322753909,72.10812377929688,114.14115996551633,59.223289489746097]},{"page":204,"text":"Elasticity and Defects","rect":[53.812843322753909,91.18268585205078,201.56806577594785,76.0426254272461]},{"page":204,"text":"8.1 Tensor of Elasticity","rect":[53.812843322753909,212.6539764404297,181.09315242671557,201.0480499267578]},{"page":204,"text":"8.1.1 Hooke’s Law","rect":[53.812843322753909,239.85919189453126,155.51166631265859,231.3250732421875]},{"page":204,"text":"Liquids have a finite and very high (as compared to gases) compressibility modulus","rect":[53.812843322753909,269.5052795410156,385.17755521880346,260.57073974609377]},{"page":204,"text":"and zero static shear modulus. For example, a boat floating on water can easily be","rect":[53.812843322753909,281.4648132324219,385.1487201519188,272.5302734375]},{"page":204,"text":"shifted just by a finger. Even very viscous liquids, for instance, polymers, rubber,","rect":[53.812843322753909,293.42437744140627,385.1328091194797,284.48980712890627]},{"page":204,"text":"and, surprisingly, stain-glass windows have no static shear modulus although they","rect":[53.812843322753909,305.3271484375,385.1646660905219,296.392578125]},{"page":204,"text":"have a dynamic shear modulus at a short time scale or at high frequencies. In fact, to","rect":[53.812843322753909,317.28668212890627,385.1965264420844,308.35211181640627]},{"page":204,"text":"shear a liquid, we should not overcome any potential barrier. In contrast to liquids,","rect":[53.812843322753909,329.2462158203125,385.1586575081516,320.3116455078125]},{"page":204,"text":"the isotropic solids, e.g., ceramics or fine polycrystalline materials have not only","rect":[53.812843322753909,341.2057800292969,385.1238640885688,332.271240234375]},{"page":204,"text":"compressibility modulus but also one shear modulus finite. As to single crystals,","rect":[53.812843322753909,353.165283203125,385.09893460776098,344.230712890625]},{"page":204,"text":"they have many elastic moduli; the lower symmetry the larger a number of their","rect":[53.812843322753909,365.1248474121094,385.1079343399204,356.1903076171875]},{"page":204,"text":"moduli.","rect":[53.812843322753909,375.05242919921877,84.66691251303439,368.14984130859377]},{"page":204,"text":"What about liquid crystals?","rect":[65.76486206054688,389.0439453125,176.34122503473129,380.109375]},{"page":204,"text":"1.","rect":[53.812843322753909,405.0,61.27850003744845,398.0801086425781]},{"page":204,"text":"2.","rect":[53.812843322753909,465.0,61.27850003744845,457.821044921875]},{"page":204,"text":"In nematic liquid crystals we see a novel feature. There is no shear modulus as in","rect":[66.27452087402344,406.95489501953127,385.25124445966255,398.02032470703127]},{"page":204,"text":"isotropic liquids, but the orientational, for example, torsional elasticity appears.","rect":[66.27452087402344,418.9143981933594,385.2711453011203,409.9798583984375]},{"page":204,"text":"Such elasticity is also characteristic of crystals but, in that case, the corresponding","rect":[66.27452087402344,430.8739318847656,385.1736992936469,421.93939208984377]},{"page":204,"text":"moduli are much smaller than the other moduli. The orientational elasticity","rect":[66.27452087402344,442.83349609375,385.17461482099068,433.89892578125]},{"page":204,"text":"determines almost all fascinating properties and applications of nematics.","rect":[66.27452087402344,454.73626708984377,358.5331692269016,445.80169677734377]},{"page":204,"text":"Other liquid crystal phases combine many types of elasticity of solid crystals","rect":[66.27452087402344,466.6958312988281,385.10190214263158,457.7413635253906]},{"page":204,"text":"with orientational elasticity and we encounter enormous variations of the mag-","rect":[66.27452087402344,478.6553649902344,385.1368344864048,469.7208251953125]},{"page":204,"text":"nitude of elasticity moduli from almost zero to that typical of three-dimensional","rect":[66.27452087402344,490.6148986816406,385.203505111428,481.68035888671877]},{"page":204,"text":"crystals.","rect":[66.27452087402344,502.574462890625,99.29263730551486,493.639892578125]},{"page":204,"text":"Everything that is discussed below is based on the Hooke law. Consider a very","rect":[65.76486206054688,520.4853515625,385.11785212567818,511.5309143066406]},{"page":204,"text":"simple example, a one-dimensional reversible extension of a long rod with cross-","rect":[53.81288146972656,532.4449462890625,385.1109555801548,523.5104370117188]},{"page":204,"text":"section A along the x-axis, see Fig. 8.1. The force per unit area","rect":[53.81288146972656,544.4044799804688,307.36450994684068,535.469970703125]},{"page":204,"text":"F","rect":[186.75909423828126,565.4482421875,192.87097924627029,558.8743286132813]},{"page":204,"text":"l","rect":[216.15792846679688,565.6275024414063,218.95506150791239,558.63525390625]},{"page":204,"text":"ux","rect":[242.2713165283203,566.9846801757813,250.37076706781765,561.005859375]},{"page":204,"text":"sx ¼ A ¼ K L ¼ K x ¼ Ku","rect":[163.25146484375,579.1412963867188,275.6862725358344,565.6185913085938]},{"page":204,"text":"(8.1)","rect":[366.09735107421877,573.9359741210938,385.16961036531105,565.4595947265625]},{"page":204,"text":"L.M. Blinov, Structure and Properties of Liquid Crystals,","rect":[53.812843322753909,623.7623291015625,251.63688919141254,616.1510620117188]},{"page":204,"text":"DOI 10.1007/978-90-481-8829-1_8, # Springer ScienceþBusiness Media B.V. 2011","rect":[53.812843322753909,633.698486328125,347.38995880274697,625.4920043945313]},{"page":204,"text":"189","rect":[372.4981994628906,622.1282958984375,385.18979400782509,616.2357177734375]},{"page":205,"text":"190","rect":[53.812843322753909,42.62403869628906,66.50444931178018,36.73143005371094]},{"page":205,"text":"Fig. 8.1 Illustration of the","rect":[53.812843322753909,67.58130645751953,146.1121153571558,59.6313591003418]},{"page":205,"text":"linear one-dimensional","rect":[53.812843322753909,75.76238250732422,132.10989097800317,69.89517211914063]},{"page":205,"text":"displacement of a solid","rect":[53.812843322753909,87.4087142944336,133.10577911524698,79.81436157226563]},{"page":205,"text":"material under an external","rect":[53.812843322753909,95.65752410888672,143.55856041159692,89.79031372070313]},{"page":205,"text":"force (Hooke’s law)","rect":[53.812843322753909,107.02202606201172,122.50827115637948,99.76632690429688]},{"page":205,"text":"8 Elasticity and Defects","rect":[303.50408935546877,44.274986267089847,385.17355807303707,36.68062973022461]},{"page":205,"text":"x","rect":[293.53680419921877,80.56692504882813,297.94908559823588,76.82376861572266]},{"page":205,"text":"p","rect":[312.59674072265627,81.26354217529297,317.00902212167338,75.78478240966797]},{"page":205,"text":".","rect":[303.7999572753906,89.88259887695313,312.5217059426559,86.3724136352539]},{"page":205,"text":"L","rect":[304.9088134765625,109.38995361328125,310.2403201670415,103.89519500732422]},{"page":205,"text":"causes a relative displacement l/L or ux/x at any point p(x) with corresponding","rect":[53.812843322753909,157.84408569335938,385.11721125653755,148.8896026611328]},{"page":205,"text":"proportionality coefficient that is elastic modulus K. Therefore, in the simplest one-","rect":[53.814205169677737,169.80361938476563,385.1181882461704,160.8690643310547]},{"page":205,"text":"dimensional case, the Hooke law relates the two vector projections, the stress sx","rect":[53.81417465209961,181.76318359375,385.1820833351628,172.82862854003907]},{"page":205,"text":"and the relative deformation (strain) ux by scalar Young modulus K (dyn/cm2 or","rect":[53.812843322753909,193.66635131835938,385.1521237930454,182.61546325683595]},{"page":205,"text":"N/m2):","rect":[53.81432342529297,205.2275848388672,81.44989640781472,194.57508850097657]},{"page":205,"text":"sx ¼ Kux","rect":[200.12696838378907,228.94021606445313,238.36197800531765,220.82989501953126]},{"page":205,"text":"(8.2)","rect":[366.09765625,229.1468963623047,385.1699155410923,220.6705322265625]},{"page":205,"text":"Theelastic energyaccumulatedinthe volumeLdue tothe deformationisgivenby","rect":[65.76622772216797,253.46441650390626,385.2844170670844,244.5298614501953]},{"page":205,"text":"l","rect":[139.85691833496095,273.61077880859377,141.8149059371223,268.7162170410156]},{"page":205,"text":"l","rect":[173.95730590820313,273.61077880859377,175.91529351036449,268.7162170410156]},{"page":205,"text":"W ¼ ð Fdl ¼ ð ALK ldl ¼ K2Ll2 A ¼ K2ux2 LA ¼ 21Ku2xV","rect":[115.89759063720703,297.59185791015627,322.5883864728328,273.873291015625]},{"page":205,"text":"0","rect":[138.5539093017578,303.95013427734377,142.03787300667129,299.1531677246094]},{"page":205,"text":"0","rect":[172.654296875,303.95013427734377,176.13826057991347,299.1531677246094]},{"page":205,"text":"and the density of elastic energy in the one-dimensional case is given by","rect":[53.813777923583987,328.49322509765627,346.3759698014594,319.47900390625]},{"page":205,"text":"g ¼ WV ¼ 21Ku2x:","rect":[186.87271118164063,363.21990966796877,252.09622132462403,342.7043151855469]},{"page":205,"text":"(8.3)","rect":[366.0968322753906,357.9257507324219,385.1690915664829,349.44940185546877]},{"page":205,"text":"The stress always causes strain and fundamental Eqs. (8.2) and (8.3) have to be","rect":[65.7653579711914,386.66473388671877,385.1531452007469,377.73016357421877]},{"page":205,"text":"generalized to describe anisotropic media.","rect":[53.813350677490237,398.6242980957031,223.79343839194065,389.68975830078127]},{"page":205,"text":"8.1.2 Stress, Strain and Elasticity Tensors","rect":[53.812843322753909,448.7912902832031,269.30004005167646,438.15350341796877]},{"page":205,"text":"8.1.2.1 Stress Tensor","rect":[53.812843322753909,474.5606384277344,149.2966235943016,467.0603942871094]},{"page":205,"text":"Generally, any vector of a surface force acting on a body can be decomposed into","rect":[53.812843322753909,500.2527160644531,385.1526726823188,491.31817626953127]},{"page":205,"text":"tangential ( fx, fy, shear) and normal (fz, pressure) components (index j ¼ x, y, z) as","rect":[53.812843322753909,513.1290283203125,385.1412393008347,503.18109130859377]},{"page":205,"text":"shown in Fig. 8.2a. In its turn, the element of surface A is a vector characterized by","rect":[53.81338119506836,524.1153564453125,385.1711358170844,515.1808471679688]},{"page":205,"text":"its area A and outward-directed unit vector s that has also three projections in the","rect":[53.814369201660159,536.074951171875,385.1751178569969,527.1404418945313]},{"page":205,"text":"Cartesian laboratory frame (index i). Therefore, the second rank stress tensor [1] is","rect":[53.81438446044922,548.0344848632813,385.18906034575658,539.0999755859375]},{"page":205,"text":"defined as","rect":[53.814414978027347,557.9620971679688,94.27031339751437,551.0595092773438]},{"page":205,"text":"si fj","rect":[224.2015380859375,583.9943237304688,236.92206841270824,574.2025146484375]},{"page":205,"text":"sij ¼A","rect":[199.90142822265626,594.6954956054688,233.88201135076248,583.2348022460938]},{"page":205,"text":"(8.4)","rect":[366.09686279296877,589.579833984375,385.16912208406105,581.1034545898438]},{"page":206,"text":"8.1 Tensor of Elasticity","rect":[53.81344223022461,44.275291442871097,134.20967620997355,36.68093490600586]},{"page":206,"text":"a","rect":[69.50361633300781,68.23306274414063,75.05887873024166,62.64430618286133]},{"page":206,"text":"F","rect":[96.59339141845703,88.4835205078125,101.47727533362964,82.7407455444336]},{"page":206,"text":"f","rect":[127.20619201660156,126.4278564453125,129.42831923168829,120.60509490966797]},{"page":206,"text":"z","rect":[129.4284210205078,128.42742919921876,132.4258947818658,125.29010009765625]},{"page":206,"text":"s","rect":[144.05419921875,76.6814956665039,148.49845364892344,72.21044921875]},{"page":206,"text":"Fig. 8.2 Vector of force F acting on a body and","rect":[53.812843322753909,199.30767822265626,228.74039215235636,191.3577423095703]},{"page":206,"text":"pressure) components (a) and the nine components","rect":[53.81344223022461,209.2159423828125,229.02106173514643,201.62158203125]},{"page":206,"text":"element vector is directed outward from the bulk","rect":[53.81344223022461,217.46473693847657,221.37140411524698,211.5975341796875]},{"page":206,"text":"b","rect":[247.80355834960938,70.06369018554688,253.90835209908759,62.75531768798828]},{"page":206,"text":"syz","rect":[306.852294921875,91.34291076660156,318.00388245276425,84.49037170410156]},{"page":206,"text":"sxz","rect":[286.7323303222656,104.55419921875,297.88294129065488,98.97938537597656]},{"page":206,"text":"191","rect":[372.49798583984377,42.62434387207031,385.1895803847782,36.73173522949219]},{"page":206,"text":"x","rect":[250.10562133789063,174.23428344726563,254.54987576806406,169.97918701171876]},{"page":206,"text":"szx","rect":[291.3284606933594,148.56561279296876,302.47903432848946,142.9907989501953]},{"page":206,"text":"syx","rect":[308.494873046875,162.22679138183595,319.9812010765363,155.3738555908203]},{"page":206,"text":"its tangential ( fx, fy, shear) and normal ( fz,","rect":[232.01820373535157,200.03880310058595,385.2064845283266,191.61172485351563]},{"page":206,"text":"of the stress tensor (b). Note that the surface","rect":[231.564453125,208.8772735595703,385.1294798591089,201.62158203125]},{"page":206,"text":"(dimension of pressure, dyn/cm2 or N/m2 in the SI system)","rect":[53.812843322753909,270.63897705078127,291.37529884187355,259.5879211425781]},{"page":206,"text":"Generally sij has 3 \u0002 3 ¼ 9 components, illustrated by Fig. 8.2b. The diagonal","rect":[65.76610565185547,283.5151062011719,385.2388749844749,273.6639404296875]},{"page":206,"text":"components (s11, s22, s33) corresponds to pressure, off-diagonal ones correspond to","rect":[53.81345748901367,294.50152587890627,385.2556084733344,285.56695556640627]},{"page":206,"text":"shear. Since sij is symmetric tensor (sij ¼ sji), only six components are different.","rect":[53.81320571899414,307.43438720703127,385.2587551644016,297.52655029296877]},{"page":206,"text":"Usually, in addition to the surface forces, the volume forces like gravitation or electric","rect":[53.813411712646487,318.4207763671875,385.2318805523094,309.4862060546875]},{"page":206,"text":"force are included into the stress tensor.","rect":[53.813411712646487,328.348388671875,211.2233089974094,321.44580078125]},{"page":206,"text":"8.1.2.2 Strain Tensor","rect":[53.813411712646487,370.4373474121094,150.42998541070785,362.9371032714844]},{"page":206,"text":"We consider a piece of soft matter in which the distortion is not uniform in space","rect":[53.813411712646487,396.1294250488281,385.0806049175438,387.19488525390627]},{"page":206,"text":"but rather local [2]. Any displacement of point p to point p’ caused by the stress","rect":[53.813411712646487,408.0889587402344,385.09653104888158,399.1544189453125]},{"page":206,"text":"tensor and shown in Fig. 8.3a–c can be considered as a combintion of four basic","rect":[53.81344223022461,420.0484924316406,385.12937200738755,411.11395263671877]},{"page":206,"text":"displacements. They are (1) translations, (2) rotation of the entire body as a solid","rect":[53.81245040893555,431.9512634277344,385.13137141278755,423.0167236328125]},{"page":206,"text":"piece without deformation, (3) shear distortion, (4) expansion or compression.","rect":[53.81245040893555,443.9107971191406,369.0224880745578,434.97625732421877]},{"page":206,"text":"Now we are going to discuss the components of the correspondent tensors","rect":[65.76447296142578,455.870361328125,385.0975686465378,446.935791015625]},{"page":206,"text":"(translation is excluded). We go back to Eq. (8.2) and instead of ux/x ¼ l/L write:","rect":[53.81245040893555,467.8299255371094,385.16840989658427,458.8770446777344]},{"page":206,"text":"ux ¼ exxx or @ux ¼ exx@x","rect":[53.813655853271487,479.18609619140627,155.18180120660629,470.5279235839844]},{"page":206,"text":"For small distortions, the coefficient exx ¼ @ux=@x will be a function of x, y, z,","rect":[65.76512145996094,492.0794982910156,385.1827663948703,482.14892578125]},{"page":206,"text":"describing an extension along the x-axis. Along y- and z-axes the extensions wil be","rect":[53.8140983581543,503.7103576660156,385.1549457378563,494.77581787109377]},{"page":206,"text":"described by coefficients","rect":[53.816078186035159,515.6698608398438,153.7603341494675,506.7353515625]},{"page":206,"text":"@uy","rect":[201.9966583251953,542.9573364257813,215.81720871820827,532.876708984375]},{"page":206,"text":"eyy ¼ @y ;ezz ¼ @z","rect":[177.5857391357422,555.7801513671875,257.60219205962377,542.357421875]},{"page":206,"text":"(8.5)","rect":[366.0970764160156,548.5429077148438,385.1693357071079,539.947021484375]},{"page":206,"text":"The diagonal tensor coefficients with equal suffixes describe compression-dila-","rect":[65.76566314697266,582.2689819335938,385.1316159805454,573.33447265625]},{"page":206,"text":"tation.","rect":[53.81364059448242,592.166748046875,79.09733243490939,585.2940063476563]},{"page":207,"text":"192","rect":[53.8128662109375,42.62428283691406,66.50447219996377,36.73167419433594]},{"page":207,"text":"a","rect":[57.26823043823242,68.23287963867188,62.82349283546627,62.64412307739258]},{"page":207,"text":"c","rect":[280.58575439453127,68.23287963867188,286.1410167917651,62.64412307739258]},{"page":207,"text":"8 Elasticity and Defects","rect":[303.5041198730469,44.275230407714847,385.17358859061519,36.68087387084961]},{"page":207,"text":".P","rect":[77.364501953125,119.06123352050781,89.57301514742211,113.42243957519531]},{"page":207,"text":"q","rect":[155.92430114746095,140.6877899169922,160.08879135990763,135.04098510742188]},{"page":207,"text":"-","rect":[155.92430114746095,142.0,160.312602734089,141.0]},{"page":207,"text":"2","rect":[155.92430114746095,148.0582275390625,159.92093282927159,142.2754669189453]},{"page":207,"text":"q","rect":[212.32876586914063,165.93760681152345,216.4932560815873,160.29080200195313]},{"page":207,"text":"-","rect":[212.32876586914063,167.0,216.7170674557687,166.0]},{"page":207,"text":"2","rect":[212.32876586914063,173.30804443359376,216.32539755095127,167.52528381347657]},{"page":207,"text":"q","rect":[273.3189697265625,141.9363250732422,277.4834599390092,136.28952026367188]},{"page":207,"text":"-","rect":[273.3189697265625,143.0,277.7072713131906,142.0]},{"page":207,"text":"2","rect":[273.3189697265625,149.3067626953125,277.31560140837316,143.5240020751953]},{"page":207,"text":"q","rect":[328.4205322265625,170.8269805908203,332.5850224390092,165.18017578125]},{"page":207,"text":"-","rect":[328.4205322265625,172.0,332.8088338131906,171.0]},{"page":207,"text":"2","rect":[328.4205322265625,178.19741821289063,332.41716390837316,172.41465759277345]},{"page":207,"text":"Fig. 8.3 A non-deformed body (a) and illustration of difference between a shear distortion (b) and","rect":[53.812843322753909,200.38458251953126,385.19403595118447,192.4346466064453]},{"page":207,"text":"pure body rotation (c)","rect":[53.81287384033203,210.29281616210938,128.79656308753185,202.69845581054688]},{"page":207,"text":"There is also shear distortion, which can be described by angles y/2 as shown in","rect":[65.76496887207031,257.9991455078125,385.14284602216255,248.7558135986328]},{"page":207,"text":"Fig. 8.3b. In this case, the displacement along the y-direction is proportional to x","rect":[53.81294250488281,269.90191650390627,385.16074407770005,260.96734619140627]},{"page":207,"text":"and vice versa:","rect":[53.81296157836914,279.9291076660156,114.44405992099832,272.9268798828125]},{"page":207,"text":"y","rect":[156.00071716308595,303.3124694824219,160.9778222184516,296.14093017578127]},{"page":207,"text":"y","rect":[193.27330017089845,303.3124694824219,198.2504052262641,296.14093017578127]},{"page":207,"text":"ux ¼ 2y;uy ¼ 2x and ux ¼ exyy;uy ¼ eyxx","rect":[134.1344757080078,316.84576416015627,304.8387836284813,303.19390869140627]},{"page":207,"text":"with the same angle for exy ¼ eyx ¼ y=2: However, we cannot yet define exy and eyx","rect":[53.81351852416992,341.3294677734375,385.1770479347722,330.8111572265625]},{"page":207,"text":"as simple derivatives of type (∂ux/∂y) or (∂uy/∂x) because similar displacements","rect":[53.812843322753909,353.2890930175781,385.15402616606908,342.3753967285156]},{"page":207,"text":"describe the pure rotation of the body without any deformation as shown in","rect":[53.813228607177737,364.2755126953125,385.1431511979438,355.3409423828125]},{"page":207,"text":"Fig. 8.3c.","rect":[53.813228607177737,376.23504638671877,91.81838651205783,367.30047607421877]},{"page":207,"text":"y","rect":[201.54344177246095,397.6860656738281,206.5205468278266,390.5145263671875]},{"page":207,"text":"y","rect":[248.16246032714845,397.6860656738281,253.1395653825141,390.5145263671875]},{"page":207,"text":"ux ¼ 2y;uy ¼ \u00032x","rect":[179.67822265625,411.2193603515625,259.29624212457505,399.8484802246094]},{"page":207,"text":"Note the opposite sign for the y angle in the uy component. To obtain the pure","rect":[65.76551055908203,435.70306396484377,385.1167377300438,425.4857482910156]},{"page":207,"text":"shear we may construct a combination","rect":[53.813716888427737,446.6894226074219,208.85849848798285,437.7548828125]},{"page":207,"text":"exy ¼ eyx ¼ 21\u0002@@uxy þ @@uyx\u0003","rect":[164.10137939453126,484.8126525878906,274.9053923640825,460.92742919921877]},{"page":207,"text":"Then the negative displacements (rotation) will be excluded and positive (shear)","rect":[65.76644134521485,508.3014221191406,385.1603025039829,499.36688232421877]},{"page":207,"text":"remains unchanged. Even more generally, we construct two combinations, one for","rect":[53.814414978027347,520.260986328125,385.1782163223423,511.32647705078127]},{"page":207,"text":"pure shear","rect":[53.814414978027347,532.1109619140625,96.95496763335422,523.2660522460938]},{"page":207,"text":"eij ¼ 12\u0002@@uxij þ @@uxji\u0003","rect":[178.66012573242188,570.6742553710938,260.34727163654346,546.4017944335938]},{"page":207,"text":"(8.6)","rect":[366.0968017578125,562.6005249023438,385.1690610489048,554.0643920898438]},{"page":207,"text":"and the other for pure rotation","rect":[53.81332015991211,594.22998046875,175.53241816571723,585.2954711914063]},{"page":208,"text":"8.1 Tensor of Elasticity","rect":[53.812843322753909,44.274620056152347,134.2090811172001,36.68026351928711]},{"page":208,"text":"193","rect":[372.49737548828127,42.62367248535156,385.1889700332157,36.73106384277344]},{"page":208,"text":"oij ¼ 12\u0002@@uxij \u0003 @@uxji\u0003","rect":[177.52590942382813,83.67259979248047,261.4234587703325,59.40016555786133]},{"page":208,"text":"(8.7)","rect":[366.0969543457031,75.59891510009766,385.1692136367954,67.12255096435547]},{"page":208,"text":"which are, in fact, symmetric (shear) and antisymmetric (rotation) parts of the","rect":[53.8134880065918,107.2284164428711,385.1155170269188,98.29386901855469]},{"page":208,"text":"common tensor describing “shear + rotation”.","rect":[53.8134880065918,119.18795013427735,238.13262601401096,110.25340270996094]},{"page":208,"text":"Finally, the total strain is described by a symmetric second rank (3 \u0002 3) strain","rect":[65.76551055908203,131.14749145507813,385.11748591474068,122.21293640136719]},{"page":208,"text":"tensor ekl with diagonal elements ∂ui/∂xi (8.5) related to expansion/compression","rect":[53.8125114440918,143.05032348632813,385.1270684342719,133.10975646972657]},{"page":208,"text":"and off-diagonal elements ∂ui/∂xj (8.6) related to shear.","rect":[53.81412887573242,155.98326110839845,279.9436916878391,145.0692901611328]},{"page":208,"text":"8.1.2.3 Tensor of Elasticity","rect":[53.812862396240237,197.238525390625,175.02329340985785,187.56690979003907]},{"page":208,"text":"For small strain, the relationship between the strain and stress is linear as in the","rect":[53.812862396240237,220.75921630859376,385.17160833551255,211.8047332763672]},{"page":208,"text":"Hooke law [3]:","rect":[53.812862396240237,232.11119079589845,114.87100083896707,223.7842254638672]},{"page":208,"text":"sij ¼ X Kijklekl ¼Kijklekl","rect":[167.95281982421876,260.950439453125,270.56852227013015,247.0157470703125]},{"page":208,"text":"k;l","rect":[195.82235717773438,268.6527404785156,202.93520623009105,262.53802490234377]},{"page":208,"text":"(8.8)","rect":[366.09765625,258.16748046875,385.1699155410923,249.69113159179688]},{"page":208,"text":"Here, the right part of the equation is abbreviation of the middle part suggested","rect":[65.76622772216797,293.1407470703125,385.17397395185005,284.2061767578125]},{"page":208,"text":"by Einstein: since symbols k and l appear twice as suffixes at Kijkl and ekl., we may","rect":[53.81417465209961,306.0174255371094,385.16704646161568,296.14581298828127]},{"page":208,"text":"remember this and remove a bulky symbol of the sum. By this convention, we","rect":[53.81426239013672,317.0604553222656,385.1670612163719,308.12591552734377]},{"page":208,"text":"always must make a summation over the repeated suffixes. The elasticity tensor","rect":[53.81426239013672,329.0199890136719,385.1173032364048,320.08544921875]},{"page":208,"text":"Kijkl is a fourth rank tensor with 9 \u0002 9 ¼ 81 components (81 mathematically","rect":[53.81426239013672,341.8963623046875,385.11162653974068,332.0452880859375]},{"page":208,"text":"possible elastic moduli!). However, even in crystals of the lowest symmetry (the","rect":[53.81264114379883,352.93939208984377,385.1446002788719,344.00482177734377]},{"page":208,"text":"triclinic system) due to physical equivalence of Kijkl ¼ Kjikl¼ Kijlk ¼ Kklij, a number","rect":[53.81264114379883,365.8156433105469,385.1252683242954,355.9075927734375]},{"page":208,"text":"of moduli reduces to 21.","rect":[53.814292907714847,374.77008056640627,152.41274686362034,367.86749267578127]},{"page":208,"text":"With enhanced phase symmetry a number of moduli further reduces and we","rect":[65.76631927490235,388.7615966796875,385.1700824566063,379.8270263671875]},{"page":208,"text":"have:","rect":[53.814292907714847,398.69915771484377,75.46470506259988,391.78656005859377]},{"page":208,"text":"–","rect":[53.814292907714847,417.0,58.7913979630805,411.0]},{"page":208,"text":"–","rect":[53.814292907714847,429.0,58.7913979630805,423.0]},{"page":208,"text":"–","rect":[53.81368637084961,441.0,58.790791426215267,435.0]},{"page":208,"text":"–","rect":[53.81368637084961,465.0,58.790791426215267,459.0]},{"page":208,"text":"–","rect":[53.81368637084961,477.0,58.790791426215267,471.0]},{"page":208,"text":"Six moduli for the tetragonal system (e.g., of symmetry D4h)","rect":[67.29627990722656,418.6320495605469,312.2574704727329,409.697509765625]},{"page":208,"text":"Five moduli for hexagonal system (D6h)","rect":[67.29627990722656,430.5920715332031,229.1023648819126,421.65753173828127]},{"page":208,"text":"Three moduli for cubic system (O): one for compression and two for shear,","rect":[67.29566955566406,442.5517883300781,385.1325954964328,433.5973205566406]},{"page":208,"text":"namely, perpendicularly to cube edge and to cube diagonals, respectively)","rect":[67.29566955566406,454.5113525390625,366.43865333406105,445.5767822265625]},{"page":208,"text":"Two for isotropic solid (compressibility and shear)","rect":[67.29566955566406,466.47088623046877,272.1602719864048,457.53631591796877]},{"page":208,"text":"One for an isotropic liquid (compressibility)","rect":[67.29566955566406,478.4304504394531,245.64327369538916,469.4759826660156]},{"page":208,"text":"The density of elastic distortion energy (a scalar quantity) is quadratic in strain:","rect":[65.7657241821289,496.3413391113281,385.1137834317405,487.40679931640627]},{"page":208,"text":"1","rect":[212.53257751464845,517.26953125,217.5096825700141,510.5362548828125]},{"page":208,"text":"gdist ¼ 2 Kijkluijukl","rect":[183.70021057128907,530.87255859375,254.76452141563792,517.4966430664063]},{"page":208,"text":"(8.9)","rect":[366.09765625,525.8142700195313,385.1699155410923,517.337890625]},{"page":208,"text":"After summation over all suffixes we have maximum 21 scalar terms in the","rect":[65.76622772216797,552.3778686523438,385.1720355816063,545.505126953125]},{"page":208,"text":"lowest symmetry case. For the isothermal processes, gdist gives a direct contribution","rect":[53.81417465209961,566.4390258789063,385.13591853192818,557.4646606445313]},{"page":208,"text":"to the free energy of the system.","rect":[53.814022064208987,578.3590087890625,184.20521207358127,569.4244995117188]},{"page":209,"text":"194","rect":[53.812843322753909,42.62525939941406,66.50444931178018,36.73265075683594]},{"page":209,"text":"8.2 Elasticity of Nematics and Cholesterics","rect":[53.812843322753909,70.66844177246094,280.76613502237958,59.06251525878906]},{"page":209,"text":"8 Elasticity and Defects","rect":[303.50408935546877,44.276206970214847,385.17355807303707,36.68185043334961]},{"page":209,"text":"8.2.1 Elementary Distortions","rect":[53.812843322753909,100.03349304199219,204.6176975223796,89.39572143554688]},{"page":209,"text":"The static continuum theory of elasticity for nematic liquid crystals has been","rect":[53.812843322753909,127.5765609741211,385.11882868817818,118.64201354980469]},{"page":209,"text":"developed by Oseen, Ericksen, Frank and others [4]. It was Oseen who introduced","rect":[53.812843322753909,139.53610229492188,385.12676325849068,130.5816192626953]},{"page":209,"text":"the concept of the vector field of the director into the physics of liquid crystals and","rect":[53.812843322753909,151.43887329101563,385.1426934342719,142.4644775390625]},{"page":209,"text":"found that a nematic is completely described by four moduli of elasticity K11, K22,","rect":[53.81280517578125,163.3984375,385.1832241585422,154.46388244628907]},{"page":209,"text":"K33, and K24 [4,5] that will be discussed below. Ericksen was the first who under-","rect":[53.814537048339847,174.94036865234376,385.13826881257668,166.36419677734376]},{"page":209,"text":"stood the importance of asymmetry of the stress tensor for the hydrostatics of","rect":[53.813331604003909,187.31805419921876,385.14827857820168,178.3834991455078]},{"page":209,"text":"nematic liquid crystals [6] and developed the theoretical basis for the general","rect":[53.813331604003909,199.277587890625,385.0875077969749,190.34303283691407]},{"page":209,"text":"continuum theory of liquid crystals based on conservation equations for mass,","rect":[53.81333923339844,211.23715209960938,385.1292995979953,202.30259704589845]},{"page":209,"text":"linear and angular momentum. Later the dynamic approach was further developed","rect":[53.81333923339844,223.19668579101563,385.09050837567818,214.2621307373047]},{"page":209,"text":"by Leslie (Chapter 9) and nowadays the continuum theory of liquid crystal is called","rect":[53.81333923339844,235.15621948242188,385.1701287370063,226.22166442871095]},{"page":209,"text":"Ericksen-Leslie theory. As to Frank, he presented a very clear description of the","rect":[53.81333923339844,247.05899047851563,385.1721881694969,238.1244354248047]},{"page":209,"text":"hydrostatic part of the problem and made a great contribution to the theory of","rect":[53.81333923339844,259.0185241699219,385.14824806062355,250.08396911621095]},{"page":209,"text":"defects. In this Chapter we shall discuss elastic properties of nematics based on the","rect":[53.81333923339844,270.97808837890627,385.17310369684068,262.04351806640627]},{"page":209,"text":"most popular version of Frank [7].","rect":[53.81333923339844,282.9376220703125,192.84579129721409,274.0030517578125]},{"page":209,"text":"8.2.1.1 Specific Features of Elasticity of Nematics","rect":[53.81333923339844,325.1660461425781,271.0182229434128,315.4944152832031]},{"page":209,"text":"As already mentioned, for the fixed direction of the nematic director n the shear","rect":[53.81333923339844,348.6867370605469,385.1372312149204,339.752197265625]},{"page":209,"text":"modulus is absent because the shear distortion is not coupled to stress due to the","rect":[53.813331604003909,360.6462707519531,385.1710590191063,351.71173095703127]},{"page":209,"text":"material “slippage” upon a translation. The compressibility modulus B is the same","rect":[53.813331604003909,372.6058349609375,385.1292194194969,363.6712646484375]},{"page":209,"text":"as for the isotropic liquid. New feature in the elastic properties originates from the","rect":[53.81329345703125,384.50860595703127,385.1720660991844,375.57403564453127]},{"page":209,"text":"spatial dependence of the orientational part of the order parameter tensor, i.e.","rect":[53.81329345703125,396.4681701660156,385.1153225472141,387.53363037109377]},{"page":209,"text":"director n(r). It is assumed that the modulus S of the order parameter QijðrÞ is","rect":[53.81329345703125,409.3039245605469,385.18979276763158,398.8180847167969]},{"page":209,"text":"unchanged. In Fig. 8.4 we can see the difference between the translation and","rect":[53.815147399902347,420.3895263671875,385.14504328778755,411.4549560546875]},{"page":209,"text":"rotation distortion of a nematic.","rect":[53.81417465209961,430.3171081542969,181.47890897299534,423.41448974609377]},{"page":209,"text":"Fig. 8.4 A difference between the translation (a) and rotation (b) distortion of a nematic: there is","rect":[53.812843322753909,574.0244750976563,385.1576278972558,566.0745239257813]},{"page":209,"text":"no elastic modulus for translation of a moving layer with respect of the immobile layer but the","rect":[53.812843322753909,583.9326782226563,385.1567015387964,576.3383178710938]},{"page":209,"text":"twist of the upper layer with respect to the bottom one is described by the twist elastic modulus","rect":[53.812843322753909,593.9086303710938,381.2722519206933,586.3142700195313]},{"page":210,"text":"8.2 Elasticity of Nematics and Cholesterics","rect":[53.812843322753909,44.274620056152347,201.7030380535058,36.68026351928711]},{"page":210,"text":"195","rect":[372.49737548828127,42.62367248535156,385.1889700332157,36.62946701049805]},{"page":210,"text":"Let us assume that a liquid is incompressible, B!1, and discuss orientational","rect":[65.76496887207031,68.2883529663086,385.12788255283427,59.35380554199219]},{"page":210,"text":"(or torsional) elasticity of a nematic. In a solid, the stress is caused by a change in","rect":[53.81294250488281,80.24788665771485,385.1408623795844,71.31333923339844]},{"page":210,"text":"the distance between neighbor points; in a nematic the stress is caused by the","rect":[53.81294250488281,92.20748138427735,385.11304510309068,83.27293395996094]},{"page":210,"text":"curvature of the director field. Now a curvature tensor dni/dxj plays the role of","rect":[53.81294250488281,105.08378601074219,385.1518491348423,95.15572357177735]},{"page":210,"text":"the strain tensor uij. Here, indices i, j ¼ 1, 2, 3 and xi correspond to the Cartesian","rect":[53.81399154663086,117.04350280761719,385.1225518327094,107.13566589355469]},{"page":210,"text":"frame axes. The linear relationship between the curvature and the torsional stress","rect":[53.813594818115237,128.02999877929688,385.09372343169408,119.09544372558594]},{"page":210,"text":"(i.e., Hooke’s law) is assumed to be valid. The stress can be caused by boundary","rect":[53.813594818115237,139.98953247070313,385.17739192060005,131.0549774169922]},{"page":210,"text":"conditions, electric or magnetic field, shear, mechanical shot, etc. We are going to","rect":[53.813594818115237,151.94906616210938,385.14153376630318,143.01451110839845]},{"page":210,"text":"write the key expression for the distortion free energy density gdist related to the","rect":[53.813594818115237,163.94847106933595,385.1743549175438,154.9740753173828]},{"page":210,"text":"“director field curvature”. To discuss a more general case, we assume that gdist","rect":[53.81362533569336,175.90830993652345,385.17034805538028,166.9339141845703]},{"page":210,"text":"depends not only on quadratic combinations of derivatives ∂ni/∂xj, but also on their","rect":[53.812843322753909,188.74476623535157,385.1120542129673,177.88771057128907]},{"page":210,"text":"linear combinations:","rect":[53.814022064208987,197.7090606689453,136.1482835538108,190.79649353027345]},{"page":210,"text":"@ni1","rect":[195.369140625,222.71144104003907,225.60994807294379,213.99050903320313]},{"page":210,"text":"@ni @nl","rect":[247.02951049804688,222.71144104003907,281.44513970177078,213.99050903320313]},{"page":210,"text":"gdist ¼ Kij @xj þ 2 Kijlm @xj \u0004 @nm","rect":[153.84884643554688,237.6744384765625,282.96073555213078,221.28256225585938]},{"page":210,"text":"(8.10)","rect":[361.0561828613281,229.6003875732422,385.10555396882668,221.1240234375]},{"page":210,"text":"As we shall see further on, the terms linear in ∂ni/∂xj, allow us to discuss not","rect":[65.76653289794922,262.15972900390627,385.135878158303,251.28932189941407]},{"page":210,"text":"only conventional nematics with D1h symmetry but also some “biased” nematic","rect":[53.81399154663086,273.1896057128906,385.1016315288719,264.2548828125]},{"page":210,"text":"phases. For example, we can discuss the phases with a spontaneous twist (choles-","rect":[53.81357955932617,285.0923767089844,385.1693051895298,276.1578369140625]},{"page":210,"text":"terics with broken mirror symmetry) or a spontaneous “splay” (uniaxial polar","rect":[53.81357955932617,297.0519104003906,385.13750587312355,288.11737060546877]},{"page":210,"text":"nematics with broken head-to-tail symmetry, n ¼6 \u0003n). For a standard nematic","rect":[53.81357955932617,309.011474609375,385.1016315288719,299.74822998046877]},{"page":210,"text":"only quadratic terms will remain.","rect":[53.81260299682617,320.97100830078127,188.1586117317844,312.03643798828127]},{"page":210,"text":"8.2.1.2 Elementary Distortions","rect":[53.81260299682617,363.1994323730469,191.04236234770969,353.5278015136719]},{"page":210,"text":"Consider elementary distortions of a nematic. The undistorted director n ¼ (0, 0, 1)","rect":[53.81260299682617,386.7201232910156,385.1504148086704,377.78558349609377]},{"page":210,"text":"is aligned along the z-axis, Fig. 8.5a. For instance, at a distance dx from the origin of","rect":[53.81260299682617,398.6796569824219,385.15044532624855,389.4462890625]},{"page":210,"text":"the Cartesian frame O the director has been turned through some angle in the zOx","rect":[53.81357955932617,410.63922119140627,385.16138494684068,401.68475341796877]},{"page":210,"text":"plane like in Fig. 8.5b. The relative distortion is then described by the ratio of dnx,","rect":[53.81362533569336,422.5987243652344,385.1832241585422,413.3653564453125]},{"page":210,"text":"an absolute change of the x-component of the director, to distance dx, over which","rect":[53.814537048339847,434.5028991699219,385.12151423505318,425.26953125]},{"page":210,"text":"the distortion occurs. In the same sketch, but in the zOy plane we see similar fan-","rect":[53.81456756591797,446.46246337890627,385.15740333406105,437.50799560546877]},{"page":210,"text":"shape or splay distortion dny. Thus for the two elementary splay distortions we","rect":[53.815574645996097,459.3522644042969,385.17078436090318,449.18865966796877]},{"page":210,"text":"write:","rect":[53.81405258178711,468.3600158691406,77.1168580777366,461.44744873046877]},{"page":210,"text":"@nx","rect":[178.65882873535157,493.35626220703127,192.42281479730984,484.6413879394531]},{"page":210,"text":"@ny","rect":[221.14273071289063,494.721923828125,234.96325058832546,484.584716796875]},{"page":210,"text":"@x ¼ a1; @y ¼ a5;","rect":[180.69801330566407,507.48809814453127,260.36646973770999,494.12213134765627]},{"page":210,"text":"When discussing the deviations of the director dnx and dny at a distance dz from","rect":[65.76561737060547,531.9602661132813,385.13773296952805,521.7966918945313]},{"page":210,"text":"O illustrated by Fig. 8.5c. we define two elementary bend distortions:","rect":[53.81284713745117,542.9901733398438,334.046766830178,533.9959106445313]},{"page":210,"text":"@nx","rect":[178.65882873535157,565.96435546875,192.42281479730984,557.24951171875]},{"page":210,"text":"@ny","rect":[221.14273071289063,567.330078125,234.96325058832546,557.1928100585938]},{"page":210,"text":"@z ¼ a3; @z ¼ a6;","rect":[181.03793334960938,578.183837890625,260.36646973770999,566.7302856445313]},{"page":211,"text":"196","rect":[53.81367874145508,42.62403869628906,66.50528473048135,36.68062973022461]},{"page":211,"text":"Fig. 8.5 Elementary","rect":[53.812843322753909,67.58130645751953,125.85039276270791,59.6313591003418]},{"page":211,"text":"distortions ofthe directorfield","rect":[53.812843322753909,75.76238250732422,155.31436676173136,69.89517211914063]},{"page":211,"text":"for particular geometry (a)","rect":[53.812843322753909,87.4087142944336,145.392914711067,79.81436157226563]},{"page":211,"text":"with n || z: splay (b), bend (c)","rect":[53.812843322753909,97.3846664428711,154.90992826087169,89.79031372070313]},{"page":211,"text":"and twist (d)","rect":[53.812835693359378,107.02202606201172,97.5853280411451,99.76632690429688]},{"page":211,"text":"8 Elasticity and Defects","rect":[303.50494384765627,44.274986267089847,385.17441256522457,36.68062973022461]},{"page":211,"text":"In the same way, in Fig. 8.5d we see two elementary twist distortions (minus","rect":[65.76496887207031,316.2102355957031,385.10300077544408,307.2159118652344]},{"page":211,"text":"appears due to different signs of dny and dnx for the same handedness of rotation","rect":[53.81296157836914,329.0865173339844,385.1180352311469,318.9364013671875]},{"page":211,"text":"about x and y axes).","rect":[53.81403732299805,340.0528259277344,134.08379788901096,331.13818359375]},{"page":211,"text":"@nx","rect":[174.80697631835938,360.0429992675781,188.57087082758327,351.38482666015627]},{"page":211,"text":"@ny","rect":[224.9945831298828,361.40863037109377,238.8151182641067,351.328125]},{"page":211,"text":"¼ a2;\u0003","rect":[191.8571014404297,367.2207336425781,223.3131029913178,360.8655700683594]},{"page":211,"text":"¼ a4;","rect":[242.1013641357422,367.2207336425781,264.2183068959131,360.8655700683594]},{"page":211,"text":"@y","rect":[176.8461456298828,374.2315368652344,187.01636541559066,364.98822021484377]},{"page":211,"text":"@x","rect":[227.0904083251953,372.3191223144531,237.20487249566879,364.98822021484377]},{"page":211,"text":"Now the components of the director field nx(x,y,z) and ny(x,y,z) can be expressed","rect":[65.76619720458985,395.6863708496094,385.17754450849068,385.8348388671875]},{"page":211,"text":"in terms of the six relative distortions mentioned (only six but not nine because of","rect":[53.81281661987305,406.7294616699219,385.1476987442173,397.794921875]},{"page":211,"text":"our assumption nz ¼ const ¼ 1). For small deviations, we get:","rect":[53.81281661987305,418.6891784667969,304.2499223477561,409.75445556640627]},{"page":211,"text":"nx ¼ a1x þ a2y þ a3z þ Oðr2Þ;","rect":[156.90769958496095,441.19012451171877,280.41920411270999,429.9156494140625]},{"page":211,"text":"ny ¼ a4x þ a5y þ a6z þ Oðr2Þ;","rect":[156.90834045410157,457.93609619140627,280.41920411270999,446.12628173828127]},{"page":211,"text":"nz ¼ 1 þ Oðr2Þ;","rect":[157.3045196533203,473.6681823730469,221.5082544056787,462.3937072753906]},{"page":211,"text":"where O(r2) describes higher order terms depending on r2 ¼ x2 + y2 + z2.","rect":[53.81362533569336,495.2649841308594,349.32683988119848,484.2140197753906]},{"page":211,"text":"8.2.1.3 Curvature Distortion Tensor","rect":[53.814205169677737,528.3507080078125,214.7220386333641,520.8504638671875]},{"page":211,"text":"From consideration of the simple case with a particular choice of the z-axis parallel","rect":[53.814205169677737,554.042724609375,385.1699662930686,545.1082153320313]},{"page":211,"text":"to the director and nz\u00051+..., ∂nz/∂xj \u00050, we construct a “reduced” curvature","rect":[53.814205169677737,566.919189453125,385.1466754741844,556.0054321289063]},{"page":211,"text":"distortion tensor ni;j ¼ @@nxji ¼ \u0002@@nnxy==@@xx @@nnxy==@@yy @@nnyx==@@zz\u0003 ¼ \u0002aa14 aa25 aa36 \u0003:","rect":[53.81382369995117,592.1092529296875,375.41245972794436,568.2239379882813]},{"page":212,"text":"8.2 Elasticity of Nematics and Cholesterics","rect":[53.812843322753909,44.274620056152347,201.7030380535058,36.68026351928711]},{"page":212,"text":"197","rect":[372.49737548828127,42.62367248535156,385.1889700332157,36.73106384277344]},{"page":212,"text":"Here, symbol “comma” between suffixes in ni,j means spatial derivatives. In the","rect":[65.76496887207031,69.20487976074219,385.1746295757469,59.35380554199219]},{"page":212,"text":"general case, with three missing elementary distortions a7, a8, and a9 the curvature","rect":[53.813899993896487,80.24788665771485,385.1553424663719,71.31333923339844]},{"page":212,"text":"distortion tensor is given by:","rect":[53.813533782958987,92.20760345458985,170.3942552334983,83.25313568115235]},{"page":212,"text":"0a1 a2 a3 1","rect":[197.974853515625,124.40707397460938,265.539065926968,106.39852905273438]},{"page":212,"text":"ni;j ¼ @aa47 aa58 aa69 A","rect":[173.44723510742188,141.69850158691407,265.539065926968,122.12586212158203]},{"page":212,"text":"(8.11)","rect":[361.0556945800781,128.3115692138672,385.10506568757668,119.83519744873047]},{"page":212,"text":"With this tensor we may turn back to Eq. (8.10) for energy density and discuss","rect":[65.7660140991211,165.212158203125,385.1100198184128,156.27760314941407]},{"page":212,"text":"two strain tensors, Kij and Kijlm.","rect":[53.81400680541992,178.08876037597657,181.82709928061252,168.23716735839845]},{"page":212,"text":"8.2.2 Frank Energy","rect":[53.812843322753909,227.28269958496095,159.96595557661264,216.64492797851563]},{"page":212,"text":"8.2.2.1 Elasticity Tensors","rect":[53.812843322753909,255.22332763671876,167.68602384673313,245.5517120361328]},{"page":212,"text":"(i) Kij tensor. Since gdist ia a scalar, the tensor of elasticity coefficients Kij is also of","rect":[53.812843322753909,279.66070556640627,385.1519712051548,269.8094482421875]},{"page":212,"text":"the second rank with nine components. This tensor has no quadrupolar symmetry","rect":[53.814144134521487,290.7038879394531,385.17192927411568,281.76934814453127]},{"page":212,"text":"like the familiar order parameter tensor Q. However due to the uniaxial symmetry,","rect":[53.814144134521487,302.6634216308594,385.1678738167453,293.649169921875]},{"page":212,"text":"particularly, polar conical symmetry, the energy gdist is invariant with respect to the","rect":[53.814144134521487,314.6628112792969,385.1732868023094,305.68841552734377]},{"page":212,"text":"rotation of our frame about the z-axis (x’ ¼ y, y’ ¼ x, z’ ¼ z). Thus, this tensor","rect":[53.81354904174805,326.5628967285156,385.1195310196079,317.64825439453127]},{"page":212,"text":"reduces to the form","rect":[53.81354904174805,336.4536437988281,132.16115521073898,329.551025390625]},{"page":212,"text":"Kij ¼ \u0002\u0003KK12 KK12 \u0003","rect":[178.94192504882813,376.6087341308594,260.0066802058794,352.7235107421875]},{"page":212,"text":"(8.12)","rect":[361.0558776855469,368.9222106933594,385.1052487930454,360.44586181640627]},{"page":212,"text":"At this stage, two moduli K1 and K2 remain finite, for a spontaneous splay and","rect":[65.7662124633789,400.0985107421875,385.1457451920844,391.163330078125]},{"page":212,"text":"twist, respectively (virtual polar cholesteric).","rect":[53.8138542175293,412.0574035644531,236.6896175911594,403.1029357910156]},{"page":212,"text":"However, when molecules have mirror symmetry (achiral molecules), the inver-","rect":[65.7668685913086,424.0169372558594,385.18560157624855,415.0823974609375]},{"page":212,"text":"sion center appears and gdist becomes invariant with respect to the following frame","rect":[53.814842224121097,436.01629638671877,385.1719440288719,427.04193115234377]},{"page":212,"text":"transformation: x’ ¼ \u0003x, y’ ¼ \u0003y, z’ ¼ \u0003z. As a result, modulus K2 vanishes and we","rect":[53.81319046020508,447.9165344238281,385.1706012554344,439.00189208984377]},{"page":212,"text":"have a tensor corresponding to a polar nematic:","rect":[53.813838958740237,459.8960876464844,247.04595313874854,450.9416198730469]},{"page":212,"text":"Kij ¼ \u0002K01 K01\u0003","rect":[182.79452514648438,497.96234130859377,256.15496511798878,474.0771179199219]},{"page":212,"text":"(8.13)","rect":[361.0554504394531,490.27581787109377,385.10482154695168,481.7994689941406]},{"page":212,"text":"(modulus K2 remains, however, in the cholesteric phase).","rect":[53.813777923583987,521.5081787109375,284.4674953987766,512.5736694335938]},{"page":212,"text":"(ii) Kijlm tensor. Generally it has 81 components, however, for n||z in Eq. (8.11)","rect":[65.7655258178711,534.3843383789063,385.15792213288918,524.4763793945313]},{"page":212,"text":"coefficients a7,8,9 ¼ 0 and due to this only 36 components remain. Now we can","rect":[53.812076568603519,545.3707885742188,385.12603083661568,536.4361572265625]},{"page":212,"text":"again","rect":[53.81302261352539,557.330322265625,75.46343318036566,548.3958129882813]},{"page":212,"text":"apply","rect":[80.54904174804688,557.330322265625,102.72702113202581,548.3958129882813]},{"page":212,"text":"the","rect":[107.79470825195313,556.0,120.0184787578758,548.3958129882813]},{"page":212,"text":"symmetry","rect":[125.07122802734375,557.330322265625,165.0373925309516,549.4117431640625]},{"page":212,"text":"arguments.","rect":[170.10308837890626,557.330322265625,214.22710843588596,549.4117431640625]},{"page":212,"text":"A","rect":[219.26991271972657,555.1888427734375,226.41703557923166,548.45556640625]},{"page":212,"text":"conventional","rect":[231.50563049316407,556.0,283.2177568204124,548.3958129882813]},{"page":212,"text":"nematic","rect":[288.2625732421875,556.0,319.95676458551255,548.3958129882813]},{"page":212,"text":"has","rect":[325.0812072753906,556.0,338.3402139102097,548.3958129882813]},{"page":212,"text":"a","rect":[343.3770446777344,556.0,347.8265765972313,550.0]},{"page":212,"text":"uniaxial","rect":[352.8932800292969,556.0,385.1150346524436,548.3958129882813]},{"page":212,"text":"symmetry. This further reduces the quantity of moduli down to 18. Among them","rect":[53.81302261352539,569.2898559570313,385.1319651472624,560.3553466796875]},{"page":212,"text":"only five coefficients are independent, namely K11, K22, K33, K24 and K12 (K15 ¼","rect":[53.81302261352539,581.2493896484375,385.14807918760689,572.3148803710938]},{"page":212,"text":"K11\u0003K22\u0003K24):","rect":[53.814231872558597,592.8110961914063,117.64617784092019,584.334716796875]},{"page":213,"text":"198","rect":[53.812843322753909,42.62367248535156,66.50444931178018,36.73106384277344]},{"page":213,"text":"8 Elasticity and Defects","rect":[303.50408935546877,44.274620056152347,385.17355807303707,36.68026351928711]},{"page":213,"text":"Kijlm","rect":[85.36425018310547,104.30449676513672,102.86870216590026,94.771728515625]},{"page":213,"text":"¼","rect":[106.15303039550781,100.0208969116211,113.81777218077092,97.69014739990235]},{"page":213,"text":"0","rect":[116.63240051269531,78.49569702148438,125.34233435958521,60.487152099609378]},{"page":213,"text":"B","rect":[116.63240051269531,84.39924621582031,125.34233435958521,78.23371124267578]},{"page":213,"text":"B","rect":[116.63240051269531,90.29383850097656,125.34233435958521,84.12830352783203]},{"page":213,"text":"B","rect":[116.63240051269531,96.18849182128906,125.34233435958521,90.02295684814453]},{"page":213,"text":"B","rect":[116.63240051269531,102.13984680175781,125.34233435958521,95.97431182861328]},{"page":213,"text":"B","rect":[116.63240051269531,108.03443908691406,125.34233435958521,101.86890411376953]},{"page":213,"text":"B","rect":[116.63240051269531,113.92909240722656,125.34233435958521,107.76355743408203]},{"page":213,"text":"B","rect":[116.63240051269531,119.88044738769531,125.34233435958521,113.71491241455078]},{"page":213,"text":"@","rect":[116.63240051269531,137.21961975097657,125.34233435958521,119.2110824584961]},{"page":213,"text":"K11","rect":[128.35787963867188,68.4708251953125,141.9533545740541,60.42303466796875]},{"page":213,"text":"K12","rect":[128.3577880859375,82.41424560546875,141.9533545740541,74.36688232421875]},{"page":213,"text":"0","rect":[131.41673278808595,95.00379943847656,136.3938378434516,88.15099334716797]},{"page":213,"text":"\u0003K12","rect":[129.54861450195313,109.337646484375,150.8466651209291,101.23309326171875]},{"page":213,"text":"K15","rect":[131.4169921875,123.36486053466797,145.01217720589004,115.1771240234375]},{"page":213,"text":"0","rect":[132.83348083496095,135.8140411376953,137.8105858903266,128.96124267578126]},{"page":213,"text":"K12","rect":[162.40162658691407,68.4708251953125,175.9970862635072,60.42315673828125]},{"page":213,"text":"K22","rect":[162.40162658691407,82.41424560546875,175.9970862635072,74.36669921875]},{"page":213,"text":"0","rect":[164.44082641601563,95.00379943847656,169.41793147138129,88.15099334716797]},{"page":213,"text":"K24","rect":[163.6478729248047,109.337646484375,177.29988167366347,101.23345947265625]},{"page":213,"text":"K12","rect":[163.30795288085938,123.28118896484375,176.9600684412416,115.17694091796875]},{"page":213,"text":"0","rect":[165.40464782714845,135.8140411376953,170.3817528825141,128.96124267578126]},{"page":213,"text":"0","rect":[197.52163696289063,67.11659240722656,202.49874201825629,60.26378631591797]},{"page":213,"text":"0","rect":[197.52163696289063,81.06013488769531,202.49874201825629,74.20732879638672]},{"page":213,"text":"K33","rect":[196.38885498046876,96.44145965576172,210.0407569178041,88.31036376953125]},{"page":213,"text":"0","rect":[194.34950256347657,107.92689514160156,199.32660761884223,101.07408905029297]},{"page":213,"text":"0","rect":[194.34950256347657,121.87037658691406,199.32660761884223,115.01757049560547]},{"page":213,"text":"0","rect":[194.35049438476563,135.8140411376953,199.32759944013129,128.96124267578126]},{"page":213,"text":"\u0003K12","rect":[226.80723571777345,68.4708251953125,248.10640022835097,60.42315673828125]},{"page":213,"text":"K24","rect":[230.6595001220703,82.41424560546875,244.2545325525697,74.36688232421875]},{"page":213,"text":"0","rect":[233.54818725585938,95.00379943847656,238.52529231122504,88.15099334716797]},{"page":213,"text":"K22","rect":[232.18844604492188,109.337646484375,245.7839514978822,101.2332763671875]},{"page":213,"text":"\u0003K12","rect":[228.33737182617188,123.28118896484375,249.6358039148744,115.1771240234375]},{"page":213,"text":"0","rect":[236.77755737304688,135.8140411376953,241.75466242841254,128.96124267578126]},{"page":213,"text":"K15","rect":[268.4980163574219,68.55449676513672,282.15010140022596,60.42315673828125]},{"page":213,"text":"K12","rect":[268.4980163574219,82.41424560546875,282.15010140022596,74.36669921875]},{"page":213,"text":"0","rect":[274.728759765625,95.00379943847656,279.70586482099068,88.15099334716797]},{"page":213,"text":"\u0003K12","rect":[260.1145324707031,109.337646484375,281.4137427576478,101.23345947265625]},{"page":213,"text":"K11","rect":[268.9512023925781,123.28118896484375,282.5465552576478,115.17694091796875]},{"page":213,"text":"0","rect":[273.48370361328127,135.8140411376953,278.4608086686469,128.96124267578126]},{"page":213,"text":"0","rect":[302.5418395996094,67.11659240722656,307.51894465497505,60.26378631591797]},{"page":213,"text":"0","rect":[302.5418395996094,81.06013488769531,307.51894465497505,74.20732879638672]},{"page":213,"text":"0","rect":[302.54180908203127,95.00379943847656,307.5189141373969,88.15099334716797]},{"page":213,"text":"0","rect":[301.805419921875,107.92689514160156,306.78252497724068,101.07408905029297]},{"page":213,"text":"0","rect":[301.805419921875,121.87037658691406,306.78252497724068,115.01757049560547]},{"page":213,"text":"K33","rect":[301.805419921875,137.3082733154297,315.4574439295228,129.12060546875]},{"page":213,"text":"1","rect":[320.8370361328125,78.49465942382813,329.5469699797024,60.486114501953128]},{"page":213,"text":"C","rect":[320.8370361328125,84.39826965332031,329.5469699797024,78.23273468017578]},{"page":213,"text":"C","rect":[320.8370361328125,90.29286193847656,329.5469699797024,84.12732696533203]},{"page":213,"text":"C","rect":[320.8370361328125,96.18751525878906,329.5469699797024,90.02198028564453]},{"page":213,"text":"C","rect":[320.8370361328125,102.13887023925781,329.5469699797024,95.97333526611328]},{"page":213,"text":"C","rect":[320.8370361328125,108.03346252441406,329.5469699797024,101.86792755126953]},{"page":213,"text":"C","rect":[320.8370361328125,113.92811584472656,329.5469699797024,107.76258087158203]},{"page":213,"text":"C","rect":[320.8370361328125,119.87947082519531,329.5469699797024,113.71393585205078]},{"page":213,"text":"A","rect":[320.8370361328125,137.21864318847657,329.5469699797024,119.2101058959961]},{"page":213,"text":"(8.14)","rect":[361.0550842285156,103.08707427978516,385.10445533601418,94.61071014404297]},{"page":213,"text":"Next, taking the head-to-tail symmetry into account we make the following","rect":[65.76541137695313,161.7532958984375,385.1672295670844,152.81874084472657]},{"page":213,"text":"transformation of the frame: x’ ¼ x, y’ ¼ y, z’ ¼ \u0003z. Now coefficients K12","rect":[53.813392639160159,173.69290161132813,385.17547254887918,164.7782745361328]},{"page":213,"text":"disappear. At this atage, only four different moduli are left, K11, K22, K33, K24.","rect":[53.812843322753909,185.67434692382813,370.85198636557348,176.7397918701172]},{"page":213,"text":"8.2.2.2 Elastic Energy of the Conventional Nematic for n||z","rect":[53.814353942871097,227.16567993164063,311.48008001520005,217.4940643310547]},{"page":213,"text":"First let us go back to the same particular case with a constraint n||z, and discuss the","rect":[53.814353942871097,250.68637084960938,385.1750873394188,241.75181579589845]},{"page":213,"text":"free energy of a conventional (uniaxial, nonpolar) nematic liquid crystal. We","rect":[53.814353942871097,262.64593505859377,385.12632024957505,253.7113800048828]},{"page":213,"text":"combine elementary distortions corresponding to splay (a1 + a5), bend (a3 + a6)","rect":[53.814353942871097,274.60546875,385.15975318757668,265.6708984375]},{"page":213,"text":"and twist (a2 + a4) and present the free energy as a sum of these combinations","rect":[53.81393051147461,286.5095520019531,385.1162454043503,277.5748291015625]},{"page":213,"text":"squared.","rect":[53.8132438659668,298.4690856933594,87.27930112387424,289.5345458984375]},{"page":213,"text":"gdist ¼ 21hK11ða1 þ a5Þ2 þ K22ða2 þ a4Þ2 þ K33ða3 þ a6Þ2i","rect":[86.8940658569336,332.99981689453127,323.513245624791,312.66351318359377]},{"page":213,"text":"(8.15a)","rect":[356.6368713378906,327.9420471191406,385.1357663711704,319.3461608886719]},{"page":213,"text":"Now, we would like to write the same in a more compact vector form. To do this,","rect":[65.76488494873047,356.5674743652344,385.1388210823703,347.6329345703125]},{"page":213,"text":"consider vector forms for each of the three contributions.","rect":[53.812870025634769,366.49505615234377,284.1791958382297,359.59246826171877]},{"page":213,"text":"Let us write the divergence of vector n ¼ (0, 0, 1) with dnz ¼ 0:","rect":[65.76488494873047,380.486572265625,325.7805619961936,371.5321044921875]},{"page":213,"text":"divn ¼ @nx þ @ny","rect":[182.79527282714845,406.4813232421875,253.99601120844265,391.74151611328127]},{"page":213,"text":"@x","rect":[215.87464904785157,412.7325134277344,225.9891284771141,405.401611328125]},{"page":213,"text":"@y","rect":[242.2713165283203,414.6449279785156,252.44153631402816,405.401611328125]},{"page":213,"text":"Evidently, this corresponds to the splay term (a1 + a5). Now we write curln under","rect":[65.76525115966797,435.1266784667969,385.11153541413918,426.192138671875]},{"page":213,"text":"condition dnz ¼ 0","rect":[53.81450653076172,446.6451721191406,125.51843348554144,438.13177490234377]},{"page":213,"text":"curln ¼ \u0003@@nzy i þ @@nzx j þ \u0002@@nxy \u0003 @@nyx\u0003k","rect":[135.89170837402345,485.2095031738281,303.1087681705768,461.32427978515627]},{"page":213,"text":"and see that it does not fit to any of terms in Eq. (8.15a). However, the scalar","rect":[53.81452178955078,508.6993103027344,385.08565650788918,499.7049865722656]},{"page":213,"text":"product","rect":[53.81354904174805,520.6588134765625,84.36601121494363,511.72430419921877]},{"page":213,"text":"n \u0004 curln ¼ \u0003nx @@nzy þ ny @@nzx þ nz\u0002@@nxy \u0003 @@nyx\u0003 \u0005 \u0002@@nxy \u0003 @@nyx\u0003","rect":[89.27342224121094,558.7244262695313,349.7326323543169,534.8391723632813]},{"page":213,"text":"with neglected the first two products of the second order of magnitude fits to the","rect":[53.812862396240237,582.2699584960938,385.17160833551255,573.33544921875]},{"page":213,"text":"twist term (a2 + a4)! Finally the vector product","rect":[53.812862396240237,594.2296142578125,243.14057786533426,585.2949829101563]},{"page":214,"text":"8.2 Elasticity of Nematics and Cholesterics","rect":[53.812843322753909,44.276206970214847,201.7030380535058,36.68185043334961]},{"page":214,"text":"199","rect":[372.49737548828127,42.62525939941406,385.1889700332157,36.73265075683594]},{"page":214,"text":"n \u0002 curln ¼ \u0003\u0002@@nzx i þ @@nzy j\u0003","rect":[157.07701110839845,83.28459167480469,281.9281889949419,59.39937210083008]},{"page":214,"text":"under condition nx, ny << nz \u0005 1) fits to the bend term. Indeed, despite sign","rect":[53.81351852416992,107.74784851074219,385.11516657880318,97.89567565917969]},{"page":214,"text":"“minus” and the vector form it contributes to the quadratic bend term of the","rect":[53.813167572021487,118.7909164428711,385.11719549371568,109.85636901855469]},{"page":214,"text":"“reduced” free energy:","rect":[53.813167572021487,130.75045776367188,145.5541673428733,121.81590270996094]},{"page":214,"text":"gdist ¼ 21\"K11\u0002@@nxx þ @@nyy\u00032 þ K22\u0002@@nyx \u0003 @@nxy\u00032 þ K33\"\u0002@@nzx\u00032 þ \u0002@@nzy\u00032##","rect":[65.8219223022461,171.7890167236328,373.15077839818209,141.92750549316407]},{"page":214,"text":"(8:15b)","rect":[343.89239501953127,186.01258850097657,373.12688575593605,177.41668701171876]},{"page":214,"text":"8.2.2.3 Frank Formula","rect":[53.81277084350586,239.39178466796876,157.80440608075629,231.8915557861328]},{"page":214,"text":"To have the free energy density in a more general form including all the nine","rect":[53.81277084350586,265.0838623046875,385.16257513238755,256.1492919921875]},{"page":214,"text":"elementary distortions a1, a2... a9 we should add the terms ∂nz/∂z, ∂nz/∂x and ∂nz/","rect":[53.81277084350586,277.0433654785156,385.208632064553,267.10333251953127]},{"page":214,"text":"∂y and rewrite the Eq. (8.15b) in the vector notations for arbitrary distortion of n","rect":[53.814083099365237,289.0034484863281,385.1867711002643,279.0628967285156]},{"page":214,"text":"with respect to the Cartesian frame. Then we obtain Frank formula for the density","rect":[53.814083099365237,300.9629821777344,385.1050957780219,291.98858642578127]},{"page":214,"text":"of elastic energy in the general vector form:","rect":[53.814083099365237,312.92254638671877,231.31057603427957,303.98797607421877]},{"page":214,"text":"gdist ¼ 1½K11ðdivnÞ2 þ K22ðn \u0004 curlnÞ2 þ K33ðn \u0002 curlnÞ2\u0006","rect":[102.92515563964844,343.0755310058594,336.1010278431787,327.11712646484377]},{"page":214,"text":"2","rect":[131.70001220703126,347.45343017578127,136.6771172623969,340.72015380859377]},{"page":214,"text":"(8.16)","rect":[361.0562438964844,342.33843994140627,385.1056150039829,333.80230712890627]},{"page":214,"text":"Here, we have splay, twist and bend terms corresponding to a particular bulk","rect":[65.76659393310547,370.9638671875,385.12856379560005,362.029296875]},{"page":214,"text":"distortion. In experiment, they can be realized in different way using variable cell","rect":[53.81456756591797,382.92340087890627,385.1205888516624,373.98883056640627]},{"page":214,"text":"geometry and boundary conditions. For example, such distortions may be created","rect":[53.81456756591797,394.8829650878906,385.17229548505318,385.94842529296877]},{"page":214,"text":"mechanically as shown in Fig. 8.6. The important condition is to anchor the director","rect":[53.81456756591797,406.8424987792969,385.10662208406105,397.907958984375]},{"page":214,"text":"firmly at the boundaries.","rect":[53.81356430053711,418.80206298828127,152.39309354330784,409.86749267578127]},{"page":214,"text":"But what about the K24 term? The so-called saddle-splay modulus K24 is","rect":[65.76558685302735,430.7423095703125,385.1323586856003,421.8077697753906]},{"page":214,"text":"important only for particular situations, in which a distortion has a two- or three-","rect":[53.81450653076172,442.7217712402344,385.16329322663918,433.7872314453125]},{"page":214,"text":"dimensional structures such as nematic droplets in the isotropic solutions [8] or blue","rect":[53.81450653076172,454.6813049316406,385.1692279644188,445.74676513671877]},{"page":214,"text":"phases [9]. The free energy term including modulus K24 is a so-called “divergence”","rect":[53.81548309326172,466.58447265625,385.07361639215318,457.6495361328125]},{"page":214,"text":"term because it has a form of divn to the first degree. Hence, if one performs the","rect":[53.81344223022461,478.5440368652344,385.17517889215318,469.6094970703125]},{"page":214,"text":"Fig. 8.6 Splay, bend and twist distortions in nematics confined between two glasses that align","rect":[53.812843322753909,574.0244750976563,385.1923880019657,566.0745239257813]},{"page":214,"text":"liquid crystal at the surfaces either homogeneously (for splay and twist) or homeotropically (for","rect":[53.812843322753909,583.9326782226563,385.18209928137949,576.3383178710938]},{"page":214,"text":"bend)","rect":[53.812843322753909,593.5700073242188,73.10408109290292,586.3142700195313]},{"page":215,"text":"200","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.73106384277344]},{"page":215,"text":"8 Elasticity and Defects","rect":[303.50408935546877,44.274620056152347,385.17355807303707,36.68026351928711]},{"page":215,"text":"minimization of the free energy using the Euler-Lagrange variation procedure","rect":[53.812843322753909,68.2883529663086,385.07907903863755,59.35380554199219]},{"page":215,"text":"(Section 8.3), the divergence term ∂n/∂z would contribute only to the boundary","rect":[53.812843322753909,80.24788665771485,385.17559138349068,70.30732727050781]},{"page":215,"text":"conditions and, in the one-dimensional geometry, the contribution of the saddle-","rect":[53.81185531616211,92.20748138427735,385.09298072663918,83.27293395996094]},{"page":215,"text":"splay elasticity vanishes. However, in a more complex geometry the coefficient K24","rect":[53.81185531616211,104.11019134521485,385.17547254887918,95.17564392089844]},{"page":215,"text":"becomes important. For example, in very thin films with tangential (planar along x)","rect":[53.812843322753909,116.07039642333985,385.15865455476418,107.13584899902344]},{"page":215,"text":"boundary conditions on one side and homeotropic (along z) on the other side, a","rect":[53.812843322753909,128.02999877929688,385.1596149273094,119.09544372558594]},{"page":215,"text":"uniform hybrid structure in the xz-plane is not stable. Instead, the absolute mini-","rect":[53.8138313293457,139.98953247070313,385.15166602937355,131.0549774169922]},{"page":215,"text":"mum of the free energy is realized for a periodic structure of linear defects (stripe","rect":[53.8138427734375,151.94906616210938,385.1218646831688,143.01451110839845]},{"page":215,"text":"domains) with distortion in the film plane xy [10].","rect":[53.8138427734375,163.90863037109376,255.32391019369846,154.9740753173828]},{"page":215,"text":"The nematic elastic moduli have dimension of force. The three moduli in Frank","rect":[65.76488494873047,173.836181640625,385.1149529557563,166.93357849121095]},{"page":215,"text":"energy (8.16) are always positive, modulus K24 may be positive or negative.","rect":[53.812870025634769,187.82827758789063,385.15667386557348,178.8931121826172]},{"page":215,"text":"Roughly, all of them are of the same order of magnitude that may be estimated","rect":[53.8138542175293,199.73104858398438,385.12578669599068,190.79649353027345]},{"page":215,"text":"from the molecular interaction energy W divided by intermolecular distance l \u00055","rect":[53.8138542175293,211.69061279296876,385.1785821061469,202.6962890625]},{"page":215,"text":"˚","rect":[55.79670715332031,214.03329467773438,59.11145912019384,212.05116271972657]},{"page":215,"text":"A. If for W we take the temperature of nematic-isotropic transition (kBT \u0005 0.033 eV","rect":[53.813838958740237,223.650146484375,385.1708350177082,214.69566345214845]},{"page":215,"text":"at T ¼ 400 K), the elastic modulus would be K\u0005W/l \u0005 1\u000410\u00036 dyn (Gauss system)","rect":[53.81307601928711,235.61026000976563,385.1226743301548,224.51931762695313]},{"page":215,"text":"or 1\u000410\u000311 N (SI system). If we take the nematic potential WN \u0005 0.15 eV the","rect":[53.813716888427737,247.56985473632813,385.17453802301255,236.5188751220703]},{"page":215,"text":"estimated modulus would be five time larger. The experimental values are in the","rect":[53.81380844116211,259.5294189453125,385.17258489801255,250.59486389160157]},{"page":215,"text":"range of 10\u00037 – 10\u00036 dyn. The data on elastic moduli for most popular liquid","rect":[53.81380844116211,271.4891662597656,385.1124199967719,260.3983154296875]},{"page":215,"text":"crystals (p-azoxyanisol (PAA), p-methoxy-benzylidene-p’-butylaniline (MBBA)","rect":[53.813419342041019,283.4486999511719,385.1691831192173,274.51416015625]},{"page":215,"text":"and pentyl-cyanobiphenyl (5CB) are collected in the Introduction to book [11].","rect":[53.814430236816409,295.3514709472656,373.2032131722141,286.3571472167969]},{"page":215,"text":"8.2.3 Cholesterics and Polar Nematics","rect":[53.812843322753909,341.4313659667969,251.88899879191085,332.8972473144531]},{"page":215,"text":"8.2.3.1 Cholesterics","rect":[53.812843322753909,369.3342590332031,143.27829374419407,361.80413818359377]},{"page":215,"text":"If molecules are chiral, the coefficient K2 from tensor (8.12) becomes finite.","rect":[53.812843322753909,394.59820556640627,385.1365932991672,386.0618896484375]},{"page":215,"text":"Formally it is possible to add it to the Frank energy introducing a scalar quantity","rect":[53.81468963623047,406.9562072753906,385.1654900651313,398.02166748046877]},{"page":215,"text":"q0 ¼ K2/K22 and obtain the following expression:","rect":[53.81468963623047,418.9159240722656,253.04080064365457,409.98138427734377]},{"page":215,"text":"gdist ¼ 1½K11ðdivnÞ2 þ K22ðn \u0004 curln þ q0Þ2 þ K33ðn \u0002 curlnÞ2\u0006","rect":[80.32329559326172,449.0687561035156,334.5716088978662,433.1102294921875]},{"page":215,"text":"2","rect":[109.09854888916016,453.446533203125,114.07565394452581,446.7132568359375]},{"page":215,"text":"(8.17)","rect":[361.0558166503906,448.3316650390625,385.10518775788918,439.8553161621094]},{"page":215,"text":"Then, expanding the second term K22ðn \u0004 curlnÞ2 þ 2K22q0ðn \u0004 curlnÞ þ K22q02","rect":[65.7661361694336,477.67669677734377,384.70588753303846,465.3410949707031]},{"page":215,"text":"we note that the last item is not interesting because independent of distortion and","rect":[53.812843322753909,488.9167175292969,385.1427849870063,479.982177734375]},{"page":215,"text":"the gradient term K22q0ðn \u0004 curlnÞ is the only one that distinguishes the free energy","rect":[53.812843322753909,501.21502685546877,385.14016047528755,491.2644958496094]},{"page":215,"text":"of a cholesteric from that of the nematic. For a cholesteric with a pure twist along","rect":[53.81326675415039,512.8358764648438,385.1760186295844,503.9013671875]},{"page":215,"text":"the z-axis, the components of the director are n ¼ (cosjz, sinjz, 0). This results in","rect":[53.81326675415039,524.79541015625,385.1431511979438,515.8609008789063]},{"page":215,"text":"ncurln ¼ \u0003dj/dz. Then, for any form of j(z), the twist contribution to elastic","rect":[53.81327438354492,536.7550048828125,385.0983356304344,527.800537109375]},{"page":215,"text":"energy density (8.17) is given by","rect":[53.81525421142578,548.7145385742188,186.80251399091254,539.780029296875]},{"page":215,"text":"gdistðzÞ ¼ 21K22\u0002djdðzzÞ \u0003 q0\u00032","rect":[157.419677734375,588.7650146484375,281.1018531092103,562.9454956054688]},{"page":215,"text":"(8.18)","rect":[361.0561828613281,581.0780639648438,385.10555396882668,572.6016845703125]},{"page":216,"text":"8.3 Variational Problem and Elastic Torques","rect":[53.8137321472168,44.275718688964847,206.4048965740136,36.68136215209961]},{"page":216,"text":"201","rect":[372.5007629394531,42.55704116821289,385.1923880019657,36.73216247558594]},{"page":216,"text":"The","rect":[65.76496887207031,66.22653198242188,81.3034900493797,59.35380554199219]},{"page":216,"text":"minimum","rect":[86.4966049194336,66.22653198242188,125.36779736161788,59.35380554199219]},{"page":216,"text":"of","rect":[130.56588745117188,66.22653198242188,138.85775123445166,59.35380554199219]},{"page":216,"text":"(8.18)","rect":[143.9911346435547,67.88993072509766,168.09525428620948,59.41356658935547]},{"page":216,"text":"corresponds","rect":[173.22067260742188,68.2883529663086,221.46275724517063,59.35380554199219]},{"page":216,"text":"to","rect":[226.58021545410157,67.0,234.35445490888129,60.369773864746097]},{"page":216,"text":"the","rect":[239.49481201171876,67.0,251.71859014703598,59.35380554199219]},{"page":216,"text":"equilibrium","rect":[256.88482666015627,68.2883529663086,303.5959544050749,59.35380554199219]},{"page":216,"text":"helical","rect":[308.7711486816406,67.0,335.5857988125999,59.35380554199219]},{"page":216,"text":"(harmonic)","rect":[340.71917724609377,67.88993072509766,385.14583717195168,59.35380554199219]},{"page":216,"text":"structure with wavevector q0 ¼ dj/dz and pitch P0 ¼ 2p/q0. Equation (8.18) is","rect":[53.81294250488281,80.24800872802735,385.18844999419408,71.29354095458985]},{"page":216,"text":"correct as long as P0>>a where a is molecular size. In the opposite case, local","rect":[53.81379318237305,92.20772552490235,385.1116777188499,83.27305603027344]},{"page":216,"text":"biaxiality becomes important (practically, P0 \u0005 1 mm, a \u0005 1 nm).","rect":[53.81364822387695,104.1104965209961,319.1134914925266,95.17594909667969]},{"page":216,"text":"8.2.3.2 Polar Nematics","rect":[53.81388473510742,140.4006805419922,156.71355070220188,132.8705596923828]},{"page":216,"text":"In precisely the same way, a spontaneously splay-deformed structure must corre-","rect":[53.81388473510742,166.006103515625,385.0999692520298,157.07154846191407]},{"page":216,"text":"spond to the equilibrium condition with finite coefficient K1 ¼6 0 in tensor (8.13).","rect":[53.81388473510742,177.96560668945313,385.1558804085422,168.70254516601563]},{"page":216,"text":"The corresponding term should be added to the splay term with (divn)2. If the","rect":[53.8140983581543,189.92535400390626,385.1753314800438,178.8744354248047]},{"page":216,"text":"molecules have, e.g., pear shape they can pack as shown in Fig. 8.7b. In this case,","rect":[53.814598083496097,201.885009765625,385.1822781136203,192.95045471191407]},{"page":216,"text":"the local symmetry is C1v (conical) with a polar rotation axis, which is compatible","rect":[53.81456756591797,213.84478759765626,385.1266559429344,204.91001892089845]},{"page":216,"text":"with existence of the spontaneous polarization. However, such packing is unstable,","rect":[53.81367874145508,225.8043212890625,385.1435513069797,216.86976623535157]},{"page":216,"text":"as seen in sketch (b), and the conventional nematic packing (a) is more probable.","rect":[53.81367874145508,237.76388549804688,385.1694607308078,228.82933044433595]},{"page":216,"text":"The splayed structure similar to that pictured in Fig. 8.7b can occur close to the","rect":[53.81367874145508,249.72341918945313,385.1743854351219,240.7888641357422]},{"page":216,"text":"interface with a solid substrate or when an external electric field reduces the overall","rect":[53.812679290771487,259.59423828125,385.13960130283427,252.69163513183595]},{"page":216,"text":"symmetry (a flexoelectric effect).","rect":[53.812679290771487,273.5857238769531,187.1234554817844,264.57147216796877]},{"page":216,"text":"8.3 Variational Problem and Elastic Torques","rect":[53.812843322753909,314.9112243652344,292.70876197550458,303.8670654296875]},{"page":216,"text":"8.3.1 Euler Equation","rect":[53.812843322753909,344.75555419921877,166.7268521199143,334.2014465332031]},{"page":216,"text":"Consider a nematic liquid crystal layer confined between two glass plates. This","rect":[53.812843322753909,372.3809814453125,385.1835976992722,363.4464111328125]},{"page":216,"text":"structureis of great technical importance. The most of liquid crystalline displays are","rect":[53.812843322753909,384.28375244140627,385.1616596050438,375.34918212890627]},{"page":216,"text":"based on it. The directors at opposite walls (z ¼ 0 and z ¼ d) are rigidly fixed at","rect":[53.812843322753909,396.2432861328125,385.179579330178,387.288818359375]},{"page":216,"text":"Fig. 8.7 Packing of conical (pear-shape) molecules in the conventional nematic phase (a) and in","rect":[53.812843322753909,595.5632934570313,385.16696685938759,587.6133422851563]},{"page":216,"text":"a hypothetical polar nematic phase (b)","rect":[53.81287384033203,605.4714965820313,185.55513089759044,597.8771362304688]},{"page":217,"text":"202","rect":[53.812843322753909,42.55807876586914,66.50444931178018,36.73320007324219]},{"page":217,"text":"8 Elasticity and Defects","rect":[303.50408935546877,44.276756286621097,385.17355807303707,36.68239974975586]},{"page":217,"text":"right angle to each other, therefore such a cell is called p/2-twist nematic cell.","rect":[53.812843322753909,68.2883529663086,385.1168484261203,59.35380554199219]},{"page":217,"text":"Along x and y the layer is infinite, the director n(z) depends only on one coordinate.","rect":[53.81282424926758,80.24788665771485,385.1775173714328,71.31333923339844]},{"page":217,"text":"How to find a director distribution along the z-axis?","rect":[53.81379318237305,92.20748138427735,262.63718450738755,83.27293395996094]},{"page":217,"text":"For simplicity we ignore the influence of external fields. The problem is to find","rect":[65.76680755615235,104.11019134521485,385.1685723405219,95.17564392089844]},{"page":217,"text":"that distribution of the director angle j(z) over cell thickness, which satisfies the","rect":[53.81478500366211,116.0697250366211,385.17652166559068,107.13517761230469]},{"page":217,"text":"minimum of the elastic free energy F for fixed boundary conditions. This is a","rect":[53.8167724609375,128.02932739257813,385.16260564996568,119.09477233886719]},{"page":217,"text":"typical variational problem although very simple in our particular case. The idea of","rect":[53.8167724609375,139.98886108398438,385.15267310945168,131.05430603027345]},{"page":217,"text":"a variational calculation is not to find a value of the integral of a function g(z, j, j’)","rect":[53.8167724609375,151.9882354736328,385.1914304336704,143.0138397216797]},{"page":217,"text":"over the interval 0 \u0007 z \u0007 d for known j(z), but to find such an unknown function","rect":[53.81679153442383,163.88803100585938,385.16155329755318,154.93356323242188]},{"page":217,"text":"j(z) that provides the minimum of the integral. Due to the great importance of this","rect":[53.817779541015628,175.86749267578126,385.14572538481908,166.9329376220703]},{"page":217,"text":"mathematical problem for liquid crystals consider it in more detail.","rect":[53.817779541015628,187.8270263671875,324.5226101448703,178.89247131347657]},{"page":217,"text":"Consider a functional F (scalar number, e.g. it might be free energy of the liquid","rect":[65.76980590820313,199.72976684570313,385.11882868817818,190.7952117919922]},{"page":217,"text":"crystal sample):","rect":[53.817779541015628,211.6893310546875,117.72082383701394,202.75477600097657]},{"page":217,"text":"b","rect":[190.44100952148438,231.81619262695313,193.92497322639785,226.9564971923828]},{"page":217,"text":"F ¼ ð gðz;fðzÞ;f0ðzÞÞdz","rect":[169.93963623046876,255.81622314453126,269.0378762637253,233.69400024414063]},{"page":217,"text":"a","rect":[189.9309539794922,261.87176513671877,193.41491768440566,258.6993713378906]},{"page":217,"text":"(8.19)","rect":[361.0566101074219,248.9851531982422,385.1059812149204,240.5087890625]},{"page":217,"text":"Here g is a function of all the three arguments z, j(z) and dj/dz. The equation is","rect":[65.76696014404297,286.3787536621094,385.18964017974096,277.38446044921877]},{"page":217,"text":"valid for any continuous function g(z) with continuous derivatives g’, g” defined","rect":[53.81494903564453,298.33831787109377,385.1149529557563,289.36395263671877]},{"page":217,"text":"within interval [a, b]. For instance, g might be density of free energy of a liquid","rect":[53.815956115722659,310.2978515625,385.1159600358344,301.323486328125]},{"page":217,"text":"crystal per unit volume, j(z) be an angle the director forms with a selected","rect":[53.815956115722659,322.21759033203127,385.1249932389594,313.28302001953127]},{"page":217,"text":"reference axis and d the thickness of the sample. The values of function g are","rect":[53.815956115722659,334.16015625,385.16773260309068,325.1658630371094]},{"page":217,"text":"fixed at both ends of the interval j(a) ¼ ja and j(b) ¼ jb. In our simplest example,","rect":[53.815940856933597,346.0810852050781,385.17144437338598,337.14532470703127]},{"page":217,"text":"infinitely strong anchoring of the director is assumed at the boundaries.","rect":[53.81370162963867,358.0406188964844,341.3689846565891,349.1060791015625]},{"page":217,"text":"Our task is to find the necessary condition for the extremum of the functional F.","rect":[65.7657241821289,370.00018310546877,385.1824001839328,361.04571533203127]},{"page":217,"text":"Let us assume that function j(z) in Fig. 8.8 corresponds to an extremum of F, i.e.","rect":[53.8137321472168,381.959716796875,385.1744656136203,373.025146484375]},{"page":217,"text":"F ¼ Fextr for j(z) (actually, for physical reasons an extremum to be found corre-","rect":[53.8127555847168,393.9197692871094,385.10064063874855,384.9852294921875]},{"page":217,"text":"sponds to a minimum). Then we introduce a new, probe function fðzÞ þ a\u0002ðzÞ","rect":[53.81254959106445,406.21795654296877,385.16830839263158,396.2674255371094]},{"page":217,"text":"where a is a small numerical parameter and Z(z) is an arbitrary function equal to","rect":[53.81450271606445,417.8388366699219,385.1433953385688,408.904296875]},{"page":217,"text":"zero at both ends of the [a, b] interval. The additional item aZ(z) is called variation","rect":[53.81550216674805,429.6519470214844,385.1204461198188,420.80706787109377]},{"page":217,"text":"of j(z) function that will result in variation dF of functional F. Now if we vary a,","rect":[53.81547546386719,441.701171875,385.1841702034641,432.4678039550781]},{"page":217,"text":"the functional F changes. Therefore, after substituting the new functions with","rect":[53.81549072265625,453.66070556640627,385.1185235123969,444.72613525390627]},{"page":217,"text":"variable a into g(z), we obtain F as a function of parameter a:","rect":[53.815486907958987,465.66009521484377,304.2634111172874,456.68572998046877]},{"page":217,"text":"Fig. 8.8 Illustration of","rect":[53.812843322753909,522.2183227539063,133.36720365149669,514.2683715820313]},{"page":217,"text":"variation procedure: one","rect":[53.812843322753909,532.1265869140625,137.37267443918706,524.5322265625]},{"page":217,"text":"searches for such a function","rect":[53.812843322753909,540.3500366210938,149.49823516993448,534.5081787109375]},{"page":217,"text":"j(z) that satisfies to an","rect":[53.812843322753909,552.0108032226563,132.00835937647745,544.484130859375]},{"page":217,"text":"extremum of functional","rect":[53.812843322753909,560.2451782226563,134.52807335105005,554.4033203125]},{"page":217,"text":"(8.19). jðzÞ þ a\u0002ðzÞ is","rect":[53.812843322753909,572.2615966796875,131.2172516155175,563.8036499023438]},{"page":217,"text":"arbitrary probe function","rect":[53.812843322753909,581.9496459960938,135.3927969375126,574.3552856445313]},{"page":217,"text":"j(z)","rect":[247.015869140625,504.9972229003906,260.6603834090703,497.46282958984377]},{"page":217,"text":"ja","rect":[251.44149780273438,539.669189453125,259.2585821597955,533.382568359375]},{"page":217,"text":"a","rect":[260.40936279296877,578.4391479492188,264.4059944747794,574.6959838867188]},{"page":217,"text":"j(z)+ah(z)","rect":[318.5899353027344,548.2423706054688,356.2534009871953,540.7080078125]},{"page":217,"text":"jb","rect":[363.97967529296877,502.2856750488281,371.7964544742486,495.99835205078127]},{"page":217,"text":"b","rect":[360.4018859863281,578.4391479492188,364.39851766813879,572.7363891601563]},{"page":217,"text":"z","rect":[377.7640686035156,570.3616943359375,381.26511795678177,566.090576171875]},{"page":218,"text":"8.3 Variational Problem and Elastic Torques","rect":[53.812843322753909,44.274620056152347,206.40399630545893,36.68026351928711]},{"page":218,"text":"203","rect":[372.4998779296875,42.55594253540039,385.19150299220009,36.73106384277344]},{"page":218,"text":"b","rect":[162.74154663085938,65.34451293945313,166.22551033577285,60.48480987548828]},{"page":218,"text":"FðaÞ ¼ ð g½z;fðzÞ þ a\u0002ðzÞ;f0ðzÞ þ a\u00020ðzÞ\u0006dz","rect":[129.0943145751953,89.34652709960938,309.93585600005346,67.22430419921875]},{"page":218,"text":"a","rect":[162.2314910888672,95.34357452392578,165.71545479378066,92.17117309570313]},{"page":218,"text":"(8.20)","rect":[361.05670166015627,82.5136489868164,385.1060727676548,74.03728485107422]},{"page":218,"text":"Next we shall explore a new idea: as we vary fðzÞ þ a\u0002ðzÞ and the functional F","rect":[65.76705169677735,120.20610809326172,385.18271630681718,110.2555923461914]},{"page":218,"text":"reaches an extremum then the varied function F(a) must have an extremum at a ¼0","rect":[53.81603240966797,131.37181091308595,385.1807488541938,122.83567810058594]},{"page":218,"text":"(due to the assumption that j(z) corresponds to Fextr). Therefore, the derivative dF/","rect":[55.968135833740237,143.72976684570313,385.1374955899436,134.77613830566407]},{"page":218,"text":"da ¼ 0 at a ¼ 0. Hence, after differentiation (8.20) with respect to a under the","rect":[53.81362533569336,155.69015502929688,385.1753314800438,146.7356719970703]},{"page":218,"text":"integral, we obtain the expression valid at a ¼ 0:","rect":[53.81364059448242,167.64968872070313,251.9721360928733,158.7151336669922]},{"page":218,"text":"b","rect":[138.61068725585938,187.77517700195313,142.09465096077285,182.9154815673828]},{"page":218,"text":"F0ð0Þ ¼ ddFa ¼ ð \u0004@@fg ðz;f;f0Þ\u0002ðzÞ þ @@fg0 ðz;f;f0Þ\u00020ðzÞ\u0005dz ¼ 0;","rect":[78.73799133300781,212.6307830810547,334.51585328263186,188.7455596923828]},{"page":218,"text":"a","rect":[138.1006317138672,217.77423095703126,141.58459541878066,214.60183715820313]},{"page":218,"text":"(8.21)","rect":[361.0567626953125,204.9442901611328,385.10613380281105,196.46792602539063]},{"page":218,"text":"where we used","rect":[53.815101623535159,240.26611328125,113.68470088056097,233.36351013183595]},{"page":218,"text":"@@ag ¼ @ðf@þga\u0002Þ \u0004 @ðf@þaa\u0002Þ ¼ @@fg \u0004 \u0002 (for a ¼ 0) and the same for the term with j’.","rect":[67.40763092041016,257.19732666015627,372.97686429526098,242.01283264160157]},{"page":218,"text":"Integrating by parts the second term in (8.21) we get:","rect":[65.76602172851563,266.160888671875,281.140272689553,257.226318359375]},{"page":218,"text":"F0ð0Þ","rect":[107.79667663574219,304.1925354003906,128.96220792876438,293.7407531738281]},{"page":218,"text":"b","rect":[142.2360382080078,286.2864685058594,145.72000191292129,281.4267578125]},{"page":218,"text":"¼ð @@fg \u0004 \u0002ðzÞdzþ\u0004@@fg0 \u0002ðzÞ\u0005ab\u0003ð \u0002ðzÞddz\u0002@@fg0\u0003dz","rect":[131.75735473632813,311.52410888671877,331.17673124419408,285.281005859375]},{"page":218,"text":"a","rect":[141.72598266601563,316.3420104980469,145.2099463709291,313.16961669921877]},{"page":218,"text":"b","rect":[197.3517608642578,327.03997802734377,200.83572456917129,322.1802673339844]},{"page":218,"text":"¼\u0004@@fg0 \u0002ðzÞ\u0005baþð \u0002ðzÞ\u0004@@fg \u0003 ddz\u0002@@fg0\u0003\u0005dz ¼0","rect":[131.75706481933595,352.2776184082031,321.39699641278755,326.0345458984375]},{"page":218,"text":"a","rect":[196.8417205810547,357.09552001953127,200.32568428596816,353.9231262207031]},{"page":218,"text":"The first term is zero because Z(z) ¼ 0at the ends of the interval [a, b]. And since","rect":[65.76593780517578,381.4735107421875,385.13184393121568,372.62860107421877]},{"page":218,"text":"Z(z) is arbitrary, the expression in the brackets under the integral must be zero.","rect":[53.81592559814453,393.5226745605469,385.1806911995578,384.588134765625]},{"page":218,"text":"Hence, we arrive at the differential Euler equation:","rect":[53.815895080566409,405.3726806640625,260.53498704502177,396.52777099609377]},{"page":218,"text":"@g d @g","rect":[186.36251831054688,428.98822021484377,234.84359065106879,419.6851501464844]},{"page":218,"text":"\u0003","rect":[200.127197265625,432.0,207.79193905088813,430.0]},{"page":218,"text":"@f dz @f0","rect":[185.5691680908203,442.755615234375,236.39228095241109,432.438720703125]},{"page":218,"text":"(8.22)","rect":[361.0562744140625,435.3517150878906,385.10564552156105,426.8753662109375]},{"page":218,"text":"What have we gained? Very much! Now, in order to find the function j(z)","rect":[65.76663970947266,466.3576965332031,385.1614011367954,457.42315673828127]},{"page":218,"text":"corresponding to gmin we have to solve a second order differential equation (8.22),","rect":[53.815574645996097,478.30029296875,385.15545316244848,469.325927734375]},{"page":218,"text":"instead of solving an integral-differential equation (8.19). Two arbitrary constants","rect":[53.81364822387695,490.2203369140625,385.15451444731908,481.2857666015625]},{"page":218,"text":"are to be found from the boundary conditions given for j(z).","rect":[53.81463623046875,502.1799011230469,299.00079007651098,493.245361328125]},{"page":218,"text":"8.3.2 Application to a Twist Cell","rect":[53.812843322753909,552.263427734375,222.28920194457587,541.7093505859375]},{"page":218,"text":"To illustrate the variation technique that is very useful for subsequent discussions","rect":[53.812843322753909,579.8895263671875,385.09393705474096,570.9550170898438]},{"page":218,"text":"of electro-optical effects, consider a simplest example. For a twist cell shown in","rect":[53.812843322753909,591.8490600585938,385.14174738935005,582.91455078125]},{"page":219,"text":"204","rect":[53.81199264526367,42.55630874633789,66.50359863428995,36.73143005371094]},{"page":219,"text":"Fig. 8.9 Twistcell in the zero","rect":[53.812843322753909,67.58130645751953,155.40067047266886,59.546695709228519]},{"page":219,"text":"field: geometry of the problem","rect":[53.812843322753909,77.4895248413086,155.4015221440314,69.89517211914063]},{"page":219,"text":"(a) and the calculated","rect":[53.812843322753909,87.07006072998047,126.5712866225712,79.81436157226563]},{"page":219,"text":"distribution of angle j(z)","rect":[53.812843322753909,97.3846664428711,138.31184476477794,89.79031372070313]},{"page":219,"text":"(i.e. the director angle) over","rect":[53.811988830566409,107.36067962646485,147.64441007239513,99.76632690429688]},{"page":219,"text":"the cell thickness (b)","rect":[53.811988830566409,116.99797821044922,124.03717893225839,109.74227905273438]},{"page":219,"text":"a","rect":[188.2318572998047,68.23336791992188,193.78711969703853,62.64461135864258]},{"page":219,"text":"x","rect":[188.81797790527345,138.02392578125,193.26223233544688,133.76882934570313]},{"page":219,"text":"z","rect":[218.77432250976563,161.16302490234376,222.77095419157627,156.90792846679688]},{"page":219,"text":"8 Elasticity and Defects","rect":[303.50323486328127,44.274986267089847,385.17270358084957,36.68062973022461]},{"page":219,"text":"b","rect":[294.64111328125,68.23336791992188,300.7459070307282,60.92499542236328]},{"page":219,"text":"f","rect":[315.4397888183594,94.03776550292969,319.6042790308061,86.55935668945313]},{"page":219,"text":"π/2","rect":[296.339111328125,99.98844909667969,307.3897805532203,93.94973754882813]},{"page":219,"text":"f(z)","rect":[323.8559265136719,118.22856140136719,337.33735484461718,110.41422271728516]},{"page":219,"text":"0","rect":[304.3243713378906,144.59654235839845,308.76862576806408,138.77378845214845]},{"page":219,"text":"Fig. 8.9a, we are interested in the coordinate dependence of the azimuthal director","rect":[53.812843322753909,215.14840698242188,385.1059506973423,206.21385192871095]},{"page":219,"text":"angle j(z), which is rigidly fixed at the two boundaries, j(0) ¼ 0, j(d) ¼ p/2. The","rect":[53.812843322753909,227.10794067382813,385.14768255426255,218.15345764160157]},{"page":219,"text":"equilibrium director distribution to be found corresponds to the minimum of the","rect":[53.8138313293457,239.06747436523438,385.17258489801255,230.13291931152345]},{"page":219,"text":"elastic free energy for the cell as a whole. First, we should write an expression for","rect":[53.8138313293457,251.02703857421876,385.18059669343605,242.0924835205078]},{"page":219,"text":"the density of Frank elastic energy. The director at any point z is given by","rect":[53.8138313293457,262.986572265625,385.17055598310005,254.05201721191407]},{"page":219,"text":"n ¼ cosfðzÞi þ sinfðzÞj: There is no z-component, nz ¼ 0, even no pretilt at the","rect":[53.81385040283203,275.2279968261719,385.17514837457505,265.2774658203125]},{"page":219,"text":"boundaries.","rect":[53.814430236816409,284.81756591796877,100.12240262533908,277.91497802734377]},{"page":219,"text":"The free energy per unit area in the x, y plane is very simple because we have no","rect":[65.7664566040039,298.8090515136719,385.1711358170844,289.87451171875]},{"page":219,"text":"derivatives over x and y, therefore","rect":[53.81444549560547,310.7486877441406,191.45829046441879,301.83404541015627]},{"page":219,"text":"curln ¼ \u0003@ny i þ @nx j ¼ \u0003cosfdfi \u0003 sinfdfj;","rect":[113.97172546386719,343.1661682128906,323.3550256459131,327.01202392578127]},{"page":219,"text":"@z","rect":[160.0225372314453,348.00299072265627,169.51984037505344,340.6720886230469]},{"page":219,"text":"@z","rect":[190.89422607421876,348.00299072265627,200.44728483306126,340.6720886230469]},{"page":219,"text":"dz","rect":[255.1859893798828,347.9728698730469,263.99545683013158,340.9806213378906]},{"page":219,"text":"dz","rect":[305.3169860839844,347.9728698730469,314.1264687930222,340.9806213378906]},{"page":219,"text":"2","rect":[295.0646057128906,368.6524963378906,298.54856941780408,363.939208984375]},{"page":219,"text":"ncurln ¼ \u0003ddfz and (ncurlnÞ2 ¼ \u0002ddfz\u0003:","rect":[137.2510986328125,389.75848388671877,301.71742188614749,365.8732604980469]},{"page":219,"text":"From here and Eq. (8.17), the density gdistand the total Frank energy are given by","rect":[65.76578521728516,413.34466552734377,385.16924372724068,404.3699951171875]},{"page":219,"text":"gdist ¼ 12K22\u0002ddfz\u00032;","rect":[131.8136749267578,460.26953125,211.02853333634278,434.4500427246094]},{"page":219,"text":"d","rect":[251.39125061035157,435.4345703125,254.87521431526504,430.5400085449219]},{"page":219,"text":"F ¼ 21 ð K22\u0002ddfz\u00032dz","rect":[222.67198181152345,460.26953125,307.2161904727097,434.4500427246094]},{"page":219,"text":"0","rect":[250.93833923339845,465.77392578125,254.4223029383119,460.9769592285156]},{"page":219,"text":"(8.23)","rect":[361.0565185546875,452.5826416015625,385.10588966218605,444.1062927246094]},{"page":219,"text":"To find j(z) we should write the Euler equation for functional (8.23). In that","rect":[65.7668685913086,492.3169860839844,385.14069993564677,483.3824462890625]},{"page":219,"text":"equation we have neither z nor j(z) given explicitly and should only use differenti-","rect":[53.81482696533203,504.2765197753906,385.18157325593605,495.34197998046877]},{"page":219,"text":"ation with respect to j’(z). Now Euler’s equation reads:","rect":[53.815818786621097,516.1792602539063,279.575331283303,507.2447509765625]},{"page":219,"text":"d @g","rect":[178.48887634277345,542.916748046875,201.53626338056098,533.6136474609375]},{"page":219,"text":"\u0003 dz @f0 ¼ \u0003K22 dz2 ¼0","rect":[166.822998046875,556.6841430664063,274.38176814130318,540.8037719726563]},{"page":219,"text":"(8.24)","rect":[361.0560607910156,549.2801513671875,385.10543189851418,540.8037719726563]},{"page":219,"text":"This equation may be considered as a balance of torques in the bulk, although in","rect":[65.76642608642578,582.2702026367188,385.14031306317818,573.335693359375]},{"page":219,"text":"this particular case, the elastic forces are balanced by the fixed boundary conditions","rect":[53.81439971923828,594.229736328125,385.11044706450658,585.2952270507813]},{"page":220,"text":"8.3 Variational Problem and Elastic Torques","rect":[53.812843322753909,44.274620056152347,206.40399630545893,36.68026351928711]},{"page":220,"text":"205","rect":[372.4998779296875,42.55594253540039,385.19150299220009,36.62946701049805]},{"page":220,"text":"and the rotation of the director is possible only together with the cell substrates.","rect":[53.812843322753909,68.2883529663086,385.15667386557348,59.35380554199219]},{"page":220,"text":"After the first integration we obtain:","rect":[53.812843322753909,80.24788665771485,199.48078782627176,71.31333923339844]},{"page":220,"text":"q ¼ df=dz ¼ const ¼C","rect":[170.4443359375,104.49571228027344,268.55699416812208,94.56511688232422]},{"page":220,"text":"(8.25)","rect":[361.0565185546875,103.7117691040039,385.10588966218605,95.11588287353516]},{"page":220,"text":"This is an important result showing a linear distribution of azimuthal angle over","rect":[65.7668685913086,128.02932739257813,385.11592994538918,119.09477233886719]},{"page":220,"text":"the layer thickness. The second integration gives us the value of the constant. It","rect":[53.814842224121097,139.98886108398438,385.1796098477561,131.05430603027345]},{"page":220,"text":"depends on the difference between the azimuthal angles fixed at the opposite","rect":[53.814842224121097,151.94839477539063,385.09790838434068,143.0138397216797]},{"page":220,"text":"boundaries. In our cell, j1 ¼ 0 at z ¼ 0 and j2 ¼ p/2 at z ¼ d. Therefore","rect":[53.814842224121097,163.82923889160157,385.1561969585594,154.95445251464845]},{"page":220,"text":"f ¼ Cz ¼ pz=2d; this linear dependence is illustrated by Fig. 8.9b. Equation","rect":[53.814369201660159,176.19715881347657,385.1133660416938,166.2665557861328]},{"page":220,"text":"(8.25) is valid for any uniform twist distortion; for instance, for nematics twisted","rect":[53.81635284423828,187.8280029296875,385.17519465497505,178.83367919921876]},{"page":220,"text":"through angles p/4 or p the functions j ¼ (pz/4d) and j ¼ (pz/d), respectively. The","rect":[53.81635284423828,199.73077392578126,385.1492084331688,190.7762908935547]},{"page":220,"text":"linear dependence remains even in the case of non-rigid boundary conditions,","rect":[53.81536102294922,211.69033813476563,385.1343044808078,202.7557830810547]},{"page":220,"text":"however, external magnetic or electric fields can easily distort such a uniform","rect":[53.81536102294922,223.64987182617188,385.1632151472624,214.71531677246095]},{"page":220,"text":"distribution.","rect":[53.81536102294922,233.57745361328126,102.41878171469455,226.6748504638672]},{"page":220,"text":"It is instructive to calculate the value of the elastic energy (per unit area) of a","rect":[65.76738739013672,247.56893920898438,385.1602252788719,238.63438415527345]},{"page":220,"text":"typical twisted cell, discussed above. Using (8.23), the free energy is given by","rect":[53.81536102294922,259.52850341796877,369.4255303483344,250.5939483642578]},{"page":220,"text":"d","rect":[185.0030975341797,279.6756591796875,188.48706123909316,274.7810974121094]},{"page":220,"text":"F ¼ 21 ð K22\u0006p=2d\u00072dz ¼ p28Kd22","rect":[156.28598022460938,303.6560363769531,280.56404182991346,279.9381408691406]},{"page":220,"text":"0","rect":[184.6066131591797,310.0149841308594,188.09057686409316,305.218017578125]},{"page":220,"text":"Taking cell thickness d ¼ 10 mm (10\u00033cm), K22 ¼ 3\u000410\u00037 dyn (or 3\u000410\u000312 N) we","rect":[65.7659683227539,334.5748291015625,385.1718219585594,323.4671936035156]},{"page":220,"text":"find F \u0005 3.7\u000410\u00034 erg/cm2 or 0.37 mJ/m2 in SI units.","rect":[53.81405258178711,346.47784423828127,265.16033597494848,335.4268493652344]},{"page":220,"text":"The example of the variational procedure considered in this section was very","rect":[65.76554107666016,358.4374084472656,385.1165703873969,349.50286865234377]},{"page":220,"text":"simple, because we operated only with one independent variable (angle j). Some-","rect":[53.81351852416992,370.3969421386719,385.1404050430454,361.46240234375]},{"page":220,"text":"time one needs to minimize the energy with respect to two variables; in fact, we met","rect":[53.813533782958987,382.3564758300781,385.1374650723655,373.42193603515627]},{"page":220,"text":"this case in Section 6.3.3 for an infinite medium. For two variables, the system of","rect":[53.813533782958987,394.3160400390625,385.1504148086704,385.3814697265625]},{"page":220,"text":"two Euler equations can be constructed using the same procedure as earlier.","rect":[53.813533782958987,406.2755432128906,385.0926785042453,397.34100341796877]},{"page":220,"text":"However, very often one deals with some constraints as, for example, in the case","rect":[53.813533782958987,418.235107421875,385.1444476909813,409.300537109375]},{"page":220,"text":"of the director that has three projections satisfying the constraint nx2 þ ny2 þ n2z ¼ 1:","rect":[53.813533782958987,432.25970458984377,385.15638672989749,419.7129821777344]},{"page":220,"text":"In such cases the Lagrange multipliers are introduced to solve the variational","rect":[53.814598083496097,442.0982360839844,385.1285844571311,433.1636962890625]},{"page":220,"text":"problem, however this, more general Euler – Lagrange approach will not be used","rect":[53.814598083496097,454.0577697753906,385.12261286786568,445.12322998046877]},{"page":220,"text":"in this book.","rect":[53.814598083496097,463.9554748535156,104.02363248373752,457.082763671875]},{"page":220,"text":"8.3.3 “Molecular Field” and Torques","rect":[53.812843322753909,518.0848999023438,247.78941993448897,507.5307922363281]},{"page":220,"text":"The director n of a nematic can be re-aligned from its equilibrium position by an","rect":[53.812843322753909,545.7108764648438,385.1507195573188,536.7763671875]},{"page":220,"text":"external magnetic (or electric) field because these fields exert torques onto n. If the","rect":[53.813846588134769,557.67041015625,385.1755756206688,548.7359008789063]},{"page":220,"text":"field is strong enough and magnetic wa or dielectric ea anisotropy is positive,","rect":[53.8138313293457,569.6301879882813,385.1348537972141,560.6954345703125]},{"page":220,"text":"the director will be aligned along the field. On the other hand, being deflected","rect":[53.81394577026367,581.5897216796875,385.0910576920844,572.63525390625]},{"page":220,"text":"from the equilibrium state by dn, the director relaxes back due to elasticity. It looks","rect":[53.81394577026367,593.5492553710938,385.1656838809128,584.31591796875]},{"page":221,"text":"206","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.68026351928711]},{"page":221,"text":"8 Elasticity and Defects","rect":[303.50408935546877,44.274620056152347,385.17355807303707,36.68026351928711]},{"page":221,"text":"like the director feels a sort of “molecular field” that causes it to rotate back to the","rect":[53.812843322753909,66.25641632080078,385.1706012554344,59.35380554199219]},{"page":221,"text":"equilibrium position. The “molecular field” should not be confused with the mean","rect":[53.812843322753909,80.24788665771485,385.13872614911568,71.31333923339844]},{"page":221,"text":"field used in the molecular theory and characterized by the nematic potential curve","rect":[53.812843322753909,92.20748138427735,385.14575994684068,83.27293395996094]},{"page":221,"text":"discussed in Chapters 3 and 6.","rect":[53.812843322753909,104.11019134521485,176.21877713705784,95.17564392089844]},{"page":221,"text":"Mathematically the “molecular” field vector h can be found using the Euler-","rect":[65.7648696899414,116.0697250366211,385.11284766999855,107.13517761230469]},{"page":221,"text":"Lagrange approach by a variation of the elastic and magnetic (or electric) parts of","rect":[53.812843322753909,128.02932739257813,385.1497739395298,119.09477233886719]},{"page":221,"text":"the free energy with respect to the director variable n(r) (with a constraint of n2 ¼","rect":[53.812843322753909,139.98886108398438,385.14807918760689,128.93870544433595]},{"page":221,"text":"1). For the elastic torque, in the absence of the external field, the splay, twist and","rect":[53.814231872558597,151.94912719726563,385.1441582780219,143.0145721435547]},{"page":221,"text":"bend terms of h are obtained [9] from the Frank energy (8.16):","rect":[53.814231872558597,163.90869140625,307.4086747891624,154.97413635253907]},{"page":221,"text":"hsplay ¼ K11rðdivnÞ","rect":[94.03317260742188,189.6658935546875,175.8066750918503,179.1799774169922]},{"page":221,"text":"hbend ¼ K33f½ðn \u0002 cirlnÞ \u0002 curln\u0006 þ curl½n \u0002 ðn \u0002 cirlnÞg","rect":[94.99342346191406,204.09417724609376,329.24880305341255,194.14366149902345]},{"page":221,"text":"htwist ¼ \u0003K22f½ðn \u0004 cirln \u0004 nÞ \u0004 curln\u0006 þ curl½ðn \u0004 cirln \u0004 nÞ\u0006 \u0004 ng","rect":[95.55882263183594,219.00128173828126,344.96181574872505,209.05076599121095]},{"page":221,"text":"In the one-constant approximation K11 ¼ K22 ¼ K33 ¼ K and the expression for","rect":[65.76708984375,243.60223388671876,385.18035255281105,234.66712951660157]},{"page":221,"text":"the molecular field becomes very simple, similar to (8.24):","rect":[53.814659118652347,255.561767578125,290.7547441250999,246.62721252441407]},{"page":221,"text":"hdist ¼ Kr2n","rect":[192.48178100585938,280.2066955566406,246.51988572428776,269.79229736328127]},{"page":221,"text":"(8.26)","rect":[361.0560302734375,280.32965087890627,385.10540138093605,271.79351806640627]},{"page":221,"text":"This approximation is useful when solving problems related to the field behavior","rect":[65.7663803100586,304.6471862792969,385.09856544343605,295.712646484375]},{"page":221,"text":"of liquid crystals. In the thermodynamic equilibrium, the director is always aligned","rect":[53.81433868408203,316.60675048828127,385.09551325849068,307.67218017578127]},{"page":221,"text":"along the molecular field vector, n||h. When there is an external electric or magnetic","rect":[53.81433868408203,328.5662841796875,385.18103826715318,319.6317138671875]},{"page":221,"text":"field (see for details Section 11.2.1), the corresponding terms given by","rect":[53.814369201660159,340.46905517578127,337.9861687760688,331.53448486328127]},{"page":221,"text":"hE ¼ \u0003 ea ðEnÞE","rect":[143.82333374023438,368.6949462890625,215.3114955108955,354.77874755859377]},{"page":221,"text":"4p","rect":[176.67620849609376,373.143310546875,187.1560093815143,366.3403015136719]},{"page":221,"text":"hH ¼ \u0003waðHnÞH","rect":[225.27865600585938,368.6952819824219,295.1359934675749,358.7447509765625]},{"page":221,"text":"(8.27)","rect":[361.0567932128906,367.95819091796877,385.10616432038918,359.4818420410156]},{"page":221,"text":"should be added to the molecular field. Such a non-zero sum of all these vectors","rect":[53.815101623535159,394.6184387207031,385.0952493106003,387.70587158203127]},{"page":221,"text":"creates a torque exerted on the director","rect":[53.815101623535159,408.5999450683594,211.30962501374854,399.6654052734375]},{"page":221,"text":"G ¼ n \u0002 Xhi","rect":[188.8001708984375,436.77410888671877,249.72387200157542,422.83941650390627]},{"page":221,"text":"(8.28)","rect":[361.0561828613281,434.0481262207031,385.10555396882668,425.57177734375]},{"page":221,"text":"which causes the director rotation. Actually the torque due to the molecular field","rect":[53.81450653076172,460.3497619628906,385.11550227216255,451.41522216796877]},{"page":221,"text":"can be balanced by other (e.g. viscous) torques. In this way, we can write a torque","rect":[53.81450653076172,472.2525329589844,385.1324848003563,463.3179931640625]},{"page":221,"text":"balance equation.","rect":[53.81450653076172,484.2120666503906,124.22562070395236,475.27752685546877]},{"page":221,"text":"n \u0002 Xhi þ viscous torque þ others ¼0","rect":[132.6080780029297,512.4430541992188,306.38616267255318,498.5083312988281]},{"page":221,"text":"(8.29)","rect":[361.0556945800781,509.7169494628906,385.10503516999855,501.2406005859375]},{"page":221,"text":"8.3.4 Director Fluctuations","rect":[53.812843322753909,552.6240844726563,197.29540137980147,544.0899658203125]},{"page":221,"text":"This is another important example of a successful application of the theory of","rect":[53.812843322753909,582.2699584960938,385.1457761367954,573.33544921875]},{"page":221,"text":"elasticity. In Section 11.1.3 we shall discuss the nature of strong light scattering by","rect":[53.812843322753909,594.2294921875,385.16863337567818,585.2949829101563]},{"page":222,"text":"8.3 Variational Problem and Elastic Torques","rect":[53.812843322753909,44.274620056152347,206.40399630545893,36.68026351928711]},{"page":222,"text":"207","rect":[372.4998779296875,42.55594253540039,385.19150299220009,36.73106384277344]},{"page":222,"text":"nematics. In fact, such scattering is caused by small values of the Frank elastic","rect":[53.812843322753909,68.2883529663086,385.09690130426255,59.35380554199219]},{"page":222,"text":"moduli. The latter results in strong thermal fluctuations of the director. Here, we","rect":[53.812843322753909,80.24788665771485,385.1686481304344,71.31333923339844]},{"page":222,"text":"consider a simple approach to calculation of the amplitude of the director fluctua-","rect":[53.812843322753909,92.20748138427735,385.1138852676548,83.27293395996094]},{"page":222,"text":"tions suggested by de Gennes [12].","rect":[53.812843322753909,104.11019134521485,194.99740262533909,95.17564392089844]},{"page":222,"text":"We again consider the director n0 oriented along the z-axis. Its fluctuating part","rect":[65.7648696899414,116.07039642333985,385.1467424161155,107.13517761230469]},{"page":222,"text":"has components (nx, ny, 0). The total Frank elastic energy, related to the fluctuations","rect":[53.81386947631836,129.0030975341797,385.0893899356003,119.09544372558594]},{"page":222,"text":"in volume V, is given by the integral of free energy density (8.15b):","rect":[53.81325149536133,139.98953247070313,327.34699876377177,130.99520874023438]},{"page":222,"text":"F ¼ 21 ð (K11\u0002@@nxx þ @@nyy\u00032 þ K22\u0002@@nyx \u0003 @@nxy\u00032 þ K33\"\u0002@@nzx\u00032 þ \u0002@@nzy\u00032#)dr","rect":[59.0820198059082,189.75682067871095,379.89368475152818,159.8953094482422]},{"page":222,"text":"V","rect":[86.8370132446289,192.27391052246095,91.11532067426264,187.5466766357422]},{"page":222,"text":"(8:30)","rect":[355.5616149902344,207.3818817138672,379.87578712312355,198.905517578125]},{"page":222,"text":"The Fourier harmonics nx and ny of the director fluctuating field are represented","rect":[65.76569366455078,235.63307189941407,385.1656426530219,225.7687530517578]},{"page":222,"text":"by volume integrals (q is wavevector):","rect":[53.81386947631836,246.6060791015625,210.49515662995948,237.502197265625]},{"page":222,"text":"Now","rect":[65.76752471923828,312.0,85.0388770098957,305.1510314941406]},{"page":222,"text":"nxðqÞ ¼ ð nxðrÞexpðiqrÞdr and nyðqÞ ¼ ð nyðrÞexpðiqrÞdr","rect":[87.85728454589844,283.02484130859377,327.0446246929344,260.9026184082031]},{"page":222,"text":"V","rect":[122.91999816894531,289.3681335449219,127.19830559857904,284.64093017578127]},{"page":222,"text":"(8.31)","rect":[361.0571594238281,276.1919250488281,385.10653053132668,267.715576171875]},{"page":222,"text":"the corresponding free energy is represented by a sum of the q harmonics:","rect":[87.68869018554688,313.8863830566406,385.16425950595927,304.95184326171877]},{"page":222,"text":"F ¼21V XnK11jnxðqÞqx þ nyðqÞqyj2","rect":[89.16191101074219,348.5962219238281,240.7705695398744,328.08062744140627]},{"page":222,"text":"q","rect":[127.73484802246094,353.46728515625,131.2188117273744,348.80279541015627]},{"page":222,"text":"þ K33q2zhjnxðqÞj2 þ jnyðqÞj2io","rect":[97.4296875,378.2842712402344,220.77559781896788,360.3753356933594]},{"page":222,"text":"þ K22jnxðqÞqy","rect":[243.51756286621095,344.574462890625,300.1617826928176,334.14532470703127]},{"page":222,"text":"\u0003 nyðqÞqxj2","rect":[304.52435302734377,346.7284240722656,351.0019690760072,334.2950134277344]},{"page":222,"text":"The q-vector consists of three components (qx, qy, qz) and the obtained quadratic","rect":[65.7656478881836,402.7280578613281,385.10215032770005,392.806640625]},{"page":222,"text":"form for nx(q) and ny(q) is complicated because it is not diagonal. However, it can","rect":[53.81411361694336,414.6744384765625,385.1236504655219,404.7667236328125]},{"page":222,"text":"be made diagonal if one takes a simplified geometry corresponding to the symmetry","rect":[53.81266403198242,425.6609191894531,385.16945735028755,416.72637939453127]},{"page":222,"text":"of a scattering experiment. To this effect, de Gennes selected new coordinate axes:","rect":[53.81266403198242,437.6204528808594,385.1276689297874,428.6859130859375]},{"page":222,"text":"the axis e2 was chosen to be perpendicular to the n0 (i.e. to z-axis) and simulta-","rect":[53.81266403198242,449.5802917480469,385.15545020906105,440.64544677734377]},{"page":222,"text":"neously perpendicular to the scattering vector q, as shown in Fig. 8.10a. The other","rect":[53.813594818115237,461.53985595703127,385.13848243562355,452.60528564453127]},{"page":222,"text":"axis e1 was chosen to be perpendicular to the z-axis and e2. Now the q-vector is","rect":[53.81260299682617,473.4996032714844,385.18878568755346,464.5648193359375]},{"page":222,"text":"resolved not into three components but only into two: qz ¼ q|| and q⊥ (|| and ⊥ to the","rect":[53.814144134521487,485.4591369628906,385.1747516460594,476.52459716796877]},{"page":222,"text":"director). Correspondingly we have two normal modes of fluctuations.","rect":[53.814022064208987,497.3619079589844,337.5150112679172,488.4273681640625]},{"page":222,"text":"In Fig. 8.10b, we see that the fluctuation mode n1(q) is a mixture of the splay and","rect":[65.76607513427735,509.3218688964844,385.1441582780219,500.38690185546877]},{"page":222,"text":"bend distortions, and the component n2(q) is a mixture of twist and bend distortions.","rect":[53.81332015991211,521.2813720703125,385.1321072151828,512.3468627929688]},{"page":222,"text":"This may be clarified as follows: the splay-bend (SB) mode on the left side of","rect":[53.81319808959961,533.2410278320313,385.14910255281105,524.3065185546875]},{"page":222,"text":"Fig. 8.10b corresponds to realignment of the molecules within the q,z-plane asq","rect":[53.81319808959961,545.2005615234375,385.17791071942818,536.2660522460938]},{"page":222,"text":"evolves and there is no twist here.In contrast, on the right side of the same figure the","rect":[53.81318283081055,557.1600952148438,385.1709369487938,548.2255859375]},{"page":222,"text":"molecules are deflected from the qz-plane of the figure; therefore, the twist and bend","rect":[53.81318283081055,569.11962890625,385.1361016373969,560.1851196289063]},{"page":222,"text":"are present but the splay is absent (TB mode).","rect":[53.81319808959961,581.0792236328125,239.74691434408909,572.1447143554688]},{"page":223,"text":"208","rect":[53.81338119506836,42.55673599243164,66.50498718409463,36.73185729980469]},{"page":223,"text":"a","rect":[80.83267974853516,68.32437133789063,86.387942145769,62.73561477661133]},{"page":223,"text":"z","rect":[138.2109375,66.3856201171875,142.20756918181065,62.13052749633789]},{"page":223,"text":"q||","rect":[136.7891082763672,85.24457550048828,144.3503428500468,77.83412170410156]},{"page":223,"text":"8 Elasticity and Defects","rect":[303.504638671875,44.275413513183597,385.1741073894433,36.68105697631836]},{"page":223,"text":"q","rect":[91.81083679199219,112.49823760986328,96.25509122216562,106.54750061035156]},{"page":223,"text":"e1","rect":[82.96758270263672,187.27236938476563,90.74522390368476,180.96971130371095]},{"page":223,"text":"q^","rect":[111.54460144042969,169.921142578125,119.93402483518148,162.53477478027345]},{"page":223,"text":"e2","rect":[183.29872131347657,145.36474609375,191.07615652087228,139.0620880126953]},{"page":223,"text":"q","rect":[209.15194702148438,158.03253173828126,213.5962014516578,152.08180236816407]},{"page":223,"text":"z","rect":[228.22866821289063,163.38580322265626,232.22529989470127,159.13070678710938]},{"page":223,"text":"2π/q","rect":[294.2666015625,137.479736328125,309.7647500356422,129.93734741210938]},{"page":223,"text":"Fig. 8.10 New coordinate axes, e1 and e2 appropriate to the normal modes of director fluctuations","rect":[53.812843322753909,208.94342041015626,385.1395920085839,200.9934844970703]},{"page":223,"text":"in a nematic liquid crystal (a) and the structure of the normal modes, namely splay-bend (SB) and","rect":[53.81338119506836,218.79495239257813,385.1971487441532,211.20059204101563]},{"page":223,"text":"twist-bend modes (TB)","rect":[53.81338119506836,228.4322052001953,132.92862027747325,221.176513671875]},{"page":223,"text":"After the transform to the new variables na(q) (a ¼ 1, 2) the equation for the free","rect":[65.76496887207031,253.63467407226563,385.1361774273094,244.7001190185547]},{"page":223,"text":"energy reads:","rect":[53.8132438659668,265.5942077636719,107.65556962558816,256.65966796875]},{"page":223,"text":"F ¼ 21V X X jnaðqÞj2\bK33qj2j þ Kaq?2","rect":[134.1925048828125,300.3042297363281,298.3907862833011,279.78863525390627]},{"page":223,"text":"q a¼1;2","rect":[173.900634765625,305.4385986328125,202.53515694221816,299.3796691894531]},{"page":223,"text":"(8.32)","rect":[361.05609130859377,295.0670166015625,385.1054624160923,286.5906677246094]},{"page":223,"text":"Here Ka is the combination of Frank elastic moduli K11 and K22.","rect":[65.76642608642578,329.4889831542969,326.3855862190891,320.992431640625]},{"page":223,"text":"The last equation has a remarkable feature: the different Fourier components of","rect":[65.76644134521485,341.88653564453127,385.14730201570168,332.95196533203127]},{"page":223,"text":"fluctuations are decoupled because they are normal modes for the system. This","rect":[53.814414978027347,353.8460998535156,385.18710722075658,344.91156005859377]},{"page":223,"text":"allows us to apply the principle of equipartition, according to which the energy of","rect":[53.814414978027347,365.8056335449219,385.1502622207798,356.83123779296877]},{"page":223,"text":"each mode is equal to kBT/2. Therefore, for each mode with a ¼ 1, 2, the final","rect":[53.814414978027347,377.76519775390627,385.181288314553,368.81072998046877]},{"page":223,"text":"equation for the mean-square magnitude of the director fluctuations reads:","rect":[53.8135871887207,389.72528076171877,352.9555803067405,380.79071044921877]},{"page":223,"text":"DjnaðqÞj2E ¼ V K33qj2jkBþTKaq?2","rect":[157.53013610839845,429.61285400390627,279.30124770908238,403.8966979980469]},{"page":223,"text":"(8.33)","rect":[361.0561828613281,419.2543640136719,385.10555396882668,410.77801513671877]},{"page":223,"text":"The latter transformation is based on the Gibbs distribution that gives us the","rect":[65.76653289794922,453.0940856933594,385.17429388238755,444.1595458984375]},{"page":223,"text":"probability of the mean square value jnaðqÞj2for a particular director fluctuation","rect":[53.81450653076172,465.4395446777344,385.1198357682563,453.4381408691406]},{"page":223,"text":"with wavevector q when the average value hjnaðqÞj2ifor all fluctuations is known:","rect":[53.81386947631836,477.3992004394531,385.14289720127177,465.3977966308594]},{"page":223,"text":"w / exp\"\u0003VðK33q2jj þkKBaTq?2 ÞjnaðqÞj2# ¼ exp\"\u0003<jjnnaaððqqÞÞjj22>#","rect":[92.27607727050781,521.9066772460938,346.6974702927133,491.2442626953125]},{"page":223,"text":"From Eq. (8.33) for the mean square amplitude of the director fluctuations, we","rect":[65.7657699584961,545.4267578125,385.17050970270005,536.4922485351563]},{"page":223,"text":"can derive the amplitude of the fluctuations of the dielectric tensor and then find the","rect":[53.81374740600586,557.3862915039063,385.1715473003563,548.4517822265625]},{"page":223,"text":"cross-section for the light scattering, see Section 11.1.3. The de Gennes description","rect":[53.81374740600586,569.3458862304688,385.1556328873969,560.411376953125]},{"page":223,"text":"of the director fluctuations in the continuous medium [12] was a strong argument","rect":[53.81277084350586,581.305419921875,385.172499252053,572.3709106445313]},{"page":223,"text":"against the so-called swarm models of liquid crystals. That model was based on the","rect":[53.81277084350586,593.2649536132813,385.1705707378563,584.3304443359375]},{"page":224,"text":"8.4 Defects in Nematics and Cholesterics","rect":[53.812843322753909,42.55594253540039,195.01965792655268,36.68026351928711]},{"page":224,"text":"209","rect":[372.4981994628906,42.62367248535156,385.1898245254032,36.73106384277344]},{"page":224,"text":"concept of rather large blocks (or swarms) with the molecules uniformly aligned","rect":[53.812843322753909,68.2883529663086,385.0949334245063,59.35380554199219]},{"page":224,"text":"within a swarm and variable orientation of swarms as a whole [13]. Such disconti-","rect":[53.812843322753909,79.64029693603516,385.17855201570168,71.31333923339844]},{"page":224,"text":"nuity was considered to be responsible for the strong light scattering by nematics.","rect":[53.812843322753909,92.20748138427735,385.1795315315891,83.27293395996094]},{"page":224,"text":"Nowadays the continuous theory is the corner-stone of the physics of liquid crystal.","rect":[53.812843322753909,104.11019134521485,385.1158108284641,95.17564392089844]},{"page":224,"text":"8.4 Defects in Nematics and Cholesterics","rect":[53.812843322753909,152.2697296142578,270.0597751590983,143.2335968017578]},{"page":224,"text":"8.4.1 Nematic Texture and Volterra Process","rect":[53.812843322753909,182.04473876953126,280.5081088993327,173.5106201171875]},{"page":224,"text":"8.4.1.1 Textures","rect":[53.812843322753909,209.91796875,128.87456144438938,202.41773986816407]},{"page":224,"text":"The concept of defects came about from crystallography. Defects are disruptions of","rect":[53.812843322753909,235.61004638671876,385.14775977937355,226.6754913330078]},{"page":224,"text":"ideal crystal lattice such as vacancies (point defects) or dislocations (linear defects).","rect":[53.812843322753909,247.569580078125,385.1009487679172,238.63502502441407]},{"page":224,"text":"In numerous liquid crystalline phases, there is variety of defects and many of them","rect":[53.812843322753909,259.5291442871094,385.12882183671555,250.59458923339845]},{"page":224,"text":"are not observed in the solid crystals. A study of defects in liquid crystals is very","rect":[53.812843322753909,271.4886474609375,385.1138848405219,262.5540771484375]},{"page":224,"text":"important from both the academic and practical points of view [7,8]. Defects in","rect":[53.812843322753909,283.44818115234377,385.1436699967719,274.51361083984377]},{"page":224,"text":"liquid crystals are very useful for (i) identification of different phases by micro-","rect":[53.81379318237305,295.3509521484375,385.13082252351418,286.4163818359375]},{"page":224,"text":"scopic observation of the characteristic defects; (ii) study of the elastic properties","rect":[53.81379318237305,307.3105163574219,385.16061796294408,298.3759765625]},{"page":224,"text":"by observation of defect interactions; (iii) understanding of the three-dimensional","rect":[53.81379318237305,319.2700500488281,385.129774642678,310.33551025390627]},{"page":224,"text":"periodic structures (e.g., the blue phase in cholesterics) using a new concept of","rect":[53.81379318237305,331.2296142578125,385.1496518692173,322.2950439453125]},{"page":224,"text":"“lattices of defects”; (iv) modelling of fundamental physical phenomena such as","rect":[53.81379318237305,343.1891174316406,385.13873685942846,334.25457763671877]},{"page":224,"text":"magnetic monopoles, interaction of quarks, etc. In the optical technology, defects","rect":[53.81379318237305,355.148681640625,385.1595803652878,346.214111328125]},{"page":224,"text":"usually play the detrimental role: examples are defect walls in the twist nematic","rect":[53.81379318237305,367.10821533203127,385.1008685894188,358.17364501953127]},{"page":224,"text":"cells, shock instability in ferroelectric smectics, Grandjean disclinations in chole-","rect":[53.81379318237305,379.0677490234375,385.1128171524204,370.1331787109375]},{"page":224,"text":"steric cells used in dye microlasers, etc. However, more recently, defect structures","rect":[53.81379318237305,390.97052001953127,385.1616250430222,382.03594970703127]},{"page":224,"text":"find their applications in three-dimensional photonic crystals (e.g. blue phases), the","rect":[53.81379318237305,402.9300842285156,385.1725543804344,393.99554443359377]},{"page":224,"text":"bistable displays and smart memory cards.","rect":[53.81379318237305,414.8896179199219,225.25216336752659,405.955078125]},{"page":224,"text":"Generally, microscopic observations reveal different types of defects, which","rect":[65.76581573486328,426.84918212890627,385.11684504560005,417.91461181640627]},{"page":224,"text":"may","rect":[53.81379318237305,438.8087463378906,71.01467219403753,431.0]},{"page":224,"text":"be","rect":[76.75526428222656,437.0,86.18190038873517,429.87420654296877]},{"page":224,"text":"0-dimensional","rect":[91.93643188476563,437.0,149.1223741299827,429.87420654296877]},{"page":224,"text":"(points),","rect":[154.86795043945313,438.8087463378906,188.42658658530002,429.87420654296877]},{"page":224,"text":"one-dimensional","rect":[194.12237548828126,437.0,260.8146196133811,429.87420654296877]},{"page":224,"text":"(lines)","rect":[266.5143737792969,438.4103088378906,292.09075294343605,429.87420654296877]},{"page":224,"text":"and","rect":[297.7825622558594,437.0,312.18630305341255,429.87420654296877]},{"page":224,"text":"two-dimensional","rect":[317.94781494140627,437.0,385.0779863125999,429.87420654296877]},{"page":224,"text":"(walls). Typical nematic textures are","rect":[53.81379318237305,450.7682800292969,201.5203327007469,441.833740234375]},{"page":224,"text":"1. The thread texture usually observed in thick layers","rect":[53.81379318237305,468.6792297363281,270.8484231387253,459.74468994140627]},{"page":224,"text":"2. Schlieren texture observed in thin cells","rect":[53.81379318237305,478.6167907714844,223.692340985405,471.7042236328125]},{"page":224,"text":"In Fig. 8.11 an example is given of a Schlieren texture in the nematic phase","rect":[65.76581573486328,498.5497131347656,385.09888494684068,489.61517333984377]},{"page":224,"text":"observed under a polarisation microscope. The polariser and analyser are always","rect":[53.81378936767578,510.5092468261719,385.08102811919408,501.57470703125]},{"page":224,"text":"crossed and their positions with respect to photos (a) and (b) differ by 45\b as shown","rect":[53.81378936767578,522.46875,385.15410700849068,513.4744873046875]},{"page":224,"text":"by small crosses. On both photos characteristic brushes (threads) are seen origi-","rect":[53.814231872558597,534.431396484375,385.1570981582798,525.4769287109375]},{"page":224,"text":"nated and terminated at some points. The points are linear singularities (disinclina-","rect":[53.814231872558597,546.3909301757813,385.14315162507668,537.4564208984375]},{"page":224,"text":"tions or just disclinations) to be discussed below. Note the difference betweena","rect":[53.814231872558597,558.3504638671875,385.1590961284813,549.4159545898438]},{"page":224,"text":"number of brushes originated or terminated in different points: only two brushes in","rect":[53.814231872558597,570.3099975585938,385.14312068036568,561.37548828125]},{"page":224,"text":"points 1 and 5 and four brushes in points 2, 3 and 4. It is evident that the pictures","rect":[53.814231872558597,582.26953125,385.11728300200658,573.2752685546875]},{"page":224,"text":"discussed are related to the local orientation of the director, i.e. to the structure of","rect":[53.814231872558597,592.140380859375,385.1501401504673,585.23779296875]},{"page":225,"text":"210","rect":[53.812381744384769,42.55636978149414,66.50398773341104,36.73149108886719]},{"page":225,"text":"8 Elasticity and Defects","rect":[303.5036315917969,44.275047302246097,385.17310030936519,36.68069076538086]},{"page":225,"text":"Fig. 8.11 Schlieren texture observed between","rect":[53.812843322753909,174.53817749023438,218.0785879531376,166.58824157714845]},{"page":225,"text":"polarizers differs in photos (a) and (b) by 45\b","rect":[53.812843322753909,184.44644165039063,209.8860590010468,176.80128479003907]},{"page":225,"text":"Fig. 8.12 Volterra process in","rect":[53.812843322753909,229.6319580078125,155.3583731093876,221.68202209472657]},{"page":225,"text":"nematic liquid crystal placed","rect":[53.812843322753909,239.48345947265626,152.58228058009073,231.88909912109376]},{"page":225,"text":"between two glasses with side","rect":[53.812843322753909,249.45941162109376,155.31691882395269,241.86505126953126]},{"page":225,"text":"view (a) and top view (b).","rect":[53.812843322753909,259.43536376953127,143.99682876660786,251.84100341796876]},{"page":225,"text":"The part of the nematic shown","rect":[53.812835693359378,269.3546142578125,155.3101553359501,261.76025390625]},{"page":225,"text":"by arrows directed from the","rect":[53.812835693359378,279.33056640625,149.05826709055425,271.7362060546875]},{"page":225,"text":"right to the left was initially","rect":[53.812835693359378,289.3064880371094,149.71146148829386,281.7121276855469]},{"page":225,"text":"removed (virtually), turned by","rect":[53.812835693359378,299.282470703125,155.3423056045048,291.6881103515625]},{"page":225,"text":"angle p about the O-axis and","rect":[53.812835693359378,309.2016906738281,152.8369650283329,301.4549255371094]},{"page":225,"text":"put back into the empty","rect":[53.81367492675781,319.1776428222656,135.02979034571573,311.5832824707031]},{"page":225,"text":"cavity. A plane wall S anda","rect":[53.81367492675781,329.1535949707031,155.3646788337183,320.839599609375]},{"page":225,"text":"linear disclination loop L are","rect":[53.813228607177737,339.1283264160156,152.50399157785894,331.5339660644531]},{"page":225,"text":"formed","rect":[53.812381744384769,347.3204040527344,78.29871887354776,341.45318603515627]},{"page":225,"text":"a","rect":[192.48196411132813,246.35784912109376,198.03722650856197,240.7690887451172]},{"page":225,"text":"x","rect":[197.5712890625,315.6336975097656,201.98357043921895,311.8905029296875]},{"page":225,"text":"x","rect":[199.7039031982422,373.0638427734375,204.11618457496113,369.3206481933594]},{"page":225,"text":"z","rect":[207.8370361328125,254.0316162109375,211.33808548607863,250.46437072753907]},{"page":225,"text":"crossed","rect":[221.88014221191407,173.0,247.2379660048954,166.8760986328125]},{"page":225,"text":"polarizer","rect":[250.9955291748047,174.470458984375,281.19308560950449,166.8760986328125]},{"page":225,"text":"and","rect":[284.9828186035156,173.0,297.2259878554813,166.8760986328125]},{"page":225,"text":"analyzer.","rect":[301.01318359375,174.470458984375,331.9129969794985,166.8760986328125]},{"page":225,"text":"Orientation","rect":[335.67987060546877,173.0,374.3385061660282,166.85916137695313]},{"page":225,"text":"n0","rect":[358.81060791015627,325.6690673828125,365.80432800940488,319.8048095703125]},{"page":225,"text":"of","rect":[378.1070556640625,173.0,385.15512174231699,166.8760986328125]},{"page":225,"text":"n0","rect":[377.2015075683594,250.29495239257813,384.19467835120175,244.4300537109375]},{"page":225,"text":"S±","rect":[377.2007141113281,262.9679260253906,385.2238128598929,256.24932861328127]},{"page":225,"text":"the director field n(r). To understand the nature of the brushes, let us form some","rect":[53.812843322753909,432.4610290527344,385.14670599176255,423.9249267578125]},{"page":225,"text":"defects artificially.","rect":[53.812843322753909,444.8190002441406,128.66850705649143,435.88446044921877]},{"page":225,"text":"8.4.1.2 Volterra Process","rect":[53.812843322753909,484.9059143066406,162.89007963286594,477.37579345703127]},{"page":225,"text":"The major part of the arrows directed to the right in Fig. 8.12a correspond to the","rect":[53.812843322753909,510.568115234375,385.17359197809068,501.633544921875]},{"page":225,"text":"initial orientation of the director n0 in the planar nematic slab. However, the part of","rect":[53.812843322753909,522.4718017578125,385.1505063614048,513.5363159179688]},{"page":225,"text":"the slab shown by arrows directed to the left is virtually taken from the sample by","rect":[53.81368637084961,534.431396484375,385.1705254655219,525.4968872070313]},{"page":225,"text":"some “mysterious force”, turned about axis O through angle p and put back into the","rect":[53.81368637084961,546.3909301757813,385.1734088726219,537.277099609375]},{"page":225,"text":"slab. After this operation called Volterra process, the director is everywhere again","rect":[53.81269454956055,558.3504638671875,385.13860407880318,549.4159545898438]},{"page":225,"text":"parallel to n0 due to the n0 ¼ \u0003n0 symmetry and such a structure in each of the two","rect":[53.81269454956055,570.310302734375,385.1514824967719,561.37548828125]},{"page":225,"text":"parts (initial and turned) is topologically stable. However, in the close proximity of","rect":[53.813655853271487,582.2698974609375,385.14754615632668,573.3353881835938]},{"page":225,"text":"the plane S , on the scale of molecular size, the director changes its orientation by","rect":[53.813655853271487,594.2296142578125,385.16628352216255,584.4482421875]},{"page":226,"text":"8.4 Defects in Nematics and Cholesterics","rect":[53.81342697143555,42.55752944946289,195.02023776053705,36.68185043334961]},{"page":226,"text":"211","rect":[372.498779296875,42.4559326171875,385.19040435938759,36.73265075683594]},{"page":226,"text":"p. The plane S parallel to the substrate and horizontal in sketch (a) is called a wall","rect":[53.812843322753909,68.2883529663086,385.15498216220927,58.507164001464847]},{"page":226,"text":"defect or just a wall.","rect":[53.81315231323242,80.24788665771485,136.4540218880344,71.29341888427735]},{"page":226,"text":"There is also a vertical wall around the block shown by a dot-line loop in the","rect":[65.76517486572266,92.20748138427735,385.1719135112938,83.27293395996094]},{"page":226,"text":"xy-projection in sketch (b) below. The wall S is not seen under microscope, but","rect":[53.81315231323242,104.11067962646485,385.1364885098655,94.3290023803711]},{"page":226,"text":"the loop L surrounding the reoriented area and called a disclination is well seen as a","rect":[53.813594818115237,116.0702133178711,385.15940130426255,107.1157455444336]},{"page":226,"text":"thin line. Very often the reoriented area surrounded by the vertical wall takes all the","rect":[53.81260299682617,128.02981567382813,385.17136419488755,119.09526062011719]},{"page":226,"text":"thickness between the glasses without formation of wall S within the bulk.","rect":[53.81260299682617,139.98934936523438,362.06975980307348,130.20816040039063]},{"page":226,"text":"The rotation angle O is not necessarily equals p, it may be 2p, 3p or, more","rect":[65.76578521728516,151.94912719726563,385.13965643121568,142.8352813720703]},{"page":226,"text":"generally O ¼ 2p \u0004 s where s is strength of a disclination. The disclinations of","rect":[53.813777923583987,163.9485321044922,385.1525815567173,154.7948455810547]},{"page":226,"text":"strength s ¼ 1/2 or 1 are observed very often, however, those of strength","rect":[53.81476593017578,175.86822509765626,385.17852107099068,166.9336700439453]},{"page":226,"text":"s ¼ 3/2 or even 2 are very rare.","rect":[53.81476593017578,187.8277587890625,196.05445523764377,178.89320373535157]},{"page":226,"text":"8.4.2 Linear Singularities in Nematics","rect":[53.812843322753909,237.85494995117188,251.88899879191085,227.30084228515626]},{"page":226,"text":"8.4.2.1 Disclination Strength","rect":[53.812843322753909,265.56060791015627,182.08578843424869,256.20770263671877]},{"page":226,"text":"Consider a disclination with its ends fixed at the opposite plates of a planar nematic","rect":[53.812843322753909,289.4000244140625,385.10092962457505,280.4654541015625]},{"page":226,"text":"cell. Such a disclination “connects” the two glass plates as in Fig. 8.13a. If we are","rect":[53.812843322753909,301.35955810546877,385.1636127300438,292.42498779296877]},{"page":226,"text":"looking at it from the top along the z-direction we can see the director distributionn","rect":[53.81282424926758,313.3191223144531,385.18646592448308,304.38458251953127]},{"page":226,"text":"(x, y) in the xy-plane around the disclination. In a polarization microscope, in the","rect":[53.81379318237305,325.27862548828127,385.1705707378563,316.34405517578127]},{"page":226,"text":"same cell, we can see different n(x, y) patterns corresponding to disclinations shown","rect":[53.81279754638672,337.2381896972656,385.1536492448188,328.30364990234377]},{"page":226,"text":"in Fig. 8.14. A point in the middle of each sketch shows the disclination under","rect":[53.81379318237305,349.14093017578127,385.11080299226418,340.20635986328127]},{"page":226,"text":"discussion that has its own strength s.","rect":[53.81379318237305,361.1004943847656,205.56174893881565,352.16595458984377]},{"page":226,"text":"The strength of a disclination is defined as follows. We traverse the disclination","rect":[65.76581573486328,373.0600280761719,385.17556086591255,364.12548828125]},{"page":226,"text":"line along the closed contour counterclockwise as shown in sketch (b) and count the","rect":[53.81380081176758,385.01959228515627,385.1715778179344,376.08502197265627]},{"page":226,"text":"angle Df the director acquires as a result of the traverse. It is evident that after the","rect":[53.81380081176758,396.9790954589844,385.1695941753563,387.7158508300781]},{"page":226,"text":"full turn Df ¼ mp where m ¼ 0, 1, 2... and, by convention, the strength s ¼ m/2. In","rect":[53.81280517578125,408.93865966796877,385.1506280045844,399.6754150390625]},{"page":226,"text":"fact, we deal with a solution of the Laplace equation, see the next paragraph. Let us","rect":[53.81282424926758,420.898193359375,385.1577187930222,411.963623046875]},{"page":226,"text":"count Df from the horizontal axis in Fig. 8.14. Then, upon the traverse in the","rect":[53.81282424926758,432.8577575683594,385.17356146051255,423.5945129394531]},{"page":226,"text":"counter-clockwise direction, for disclinations of strength s ¼ 1/2 and s ¼ 1, the","rect":[53.812843322753909,444.7605285644531,385.1725543804344,435.82598876953127]},{"page":226,"text":"Fig. 8.13 Two disclinations","rect":[53.812843322753909,474.03961181640627,151.64821322440424,466.08966064453127]},{"page":226,"text":"fixed by their end at the two","rect":[53.812843322753909,483.8911437988281,150.50089019923136,476.2967834472656]},{"page":226,"text":"glasses limiting a layer ofa","rect":[53.812843322753909,493.8670959472656,148.2265410163355,486.2727355957031]},{"page":226,"text":"nematic liquid crystal. They","rect":[53.812843322753909,503.8430480957031,149.93143219141886,496.2486877441406]},{"page":226,"text":"interact with each other by the","rect":[53.812843322753909,513.7622680664063,155.3456816413355,506.16790771484377]},{"page":226,"text":"elastic force proportional to","rect":[53.812843322753909,523.7382202148438,148.73082489161417,516.1438598632813]},{"page":226,"text":"1/r12 (a). The structure of the","rect":[53.812843322753909,533.6295166015625,155.34459826731206,526.1190795898438]},{"page":226,"text":"director field n(r) near the","rect":[53.81258773803711,543.3507690429688,143.95852038645269,536.0950317382813]},{"page":226,"text":"two disclinations of positive","rect":[53.81258773803711,553.608642578125,150.46340319895269,546.0142822265625]},{"page":226,"text":"andnegative strength andfour","rect":[53.81258773803711,563.5845947265625,155.31158536536388,555.990234375]},{"page":226,"text":"dark brushes corresponding to","rect":[53.81258773803711,573.560546875,155.3589834609501,565.9661865234375]},{"page":226,"text":"the s ¼ 1 disclinations (b)","rect":[53.81258773803711,583.1978759765625,149.81187528235606,575.942138671875]},{"page":226,"text":"a","rect":[188.8549346923828,470.893798828125,194.41019708961665,465.3050537109375]},{"page":226,"text":"·","rect":[210.2893524169922,523.2175903320313,213.96625356425799,520.3622436523438]},{"page":227,"text":"212","rect":[53.81281280517578,42.455078125,66.50441879420205,36.73179626464844]},{"page":227,"text":"8 Elasticity and Defects","rect":[303.5040588378906,44.275352478027347,385.1735275554589,36.68099594116211]},{"page":227,"text":"Fig. 8.14 Structure of the director field n(r) around a disclination of different strength s. The","rect":[53.812843322753909,299.4626770019531,385.1957335212183,291.5127258300781]},{"page":227,"text":"values of s ¼ 1/2 and 1 are shown under each sketch","rect":[53.81281280517578,308.44805908203127,247.78790802149698,301.77655029296877]},{"page":227,"text":"director angle changes through p and 2p, respectively. The presence of dis-","rect":[53.812843322753909,337.2388610839844,385.15166602937355,328.3043212890625]},{"page":227,"text":"clinations of strength s ¼ 1/2 is a characteristic feature of any liquid crystalline","rect":[53.813812255859378,349.1983947753906,385.17356146051255,340.26385498046877]},{"page":227,"text":"phases with head-to-tail symmetry, n ¼ \u0003n.","rect":[53.81280517578125,361.157958984375,232.46697659994846,352.223388671875]},{"page":227,"text":"Note that the dark brushes in Fig. 8.11 mark the areas where the director is either","rect":[65.76480865478516,373.11749267578127,385.14165626374855,364.18292236328127]},{"page":227,"text":"parallel or perpendicular to a polariser crossed with analyser. Therefore, a number","rect":[53.81278610229492,385.020263671875,385.12078224031105,376.085693359375]},{"page":227,"text":"of brushes attached to a disclination (either 2 or 4 in the photo) is N ¼ 4s. In","rect":[53.81278610229492,396.97979736328127,385.1505974870063,388.04522705078127]},{"page":227,"text":"Fig. 8.13b the scheme is shown of the four brushes attached to two disclinations of","rect":[53.812767028808597,408.9393310546875,385.14864478913918,400.0047607421875]},{"page":227,"text":"opposite sign (s ¼ 1) corresponding to Fig. 8.14a, b. As to the sign of s, it can be","rect":[53.811771392822269,420.89886474609377,385.1515582866844,411.96429443359377]},{"page":227,"text":"established by a rotating of a pair of crossed polarisers: for their clockwise rotation","rect":[53.812767028808597,432.8584289550781,385.1158379655219,423.92388916015627]},{"page":227,"text":"the brushes rotate either clockwise (sign +) or anticlockwise (sign \u0003). Note the","rect":[53.812767028808597,444.8179626464844,385.17346990777818,435.8834228515625]},{"page":227,"text":"analogy with the electric charges: s ¼ +1 corresponds to a source and s ¼ \u00031 to a","rect":[53.812747955322269,456.7774963378906,385.16053045465318,447.84295654296877]},{"page":227,"text":"drain. Correspondingly the director lines are divergent or convergent. The lines of","rect":[53.81375503540039,468.7370300292969,385.14962135163918,459.802490234375]},{"page":227,"text":"n(r) are similar to the electric field lines, see Fig. 8.13b. Defects of the same strength","rect":[53.81375503540039,480.6398010253906,385.2560967545844,471.70526123046877]},{"page":227,"text":"but opposite sign may annihilate with each other as the electric charges of opposite","rect":[53.81473922729492,492.599365234375,385.17847479059068,483.664794921875]},{"page":227,"text":"sign do. It happens, e.g., at temperatures close to the nematic-isotropic transition.","rect":[53.81473922729492,504.5589294433594,381.8289455940891,495.6243896484375]},{"page":227,"text":"8.4.2.2 The Director Field Around Disclination","rect":[53.81473922729492,544.6458740234375,260.9499089176471,537.11572265625]},{"page":227,"text":"The problem is to find the distribution of the director around a disclination [14]. To","rect":[53.81473922729492,570.3079833984375,385.1734551530219,561.3734741210938]},{"page":227,"text":"solve it we can use the elasticity theory discussed in Section 8.3. Let a liquid crystal","rect":[53.814720153808597,582.2675170898438,385.1246782071311,573.3330078125]},{"page":227,"text":"layer is situated in the x, y plane of drawing, and singularity L is parallel to the","rect":[53.814720153808597,594.2271118164063,385.1735309429344,585.2926025390625]},{"page":228,"text":"8.4 Defects in Nematics and Cholesterics","rect":[53.81285095214844,42.55716323852539,195.01965792655268,36.68148422241211]},{"page":228,"text":"Fig. 8.15 Geometry for","rect":[53.812843322753909,67.58130645751953,137.09094327552013,59.6313591003418]},{"page":228,"text":"calculation of the director","rect":[53.812843322753909,75.76238250732422,142.1413277970045,69.89517211914063]},{"page":228,"text":"distribution arounda","rect":[53.812843322753909,85.68157196044922,125.00259539865971,79.81436157226563]},{"page":228,"text":"disclination L. C is the","rect":[53.812843322753909,96.0,133.2538389899683,89.69718170166016]},{"page":228,"text":"azimuthal angle for an","rect":[53.812843322753909,107.36067962646485,130.81958526759073,99.76632690429688]},{"page":228,"text":"arbitrary point r in the x,y","rect":[53.812843322753909,117.33663177490235,145.2245421149683,109.74227905273438]},{"page":228,"text":"plane of the nematic layer; j","rect":[53.81285095214844,127.25582122802735,153.53502070631365,119.66146850585938]},{"page":228,"text":"is the director angle in pointr","rect":[53.81285095214844,137.23178100585938,155.3643126227808,129.63742065429688]},{"page":228,"text":"y","rect":[272.1451110839844,73.0837631225586,276.141742765795,67.197021484375]},{"page":228,"text":"ρ","rect":[296.1081848144531,111.94961547851563,300.4964864010812,106.54276275634766]},{"page":228,"text":"213","rect":[372.49822998046877,42.55716323852539,385.1898245254032,36.73228454589844]},{"page":228,"text":"normal z to the layer, see Fig. 8.15. Point L is chosen as the reference system center","rect":[53.812843322753909,227.16473388671876,385.0959409317173,218.17041015625]},{"page":228,"text":"and r is radius-vector of an arbitrary point r characterized by length r and angle C.","rect":[53.812843322753909,239.124267578125,385.1825222542453,230.0801544189453]},{"page":228,"text":"We are going to relate the director angle f(r) to the radius-vector angle C(r). The","rect":[53.81386947631836,251.08383178710938,385.14771307184068,241.82058715820313]},{"page":228,"text":"components of the director field n(r) in the x,y-plane are independent of z:","rect":[53.81388854980469,263.0433654785156,354.33747727939677,254.1088104248047]},{"page":228,"text":"nðrÞ ¼ ½cosjðx;yÞ; sinjðx;yÞ;","rect":[146.42884826660157,289.28521728515627,272.2112573353662,279.3346862792969]},{"page":228,"text":"0\u0006:","rect":[282.21221923828127,289.28521728515627,292.5476831166162,279.3346862792969]},{"page":228,"text":"In the one-constant approximation, the distortion free energy per unit volume is","rect":[65.77286529541016,314.8497619628906,385.1915627871628,305.91522216796877]},{"page":228,"text":"given by","rect":[53.82084274291992,326.809326171875,88.8118828873969,317.874755859375]},{"page":228,"text":"gdistðrÞ ¼ 12K\"\u0002@@jx\u00032 þ \u0002@@jy\u00032#","rect":[153.006591796875,372.83587646484377,285.9739290329477,342.974365234375]},{"page":228,"text":"(8.34)","rect":[361.0555725097656,362.1771545410156,385.10494361726418,353.7008056640625]},{"page":228,"text":"Now we introduce a cylindrical coordinate frame: x ¼ rcosC;y ¼ rsinC and","rect":[65.76590728759766,398.3974304199219,385.14675227216255,389.3533020019531]},{"page":228,"text":"z, write down the gradient of f in that frame","rect":[53.814903259277347,410.35699462890627,234.13640630914535,401.09375]},{"page":228,"text":"@j * 1 @j ~ @j~","rect":[179.45187377929688,435.7271728515625,290.0451698697259,426.5436096191406]},{"page":228,"text":"rj ¼","rect":[151.35720825195313,442.5283203125,176.63791683409125,433.6634826660156]},{"page":228,"text":"\u0004rþ\u0004","rect":[194.00990295410157,442.50823974609377,226.9452356068506,434.6600341796875]},{"page":228,"text":"\u0004cþ","rect":[245.27410888671876,442.5381164550781,266.70352962706,433.3645324707031]},{"page":228,"text":"k","rect":[282.94085693359377,440.6257019042969,287.39038885309068,433.65338134765627]},{"page":228,"text":"@r","rect":[180.01927185058595,449.3102722167969,191.23468369792057,440.1466369628906]},{"page":228,"text":"r @C","rect":[216.55404663085938,449.3102722167969,243.03340441219303,440.1466369628906]},{"page":228,"text":"@z","rect":[270.36669921875,447.4775390625,279.91974271880346,440.1466369628906]},{"page":228,"text":"and substitute it in 8.34. Then, neglecting the z-dependence, we get the free energy","rect":[53.81187057495117,474.8020935058594,385.16860285810005,465.8675537109375]},{"page":228,"text":"density","rect":[53.8128776550293,486.7616271972656,82.62036219647894,477.82708740234377]},{"page":228,"text":"gdistðr;C;zÞ ¼ 12K\"\u0002@@jr\u00032 þ r12 \u0002@@Cj\u00032#:","rect":[134.3623504638672,532.7891845703125,304.60597169083499,502.9277038574219]},{"page":228,"text":"Further, if we consider the most important practical cases (see Figs. 8.11 and","rect":[65.76561737060547,558.3507080078125,385.1454705338813,549.4161987304688]},{"page":228,"text":"8.14), we find that the director angle f does not change too much with distancer","rect":[53.8136100769043,570.310302734375,385.1862828190143,561.0470581054688]},{"page":228,"text":"from the disclination, but changes very strongly with angle C. Therefore we can","rect":[53.8136100769043,582.2698364257813,385.1275567155219,573.2257690429688]},{"page":228,"text":"leave only the second term, that is the angular part, especially important for small r:","rect":[53.8136100769043,594.2293701171875,385.1514116055686,585.2948608398438]},{"page":229,"text":"214","rect":[53.812843322753909,42.454345703125,66.50444931178018,36.73106384277344]},{"page":229,"text":"gdistðCÞ ¼ 21K r12 \u0002@@Cj\u00032","rect":[169.4257049560547,85.26921081542969,269.09309456428846,59.44980239868164]},{"page":229,"text":"8 Elasticity and Defects","rect":[303.50408935546877,44.274620056152347,385.17355807303707,36.68026351928711]},{"page":229,"text":"(8.35)","rect":[361.0561828613281,77.5823745727539,385.10555396882668,68.98648834228516]},{"page":229,"text":"The corresponding Euler equation (8.24)","rect":[65.76653289794922,108.7586898803711,229.95325599519385,99.82414245605469]},{"page":229,"text":"@2j","rect":[206.5848846435547,133.3916473388672,222.900728909333,122.98892211914063]},{"page":229,"text":"K","rect":[197.8628692626953,138.13214111328126,204.49237319644238,131.5582275390625]},{"page":229,"text":"¼0","rect":[225.67430114746095,138.2510528564453,241.13118830731879,131.39825439453126]},{"page":229,"text":"@c2","rect":[206.58477783203126,148.06138610839845,222.4175421961244,136.93252563476563]},{"page":229,"text":"(8.36)","rect":[361.0565185546875,139.87461853027345,385.10588966218605,131.33848571777345]},{"page":229,"text":"has the general solution j ¼ AC þ j0: As follows from Fig. 8.14, the change of","rect":[53.81386947631836,171.94029235839845,385.1508725723423,162.5167694091797]},{"page":229,"text":"angle f by 2p corresponds to the same director field, and the first arbitrary constant","rect":[53.81399154663086,183.52041625976563,385.17378099033427,174.25717163085938]},{"page":229,"text":"A must satisfy condition A ¼ 1, 2, ... for any nematic liquid crystals (a polar","rect":[53.8139762878418,195.47998046875,385.1408628067173,186.54542541503907]},{"page":229,"text":"nematic included). For unpolar ones (e.g., conventional nematics) A ¼ 1/2, 1,","rect":[53.8139762878418,207.38275146484376,385.17474027182348,198.4481964111328]},{"page":229,"text":"3/2 ... Therefore, A ¼ s (the disclination strength) and the director angle at any","rect":[61.51657485961914,219.34228515625,385.1448906998969,210.40773010253907]},{"page":229,"text":"azimuth C is found:","rect":[53.81301498413086,229.2798614501953,135.34496934482645,222.2577362060547]},{"page":229,"text":"j ¼ sC þ j0","rect":[192.1396942138672,255.60118103027345,246.32168648323379,246.1768341064453]},{"page":229,"text":"(8.39)","rect":[361.0561828613281,254.82334899902345,385.10555396882668,246.34698486328126]},{"page":229,"text":"For instance, s ¼ 0 corresponds to a uniform state with the director oriented at","rect":[65.76653289794922,279.140869140625,385.181288314553,270.206298828125]},{"page":229,"text":"angle f0 with respect to the x-axis. In the case of s ¼ +1/2 shown in Fig. 8.14a,","rect":[53.81450653076172,291.1006774902344,385.1811184456516,281.837158203125]},{"page":229,"text":"counting C counterclockwise from the horizontal line x where f0 ¼ 0 and C0 ¼ 0,","rect":[53.81244659423828,303.0602111816406,385.17547269369848,293.7969665527344]},{"page":229,"text":"we find from (8.39) that the director changes its direction from 0 to p as shown in","rect":[53.813777923583987,314.9631042480469,385.14360896161568,306.028564453125]},{"page":229,"text":"the figure by two arrows.","rect":[53.813777923583987,326.92266845703127,155.0242123177219,317.98809814453127]},{"page":229,"text":"8.4.2.3 Energy of a Disclination","rect":[53.813777923583987,369.15106201171877,194.90079087565494,359.47943115234377]},{"page":229,"text":"We are interested in the elastic energy stored around the disclination per its unit","rect":[53.813777923583987,392.6717529296875,385.15559251377177,383.7371826171875]},{"page":229,"text":"length, l ¼ 1, see Fig. 8.16. The free energy is given by the same Eq. (8.35) and the","rect":[53.813777923583987,404.6313171386719,385.17453802301255,395.6369934082031]},{"page":229,"text":"limits for integration correspond to the sample radius rmax and a core of the","rect":[53.81380844116211,416.5908508300781,385.1186908550438,407.65631103515627]},{"page":229,"text":"disclination a that is excluded from consideration:","rect":[53.814720153808597,426.5295715332031,256.2295976407249,419.61700439453127]},{"page":229,"text":"Fdiscl ¼ 21K ð dc ð r12\u0002ddjc\u0003rdr","rect":[150.05401611328126,473.4193420410156,288.9470250065143,449.53411865234377]},{"page":229,"text":"0","rect":[200.12710571289063,478.923828125,203.6110694178041,474.1268615722656]},{"page":229,"text":"(8.40)","rect":[361.0563049316406,465.7326354980469,385.10567603913918,457.25628662109377]},{"page":229,"text":"As dj/dC ¼ s, the energy of a disclination per unit length","rect":[65.76665496826172,503.4838562011719,301.46144953778755,494.4397277832031]},{"page":229,"text":"Fdiscl ¼ pKs2 ln\u0006rmax=a\u0007","rect":[171.41067504882813,529.9945678710938,267.56519354059068,517.7142333984375]},{"page":229,"text":"(8.41)","rect":[361.0551452636719,528.3082275390625,385.1045163711704,519.8318481445313]},{"page":229,"text":"diverges logarithmically when r!1. However, this condition is not realistic","rect":[53.8134880065918,553.5331420898438,385.0925983257469,544.5986328125]},{"page":229,"text":"because all preparations have finite limits and there are alsoadditionalconfinements","rect":[53.81446075439453,565.49267578125,385.16428007231908,556.5581665039063]},{"page":229,"text":"due, for instance, to other defects, etc. In practice, rmax \u0005 10 – 100 mm, a \u0005 10 nm,","rect":[53.81446075439453,577.4522094726563,385.1833157112766,568.5177001953125]},{"page":229,"text":"ln(rmax/a) \u0005 10 and Fdiscl \u0005 30 K \u0005 3.10\u00035 erg/cm (or 3.10\u000310 J/m).","rect":[53.814598083496097,589.3551635742188,332.4352078011203,578.2244873046875]},{"page":230,"text":"8.4 Defects in Nematics and Cholesterics","rect":[53.81284713745117,42.55630874633789,195.01965792655268,36.68062973022461]},{"page":230,"text":"Fig. 8.16 Geometry for","rect":[53.812843322753909,67.58130645751953,137.09094327552013,59.6313591003418]},{"page":230,"text":"calculation of the energy ofa","rect":[53.812843322753909,77.4895248413086,154.4014219977808,69.89517211914063]},{"page":230,"text":"linear disclination with radius","rect":[53.812843322753909,85.68157196044922,155.3414657878808,79.81436157226563]},{"page":230,"text":"a (l ¼ 1 is unit length)","rect":[53.812843322753909,97.3846664428711,131.1064156876295,89.77337646484375]},{"page":230,"text":"z","rect":[301.6099853515625,66.96593475341797,305.1110347048286,62.694847106933597]},{"page":230,"text":"r","rect":[329.8837890625,90.78005981445313,334.2720906491281,85.37320709228516]},{"page":230,"text":"dr","rect":[334.2370910644531,134.80740356445313,342.62203571748747,127.35298919677735]},{"page":230,"text":"l=1","rect":[353.3634948730469,167.8138427734375,364.85780721892,162.11105346679688]},{"page":230,"text":"215","rect":[372.4981994628906,42.55630874633789,385.1898245254032,36.62983322143555]},{"page":230,"text":"x","rect":[371.35919189453127,111.40841674804688,375.7714732712502,107.66521453857422]},{"page":230,"text":"2a","rect":[324.7004699707031,222.27261352539063,332.6937142013419,216.60182189941407]},{"page":230,"text":"If there are two disclinations separated by distance r12, then the energy of their","rect":[65.76496887207031,280.1046142578125,385.11309181062355,271.1700439453125]},{"page":230,"text":"interaction per unit thickness of the sample L ¼ 1 (see Fig. 8.13a) follows from","rect":[53.81405258178711,292.00738525390627,385.13797711015305,283.07281494140627]},{"page":230,"text":"(8.41):","rect":[53.81306076049805,303.5684814453125,80.71331651279519,295.0921325683594]},{"page":230,"text":"W12 \u0005 \u00032pKs1s2 lnr12=a","rect":[169.36851501464845,329.0653991699219,269.6227959733344,318.2464294433594]},{"page":230,"text":"(8.42)","rect":[361.0552062988281,327.7147521972656,385.10457740632668,319.2384033203125]},{"page":230,"text":"The force of interaction between them dW/dr is proportional to 1/r12. Here we","rect":[65.76555633544922,352.54229736328127,385.1710895366844,343.60772705078127]},{"page":230,"text":"see an analogy with the force of interaction between two infinite parallel wires","rect":[53.814353942871097,364.4454650878906,385.1452676211472,355.51092529296877]},{"page":230,"text":"carrying electric currents. For disclinations of opposite sign s1s2 < 0 the interaction","rect":[53.814353942871097,376.4049987792969,385.08986750653755,367.470458984375]},{"page":230,"text":"energy is positive and decreases with decreasing distance. Therefore such disclina-","rect":[53.81374740600586,388.3648376464844,385.0779965957798,379.4302978515625]},{"page":230,"text":"tions attract each other.","rect":[53.81374740600586,398.2625732421875,148.13287015463596,391.38983154296877]},{"page":230,"text":"8.4.3 Point Singularities and Walls","rect":[53.812843322753909,450.4080505371094,235.85753516886397,439.85394287109377]},{"page":230,"text":"8.4.3.1 Point Singularities in the Bulk (Hedgehogs)","rect":[53.812843322753909,478.11370849609377,276.94442115632668,468.76080322265627]},{"page":230,"text":"There are some military applications of hedgehogs like a barbed wire or hedgehogs","rect":[53.812843322753909,501.8963623046875,385.1487161074753,492.9617919921875]},{"page":230,"text":"against tanks, Fig. 8.17a. In the peaceful field of liquid crystals, at a certain","rect":[53.812843322753909,513.8558959960938,385.17656794599068,504.92138671875]},{"page":230,"text":"temperature, hedgehogs are observed in spherical drops of nematics floating in an","rect":[53.81285095214844,525.8154296875,385.1486748795844,516.8809204101563]},{"page":230,"text":"isotropic liquid. A conoscopic image of such a drop is shown in the same figure (b).","rect":[53.81285095214844,537.7749633789063,385.1756252815891,528.8404541015625]},{"page":230,"text":"The liquid provides an alignment of the director perpendicular to the boundary.","rect":[53.81285095214844,549.7344970703125,385.1716579964328,540.7999877929688]},{"page":230,"text":"Then, in the centre of the drop, appears a point defect (c) called hedgehog that has","rect":[53.81285095214844,561.73388671875,385.1327248965378,552.7395629882813]},{"page":230,"text":"radial distribution of the director around it. Two such hedgehogs interact with each","rect":[53.812843322753909,573.653564453125,385.0989617448188,564.7190551757813]},{"page":230,"text":"other very specifically: their interaction energy is proportional to the distance","rect":[53.812843322753909,585.6130981445313,385.1566852398094,576.6785888671875]},{"page":231,"text":"216","rect":[53.81452560424805,42.55728530883789,66.50613159327432,36.68160629272461]},{"page":231,"text":"8 Elasticity and Defects","rect":[303.5057678222656,44.275962829589847,385.1752365398339,36.68160629272461]},{"page":231,"text":"Fig. 8.17 Hedgehogs. Some military applications (a), a conoscopic image of a spherical nematic","rect":[53.812843322753909,251.68072509765626,385.1643919684839,243.7307891845703]},{"page":231,"text":"drop floating in an isotropic liquid (b), and the structure of the director inside the drop with a point","rect":[53.812828063964847,261.5889587402344,385.18893150534697,253.99459838867188]},{"page":231,"text":"defect in the center called a hedgehog (c)","rect":[53.81366729736328,271.5649108886719,195.58227628333263,263.9705505371094]},{"page":231,"text":"a","rect":[99.4122085571289,300.6812744140625,104.96747095436275,295.092529296875]},{"page":231,"text":"b","rect":[157.16494750976563,300.728271484375,163.26974125924384,293.4198913574219]},{"page":231,"text":"c","rect":[253.03118896484376,300.7422790527344,258.5864513620776,295.1535339355469]},{"page":231,"text":"Fig. 8.18 Boodjooms. Structure of the director with two boodjooms in a nematic drop with","rect":[53.812843322753909,384.76727294921877,385.1805471816532,376.81732177734377]},{"page":231,"text":"tangential alignment of molecules at the surfaces (a), linear disclination with a point defect at","rect":[53.812843322753909,394.6755065917969,385.1720552846438,387.0811462402344]},{"page":231,"text":"the boundary of a nematic layer (b), and the same point defect (boodjoom) after annihilation of the","rect":[53.812843322753909,404.6514587402344,385.15514514231207,397.0570983886719]},{"page":231,"text":"linear disclination (c)","rect":[53.813690185546878,414.28875732421877,127.04171079260995,407.0330505371094]},{"page":231,"text":"between them, as between two","rect":[53.812843322753909,443.0,185.98684016278754,435.88446044921877]},{"page":231,"text":"discussed in the literature [15].","rect":[53.812843322753909,456.1709289550781,178.62669034506565,447.7842102050781]},{"page":231,"text":"quarks.","rect":[190.94802856445313,444.8190002441406,220.04018826987034,435.88446044921877]},{"page":231,"text":"Such","rect":[224.93466186523438,443.0,244.8968438248969,435.88446044921877]},{"page":231,"text":"an","rect":[249.85800170898438,443.0,259.2846612077094,437.0]},{"page":231,"text":"interesting","rect":[264.1890869140625,444.8190002441406,306.4547051530219,435.88446044921877]},{"page":231,"text":"analogy","rect":[311.4308166503906,444.8190002441406,343.03542414716255,435.88446044921877]},{"page":231,"text":"has","rect":[348.02349853515627,443.0,361.2825051699753,435.88446044921877]},{"page":231,"text":"been","rect":[366.2626037597656,443.0,385.11586848310005,435.88446044921877]},{"page":231,"text":"8.4.3.2 Point Singularities at the Surfaces (Boodjooms)","rect":[53.812843322753909,498.69818115234377,294.0646909317173,489.3353271484375]},{"page":231,"text":"A variation of temperature or the chemical content of the isotropic solvent for the","rect":[53.812843322753909,522.4708251953125,385.17160833551255,513.5363159179688]},{"page":231,"text":"nematic drops results in a change of the alignment of the liquid crystal at the drop","rect":[53.812843322753909,534.430419921875,385.13381281903755,525.4959106445313]},{"page":231,"text":"boundary from perpendicular (homeotropic) to parallel (homogeneous). Then the","rect":[53.812843322753909,546.3899536132813,385.17258489801255,537.4554443359375]},{"page":231,"text":"director pattern within the drop changes from that containing one singular point in","rect":[53.812843322753909,558.3494873046875,385.13979426435005,549.4149780273438]},{"page":231,"text":"the centre (hedgehog, Fig. 8.17c) to the new pattern with two singular points at the","rect":[53.812843322753909,570.3090209960938,385.17261541559068,561.37451171875]},{"page":231,"text":"“north” and “south” poles. These are defects with a funny name boodjooms, coming","rect":[53.812843322753909,582.2685546875,385.1128777604438,573.3340454101563]},{"page":231,"text":"from L. Carrol’s story “Alice in the Wonderful Land”, are seen in Fig. 8.18a.","rect":[53.812843322753909,594.2280883789063,365.8653835823703,585.2935791015625]},{"page":232,"text":"8.4 Defects in Nematics and Cholesterics","rect":[53.81281280517578,42.55752944946289,195.01962740897455,36.68185043334961]},{"page":232,"text":"217","rect":[372.4981689453125,42.55752944946289,385.18979400782509,36.73265075683594]},{"page":232,"text":"Another example is formation of boodjooms at the cell surfaces. Now we are","rect":[65.76496887207031,68.2883529663086,385.1618121929344,59.35380554199219]},{"page":232,"text":"interested not in the linear disclinations responsible for the Schlieren texture but in","rect":[53.812950134277347,80.24788665771485,385.1379021745063,71.31333923339844]},{"page":232,"text":"their nuclei at the solid substrates limiting a liquid crystal cell. The linear disclina-","rect":[53.812950134277347,92.20748138427735,385.07821021882668,83.27293395996094]},{"page":232,"text":"tions of strength s ¼ 1 may annihilate within the bulk due to some reconstruction","rect":[53.812950134277347,104.11019134521485,385.17669001630318,95.17564392089844]},{"page":232,"text":"of the director field induced, for instance, by temperature or a flow of the material.","rect":[53.81294250488281,116.0697250366211,385.0980495979953,107.13517761230469]},{"page":232,"text":"For example, a bulk disclination of strength s ¼ +1 shown by the solid vertical line","rect":[53.81294250488281,128.02932739257813,385.13782537652818,119.09477233886719]},{"page":232,"text":"in Fig. 8.18b disappears but its nuclei localized at the surfaces transform into new,","rect":[53.81294250488281,139.98886108398438,385.1548122933078,131.05430603027345]},{"page":232,"text":"surface defects. Fig. 8.18c illustrates the situation at one of the two surfaces. The","rect":[53.81294250488281,151.94839477539063,385.1458209819969,143.0138397216797]},{"page":232,"text":"escaped line leaves behind it a boodjoom. We meet such a situation in thick planar","rect":[53.81295394897461,163.907958984375,385.1100095352329,154.97340393066407]},{"page":232,"text":"cells where the Schlieren textures with four brushes are observed.","rect":[53.81295394897461,173.8455047607422,319.0030788948703,166.9329376220703]},{"page":232,"text":"8.4.3.3 Walls","rect":[53.81295394897461,215.92453002929688,115.55594266997531,208.42430114746095]},{"page":232,"text":"Walls are two-dimensional defects or planes separated area of the liquid crystals","rect":[53.81295394897461,241.61660766601563,385.1001016055222,232.6820526123047]},{"page":232,"text":"with different director alignment. We met them once when having discussed the","rect":[53.81295394897461,253.51931762695313,385.1707233257469,244.5847625732422]},{"page":232,"text":"Volterra process. Another well known example is the so-called hybrid cell, in which","rect":[53.81295394897461,265.4788818359375,385.1180047135688,256.5443115234375]},{"page":232,"text":"the initial alignment of the director is parallel to one boundary surface W(0) ¼ p/2","rect":[53.81295394897461,277.43841552734377,384.1234978776313,268.2050476074219]},{"page":232,"text":"and perpendicular to the other W(d) ¼ 0. In such a cell, the structure of the director","rect":[53.81296157836914,289.39801025390627,385.10698829499855,280.1646423339844]},{"page":232,"text":"field in the bulk is degenerate in the sense that the elastic energy is the same for the","rect":[53.81295394897461,301.3575439453125,385.17075384332505,292.4229736328125]},{"page":232,"text":"two director patterns shown in Fig. 8.19. Between the two orientations with +W(z) or","rect":[53.81295394897461,313.3171081542969,385.15276466218605,304.083740234375]},{"page":232,"text":"\u0003W(z), there is a defect plane (a wall) where the director changes its orientation","rect":[53.81491470336914,325.2766418457031,385.1209649186469,316.04327392578127]},{"page":232,"text":"within a very narrow layer. When we look at the texture in a polarization micro-","rect":[53.81491470336914,337.2362060546875,385.13289771882668,328.3016357421875]},{"page":232,"text":"scope (top view in the same figure) we can see thin lines separating the area with","rect":[53.81491470336914,349.13897705078127,385.11696711591255,340.20440673828127]},{"page":232,"text":"W(z) tilt. The total areas occupied by the +W and \u0003W domains are approximately","rect":[61.51847839355469,361.0985107421875,385.1756524186469,351.8651428222656]},{"page":232,"text":"equal. At the normal incidence of light, the areas of different tilt look similarly but","rect":[53.814876556396487,373.0580749511719,385.1348710782249,364.12353515625]},{"page":232,"text":"the walls between them are well seen. Such degeneracy in hybrid cells can be","rect":[53.814876556396487,385.01763916015627,385.1518024273094,376.08306884765627]},{"page":232,"text":"removed by special treatment of a planar surface providing a small pretilt angle W(0)","rect":[53.814876556396487,396.9771728515625,385.18258033601418,387.7438049316406]},{"page":232,"text":"< p/2.","rect":[53.81588363647461,407.85101318359377,80.05318875815158,400.00213623046877]},{"page":232,"text":"Another example is a twist nematic cell with a planar orientation of the director","rect":[65.76890563964844,420.896240234375,385.1090024551548,411.961669921875]},{"page":232,"text":"at both boundaries W ¼ p/2 differing by their azimuth, j ¼ 0 and p/2. In such cells,","rect":[53.81687927246094,432.8557434082031,385.18755765463598,423.62237548828127]},{"page":232,"text":"the areas with the director twist in the bulk by angle +p/2 and \u0003p/2 have the same","rect":[53.81685256958008,444.7585144042969,385.1337360210594,435.823974609375]},{"page":232,"text":"a","rect":[109.0418930053711,470.0552673339844,114.59715540260494,464.4665222167969]},{"page":232,"text":"z","rect":[123.0828628540039,477.9408874511719,126.58391220727003,473.6697998046875]},{"page":232,"text":"-J","rect":[116.25992584228516,497.4953308105469,123.0022469042683,492.5403747558594]},{"page":232,"text":"+J","rect":[135.65737915039063,496.781494140625,144.73571400631909,491.8265380859375]},{"page":232,"text":"b","rect":[257.02471923828127,470.0433044433594,263.1295129877594,462.73492431640627]},{"page":232,"text":"d","rect":[243.3621826171875,512.3229370117188,247.35881429899815,506.62017822265627]},{"page":232,"text":"Top view","rect":[281.0799865722656,477.8727111816406,311.50554841158228,470.57025146484377]},{"page":232,"text":"+J","rect":[276.26165771484377,504.4374694824219,286.63691934846067,498.7746887207031]},{"page":232,"text":"-J","rect":[292.3233337402344,520.1077270507813,300.0288260867419,514.4448852539063]},{"page":232,"text":"x","rect":[154.61068725585938,551.0228881835938,159.0229686325783,547.2797241210938]},{"page":232,"text":"wall","rect":[214.01893615722657,552.5121459960938,228.0950684145101,546.6813354492188]},{"page":232,"text":"Fig. 8.19 Walls. Hybrid nematic cell with planar and homeotropic director alignment at opposite","rect":[53.812843322753909,574.0244750976563,385.1847471930933,565.9898681640625]},{"page":232,"text":"boundaries with a wall between the two degenerate patterns differing by a tilt angle sign (a), and","rect":[53.812843322753909,583.9326782226563,385.1939749160282,576.3383178710938]},{"page":232,"text":"the top view on the different tilt areas surrounded by walls (b)","rect":[53.81281280517578,593.9086303710938,267.4632577286451,586.3142700195313]},{"page":233,"text":"218","rect":[53.8128547668457,42.55722427368164,66.50446075587198,36.73234558105469]},{"page":233,"text":"8 Elasticity and Defects","rect":[303.5041198730469,44.275901794433597,385.17358859061519,36.68154525756836]},{"page":233,"text":"energy. They are separated by walls, which scatter light. In the display technology,","rect":[53.812843322753909,68.2883529663086,385.1537136604953,59.35380554199219]},{"page":233,"text":"a small amount of a chiral impurity is added to a nematic material in order to","rect":[53.812843322753909,80.24788665771485,385.14077082685005,71.31333923339844]},{"page":233,"text":"remove the degeneracy.","rect":[53.812843322753909,92.20748138427735,149.7355923226047,83.27293395996094]},{"page":233,"text":"8.4.4 Defects in Cholesterics","rect":[53.812843322753909,142.23443603515626,203.32164649698897,131.68032836914063]},{"page":233,"text":"8.4.4.1 Singular t- and l-Lines in the Planar Texture","rect":[53.812843322753909,169.94015502929688,288.6107562847313,160.58726501464845]},{"page":233,"text":"We have briefly discussed the cholesteric Grandjean texture in Section 4.7.4. In that","rect":[53.812835693359378,193.77957153320313,385.1387162930686,184.8450164794922]},{"page":233,"text":"case the defects appear in the wedge-type cells having continuously changing","rect":[53.812843322753909,205.73910522460938,385.16860285810005,196.80455017089845]},{"page":233,"text":"thickness. But even in the planar cholesteric texture observed along the helical","rect":[53.812843322753909,217.69866943359376,385.1089006192405,208.7641143798828]},{"page":233,"text":"z-axis, we may see some defects, namely, singular lines which are more complex","rect":[53.812843322753909,229.65817260742188,385.16164485028755,220.72361755371095]},{"page":233,"text":"than the corresponding disclinations in the nematic phase. A cholesteric may be","rect":[53.812843322753909,241.61773681640626,385.15076482965318,232.6831817626953]},{"page":233,"text":"represented by a stuck of quasi-layers with interlayer distance P0/2. Therefore, we","rect":[53.812843322753909,253.52047729492188,385.17063177301255,244.58592224121095]},{"page":233,"text":"can use some concepts of dislocation theory from the solid state physics. To","rect":[53.813899993896487,265.4808349609375,385.1736687760688,256.5462646484375]},{"page":233,"text":"understand the appearance of such defects consider again the Volterra process [16].","rect":[53.813899993896487,277.44036865234377,385.15472074057348,268.50579833984377]},{"page":233,"text":"In Fig. 8.20a we see a stack of the quasi-layers with vertical helical axis. The","rect":[65.7649154663086,289.3999328613281,385.1458209819969,280.46539306640627]},{"page":233,"text":"dash and dot lines show the orientation of the director parallel and perpendicular to","rect":[53.81289291381836,301.3594665527344,385.13982478192818,292.4249267578125]},{"page":233,"text":"the plane of drawing, respectively. We make a virtual cut S along a dash line","rect":[53.81289291381836,313.3190002441406,385.1378558941063,304.38446044921877]},{"page":233,"text":"terminated by point L and then separate two lips as shown by two arrows in the","rect":[53.81289291381836,325.2785339355469,385.1716693706688,316.343994140625]},{"page":233,"text":"figure. The borders of the gap S1 and S2 are turned correspondingly up and down","rect":[53.81289291381836,337.2388610839844,385.1153191666938,328.30352783203127]},{"page":233,"text":"through angles p/2. Then we add some cholesteric material to the right of the","rect":[53.81330490112305,349.1416015625,385.1720660991844,340.20703125]},{"page":233,"text":"S1–S2 line with layers orientated parallel to the initial helical axis. Note that the","rect":[53.81330490112305,361.1014709472656,385.1738666362938,352.16693115234377]},{"page":233,"text":"director is not discontinuous at S1–S2 line, Fig. 8.20b. Finally, the structure relaxes","rect":[53.813053131103519,373.0611877441406,385.0765725527878,364.12646484375]},{"page":233,"text":"and we arrive at a new situation with linear defect t\u0003 shown in sketch (c). The","rect":[53.81342697143555,384.6223449707031,385.1481403179344,376.0]},{"page":233,"text":"director is discontinuous at the core of line L where n⊥L. This singular line","rect":[53.81433868408203,396.9803161621094,385.14124334527818,388.0457763671875]},{"page":233,"text":"\u0003","rect":[155.71754455566407,401.0,161.46598578191454,400.0]},{"page":233,"text":"(disclination) is called t -line. If, from the beginning, we had made a cutS","rect":[53.815345764160159,408.9400634765625,385.1839634830768,400.00531005859377]},{"page":233,"text":"along the neighbour quasi-layer with the director perpendicular to the cut (shown","rect":[53.813236236572269,420.89959716796877,385.12224665692818,411.96502685546877]},{"page":233,"text":"a","rect":[56.19194030761719,450.53472900390627,61.747202704851037,444.94598388671877]},{"page":233,"text":"b","rect":[179.64266967773438,450.4887390136719,185.74746342721259,443.18035888671877]},{"page":233,"text":"S1","rect":[247.5986328125,452.7708740234375,256.26333608630196,444.8766174316406]},{"page":233,"text":"c","rect":[285.29754638671877,450.53472900390627,290.8528087839526,444.94598388671877]},{"page":233,"text":"L","rect":[113.85494995117188,504.1693115234375,118.73883386634448,498.4265441894531]},{"page":233,"text":"S1S","rect":[146.7138671875,499.37261962890627,164.9616793930276,488.3997497558594]},{"page":233,"text":"S2","rect":[146.71434020996095,505.93212890625,155.3789671898176,498.0378112792969]},{"page":233,"text":"L","rect":[244.74403381347657,500.83709716796877,249.62791772864916,495.0943298339844]},{"page":233,"text":"L","rect":[342.48486328125,501.39300537109377,347.3687471964226,495.6502380371094]},{"page":233,"text":"–","rect":[378.4316711425781,488.0,382.87592557275158,486.0]},{"page":233,"text":"t","rect":[373.01629638671877,491.9603271484375,376.5253390033485,487.7212219238281]},{"page":233,"text":"S2","rect":[247.5986328125,542.400390625,256.26333608630196,534.5064697265625]},{"page":233,"text":"Fig. 8.20 Volterra process. A stack of the cholesteric quasi-layers with vertical helical axis anda","rect":[53.812843322753909,564.1054077148438,385.1746153571558,556.1554565429688]},{"page":233,"text":"cut S shown by the solid line terminated in point L (a). The cut is open up-down and the cholesteric","rect":[53.812843322753909,573.9569091796875,385.16442248606207,566.362548828125]},{"page":233,"text":"material is added on the right of the cut (b). The final structure of the quasi-layers after relaxation","rect":[53.81285858154297,583.932861328125,385.13315338282509,576.3385009765625]},{"page":233,"text":"\u0003","rect":[129.37754821777345,587.0,134.26370052579817,586.0]},{"page":233,"text":"leaving a line defect t (c)","rect":[53.81285858154297,593.9088134765625,146.07319730384044,586.314453125]},{"page":234,"text":"8.4 Defects in Nematics and Cholesterics","rect":[53.812862396240237,42.55636978149414,195.01967318534174,36.68069076538086]},{"page":234,"text":"219","rect":[372.49822998046877,42.62409973144531,385.1898550429813,36.73149108886719]},{"page":234,"text":"Fig. 8.21 Structure of the director field around different singular lines (disclinations) ina","rect":[53.812843322753909,266.984619140625,385.1754698493433,259.03466796875]},{"page":234,"text":"cholesteric liquid crystal: t\u0003, l\u0003 and t+, l+. Signs (\u0003) and (+) correspond to different Volterra","rect":[53.812843322753909,276.83612060546877,385.1246885993433,268.3851623535156]},{"page":234,"text":"processes","rect":[53.812862396240237,286.8119201660156,86.21030123709955,281.114013671875]},{"page":234,"text":"by dots) and repeated exactly the same procedure, we would arrive at another","rect":[53.812843322753909,313.319580078125,385.1039060196079,304.385009765625]},{"page":234,"text":"structure with a new distribution of the director and a singular line called l\u0003-","rect":[53.812843322753909,325.27911376953127,385.15975318757668,316.0]},{"page":234,"text":"line. The latter has no core because the director n||L is continuous at this disclina-","rect":[53.81393051147461,335.2068176269531,385.0810788711704,328.30419921875]},{"page":234,"text":"tion. The structures of the director around the t\u0003- and l\u0003-lines are compared in","rect":[53.81393051147461,349.198486328125,385.14406672528755,339.9451904296875]},{"page":234,"text":"Fig. 8.21 (lower plot). There are also t+- and l+ singular lines shown in Fig. 8.21","rect":[53.814205169677737,361.1581726074219,385.1596306901313,351.19610595703127]},{"page":234,"text":"(upper plot) which can be obtained using another type of the Volterra process, see","rect":[53.81386947631836,373.1177062988281,385.1139301128563,364.18316650390627]},{"page":234,"text":"[16] for details.","rect":[53.81386947631836,384.4128723144531,116.31234403158908,376.0859375]},{"page":234,"text":"8.4.4.2 Defects in the Polygonal or Fingerprint Textures","rect":[53.812862396240237,427.2489013671875,299.91183866606908,417.5772705078125]},{"page":234,"text":"When the limiting glasses are treated by a surfactant, the director aligns perpendic-","rect":[53.812862396240237,450.7695617675781,385.13582740632668,441.83502197265627]},{"page":234,"text":"ular to boundaries. Such an alignment (homeotropic) is, in principle, hardly com-","rect":[53.812862396240237,462.7291259765625,385.13582740632668,453.7945556640625]},{"page":234,"text":"patible with the helical structure shown in the Inset to Fig. 8.22 and a number of","rect":[53.812862396240237,474.6886901855469,385.15068946687355,465.754150390625]},{"page":234,"text":"defects form. A typical polygonal texture (another name is finger-print texture) is","rect":[53.8128776550293,486.68804931640627,385.1875344668503,477.673828125]},{"page":234,"text":"shown in the photo (same figure). By measuring the distance l between neighbour","rect":[53.812862396240237,498.6077575683594,385.1088803848423,489.6532897949219]},{"page":234,"text":"stripes we can determine the pitch of the helix from the microscopic observations","rect":[53.813838958740237,510.5672912597656,385.1696816836472,501.63275146484377]},{"page":234,"text":"(P0 ¼ 2 l). Another type of defects in this geometry is focal-conic domains related","rect":[53.813838958740237,522.4718017578125,385.1155938248969,513.517333984375]},{"page":234,"text":"to the quasi-lamellar structure of a cholesteric. They are not so well pronounced as","rect":[53.81253433227539,534.431396484375,385.1384622012253,525.4968872070313]},{"page":234,"text":"similar domains in the genuine lamellarphases, such as the smectic A phase, and we","rect":[53.81253433227539,546.3909301757813,385.16831243707505,537.4564208984375]},{"page":234,"text":"shall see their features in the next Section.","rect":[53.81253433227539,556.3284912109375,224.88062711264377,549.4159545898438]},{"page":234,"text":"Earlier in Section 4.8 we discussed the blue phases observed in cholesterics close","rect":[65.76455688476563,570.3099975585938,385.1265338726219,561.37548828125]},{"page":234,"text":"to the transition to the isotropic phase. The whole appearance of the blue phase is","rect":[53.81253433227539,582.26953125,385.1872598086472,573.3350219726563]},{"page":234,"text":"owed to the defects, which form a three dimensional lattice.","rect":[53.81253433227539,592.1972045898438,295.7476772835422,585.2946166992188]},{"page":235,"text":"220","rect":[53.81285095214844,42.55630874633789,66.50445694117471,36.73143005371094]},{"page":235,"text":"Fig. 8.22 Microscopic","rect":[53.812843322753909,67.58130645751953,133.27160022043706,59.6313591003418]},{"page":235,"text":"finger-print texture of the","rect":[53.812843322753909,77.4895248413086,140.6174864509058,69.89517211914063]},{"page":235,"text":"cholesteric phase observed in","rect":[53.812843322753909,87.4087142944336,154.28210968165323,79.81436157226563]},{"page":235,"text":"geometry shown in the Inset;","rect":[53.812843322753909,97.3846664428711,153.2803316518313,89.79031372070313]},{"page":235,"text":"arrows indicate the direction","rect":[53.81285095214844,105.71820068359375,152.38939422755167,99.76632690429688]},{"page":235,"text":"of the incident light on the","rect":[53.81285095214844,117.33663177490235,144.97919604563237,109.74227905273438]},{"page":235,"text":"texture","rect":[53.81285095214844,125.50328063964844,77.41077563547612,120.52503967285156]},{"page":235,"text":"8","rect":[303.50408935546877,42.55630874633789,307.7346243300907,36.73143005371094]},{"page":235,"text":"Elasticity","rect":[310.13165283203127,44.274986267089847,342.2667288222782,36.68062973022461]},{"page":235,"text":"k0","rect":[313.43072509765627,160.06484985351563,320.76055898669258,152.2149200439453]},{"page":235,"text":"z","rect":[310.0415954589844,192.073974609375,314.038227140795,187.890869140625]},{"page":235,"text":"and","rect":[344.62823486328127,43.0,356.8713430800907,36.68062973022461]},{"page":235,"text":"Defects","rect":[359.2996826171875,43.0,385.17358859061519,36.68062973022461]},{"page":235,"text":"8.5 Smectic Phases","rect":[53.812843322753909,283.9392395019531,158.43341712687178,274.90313720703127]},{"page":235,"text":"8.5.1 Elasticity of SmecticA","rect":[53.812843322753909,315.81793212890627,202.41980840250234,305.1801452636719]},{"page":235,"text":"8.5.1.1 Free Energy","rect":[53.812843322753909,343.7588806152344,144.57334223798285,334.0872497558594]},{"page":235,"text":"SmA is a one-dimensional lamellar crystal with the interlayer distance almost","rect":[53.812843322753909,367.2795715332031,385.07502610752177,358.34503173828127]},{"page":235,"text":"rigidly fixed. In order to discuss elasticity we need an additional variable that","rect":[53.812843322753909,379.2391052246094,385.1347795254905,370.3045654296875]},{"page":235,"text":"would describe the lamellar structure. Consider a small distortion of smectic layers","rect":[53.812843322753909,391.19866943359377,385.09796537505346,382.26409912109377]},{"page":235,"text":"[17]. In Fig. 8.23 dash and solid lines indicate undisturbed and distorted layers,","rect":[53.812843322753909,403.1581726074219,385.0959744026828,394.2236328125]},{"page":235,"text":"respectively. Short rods perpendicular to the lowest solid line indicate local direc-","rect":[53.81184387207031,415.0609436035156,385.1586850723423,406.12640380859377]},{"page":235,"text":"tors, which are always perpendicular to the layers. Now, we introduce a layer","rect":[53.81184387207031,427.0205078125,385.1158383926548,418.0859375]},{"page":235,"text":"displacement along the z-axis, u ¼ uz. In fact, it is a scalar field uz(x, y, z), depending","rect":[53.81184387207031,438.9809875488281,385.11171809247505,430.04547119140627]},{"page":235,"text":"generally on all the three co-ordinates. Its derivatives describe two types of elasticity:","rect":[53.81368637084961,450.9405212402344,385.2063432461936,442.0059814453125]},{"page":235,"text":"1.","rect":[53.81368637084961,467.0,61.279343085544159,459.9766540527344]},{"page":235,"text":"2.","rect":[53.81368637084961,491.0,61.279343085544159,483.895751953125]},{"page":235,"text":"Elasticity ∂u/∂z corresponding to a change in the interlayer distance due to","rect":[66.27536010742188,468.8514404296875,385.1425713639594,458.910888671875]},{"page":235,"text":"compressibility alongz","rect":[66.2753677368164,480.8110046386719,159.60704435454563,471.87646484375]},{"page":235,"text":"Elasticity due to layer curvature ∂u/∂x and ∂u/∂y. As seen from the figure, the","rect":[66.27536010742188,492.7705383300781,385.17343939020005,482.8299865722656]},{"page":235,"text":"director angle W in the x, z plane for small distortions is given by # \u0005 @u=@x","rect":[66.27538299560547,505.0587463378906,385.16150701715318,495.128173828125]},{"page":235,"text":"¼ \u0003dn \u0005 \u0003nx:","rect":[66.27637481689453,516.0853271484375,126.5706323353662,507.45623779296877]},{"page":235,"text":"Due to uniaxiality, the same is true for ny and all the three components are","rect":[65.76532745361328,535.53125,367.37610662652818,525.6668090820313]},{"page":235,"text":"@u","rect":[169.2557373046875,556.1917114257813,179.89778986981879,548.8607788085938]},{"page":235,"text":"nx ¼ \u0003@x <<1;ny ¼ \u0003@y<<1;nz \u00051","rect":[138.04502868652345,571.7071533203125,300.94851008466255,555.9935302734375]},{"page":235,"text":"(8.43)","rect":[361.0559997558594,564.4707641601563,385.10540138093605,555.994384765625]},{"page":236,"text":"8.5 Smectic Phases","rect":[53.813697814941409,42.55679702758789,119.50459750174798,36.63032150268555]},{"page":236,"text":"Fig. 8.23 Distortion of","rect":[53.812843322753909,67.58130645751953,134.66937345130138,59.6313591003418]},{"page":236,"text":"smectic layers (solid lines)","rect":[53.812843322753909,77.4895248413086,145.3362283096998,69.87823486328125]},{"page":236,"text":"from their equilibrium","rect":[53.81285095214844,87.4087142944336,130.27301262254702,79.81436157226563]},{"page":236,"text":"position (dash lines). In the","rect":[53.81285095214844,97.3846664428711,147.81025073313237,89.77337646484375]},{"page":236,"text":"SmA phase, the director","rect":[53.81284713745117,107.36067962646485,136.53334134680919,99.76632690429688]},{"page":236,"text":"shown by short rods crossing","rect":[53.81284713745117,117.33663177490235,153.42671722559855,109.74227905273438]},{"page":236,"text":"the lower layer is always","rect":[53.81284713745117,127.25582122802735,139.17404635428705,119.66146850585938]},{"page":236,"text":"parallel to the layer normal","rect":[53.81284713745117,137.23178100585938,146.90916160788599,129.63742065429688]},{"page":236,"text":"z","rect":[218.70677185058595,183.74411010742188,222.70340353239659,179.56100463867188]},{"page":236,"text":"l","rect":[231.0411834716797,113.48468017578125,232.8156879384036,107.74190521240235]},{"page":236,"text":"x","rect":[258.8025817871094,202.40493774414063,262.79921346892,198.22183227539063]},{"page":236,"text":"q = –¶u–","rect":[332.8921813964844,150.75633239746095,356.3923440297828,142.37496948242188]},{"page":236,"text":"¶x","rect":[346.4007568359375,155.20257568359376,355.94472128142,149.539794921875]},{"page":236,"text":"221","rect":[372.4990234375,42.4552001953125,385.19064850001259,36.73191833496094]},{"page":236,"text":"u","rect":[370.05841064453127,85.4106216430664,374.5026650747047,81.11553955078125]},{"page":236,"text":"a","rect":[99.18651580810547,246.2677001953125,104.74177820533932,240.67893981933595]},{"page":236,"text":"Splay","rect":[165.70632934570313,248.32125854492188,187.0323586782203,240.72286987304688]},{"page":236,"text":"b","rect":[270.0648193359375,246.2677001953125,276.1696130854157,238.95932006835938]},{"page":236,"text":"l","rect":[318.815673828125,281.826904296875,321.03780104321177,276.0841369628906]},{"page":236,"text":"l’’","rect":[316.7494201660156,309.5914001464844,323.415791765868,303.8486328125]},{"page":236,"text":"Fig. 8.24 A splay (a) and bend (b) distortion of smecticA","rect":[53.812843322753909,342.9368896484375,256.78630615812747,334.9869384765625]},{"page":236,"text":"The distortion free energy density can be expanded around its equilibrium value","rect":[65.76496887207031,367.7898254394531,385.14087713434068,358.85528564453127]},{"page":236,"text":"g0, but, in the simplest case, only quadratic components are held:","rect":[53.812950134277347,379.732421875,317.0057817227561,370.75823974609377]},{"page":236,"text":"gdist ¼ g0 þ 21B\u0002@@uz\u00032 þ 21K11\u0002@@2xu2 þ @@2yu2\u00032","rect":[130.6230926513672,420.4229736328125,307.89500496467908,393.92333984375]},{"page":236,"text":"(8.44)","rect":[361.0561828613281,412.7360534667969,385.10555396882668,404.25970458984377]},{"page":236,"text":"Here B and K11 are moduli of layer compressibility and layer curvature, respec-","rect":[53.81450653076172,443.9689025878906,385.16387306062355,435.03436279296877]},{"page":236,"text":"tively.","rect":[53.81303787231445,455.8716735839844,79.09672971274142,446.9371337890625]},{"page":236,"text":"Why does the free energy density acquire this particular form? First, in the","rect":[65.76506042480469,467.8312072753906,385.17078436090318,458.89666748046877]},{"page":236,"text":"curvature term with modulus K11, we must use the second derivatives because the","rect":[53.813045501708987,479.2934265136719,385.1738971538719,470.856201171875]},{"page":236,"text":"first derivatives correspond to a pure rotation of all the layers that does not cost","rect":[53.81318283081055,491.7507019042969,385.1550126797874,482.816162109375]},{"page":236,"text":"energy. The higher derivatives are ignored for small distortions. For the compress-","rect":[53.81318283081055,503.7102355957031,385.09429298249855,494.77569580078127]},{"page":236,"text":"ibility term, the first derivative (∂u/∂z) is sufficient. Second, both the compressibil-","rect":[53.81318283081055,515.6697387695313,385.1380857071079,505.7292175292969]},{"page":236,"text":"ity and the curvature terms must be squared due to head-to-tail symmetry and","rect":[53.813167572021487,527.6292724609375,385.1451043229438,518.6947631835938]},{"page":236,"text":"parabolic form of the density increment gdist-g0 as a function of distortion (Hooke’s","rect":[53.813167572021487,539.58935546875,385.1463967715378,530.6543579101563]},{"page":236,"text":"law). However, the question arises why is only splay modulus K11 taken into","rect":[53.81350326538086,551.4921264648438,385.1578301530219,542.5576171875]},{"page":236,"text":"account in (8.44) and not the other two Frank moduli K22 and K33. Considering","rect":[53.81405258178711,563.4520263671875,385.14080134442818,554.5173950195313]},{"page":236,"text":"the splay and bend distortions of the SmA phase in Fig. 8.24 we can see that","rect":[53.81393051147461,575.4115600585938,385.139784408303,566.47705078125]},{"page":236,"text":"only the splay distortion is allowed because it leaves the interlayer distance and the","rect":[53.813899993896487,587.37109375,385.17371404840318,578.4365844726563]},{"page":237,"text":"222","rect":[53.81283950805664,42.4560546875,66.50444549708291,36.73277282714844]},{"page":237,"text":"8 Elasticity and Defects","rect":[303.50408935546877,44.276329040527347,385.17355807303707,36.68197250366211]},{"page":237,"text":"layer thickness unchanged. The bend would evidently change an interlayer dis-","rect":[53.812843322753909,68.2883529663086,385.1486753067173,59.35380554199219]},{"page":237,"text":"tance, l 6¼ l’ or l”, that would require too much energy. The absence of twist","rect":[53.812843322753909,80.24788665771485,385.12483079502177,70.9846420288086]},{"page":237,"text":"corresponds to scalar product n\u0004curln ¼ 0 in the Frank energy (8.16). This condition","rect":[53.812843322753909,92.20748138427735,385.15862361005318,83.27293395996094]},{"page":237,"text":"is fulfilled for the director components nx ¼ \u0003∂u/∂x and ny ¼ \u0003∂u/∂y, nz ¼1","rect":[53.8138313293457,105.08378601074219,385.1793145280219,94.17012023925781]},{"page":237,"text":"because curln ¼ (∂2u/∂z∂y)i \u0003 (∂2u/∂z∂x)j and n ¼ k. Thus, in (8.44) we may","rect":[53.81456756591797,116.16004180908203,385.1670769791938,105.01939392089844]},{"page":237,"text":"hold the single curvature modulus K11.","rect":[53.81332015991211,128.02999877929688,210.48955960776096,119.09544372558594]},{"page":237,"text":"The order of magnitude of the splay modulus is the same as that in nematics K11","rect":[65.76529693603516,139.98953247070313,385.17547254887918,131.0549774169922]},{"page":237,"text":"\u0005 10\u00037 – 10\u00036 dyn (or 10\u000312 – 10\u000311 N in SI system). Modulus B found for a liquid","rect":[53.812843322753909,151.94937133789063,385.2004326920844,140.8583984375]},{"page":237,"text":"crystal 8OCB at temperature 60\bC is B ¼ 8\u0004106 erg/cm3 (or 8\u0004105 J/m3 in the SI","rect":[53.8138542175293,163.908935546875,385.15935645906105,152.7782745361328]},{"page":237,"text":"system) [18]. In that experiment, the compression-dilatation distortion of smectic","rect":[53.814537048339847,175.86846923828126,385.18128240777818,166.9339141845703]},{"page":237,"text":"layers was induced by an external force from a piezoelectric driver.","rect":[53.813533782958987,187.8280029296875,326.5937771370578,178.89344787597657]},{"page":237,"text":"8.5.1.2 Wave-Like Distortion","rect":[53.813533782958987,224.76742553710938,184.34012193034244,217.26719665527345]},{"page":237,"text":"It is very instructive to consider a behaviour of the smectic layers attached toa","rect":[53.813533782958987,250.45947265625,385.16040838434068,241.52491760253907]},{"page":237,"text":"corrugated surface. This would explain why the uniform smectic phase is much","rect":[53.813533782958987,262.41900634765627,385.15737238935005,253.4844512939453]},{"page":237,"text":"more transparent than the nematic phase. The geometry is shown in Fig. 8.25a.","rect":[53.813533782958987,274.3785705566406,385.1822170784641,265.3842468261719]},{"page":237,"text":"A solid substrate is assumed to have a one-dimensional cosine-form relief:","rect":[53.813533782958987,284.3161315917969,354.95234544345927,277.403564453125]},{"page":237,"text":"zðxÞ ¼ acosqx","rect":[189.081298828125,310.5390930175781,249.88660467340316,300.58856201171877]},{"page":237,"text":"with a period L ¼ 2p/q. We are interested in a distance L (penetration length) along","rect":[53.814537048339847,334.1593322753906,385.16130915692818,324.8861389160156]},{"page":237,"text":"z, at which the distortion is smoothed out. In other words, how far do smectic layers","rect":[53.815513610839847,346.0790710449219,385.10370267974096,337.14453125]},{"page":237,"text":"“feel” the influence of the surface? The distortion of the layers is given by:","rect":[53.815513610839847,358.0386047363281,356.92524583408427,349.10406494140627]},{"page":237,"text":"uðx;zÞ ¼ u0ðzÞcosqx","rect":[65.76757049560547,370.3390197753906,150.87044561578598,360.38629150390627]},{"page":237,"text":"with","rect":[160.87342834472657,367.93853759765627,178.59192744306098,361.0657958984375]},{"page":237,"text":"u0ðz ¼ 0Þ ¼ a.","rect":[188.62974548339845,370.3390197753906,248.55677457358127,360.38848876953127]},{"page":237,"text":"a","rect":[101.33818817138672,394.4884948730469,106.89345056862057,388.8997497558594]},{"page":237,"text":"z","rect":[142.7771759033203,400.21417236328127,146.77380758513096,395.9590759277344]},{"page":237,"text":"b","rect":[233.1767578125,394.4884948730469,239.2815515619782,387.18011474609377]},{"page":237,"text":"d","rect":[231.65985107421876,456.2326354980469,236.54373498939135,450.3858947753906]},{"page":237,"text":"2p /q","rect":[156.3856964111328,527.418212890625,173.5232424356804,519.8758544921875]},{"page":237,"text":"x","rect":[208.24520874023438,542.3926391601563,212.6894631704078,538.1375732421875]},{"page":237,"text":"z","rect":[232.87005615234376,507.54730224609377,236.8666878341544,503.2922058105469]},{"page":237,"text":"x","rect":[271.4023742675781,522.212158203125,275.84662869775158,517.9570922851563]},{"page":237,"text":"Fig. 8.25 Distortion of a homeotropically aligned SmA liquid crystal by a corrugated surface of","rect":[53.812843322753909,564.1054077148438,385.2109383927076,556.1554565429688]},{"page":237,"text":"solid boundary plate with the dotted line pictured an exponential decay of the distortion (a) and the","rect":[53.812843322753909,573.9569091796875,385.1542296149683,566.362548828125]},{"page":237,"text":"wave-like splay distortions in a thin layer with the arrows indicating the direction of the induced","rect":[53.812843322753909,583.932861328125,385.14410919337197,576.3385009765625]},{"page":237,"text":"local pressure (b)","rect":[53.812843322753909,593.9088134765625,113.61567777259042,586.314453125]},{"page":238,"text":"8.5 Smectic Phases","rect":[53.812843322753909,42.55594253540039,119.50374300956048,36.62946701049805]},{"page":238,"text":"The elastic energy density averaged over the x,y-plane reads:","rect":[65.76496887207031,68.2883529663086,311.9455400235374,59.35380554199219]},{"page":238,"text":"223","rect":[372.4981689453125,42.55594253540039,385.18979400782509,36.73106384277344]},{"page":238,"text":"gdist","rect":[124.33556365966797,101.08924102783203,139.40349846641917,94.30615997314453]},{"page":238,"text":"(8.45)","rect":[361.0552062988281,133.35679626464845,385.10457740632668,124.76090240478516]},{"page":238,"text":"Here <cos2qx> ¼ 1/2 and","rect":[65.76556396484375,167.48365783691407,173.06739131024848,156.54222106933595]},{"page":238,"text":"ls ¼ \bK11=B 1=2","rect":[187.3258056640625,202.2332305908203,251.13644478401504,181.88156127929688]},{"page":238,"text":"(8.46)","rect":[361.0561828613281,197.51905822753907,385.10555396882668,188.98292541503907]},{"page":238,"text":"is a smectic characteristic length (l \u0005 1 nm).","rect":[53.81450653076172,225.7311248779297,238.4531673958469,216.43798828125]},{"page":238,"text":"Now","rect":[65.76750946044922,236.0,85.03886175110664,228.91546630859376]},{"page":238,"text":"we","rect":[94.42965698242188,236.0,106.02631414605939,230.0]},{"page":238,"text":"make","rect":[115.38824462890625,236.0,137.11429632379379,228.7162628173828]},{"page":238,"text":"minimisation","rect":[146.48619079589845,236.0,199.34206477216254,228.7162628173828]},{"page":238,"text":"of","rect":[208.68209838867188,236.0,216.97396217195166,228.7162628173828]},{"page":238,"text":"(8.45)","rect":[226.35479736328126,237.2523956298828,250.45891700593604,228.656494140625]},{"page":238,"text":"@@ug0 \u0003 ddz \b@@ug00 ¼ 2l2sq4u0 \u0003 ddz 2u00 ¼ 0 and get","rect":[55.455509185791019,257.83758544921877,240.61704881259989,239.9286346435547]},{"page":238,"text":"writing","rect":[259.77606201171877,237.65081787109376,288.6571892838813,228.7162628173828]},{"page":238,"text":"the","rect":[298.01116943359377,236.0,310.2349323101219,228.7162628173828]},{"page":238,"text":"Euler","rect":[319.6496276855469,236.0,341.3000424942173,228.7162628173828]},{"page":238,"text":"equation","rect":[350.69085693359377,237.65081787109376,385.1791924577094,228.7162628173828]},{"page":238,"text":"dd2zu20 \u0003 l2sq4u0 ¼ 0","rect":[182.34071350097657,293.7296142578125,258.3513878922797,271.6060791015625]},{"page":238,"text":"(8.47)","rect":[361.0559387207031,288.4350891113281,385.1053403457798,279.958740234375]},{"page":238,"text":"This equation has a solution: u0ðzÞ ¼ aexpð\u0003z=LpÞ: Thus, the distortion “pro-","rect":[65.76627349853516,317.9851379394531,385.14788184968605,307.61865234375]},{"page":238,"text":"pagates” into the bulk over a penetration distance","rect":[53.81405258178711,329.1900939941406,253.57314336969223,320.25555419921877]},{"page":238,"text":"1","rect":[209.3605499267578,352.4416198730469,214.33765498212348,345.7083435058594]},{"page":238,"text":"L¼","rect":[180.3022003173828,359.2428283691406,200.20166805479438,352.6689147949219]},{"page":238,"text":"¼","rect":[223.4084930419922,357.9186706542969,231.0732348272553,355.5879211425781]},{"page":238,"text":"p","rect":[185.79611206054688,362.13946533203127,189.28007576546035,357.5516662597656]},{"page":238,"text":"q2ls","rect":[203.01673889160157,368.07659912109377,220.1433759297365,358.497802734375]},{"page":238,"text":"4p2ls","rect":[233.88787841796876,367.72186279296877,256.56627326860368,358.497802734375]},{"page":238,"text":"(8.48a)","rect":[356.6379089355469,360.9864807128906,385.13680396882668,352.5101318359375]},{"page":238,"text":"This distance is quite large. For instance, for period of the surface relief L ¼1","rect":[65.76592254638672,391.5960388183594,385.17861262372505,382.3626708984375]},{"page":238,"text":"mm and smectic length l ¼ 1 nm, the penetration length is 2.5 mm and the larger the","rect":[53.813899993896487,403.5555725097656,385.1736530132469,394.3022766113281]},{"page":238,"text":"period of a distortion the longer the penetration length.","rect":[53.813899993896487,415.51513671875,275.4722866585422,406.58056640625]},{"page":238,"text":"The result obtained has very interesting consequences: (i) to have well aligned","rect":[65.76592254638672,427.47467041015627,385.09710017255318,418.54010009765627]},{"page":238,"text":"SmA samples, very flat glasses without corrugation are needed; (ii) even small dust","rect":[53.813899993896487,439.4341735839844,385.17268235752177,430.4996337890625]},{"page":238,"text":"particles or other inhomogeneities create characteristic defects in the form of semi-","rect":[53.813899993896487,451.3369445800781,385.1457761367954,442.40240478515627]},{"page":238,"text":"spheres (see Fig. 8.29b below) and well seen under an optical microscope; (iii)","rect":[53.813899993896487,463.2965087890625,385.1378110489048,454.3619384765625]},{"page":238,"text":"layers are often broken(not bent) by external factors: in particular, strongmolecular","rect":[53.813899993896487,475.2560729980469,385.1517575821079,466.321533203125]},{"page":238,"text":"chirality may result in the formation of defect phases like twist-grain-boundary","rect":[53.813899993896487,487.2156066894531,385.1140069108344,478.28106689453127]},{"page":238,"text":"phase; (iv) the thermal fluctuations of director in smectic A phase are weak and the","rect":[53.813899993896487,499.1751403808594,385.17466009332505,490.2406005859375]},{"page":238,"text":"smectic samples are not as opaque as nematic samples. In fact there is a critical cell","rect":[53.813899993896487,511.1346740722656,385.12190110752177,502.20013427734377]},{"page":238,"text":"thickness for short-wave fluctuations.","rect":[53.813899993896487,521.032470703125,204.48281522299534,514.1597290039063]},{"page":238,"text":"Consider a SmA layer of thickness d between two glasses, Fig. 8.25b. Flat","rect":[65.76592254638672,535.0537109375,385.1477494961936,526.0594482421875]},{"page":238,"text":"surfaces stabilise the parallel arrangement of the layers while thermal fluctuations","rect":[53.813899993896487,546.95654296875,385.0980264102097,538.0220336914063]},{"page":238,"text":"excite wave-like splay distortions. In thin cells these fluctuations are markedly","rect":[53.813899993896487,558.9160766601563,385.17669001630318,549.9815673828125]},{"page":238,"text":"suppressed: for d>>Lp they are very strong, for d<<Lp they are quenched for any","rect":[53.813899993896487,571.7340087890625,385.14482966474068,561.921142578125]},{"page":238,"text":"wavevector. For a fixed cell thickness, we can find a critical wavevector qc for","rect":[53.8139762878418,582.7272338867188,385.18135963288918,573.9022827148438]},{"page":239,"text":"224","rect":[53.812843322753909,42.454345703125,66.50444931178018,36.73106384277344]},{"page":239,"text":"8 Elasticity and Defects","rect":[303.50408935546877,44.274620056152347,385.17355807303707,36.68026351928711]},{"page":239,"text":"surviving fluctuations. Assume that the critical penetration length Lpc ¼ d/p. Then,","rect":[53.812843322753909,69.14512634277344,385.1371731331516,59.333885192871097]},{"page":239,"text":"from (8.46), the critical wave vector is","rect":[53.814292907714847,79.8494644165039,209.53198637114719,71.31333923339844]},{"page":239,"text":"qc ¼ \u00061=Lcpls\u00071=2 ¼ ðp=dlsÞ1=2","rect":[154.8117218017578,108.91228485107422,283.7071692713197,94.53634643554688]},{"page":239,"text":"(8.48b)","rect":[356.0715026855469,107.22647857666016,385.1596921524204,98.69035339355469]},{"page":239,"text":"For typical values of d \u0005 10 mm and l \u0005 1 nm, qc \u0005 1.8\u0004107 m\u00031 and only those","rect":[65.76595306396485,132.45138549804688,385.1478961773094,121.40013122558594]},{"page":239,"text":"fluctuations survive whose period is less than L < Lc ¼ 2p/qc \u0005 0.35 mm.","rect":[53.814022064208987,144.41061401367188,354.3256802132297,135.17724609375]},{"page":239,"text":"8.5.2 Peierls Instability of the SmA Structure","rect":[53.812843322753909,194.5211639404297,285.8678415629407,183.88339233398438]},{"page":239,"text":"In Section 5.7.2 we discussed a general problem of stability of one, two- and three-","rect":[53.812843322753909,222.06338500976563,385.1606687149204,213.06906127929688]},{"page":239,"text":"dimensional phases. Here, we shall analyze stability of the smectic A liquid crystal,","rect":[53.812843322753909,234.02294921875,385.1168179085422,225.08839416503907]},{"page":239,"text":"which is three-dimensional structure with one-dimensional periodicity. The ques-","rect":[53.812843322753909,245.98248291015626,385.1596921524204,237.0479278564453]},{"page":239,"text":"tion of stability is tightly related to the elastic properties of the smectic A phase.","rect":[53.812843322753909,257.9420471191406,385.1815456917453,249.0074920654297]},{"page":239,"text":"Consider a stack of smectic layers (each of thickness l) with their normal along the","rect":[53.812843322753909,269.9015808105469,385.1715778179344,260.9471130371094]},{"page":239,"text":"z-direction. The size of the sample along z is L, along x and y it is L⊥, the volume is","rect":[53.81181716918945,281.8611145019531,385.1893350039597,272.92657470703127]},{"page":239,"text":"V ¼ L⊥2L. Fluctuations of layer displacement u(r) ¼ u(z, r⊥) along z and in both","rect":[53.81368637084961,293.8214416503906,385.12810603192818,282.7705993652344]},{"page":239,"text":"directions perpendicular to z can be expanded in the Fourier series with wavevec-","rect":[53.81313705444336,305.7809753417969,385.1301206192173,296.846435546875]},{"page":239,"text":"tors qz and q⊥ (normal modes):","rect":[53.814144134521487,317.5744323730469,180.27574021884989,308.74945068359377]},{"page":239,"text":"uq ¼ ð uðrÞexpðiqrÞdr","rect":[173.22019958496095,354.1025695800781,265.8114703960594,331.9803466796875]},{"page":239,"text":"Then,onaccountof(8.44),thefreeenergyofdistortioninthevolumeO ¼ (2p)3/V","rect":[65.76665496826172,377.6521301269531,385.1807936993953,366.6012878417969]},{"page":239,"text":"in the wavevector space (qx, qy, qz), which is a sum (or an integral in the q-space) of","rect":[55.85270690917969,390.541748046875,385.1509946426548,380.6773681640625]},{"page":239,"text":"the energy of all normal modes, reads:","rect":[53.81313705444336,401.5715637207031,209.16252000400614,392.63702392578127]},{"page":239,"text":"F ¼ 2ð21pÞ3 ð d3q\u0006Bq2z þ K11q4?\u0007juðqÞj2","rect":[139.5736541748047,440.169921875,299.05830452522596,415.8675231933594]},{"page":239,"text":"O","rect":[189.36480712890626,444.484375,194.36777900916199,439.6107177734375]},{"page":239,"text":"(8.49)","rect":[361.0561828613281,431.1573181152344,385.10555396882668,422.68096923828127]},{"page":239,"text":"The Gibbs distribution gives us probability of the mean square value juðqÞj2 for a","rect":[65.76653289794922,469.4075012207031,385.16138494684068,457.4057922363281]},{"page":239,"text":"particular fluctuation of the layer distortion when the average value hjuðqÞj2i for all","rect":[53.814598083496097,481.3667297363281,385.14839036533427,469.3654479980469]},{"page":239,"text":"fluctuations is known:","rect":[53.814537048339847,490.9189758300781,142.59016282627176,484.00640869140627]},{"page":239,"text":"w / exp\"\u0003VðBqz2 þ KkB11Tq?4 ÞjuðqÞj2# ¼ exp\"\u0003<jjuuððqqÞÞjj22>#","rect":[99.24353790283203,536.983642578125,339.78676472630709,507.121826171875]},{"page":239,"text":"From this equality and using Eq. (8.49), the mean-square average of the fluctua-","rect":[65.7662582397461,560.5607299804688,385.1132749160923,551.626220703125]},{"page":239,"text":"tion amplitude is found","rect":[53.81426239013672,572.520263671875,147.72925654462348,563.5857543945313]},{"page":240,"text":"8.5 Smectic Phases","rect":[53.812843322753909,42.55630874633789,119.50374300956048,36.62983322143555]},{"page":240,"text":"Fig. 8.26 Cylindrical","rect":[53.812843322753909,67.58130645751953,129.134144823218,59.6313591003418]},{"page":240,"text":"coordinates selected fora","rect":[53.812843322753909,75.76238250732422,141.14716479563237,69.89517211914063]},{"page":240,"text":"stack of smectic layers that","rect":[53.812843322753909,87.4087142944336,146.84652427878442,79.81436157226563]},{"page":240,"text":"becomes unstable in the","rect":[53.812843322753909,95.63212585449219,135.91652819895269,89.79031372070313]},{"page":240,"text":"infinitely thick sample","rect":[53.812843322753909,107.36067962646485,130.30430743479253,99.76632690429688]},{"page":240,"text":"x","rect":[268.89459228515627,145.57298278808595,274.40988876586325,140.89402770996095]},{"page":240,"text":"z","rect":[298.3114929199219,71.8916015625,302.6877607796133,67.4325942993164]},{"page":240,"text":"qz","rect":[291.6721496582031,102.61772155761719,300.1688960329536,95.939208984375]},{"page":240,"text":"j","rect":[348.6845397949219,102.67454528808594,354.70940171134637,95.9260482788086]},{"page":240,"text":"r","rect":[334.94622802734377,136.80015563964845,340.4315500706556,130.04165649414063]},{"page":240,"text":"225","rect":[372.4981689453125,42.55630874633789,385.18979400782509,36.62983322143555]},{"page":240,"text":"u2ðqÞ\u000B ¼ ðk2BpTÞ3 ð Bq2z þd3Kq11q?4","rect":[159.3994598388672,269.50396728515627,283.83295547275426,243.60574340820313]},{"page":240,"text":"O","rect":[224.9945831298828,273.76177978515627,229.99755501013855,268.88812255859377]},{"page":240,"text":"(8.50)","rect":[361.0561828613281,260.4346923828125,385.10555396882668,251.83880615234376]},{"page":240,"text":"In fact, we used the equipartition theorem showing that each normal mode has","rect":[65.76653289794922,298.3552551269531,385.1315041934128,289.42071533203127]},{"page":240,"text":"the same energy kBT/2. Factor 1/2 disappeared from (8.50) because each u(q)","rect":[53.81450653076172,310.31549072265627,385.1584409317173,301.3211669921875]},{"page":240,"text":"corresponds to two fluctuation modes with wavevectors","rect":[53.81260299682617,322.2750244140625,275.1175881777878,313.3404541015625]},{"page":240,"text":"q. The integration should","rect":[285.2071838378906,322.2750244140625,385.1783379655219,313.3404541015625]},{"page":240,"text":"be","rect":[53.81258773803711,333.0,63.23922384454573,325.24322509765627]},{"page":240,"text":"made","rect":[68.48310089111328,333.0,90.20916021539533,325.24322509765627]},{"page":240,"text":"in","rect":[95.44607543945313,333.0,103.22031489423284,325.24322509765627]},{"page":240,"text":"cylindrical","rect":[108.4741439819336,334.17779541015627,151.34593064853739,325.24322509765627]},{"page":240,"text":"co-ordinates","rect":[156.6216583251953,333.0,206.01543058501438,325.24322509765627]},{"page":240,"text":"according","rect":[211.33993530273438,334.17779541015627,250.8054131608344,325.24322509765627]},{"page":240,"text":"to","rect":[256.0323791503906,333.0,263.8066033463813,326.25921630859377]},{"page":240,"text":"the","rect":[269.0604248046875,333.0,281.2842181987938,325.24322509765627]},{"page":240,"text":"symmetry","rect":[286.5639343261719,334.17779541015627,326.54103938153755,326.25921630859377]},{"page":240,"text":"of","rect":[331.82275390625,333.0,340.1146176895298,325.24322509765627]},{"page":240,"text":"SmA,","rect":[345.3604736328125,333.0,368.2651333382297,325.3030090332031]},{"page":240,"text":"for","rect":[373.5697326660156,333.0,385.17632423249855,325.24322509765627]},{"page":240,"text":"geometry see Fig. 8.26:","rect":[53.81258773803711,346.1373291015625,148.82751329013895,337.2027587890625]},{"page":240,"text":"ffiffiffiffiffiffiffiffiffiffiffiffiffiffi","rect":[123.99573516845703,362.0,153.08591759756338,360.0]},{"page":240,"text":"ffiffi","rect":[186.36251831054688,364.0,190.50347436025869,363.0]},{"page":240,"text":"ffiffi","rect":[236.21022033691407,364.0,240.35117638662588,363.0]},{"page":240,"text":"r ¼ qq2x þ q2y ¼ q?p2; dr ¼ p2dq? and rdr ¼ 2q?dq?;","rect":[95.27684783935547,378.8043518066406,342.04884278458499,360.8854675292969]},{"page":240,"text":"Now the limits of integration are changed:","rect":[65.7657699584961,402.2518005371094,237.07273728916239,393.3172607421875]},{"page":240,"text":"2p","rect":[123.42977142333985,421.26531982421877,133.90956467936588,414.4623107910156]},{"page":240,"text":"2p 2p","rect":[168.57601928710938,421.26531982421877,197.63536423991276,414.4623107910156]},{"page":240,"text":"2p","rect":[235.41729736328126,421.26531982421877,245.8970982487018,414.4623107910156]},{"page":240,"text":"L \u0007 qz \u0007 l ; L? \u0007 q? \u0007 a0 and 0 \u0007 j \u0007 2p","rect":[125.92230987548828,436.3563537597656,317.21222276042058,421.20416259765627]},{"page":240,"text":"Here a0 is radius of a rod-like molecule. However, we can assume infinitely large","rect":[65.7648696899414,457.9123229980469,385.1174701519188,448.977783203125]},{"page":240,"text":"sample in the x, y direction, L⊥!1 and, in addition, a0!0. Then, after introducing","rect":[53.81343460083008,469.8719787597656,385.16774836591255,460.93731689453127]},{"page":240,"text":"ls ¼ (K11/B)1/2 and integrating over j we obtain the double integral:","rect":[53.81296920776367,481.8316345214844,331.6736894375999,470.7408447265625]},{"page":240,"text":"2p=l","rect":[250.93801879882813,502.10772705078127,263.77152519981765,495.1563415527344]},{"page":240,"text":"1","rect":[281.4131164550781,501.2418518066406,288.38104386490508,498.0973205566406]},{"page":240,"text":"u2ðqÞ\u000B ¼ k4BpT2 ðð B2qq2z?þdqKzd11qq??4 ¼ 4pk2BKT11 ð dqz ð l\u0003s22qq?2zdþq?q?4","rect":[81.2854995727539,527.5050659179688,338.0422755411136,503.0881042480469]},{"page":240,"text":"2p=L","rect":[249.5214385986328,533.8491821289063,264.1868110669432,526.8977661132813]},{"page":240,"text":"(8.51)","rect":[361.0561828613281,518.44580078125,385.10555396882668,509.8498840332031]},{"page":240,"text":"1","rect":[209.2472381591797,544.3192749023438,216.21516556900662,541.1748046875]},{"page":240,"text":"The integral over q⊥ has a form Ð k2dþmm2 ¼ 1k arctanmk j01 ¼ 2pk where we put","rect":[65.76653289794922,558.2546997070313,385.1364579922874,545.8287353515625]},{"page":240,"text":"0","rect":[209.5872802734375,563.604736328125,213.07124397835097,558.8078002929688]},{"page":240,"text":"k ¼ qz=ls and m ¼ q2?;dm ¼ 2q?dq?: Now, integrating over qz using dqz ¼ lsdk,","rect":[53.813533782958987,573.7235107421875,385.1829800179172,562.60546875]},{"page":240,"text":"we arrive at the final result:","rect":[53.81426239013672,583.3092041015625,165.04858262607645,576.3966674804688]},{"page":241,"text":"226","rect":[53.813682556152347,42.55740737915039,66.50528854517862,36.68172836303711]},{"page":241,"text":"8 Elasticity and Defects","rect":[303.50494384765627,44.276084899902347,385.17441256522457,36.68172836303711]},{"page":241,"text":"2pl=l","rect":[154.92454528808595,67.47931671142578,171.5535076216926,60.52791976928711]},{"page":241,"text":"u2ðqÞ\u000B ¼ 4pk2BKT11 ð p2lksdk ¼ 8kBpTKl11s lnLl ¼ 8pðBkKBT11Þ1=2 lnLl","rect":[81.45574188232422,93.45342254638672,336.6989476139362,68.1607437133789]},{"page":241,"text":"2pl=L","rect":[153.45152282714845,99.22077178955078,171.96879348881823,92.26937866210938]},{"page":241,"text":"(8.52)","rect":[361.055908203125,83.81668853759766,385.10527931062355,75.2208023071289]},{"page":241,"text":"This formula shows that, when the size of the sample along the layer normal","rect":[65.7662582397461,123.7213363647461,385.1829667813499,114.78678894042969]},{"page":241,"text":"increases to infinity (L!1) the average mean square magnitude of the fluctuation","rect":[53.814231872558597,135.68087768554688,385.12017146161568,126.74632263183594]},{"page":241,"text":"in the interlayer distance diverges, <u2(q)> !1. Formally, the ideal smectic A","rect":[53.815208435058597,147.64157104492188,385.1386084552082,136.59056091308595]},{"page":241,"text":"structure becomes unstable. However, the divergence is logarithmic that is very","rect":[53.813716888427737,159.60113525390626,385.1158074479438,150.6665802001953]},{"page":241,"text":"smooth. Such fluctuations destroy the true long-range positional order along z.","rect":[53.813716888427737,171.50387573242188,385.1833462288547,162.56932067871095]},{"page":241,"text":"Instead, the quasi-long range positional order forms that was discussed in Sec-","rect":[53.81468963623047,183.46343994140626,385.10970435945168,174.5288848876953]},{"page":241,"text":"tion 5.7.2. In experiments, the quasi-long range order manifests itself by deviation","rect":[53.81468963623047,195.4229736328125,385.1127862077094,186.42864990234376]},{"page":241,"text":"of the X-ray Bragg reflections from the d-function form.","rect":[53.81468963623047,207.38253784179688,281.2485012581516,198.149169921875]},{"page":241,"text":"8.5.3 Defects in SmecticA","rect":[53.812843322753909,257.46649169921877,192.9593286661742,246.91238403320313]},{"page":241,"text":"There are many types of defects originated from the layered structure of the smectic","rect":[53.812843322753909,285.0925598144531,385.17954290582505,276.15802001953127]},{"page":241,"text":"A phase. Here, we shall only present a brief survey of the most important cases.","rect":[53.812843322753909,297.0520935058594,376.7941860726047,288.1175537109375]},{"page":241,"text":"8.5.3.1 Steps and Dislocations","rect":[53.812843322753909,338.8123779296875,186.9205666934128,329.60888671875]},{"page":241,"text":"Steps are observed at the edge of the drop of a smectic preparation on a surfactant","rect":[53.812843322753909,362.7444152832031,385.17158372470927,353.80987548828127]},{"page":241,"text":"covered glass, as shown in Fig. 8.27. In the blown part of the same figure, the","rect":[53.812843322753909,374.7039794921875,385.17456854059068,365.7694091796875]},{"page":241,"text":"structure of each step containing a single p-disclination is seen. In the three","rect":[53.81285095214844,386.66351318359377,385.12180364801255,377.72894287109377]},{"page":241,"text":"dimensional picture of Fig. 8.28, we can see a difference between the p-disclination","rect":[53.812843322753909,398.6230773925781,385.1566094498969,389.68853759765627]},{"page":241,"text":"and another defect, called an edge dislocation which is typical of solid crystals. In","rect":[53.81282424926758,410.6224060058594,385.1486748795844,401.62811279296877]},{"page":241,"text":"the smectic A, and additional smectic layer is incorporated between two other","rect":[53.81282424926758,422.5421447753906,385.13677345124855,413.60760498046877]},{"page":241,"text":"molecular layers, and the edge of the irregular layer forms such a dislocation.","rect":[53.81282424926758,434.5016784667969,366.9106411507297,425.567138671875]},{"page":241,"text":"Fig. 8.27","rect":[53.812843322753909,584.0003662109375,85.61292785792276,576.0504150390625]},{"page":241,"text":"containing","rect":[53.812843322753909,593.9085693359375,89.66239685206338,586.314208984375]},{"page":241,"text":"Free surface","rect":[95.05758666992188,496.8673400878906,142.5935159047828,490.9486083984375]},{"page":241,"text":"SmA layers","rect":[157.55612182617188,500.6497497558594,201.09542325341563,493.0513610839844]},{"page":241,"text":"Zoom","rect":[207.12074279785157,519.2274169921875,228.87841103260309,513.380615234375]},{"page":241,"text":"p-disclination","rect":[288.8614501953125,515.2914428710938,340.3300509073601,509.16473388671877]},{"page":241,"text":"step","rect":[332.5902099609375,540.1293334960938,349.02435632728199,533.0748291015625]},{"page":241,"text":"glass","rect":[120.88141632080078,564.057861328125,141.32018521630625,556.58740234375]},{"page":241,"text":"Steps at the edge of a drop of the smectic A phase (left); the structure of","rect":[91.59490966796875,583.9326171875,349.8072823868482,576.304443359375]},{"page":241,"text":"a single p-disclination is seen in the blown part (right)","rect":[92.04841613769531,593.9424438476563,278.8493661270826,586.2973022460938]},{"page":241,"text":"each","rect":[352.84228515625,582.1123657226563,368.4191030898563,576.3382568359375]},{"page":241,"text":"step","rect":[371.5353088378906,583.9326171875,385.1830191054813,577.2018432617188]},{"page":242,"text":"8.5 Smectic Phases","rect":[53.812843322753909,42.55643081665039,119.50374300956048,36.62995529174805]},{"page":242,"text":"a","rect":[98.78913879394531,68.44943237304688,104.34440119117916,62.86067581176758]},{"page":242,"text":"Fig. 8.28 Illustration","rect":[53.812843322753909,167.39639282226563,128.62223571925089,159.4464569091797]},{"page":242,"text":"smectic A","rect":[53.812843322753909,175.55210876464845,88.26716400724857,169.71029663085938]},{"page":242,"text":"a","rect":[58.59321594238281,206.3646240234375,64.14847833961666,200.77586364746095]},{"page":242,"text":"L","rect":[115.5306396484375,225.6917724609375,120.4145235636101,219.94898986816407]},{"page":242,"text":"of","rect":[131.5294647216797,165.5761260986328,138.57753079993419,159.73431396484376]},{"page":242,"text":"the","rect":[141.4424591064453,165.5761260986328,151.8326506354761,159.73431396484376]},{"page":242,"text":"b","rect":[215.0899658203125,68.23342895507813,221.1947595697907,60.92505645751953]},{"page":242,"text":"π–disclination","rect":[177.9134521484375,145.47486877441407,229.17606595361094,139.5921173095703]},{"page":242,"text":"dislocation","rect":[278.2283020019531,142.25875854492188,315.98847440087658,136.4040069580078]},{"page":242,"text":"difference between a p-disclination and an edge","rect":[154.69757080078126,167.36253356933595,321.8518919684839,159.71737670898438]},{"page":242,"text":"c","rect":[239.93455505371095,206.3646240234375,245.48981745094478,200.77586364746095]},{"page":242,"text":"d","rect":[315.3662109375,206.3646240234375,321.4710046869782,199.05624389648438]},{"page":242,"text":"dislocation","rect":[324.7464294433594,166.0,362.44894928126259,159.71737670898438]},{"page":242,"text":"227","rect":[372.4981689453125,42.55643081665039,385.18979400782509,36.73155212402344]},{"page":242,"text":"in the","rect":[365.3037109375,165.5761260986328,385.1533751227808,159.73431396484376]},{"page":242,"text":"Smectic layers","rect":[56.701812744140628,312.5204772949219,108.66601949919346,304.9220886230469]},{"page":242,"text":"Fig. 8.29 Structure of some defects in the SmA phase, namely, cylinder (a), semi-sphere (b), and","rect":[53.812843322753909,332.4508972167969,385.19403595118447,324.4162902832031]},{"page":242,"text":"radial hedgehog (c) and a monopole in SmC (d)","rect":[53.81200408935547,342.3591613769531,218.40857022864513,334.7648010253906]},{"page":242,"text":"8.5.3.2 Cylinders, Tores, Hedgehogs","rect":[53.812843322753909,367.6214599609375,214.22592558013157,357.9498291015625]},{"page":242,"text":"Many structural defects compatible with the incompressible smectic layers can be","rect":[53.812843322753909,391.14215087890627,385.2612994976219,382.20758056640627]},{"page":242,"text":"observed under a microscope. Among them are cylinders, tores and hemispheres","rect":[53.812843322753909,403.1017150878906,385.16659940825658,394.16717529296877]},{"page":242,"text":"observed at the surfaces, radial hedgehogs observed in smectic drops, etc. Three of","rect":[53.812843322753909,415.0044860839844,385.2612241348423,406.0699462890625]},{"page":242,"text":"them are presented in Fig. 8.29a–c. Note that in all defect structures of this type, the","rect":[53.812843322753909,426.9640197753906,385.1586383648094,418.02947998046877]},{"page":242,"text":"splay distortion plays the fundamental rolebut bend and twistare absent. Other,more","rect":[53.81282424926758,438.9235534667969,385.24533880426255,429.9690856933594]},{"page":242,"text":"special defects, namely, the walls composed of screw dislocations, are observed in","rect":[53.81282424926758,450.8830871582031,385.25429621747505,441.94854736328127]},{"page":242,"text":"the TGBA phase.","rect":[53.81282424926758,462.8426208496094,123.11602445151095,453.9080810546875]},{"page":242,"text":"8.5.3.3 Focal-Conics","rect":[53.81282424926758,496.8645935058594,146.68957914458469,489.33447265625]},{"page":242,"text":"These are the most striking features of smectic textures [19]. Smectic layers of","rect":[53.81282424926758,522.5267333984375,385.15068946687355,513.5922241210938]},{"page":242,"text":"constant thickness (incompressible, modulus B!1) form surfaces called Dupin","rect":[53.812843322753909,534.4295654296875,385.13872614911568,525.4950561523438]},{"page":242,"text":"cyclides. We have seen some of them, which have the form of tori including","rect":[53.8138313293457,546.3890991210938,385.16960993817818,537.4346313476563]},{"page":242,"text":"disclinations, see Fig. 4.7b. Such cyclides can fill any volume of a liquid crystal","rect":[53.8138313293457,558.3486328125,385.1227861172874,549.4141235351563]},{"page":242,"text":"by cones of different size. An example is a focal-conic pair, namely, two cones with","rect":[53.8138313293457,570.3081665039063,385.1168145280219,561.3338012695313]},{"page":242,"text":"a common base. The common base is an ellipse with apices at A and C and foci at O","rect":[53.8138313293457,582.2677612304688,385.1357703204426,573.3132934570313]},{"page":242,"text":"and O’, see Fig. 8.30a. The hyperbola B–B’ passes through focus O. The focus of","rect":[53.8138313293457,594.227294921875,385.1487058242954,585.2728271484375]},{"page":243,"text":"228","rect":[53.812843322753909,42.55630874633789,66.50444931178018,36.73143005371094]},{"page":243,"text":"8 Elasticity and Defects","rect":[303.50408935546877,44.274986267089847,385.17355807303707,36.68062973022461]},{"page":243,"text":"Fig. 8.30 Focal-conic defect structure in SmA: A pair of cones with a common elliptical base and","rect":[53.812843322753909,182.30343627929688,385.1948599257938,174.35350036621095]},{"page":243,"text":"a hyperbola connecting cone apices (a); cross-section of the upper cone by plane ABC with gaps","rect":[53.812843322753909,192.21170043945313,385.16525729178707,184.61734008789063]},{"page":243,"text":"between lines (Dupin cyclides) indicating the smectic layers (b); filling the space of the sample by","rect":[53.812843322753909,202.13088989257813,385.2083792129032,194.53652954101563]},{"page":243,"text":"cones of different size (c)","rect":[53.81285858154297,211.76820373535157,141.54145902259044,204.51251220703126]},{"page":243,"text":"Fig. 8.31 Photos of some focal conic textures: polygonal (a); and fan-shape (b) textures","rect":[53.812843322753909,356.6536560058594,358.17187960624019,348.7037048339844]},{"page":243,"text":"the hyperbola C coincides with the apex of the ellipse C. Using multiple lines such","rect":[53.812843322753909,381.506591796875,385.12184992841255,372.572021484375]},{"page":243,"text":"as BD and B’D one can build two conical surfaces with apices at B and B’.","rect":[53.812843322753909,393.4661560058594,358.5301785042453,384.5316162109375]},{"page":243,"text":"The bulk of the cones is filled by smectic layers. Fig. 8.30b represents the cross-","rect":[65.76486206054688,405.3689270019531,385.1148618301548,396.43438720703127]},{"page":243,"text":"section of the upper cone by plane ABC. Note that along the line OB smectic layers","rect":[53.812843322753909,417.3284606933594,385.1009560977097,408.3739929199219]},{"page":243,"text":"are continuous although their slope is changed. Using such cones all the space","rect":[53.812843322753909,429.2879943847656,385.0790485210594,420.35345458984377]},{"page":243,"text":"occupied by a liquid crystal can be filled, Fig. 8.30c. It is very fascinating that such","rect":[53.812843322753909,441.2475280761719,385.12481013349068,432.31298828125]},{"page":243,"text":"“mathematical","rect":[53.81282424926758,452.0,112.32865769931863,444.27252197265627]},{"page":243,"text":"structures”","rect":[118.50025939941406,452.0,161.7851489849266,445.28851318359377]},{"page":243,"text":"are","rect":[167.95079040527345,452.0,180.16460455133285,446.0]},{"page":243,"text":"indeed","rect":[186.30335998535157,452.0,212.93085566571723,444.27252197265627]},{"page":243,"text":"observed","rect":[219.10047912597657,452.0,255.05510798505316,444.27252197265627]},{"page":243,"text":"in","rect":[261.24365234375,452.0,269.01787653974068,444.27252197265627]},{"page":243,"text":"experiment!","rect":[275.1217956542969,453.20709228515627,323.4933103164829,444.27252197265627]},{"page":243,"text":"A","rect":[329.6708984375,451.0655822753906,336.8180212970051,444.3323059082031]},{"page":243,"text":"variety","rect":[342.9259338378906,453.20709228515627,370.6882256608344,444.27252197265627]},{"page":243,"text":"of","rect":[376.8558654785156,452.0,385.1477292617954,444.27252197265627]},{"page":243,"text":"observed focal conic textures is very large. The microscopic photos in Fig. 8.31,","rect":[53.81282424926758,465.1666259765625,385.1815151741672,456.2320556640625]},{"page":243,"text":"illustrate two of them. Photo (a) shows rather large but short polygons (polygonal","rect":[53.812843322753909,477.166015625,385.10090501377177,468.1717224121094]},{"page":243,"text":"texture) and photo (b) demonstrates the so-called fan-shape texture with very","rect":[53.812843322753909,489.0857238769531,385.11785212567818,480.111328125]},{"page":243,"text":"narrow, elongated polygons.","rect":[53.81385040283203,500.9884948730469,168.14095731283909,492.053955078125]},{"page":243,"text":"8.5.4 Smectic C Elasticity and Defects","rect":[53.812843322753909,544.18505859375,250.8760288212077,533.5473022460938]},{"page":243,"text":"8.5.4.1 Elastic Energy","rect":[53.812843322753909,572.1258544921875,153.9193505387641,562.4542236328125]},{"page":243,"text":"In the smectic C phase the director is free to rotate about the normal z to the smectic","rect":[53.812843322753909,595.6465454101563,385.23731268121568,586.7120361328125]},{"page":243,"text":"layers. In the general case, the smectic layers are considered compressible. The elastic","rect":[53.81187438964844,607.6060791015625,385.23731268121568,598.651611328125]},{"page":244,"text":"8.5 Smectic Phases","rect":[53.81318283081055,42.55588150024414,119.50408633231439,36.6294059753418]},{"page":244,"text":"229","rect":[372.4985046386719,42.62361145019531,385.19012970118447,36.73100280761719]},{"page":244,"text":"Fig. 8.32","rect":[53.812843322753909,177.31549072265626,84.93350738673135,169.3655548095703]},{"page":244,"text":"a","rect":[111.30769348144531,68.23306274414063,116.86295587867916,62.64430618286133]},{"page":244,"text":"Jn","rect":[176.57237243652345,78.5166015625,194.54602380775072,72.82027435302735]},{"page":244,"text":"c","rect":[191.0433807373047,94.30045318603516,195.48763516747813,89.82940673828125]},{"page":244,"text":"z","rect":[118.47015380859375,124.21246337890625,122.46678549040439,120.02935791015625]},{"page":244,"text":"W","rect":[157.5990753173828,121.56268310546875,163.73790158064396,116.10784912109375]},{"page":244,"text":"y","rect":[163.7379150390625,124.83998107910156,166.73538880042049,120.4249267578125]},{"page":244,"text":"y","rect":[128.6890411376953,144.14785766601563,132.68567281950596,138.2611083984375]},{"page":244,"text":"x","rect":[172.53610229492188,155.64617919921876,176.53273397673252,151.46307373046876]},{"page":244,"text":"Smectic C. Definition of rotation","rect":[90.91548919677735,175.52908325195313,203.38342804102823,169.65341186523438]},{"page":244,"text":"b","rect":[234.9072723388672,70.08670043945313,241.0120660883454,62.77832794189453]},{"page":244,"text":"W","rect":[217.18896484375,106.13104248046875,223.32779110701115,100.67620849609375]},{"page":244,"text":"x","rect":[223.3276824951172,108.130615234375,226.32515625647518,104.9932861328125]},{"page":244,"text":"axes Ox, Oy","rect":[205.79229736328126,178.0466766357422,245.56103968064827,169.50100708007813]},{"page":244,"text":"y","rect":[252.26808166503907,136.50784301757813,256.2647133468497,130.62109375]},{"page":244,"text":"and","rect":[247.9358367919922,175.52061462402345,260.1790060439579,169.65341186523438]},{"page":244,"text":"O","rect":[262.5506286621094,175.4190216064453,268.6256768856665,169.50100708007813]},{"page":244,"text":"z","rect":[268.6112976074219,176.84457397460938,271.05437376143427,174.39520263671876]},{"page":244,"text":"(a)","rect":[273.4261779785156,176.9091033935547,283.32391446692636,169.70420837402345]},{"page":244,"text":"and","rect":[285.7175598144531,176.0,297.9607290664188,169.65341186523438]},{"page":244,"text":"W","rect":[318.6218566894531,129.18988037109376,324.76068295271429,123.73504638671875]},{"page":244,"text":"y","rect":[324.76068115234377,132.46705627441407,327.75815491370175,128.052001953125]},{"page":244,"text":"x","rect":[304.14593505859377,153.5322265625,308.1425667404044,149.34912109375]},{"page":244,"text":"layer distortion","rect":[300.3323669433594,177.24777221679688,352.21225494532509,169.65341186523438]},{"page":244,"text":"u","rect":[354.5982971191406,175.50369262695313,358.82883209376259,171.65147399902345]},{"page":244,"text":"¼","rect":[361.1691589355469,174.30145263671876,367.6841827964647,172.32032775878907]},{"page":244,"text":"u(x)","rect":[370.06256103515627,176.9091033935547,383.7559823380201,169.70420837402345]},{"page":244,"text":"energy of the smectic C phase may be analysed in terms of four variables, the","rect":[53.812843322753909,202.11187744140626,385.20740545465318,193.1773223876953]},{"page":244,"text":"compressibility g ¼ ∂u/∂z (u ¼ uz) as in smectics A, and three axial vectors Oi(r)","rect":[53.812843322753909,214.0714111328125,385.1604245742954,204.13084411621095]},{"page":244,"text":"shown in Fig. 8.32a; the x-axis coincides with the c-director. Then, according to","rect":[53.814598083496097,226.03106689453126,385.25393000653755,217.0965118408203]},{"page":244,"text":"Fig. 8.32b, the derivatives of layer distortion u ¼ u(x) are given by:","rect":[53.813594818115237,237.9906005859375,318.9897905118186,229.05604553222657]},{"page":244,"text":"\u0003 @@ux ¼ Oy and \u0003 @@Oxy ¼ @@2xu2","rect":[158.66427612304688,274.37469482421877,280.83901301435005,252.22122192382813]},{"page":244,"text":"The sign","rect":[65.76496887207031,297.90234375,101.72156611493597,288.9677734375]},{"page":244,"text":"analogy","rect":[53.81294250488281,309.8619079589844,85.41756526044378,300.9273681640625]},{"page":244,"text":"minus","rect":[105.5867919921875,296.0,129.94474424468235,288.9677734375]},{"page":244,"text":"takes","rect":[133.85276794433595,296.0,154.3584329043503,288.9677734375]},{"page":244,"text":"into account the sense of rotation","rect":[158.209716796875,296.0,297.31682673505318,288.9677734375]},{"page":244,"text":"@@uy ¼ Ox and @@Oyx ¼ @@2yu2","rect":[168.57601928710938,348.15838623046877,270.41628352216255,324.0924987792969]},{"page":244,"text":"about","rect":[301.2358093261719,296.0,323.41377122470927,288.9677734375]},{"page":244,"text":"the","rect":[327.2352294921875,296.0,339.45899236871568,288.9677734375]},{"page":244,"text":"y-axis.","rect":[343.3251647949219,297.8824157714844,369.6600307991672,288.9677734375]},{"page":244,"text":"By","rect":[373.57403564453127,297.90234375,385.1806573014594,289.1669921875]},{"page":244,"text":"and the rotation through angle Oz about the z-axis does not create distortion u.","rect":[53.812843322753909,371.64410400390627,369.83172269369848,362.5302734375]},{"page":244,"text":"The free energy density of distortion must be a quadratic function of rV. It","rect":[65.76544952392578,383.6036682128906,385.18012864658427,374.3005676269531]},{"page":244,"text":"consists of three groups of terms describing (1) the nematic-like distortion of the","rect":[53.81241989135742,395.5632019042969,385.1721881694969,386.628662109375]},{"page":244,"text":"c-director (gc), (2) distortion of the smectic layers (gl) and (3) the cross terms (gcl):","rect":[53.81241989135742,407.5628967285156,385.1812577969749,398.58819580078127]},{"page":244,"text":"g ¼ gc þ gl þ gcl","rect":[184.94735717773438,431.4255065917969,253.51838188927074,423.4372253417969]},{"page":244,"text":"Totally we obtain 4 + 4 + 2 ¼ 10 elastic moduli [9]. When the interlayer distance","rect":[65.76496887207031,455.3049621582031,385.12586248590318,446.37042236328127]},{"page":244,"text":"is fixed, only four nematic-like moduli are left (with dimension [energy/length]).","rect":[53.81393051147461,467.2645263671875,379.57049222494848,458.3299560546875]},{"page":244,"text":"8.5.4.2 Defects in Smectics C","rect":[53.81393051147461,507.3514099121094,183.31124639466135,499.8212890625]},{"page":244,"text":"Like in the nematic phase, the textures of SmC reveal blurred Schlieren patterns","rect":[53.81393051147461,533.0135498046875,385.11298002349096,524.0790405273438]},{"page":244,"text":"with linear singularities of strength s ¼ 1. The singularities of s ¼ 1/2 are not","rect":[53.81393051147461,544.97314453125,385.136854720803,536.0386352539063]},{"page":244,"text":"observed due to the reduced symmetry (C2h) of the SmC phase. Chiral smectics C*","rect":[53.81394958496094,556.9337158203125,385.1787041764594,547.9981689453125]},{"page":244,"text":"are periodic structures and the helical pitch can be measured under a microscope","rect":[53.81399154663086,568.83642578125,385.0742572612938,559.9019165039063]},{"page":244,"text":"either from the Grandjean lines or as a distance between the lines indicating","rect":[53.81399154663086,580.7960205078125,385.16680232099068,571.8615112304688]},{"page":244,"text":"periodicity, like in Fig. 8.22 for the cholesteric phase. On the other hand, like in","rect":[53.81399154663086,592.7555541992188,385.14287653974068,583.8010864257813]},{"page":245,"text":"230","rect":[53.812843322753909,42.55655288696289,66.50444931178018,36.73167419433594]},{"page":245,"text":"8 Elasticity and Defects","rect":[303.50408935546877,44.275230407714847,385.17355807303707,36.68087387084961]},{"page":245,"text":"Fig. 8.33 Smectic C phase: the uniform structure (a) and the structure with chevrons anda","rect":[53.812843322753909,164.9591064453125,385.1737303473902,157.00917053222657]},{"page":245,"text":"disclination between them (b)","rect":[53.812843322753909,174.4719696044922,156.2129372940748,167.21627807617188]},{"page":245,"text":"Fig. 8.34 A typical","rect":[53.812843322753909,295.32501220703127,122.51335624899927,287.37506103515627]},{"page":245,"text":"appearance of the zigzag","rect":[53.812843322753909,305.2332458496094,138.81697601466105,297.6388854980469]},{"page":245,"text":"defect in the chiral SmC*","rect":[53.812843322753909,313.4905090332031,141.62860626368448,307.56402587890627]},{"page":245,"text":"phase","rect":[53.812843322753909,325.12841796875,73.09562060617924,317.5340576171875]},{"page":245,"text":"SmA phase, the stepped drops as well as polygonal and fan-shape textures are also","rect":[53.812843322753909,367.10968017578127,385.1536492448188,358.17510986328127]},{"page":245,"text":"observed.","rect":[53.812843322753909,377.0372619628906,92.35056729819064,370.1346435546875]},{"page":245,"text":"Normally, the smectic C phase should form the lowest energy uniform texture","rect":[65.76486206054688,391.0287780761719,385.1576618023094,382.09423828125]},{"page":245,"text":"shown in Fig. 8.33a. However, more often we see so-called chevrons, see","rect":[53.812843322753909,402.9883117675781,385.11292303277818,394.05377197265627]},{"page":245,"text":"Fig. 8.33b, which usually form on cooling from the smectic A phase. In both","rect":[53.812835693359378,414.9478454589844,385.1258172135688,406.0133056640625]},{"page":245,"text":"cases, at the boundaries, molecules are aligned parallel to the surface without any","rect":[53.812835693359378,426.8506164550781,385.14077082685005,417.91607666015627]},{"page":245,"text":"tilt. The apex of a particular chevron can be oriented either to the left or to the right.","rect":[53.812835693359378,438.8101806640625,385.1994900276828,429.8756103515625]},{"page":245,"text":"The areas of the left and right chevrons are separated by a disclination line having a","rect":[53.812835693359378,450.76971435546877,385.1597064800438,441.83514404296877]},{"page":245,"text":"form of a zigzag. In the same figure, the core of such a line is pictured by the point.","rect":[53.812835693359378,462.729248046875,385.1407436897922,453.794677734375]},{"page":245,"text":"When observed from the top the zigzag defects are seen. One of such a zigzag is","rect":[53.812835693359378,474.6888122558594,385.18552030669408,465.7343444824219]},{"page":245,"text":"demonstrated by photo, Fig. 8.34. Such zigzag defects play the detrimental role in","rect":[53.812835693359378,486.6483459472656,385.13970271161568,477.71380615234377]},{"page":245,"text":"the displays based on SmC* ferroelectric liquid crystals.","rect":[53.812843322753909,498.6078796386719,281.3252834847141,489.67333984375]},{"page":245,"text":"Upon transition from the SmA to the SmC phase, due to appearance of the","rect":[65.76486206054688,510.5674133300781,385.1726459331688,501.63287353515627]},{"page":245,"text":"molecular tilt, radial hedgehogs discussed in Section 8.5.3 transform into other","rect":[53.812843322753909,522.4701538085938,385.13872657624855,513.4758911132813]},{"page":245,"text":"defects, called monopoles. Their characteristic feature is a disclination line going","rect":[53.812843322753909,534.4296875,385.11687556317818,525.4951782226563]},{"page":245,"text":"from the centre along a radius, Fig. 8.29d. The name “monopole” was inherited","rect":[53.812843322753909,546.3892822265625,385.1745537858344,537.4547729492188]},{"page":245,"text":"from the Dirac magnetic monopole, an isolated magnetic charge in the form of the","rect":[53.812862396240237,558.3488159179688,385.1726459331688,549.414306640625]},{"page":245,"text":"hedgehog with an adjacent singularity in the field of the magnetic vector-potential","rect":[53.812862396240237,570.308349609375,385.1596513516624,561.3738403320313]},{"page":245,"text":"A(r). The mathematical treatment of the magnetic monopole (not discovered yet)","rect":[53.812862396240237,582.2679443359375,385.1437924942173,573.3334350585938]},{"page":245,"text":"and SmC monopole observed in smectic drops is very similar [15].","rect":[53.81385803222656,594.2274780273438,324.0347866585422,585.2332153320313]},{"page":246,"text":"References","rect":[53.812129974365237,42.52726364135742,91.48080904959955,36.68545150756836]},{"page":246,"text":"References","rect":[53.812843322753909,68.09864807128906,109.59614448282879,59.31352233886719]},{"page":246,"text":"231","rect":[372.49749755859377,42.56113052368164,385.1890921035282,36.73625183105469]},{"page":246,"text":"1.","rect":[58.06126022338867,94.0,64.40706131055318,87.80046081542969]},{"page":246,"text":"2.","rect":[58.06126022338867,124.0,64.40706131055318,117.67155456542969]},{"page":246,"text":"3.","rect":[58.06126022338867,144.0,64.40706131055318,137.62351989746095]},{"page":246,"text":"4.","rect":[58.06126022338867,164.0,64.40706131055318,157.51866149902345]},{"page":246,"text":"5.","rect":[58.06126022338867,184.0,64.40706131055318,177.36900329589845]},{"page":246,"text":"6.","rect":[58.06126022338867,193.27142333984376,64.40706131055318,187.395751953125]},{"page":246,"text":"7.","rect":[58.061302185058597,213.16659545898438,64.4071032722231,207.4602508544922]},{"page":246,"text":"8.","rect":[58.0612907409668,223.14254760742188,64.4070918281313,217.3176727294922]},{"page":246,"text":"9.","rect":[58.0612907409668,233.18626403808595,64.4070918281313,227.2936553955078]},{"page":246,"text":"10.","rect":[53.812992095947269,253.01370239257813,64.38933059766255,247.18882751464845]},{"page":246,"text":"11.","rect":[53.812992095947269,283.0,64.38933059766255,277.0599365234375]},{"page":246,"text":"12.","rect":[53.812992095947269,303.0,64.38933059766255,297.0118713378906]},{"page":246,"text":"13.","rect":[53.81215286254883,323.0,64.38849136426411,316.9071044921875]},{"page":246,"text":"14.","rect":[53.81130599975586,343.0,64.38764450147115,336.8590087890625]},{"page":246,"text":"15.","rect":[53.81130599975586,363.0,64.38764450147115,356.652587890625]},{"page":246,"text":"16.","rect":[53.81128692626953,392.4502258300781,64.38762542798482,386.5745544433594]},{"page":246,"text":"17.","rect":[53.81128692626953,403.0,64.38762542798482,396.6012878417969]},{"page":246,"text":"18.","rect":[53.81128692626953,423.0,64.38762542798482,416.4964599609375]},{"page":246,"text":"19.","rect":[53.812129974365237,442.3409729003906,64.38846847608052,436.4483337402344]},{"page":246,"text":"Dunmur, D., Toriyama, K.: Elastic Properties. In: Demus, D., Goodby, J., Gray, G.W., Spiess,","rect":[68.59698486328125,95.3440170288086,385.14502212598287,87.74966430664063]},{"page":246,"text":"H.-W., Vill, V. (eds.) Physical Properties of Liquid Crystals, pp. 151–178. Wiley-VCH,","rect":[68.59698486328125,105.3199691772461,385.1246974189516,97.67481231689453]},{"page":246,"text":"Weinheim (1999)","rect":[68.59698486328125,114.95726776123047,128.99295133216075,107.70156860351563]},{"page":246,"text":"Feynman, R.P., Leighton, R.B., Sands, M.: The Feynman Lectures on Physics, vol. 2.","rect":[68.59698486328125,125.2151107788086,385.1881739814516,117.62075805664063]},{"page":246,"text":"Addison-Westley, Reading, MA (1964)","rect":[68.59698486328125,135.19107055664063,203.70843595130138,127.59671020507813]},{"page":246,"text":"Landau, L.D., Lefshitz, E.M.: Theory of Elasticity, 3rd edn. Nauka, Moscow (1969) (in","rect":[68.59698486328125,145.16708374023438,385.1518606582157,137.57272338867188]},{"page":246,"text":"Russian) (see also Theory of Elasticity. Pergamon, London (1959))","rect":[68.59698486328125,155.14303588867188,298.5731515274732,147.4978790283203]},{"page":246,"text":"Sluckin, T.J., Dunmur, D.A., Stegemeyer, H.: Crystals that flow: Classic Papers in the History","rect":[68.59698486328125,165.06222534179688,385.1780142226688,157.46786499023438]},{"page":246,"text":"of Liquid Crystals. Taylor & Francis, London (2004) (Sect. C, pp. 139–161; 335–363)","rect":[68.59698486328125,175.0382080078125,364.20309537512949,167.39305114746095]},{"page":246,"text":"Oseen, C.W.: The theory of liquid crystals. Trans. Faraday Soc. 29, 883–899 (1933)","rect":[68.59698486328125,185.01416015625,356.76324552161386,177.04727172851563]},{"page":246,"text":"Ericksen, J.L.: Hydrostatic theory of liquid crystals. Arch. Rat. Mech. Anal. 9, 371–378","rect":[68.59698486328125,194.9901123046875,385.1747183242313,187.02322387695313]},{"page":246,"text":"(1961)","rect":[68.5970230102539,204.57066345214845,91.15423673255136,197.31497192382813]},{"page":246,"text":"Frank, F.C.: On the theory of liquid crystals. Disc. Faraday Soc. 25, 19–28 (1958)","rect":[68.59703063964844,214.88528442382813,349.5688485489576,207.24012756347657]},{"page":246,"text":"Kleman, M., Lavrentovich, O.: Soft Matter Physics. Springer, New York (2003)","rect":[68.59701538085938,224.86123657226563,342.9413155899732,217.24993896484376]},{"page":246,"text":"De Gennes, P.G., Prost, J.: The Physics of Liquid Crystals, 2nd edn. Clarendon Press, Oxford","rect":[68.59701538085938,234.83721923828126,385.1746878066532,227.22592163085938]},{"page":246,"text":"(1995)","rect":[68.59701538085938,244.41773986816407,91.15422910315682,237.1112518310547]},{"page":246,"text":"Lavrentovich, O.D., Pergamenshchik, V.M.: Patterns in thin liquid crystal films and the","rect":[68.5970230102539,254.73239135742188,385.15343615793707,247.12109375]},{"page":246,"text":"divergence (“surfacelike”) elasticity. In: Kumar, S. (ed.) Liquid Crystals in the Nineties and","rect":[68.59701538085938,264.7083435058594,385.19577545313759,257.1139831542969]},{"page":246,"text":"Beyond” S, pp. 251–298. World Scientific, Singapore (1995)","rect":[68.59701538085938,274.6275634765625,277.2910165177076,266.9823913574219]},{"page":246,"text":"Blinov, L.M., Chigrinov, V.G.: Electrooptic Effects in Liquid Crystalline Materials. Springer,","rect":[68.5970230102539,284.603515625,385.190706940436,277.0091552734375]},{"page":246,"text":"New York (1993)","rect":[68.59701538085938,294.2408142089844,129.21973508460216,286.985107421875]},{"page":246,"text":"De Gennes, P.G.: Fluctuations d’orientation et diffusion Rayleigh dans un cristal ne´matique.","rect":[68.5970230102539,304.5554504394531,385.17810318067037,296.0]},{"page":246,"text":"C.R. Acad. Sci. Paris 266, 15–17 (1968)","rect":[68.5970230102539,314.13604736328127,206.76629727942638,306.8210754394531]},{"page":246,"text":"Ornstein, L.S., Kast, W.: New arguments for the swarm theory of liquid crystals. Trans. Farad.","rect":[68.59618377685547,324.45068359375,385.1737696845766,316.8393859863281]},{"page":246,"text":"Soc. 29, 930–944 (1933)","rect":[68.59617614746094,334.0879821777344,153.2931832657545,326.4597473144531]},{"page":246,"text":"Vertogen, G., de Jeu, W.H.: Thermotropic Liquid Crystals. Fundamentals. Springer, Berlin","rect":[68.5953369140625,344.402587890625,385.16537994532509,336.8082275390625]},{"page":246,"text":"(1987)","rect":[68.59532928466797,353.983154296875,91.15254300696542,346.7782287597656]},{"page":246,"text":"Kurik, M.V., Lavrentovich, O.D.: Defects in liquid crystals: homotopic theory and experi-","rect":[68.5953369140625,364.2977600097656,385.1493234024732,356.68646240234377]},{"page":246,"text":"mental investigations. Usp. Fiz. Nauk. 154, 381–431 (1988) [Sov. Phys. Uspekhi 31, 196","rect":[68.59532928466797,374.27374267578127,385.1874136367313,366.61163330078127]},{"page":246,"text":"(1988)]","rect":[68.5953140258789,383.85430908203127,93.97006314856698,376.6493835449219]},{"page":246,"text":"Kle´man, M.: Points, Lines and Walls. Wiley, Chichester (1983)","rect":[68.5953140258789,394.1689147949219,286.69467252356699,386.26806640625]},{"page":246,"text":"De Gennes, P.G.: Conjectures sur l’etat smectique. J. Physique (Paris) 30, Colloq. C.4,","rect":[68.5953140258789,404.1448669433594,385.1764552314516,396.5505065917969]},{"page":246,"text":"C4-62–C4-68 (1969)","rect":[68.5953140258789,413.78216552734377,140.09390348059825,406.5264587402344]},{"page":246,"text":"Bartolino, R., Durand, G.: Plasticity in a smectic A liquid crystal. Phys. Rev. Lett. 39,","rect":[68.5953140258789,424.0400390625,385.2042872627016,416.0731506347656]},{"page":246,"text":"1346–1349 (1977)","rect":[68.59615325927735,433.67730712890627,131.65481656653575,426.4216003417969]},{"page":246,"text":"Bouligand, Y.: Defects and textures. In: Demus, D., Goodby, J., Gray, G.W., Spiess, H.-W.,","rect":[68.59616088867188,443.9919128417969,385.15515396192037,436.3975524902344]},{"page":246,"text":"Vill, V. (eds.) Physical Properties of Liquid Crystals, pp. 304–374. Wiley-VCH, Weinheim","rect":[68.59615325927735,453.9678649902344,385.1264061772345,446.3735046386719]},{"page":246,"text":"(1999)","rect":[68.59615325927735,463.5484619140625,91.15336698157479,456.3435363769531]},{"page":247,"text":"Chapter9","rect":[53.812843322753909,72.10812377929688,114.14115996551633,59.08384323120117]},{"page":247,"text":"Elements of Hydrodynamics","rect":[53.812843322753909,91.18268585205078,247.4788934126666,76.0426254272461]},{"page":247,"text":"9.1 Hydrodynamic Variables","rect":[53.812843322753909,212.6539764404297,210.2927768680827,200.92852783203126]},{"page":247,"text":"We shall discuss here the macroscopic dynamics of liquid crystals that is an area of","rect":[53.812843322753909,239.63455200195313,385.1487058242954,230.6999969482422]},{"page":247,"text":"hydrodynamics or macroscopic properties related to elasticity and viscosity. With","rect":[53.812843322753909,251.5941162109375,385.1287774186469,242.65956115722657]},{"page":247,"text":"respect to the molecular dynamics, which deals, for example, with NMR, molecular","rect":[53.812843322753909,263.49688720703127,385.15169654695168,254.5623321533203]},{"page":247,"text":"diffusion or dipolar relaxation of molecules, the area of hydrodynamics is a long","rect":[53.812843322753909,275.4564208984375,385.1248406510688,266.5218505859375]},{"page":247,"text":"scale, both in space and time. The molecular dynamics deals with distances of about","rect":[53.812843322753909,287.4159851074219,385.15571458408427,278.4814453125]},{"page":247,"text":"molecular size, a \u0001 10 A˚, i.e., with wavevectors about 107 cm\u00031, however, in the","rect":[53.812843322753909,297.3535461425781,385.1747516460594,287.72076416015627]},{"page":247,"text":"vicinity of phase transitions, due to critical behaviour, characteristic lengths of","rect":[53.814022064208987,311.335693359375,385.1499265274204,302.401123046875]},{"page":247,"text":"short-range correlations can be one or two orders of magnitude larger. Therefore, as","rect":[53.814022064208987,323.29522705078127,385.1379434023972,314.36065673828127]},{"page":247,"text":"a limit of the hydrodynamic approach we may safely take the range of wavevectors","rect":[53.814022064208987,335.2547912597656,385.15292753325658,326.32025146484377]},{"page":247,"text":"q \u0004 106 cm\u00031 and corresponding frequencies o \u0004 csq \u0001 105\u0005 106 ¼ 1011s\u00031 (cs is","rect":[53.814022064208987,347.2147216796875,385.1891213809128,336.02734375]},{"page":247,"text":"sound velocity).","rect":[53.81450653076172,359.11749267578127,118.48104520346408,350.18292236328127]},{"page":247,"text":"In the hydrodynamic limit one considers only those variables whose relaxation","rect":[65.76653289794922,371.0770263671875,385.16835871747505,362.1424560546875]},{"page":247,"text":"times","rect":[53.81450653076172,382.0,75.46490873442844,374.10205078125]},{"page":247,"text":"decrease","rect":[80.83421325683594,382.0,115.2061008281883,374.10205078125]},{"page":247,"text":"with","rect":[120.59829711914063,382.0,138.31679621747504,374.10205078125]},{"page":247,"text":"increasing","rect":[143.65225219726563,383.0365905761719,184.78203669599066,374.10205078125]},{"page":247,"text":"wavevector","rect":[190.10055541992188,382.0,236.19555030671729,375.1180114746094]},{"page":247,"text":"of","rect":[241.53298950195313,382.0,249.8248532852329,374.10205078125]},{"page":247,"text":"the","rect":[255.12747192382813,382.0,267.3512348003563,374.10205078125]},{"page":247,"text":"corresponding","rect":[272.6309509277344,383.0365905761719,329.79204646161568,374.10205078125]},{"page":247,"text":"visco-elastic","rect":[335.1105651855469,382.0,385.14539373590318,374.10205078125]},{"page":247,"text":"modes. For instance, a small vortex made by a spoon in a glass of tea relaxes faster","rect":[53.81450653076172,394.9961242675781,385.13643775788918,386.06158447265627]},{"page":247,"text":"than a whirl in a river, or, after a tempest, short waves at the sea surface disappear","rect":[53.81450653076172,406.9556884765625,385.14742408601418,398.0211181640625]},{"page":247,"text":"faster then waves with a large period. The relaxation of cyclones in atmosphere","rect":[53.81450653076172,418.9151916503906,385.16333807184068,409.98065185546877]},{"page":247,"text":"takes days or weeks. As a rule, the hydrodynamic relaxation times follow the law","rect":[53.81450653076172,430.8747253417969,385.13448858216136,421.940185546875]},{"page":247,"text":"t~Aq\u00032. The strings of a guitar also obey the same law.","rect":[53.81450653076172,442.8352355957031,278.1285671761203,431.72760009765627]},{"page":247,"text":"For the isotropic liquid, one introduces five variables related to the corresponding","rect":[65.76544189453125,454.7380065917969,385.17226496747505,445.803466796875]},{"page":247,"text":"conservation laws. The variables are density of mass r, three components of the","rect":[53.813419342041019,466.6975402832031,385.1741412944969,457.76300048828127]},{"page":247,"text":"vector of linear momentum density mv, and density of energy E. When electric","rect":[53.81339645385742,478.6570739746094,385.15222967340318,469.7225341796875]},{"page":247,"text":"charges enter the problem, the conservation of charge must be taken into account.","rect":[53.81438064575195,490.61663818359377,385.1044582894016,481.68206787109377]},{"page":247,"text":"Then we are in the realm of electrohydrodynamics.","rect":[53.81438064575195,502.5562438964844,260.3383450081516,493.6217041015625]},{"page":247,"text":"For nematic liquid crystals, the symmetry is reduced and we need additional","rect":[65.76637268066406,514.5357055664063,385.1701799161155,505.6011962890625]},{"page":247,"text":"variables. The nematic is degenerate in the sense that all equilibrium orientations of","rect":[53.81435012817383,526.4952392578125,385.14727149812355,517.5607299804688]},{"page":247,"text":"the director are equivalent. According to the Goldstone theorem the parameter of","rect":[53.81435012817383,538.4547729492188,385.1502622207798,529.520263671875]},{"page":247,"text":"degeneracy is also a hydrodynamic variable; for a long distance process q!0 and","rect":[53.81435012817383,550.3575439453125,385.1462029557563,541.4230346679688]},{"page":247,"text":"the relaxation time should diverge, t!1. In nematics, this parameter is the director","rect":[53.81430435180664,562.3170776367188,385.10833106843605,553.382568359375]},{"page":247,"text":"n(r), the orientational part of the order parameter tensor. For a finite distortion of","rect":[53.81430435180664,574.276611328125,385.14922462312355,565.1727294921875]},{"page":247,"text":"the director over a large distance (L!1), the distortion wavevector q!0 and the","rect":[53.81430435180664,586.2361450195313,385.17603338434068,577.3016357421875]},{"page":247,"text":"L.M. Blinov, Structure and Properties of Liquid Crystals,","rect":[53.812843322753909,623.7623291015625,251.63688919141254,616.1510620117188]},{"page":247,"text":"DOI 10.1007/978-90-481-8829-1_9, # Springer ScienceþBusiness Media B.V. 2011","rect":[53.812843322753909,633.698486328125,347.38995880274697,625.4920043945313]},{"page":247,"text":"233","rect":[372.4981994628906,622.0606079101563,385.18979400782509,616.2357177734375]},{"page":248,"text":"234","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.73106384277344]},{"page":248,"text":"9 Elements of Hydrodynamics","rect":[281.0162658691406,44.274620056152347,385.17368014334957,36.68026351928711]},{"page":248,"text":"orientational relaxation requires infinite time. By the way, the magnitude S of the","rect":[53.812843322753909,68.2883529663086,385.17160833551255,59.35380554199219]},{"page":248,"text":"order parameter tensor is not a hydrodynamic variable. The director field n(r) is","rect":[53.812843322753909,80.24788665771485,385.18744291411596,71.14401245117188]},{"page":248,"text":"indeed an independent variable, because it may rotate even in the immobile nematic","rect":[53.812843322753909,92.20748138427735,385.10092962457505,83.27293395996094]},{"page":248,"text":"phase, for example, in the case of the pure twist distortion induced by a magnetic","rect":[53.812843322753909,104.11019134521485,385.17856634332505,95.17564392089844]},{"page":248,"text":"field. On the other hand a flow of the nematic can influence the vector field n(r) and","rect":[53.812843322753909,115.67130279541016,385.14568415692818,106.96585083007813]},{"page":248,"text":"vice versa. Thus, two variables v(r) and n(r) are coupled. Together with two","rect":[53.81385040283203,128.02932739257813,385.1526726823188,118.92544555664063]},{"page":248,"text":"components of the director (due to the constraint n2 ¼ 1) the number of hydrody-","rect":[53.812862396240237,139.98959350585938,385.15032325593605,128.93870544433595]},{"page":248,"text":"namic variables for a nematic becomes seven [1,2].","rect":[53.813472747802737,151.34153747558595,261.2731594612766,143.0145721435547]},{"page":248,"text":"For discussion of dynamics of lamellar smectic phases it is important to include","rect":[65.76648712158203,163.90869140625,385.0995258159813,154.95420837402345]},{"page":248,"text":"another variable, the layer displacement u (r) [3] or, more generally, the phase of","rect":[53.814476013183597,175.86822509765626,385.15230689851418,166.9336700439453]},{"page":248,"text":"the density wave [4]. This variable is also hydrodynamic: for a weak compression","rect":[53.81548309326172,187.8277587890625,385.17424861005318,178.89320373535157]},{"page":248,"text":"or dilatation of a very thick stack of smectic layers (L!1) the relaxation would","rect":[53.81548309326172,199.73052978515626,385.1393975358344,190.7959747314453]},{"page":248,"text":"require infinite time. On the other hand, the director in the smectic A phase is no","rect":[53.81548309326172,211.69009399414063,385.1713494401313,202.73561096191407]},{"page":248,"text":"longer independent variable because it must always be perpendicular to the smectic","rect":[53.81548309326172,223.64959716796876,385.1821979351219,214.7150421142578]},{"page":248,"text":"layers. Therefore, total number of hydrodynamic variables for a SmA is six. For the","rect":[53.81548309326172,235.60916137695313,385.17527044488755,226.6746063232422]},{"page":248,"text":"smectic C phase, the director acquires a degree of freedom for rotation about the","rect":[53.81548309326172,247.56869506835938,385.1732562847313,238.63414001464845]},{"page":248,"text":"normal to the layers and the number of variables again becomes seven.","rect":[53.81548309326172,259.52825927734377,340.4042629769016,250.5937042236328]},{"page":248,"text":"Why we are interested in hydrodynamics? Because we are interested in variety","rect":[65.76750946044922,271.48779296875,385.17632380536568,262.55322265625]},{"page":248,"text":"of flow phenomena in different geometry, the variety of viscosities of liquid crystals","rect":[53.81548309326172,283.44732666015627,385.10556425200658,274.51275634765627]},{"page":248,"text":"in different regimes, enormous viscosity of helical and layered structures, under-","rect":[53.81548309326172,295.3500671386719,385.1384519180454,286.41552734375]},{"page":248,"text":"standing of thermal convection, flow instabilities, etc. Moreover, in an external","rect":[53.81548309326172,307.30963134765627,385.107557845803,298.37506103515627]},{"page":248,"text":"electric field, the electrohydrodynamic instabilities arise which need a background","rect":[53.81548309326172,319.2691650390625,385.1544121842719,310.3345947265625]},{"page":248,"text":"for their interpretation. At first, however, we recall hydrodynamics of an isotropic","rect":[53.81548309326172,331.2287292480469,385.17627752496568,322.294189453125]},{"page":248,"text":"liquid.","rect":[53.81548309326172,343.18829345703127,79.69842191244845,334.25372314453127]},{"page":248,"text":"9.2 Hydrodynamics of an Isotropic Liquid","rect":[53.812843322753909,393.9192810058594,278.53475922440648,382.1938171386719]},{"page":248,"text":"Our task is to derive the equation for motion of the isotropic liquid in order to","rect":[53.812843322753909,420.89971923828127,385.1387566666938,411.94525146484377]},{"page":248,"text":"prepare a soil for discussion the dynamic properties of nematics. In this Section, we","rect":[53.812843322753909,432.8592834472656,385.16663397027818,423.92474365234377]},{"page":248,"text":"follow the approach [5] using two conservation laws.","rect":[53.812843322753909,444.7620544433594,268.3658718636203,435.7677307128906]},{"page":248,"text":"9.2.1 Conservation of Mass Density","rect":[53.812843322753909,494.929443359375,238.30985816450326,484.2916564941406]},{"page":248,"text":"Consider conservation of mass density, r(x, y, z, t). The mass continuity equation","rect":[53.812843322753909,522.4718017578125,385.17556086591255,513.5372924804688]},{"page":248,"text":"comes from consideration of the balance of the mass density in the volume V and its","rect":[53.81185531616211,534.431396484375,385.1506692324753,525.4968872070313]},{"page":248,"text":"flux through the surface surrounding the volume with subsequent application of the","rect":[53.81185531616211,546.3909301757813,385.17163885309068,537.4564208984375]},{"page":248,"text":"Gauss theorem:","rect":[53.81185531616211,556.3284912109375,116.50048692295144,549.4159545898438]},{"page":248,"text":"@@rt ¼ \u0003rrv þ sources","rect":[173.900634765625,593.4873046875,266.66391386138158,572.5533447265625]},{"page":248,"text":"(9.1)","rect":[366.09454345703127,588.21923828125,385.16680274812355,579.7428588867188]},{"page":249,"text":"9.2 Hydrodynamics of an Isotropic Liquid","rect":[53.812843322753909,44.274620056152347,198.24075836329386,36.68026351928711]},{"page":249,"text":"235","rect":[372.49737548828127,42.55594253540039,385.1889700332157,36.62946701049805]},{"page":249,"text":"Here the velocity v(x, y, z, t) of a liquid is “measured” in a particular fixed point","rect":[65.76496887207031,68.2883529663086,385.14680345127177,59.35380554199219]},{"page":249,"text":"in space x, y, z. It is not the velocity of a small unit volume of moving liquid.","rect":[53.81294250488281,80.24788665771485,385.17864652182348,71.31333923339844]},{"page":249,"text":"“Sources” mean the presence of sources and sinks in the volume discussed. If we","rect":[53.81294631958008,92.20748138427735,385.1677631206688,83.27293395996094]},{"page":249,"text":"are not interested in propagation of sound i.e. ignore a local compression and","rect":[53.81294631958008,104.11019134521485,385.1429070573188,95.17564392089844]},{"page":249,"text":"dilatation we may put dr ¼ 0 and r ¼ const. Then, it is the case of incompressible","rect":[53.81294631958008,116.0697250366211,385.1398395366844,106.83636474609375]},{"page":249,"text":"liquid:","rect":[53.81393051147461,127.91975402832031,80.00544602939675,119.0748519897461]},{"page":249,"text":"@r=@t ¼ \u0003rrv þ sources","rect":[166.31044006347657,152.27708435058595,272.64532865630346,142.3464813232422]},{"page":249,"text":"and, in the stationary regime, divv ¼ sources.","rect":[53.813968658447269,175.86749267578126,237.9180874397922,166.9329376220703]},{"page":249,"text":"For the subsequent discussion let us write down the continuity equation in the","rect":[65.76598358154297,187.8270263671875,385.1717914409813,178.89247131347657]},{"page":249,"text":"tensor form:","rect":[53.81396484375,197.7078094482422,103.18087632724832,190.7952423095703]},{"page":249,"text":"@r","rect":[169.53903198242188,223.60757446289063,180.69868821452213,214.44393920898438]},{"page":249,"text":"@ðrvjÞ","rect":[203.29945373535157,224.5245361328125,229.11020292388157,214.03871154785157]},{"page":249,"text":"¼\u0003","rect":[183.47293090820313,227.04193115234376,201.6174700567475,224.711181640625]},{"page":249,"text":"@t","rect":[170.842041015625,235.37786865234376,179.35986192295145,228.04696655273438]},{"page":249,"text":"(9.2)","rect":[366.0979309082031,230.11048889160157,385.17022071687355,221.63412475585938]},{"page":249,"text":"9.2.2 Conservation of Momentum Density","rect":[53.812843322753909,299.8907165527344,269.6906870219251,289.2529296875]},{"page":249,"text":"9.2.2.1 Ideal Liquid","rect":[53.812843322753909,327.3634033203125,144.31952256510807,318.0603332519531]},{"page":249,"text":"Consider now the conservation of momentum density (or linear momentum vector","rect":[53.812843322753909,351.35223388671877,385.11187110749855,342.41766357421877]},{"page":249,"text":"rv). First we write this law for the ideal (without viscosity) liquid in two different","rect":[53.812843322753909,363.2550048828125,385.1716447598655,354.3204345703125]},{"page":249,"text":"presentations. The Lagrange form of the equation of motion of the element of liquid","rect":[53.812843322753909,375.2543640136719,385.1148308854438,366.2401428222656]},{"page":249,"text":"coincide with the Newton form (mdv/dt¼F):","rect":[53.812843322753909,386.7756652832031,233.76806504795145,378.2196350097656]},{"page":249,"text":"dv","rect":[186.36251831054688,408.39495849609377,196.66413203290473,401.4027099609375]},{"page":249,"text":"r ¼ \u0003grad p þ f","rect":[179.22494506835938,417.1581726074219,259.0670865616001,408.0941162109375]},{"page":249,"text":"dt","rect":[187.60879516601563,421.99798583984377,195.38304002353739,415.0057373046875]},{"page":249,"text":"(9.3)","rect":[366.0967712402344,416.7597351074219,385.16903053132668,408.28338623046877]},{"page":249,"text":"Here, p is scalar pressure, vector f is the volume force in (dyn/cm3) coming, e.g.,","rect":[65.76529693603516,445.499267578125,385.1424831917453,434.4482116699219]},{"page":249,"text":"from the gravity, electric or magnetic field (e.g., for gravity force f ¼ rg) and vector","rect":[53.81362533569336,457.53851318359377,385.1146176895298,448.394775390625]},{"page":249,"text":"v is velocity of moving liquid particle as if the measuring device is placed on the","rect":[53.814598083496097,469.45819091796877,385.17432439996568,460.4439697265625]},{"page":249,"text":"particle. However, in hydrodynamics, the velocity is usually considered as a vector","rect":[53.814598083496097,481.3211364746094,385.10565580474096,472.3865966796875]},{"page":249,"text":"field defined in each point of the space. The change of velocity dv within time","rect":[53.814598083496097,493.2806701660156,385.1623920269188,484.3062744140625]},{"page":249,"text":"interval dt consists of two terms: one of them, namely, (∂v/∂t)dt is taken at fixed","rect":[53.815574645996097,505.2402038574219,385.1474236588813,495.2996520996094]},{"page":249,"text":"coordinates x, y, z of the reference point and the other part is delivered by different","rect":[53.81456756591797,517.19970703125,385.1742997891624,508.26519775390627]},{"page":249,"text":"particles arriving from the neighbor points located at a distance dr ¼ dxi þ dyj þ","rect":[53.814537048339847,529.2489624023438,385.1483843633881,520.1450805664063]},{"page":249,"text":"dzk from the reference point (the so-called convective term):","rect":[53.814537048339847,541.1187744140625,299.628798080178,532.164306640625]},{"page":249,"text":"dxð@v=@xÞ þ dyð@v=@yÞ þ dzð@v=@zÞ ¼ ðdrrÞv:","rect":[120.25887298583985,565.3765869140625,318.7129663197412,555.4260864257813]},{"page":249,"text":"Therefore, the total velocity in the reference point satisfies the equation","rect":[65.76851654052735,588.9002075195313,354.81887141278755,579.9258422851563]},{"page":250,"text":"236","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.68026351928711]},{"page":250,"text":"9 Elements of Hydrodynamics","rect":[281.0162658691406,44.274620056152347,385.17368014334957,36.68026351928711]},{"page":250,"text":"dv @v","rect":[182.05755615234376,66.7528076171875,216.26303950360785,59.421897888183597]},{"page":250,"text":"dt ¼ @t þ ðvrÞv;","rect":[183.30381774902345,80.35577392578125,256.9110559193506,65.87430572509766]},{"page":250,"text":"(9.4)","rect":[366.0968017578125,75.0877456665039,385.1690610489048,66.61138153076172]},{"page":250,"text":"where (vrv) is called convection term. Correspondingly the equation of the motion","rect":[53.81332015991211,103.94026947021485,385.17607966474068,94.99575805664063]},{"page":250,"text":"of an ideal liquid in the Euler form is given by:","rect":[53.81332015991211,115.8998031616211,245.0516038663108,106.9254150390625]},{"page":250,"text":"@v","rect":[164.44091796875,137.43377685546876,175.08297053388129,130.10287475585938]},{"page":250,"text":"r @t þ rðvrÞv ¼ \u0003grad p þf","rect":[157.24652099609376,151.0367431640625,280.99178443757668,136.55528259277345]},{"page":250,"text":"(9.5)","rect":[366.0992736816406,145.7687225341797,385.1715329727329,137.17282104492188]},{"page":250,"text":"The same equation can be written in the tensor form:","rect":[65.76785278320313,174.62124633789063,279.857069564553,165.6866912841797]},{"page":250,"text":"@vi","rect":[171.91810607910157,197.60191345214845,184.0155437545051,188.82427978515626]},{"page":250,"text":"r @t ¼ \u0003rvj @xj \u0003 @xi þ fi","rect":[164.72662353515626,212.56488037109376,273.7414348677864,195.914794921875]},{"page":250,"text":"(9.6)","rect":[366.09765625,204.4907684326172,385.1699155410923,195.9546356201172]},{"page":250,"text":"Now we define the rate of the momentum change:","rect":[65.76622772216797,236.0634765625,268.022841048928,227.12892150878907]},{"page":250,"text":"@rv","rect":[181.7176513671875,259.429931640625,197.91016474774848,250.26629638671876]},{"page":250,"text":"@v @r","rect":[218.31039428710938,259.429931640625,252.29722947428776,250.26629638671876]},{"page":250,"text":"@t ¼ r @t þ @t v","rect":[185.5689239501953,271.2001953125,258.91577998212349,258.382080078125]},{"page":250,"text":"or in tensor notations","rect":[53.812252044677737,292.7228088378906,139.27710355864719,285.85009765625]},{"page":250,"text":"@","rect":[177.97911071777345,316.3184509277344,183.15529997535374,308.987548828125]},{"page":250,"text":"@vi @r","rect":[219.55665588378907,318.15118408203127,255.4702641422565,308.987548828125]},{"page":250,"text":"@trvi ¼ r @t þ @t vi","rect":[176.56263732910157,329.921875,263.601939018177,317.103759765625]},{"page":250,"text":"and rewrite this rate on account of Eqs. (9.6) and (9.2):","rect":[53.812843322753909,353.5062561035156,276.36408860752177,344.57171630859377]},{"page":250,"text":"@@rtvi ¼ \u0003rvj @@vxij \u0003 vi @ð@rxvjjÞ \u0003 @@xpi þ fi","rect":[143.1422576904297,391.9029541015625,297.022593315052,367.813720703125]},{"page":250,"text":"Using identity@p=@xi \u0006 dij@p=@xj with Kronecker symbol dij we","rect":[65.76622772216797,416.3748779296875,331.6417926616844,405.79962158203127]},{"page":250,"text":"the result in the compact form of the law of momentum conservation","rect":[53.81380844116211,427.3612365722656,332.7964715104438,418.3868408203125]},{"page":250,"text":"liquid:","rect":[53.81380844116211,439.3207702636719,80.00532395908425,430.38623046875]},{"page":250,"text":"@@trvi ¼ \u0003@@Pxjij þ fi","rect":[179.90469360351563,477.3209533691406,260.259898002552,453.580322265625]},{"page":250,"text":"where a symmetric second rank tensor","rect":[53.81417465209961,500.8196105957031,208.78431068269385,491.88507080078127]},{"page":250,"text":"(9.7)","rect":[366.09765625,383.82879638671877,385.1699155410923,375.3524475097656]},{"page":250,"text":"may present","rect":[335.3955078125,415.4017028808594,385.1745439297874,407.4831237792969]},{"page":250,"text":"for the ideal","rect":[335.7926940917969,425.32928466796877,385.11482102939677,418.42669677734377]},{"page":250,"text":"(9.8)","rect":[366.09765625,469.24688720703127,385.1699155410923,460.7705383300781]},{"page":250,"text":"Pij ¼ pdij þ rvivj;","rect":[181.4351348876953,525.5562744140625,255.8914331166162,515.5057373046875]},{"page":250,"text":"(9.9)","rect":[366.09747314453127,524.3406982421875,385.16973243562355,515.8643188476563]},{"page":250,"text":"is called tensor of momentum density flux (in units dyn/cm2). It includes only the","rect":[53.81400680541992,549.1117553710938,385.1748737163719,538.0608520507813]},{"page":250,"text":"reversible part of the momentum transfer, because there is no energy dissipation by","rect":[53.81412887573242,561.0712890625,385.1679619889594,552.1367797851563]},{"page":250,"text":"the flow of the ideal liquid. Note that the form of Eq. (9.8) is very similar to the form","rect":[53.81412887573242,573.0308227539063,385.1380686628874,564.0963134765625]},{"page":250,"text":"of density conservation law (9.2).","rect":[53.81412887573242,584.9903564453125,189.27698941733127,576.0558471679688]},{"page":251,"text":"9.2 Hydrodynamics of an Isotropic Liquid","rect":[53.8132209777832,44.275657653808597,198.2411398330204,36.68130111694336]},{"page":251,"text":"237","rect":[372.4977722167969,42.55698013305664,385.1893667617313,36.73210144042969]},{"page":251,"text":"9.2.2.2 Viscous Liquid","rect":[53.812843322753909,68.2186279296875,155.39853257487369,58.91554641723633]},{"page":251,"text":"For viscous liquids the law for the mass conservation remains unchanged. As to the","rect":[53.812843322753909,92.20748138427735,385.1756061382469,83.27293395996094]},{"page":251,"text":"momentum density conservation, it keeps the same form (9.8) but tensor Pij should","rect":[53.812843322753909,105.08378601074219,385.15837946942818,95.15572357177735]},{"page":251,"text":"be changed to take the dissipation into account. Now we write","rect":[53.81456756591797,116.0702133178711,305.3157123394188,107.13566589355469]},{"page":251,"text":"Pij ¼ pdij þ rvivj \u0003 s0ij ¼ \u0003sij þ rvivj","rect":[140.31166076660157,142.62078857421876,298.15555840294265,130.32997131347657]},{"page":251,"text":"(9.10)","rect":[361.0561828613281,140.1013641357422,385.10555396882668,131.625]},{"page":251,"text":"The new tensor","rect":[65.76653289794922,164.05731201171876,127.89573036042822,157.18458557128907]},{"page":251,"text":"sij ¼ \u0003pdij þ s0ij","rect":[184.72132873535157,192.61349487304688,253.7777965621223,180.37916564941407]},{"page":251,"text":"(9.11)","rect":[361.0561828613281,190.0938262939453,385.10555396882668,181.61746215820313]},{"page":251,"text":"called stress tensor, includes the pressure term -pdij and the term si0j called viscous","rect":[53.81450653076172,217.0851593017578,385.10272611724096,206.5091094970703]},{"page":251,"text":"stress tensor. The latter describes the irreversible transfer of momentum in a","rect":[53.81368637084961,226.19586181640626,385.15952337457505,219.1936492919922]},{"page":251,"text":"moving liquid.","rect":[53.81368637084961,240.08773803710938,113.01734586264377,231.15318298339845]},{"page":251,"text":"Now let us try to imagine the form of tensor si0j. In Fig. 9.1, the upper part of the","rect":[65.76570892333985,252.96409606933595,385.1752094097313,242.38804626464845]},{"page":251,"text":"liquid is moving, the lower part is immobile. The components of s’ij are the","rect":[53.813472747802737,264.9237365722656,385.11887396051255,255.0159149169922]},{"page":251,"text":"tangential shear forces acting on a unit area having its normal along the xj-axis","rect":[53.81393051147461,276.82672119140627,385.12863554106908,266.9755859375]},{"page":251,"text":"while the liquid moves along the xi-axis. This force is caused by the gradients of","rect":[53.81368637084961,287.8699035644531,385.1506589492954,278.93524169921877]},{"page":251,"text":"momentum ∂rvi/∂xj (or just velocity gradients ∂vi/∂xj in case of incompressible","rect":[53.81382369995117,300.7592468261719,385.13840521051255,289.8888854980469]},{"page":251,"text":"liquid with constant r) otherwise there is no friction force. The momentum is","rect":[53.81351852416992,311.7890930175781,385.1305276309128,302.85455322265627]},{"page":251,"text":"transferred from upper to lower layers (momentum flux). The correct form of this","rect":[53.81351852416992,323.7486572265625,385.1434365664597,314.8140869140625]},{"page":251,"text":"tensor should exclude the rotation of a liquid as a whole because such a rotation","rect":[53.81351852416992,335.70819091796877,385.1155938248969,326.77362060546877]},{"page":251,"text":"does not result in friction at all. Therefore we write a symmetric shear rate tensor for","rect":[53.81351852416992,347.5910339355469,385.18023048249855,338.6763916015625]},{"page":251,"text":"the incompressible liquid as we did earlier Eq. (8.6) when we discussed the shear","rect":[53.81351852416992,359.5705261230469,385.13543067781105,350.635986328125]},{"page":251,"text":"distortion of the solid (soft) matter [6]:","rect":[53.81351852416992,371.1316223144531,210.54455430576395,362.59552001953127]},{"page":251,"text":"Aij ¼ 12\u0001@@xvij þ @@xvij\u0003","rect":[178.31980895996095,409.984130859375,260.63067312580128,385.76837158203127]},{"page":251,"text":"(9.12)","rect":[361.05572509765627,401.9670715332031,385.1050962051548,393.49072265625]},{"page":251,"text":"Then, for not very strong gradients, there should be linear relationship between","rect":[65.76607513427735,433.5398254394531,385.1788262467719,424.60528564453127]},{"page":251,"text":"s’ij and Aij and we may write","rect":[53.81405258178711,446.415771484375,173.33335149957504,436.564697265625]},{"page":251,"text":"s0ij ¼ \u0002\u0001@@xvji þ @@xvji\u0003 where i;j ¼ x;y;z","rect":[134.3626251220703,483.95263671875,304.62030424224096,459.6802062988281]},{"page":251,"text":"(9.13)","rect":[361.0577087402344,475.8788757324219,385.1070798477329,467.40252685546877]},{"page":251,"text":"xj","rect":[239.1757354736328,508.518798828125,244.503256890082,501.0823974609375]},{"page":251,"text":"Fig. 9.1 Geometry of shear","rect":[53.812843322753909,553.506103515625,150.24957364661388,545.4714965820313]},{"page":251,"text":"of isotropic liquid and","rect":[53.812843322753909,563.414306640625,129.78225464014933,555.8199462890625]},{"page":251,"text":"illustration of the component","rect":[53.812843322753909,573.3335571289063,152.91819481101099,565.7391967773438]},{"page":251,"text":"s0ij of the viscous stress","rect":[53.812843322753909,584.1079711914063,136.1749618816308,575.1212768554688]},{"page":251,"text":"tensor","rect":[53.8132209777832,591.5325317382813,74.50899594885995,586.5542602539063]},{"page":251,"text":"σ′ij","rect":[207.14688110351563,549.759765625,214.62869939496484,542.0]},{"page":251,"text":"vi","rect":[348.4150390625,534.00927734375,353.74274358441797,527.8262939453125]},{"page":251,"text":"unit area","rect":[320.189208984375,573.862548828125,351.29178860986095,568.0077514648438]},{"page":251,"text":"v=0","rect":[307.6997375488281,588.6412963867188,324.14188626611095,582.87451171875]},{"page":251,"text":" ","rect":[318.03106689453127,589.0,319.6976623058463,588.0]},{"page":251,"text":"xi","rect":[379.3890380859375,554.7546997070313,384.7164679496523,548.572021484375]},{"page":252,"text":"238","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.73106384277344]},{"page":252,"text":"9 Elements of Hydrodynamics","rect":[281.0162658691406,44.274620056152347,385.17368014334957,36.68026351928711]},{"page":252,"text":"Here Z is shear viscosity coefficient. For the case shown in the Fig. 9.1, xi ¼ x,","rect":[65.76496887207031,68.2883529663086,385.1582302620578,59.31396484375]},{"page":252,"text":"xj ¼ z, the velocity has only component vx and the gradient of velocity has a simple","rect":[53.814414978027347,81.16453552246094,385.13477361871568,71.31346130371094]},{"page":252,"text":"form ∂vx/∂z. Then s’xz ¼ Z∂vx/∂z. The dynamic viscosity coefficient Z is","rect":[53.81386947631836,92.20772552490235,385.18735136138158,82.26704406738281]},{"page":252,"text":"measured","rect":[53.812740325927737,103.0,92.12748805097113,95.17594909667969]},{"page":252,"text":"in","rect":[97.42810821533203,103.0,105.20234767011175,95.17594909667969]},{"page":252,"text":"Poise","rect":[110.51292419433594,103.0,132.12749517633285,95.17594909667969]},{"page":252,"text":"(g\u0005s\u00031cm\u00031).","rect":[137.41915893554688,104.1104965209961,189.33364530112034,93.05973815917969]},{"page":252,"text":"Sometimes","rect":[194.6890106201172,102.08869934082031,239.13554777251438,95.17613220214844]},{"page":252,"text":"one","rect":[244.4232177734375,102.04885864257813,258.8269733257469,97.40727996826172]},{"page":252,"text":"uses","rect":[264.13555908203127,102.04885864257813,281.2269326602097,97.40727996826172]},{"page":252,"text":"the","rect":[286.62310791015627,102.04885864257813,298.8468707866844,95.17613220214844]},{"page":252,"text":"so-called","rect":[304.1265869140625,102.07874298095703,340.25042048505318,95.17613220214844]},{"page":252,"text":"kinematic","rect":[345.58984375,102.04885864257813,385.1339801616844,95.17613220214844]},{"page":252,"text":"viscosity Z/r measured in Stokes (cm2s\u00031). The SI unit for dynamic viscosity is","rect":[53.813045501708987,116.07039642333985,385.1875039492722,105.01939392089844]},{"page":252,"text":"Pa\u0005s (N\u0005m\u00032s). Numerically 1 Pa\u0005s ¼10 P.","rect":[53.8138542175293,128.02999877929688,224.4232143929172,116.97911071777344]},{"page":252,"text":"A more general form of the viscous stress tensor (for compressible liquid) has","rect":[65.76404571533203,139.98953247070313,385.13202299224096,131.0549774169922]},{"page":252,"text":"two different terms, one is the same corresponding to shear (9.13) with viscosityZ","rect":[53.81202697753906,151.94906616210938,385.1866633575752,143.01451110839845]},{"page":252,"text":"and the other is compressibility with viscosity coefficient called second viscosity z.","rect":[53.81203842163086,163.90863037109376,385.1817287972141,154.65533447265626]},{"page":252,"text":"Both the coefficients are positive scalars.","rect":[53.813045501708987,175.8681640625,219.14748807455784,166.93360900878907]},{"page":252,"text":"By defining the shape of a selected small volume in the moving liquid, e.g., a","rect":[65.76506805419922,187.82766723632813,385.15686834527818,178.8931121826172]},{"page":252,"text":"cube, we clearly see that the viscous force acting on the cube arises from the","rect":[53.813045501708987,199.73043823242188,385.11411321832505,190.79588317871095]},{"page":252,"text":"differences in stress tensors on opposite faces of the cube. Consequently, the","rect":[53.813045501708987,211.69000244140626,385.11606634332505,202.7554473876953]},{"page":252,"text":"force is determined by the spatial derivatives of tensor Pij including s’ij as seen","rect":[53.813045501708987,224.62376403808595,385.1093377213813,214.69505310058595]},{"page":252,"text":"from Eqs. 9.8 and 9.10. The viscous tensor in turn, according to Eq. (9.13), is","rect":[53.814292907714847,235.61013793945313,385.18893827544408,226.6755828857422]},{"page":252,"text":"proportional to spatial derivatives of the velocity v. Hence, in many simple cases,","rect":[53.814292907714847,247.56967163085938,385.1252712776828,238.63511657714845]},{"page":252,"text":"the viscous force is given by vector Zr2v.","rect":[53.814292907714847,259.52923583984377,226.01064725424534,248.4784698486328]},{"page":252,"text":"9.2.3 Navier-Stokes Equation","rect":[53.812843322753909,303.3779602050781,207.89303559159399,292.8238525390625]},{"page":252,"text":"In the next step, the equation","rect":[53.812843322753909,331.00396728515627,180.06508723310004,322.06939697265627]},{"page":252,"text":"viscous liquid may be cast in","rect":[53.812843322753909,342.9635314941406,179.74652949384223,334.02899169921877]},{"page":252,"text":"Reynolds number","rect":[53.812843322753909,354.9230651855469,124.84907661286963,345.988525390625]},{"page":252,"text":"of motion for an isotropic, incompressible, and","rect":[184.71768188476563,331.00396728515627,385.1427849870063,322.06939697265627]},{"page":252,"text":"different forms depending on the dimensionless","rect":[184.32049560546876,342.9635314941406,385.08893217192846,334.02899169921877]},{"page":252,"text":"Re ¼ rvl=Z","rect":[195.36868286132813,379.17083740234377,243.57689162905957,369.2402648925781]},{"page":252,"text":"(9.14)","rect":[361.0564880371094,378.4437255859375,385.1058591446079,369.9673767089844]},{"page":252,"text":"where l is a characteristic dimension of the flow structure, for instance a tube","rect":[53.81483459472656,400.7422180175781,385.10691106988755,393.7499694824219]},{"page":252,"text":"diameter. In the case of relatively low velocity of a viscous liquid in narrow","rect":[53.8148307800293,414.6640319824219,385.0979285235676,405.7294921875]},{"page":252,"text":"capillaries (Re \u0004 1), the convection term in (9.5) is disregarded, and Eq. (9.8) on","rect":[53.8148307800293,426.62353515625,385.1725701432563,417.62921142578127]},{"page":252,"text":"account of (9.10) becomes the well known Navier-Stokes equation:","rect":[53.81584167480469,438.4735412597656,325.2223344571311,429.6286315917969]},{"page":252,"text":"r@@vt ¼ \u0003r~p þ f þ \u0002r2v","rect":[167.78456115722657,473.7774963378906,271.2093133073188,452.7867736816406]},{"page":252,"text":"(9.15)","rect":[361.0560302734375,468.4532470703125,385.10540138093605,459.85736083984377]},{"page":252,"text":"~","rect":[81.34375,487.5873107910156,88.39133075839777,485.6051940917969]},{"page":252,"text":"where rp is vector of pressure gradient and f is an external volume force.","rect":[53.81438446044922,497.3057861328125,352.9831814339328,488.24176025390627]},{"page":252,"text":"As a simple example, consider a spherical particle of radius Rp moving by an","rect":[65.76641082763672,510.12188720703127,385.15358820966255,500.33062744140627]},{"page":252,"text":"external force f with given velocity in a viscous liquid with Re \u0004 1. Then dv/dt ¼ 0,","rect":[53.813777923583987,521.2247314453125,385.1755337288547,512.1607055664063]},{"page":252,"text":"gradient of pressure is absent and the external force is equal to the friction force:","rect":[53.81380844116211,533.1842651367188,380.2213578946311,524.249755859375]},{"page":252,"text":"f ¼ \u0003\u0002r2v:","rect":[194.5195770263672,558.3504028320313,244.4494622914209,547.4149169921875]},{"page":252,"text":"(9.16)","rect":[361.0560607910156,557.9523315429688,385.10543189851418,549.4161987304688]},{"page":252,"text":"If the particle is moving in the z-direction with velocity v0, the velocity of the","rect":[65.76641082763672,582.2699584960938,385.1748737163719,573.3352661132813]},{"page":252,"text":"liquid vz(r) decreases from v0 to 0 as a function of the polar transverse coordinate r.","rect":[53.81411361694336,594.2296142578125,385.1828579476047,585.2949829101563]},{"page":253,"text":"9.3 Viscosity of Nematics","rect":[53.812843322753909,44.274620056152347,142.61345370291986,36.68026351928711]},{"page":253,"text":"239","rect":[372.4981689453125,42.62367248535156,385.18979400782509,36.73106384277344]},{"page":253,"text":"By integrating the right part of the","rect":[53.812843322753909,68.2883529663086,202.32370794488754,59.35380554199219]},{"page":253,"text":"relationship between the acting force","rect":[53.8138313293457,80.24788665771485,202.9328540630516,71.31333923339844]},{"page":253,"text":"last equation over r, Stokes has","rect":[206.69656372070313,68.2883529663086,340.89426054106908,59.35380554199219]},{"page":253,"text":"and the velocity of the particle:","rect":[205.7339630126953,80.24788665771485,332.0828080899436,71.31333923339844]},{"page":253,"text":"found","rect":[345.3079833984375,66.25641632080078,368.53115168622505,59.35380554199219]},{"page":253,"text":"the","rect":[372.8940734863281,66.22653198242188,385.1178363628563,59.35380554199219]},{"page":253,"text":"f ¼ 6pRpZvz","rect":[193.33006286621095,104.92186737060547,245.18057685747088,95.13580322265625]},{"page":253,"text":"(9.17)","rect":[361.0561828613281,103.7122573852539,385.10555396882668,95.23589324951172]},{"page":253,"text":"By measuring velocity of a spherical particle sinking in a liquid under gravity","rect":[65.76653289794922,128.42617797851563,385.1742791276313,119.49162292480469]},{"page":253,"text":"force the viscosity of the liquid can be found (the buoyancy effect should be taken","rect":[53.81450653076172,140.38571166992188,385.1393670182563,131.45115661621095]},{"page":253,"text":"into account). Note that in Section 7.3.3, using an electric field as an action force,","rect":[53.81450653076172,152.34530639648438,385.1244167854953,143.41075134277345]},{"page":253,"text":"the same Stokes’ law has been applied (with some precautions) to evaluation of","rect":[53.814476013183597,164.30484008789063,385.15044532624855,155.3702850341797]},{"page":253,"text":"velocity and mobility of spherical ions in isotropic liquids or nematic liquid crystals","rect":[53.814476013183597,176.20761108398438,385.1025735293503,167.27305603027345]},{"page":253,"text":"For large Reynolds numbers, Re ¼ rvl/Z>1 the flow in no longer laminar and","rect":[65.7665023803711,188.16714477539063,385.1444024186469,179.21266174316407]},{"page":253,"text":"even becomes turbulent. Then, the convective term (vr)v should be added to the","rect":[53.81450653076172,199.72825622558595,385.1752399273094,191.18215942382813]},{"page":253,"text":"left part of the Navier-Stokes equation","rect":[53.81450653076172,212.08624267578126,208.86922541669379,203.1516876220703]},{"page":253,"text":"r@@vt þ rðvrÞv ¼ \u0003r~p þ f þ Zr2v","rect":[145.74960327148438,247.28094482421876,293.2443780045844,226.34701538085938]},{"page":253,"text":"(9.18)","rect":[361.05645751953127,241.95677185058595,385.1058286270298,233.48040771484376]},{"page":253,"text":"This situation is encountered in the physics of electrohydrodynamic instabilities.","rect":[65.76680755615235,270.8092956542969,385.1427273323703,261.874755859375]},{"page":253,"text":"Resuming the discussion of isotropic liquids note that the four basic equations","rect":[65.76680755615235,282.7688293457031,385.17160429106908,273.83428955078127]},{"page":253,"text":"for conservation of mass and momentum include only four material parameters:","rect":[53.81476593017578,294.7283935546875,385.1556230313499,285.7938232421875]},{"page":253,"text":"mass density r, compressibility b, viscosity Z and second viscosity z. Other two","rect":[53.81476593017578,306.68792724609377,385.15456477216255,297.43463134765627]},{"page":253,"text":"parameters, namely, thermal conductivity k and specific heat capacity Cp (or CV)","rect":[53.81574249267578,319.577392578125,385.15975318757668,309.7029724121094]},{"page":253,"text":"would come about as soon as the energy conservation law is applied to thermal","rect":[53.81393051147461,330.55029296875,385.1806474454124,321.61572265625]},{"page":253,"text":"processes. So, the isotropic liquid is completely described by six parameters.","rect":[53.81393051147461,342.5098571777344,363.6049465706516,333.5753173828125]},{"page":253,"text":"9.3 Viscosity of Nematics","rect":[53.812843322753909,387.9677429199219,190.35411720011397,376.2422790527344]},{"page":253,"text":"9.3.1 Basic Equations","rect":[53.812843322753909,417.1929016113281,170.70336036417647,406.71051025390627]},{"page":253,"text":"Here the discussion of viscous properties of nematic liquid crystals is based on the","rect":[53.812843322753909,444.8190002441406,385.17063177301255,435.88446044921877]},{"page":253,"text":"approach developed by F. Leslie [7]. For the nematic phase we have the equation","rect":[53.812843322753909,456.7785339355469,385.1755303483344,447.843994140625]},{"page":253,"text":"for conservation of mass, the modified equations for conservation of momentum","rect":[53.8138313293457,468.73809814453127,385.0870127546843,459.80352783203127]},{"page":253,"text":"and energy E and one additional equation for conservation of the angular momen-","rect":[53.8138313293457,480.640869140625,385.0880063614048,471.706298828125]},{"page":253,"text":"tum of the director [8,9]. Totally there are seven equations: two for scalar quantities","rect":[53.8138313293457,492.6004333496094,385.1487161074753,483.6658935546875]},{"page":253,"text":"(r and E), three for momentum (mv) and two for director (due to condition n2 ¼ 1),","rect":[53.81482696533203,504.4603576660156,385.1478542854953,493.4764709472656]},{"page":253,"text":"which completely describe the hydrodynamics of nematics. In this case","rect":[53.81399154663086,516.5202026367188,341.47180975152818,507.585693359375]},{"page":253,"text":"1.","rect":[53.81399154663086,533.0,61.279648261325409,525.556396484375]},{"page":253,"text":"2.","rect":[53.81399154663086,557.0,61.279648261325409,549.4754638671875]},{"page":253,"text":"The equation for mass conservation for an incompressible nematic can still be","rect":[66.27566528320313,534.43115234375,385.14890325738755,525.4966430664063]},{"page":253,"text":"used in the form of divv ¼ 0.","rect":[66.2756576538086,544.3687133789063,184.7656216194797,537.4561767578125]},{"page":253,"text":"The dissipation related to the pure director rotation has to be taken into account","rect":[66.27566528320313,558.3502197265625,385.20661790439677,549.4157104492188]},{"page":253,"text":"when writing the conservation energy equation. In addition, the heat transfer","rect":[66.27568817138672,570.3097534179688,385.10213600007668,561.375244140625]},{"page":253,"text":"becomes anisotropic and the thermal conductivity is described by two coeffi-","rect":[66.27568817138672,582.269287109375,385.12700782624855,573.3347778320313]},{"page":253,"text":"cients k|| and k⊥.","rect":[66.27568817138672,593.73193359375,134.81161160971409,585.2943115234375]},{"page":254,"text":"240","rect":[53.81455993652344,42.55734634399414,66.50616592554971,36.73246765136719]},{"page":254,"text":"9 Elements of Hydrodynamics","rect":[281.01800537109377,44.276023864746097,385.17541964530269,36.68166732788086]},{"page":254,"text":"3. Instead of the equation for conservation of the linear momentum for the director,","rect":[53.812843322753909,68.2883529663086,385.10393949057348,59.35380554199219]},{"page":254,"text":"the conservation of the angular momentum is used. Leslie had taken into account","rect":[66.27452087402344,80.24788665771485,385.1059709317405,71.31333923339844]},{"page":254,"text":"not only the velocity gradients, but also the orientation of the director n and its","rect":[66.27452087402344,92.20748138427735,385.1516457949753,83.27293395996094]},{"page":254,"text":"relative rotation rate N.In fact, vector N islinear director velocity with respect to","rect":[66.2745361328125,104.11019134521485,385.14174738935005,95.17564392089844]},{"page":254,"text":"the liquid that may rotate itself:","rect":[66.27552032470703,116.0697250366211,193.79594285556864,107.13517761230469]},{"page":254,"text":"dn","rect":[203.4127655029297,137.29055786132813,214.17525071452213,130.2982940673828]},{"page":254,"text":"N¼","rect":[183.0205078125,143.99197387695313,200.5976488897553,137.12921142578126]},{"page":254,"text":"\u0003 ½o \u0007 n\b","rect":[216.38409423828126,146.39251708984376,255.9470971791162,136.44200134277345]},{"page":254,"text":"dt","rect":[204.88497924804688,150.89358520507813,212.65922410556864,143.9013214111328]},{"page":254,"text":"(9.19)","rect":[361.0545959472656,145.6554412841797,385.10396705476418,137.1790771484375]},{"page":254,"text":"Figure 9.2 shows the case in that the director n(t) rotates faster than liquid","rect":[65.76496124267578,180.34573364257813,385.11296931317818,171.4111785888672]},{"page":254,"text":"particles. In the figure, r(t) and v are radius-vector and linear velocity of a liquid","rect":[53.81193161010742,192.30526733398438,385.1109551530219,183.37071228027345]},{"page":254,"text":"particle, v ¼ (1/2)curlv is angular velocity of liquid, (v \u0007 n) is that component of","rect":[53.81293869018555,204.26483154296876,385.1497739395298,195.16094970703126]},{"page":254,"text":"director linear velocity, which is solely caused by rotating liquid and dn/dt ¼ (V \u0007","rect":[53.81293869018555,216.224365234375,385.1457903692475,206.92127990722657]},{"page":254,"text":"n) is total linear velocity of the director with respect to immobile laboratory frame","rect":[53.811946868896487,228.18389892578126,385.1706928081688,219.08001708984376]},{"page":254,"text":"(in the figure V > v is angular velocity of the director in the laboratory frame).","rect":[53.811946868896487,240.14346313476563,377.6732143929172,230.8403778076172]},{"page":254,"text":"The second rank viscous stress tensor found by Leslie for the incompressible","rect":[65.76297760009766,252.10299682617188,385.13388860895005,243.16844177246095]},{"page":254,"text":"nematic phase consists of nine matrix elements, each of them having the form:","rect":[53.81095504760742,264.0625305175781,371.67280442783427,255.1279754638672]},{"page":254,"text":"s0ij ¼ a1ninjnknlAkl þ a2njNi þ a3niNj","rect":[143.0265655517578,291.0679016113281,295.4365337447395,278.7770690917969]},{"page":254,"text":"þa4Aij þ a5njnpApi þ a6ninpApj","rect":[168.009521484375,306.42828369140627,295.46512871544265,296.77606201171877]},{"page":254,"text":"(9.20)","rect":[361.0561828613281,305.21258544921877,385.10555396882668,296.7362365722656]},{"page":254,"text":"with i,j ¼x, y, z. The corresponding six viscosity coefficients ai are called Leslie","rect":[53.81450653076172,330.4932861328125,385.1824115581688,321.5393981933594]},{"page":254,"text":"coefficients. In fact, only five of them are independent, because due to Parodi’s","rect":[53.813716888427737,342.4533996582031,385.11371244536596,333.47900390625]},{"page":254,"text":"relationship a6 \u0003 a5 ¼ a2 þ a3 [10].","rect":[53.814720153808597,354.30340576171877,203.55074735190159,345.45849609375]},{"page":254,"text":"Let us look more carefully at each term in a particular tensor component s’ij.","rect":[65.76455688476563,367.28936767578127,385.1832241585422,357.43817138671877]},{"page":254,"text":"The three terms including the velocity gradient tensor Aij are related to shear due","rect":[53.814537048339847,379.24896240234377,385.1450580425438,369.39788818359377]},{"page":254,"text":"to the mass flow in different director configurations. Among them the term with a4","rect":[53.81417465209961,390.29205322265627,385.18130140630105,381.35748291015627]},{"page":254,"text":"is the only one that is independent of n and N. Therefore, it exists even in the","rect":[53.812843322753909,402.25177001953127,385.1715778179344,393.31719970703127]},{"page":254,"text":"isotropic phase and a4 ¼ 2Z. The terms with a2 and a3 depend only on the director","rect":[53.81181716918945,414.1547546386719,385.1074460586704,405.22003173828127]},{"page":254,"text":"components and director rate (velocity) components N but do not contain velocity","rect":[53.8133659362793,426.11431884765627,385.1203545670844,417.17974853515627]},{"page":254,"text":"gradients;they describe a physical situation involving pure director rotation without","rect":[53.813350677490237,438.0738525390625,385.2039628750999,429.1392822265625]},{"page":254,"text":"dn/dt","rect":[156.7371063232422,492.3670654296875,174.43729279530698,486.519287109375]},{"page":254,"text":"v(t1)","rect":[187.65684509277345,472.9760437011719,202.14094371180463,465.24169921875]},{"page":254,"text":"r","rect":[219.4124755859375,511.05364990234377,222.4897281717502,507.25445556640627]},{"page":254,"text":"n(t1)","rect":[237.55349731445313,476.47430419921877,252.532880967664,468.7399597167969]},{"page":254,"text":"Ω","rect":[257.351806640625,494.69439697265627,263.4897888519131,489.280029296875]},{"page":254,"text":"n(t2)","rect":[249.86231994628907,513.5101318359375,264.841474717664,505.7757568359375]},{"page":254,"text":"v(t2)","rect":[262.8431701660156,537.357421875,277.3268873153203,529.6231079101563]},{"page":254,"text":"Fig. 9.2 Mutual rotation of a small spherical volume of a nematic liquid crystal and the director n","rect":[53.812843322753909,560.1944580078125,385.2118008951858,552.1598510742188]},{"page":254,"text":"rotating within this volume (v and v are linear and angular velocity of small liquid volume; dn/dt","rect":[53.81287384033203,570.0459594726563,385.16610435690947,562.4346923828125]},{"page":254,"text":"and V are linear and angular velocities of the director with respect to immobile laboratory frame","rect":[53.813716888427737,580.0219116210938,385.1703734137964,572.1143188476563]},{"page":254,"text":"(here V > v)","rect":[53.813716888427737,589.6592407226563,101.83451932532479,582.0902709960938]},{"page":255,"text":"9.3 Viscosity of Nematics","rect":[53.812843322753909,44.274620056152347,142.61345370291986,36.68026351928711]},{"page":255,"text":"241","rect":[372.4981689453125,42.454345703125,385.18979400782509,36.73106384277344]},{"page":255,"text":"flow. For example, it could be a famous Frederiks transition used in display","rect":[53.812843322753909,68.2883529663086,385.16960993817818,59.35380554199219]},{"page":255,"text":"technology, if the director reorientation is so slow that a relatively weak effect of","rect":[53.812843322753909,80.24788665771485,385.14974342195168,71.31333923339844]},{"page":255,"text":"the mass transfer (i.e. backflow effect) is neglected.","rect":[53.812843322753909,92.20748138427735,261.91460843588598,83.27293395996094]},{"page":255,"text":"According to the form of tensor s’ij (9.20), a4 is independent of the nematic","rect":[65.76486206054688,105.08378601074219,385.1022113628563,95.17564392089844]},{"page":255,"text":"order parameter Q, coefficients a2, a3, a5, a6 are proportional to Q, and a1 / Q2 (the","rect":[53.814144134521487,116.07039642333985,385.14807928277818,105.01939392089844]},{"page":255,"text":"latter is usually smaller than the others). The values of the Leslie coefficients for a","rect":[53.814231872558597,128.02999877929688,385.1591266460594,119.09544372558594]},{"page":255,"text":"popular liquid crystal E7 at 25 C are (in Poise, 1 P ¼ 0.1 Pa\u0005s): a1 ¼ \u00030.18, a2 ¼","rect":[53.814231872558597,139.98953247070313,385.14807918760689,130.99520874023438]},{"page":255,"text":"\u00031.746, a3 ¼ \u00030.214, a4 ¼ 1.736, a5 ¼ 1.716, a6 ¼ a2 þ a3 þ a5 ¼ 0.244.","rect":[53.814231872558597,151.53128051757813,362.4901089241672,143.0145721435547]},{"page":255,"text":"l","rect":[53.812843322753909,166.0723114013672,57.35614695443364,162.8584747314453]},{"page":255,"text":"l","rect":[53.812843322753909,189.99159240722657,57.35614695443364,186.7777557373047]},{"page":255,"text":"4. Finally, the equation of motion for the director of a nematic has no analogy in","rect":[65.8216323852539,169.8604736328125,385.1418389420844,160.92591857910157]},{"page":255,"text":"a system of equations for isotropic liquid and is given by","rect":[65.8216323852539,181.82000732421876,295.8395623307563,172.8854522705078]},{"page":255,"text":"@O","rect":[191.4606475830078,198.87582397460938,204.32845830872385,191.544921875]},{"page":255,"text":"I @t ¼ ½n \u0007 h\b \u0003G","rect":[186.1925811767578,212.53561401367188,264.81822463687208,197.99729919433595]},{"page":255,"text":"Here I is “moment of inertia for the director” and V is vector of total angular","rect":[65.7658462524414,242.0146484375,385.10787330476418,232.71156311035157]},{"page":255,"text":"velocity of the director, dn/dt ¼ (V \u0007 n) as shown in Fig. 9.2. Equation (9.21) is","rect":[53.81382369995117,253.97418212890626,385.1875344668503,244.6710968017578]},{"page":255,"text":"formally analogous to the Newton equation","rect":[53.81283187866211,265.9337463378906,228.8914727311469,256.99920654296877]},{"page":255,"text":"Ido=dt ¼ M\u0003G","rect":[186.41883850097657,290.1815185546875,252.59444378495773,280.2509460449219]},{"page":255,"text":"for rotational motion of a solid body in viscous medium with angular frequency v,","rect":[53.813838958740237,313.7719421386719,385.18255277182348,304.83740234375]},{"page":255,"text":"a torque of external force M and a frictional torque G.","rect":[53.8138542175293,325.7314453125,273.8765835335422,316.7072448730469]},{"page":255,"text":"In our case, the first term (n \u0007 h) on the right side of (9.21) describes the torque","rect":[65.76687622070313,337.63421630859377,385.13282049371568,328.53033447265627]},{"page":255,"text":"exerted on the director due to both an external field and the elastic forces of the","rect":[53.81584930419922,347.5617980957031,385.1736530132469,340.6591796875]},{"page":255,"text":"nematic, that is due to the molecular field h discussed in Sections 8.3.3 and 11.2.1.","rect":[53.81584930419922,359.5313415527344,385.18450589682348,352.6187744140625]},{"page":255,"text":"This torque has the same form as the external torque M \u0007 H exerted by the","rect":[53.81584930419922,373.5128479003906,385.12082708551255,364.57830810546877]},{"page":255,"text":"magnetic field H on the magnetization M of substance. Vector G in (9.21) describes","rect":[53.81682586669922,385.472412109375,385.15863432036596,376.3187255859375]},{"page":255,"text":"a frictional torque consisting of two parts related to the director velocity N rela-","rect":[53.81684112548828,397.43194580078127,385.10491309968605,388.49737548828127]},{"page":255,"text":"tively liquid and to the liquid velocity gradient or shear rate tensor Aij given by","rect":[53.815834045410159,410.31005859375,385.17180720380318,400.45697021484377]},{"page":255,"text":"Eq. (9.12):","rect":[53.81405258178711,421.3531494140625,97.08501298496316,412.4783630371094]},{"page":255,"text":"G ¼ n \u0007 ½g1N þ g2Aijn\b","rect":[171.4091033935547,446.0898132324219,267.56056153458499,435.6036071777344]},{"page":255,"text":"(9.22)","rect":[361.0564880371094,444.8174133300781,385.1058591446079,436.341064453125]},{"page":255,"text":"The coefficients of friction for the director have the dimensions of viscosity and","rect":[65.7668228149414,469.5881042480469,385.14470759442818,460.653564453125]},{"page":255,"text":"are particular combinations of Leslie coefficients, g1 ¼ a3 – a2, g2 ¼ a3 þ a2.","rect":[53.814796447753909,481.54766845703127,369.1526150276828,472.61309814453127]},{"page":255,"text":"It is significant that only two coefficients of viscosity enter the equation for","rect":[65.76619720458985,493.5077209472656,385.1779721817173,484.57318115234377]},{"page":255,"text":"motion of the director. One (g2) describes the director coupling to fluid motion. For","rect":[53.81417465209961,505.4676208496094,385.15007911531105,496.5127868652344]},{"page":255,"text":"example, if the director turnes rapidly under the influence of the magnetic field,","rect":[53.81318283081055,517.4271240234375,385.1092190315891,508.49261474609377]},{"page":255,"text":"then, due to friction,this rotation drags the liquid and creates flow. It is the backflow","rect":[53.81318283081055,529.3866577148438,385.06149053528636,520.4521484375]},{"page":255,"text":"effect that will be described in more details in Section 11.2.5. The other coefficient","rect":[53.81318283081055,539.2674560546875,385.08430345127177,532.295166015625]},{"page":255,"text":"(g1) describes rather a slow director motion in an immobile liquid. Therefore, the","rect":[53.81320571899414,553.2494506835938,385.1715778179344,544.31494140625]},{"page":255,"text":"kinetics of all optical effects caused by pure realignment of the director is deter-","rect":[53.81277847290039,565.208984375,385.1874936660923,556.2744750976563]},{"page":255,"text":"mined by the same coefficient g1. However a description of flow demands for all the","rect":[53.81277847290039,577.1687622070313,385.1726459331688,568.2340698242188]},{"page":255,"text":"five viscosity coefficients.","rect":[53.81391525268555,589.1282958984375,158.4834408333469,580.1937866210938]},{"page":256,"text":"242","rect":[53.8137092590332,42.455810546875,66.50531524805948,36.73252868652344]},{"page":256,"text":"9.3.2","rect":[53.812843322753909,68.0,77.70290889644213,59.370697021484378]},{"page":256,"text":"Measurements of Leslie coefficients","rect":[89.66944885253906,69.85308837890625,267.2084567997233,59.298980712890628]},{"page":256,"text":"9 Elements of Hydrodynamics","rect":[281.01715087890627,44.276084899902347,385.17456515311519,36.68172836303711]},{"page":256,"text":"9.3.2.1 Laminar Shear Flow","rect":[53.812843322753909,95.65619659423828,180.48812627747385,88.10615539550781]},{"page":256,"text":"When the nematic flows through a capillary its apparent viscosity depends on the","rect":[53.812843322753909,121.3980941772461,385.17063177301255,112.46354675292969]},{"page":256,"text":"velocity of flow, more precisely, on shear rate. M. Miesowicz made first experi-","rect":[53.812843322753909,133.35763549804688,385.1776059707798,124.42308044433594]},{"page":256,"text":"ments on properly aligned nematics by a strong magnetic field and found different","rect":[53.812843322753909,145.31716918945313,385.17158372470927,136.3826141357422]},{"page":256,"text":"viscosity coefficients for differently aligned preparations [11]. The idea is illu-","rect":[53.812843322753909,157.27670288085938,385.1576169571079,148.34214782714845]},{"page":256,"text":"strated in Fig. 9.3. The liquid crystal layer of thickness d is placed between two","rect":[53.81181716918945,169.23623657226563,385.15065852216255,160.28175354003907]},{"page":256,"text":"plates. The upper plate is moving along x with velocity v0, but the lower plate is","rect":[53.81183624267578,181.13992309570313,385.18747343169408,172.20445251464845]},{"page":256,"text":"immobile. This creates a gradient of velocity or shear rate dvx/dy ¼ (v0/d), hence","rect":[53.81379318237305,193.0994873046875,385.09531439020005,184.14500427246095]},{"page":256,"text":"vx(y) ¼ (v0/d)y. The correspondent component of the viscous tensor is given by","rect":[53.81417465209961,205.05923461914063,373.89580622724068,196.10475158691407]},{"page":256,"text":"s0xy ¼ Zdvx=dy","rect":[190.78009033203126,234.22979736328126,248.1864703960594,219.28721618652345]},{"page":256,"text":"(9.23)","rect":[361.0562744140625,230.8472137451172,385.10564552156105,222.370849609375]},{"page":256,"text":"where Z is an apparent viscosity coefficient independent of shear rate. In fact,","rect":[53.81462860107422,257.7156066894531,385.1246609261203,248.7810516357422]},{"page":256,"text":"Miesowicz used slowly oscillating upper plate in the x-direction and measured","rect":[53.81462860107422,269.6751708984375,385.1555108170844,260.7406005859375]},{"page":256,"text":"damping of the oscillations. The director was fixed by a strong magnetic field either","rect":[53.81462860107422,281.63470458984377,385.1425717910923,272.70013427734377]},{"page":256,"text":"along z (geometry a) or along y (geometry c). Without field, the shear itself orients","rect":[53.81462860107422,293.59423828125,385.14252103911596,284.65966796875]},{"page":256,"text":"the director along x (geometry b). Using this technique the three flow viscosities","rect":[53.81560516357422,305.55377197265627,385.09372343169408,296.61920166015627]},{"page":256,"text":"coefficients Za ¼ 3.4, Zb ¼ 2.4 and Zc ¼ 9.2 cP have been measured for p-","rect":[53.816612243652347,317.42425537109377,385.15999732820168,308.5787353515625]},{"page":256,"text":"azoxyanisole at 122 C and, nowadays they are called Miesowicz coefficients.","rect":[53.814144134521487,329.4736022949219,366.8261379769016,320.5389404296875]},{"page":256,"text":"It would be very instructive to relate the experimental (Miesowicz) and theoreti-","rect":[65.76499938964844,341.4331359863281,385.14690528718605,332.49859619140627]},{"page":256,"text":"cal (Leslie) coefficients of viscosity. Our task now is to use the viscous tensor (9.20)","rect":[53.8129768371582,353.3359069824219,385.15877662507668,344.3814392089844]},{"page":256,"text":"and find the relationships between the coefficients for each of the three basic","rect":[53.81296157836914,365.29547119140627,385.1269000835594,356.36090087890627]},{"page":256,"text":"orientations of the director, namely nx ¼ 1, ny ¼ 1, or nz ¼ 1. At first, we shall","rect":[53.81296157836914,378.2286682128906,385.1202836758811,368.3204345703125]},{"page":256,"text":"prepare some combinations of parameters useful in all the geometries mentioned:","rect":[53.81330490112305,389.215087890625,382.4794755704124,380.280517578125]},{"page":256,"text":"1. As we have only one component of shear dvx/dy the tensor of shear (9.12) for our","rect":[53.81330490112305,407.1259765625,385.14339576570168,398.1715087890625]},{"page":256,"text":"geometry becomes very simple:","rect":[66.27521514892578,419.0859069824219,194.65864427158426,410.1513671875]},{"page":256,"text":"y","rect":[145.7578887939453,474.200439453125,149.753870978333,468.84259033203127]},{"page":256,"text":"vx(y)","rect":[133.62637329101563,500.38671875,149.7546621474975,492.58746337890627]},{"page":256,"text":"v0","rect":[224.1946563720703,472.79595947265627,230.6925158631932,467.07586669921877]},{"page":256,"text":"z","rect":[145.36068725585938,540.96728515625,148.861167649383,536.6969604492188]},{"page":256,"text":"x","rect":[291.40826416015627,540.3515014648438,295.8198284917203,536.6090087890625]},{"page":256,"text":"Fig. 9.3 Miesowicz’s experiment. Upper plate oscillates in the x-direction and one measures","rect":[53.812843322753909,561.8948974609375,385.1542404460839,553.8602905273438]},{"page":256,"text":"damping of the oscillations. The director is fixed by a strong magnetic field either along z (nz,","rect":[53.813716888427737,571.8031005859375,385.2064845283266,564.208740234375]},{"page":256,"text":"geometry a) or along y (ny, geometry c). Without field, the shear itself aligns the director alongx","rect":[53.81344223022461,582.57763671875,385.1737303473902,574.184326171875]},{"page":256,"text":"(geometry b)","rect":[53.8137092590332,591.754638671875,98.15226071936776,584.1602783203125]},{"page":257,"text":"9.3 Viscosity of Nematics","rect":[53.812843322753909,44.274620056152347,142.61345370291986,36.68026351928711]},{"page":257,"text":"243","rect":[372.4981689453125,42.55594253540039,385.18979400782509,36.73106384277344]},{"page":257,"text":"2Aij","rect":[70.63645935058594,70.97618865966797,85.65498619835278,61.22747039794922]},{"page":257,"text":"¼","rect":[88.93285369873047,66.6360092163086,96.59759548399357,64.30525970458985]},{"page":257,"text":"¼","rect":[88.93285369873047,102.2881088256836,96.59759548399357,99.95735931396485]},{"page":257,"text":"¼","rect":[88.93402099609375,149.55960083007813,96.59876278135686,147.22885131835938]},{"page":257,"text":"2Axy","rect":[99.41264343261719,70.97618865966797,116.74398941889186,61.22747039794922]},{"page":257,"text":"2 @vx=@x","rect":[99.41230773925781,95.45730590820313,133.59954107477035,77.3899154663086]},{"page":257,"text":"66 @vy\u0004@x","rect":[99.41230773925781,107.26541137695313,133.93961370660629,93.48690032958985]},{"page":257,"text":"4 @vz=@x","rect":[99.41230773925781,124.75105285644531,133.37302435113754,106.84211730957031]},{"page":257,"text":"42v000=d v000=d","rect":[99.41232299804688,166.0713653564453,148.0623711686469,130.77145385742188]},{"page":257,"text":"00 3","rect":[158.3231658935547,151.00416564941407,171.5375454508038,130.77145385742188]},{"page":257,"text":"05","rect":[158.32333374023438,166.0713653564453,171.5375454508038,148.1624298095703]},{"page":257,"text":"@@@vvvyxz\u0004==@@@zzz3757 þ 264@@@vvvxxx===@@@xyz","rect":[180.2451171875,124.75105285644531,260.0891345806297,77.38973236083985]},{"page":257,"text":"@vy\u0004@x","rect":[270.08416748046877,92.58824920654297,296.3988727398094,79.03328704833985]},{"page":257,"text":"@vy\u0004@y","rect":[270.08416748046877,108.7388916015625,296.3988727398094,93.4869613647461]},{"page":257,"text":"@vy\u0004@z","rect":[270.3663330078125,123.19591522216797,296.1206399356003,109.64095306396485]},{"page":257,"text":"@@@vvvzzz===@@@yxz 573","rect":[306.3938903808594,122.98004150390625,339.77315641271789,79.03304290771485]},{"page":257,"text":"(9.24a)","rect":[356.6370544433594,152.6277313232422,385.13594947663918,144.1513671875]},{"page":257,"text":"2. Vectors npApi and npApj:","rect":[53.81306457519531,190.50181579589845,164.3490892178733,180.5941925048828]},{"page":257,"text":"npApi ¼ npApx ¼ \u00050;nyðv0=2dÞ;0\u0006; npApj ¼ npApy ¼ ½nxðv0=2dÞ;0;0\b; (9.24b)","rect":[61.970035552978519,215.6985321044922,385.1600278457798,203.76588439941407]},{"page":257,"text":"(here, multiplying the corresponding matrices we used a column np¼(nx, ny, nz)","rect":[66.27591705322266,240.1542510986328,385.15975318757668,230.30259704589845]},{"page":257,"text":"and the first and second rows of (9.24a).","rect":[66.27660369873047,250.7988739013672,228.79204984213596,242.2627410888672]},{"page":257,"text":"3. Scalar products:","rect":[53.814903259277347,263.1000671386719,131.27061326572489,254.16551208496095]},{"page":257,"text":"njnpApi ¼ ny2ðv0=2dÞ and ninpApj ¼ n2xðv0=2dÞ","rect":[123.65762329101563,289.14068603515627,315.32440580474096,277.33087158203127]},{"page":257,"text":"(9.24c)","rect":[356.6373596191406,287.9249572753906,385.1362546524204,279.4486083984375]},{"page":257,"text":"4. Vector v ¼ (1/2)curlv for v ¼ (v0y/d, 0, 0): v ¼ (0, 0, \u0003v0/2d).","rect":[53.813350677490237,312.6761474609375,325.3954739144016,303.7416076660156]},{"page":257,"text":"5. Director velocity N (9.19) in the steady state conditions (dn/dt¼0):","rect":[53.81374740600586,324.6556396484375,337.1374650723655,315.66131591796877]},{"page":257,"text":"N ¼ \u0003½o \u0007 n\b ¼ ð\u0003nyv0=2d;nxv0=2d;0Þ ¼ ðv0=2dÞð\u0003nyi þ nxjÞ","rect":[74.71456909179688,349.3923034667969,335.1503945742722,338.90606689453127]},{"page":257,"text":"(9.24d)","rect":[356.0721435546875,348.1199035644531,385.16033302156105,339.58380126953127]},{"page":257,"text":"Now we are ready to consider the three geometries with different director","rect":[65.76656341552735,378.8988037109375,385.1066831192173,369.9642333984375]},{"page":257,"text":"alignment marked by symbols with letters nx, ny, nz in Fig. 9.3.","rect":[53.814537048339847,391.7754211425781,308.1451687386203,381.86700439453127]},{"page":257,"text":"Geometry (a), nz ¼ 1 (Director Perpendicular to the Shear Plane)","rect":[53.813106536865237,432.6894226074219,316.86284766999855,423.7548828125]},{"page":257,"text":"In this case nx ¼ ny ¼0. Hence, N ¼ 0 and terms with coefficients a2 and a3 in","rect":[53.81386184692383,457.52520751953127,385.1442498307563,447.6171875]},{"page":257,"text":"viscous tensor (9.20) vanish. Terms with a5, a6 also disappear because vector npAxy","rect":[53.81438446044922,469.42816162109377,385.1770479347722,459.5770263671875]},{"page":257,"text":"has only x- and y-components and forms zero scalar product with nz ¼ (0, 0, 1). The","rect":[53.812843322753909,480.4713439941406,385.1479877300438,471.53680419921877]},{"page":257,"text":"term with a1 also vanish because nknlAkl is a scalar, in front of which there isa","rect":[53.81411361694336,492.0527038574219,385.15979803277818,483.496337890625]},{"page":257,"text":"product ninj ¼ nxny ¼ 0. It is only finite when the director has both x and y","rect":[53.81400680541992,505.3070983886719,385.1607135601219,495.45599365234377]},{"page":257,"text":"projections finite.","rect":[53.81294631958008,516.3501586914063,124.05881925131564,507.4156494140625]},{"page":257,"text":"Therefore, in (9.20) we have only one finite term for the viscous stress tensor","rect":[65.76496887207031,528.3096923828125,385.11797462312355,519.3751831054688]},{"page":257,"text":"component:","rect":[53.81294631958008,540.2124633789063,101.05662400791238,532.2938842773438]},{"page":257,"text":"s0xy ¼ Zddvyx ¼ a4Axy ¼ a24 \u0005 vd0","rect":[157.70005798339845,576.9788208007813,279.11924812874158,554.4408569335938]},{"page":257,"text":"(9.25)","rect":[361.0562438964844,569.79833984375,385.1056150039829,561.2024536132813]},{"page":258,"text":"244","rect":[53.812843322753909,42.454345703125,66.50444931178018,36.73106384277344]},{"page":258,"text":"and","rect":[53.812843322753909,66.25641632080078,68.21658412030706,59.35380554199219]},{"page":258,"text":"Za ¼ a4=2:","rect":[196.50146484375,92.53642272949219,242.52394044083497,82.60582733154297]},{"page":258,"text":"We see that a nematic liquid crystal behaves as an","rect":[65.76604461669922,116.0699691772461,277.80864802411568,107.13542175292969]},{"page":258,"text":"plane of the shear (y,z) is perpendicular to the director.","rect":[53.814022064208987,128.02957153320313,275.14287992026098,119.09501647949219]},{"page":258,"text":"9 Elements of Hydrodynamics","rect":[281.0162658691406,44.274620056152347,385.17368014334957,36.68026351928711]},{"page":258,"text":"isotropic liquid when the","rect":[281.580322265625,116.0699691772461,385.1708453960594,107.13542175292969]},{"page":258,"text":"Geometry (b), nx ¼ 1 (Director in the Shear Plane Parallel to the Velocity of Upper","rect":[53.814022064208987,169.8604736328125,385.1258481582798,160.92503356933595]},{"page":258,"text":"Plate)","rect":[53.812923431396487,181.4215850830078,77.11573158113134,172.8854522705078]},{"page":258,"text":"In this case, ny ¼ nz ¼ 0 and N ¼ Ny ¼ (v0/2d)nx. The term with a1vanishes due to","rect":[53.812923431396487,206.65586853027345,385.14324275067818,196.78489685058595]},{"page":258,"text":"nxny ¼ 0. The term a2njNi ¼ a2nyNx also vanishes due to ny ¼ 0. Then, according to","rect":[53.814353942871097,218.6155242919922,385.14373103192818,208.76451110839845]},{"page":258,"text":"(9.24c), the term","rect":[53.8138542175293,229.2601776123047,121.03362988114913,220.7240447998047]},{"page":258,"text":"a5njnpApi ¼ a5n2yðv0=2dÞ ¼ 0:","rect":[156.6239013671875,255.69903564453126,282.3448022572412,243.88922119140626]},{"page":258,"text":"What is left? The “isotropic” term a4v0/2d is always finite. The term a3niNj ¼","rect":[65.76607513427735,280.1142272949219,385.14807918760689,270.2432556152344]},{"page":258,"text":"a3nxNy ¼ a3v0/2d, and, according to (9.24c), the term a6 ninpApj ¼ a6 n2x(v0/2d) ¼","rect":[53.814231872558597,292.0738830566406,385.14817074034127,280.1064453125]},{"page":258,"text":"a6v0/2d . Therefore, three terms contribute to s’xy:","rect":[53.814292907714847,304.0335388183594,258.1535325772483,294.16253662109377]},{"page":258,"text":"sx0y ¼ a3Ny þ ða4 þ a6ÞAxy ¼ ða3 þ a4 þ a6Þv0=2d","rect":[117.42588806152344,329.6110534667969,321.2835625748969,317.376708984375]},{"page":258,"text":"(9.26)","rect":[361.0556335449219,327.0912170410156,385.1050046524204,318.55511474609377]},{"page":258,"text":"and","rect":[53.813961029052737,351.0770568847656,68.21770564130316,344.1744384765625]},{"page":258,"text":"\u0002b ¼ 21ða3 þ a4 þ a6Þ","rect":[176.39309692382813,379.08135986328127,262.58737577544408,367.3399658203125]},{"page":258,"text":"Geometry (c), ny ¼ 1 (Director in the Shear Plane Perpendicular to the Upper Plate","rect":[53.81374740600586,433.49249267578127,385.1195758648094,423.58477783203127]},{"page":258,"text":"Velocity)","rect":[53.813541412353519,444.4788818359375,91.61664710847509,435.5443115234375]},{"page":258,"text":"In this case, nx ¼ nz ¼0 and N¼ Nx ¼ (\u0003nyv0/2d, 0, 0). The term with a1 is again","rect":[53.813541412353519,469.31475830078127,385.1423272233344,459.4438171386719]},{"page":258,"text":"absent, and a6, a3 vanish because a3niNj ¼ a3nxNy ¼ 0 and a6ninpApj ¼ a6 n2x(v0/","rect":[53.814414978027347,481.2744140625,385.151991439553,469.3070068359375]},{"page":258,"text":"2d) ¼ 0.","rect":[53.814144134521487,491.9190368652344,87.84758420492892,483.3630065917969]},{"page":258,"text":"Now, the terms a2njNi ¼ a2nyNx ¼ \u0003a2v0/2d, and a5njnpApj ¼ a5 ny2(v0/2d) ¼ a5","rect":[65.76717376708985,505.1937255859375,385.18130140630105,493.0830993652344]},{"page":258,"text":"(v0/2d) contribute to s’xy, as well as the “isotropic” term a4v0/2d. Hence,","rect":[53.812843322753909,517.1533813476563,348.8791164925266,507.2257385253906]},{"page":258,"text":"s0xy ¼ a2Nx þ ða4 þ a5ÞAxy ¼ ð\u0003a2 þ a4 þ a5Þv0=2d","rect":[113.57455444335938,542.730712890625,325.1353692155219,530.4398803710938]},{"page":258,"text":"(9.27)","rect":[361.05615234375,540.2112426757813,385.10552345124855,531.73486328125]},{"page":258,"text":"and","rect":[53.814476013183597,564.197021484375,68.21821681073675,557.29443359375]},{"page":258,"text":"\u0002c ¼ 12ð\u0003a2 þ a4 þ a5Þ","rect":[172.76927185058595,592.2012329101563,266.21274198638158,580.4598999023438]},{"page":259,"text":"9.3 Viscosity of Nematics","rect":[53.812843322753909,44.274620056152347,142.61345370291986,36.68026351928711]},{"page":259,"text":"245","rect":[372.4981689453125,42.55594253540039,385.18979400782509,36.62946701049805]},{"page":259,"text":"Therefore, the viscous tensor component s’xy corresponding to shear dvx/dy has","rect":[65.76496887207031,69.20487976074219,385.1340371523972,59.333885192871097]},{"page":259,"text":"been found for all the three geometries.","rect":[53.814144134521487,80.24788665771485,213.16213651205784,71.31333923339844]},{"page":259,"text":"With the Miesowicz technique one can measure three combinations of the","rect":[65.76616668701172,92.20748138427735,385.16297186090318,83.27293395996094]},{"page":259,"text":"Leslie viscosity coefficients from Eqs. (9.25) to (9.27). On account of the Parodi","rect":[53.814144134521487,104.11019134521485,385.16987474033427,95.11588287353516]},{"page":259,"text":"relationship, to find all five coefficients, one needs, at least, two additional mea-","rect":[53.81511688232422,116.0697250366211,385.1330808242954,107.13517761230469]},{"page":259,"text":"surements.","rect":[53.81511688232422,127.0,97.25827451254611,120.1107406616211]},{"page":259,"text":"In","rect":[102.58676147460938,127.0,110.96224299481878,119.29398345947266]},{"page":259,"text":"particular,","rect":[116.29470825195313,128.02932739257813,157.67037625815159,119.09477233886719]},{"page":259,"text":"the","rect":[163.02572631835938,127.0,175.34108770562973,119.09477233886719]},{"page":259,"text":"ratio","rect":[180.6984405517578,127.0,199.2700127702094,119.09477233886719]},{"page":259,"text":"of","rect":[204.60247802734376,127.0,212.95903907624854,119.09477233886719]},{"page":259,"text":"coefficients","rect":[218.31143188476563,127.0,264.8762551699753,119.09477233886719]},{"page":259,"text":"a3/a2","rect":[270.2554931640625,127.66854095458985,291.66009168950418,119.09544372558594]},{"page":259,"text":"can","rect":[296.9905090332031,127.0,310.97417536786568,121.0]},{"page":259,"text":"be","rect":[316.306640625,127.0,325.79798162652818,119.09544372558594]},{"page":259,"text":"measured","rect":[331.14837646484377,127.0,369.8284539323188,119.09544372558594]},{"page":259,"text":"by","rect":[375.1609191894531,128.02999877929688,385.17983332685005,119.09544372558594]},{"page":259,"text":"observation of the director field distortion due to capillary flow of a nematic.","rect":[53.81412887573242,139.98953247070313,385.18279691244848,131.0549774169922]},{"page":259,"text":"The last combination g1 ¼ a3 \u0003 a2 can be found from the dynamics of director","rect":[53.81412887573242,151.94937133789063,385.1877683242954,143.01451110839845]},{"page":259,"text":"relaxation.","rect":[53.81313705444336,161.84710693359376,96.90989347006564,154.97438049316407]},{"page":259,"text":"9.3.2.2 Poiseuille Flow in Magnetic Field","rect":[53.81313705444336,205.818603515625,234.01022691081119,196.36610412597657]},{"page":259,"text":"The mass of an isotropic liquid with density r and viscosity Z flowing out from the","rect":[53.81313705444336,229.65802001953126,385.1748737163719,220.7234649658203]},{"page":259,"text":"cylindrical capillary of radius R and length L in a time unit is governed by the","rect":[53.81412887573242,241.61758422851563,385.17383611871568,232.6830291748047]},{"page":259,"text":"Poiseuille-Stokes law,","rect":[53.81315231323242,251.4983367919922,142.86053128744846,244.5857696533203]},{"page":259,"text":"Q ¼ pR2rv ¼ prp0R4","rect":[174.80657958984376,284.936279296875,262.0124976404603,267.7518005371094]},{"page":259,"text":"8LZ","rect":[239.72225952148438,291.7478942871094,256.8315608185127,282.9627685546875]},{"page":259,"text":"(9.28a)","rect":[356.637451171875,284.6370849609375,385.1363462051548,276.1607360839844]},{"page":259,"text":"where v is the flow velocity and p’ is the fixed difference in pressure between the","rect":[53.81345748901367,324.2021484375,385.17322576715318,315.267578125]},{"page":259,"text":"open ends of the capillary. From this law the velocity of liquid is given by","rect":[53.81345748901367,336.16168212890627,354.29628840497505,327.22711181640627]},{"page":259,"text":"v ¼ p0R2\u00048LZ ¼ rpR2\u00048Z","rect":[166.8763885498047,371.7248840332031,272.1254145782783,355.38031005859377]},{"page":259,"text":"(9.28b)","rect":[356.07122802734377,368.41204833984377,385.1594174942173,359.87591552734377]},{"page":259,"text":"where rp ¼ p0=L is the pressure gradient.","rect":[53.81367111206055,400.5400390625,223.90194364096409,390.5516052246094]},{"page":259,"text":"In a strong magnetic field, the director of overwhelming majority of liquid","rect":[65.76538848876953,412.1708984375,385.1133965592719,403.236328125]},{"page":259,"text":"crystals aligns parallel to the field. This is widely used in viscosimetry of liquid","rect":[53.81337356567383,424.1304626464844,385.1124199967719,415.1959228515625]},{"page":259,"text":"crystals. The Poiseuille flow in nematic liquid crystals has carefully been studied by","rect":[53.81337356567383,436.0332336425781,385.1672295670844,427.09869384765627]},{"page":259,"text":"€","rect":[60.72358322143555,440.0,65.7006882768012,438.0]},{"page":259,"text":"Gahwiller [12]. In the experimental scheme of Fig. 9.4, a flat capillary is placed","rect":[53.81337356567383,447.9927673339844,385.11541071942818,439.0582275390625]},{"page":259,"text":"between poles of a magnet. The flow velocity is directed along z. The cross-section","rect":[53.8133430480957,459.9523010253906,385.0974053483344,451.01776123046877]},{"page":259,"text":"of the capillary is not a square; in the ideal case, a \u0004 b and one deals with the well-","rect":[53.81428909301758,471.9118347167969,385.1312192520298,462.977294921875]},{"page":259,"text":"defined shear rate ∂vz/∂x (in the discussed experiment, a ¼ 0.4 mm, b ¼ 40 mm). In","rect":[53.8153076171875,483.8722229003906,385.1506280045844,473.9308166503906]},{"page":259,"text":"the absence of the field, the director is solely oriented by shear flow n ¼ nz. This","rect":[53.81283187866211,495.8317565917969,385.1888772402878,486.897216796875]},{"page":259,"text":"corresponds to case b with Miesowicz viscosity coefficient Z2 ¼ Zb ¼ (1/2)( a3 + a4","rect":[53.814205169677737,507.7914123535156,385.18130140630105,498.85687255859377]},{"page":259,"text":"+ a6). When the director is oriented by field H in the y-direction perpendicularly to","rect":[53.812843322753909,519.7510986328125,385.14104548505318,510.81658935546877]},{"page":259,"text":"side a, i.e., to the shear plane, as shown in the figure, it does not interact with the","rect":[53.81315231323242,531.6538696289063,385.16895330621568,522.7193603515625]},{"page":259,"text":"shear (case a, Z3 ¼ Za ¼ a4/2.). When the capillary is turned by 90 degrees so that","rect":[53.81315231323242,543.6137084960938,385.13850267002177,534.6788940429688]},{"page":259,"text":"its short side a||Hy, the director is oriented along the shear rate (case c) and the","rect":[53.8136100769043,556.559814453125,385.17466009332505,546.6387939453125]},{"page":259,"text":"Miesowicz coefficient Z1 ¼ Zc ¼ (1/2)(\u0003a2 + a4 + a5) is measured. Therefore, for","rect":[53.813907623291019,567.4434204101563,385.1798337539829,558.5984497070313]},{"page":259,"text":"properly selected parameters of a capillary, both experiments discussed give the","rect":[53.81412887573242,579.4925537109375,385.17285955621568,570.5580444335938]},{"page":259,"text":"same results.","rect":[53.81412887573242,589.390380859375,105.72036405112033,582.5176391601563]},{"page":260,"text":"246","rect":[53.81351852416992,42.55685806274414,66.5051245131962,36.68117904663086]},{"page":260,"text":"N","rect":[173.6186981201172,138.6207275390625,179.38983426865176,132.87794494628907]},{"page":260,"text":"a","rect":[184.5376739501953,75.7381362915039,188.98192838036875,71.26708984375]},{"page":260,"text":"h1","rect":[208.81590270996095,92.38055419921875,216.96813955309885,86.19757080078125]},{"page":260,"text":"h3","rect":[208.0660400390625,127.35584259033203,216.2187651634504,121.0653076171875]},{"page":260,"text":"b","rect":[235.04312133789063,66.48966217041016,239.92700525306322,60.642906188964847]},{"page":260,"text":"h2","rect":[229.3677978515625,115.75042724609375,237.52102651598947,109.5677490234375]},{"page":260,"text":"9 Elements of Hydrodynamics","rect":[281.0169372558594,44.275535583496097,385.1743515300683,36.68117904663086]},{"page":260,"text":"S","rect":[276.0083923339844,138.0144500732422,281.33989899751978,131.97573852539063]},{"page":260,"text":"z","rect":[156.10992431640626,174.61227416992188,160.1065559982169,170.357177734375]},{"page":260,"text":"vz","rect":[229.79713439941407,173.86257934570313,237.2394476383111,167.608154296875]},{"page":260,"text":"x","rect":[147.61709594726563,212.60415649414063,152.06135037743906,208.34906005859376]},{"page":260,"text":"y","rect":[183.08718872070313,198.81112670898438,187.53144315087656,192.85238647460938]},{"page":260,"text":"H","rect":[225.55120849609376,195.35791015625,231.32234464462833,189.61512756347657]},{"page":260,"text":"y","rect":[231.32223510742188,198.63511657714845,234.65542593005197,194.16607666015626]},{"page":260,"text":"Fig. 9.4 Geometry of the G€ahwiller experiment. A flat capillary is placed between S and N poles","rect":[53.812843322753909,234.27972412109376,385.1653183269433,226.0]},{"page":260,"text":"of a magnet. The flow velocity is directed along –z and due to condition a\u0004b the shear rate has","rect":[53.812843322753909,244.18795776367188,385.1847275066308,236.59359741210938]},{"page":260,"text":"only ∂vz/∂x component. The magnetic field is fixed along y and the cell with a liquid crystal may","rect":[53.813697814941409,254.107177734375,385.19247955470009,245.65753173828126]},{"page":260,"text":"be rotated about the z-axis by 90","rect":[53.81296920776367,264.08294677734377,166.16075653223917,256.48858642578127]},{"page":260,"text":"Fig. 9.5 Shear (a) andPoiseuille (b) flow in thin planar capillary filled with nematic liquid crystal.","rect":[53.812843322753909,423.3669128417969,385.1729457099672,415.3323059082031]},{"page":260,"text":"Dash lines show distribution of the vx velocity component while the solid lines represent the","rect":[53.812835693359378,433.27508544921877,385.1549620368433,425.68072509765627]},{"page":260,"text":"director profile at high shear rate. Non-distorted close-to-surface layers are marked bye","rect":[53.81351852416992,443.1943054199219,354.5292601325464,435.5999450683594]},{"page":260,"text":"9.3.2.3 Capillary Flow and Determination of a2 and a3","rect":[53.812843322753909,475.0882568359375,295.5119288477073,465.3170471191406]},{"page":260,"text":"For a study of liquid crystals, flat plane capillaries with transparent plates are very","rect":[53.812843322753909,498.6091613769531,385.1129082780219,489.67462158203127]},{"page":260,"text":"convenient, because in this case we can create and observe a proper orientation of","rect":[53.812843322753909,510.5686950683594,385.1487973770298,501.6341552734375]},{"page":260,"text":"the director. There can be distinguished a simple shear flow and the Poiseuille flow,","rect":[53.812843322753909,522.471435546875,385.1268276741672,513.5369262695313]},{"page":260,"text":"both shown in Fig. 9.5. As discussed above, the shear flow occurs when the upper","rect":[53.812843322753909,534.4309692382813,385.1098569473423,525.4367065429688]},{"page":260,"text":"plate is moving with constant velocity v0 and the lower plate is fixed. Then, for","rect":[53.812843322753909,546.3910522460938,385.17949806062355,537.4559936523438]},{"page":260,"text":"small v0, the profile of velocity of isotropic liquid is linear (the dash line in","rect":[53.81379318237305,558.3507080078125,385.14421931317818,549.4160766601563]},{"page":260,"text":"Fig. 9.5a). The Poiseuille flow occurs when the liquid is moved between two","rect":[53.81331253051758,570.3102416992188,385.15117732099068,561.3159790039063]},{"page":260,"text":"immobile plates under an external pressure, as discussed in the previous paragraph.","rect":[53.813316345214847,582.269775390625,385.1711086800266,573.3352661132813]},{"page":260,"text":"In this case the profile has a form of parabola (the dash line in Fig. 9.5b). In both","rect":[53.813316345214847,594.2293090820313,385.12822810224068,585.2350463867188]},{"page":261,"text":"9.3 Viscosity of Nematics","rect":[53.81261444091797,44.276206970214847,142.61322482108393,36.68185043334961]},{"page":261,"text":"247","rect":[372.4979553222656,42.55752944946289,385.1895803847782,36.73265075683594]},{"page":261,"text":"cases, for an isotropic liquid, the viscosity is independent of the shear rate ∂vx/∂z","rect":[53.812843322753909,68.2883529663086,385.16693510161596,58.34779739379883]},{"page":261,"text":"that is the flow is Newtonian.","rect":[53.81313705444336,78.22590637207031,172.3319515755344,71.31333923339844]},{"page":261,"text":"The latter is not true for liquid crystals. Imagine that initially we have a uniform","rect":[65.7651596069336,92.20748138427735,385.15006207109055,83.27293395996094]},{"page":261,"text":"alignment of the director. In the case, shown in Fig. 9.5, such alignment is home-","rect":[53.81313705444336,104.11019134521485,385.13005958406105,95.11588287353516]},{"page":261,"text":"otropic that is perpendicular to the plates, although it is not important, as we shall","rect":[53.81313705444336,116.0697250366211,385.1181474454124,107.13517761230469]},{"page":261,"text":"see below. In the absence of external fields,the shear and Poiseuille flows distort the","rect":[53.81313705444336,126.0,385.17090643121568,119.09477233886719]},{"page":261,"text":"initial uniform alignment (solid lines in the figure). This is a result of coupling","rect":[53.81313705444336,139.98886108398438,385.10524836591255,131.05430603027345]},{"page":261,"text":"between the flow and the director. The flow causes realignment of the director","rect":[53.81313705444336,151.94839477539063,385.10723243562355,143.0138397216797]},{"page":261,"text":"everywhere except thin boundary layer. With increasing shear rate the director in","rect":[53.81313705444336,163.907958984375,385.1370781998969,154.97340393066407]},{"page":261,"text":"the bulk becomes more and more parallel to the limiting plates. For a simple shear","rect":[53.81313705444336,175.86749267578126,385.1350339492954,166.9329376220703]},{"page":261,"text":"flow, in the limit of infinite shear rate the direction of the director saturates at angle","rect":[53.81313705444336,187.8270263671875,385.14011419488755,178.89247131347657]},{"page":261,"text":"Ws depending on three Leslie coefficients [8]:","rect":[53.81313705444336,199.73126220703126,235.69460923740457,190.49639892578126]},{"page":261,"text":"cos2ys ¼ \u0003g1 ¼ \u0003a2 \u0003 a3","rect":[162.8546905517578,227.15353393554688,275.6639563806947,214.0203857421875]},{"page":261,"text":"g2","rect":[213.49559020996095,234.65896606445313,221.96432564339004,227.6636199951172]},{"page":261,"text":"(9.29a)","rect":[356.6379089355469,227.2197723388672,385.13680396882668,218.743408203125]},{"page":261,"text":"Using the Parodi relationship we get","rect":[65.76592254638672,258.22576904296877,212.973768783303,249.2912139892578]},{"page":261,"text":"cos2ys ¼ 11 \u0003þ aa33==aa22 ¼ 11 \u0003þ ttaann22yyss ;","rect":[148.75119018554688,296.8139953613281,290.21852051895999,272.456298828125]},{"page":261,"text":"from which we arrive at the saturation angle ys:","rect":[53.81399154663086,320.291259765625,246.3713517911155,311.0479431152344]},{"page":261,"text":"tanWs ¼ a3=a2","rect":[192.08407592773438,346.7736511230469,248.07771370491347,334.58038330078127]},{"page":261,"text":"(9.29b)","rect":[356.0715026855469,344.0388488769531,385.1596921524204,335.50274658203127]},{"page":261,"text":"As a rule, a2 is large and negative but a3 is one or two orders of magnitude","rect":[65.76595306396485,370.34051513671877,385.1061481304344,361.40594482421877]},{"page":261,"text":"smaller and usually also negative. Therefore angle ys is small (in 5CB ys \u0001 1.5 ).","rect":[53.814083099365237,382.3000793457031,385.1559109261203,373.0567626953125]},{"page":261,"text":"However, the measurements of Ws by optical methods allows the determination of","rect":[53.81411361694336,394.2597961425781,385.14861427156105,385.02630615234377]},{"page":261,"text":"the ratio a3/a2.","rect":[53.813777923583987,405.8013916015625,113.79624600912814,397.22802734375]},{"page":261,"text":"It is interesting that, in some materials, in a certain temperature range, a3 can","rect":[65.76519012451172,418.1222839355469,385.1787957291938,409.187744140625]},{"page":261,"text":"change sign and become positive. Then the flow is no longer laminar and a flow","rect":[53.814083099365237,430.0819396972656,385.09631109192699,421.14739990234377]},{"page":261,"text":"instability in the form of director tumbling is observed. How can we explain the","rect":[53.814083099365237,442.04150390625,385.16407049371568,433.10693359375]},{"page":261,"text":"sign inversion of nematic viscosity? We should remember that a3 is a special","rect":[53.814083099365237,454.0013732910156,385.129530502053,445.06646728515627]},{"page":261,"text":"coefficient that describes coupling a flow with the director. For the flow rate","rect":[53.813594818115237,465.9609069824219,385.10166204645005,457.0263671875]},{"page":261,"text":"fixed, this coefficient defines the direction of director rotation and depends on","rect":[53.813594818115237,477.9204406738281,385.18532649091255,468.98590087890627]},{"page":261,"text":"both a molecular form and, short-range smectic-like fluctuations [13]. Figure 9.6","rect":[53.813594818115237,489.8799743652344,385.1783074479438,480.9454345703125]},{"page":261,"text":"Fig. 9.6 A picture of","rect":[53.812843322753909,533.951171875,128.15518277747325,525.9165649414063]},{"page":261,"text":"collisions of ellipsoidal","rect":[53.812843322753909,543.859375,133.5101900502688,536.2650146484375]},{"page":261,"text":"molecules that may","rect":[53.812843322753909,553.7786254882813,120.43362182764932,546.1842651367188]},{"page":261,"text":"qualitative explain different","rect":[53.812843322753909,563.7545776367188,148.81118492331567,556.1602172851563]},{"page":261,"text":"signs of the director rotation","rect":[53.812843322753909,573.7305297851563,150.9865469375126,566.1361694335938]},{"page":261,"text":"for negative a2 and positive","rect":[53.812843322753909,583.7064819335938,148.93453357004644,576.1115112304688]},{"page":261,"text":"a3 coefficients","rect":[53.81264877319336,593.28955078125,103.09919436454095,586.0307006835938]},{"page":261,"text":"a","rect":[179.20347595214845,518.3922729492188,184.75873834938228,512.8035278320313]},{"page":261,"text":"a2<0","rect":[201.87265014648438,589.315673828125,219.2376644474983,581.7357177734375]},{"page":261,"text":"b","rect":[279.0777893066406,518.3922729492188,285.1825830561188,511.08392333984377]},{"page":261,"text":"a3>0","rect":[310.1783752441406,589.3816528320313,327.5433895451546,581.7357177734375]},{"page":262,"text":"248","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.73106384277344]},{"page":262,"text":"9 Elements of Hydrodynamics","rect":[281.0162658691406,44.274620056152347,385.17368014334957,36.68026351928711]},{"page":262,"text":"may help to understand this. Imagine that dark molecules move to the left alongx","rect":[53.812843322753909,68.2883529663086,385.16062200738755,59.35380554199219]},{"page":262,"text":"and the bright ones are immobile. Then a shear appears. When the director is","rect":[53.812843322753909,80.24788665771485,385.1984902773972,71.31333923339844]},{"page":262,"text":"perpendicular to flow (case a2) then, after collisions with dark molecules, the","rect":[53.812843322753909,92.20748138427735,385.1347125835594,83.27293395996094]},{"page":262,"text":"bright ones rotate anti-clockwise (this corresponds to negative a2). When the","rect":[53.81368637084961,104.1104965209961,385.1402057476219,95.17594909667969]},{"page":262,"text":"director is parallel to the flow (case a3), collisions with dark molecules may","rect":[53.81432342529297,116.07039642333985,385.21856013349068,107.13566589355469]},{"page":262,"text":"result in the clockwise rotation of the bright molecules (then a3 is positive). Note,","rect":[53.81400680541992,128.02999877929688,385.2186550667453,119.09544372558594]},{"page":262,"text":"that flow independent coefficient for the director rotation g1 ¼ a3 \u0003 a2 is always","rect":[53.81411361694336,139.98959350585938,385.1245156680222,131.0549774169922]},{"page":262,"text":"positive.","rect":[53.81456756591797,151.94912719726563,89.14205594565158,143.0145721435547]},{"page":262,"text":"9.3.2.4 Determination of g1","rect":[53.81456756591797,189.2958984375,177.06696729985573,180.04261779785157]},{"page":262,"text":"In the simplest case, one follows the relaxation of the director without flow ofa","rect":[53.812843322753909,213.27801513671876,385.15866888238755,204.3434600830078]},{"page":262,"text":"liquid. It is sufficient to consider only the equation for the director motion (9.21). In","rect":[53.812843322753909,225.237548828125,385.15163508466255,216.30299377441407]},{"page":262,"text":"the simplest geometry of Fig. 9.7, a planar nematic layer of thickness d is confined","rect":[53.81385040283203,237.19711303710938,385.1048516373969,228.2426300048828]},{"page":262,"text":"between two glasses. The boundary conditions on both glasses correspond to the","rect":[53.812843322753909,249.15664672851563,385.1715778179344,240.2220916748047]},{"page":262,"text":"fixed alignment of the director n parallel to the y-axis. We create a distortion of the","rect":[53.812843322753909,261.1161804199219,385.1736224956688,252.18162536621095]},{"page":262,"text":"liquid crystal by a magnetic field applied along the x-axis. The distortion occurs","rect":[53.812862396240237,273.0757141113281,385.1287881289597,264.14117431640627]},{"page":262,"text":"only within the xy-plane and is described by the azimuthal angle j between the","rect":[53.812843322753909,285.0352478027344,385.17359197809068,276.1007080078125]},{"page":262,"text":"director and the y-axis. The director components are (sinj, cosj, 0). In the field","rect":[53.812843322753909,296.9380187988281,385.11782160810005,288.00347900390627]},{"page":262,"text":"slightly exceeding the threshold for distortion, the distortion angle follows a","rect":[53.8138313293457,308.8975830078125,385.1607135601219,299.9630126953125]},{"page":262,"text":"harmonic law j ¼ j0cos(pz/d) shown by the dash line in the figure. When the","rect":[53.8138313293457,320.8581848144531,385.17298162652818,311.9037170410156]},{"page":262,"text":"field is switched off, the distortion relaxes and we can follow the dynamics of the","rect":[53.81325149536133,332.8177490234375,385.1730426616844,323.8831787109375]},{"page":262,"text":"director relaxation by, e.g., an optical technique.","rect":[53.81325149536133,344.77728271484377,249.4254116585422,335.84271240234377]},{"page":262,"text":"To describe this effect we should write Eq.(9.21) for the director motion, i.e., the","rect":[65.76526641845703,356.73681640625,385.17298162652818,347.80224609375]},{"page":262,"text":"balance of torques. However, up to now, nobody has observed any effect related to","rect":[53.81325149536133,368.6963806152344,385.1421746354438,359.7618408203125]},{"page":262,"text":"the inertia of the director. Such effects would result in oscillatory character of the","rect":[53.81325149536133,380.6558837890625,385.1720355816063,371.7213134765625]},{"page":262,"text":"director relaxation. The inertial term for the director in a unit volume can be","rect":[53.81325149536133,390.5267028808594,385.15012396051255,383.62408447265627]},{"page":262,"text":"estimated as a sum of the inertia moments of the molecules in this volume. Let","rect":[53.81325149536133,402.48626708984377,385.1152787930686,395.58367919921877]},{"page":262,"text":"consider a spherical ball of 1 cm3 volume (diameter D \u0001 1.2 cm). The typical","rect":[53.81325149536133,416.4786682128906,385.1267228848655,405.427734375]},{"page":262,"text":"molecular volume is Vm ¼ M/rNav \u0001 10\u000321 cm3, the molecular diameter a is about","rect":[53.81277084350586,428.3387145996094,385.1597734219749,417.3873596191406]},{"page":262,"text":"y","rect":[282.1390075683594,458.85107421875,286.63521807614009,452.8226318359375]},{"page":262,"text":"Fig. 9.7 Geometry of the","rect":[53.812843322753909,523.5786743164063,142.7699980475855,515.5440673828125]},{"page":262,"text":"twist effect that may be used","rect":[53.812843322753909,533.4868774414063,152.4680685439579,525.8925170898438]},{"page":262,"text":"for observation of director","rect":[53.812843322753909,541.6790161132813,143.84117215735606,535.811767578125]},{"page":262,"text":"relaxation and determination","rect":[53.812843322753909,551.6549682617188,152.05518097071573,545.7877197265625]},{"page":262,"text":"of viscosity coefficient g1.","rect":[53.812843322753909,563.3580322265625,143.50162002637348,555.763671875]},{"page":262,"text":"Dash line shows the distortion","rect":[53.81342697143555,571.6063842773438,155.3149618545048,565.7391357421875]},{"page":262,"text":"profile in the magnetic","rect":[53.81342697143555,583.252685546875,131.20682666086675,575.6583251953125]},{"page":262,"text":"field Hx","rect":[53.81342697143555,592.8757934570313,80.15714716355842,585.63427734375]},{"page":262,"text":"–d/2","rect":[260.54071044921877,589.7838134765625,275.7918585674308,583.9453735351563]},{"page":262,"text":"ϕ0","rect":[297.7361145019531,565.2792358398438,306.6553472764987,559.0637817382813]},{"page":262,"text":"0","rect":[296.11712646484377,589.767822265625,300.1137579814933,584.0973510742188]},{"page":262,"text":"+d/2","rect":[318.8251647949219,589.7838134765625,334.81169499321205,583.9453735351563]},{"page":262,"text":"H","rect":[364.1059265136719,498.6338806152344,371.5066892796721,492.45220947265627]},{"page":262,"text":"x","rect":[371.5066833496094,500.6334533691406,375.00373607119368,497.3861389160156]},{"page":262,"text":"Light","rect":[358.7374572753906,519.6945190429688,377.3057995588386,512.2324829101563]},{"page":262,"text":"z","rect":[358.6075439453125,574.671142578125,362.54622435012836,569.8663940429688]},{"page":263,"text":"9.3 Viscosity of Nematics","rect":[53.812843322753909,44.274620056152347,142.61345370291986,36.68026351928711]},{"page":263,"text":"249","rect":[372.4981689453125,42.62367248535156,385.18979400782509,36.73106384277344]},{"page":263,"text":"10\u00037 cm and the moment of inertia of a molecule is rVma2. We have a sum of n ¼","rect":[53.812843322753909,68.18875122070313,385.14737728331,57.23747253417969]},{"page":263,"text":"1/Vm cm\u00033 molecular moments of inertia in the ball, i.e. S ¼ nrVma2 ¼ ra2 ¼","rect":[53.81351852416992,80.14840698242188,385.14807918760689,69.19712829589844]},{"page":263,"text":"10\u000314 g/cm. At the same time, the moment of inertia of the ball as a whole is of the","rect":[53.814231872558597,92.20772552490235,385.1720355816063,81.15672302246094]},{"page":263,"text":"order of rD2 \u0001 1 g/cm. The difference between the two estimations is 14 orders of","rect":[53.8132438659668,104.11067962646485,385.1487973770298,93.05973815917969]},{"page":263,"text":"magnitude! For a particle of 1 mm3 volume, the ratio would still be as high as 106.","rect":[53.812950134277347,116.07039642333985,385.1832241585422,104.97955322265625]},{"page":263,"text":"Indeed the director has almost no inertia and its motion in rather viscous liquid","rect":[53.814537048339847,128.02999877929688,385.11357966474068,119.09544372558594]},{"page":263,"text":"crystals is strongly overdamped. Therefore, one can always assume I \u0001 0. Then","rect":[53.814537048339847,139.98953247070313,385.14043513349068,131.0549774169922]},{"page":263,"text":"Eq. (9.21) becomes simpler:","rect":[53.81356430053711,151.94906616210938,167.694716048928,143.01451110839845]},{"page":263,"text":"½n \u0007 h\b \u0003 G ¼0","rect":[186.75901794433595,176.206787109375,252.23183527997504,166.2562713623047]},{"page":263,"text":"(9.30)","rect":[361.05426025390627,175.46971130371095,385.1036313614048,166.99334716796876]},{"page":263,"text":"As during relaxation the external field is switched off, the molecular field","rect":[65.76457977294922,199.73043823242188,385.11464777997505,190.79588317871095]},{"page":263,"text":"includes only the elastic torque. For the pure twist distortion and our geometry,","rect":[53.812557220458987,211.69000244140626,385.13946195151098,202.7554473876953]},{"page":263,"text":"the molecular field vector h is opposite to the magnetic field, i.e. directed opposite","rect":[53.812557220458987,223.6495361328125,385.17627752496568,214.71498107910157]},{"page":263,"text":"to x- axis. Its absolute value is K22∂2j/∂z2 (Section 8.3.3). The torque (n \u0007 h) is","rect":[53.811580657958987,235.61026000976563,385.1883584414597,224.5591583251953]},{"page":263,"text":"directed along z and has the same absolute value.","rect":[53.81374740600586,247.56979370117188,252.60825772787815,238.63523864746095]},{"page":263,"text":"The viscous torque is given by Eq. (9.22). Due to the absence of flow it contains","rect":[65.7657699584961,259.52935791015627,385.1535684023972,250.5948028564453]},{"page":263,"text":"only one term, namely,","rect":[53.81376266479492,271.4888916015625,147.27285428549534,262.5543212890625]},{"page":263,"text":"G ¼ g1½n \u0007 N\b","rect":[189.9883575439453,292.74298095703127,248.9795373646631,282.7924499511719]},{"page":263,"text":"(9.31)","rect":[361.0550231933594,292.0058898925781,385.1043943008579,283.529541015625]},{"page":263,"text":"where the angular velocity of the director is (n \u0007 N) ¼ (n \u0007 dn/dt) ¼ (n \u0007 [V \u0007 n])","rect":[53.813350677490237,316.32342529296877,385.19197975007668,307.02032470703127]},{"page":263,"text":"¼ V ¼ dw/dt directed along z (here the formula for the double vector product a \u0007 b","rect":[53.817344665527347,328.2261962890625,385.1929966862018,318.923095703125]},{"page":263,"text":"\u0007 c ¼ b(ac) \u0003 c(ab) was used).","rect":[53.820316314697269,339.78729248046877,183.61626859213596,331.08184814453127]},{"page":263,"text":"Then the equation of motion reads [14]:","rect":[65.77334594726563,352.1452941894531,226.97880418369364,343.21075439453127]},{"page":263,"text":"@2j","rect":[200.2972869873047,375.75885009765627,216.613131253083,365.4128112792969]},{"page":263,"text":"K22 @z2 ¼ g1 @t","rect":[184.50119018554688,387.50921630859377,250.90160996982645,373.923828125]},{"page":263,"text":"(9.32)","rect":[361.0549011230469,382.2410888671875,385.1042722305454,373.7647399902344]},{"page":263,"text":"The solution has the following form","rect":[65.76526641845703,411.0936584472656,211.2928538191374,402.15911865234377]},{"page":263,"text":"j ¼ j0 cospz=dexpð\u0003t=tÞ","rect":[167.61276245117188,437.075439453125,271.36618436919408,425.3699035644531]},{"page":263,"text":"(9.33)","rect":[361.0545959472656,435.6351623535156,385.10396705476418,427.1588134765625]},{"page":263,"text":"Substituting this form into (9.32) we find the characteristic relaxation time:","rect":[65.76494598388672,460.5194396972656,369.0936113125999,451.58489990234377]},{"page":263,"text":"g1d2","rect":[200.18399047851563,485.4148864746094,217.94253609260879,474.7500915527344]},{"page":263,"text":"t¼","rect":[179.90489196777345,489.96160888671877,194.70681026670844,485.2901306152344]},{"page":263,"text":"p2K22","rect":[197.46498107910157,498.2247619628906,220.5767981775697,489.0904235839844]},{"page":263,"text":"K22q2","rect":[234.3409881591797,498.7263488769531,256.9143683679994,489.0904235839844]},{"page":263,"text":"(9.34)","rect":[361.0561828613281,491.6357116699219,385.10555396882668,483.15936279296877]},{"page":263,"text":"In conclusion, from the measurements of t the coefficient g1 ¼ a3 \u0003 a2 could be","rect":[65.76653289794922,522.1556396484375,385.15381658746568,513.3104858398438]},{"page":263,"text":"found if the cell thickness and elastic modulus are known. Note that g1 coefficient is","rect":[53.814022064208987,534.1149291992188,385.1889993106003,525.27001953125]},{"page":263,"text":"the most important for applications. Then, using data on the ratio ofa3/a2 we can find","rect":[53.81432342529297,546.1641845703125,385.17836848310005,537.2296752929688]},{"page":263,"text":"a3 and a2 separately. Further, using the known coefficient for the isotropic phase","rect":[53.813655853271487,558.0672607421875,385.15613592340318,549.1327514648438]},{"page":263,"text":"viscosity a4 ¼ 2Za, the coefficients a5 ¼ 2Zc \u0003 2Za þ a2 and a6 ¼ 2Zb \u0003 2Za \u0003 a3","rect":[53.81324005126953,570.0267944335938,385.18130140630105,561.09228515625]},{"page":263,"text":"can be calculated and, for the particular nematic liquid crystal, the applicability of","rect":[53.812843322753909,581.9866333007813,385.14775977937355,573.0521240234375]},{"page":263,"text":"the Parodi relationship a6 \u0003 a5 ¼ a2þ a3 verified. As to a1 it can be found from","rect":[53.812843322753909,593.9461669921875,385.1384348738249,585.0116577148438]},{"page":264,"text":"250","rect":[53.813716888427737,42.55740737915039,66.505322877454,36.63093185424805]},{"page":264,"text":"9 Elements of Hydrodynamics","rect":[281.01715087890627,44.276084899902347,385.17456515311519,36.68172836303711]},{"page":264,"text":"the Poiseuille flow in the magnetic field oriented at some angle j with respect to the","rect":[53.812843322753909,68.2883529663086,385.17359197809068,59.35380554199219]},{"page":264,"text":"y-axis in the xy plane (Fig. 9.4) in order to have finite both ni ¼ nx and nj ¼ ny","rect":[53.812862396240237,81.16453552246094,385.1820833351628,71.31333923339844]},{"page":264,"text":"director components in tensor (9.20). However a1 is usually smaller than the other","rect":[53.812843322753909,92.20772552490235,385.1391538223423,83.27317810058594]},{"page":264,"text":"coefficients and often can be ignored.","rect":[53.813289642333987,104.1104965209961,205.21583219076877,95.17594909667969]},{"page":264,"text":"9.4 Flow in Cholesterics and Smectics","rect":[53.812843322753909,152.2697296142578,255.49874000284835,143.11407470703126]},{"page":264,"text":"9.4.1 Cholesterics","rect":[53.812843322753909,182.04473876953126,150.07547645304366,173.5106201171875]},{"page":264,"text":"The equations for hydrodynamics of cholesterics are basically the same as for","rect":[53.812843322753909,211.69094848632813,385.11992774812355,202.7563934326172]},{"page":264,"text":"nematics, but there are some specific features related to the helical structure.","rect":[53.812843322753909,223.65048217773438,362.6920437386203,214.71592712402345]},{"page":264,"text":"9.4.1.1 Shear","rect":[53.812843322753909,263.4884033203125,116.14511907770002,256.1076965332031]},{"page":264,"text":"Consider shear in three basic geometries shown in Fig. 9.8. In each sketch the helix","rect":[53.812843322753909,289.39959716796877,385.13079157880318,280.46502685546877]},{"page":264,"text":"axis h is aligned differently with respect to plane xy of the shear rate [15].","rect":[53.812862396240237,301.359130859375,353.2623562386203,292.36480712890627]},{"page":264,"text":"Geometry I, h||x, syx ¼ @vy=@xjjh and v⊥h","rect":[53.81185531616211,344.0076599121094,229.8096959049518,333.5885009765625]},{"page":264,"text":"In the left part of the figure, the helical axis is parallel to the velocity gradient","rect":[53.813289642333987,367.1094970703125,385.17207200595927,358.1749267578125]},{"page":264,"text":"(shear) shown by two arrows. When cell thickness is less than the cholesteric pitch,","rect":[53.813289642333987,379.06903076171877,385.1233181526828,370.13446044921877]},{"page":264,"text":"d \u0004 P0, and the rate of shear is small, then an effective viscosity is given by","rect":[53.813289642333987,391.0289611816406,385.16765681317818,382.0741271972656]},{"page":264,"text":"averaging two Miesowicz coefficients:","rect":[53.812835693359378,402.9317321777344,209.37425096103739,393.9971923828125]},{"page":264,"text":"ZI ¼ ZaZb=Za þ Zb:","rect":[176.84588623046876,427.2366027832031,262.12245118302249,417.3060302734375]},{"page":264,"text":"Fig. 9.8 Three different geometries of a shear of a cholesteric liquid crystal: helical axis h||x (I),","rect":[53.812843322753909,584.0003662109375,385.16186782910787,575.9657592773438]},{"page":264,"text":"h||z (II) and h||y (III)","rect":[53.81287384033203,593.8916625976563,124.03636258704354,586.314208984375]},{"page":265,"text":"9.4 Flow in Cholesterics and Smectics","rect":[53.812843322753909,42.62367248535156,185.05249484061518,36.68026351928711]},{"page":265,"text":"251","rect":[372.4981689453125,42.55594253540039,385.18979400782509,36.62946701049805]},{"page":265,"text":"For higher shear rates the helix is unwound, the director becomes almost parallel","rect":[65.76496887207031,68.2883529663086,385.16682298252177,59.35380554199219]},{"page":265,"text":"to the flow lines (n||v) and ZI\u0001Zb as in nematics.","rect":[53.812950134277347,80.15836334228516,252.43503995444065,71.31333923339844]},{"page":265,"text":"Geometry II, h||z, syx ¼ @vy=@x?h and v⊥h","rect":[53.81370162963867,121.93230438232422,235.92715806315494,111.51311492919922]},{"page":265,"text":"In the middle part of the figure, the helical axis is perpendicular to the flow direction","rect":[53.81387710571289,145.03408813476563,385.09304133466255,136.0995330810547]},{"page":265,"text":"and the effective viscosity at low shear rate equals","rect":[53.81387710571289,156.99365234375,257.3287240420456,148.05909729003907]},{"page":265,"text":"\u0002II ¼ 21ð\u0002b þ \u0002cÞ:","rect":[185.7967529296875,182.9093017578125,253.17286622208497,171.22439575195313]},{"page":265,"text":"At high shear rate, due to helix unwinding, we again have ZII \u0001 ZI \u0001 Zb. For","rect":[65.76593780517578,206.47650146484376,385.1526120742954,197.5419464111328]},{"page":265,"text":"small distortions, in both cases, the disturbed helical structure relaxes with a rate of","rect":[53.813777923583987,216.40383911132813,385.15071998445168,209.50123596191407]},{"page":265,"text":"the general hydrodynamic form","rect":[53.813777923583987,230.39532470703126,181.10323284745773,221.4607696533203]},{"page":265,"text":"t ¼ g1\u0004K22q02 ¼ g1P20\u0004K22p2","rect":[164.32839965820313,261.83587646484377,274.1912085779603,244.62603759765626]},{"page":265,"text":"(9.35)","rect":[361.0561828613281,257.7139587402344,385.10555396882668,249.11807250976563]},{"page":265,"text":"Geometry III, h||y, syx ¼ @vy=@xjjh, and v||h","rect":[53.81450653076172,315.1004638671875,236.549976056319,304.6812744140625]},{"page":265,"text":"In the right part of the figure, the direction of the flow coincides with the helical","rect":[53.81357955932617,338.2022705078125,385.1096025235374,329.2677001953125]},{"page":265,"text":"axis. This case isespecially interesting because it results in the so-called permeation","rect":[53.81357955932617,350.16180419921877,385.15544978192818,341.22723388671877]},{"page":265,"text":"effect.","rect":[53.81357955932617,360.0595397949219,79.07736630942111,353.18682861328127]},{"page":265,"text":"9.4.1.2 Permeation Effect","rect":[53.81357955932617,401.2450866699219,168.9798062881626,393.6153869628906]},{"page":265,"text":"In experiments with cholesteric liquid crystals (geometry III), extraordinary high","rect":[53.81357955932617,426.8504943847656,385.1255730729438,417.91595458984377]},{"page":265,"text":"viscosity ZIII is observed, few orders of magnitude higher than the viscosity of the","rect":[53.81357955932617,438.8108825683594,385.1731647319969,429.87548828125]},{"page":265,"text":"isotropic phase or a non-twisted nematic. It seems surprising because the local","rect":[53.81340408325195,450.7704162597656,385.1134477383811,441.83587646484377]},{"page":265,"text":"structure of nematics and cholesterics is the same. In addition such a flow is","rect":[53.81340408325195,460.6980285644531,385.12936796294408,453.79541015625]},{"page":265,"text":"strongly non-Newtonian: with increasing shear rate (s) ZIII decreases, as schema-","rect":[53.81340408325195,474.68951416015627,385.06939063874855,465.75494384765627]},{"page":265,"text":"tically shown in Fig. 9.9. In the case of the Poiseuille flow, the viscosity depends","rect":[53.814205169677737,486.6494140625,385.16498197661596,477.71484375]},{"page":265,"text":"also on the radius of a capillary.","rect":[53.81417465209961,498.6089782714844,183.58620877768284,489.6744384765625]},{"page":265,"text":"The explanation of these observations has been given in terms of the so-called","rect":[65.7662124633789,510.5685119628906,385.09136286786568,501.63397216796877]},{"page":265,"text":"permeation effect [16]. Helfrich assumed that the helical structure with wavevector","rect":[53.81417465209961,522.4712524414063,385.16997657624855,513.5367431640625]},{"page":265,"text":"q0 is fixed by the boundary conditions at the walls of a cylindrical capillary of","rect":[53.81319808959961,534.431396484375,385.14592872468605,525.4968872070313]},{"page":265,"text":"radius R. Schematically it is shown by “fixation points” at each period of the helix","rect":[53.81302261352539,546.3909301757813,385.1299981217719,537.4564208984375]},{"page":265,"text":"in Fig. 9.10a. The liquid crystal flows out of the capillary with a constant velocity","rect":[53.81302261352539,558.3504638671875,385.11806574872505,549.4159545898438]},{"page":265,"text":"v||q0. Therefore, the mass of the liquid of density r escaping the capillary in a time","rect":[53.81203079223633,570.310302734375,385.1594623394188,561.3755493164063]},{"page":265,"text":"unit is given by Q ¼ pR2rv. The flow velocity is considered to be uniform along the","rect":[53.81264877319336,582.2699584960938,385.17237127496568,571.7877197265625]},{"page":265,"text":"capillary radius (except the boundary layer l \u0004 R). At the same time molecules","rect":[53.81361770629883,594.2294921875,385.1604348574753,585.2750244140625]},{"page":266,"text":"252","rect":[53.812843322753909,42.55630874633789,66.50444931178018,36.62983322143555]},{"page":266,"text":"Fig. 9.9 Comparison of the","rect":[53.812843322753909,67.58130645751953,150.24703357004644,59.546695709228519]},{"page":266,"text":"dependencies of the viscosity","rect":[53.812843322753909,77.4895248413086,154.2305197158329,69.89517211914063]},{"page":266,"text":"coefficient on shear rate for","rect":[53.812843322753909,85.65617370605469,148.19268888098888,79.81436157226563]},{"page":266,"text":"the cholesteric and isotropic","rect":[53.812843322753909,97.3846664428711,150.1091398932886,89.79031372070313]},{"page":266,"text":"phases","rect":[53.812843322753909,107.36067962646485,76.3531387615136,99.76632690429688]},{"page":266,"text":"9 Elements of Hydrodynamics","rect":[281.0162658691406,44.274986267089847,385.17368014334957,36.68062973022461]},{"page":266,"text":"Fig. 9.10 Poiseuille flow ina","rect":[53.812843322753909,235.30001831054688,155.36428210520269,227.26541137695313]},{"page":266,"text":"cylindrical capillary with","rect":[53.812843322753909,245.208251953125,139.79340118555948,237.6138916015625]},{"page":266,"text":"permeation effect in the","rect":[53.812843322753909,255.12747192382813,135.5763945319605,247.53311157226563]},{"page":266,"text":"cholesteric (a) and smectic A","rect":[53.812843322753909,264.7647705078125,154.31513001066655,257.5090637207031]},{"page":266,"text":"(b) phases","rect":[53.81284713745117,275.07940673828127,89.04135592459955,267.48504638671877]},{"page":266,"text":"a","rect":[198.13900756835938,235.95166015625,203.6942699655932,230.36289978027345]},{"page":266,"text":"R","rect":[231.7476806640625,254.05111694335938,237.51881681259708,248.30833435058595]},{"page":266,"text":"q0 Z","rect":[254.55885314941407,246.82827758789063,272.18939538978199,239.14488220214845]},{"page":266,"text":"b","rect":[303.85662841796877,235.95166015625,309.9614221674469,228.64328002929688]},{"page":266,"text":"R","rect":[326.83404541015627,250.35171508789063,332.6051815586908,244.6089324951172]},{"page":266,"text":"P","rect":[197.69097900390626,325.5326843261719,203.02248566744167,319.7899169921875]},{"page":266,"text":"0","rect":[203.0229949951172,328.640625,206.35618581774728,324.2735595703125]},{"page":266,"text":"v = const","rect":[229.00479125976563,375.2965087890625,261.8249341903203,369.7856750488281]},{"page":266,"text":"v","rect":[348.5709228515625,359.74359130859377,352.0719722048286,356.0083923339844]},{"page":266,"text":"are free in the bulk and may rotate about the z-axis. In such a situation the director","rect":[53.812843322753909,420.7297668457031,385.10686622468605,411.79522705078127]},{"page":266,"text":"j(z) ¼ q0z must rotate like a screw with angular velocity O:","rect":[53.812843322753909,432.6894226074219,297.0109697110374,423.5755920410156]},{"page":266,"text":"dj dj dz","rect":[187.66543579101563,455.79217529296877,240.65806974517063,446.9173889160156]},{"page":266,"text":"O ¼ dt ¼ dz \u0005 dt ¼ q0v;","rect":[167.2727813720703,467.5126647949219,270.0532983509912,453.5596008300781]},{"page":266,"text":"(9.36)","rect":[361.05670166015627,462.27508544921877,385.1060727676548,453.73895263671877]},{"page":266,"text":"and this rotation exerts a friction torque from the capillary walls on the director:","rect":[53.815040588378909,491.0140686035156,385.142958236428,482.07952880859377]},{"page":266,"text":"G ¼ g1O.","rect":[53.815040588378909,503.156005859375,92.32770963217502,493.8597106933594]},{"page":266,"text":"The work rpv made by the external pressure gradient rp must be equal to the","rect":[65.764892578125,514.9330444335938,385.1715778179344,505.9885559082031]},{"page":266,"text":"energy dissipated in unit volume and unit time due to the friction g1(dj/dt)2 ¼","rect":[53.810882568359378,526.892578125,385.14807918760689,515.7852172851563]},{"page":266,"text":"g1O2. Therefore, rpv ¼ g1O2 ¼ g1v2q02:","rect":[53.814231872558597,539.5154418945313,216.80636536759278,527.6900024414063]},{"page":266,"text":"From here, we obtain the relationship between the pressure gradient and flow","rect":[65.76616668701172,550.75537109375,385.1311316485676,541.8208618164063]},{"page":266,"text":"velocity of a cholesteric:","rect":[53.814144134521487,562.7149047851563,153.37417466709207,553.7803955078125]},{"page":266,"text":"rp ¼ g1q02v","rect":[195.08628845214845,588.65771484375,243.88863409234848,576.9456176757813]},{"page":266,"text":"(9.37a)","rect":[356.6369934082031,587.5397338867188,385.1358884414829,579.0633544921875]},{"page":267,"text":"9.4 Flow in Cholesterics and Smectics","rect":[53.812843322753909,42.62367248535156,185.05249484061518,36.68026351928711]},{"page":267,"text":"253","rect":[372.4981689453125,42.55594253540039,385.18979400782509,36.62946701049805]},{"page":267,"text":"Note that the Poiseuille-Stokes equation for an isotropic liquid (9.28b) or for a","rect":[65.76496887207031,68.2883529663086,385.1597369976219,59.35380554199219]},{"page":267,"text":"nematic would give us","rect":[53.81393051147461,80.24788665771485,144.92680753813938,71.31333923339844]},{"page":267,"text":"8vZ","rect":[223.80496215820313,103.22745513916016,239.89496108706738,94.44231414794922]},{"page":267,"text":"rp ¼ R2","rect":[197.40838623046876,114.778564453125,236.4088752039369,101.11709594726563]},{"page":267,"text":"(9.37b)","rect":[356.0715026855469,109.66373443603516,385.1596921524204,101.12760925292969]},{"page":267,"text":"From comparison of the last two formulas we can find the apparent viscosity for","rect":[65.76595306396485,144.29733276367188,385.1777585586704,135.36277770996095]},{"page":267,"text":"a cholesteric:","rect":[53.81393051147461,154.2349090576172,107.22822434970925,147.3223419189453]},{"page":267,"text":"Zapp ¼ g1ðq0RÞ2.8 ¼ g1kp","rect":[166.02772521972657,189.42384338378907,272.43516609749158,170.48757934570313]},{"page":267,"text":"(9.38)","rect":[361.0561828613281,184.7091522216797,385.10555396882668,176.2327880859375]},{"page":267,"text":"According to this simplest theory, the amplification factor due to chirality can be","rect":[65.76653289794922,218.8895263671875,385.1514362163719,209.95497131347657]},{"page":267,"text":"as high as kp \u0001 107 (R ¼ 0.1 cm, q0 ¼ 2p/P0 \u0001 105 cm\u00031). The coefficient kp is","rect":[53.81450653076172,231.7787322998047,385.1891213809128,219.71836853027345]},{"page":267,"text":"called permeation coefficient because the molecules permeate through the fixed","rect":[53.81450653076172,242.80859375,385.1444024186469,233.83419799804688]},{"page":267,"text":"cholesteric quasi-layers. In reality kp is smaller than 107 due to a non-ideal helical","rect":[53.814537048339847,255.69798278808595,385.1101518399436,243.81031799316407]},{"page":267,"text":"structure, surface defects, non-uniform velocity profile etc., but, nevertheless, the","rect":[53.81415939331055,266.7278747558594,385.1719440288719,257.7933349609375]},{"page":267,"text":"effect is very remarkable.","rect":[53.81415939331055,278.6306457519531,156.97262235190159,269.69610595703127]},{"page":267,"text":"9.4.2 Smectic A Phase","rect":[53.812843322753909,329.0537109375,172.8690012309095,320.7108154296875]},{"page":267,"text":"9.4.2.1 Flow and Viscosity","rect":[53.812843322753909,359.2893371582031,172.07283869794379,349.51812744140627]},{"page":267,"text":"For the smectic A phase the permeation effect is even more important [16]. In fact,","rect":[53.812843322753909,382.8100280761719,385.1247829964328,373.87548828125]},{"page":267,"text":"with the layers fixed at the walls of a capillary, a smectic may flow only as a whole,","rect":[53.81282424926758,394.76959228515627,385.1138577034641,385.83502197265627]},{"page":267,"text":"like a plug, without velocity gradients, Fig. 9.10b. The velocity is again given by","rect":[53.81282424926758,406.7291259765625,385.17156306317818,397.7945556640625]},{"page":267,"text":"equation","rect":[53.813812255859378,418.68865966796877,88.30216303876409,409.75408935546877]},{"page":267,"text":"rp ¼ kpv,","rect":[93.52214813232422,419.4437255859375,135.3785366585422,409.73419189453127]},{"page":267,"text":"where","rect":[140.5935516357422,416.6273498535156,164.9315875835594,409.754638671875]},{"page":267,"text":"k","rect":[170.21926879882813,416.70703125,174.66880071832504,409.7347106933594]},{"page":267,"text":"p","rect":[174.69366455078126,419.6189270019531,178.42641859380104,415.1504821777344]},{"page":267,"text":"is","rect":[183.64361572265626,417.0,190.27312101470188,409.754638671875]},{"page":267,"text":"the","rect":[195.5388946533203,417.0,207.76267278863754,409.754638671875]},{"page":267,"text":"permeation","rect":[212.9856414794922,418.6891784667969,258.0135125260688,409.754638671875]},{"page":267,"text":"coefficient","rect":[263.343017578125,417.0,305.6135392911155,409.754638671875]},{"page":267,"text":"depending","rect":[310.8106384277344,418.6891784667969,352.4500664811469,409.754638671875]},{"page":267,"text":"on","rect":[357.7118835449219,417.0,367.66606989911568,411.0]},{"page":267,"text":"the","rect":[372.89306640625,417.0,385.11682928277818,409.754638671875]},{"page":267,"text":"smectic characteristic length ls given by Eq. (8.46), conventional nematic viscosity","rect":[53.813838958740237,430.6488342285156,385.10540095380318,421.3954162597656]},{"page":267,"text":"Z and temperature:","rect":[53.81340408325195,442.6083679199219,131.15066392490457,433.673828125]},{"page":267,"text":"Z TNA \u0003 T","rect":[206.07501220703126,465.4745178222656,253.82660264323307,456.8487548828125]},{"page":267,"text":"kp ¼ l2 \u0005 TNA","rect":[182.9633026123047,478.5696105957031,244.789354792915,463.35406494140627]},{"page":267,"text":"s","rect":[210.15347290039063,481.02685546875,212.836124953174,477.7149963378906]},{"page":267,"text":"(9.39)","rect":[361.0561828613281,471.9107971191406,385.10555396882668,463.4344482421875]},{"page":267,"text":"The smaller the temperature difference TNA \u0003 T, the smaller is the smectic order","rect":[65.76653289794922,510.5684509277344,385.1475766739048,501.6339111328125]},{"page":267,"text":"parameter, that is the amplitude of the density wave. Consequently, the permeation","rect":[53.81376266479492,522.5283203125,385.15261164716255,513.5938110351563]},{"page":267,"text":"coefficient","rect":[53.81376266479492,533.0,96.05544908115457,525.49658203125]},{"page":267,"text":"in","rect":[101.28240203857422,533.0,109.05664149335394,525.49658203125]},{"page":267,"text":"SmA","rect":[114.31047058105469,533.0,134.72656011536447,525.5563354492188]},{"page":267,"text":"should","rect":[139.97044372558595,533.0,166.50837031415473,525.49658203125]},{"page":267,"text":"decrease","rect":[171.74826049804688,533.0,206.20574224664535,525.49658203125]},{"page":267,"text":"upon","rect":[211.45559692382813,534.4310913085938,231.3640069108344,527.0]},{"page":267,"text":"approaching","rect":[236.60589599609376,534.4310913085938,286.0414971452094,525.49658203125]},{"page":267,"text":"the","rect":[291.2684631347656,533.0,303.4922260112938,525.49658203125]},{"page":267,"text":"SmA-N","rect":[308.7152099609375,533.0,339.5931677813801,525.5563354492188]},{"page":267,"text":"transition.","rect":[344.8539733886719,533.0,385.1525845101047,525.49658203125]},{"page":267,"text":"Indeed, in experiment, very close to TNA the Poiseuille flow is observed, as in the","rect":[53.81376266479492,546.390625,385.1731647319969,537.4561157226563]},{"page":267,"text":"nematic phase, but already at TNA \u0003 T > 0.3 K the plug flow occurs with apparent","rect":[53.81345748901367,558.3507080078125,385.174940658303,549.4160766601563]},{"page":267,"text":"viscosity two orders of magnitude larger than Z.","rect":[53.813228607177737,570.3102416992188,249.12081571127659,561.375732421875]},{"page":267,"text":"If","rect":[65.76524353027344,580.1282958984375,72.39474616853369,573.3352661132813]},{"page":267,"text":"both","rect":[77.6605224609375,581.0,95.38897028974066,573.3352661132813]},{"page":267,"text":"the","rect":[100.7144775390625,581.0,112.93824041559064,573.3352661132813]},{"page":267,"text":"compressibility","rect":[118.27469635009766,582.269775390625,179.91814509442816,573.3352661132813]},{"page":267,"text":"and","rect":[185.2287139892578,581.0,199.63245478681098,573.3352661132813]},{"page":267,"text":"the","rect":[204.9410400390625,581.0,217.16480291559066,573.3352661132813]},{"page":267,"text":"permeation","rect":[222.5012664794922,582.269775390625,267.4644707780219,573.3352661132813]},{"page":267,"text":"effect","rect":[272.8576354980469,581.0,295.63288743564677,573.3352661132813]},{"page":267,"text":"are","rect":[300.95343017578127,581.0,313.16724432184068,575.0]},{"page":267,"text":"disregarded,","rect":[318.5136413574219,582.269775390625,367.6088528206516,573.3352661132813]},{"page":267,"text":"the","rect":[372.89154052734377,581.0,385.1153034038719,573.3352661132813]},{"page":267,"text":"structure of the viscous stress tensor sij is identical for the SmA and nematic phases","rect":[53.813228607177737,595.159423828125,385.1360207949753,585.2947998046875]},{"page":268,"text":"254","rect":[53.812843322753909,42.55679702758789,66.50444931178018,36.63032150268555]},{"page":268,"text":"a","rect":[103.32901000976563,68.23306274414063,108.88427240699947,62.64430618286133]},{"page":268,"text":"b","rect":[234.02154541015626,68.60800170898438,240.12633915963446,61.29962921142578]},{"page":268,"text":"9 Elements of Hydrodynamics","rect":[281.0162658691406,44.275474548339847,385.17368014334957,36.68111801147461]},{"page":268,"text":"Fig. 9.11","rect":[53.812843322753909,205.93939208984376,84.87682098536416,197.90478515625]},{"page":268,"text":"plane but","rect":[53.812843322753909,215.84762573242188,85.55370048728052,208.25326538085938]},{"page":268,"text":"along the","rect":[53.813697814941409,225.766845703125,85.44625232004643,218.1724853515625]},{"page":268,"text":"z","rect":[212.69186401367188,140.87872314453126,216.6877465391181,136.624267578125]},{"page":268,"text":"y","rect":[202.7405242919922,181.0817108154297,207.18394566028844,175.12387084960938]},{"page":268,"text":"x","rect":[268.3537292480469,184.08465576171876,272.7971506163431,179.8302001953125]},{"page":268,"text":"Two geometries for easy flow in Smectic A: flow velocity in both cases is in the layer","rect":[90.8587875366211,205.87167358398438,385.1855782852857,198.27731323242188]},{"page":268,"text":"shear is either perpendicular to layers (a) or parallel to them (b). For the flow velocity","rect":[88.08355712890625,215.84762573242188,385.1593069472782,208.25326538085938]},{"page":268,"text":"layer normal the permeation effect is observed, see Fig. 9.10b","rect":[87.80096435546875,225.766845703125,299.88446563868447,218.1724853515625]},{"page":268,"text":"pressure","rect":[318.7559509277344,267.2682189941406,349.85284111397137,261.3168640136719]},{"page":268,"text":"Fig. 9.12 Undulation or","rect":[53.812843322753909,324.288818359375,138.46499723059825,316.25421142578127]},{"page":268,"text":"wave-like instability in the","rect":[53.812843322753909,334.19708251953127,145.43270251535894,326.60272216796877]},{"page":268,"text":"smectic A layer subjected toa","rect":[53.812843322753909,344.11627197265627,155.36513659739019,336.52191162109377]},{"page":268,"text":"dilatation-compression","rect":[53.812843322753909,354.0922546386719,131.57854980616495,346.4978942871094]},{"page":268,"text":"distortion","rect":[53.812843322753909,362.341064453125,86.32027191309854,356.4738464355469]},{"page":268,"text":"2π/qx","rect":[311.9698791503906,360.4564208984375,330.46811023852276,352.56201171875]},{"page":268,"text":"x","rect":[379.4037170410156,354.8750915527344,383.4007485375333,350.6915588378906]},{"page":268,"text":"due to the same point group symmetry, D1h. There are five independent viscosity","rect":[53.812843322753909,403.2720947265625,385.1051873307563,394.3375244140625]},{"page":268,"text":"coefficients for SmA as for a nematic [17]. However, tensor (9.20) can only be used","rect":[53.81315231323242,415.2316589355469,385.1251153092719,406.297119140625]},{"page":268,"text":"when the velocity has no component along the layer normal z as shown by examples","rect":[53.81315231323242,427.1911926269531,385.15497221099096,418.25665283203127]},{"page":268,"text":"in Fig. 9.11 for velocity vy and gradient ∂vy/∂z in case (a) and velocity vy and","rect":[53.81315231323242,440.0675048828125,385.14626399091255,429.2104797363281]},{"page":268,"text":"gradient ∂vy /∂x in case (b). In these two examples of the Poiseuille flow the","rect":[53.814414978027347,452.02716064453127,385.17112005426255,441.1700134277344]},{"page":268,"text":"viscosities are given by","rect":[53.812347412109378,463.013427734375,148.27979365399848,454.078857421875]},{"page":268,"text":"Za ¼ 1=2ðm0 þ m2 \u0003 2l1 þ l4Þ and Zb ¼ 1=2m0","rect":[118.6142578125,492.95343017578127,319.90376350960096,480.21197509765627]},{"page":268,"text":"New viscosity coefficients mi and li are related to Leslie coefficients. In particu-","rect":[65.76496887207031,522.4718017578125,385.10808692781105,513.2185668945313]},{"page":268,"text":"lar, m0 ¼ a4 (viscosity of an isotropic liquid). Viscosimetry of SmA liquid crystals is","rect":[53.813045501708987,534.431396484375,385.18564237700658,525.4968872070313]},{"page":268,"text":"difficult. For instance, in geometry (a), the upper and lower plates should be parallel","rect":[53.81290054321289,546.3909301757813,385.163710189553,537.4564208984375]},{"page":268,"text":"with a great accuracy (few nanometers); otherwise defects appear. However, for","rect":[53.81290054321289,558.3504638671875,385.1756833633579,549.4159545898438]},{"page":268,"text":"several compounds the correspondent viscosities have been measured. In geometry","rect":[53.81290054321289,570.31005859375,385.1428155045844,561.3755493164063]},{"page":268,"text":"(b) there was found a shear rate threshold: above the threshold the isotropic","rect":[53.81290054321289,582.2695922851563,385.0950092144188,573.3350830078125]},{"page":268,"text":"behaviour (a4) was observed. At lower rates, defects control a flow.","rect":[53.81290054321289,593.8312377929688,327.0091213753391,585.2946166992188]},{"page":269,"text":"References","rect":[53.813838958740237,42.52750778198242,91.48251803397455,36.68569564819336]},{"page":269,"text":"255","rect":[372.49920654296877,42.56137466430664,385.1908010879032,36.6348991394043]},{"page":269,"text":"9.4.2.2 Undulation Instability","rect":[53.812843322753909,68.68677520751953,185.83655634931098,58.91554641723633]},{"page":269,"text":"In the experiment, it is possible to create a dilatation of the smectic layers with","rect":[53.812843322753909,92.20748138427735,385.11589900067818,83.27293395996094]},{"page":269,"text":"piezoelectric drivers. Evidently, an increase of the interlayer distance would cost","rect":[53.812843322753909,104.11019134521485,385.152723861428,95.17564392089844]},{"page":269,"text":"a lot of energy. Instead, at a certain critical dilatation wc ¼ 2pls=d, where","rect":[53.812843322753909,116.39909362792969,385.11093939020005,106.46849822998047]},{"page":269,"text":"ls ¼ ðK11=BÞ1=2, a wave-like or undulation distortion is observed as illustrated by","rect":[53.81393051147461,130.9193115234375,385.1691521745063,118.45559692382813]},{"page":269,"text":"Fig. 9.12. The wavevector of the distortion is proportional to inverse geometrical","rect":[53.8133430480957,142.54019165039063,385.14924485752177,133.6056365966797]},{"page":269,"text":"ffiffiffiffiffiffiffi","rect":[338.6252136230469,145.0,352.5909835155321,144.0]},{"page":269,"text":"average of cell thickness and smectic characteristic length, qx ¼ p=plsd. There-","rect":[53.8133430480957,154.82859802246095,385.1243833145298,144.184814453125]},{"page":269,"text":"fore, a typical undulation period is about 0.3 mm (d \u0001 10 mm, ls \u0001 0.01 mm) and","rect":[53.814414978027347,166.45947265625,385.1459588151313,157.2061767578125]},{"page":269,"text":"may be observed optically. A similar instability arises in cholesterics under the","rect":[53.81411361694336,178.36236572265626,385.1719135112938,169.4278106689453]},{"page":269,"text":"influence of the magnetic or electric field, see Section 12.2.3.","rect":[53.81411361694336,190.3218994140625,301.34942288901098,181.38734436035157]},{"page":269,"text":"References","rect":[53.812843322753909,232.5298309326172,109.59614448282879,223.7447052001953]},{"page":269,"text":"1.","rect":[58.06126022338867,258.0,64.40706131055318,252.1748504638672]},{"page":269,"text":"2.","rect":[58.06126022338867,268.0,64.40706131055318,262.1507873535156]},{"page":269,"text":"3.","rect":[58.06126022338867,278.0,64.40706131055318,272.12677001953127]},{"page":269,"text":"4.","rect":[58.06126022338867,298.0,64.40706131055318,292.0219421386719]},{"page":269,"text":"5.","rect":[58.06126022338867,318.0,64.40706131055318,311.87225341796877]},{"page":269,"text":"6.","rect":[58.06126022338867,338.0,64.40706131055318,331.8182373046875]},{"page":269,"text":"7.","rect":[58.06126022338867,357.64581298828127,64.40706131055318,351.939453125]},{"page":269,"text":"8.","rect":[58.06126022338867,378.0,64.40706131055318,371.71612548828127]},{"page":269,"text":"9.","rect":[58.06126022338867,398.0,64.40706131055318,391.6679992675781]},{"page":269,"text":"10.","rect":[53.812957763671878,418.0,64.38929626538716,411.5632019042969]},{"page":269,"text":"11.","rect":[53.812957763671878,428.0,64.38929626538716,421.5391845703125]},{"page":269,"text":"12.","rect":[53.81296157836914,457.13360595703127,64.38930008008443,451.4103088378906]},{"page":269,"text":"13.","rect":[53.81379699707031,487.10638427734377,64.3901354987856,481.281494140625]},{"page":269,"text":"14.","rect":[53.81295394897461,507.0,64.3892924506899,501.2334289550781]},{"page":269,"text":"15.","rect":[53.812103271484378,527.0,64.38844177319966,521.0270385742188]},{"page":269,"text":"16.","rect":[53.8129997253418,547.0,64.38933822705708,541.0296630859375]},{"page":269,"text":"17.","rect":[53.813838958740237,567.0,64.39017746045552,560.9757080078125]},{"page":269,"text":"De Gennes, P.G.: The Physics of Liquid Crystals. Clarendon Press, Oxford (1975)","rect":[68.59698486328125,259.7184143066406,350.3058175430982,252.07325744628907]},{"page":269,"text":"Chandrasekhar, S.: Liquid Crystals, 2nd edn. Cambridge University Press, Cambridge (1992)","rect":[68.59698486328125,269.6943664550781,385.20089810950449,262.1000061035156]},{"page":269,"text":"De Gennes, P.G., Prost, J.: The Physics of Liquid Crystals, 2nd edn. Clarendon Press, Oxford","rect":[68.59698486328125,279.67034912109377,385.17465728907509,272.0590515136719]},{"page":269,"text":"(1995)","rect":[68.59698486328125,289.3076171875,91.1541985855787,282.0010986328125]},{"page":269,"text":"Kats, E.I., Lebedev, V.V.: Dynamics of Liquid Crystals. Nauka, Moscow (1988) (in Russian)","rect":[68.59698486328125,299.5655212402344,385.1501473770826,291.9711608886719]},{"page":269,"text":"(Fluctuation Effects in the Dynamics of Liquid Crystals. Springer, New York (1993)).","rect":[68.59698486328125,309.5414733886719,363.338412018561,301.9471130371094]},{"page":269,"text":"Landau, L., Lifshitz, E.: Hydrodynamics, 3rd edn. Nauka, Moscow (1986) (in Russian) (see","rect":[68.59698486328125,319.5174255371094,385.1754698493433,311.9230651855469]},{"page":269,"text":"also Fluid Mechanics, 2nd edn. Pergamon, Oxford (1987)).","rect":[68.59698486328125,329.4933776855469,271.1194789130922,321.882080078125]},{"page":269,"text":"Feynman, R.P., Leighton, R.B., Sands, M.: The Feynman Lectures on Physics, vol. 2.","rect":[68.59698486328125,339.41259765625,385.1881739814516,331.8182373046875]},{"page":269,"text":"Addison-Westley, Reading, MA (1964).","rect":[68.59698486328125,349.3885803222656,205.82370254590473,341.7942199707031]},{"page":269,"text":"Atkin, R.J., Sluckin, T.J., Stewart, I.W.: Reflection on the life and work of Frank Matthews","rect":[68.59698486328125,357.64581298828127,385.1373947429589,351.7701416015625]},{"page":269,"text":"Leslie. J. Non-Newtonian Fluid Mech. 119, 7–23 (2004)","rect":[68.59698486328125,368.9450988769531,261.6557626114576,361.3168640136719]},{"page":269,"text":"Leslie, F.: Some constitutive equations for liquid crystals. Arch. Ration. Mech. Anal.","rect":[68.59698486328125,379.25970458984377,385.1882044990297,371.66534423828127]},{"page":269,"text":"(Germany) 28, 265–283 (1968)","rect":[68.59698486328125,389.2356262207031,175.669281898567,381.3533935546875]},{"page":269,"text":"Leslie, F.M.: Introduction to nematodynamics. In: Dunmur, D., Fukuda, A., Luckhurst, G.,","rect":[68.59698486328125,399.2115783691406,385.18396255567037,391.6172180175781]},{"page":269,"text":"INSPEC (eds.) Physical Properties of Liquid crystals: Nematics, pp. 377–386, London (2001).","rect":[68.59698486328125,409.1308288574219,385.162874909186,401.5364685058594]},{"page":269,"text":"Parodi, O.: Stress tensor for nematic liquid crystals. J. Phys. (Paris) 31, 581–584 (1970)","rect":[68.59698486328125,419.1067810058594,369.05548184973886,411.2414855957031]},{"page":269,"text":"Miesowicz,","rect":[68.59698486328125,428.0,107.86480972363911,421.4884033203125]},{"page":269,"text":"M.:","rect":[112.55393981933594,428.0,124.53481773581568,421.65771484375]},{"page":269,"text":"The","rect":[129.15118408203126,428.0,142.35891101145269,421.4884033203125]},{"page":269,"text":"three","rect":[146.99472045898438,428.0,163.9845518805933,421.4884033203125]},{"page":269,"text":"coefficients","rect":[168.633056640625,428.0,207.72321017264643,421.4884033203125]},{"page":269,"text":"of","rect":[212.36328125,428.0,219.4113473282545,421.4884033203125]},{"page":269,"text":"viscosity","rect":[224.0319366455078,429.082763671875,254.15335601954386,421.4884033203125]},{"page":269,"text":"of","rect":[258.81201171875,428.0,265.86007779700449,421.4884033203125]},{"page":269,"text":"anisotropic","rect":[270.480712890625,429.082763671875,308.17479846262457,421.4884033203125]},{"page":269,"text":"liquids.","rect":[312.8512268066406,429.082763671875,338.0482508857485,421.4884033203125]},{"page":269,"text":"Nature","rect":[342.6468505859375,428.0,365.71166369699957,421.65771484375]},{"page":269,"text":"158,","rect":[370.4024963378906,428.45623779296877,385.2059657294985,421.2005310058594]},{"page":269,"text":"27 (1946); Influence of the magnetic field on the viscosity of para-azoxyanisole. Nature","rect":[68.59783172607422,439.0587158203125,385.14244982981207,431.46435546875]},{"page":269,"text":"136, 261 (1936).","rect":[68.59783172607422,448.6392822265625,125.55775711133443,441.1126403808594]},{"page":269,"text":"G€ahwiller, C.: The viscosity coefficients of a room-temperature liquid crystal (MBBA). Phys.","rect":[68.59699249267578,458.9538879394531,385.1687953193422,451.0]},{"page":269,"text":"Lett. 36A, 311–312 (1971); Direct determination of the five independent viscosity coefficients","rect":[68.59698486328125,468.92987060546877,385.1366318035058,461.0645751953125]},{"page":269,"text":"of nematic liquid crystals. Mol. Cryst. Liq. Cryst. 20, 301–318 (1973).","rect":[68.59783172607422,478.8490905761719,310.1080653388735,471.1869812011719]},{"page":269,"text":"Helfrich, W.: Molecular theory of flow alignment of nematic liquid crystals. J. Chem. Phys.","rect":[68.59782409667969,488.8250732421875,385.1703822334047,481.230712890625]},{"page":269,"text":"50, 100–106 (1969)","rect":[68.59782409667969,498.4623718261719,136.1315621720045,491.138916015625]},{"page":269,"text":"Brochard, F., Pieranski, P., Guyon, E.: Dynamics of the orientation of a nematic-liquid-crystal","rect":[68.59698486328125,508.7770080566406,385.10094932761259,501.1826477050781]},{"page":269,"text":"film in a variable magnetic field. Phys. Rev. Lett. 28, 1681–1683 (1972)","rect":[68.59697723388672,518.6962280273438,315.9216622696607,510.81402587890627]},{"page":269,"text":"Leslie, F.M.: Continuum theory of cholesteric liquid crystals. Mol. Cryst. Liq. Cryst. 7,","rect":[68.59613037109375,528.6721801757813,385.2051722724672,521.0778198242188]},{"page":269,"text":"407–420 (1969)","rect":[68.5970230102539,538.3095092773438,123.1590813980787,531.0537719726563]},{"page":269,"text":"Helfrich, W.: Capillary flow of cholesteric and smectic liquid crystals. Phys. Rev. Lett. 23,","rect":[68.59703063964844,548.6240234375,385.2059962470766,540.7587890625]},{"page":269,"text":"372–374 (1969)","rect":[68.59786224365235,558.2046508789063,123.15992063147714,550.9489135742188]},{"page":269,"text":"Schneider, F., Kneppe, H.: Flow phenomena and viscosity. In: Demus, D., Goodby, J., Gray,","rect":[68.59786987304688,568.5192260742188,385.1756007392641,560.9248657226563]},{"page":269,"text":"G.W., Spiess, H.-W., Vill, V. (eds.) Physical Properties of Liquid Crystals, pp. 352–374.","rect":[68.59786224365235,578.4951782226563,385.1942469794985,570.8500366210938]},{"page":269,"text":"Wiley-VCH, Weinheim (1999).","rect":[68.59786224365235,588.4711303710938,176.99179336621723,580.8767700195313]},{"page":270,"text":"Chapter 10","rect":[53.812843322753909,72.10812377929688,121.10908599090695,59.571903228759769]},{"page":270,"text":"Liquid Crystal – Solid Interface","rect":[53.812843322753909,91.18268585205078,272.3518883300605,76.0426254272461]},{"page":270,"text":"10.1 General Properties","rect":[53.812843322753909,212.0922088623047,184.5524204227702,201.34686279296876]},{"page":270,"text":"10.1.1 Symmetry","rect":[53.812843322753909,241.9628448486328,144.93792518110483,231.39678955078126]},{"page":270,"text":"Now we are interested in phenomena at an interface between a liquid crystal and","rect":[53.812843322753909,269.5052795410156,385.14174738935005,260.57073974609377]},{"page":270,"text":"another phase (gas, liquid or solid) [1, 2]. Why is it important? First, the structure of","rect":[53.812843322753909,281.4648132324219,385.14971290437355,272.5302734375]},{"page":270,"text":"a liquid crystal in a thin interfacial layer is different from that in the bulk and","rect":[53.812862396240237,293.42437744140627,385.14080134442818,284.48980712890627]},{"page":270,"text":"manifests many novel features. Second, the interface plays a decisive role in","rect":[53.812862396240237,305.3271484375,385.1397332291938,296.392578125]},{"page":270,"text":"applications, because liquid crystals are always used in a confined geometry.","rect":[53.812862396240237,317.28668212890627,385.13982816244848,308.35211181640627]},{"page":270,"text":"There are two approaches to the surface problems, microscopic and macroscopic.","rect":[53.812862396240237,329.2462158203125,385.1347622444797,320.3116455078125]},{"page":270,"text":"In the first approach, we are interested in a structure and properties of interfacial","rect":[53.812862396240237,341.2057800292969,385.1805864102561,332.271240234375]},{"page":270,"text":"liquid crystal layers at the molecular level; in the second one, we ignore the","rect":[53.812862396240237,353.165283203125,385.11692083551255,344.230712890625]},{"page":270,"text":"microscopic details and use only symmetry properties and the concept of the","rect":[53.812862396240237,365.1248474121094,385.1149066753563,356.1903076171875]},{"page":270,"text":"director.","rect":[53.812862396240237,375.05242919921877,87.48496671225314,368.14984130859377]},{"page":270,"text":"What does occur at the interface? Consider, for example, a contact of a liquid","rect":[65.76488494873047,389.0439453125,385.1109551530219,380.109375]},{"page":270,"text":"crystal with a solid substrate shown in Fig. 10.1. We notice that","rect":[53.812862396240237,400.94671630859377,311.33027513095927,392.01214599609377]},{"page":270,"text":"(i)","rect":[61.06350326538086,418.5159606933594,70.4901364883579,409.9798583984375]},{"page":270,"text":"(ii)","rect":[58.28824234008789,454.33782958984377,70.512009962479,445.80169677734377]},{"page":270,"text":"(iii)","rect":[55.45624923706055,526.0951538085938,70.55280433503759,517.5590209960938]},{"page":270,"text":"There is a change in symmetry; the interface is not a mirror plane, there is a","rect":[75.62153625488281,418.9143981933594,383.06137884332505,409.9798583984375]},{"page":270,"text":"new, polar vector, namely the normal h to the interface. Therefore, an","rect":[75.62152862548828,430.8739318847656,385.15065852216255,421.93939208984377]},{"page":270,"text":"interfacial layer has properties of a polar phase.","rect":[75.62150573730469,442.83349609375,267.81243558432348,433.89892578125]},{"page":270,"text":"The properties change continuously along the surface normal for the phase of","rect":[75.62150573730469,454.73626708984377,385.1476987442173,445.80169677734377]},{"page":270,"text":"the same symmetry. Macroscopically we can consider a change of order","rect":[75.62150573730469,466.6958312988281,385.0889829239048,457.76129150390627]},{"page":270,"text":"parameters with distance. In a particular case of the nematic phase, both the","rect":[75.62150573730469,478.6553649902344,385.17163885309068,469.7208251953125]},{"page":270,"text":"absolute value of the orientational order |Q| ¼ S and the direction of the","rect":[75.62150573730469,490.5053405761719,385.1167682476219,481.60064697265627]},{"page":270,"text":"director n can depend on distance from the interface. The positional order","rect":[75.62149047851563,502.574462890625,385.1426938614048,493.639892578125]},{"page":270,"text":"can also change.","rect":[75.6214828491211,514.5339965820313,142.06086392905002,505.5994873046875]},{"page":270,"text":"The molecules at the surface are in different surrounding in comparison with","rect":[75.62149047851563,526.4935302734375,385.11480036786568,517.5590209960938]},{"page":270,"text":"those","rect":[75.6214828491211,536.391357421875,96.65473211480939,529.5186157226563]},{"page":270,"text":"in","rect":[101.90457916259766,536.3116455078125,109.67881861737738,529.5186157226563]},{"page":270,"text":"the","rect":[114.93264770507813,536.391357421875,127.1564182110008,529.5186157226563]},{"page":270,"text":"bulk.","rect":[132.37939453125,536.391357421875,152.59640164877659,529.5186157226563]},{"page":270,"text":"Therefore,","rect":[157.81240844726563,536.391357421875,199.75147672201877,529.5186157226563]},{"page":270,"text":"the","rect":[204.99734497070313,536.391357421875,217.22110784723129,529.5186157226563]},{"page":270,"text":"molecular","rect":[222.444091796875,536.391357421875,262.51677833406105,529.5186157226563]},{"page":270,"text":"interactions","rect":[267.7596435546875,536.391357421875,314.47775663481908,529.5186157226563]},{"page":270,"text":"and,","rect":[319.70172119140627,536.4212036132813,336.5940212776828,529.5186157226563]},{"page":270,"text":"hence,","rect":[341.849853515625,536.391357421875,367.7327847054172,529.5186157226563]},{"page":270,"text":"the","rect":[372.8910827636719,536.391357421875,385.11484564020005,529.5186157226563]},{"page":270,"text":"thermodynamic properties are also different. Even new liquid crystal phases","rect":[75.6214828491211,550.3558349609375,385.1496926699753,541.4213256835938]},{"page":270,"text":"can form at the surface. For instance, if in the bulk a uniaxial nematic is stable,","rect":[75.6214828491211,560.253662109375,385.1656155159641,553.3809204101563]},{"page":270,"text":"at the surface it could be transformed in either a uniaxial smectic A or biaxial","rect":[75.6214828491211,572.2430419921875,385.12580735752177,565.3404541015625]},{"page":270,"text":"nematic.","rect":[75.6214828491211,584.1727294921875,109.80424161459689,577.2999877929688]},{"page":270,"text":"L.M. Blinov, Structure and Properties of Liquid Crystals,","rect":[53.812843322753909,623.7623291015625,251.63688919141254,616.1510620117188]},{"page":270,"text":"DOI 10.1007/978-90-481-8829-1_10, # Springer ScienceþBusiness Media B.V. 2011","rect":[53.812843322753909,633.698486328125,351.58160919337197,625.4920043945313]},{"page":270,"text":"257","rect":[372.4981994628906,622.0606079101563,385.18979400782509,616.1340942382813]},{"page":271,"text":"258","rect":[53.812530517578128,42.55624771118164,66.5041365066044,36.6297721862793]},{"page":271,"text":"Fig. 10.1 A liquid crystal","rect":[53.812843322753909,67.58130645751953,144.18554405417505,59.85148620605469]},{"page":271,"text":"phase at the interface with an","rect":[53.812843322753909,77.4895248413086,154.44034332423136,69.89517211914063]},{"page":271,"text":"isotropic phase (gas, liquid,","rect":[53.812843322753909,87.4087142944336,148.08862564160786,79.81436157226563]},{"page":271,"text":"amorphous solid) witha","rect":[53.812843322753909,97.3846664428711,136.72794482492925,89.79031372070313]},{"page":271,"text":"surface layer of thickness x","rect":[53.812843322753909,107.36067962646485,147.6554159560673,99.49539947509766]},{"page":271,"text":"and a qualitative dependence","rect":[53.812843322753909,117.33663177490235,153.0138029792261,109.74227905273438]},{"page":271,"text":"of an order parameter (e.g.","rect":[53.812843322753909,127.25582122802735,145.34302017285786,119.66146850585938]},{"page":271,"text":"orientational) on the distance","rect":[53.812843322753909,136.8931121826172,153.45885607981206,129.63742065429688]},{"page":271,"text":"z from the surface (Ss) to the","rect":[53.812843322753909,146.8690643310547,152.00307605051519,139.61251831054688]},{"page":271,"text":"bulk (SB) values (n is","rect":[53.81320571899414,156.8441925048828,127.2520874309472,149.58843994140626]},{"page":271,"text":"director)","rect":[53.812530517578128,166.7633819580078,83.04552548987557,159.5076904296875]},{"page":271,"text":"10 Liquid Crystal","rect":[265.381591796875,44.274925231933597,325.71578697409697,36.68056869506836]},{"page":271,"text":"h","rect":[244.30760192871095,145.5693359375,249.19148584388354,139.82655334472657]},{"page":271,"text":"Symm 1","rect":[248.83740234375,180.4895782470703,275.659779755308,173.84100341796876]},{"page":271,"text":"x","rect":[296.7209167480469,114.3245849609375,300.66159558631218,106.33428955078125]},{"page":271,"text":"Symm 1–2","rect":[284.50103759765627,180.4902801513672,319.1008808295268,173.84170532226563]},{"page":271,"text":"– Solid Interface","rect":[328.0874328613281,43.0,385.1267637946558,36.68056869506836]},{"page":271,"text":"S","rect":[356.9638671875,96.43101501464844,362.2953738510354,90.39230346679688]},{"page":271,"text":"s","rect":[362.2945556640625,98.3704833984375,365.2920294254205,95.05918884277344]},{"page":271,"text":"Symm 2","rect":[331.7720642089844,180.4895782470703,358.5944721381205,173.84100341796876]},{"page":271,"text":"(iv) The elastic moduli such as K24 and K13 often neglected in the description of","rect":[56.0787353515625,225.12423706054688,382.54412208406105,216.18968200683595]},{"page":271,"text":"bulk properties becomes important at the interface.","rect":[75.62197875976563,237.08377075195313,281.2381863167453,228.1492156982422]},{"page":271,"text":"10.1.2 Surface Properties of a Liquid","rect":[53.812843322753909,287.1673278808594,246.5351995458562,276.61322021484377]},{"page":271,"text":"As throughout the book, at first we review","rect":[53.812843322753909,314.7932434082031,233.3778662677082,305.85870361328127]},{"page":271,"text":"interface and then switch to liquid crystals.","rect":[53.812843322753909,326.7528076171875,227.59148831869846,317.8182373046875]},{"page":271,"text":"properties","rect":[237.56658935546876,314.7932434082031,277.53375639067846,305.85870361328127]},{"page":271,"text":"of","rect":[281.6358947753906,313.0,289.9277585586704,305.85870361328127]},{"page":271,"text":"isotropic","rect":[293.98406982421877,314.7932434082031,328.97910345270005,305.85870361328127]},{"page":271,"text":"liquids","rect":[333.0693054199219,314.7932434082031,360.2960549746628,305.85870361328127]},{"page":271,"text":"at","rect":[364.3941955566406,313.0,371.6408830411155,306.8746643066406]},{"page":271,"text":"an","rect":[375.7231140136719,313.0,385.1497429948188,307.0]},{"page":271,"text":"10.1.2.1 Surface Tension","rect":[53.812843322753909,366.8397216796875,165.45331162760807,359.518798828125]},{"page":271,"text":"Due to a difference in molecular forces in the bulk and at the interface, there is an","rect":[53.812843322753909,390.469970703125,385.14870539716255,383.5673828125]},{"page":271,"text":"excess of energy in a surface layer. For instance, one should make a work to","rect":[53.812843322753909,404.4614562988281,385.13872614911568,395.52691650390627]},{"page":271,"text":"increase an interface A between a gas and a liquid. When a chemical composition","rect":[53.812843322753909,416.3642272949219,385.15667048505318,407.4296875]},{"page":271,"text":"of the contacting phases is fixed, the surface tension is","rect":[53.81285095214844,428.3237609863281,273.5381204043503,419.38922119140627]},{"page":271,"text":"dF","rect":[223.06858825683595,449.545166015625,234.15758508123123,442.55291748046877]},{"page":271,"text":"s¼","rect":[203.18572998046876,456.2459411621094,220.25322750303656,451.56451416015627]},{"page":271,"text":"dA","rect":[223.06858825683595,463.148193359375,234.15758508123123,456.15594482421877]},{"page":271,"text":"(10.1)","rect":[361.0558776855469,457.9099426269531,385.1052487930454,449.43359375]},{"page":271,"text":"where the free energy of the surface F ¼ E \u0002 TS takes into account not only a","rect":[53.814205169677737,486.6489562988281,385.16101873590318,477.71441650390627]},{"page":271,"text":"change in the internal energy E but also entropy S.","rect":[53.81421661376953,498.6084899902344,258.18511624838598,489.6739501953125]},{"page":271,"text":"The surface tension determines capillaryeffects, wetting phenomena and a shape","rect":[65.76624298095703,510.5680236816406,385.0973590679344,501.63348388671877]},{"page":271,"text":"of liquid drops, in particular, the spherical shape of small radius drops when the","rect":[53.81421661376953,522.4707641601563,385.17298162652818,513.5362548828125]},{"page":271,"text":"gravity is not essential. The corresponding excess pressure in a drop of radius r is","rect":[53.81421661376953,534.4302978515625,385.1888772402878,525.4957885742188]},{"page":271,"text":"Dp ¼ 2s/r (Laplace-Young formula). Small drops of the nematic phase are,","rect":[53.81426239013672,546.3898315429688,385.1591457894016,537.1265869140625]},{"page":271,"text":"strictly speaking, not spherical due to anisotropy of the surface tension but practi-","rect":[53.81427764892578,558.349365234375,385.11528907624855,549.4148559570313]},{"page":271,"text":"cally they may be considered spherical. The surface tension of both a liquid crystal","rect":[53.81427764892578,570.3089599609375,385.122206283303,561.3744506835938]},{"page":271,"text":"and a solid substrate determines orientation of the liquid crystal director on the","rect":[53.81427764892578,582.2684936523438,385.17298162652818,573.333984375]},{"page":271,"text":"substrate.","rect":[53.81427764892578,592.166259765625,91.81547208334689,585.2935180664063]},{"page":272,"text":"10.1 General Properties","rect":[53.812835693359378,44.277000427246097,134.60504611014643,36.68264389038086]},{"page":272,"text":"259","rect":[372.4981994628906,42.62605285644531,385.1898245254032,36.6318473815918]},{"page":272,"text":"10.1.2.2 Adsorption","rect":[53.812843322753909,68.2186279296875,144.23590440104557,59.27412033081055]},{"page":272,"text":"The adsorption is a non-uniform spatial distribution of different chemical species at","rect":[53.812843322753909,92.20748138427735,385.177626205178,83.27293395996094]},{"page":272,"text":"an interface between different media [3]. The situation at the liquid–gas interface is","rect":[53.812843322753909,104.11019134521485,385.1865273867722,95.17564392089844]},{"page":272,"text":"illustrated by Fig. 10.2. If there is some concentration of surface-active (surfactant)","rect":[53.81285095214844,116.0697250366211,385.1706479629673,107.13517761230469]},{"page":272,"text":"molecules in water, they “prefer” to leave the bulk and go to the surface. The reason","rect":[53.81285095214844,128.02932739257813,385.1258477311469,119.09477233886719]},{"page":272,"text":"is that water does not like such guests, because surfactant molecules destroy a","rect":[53.81285095214844,139.98886108398438,385.15772283746568,131.05430603027345]},{"page":272,"text":"network of hydrogen bonds formed by water molecules. The break of the network","rect":[53.81285095214844,151.94839477539063,385.1756524186469,143.0138397216797]},{"page":272,"text":"would cost considerable decrease in entropy. Therefore, water pushes its guests out","rect":[53.81285095214844,163.907958984375,385.20442063877177,154.97340393066407]},{"page":272,"text":"to the surface. Then, at the water surface, we see a peak of concentration of","rect":[53.81285095214844,175.86749267578126,385.1497739395298,166.9329376220703]},{"page":272,"text":"surfactant molecules. In this way the total surface energy is reduced and the surface","rect":[53.81285095214844,187.8270263671875,385.08106268121568,178.89247131347657]},{"page":272,"text":"tension decreases. For instance, the excess of foreign molecules on the water","rect":[53.81285095214844,199.72976684570313,385.08098731843605,190.7952117919922]},{"page":272,"text":"surface creates a certain pressure on a floating barrier in a Langmuir trough and","rect":[53.81285095214844,211.6893310546875,385.14275446942818,202.75477600097657]},{"page":272,"text":"the barrier shifts in the direction of the pure water surface. The so-called surface","rect":[53.81285095214844,223.64886474609376,385.0830768413719,214.7143096923828]},{"page":272,"text":"pressure exerted onto the barrier is p ¼ s0 \u0002 s where s0 is surface tension of pure","rect":[53.81285095214844,235.61026000976563,385.11829412652818,226.6738739013672]},{"page":272,"text":"water. Measuring the temperature dependence p(T) one can study single mono-","rect":[53.81427764892578,247.56979370117188,385.14910255281105,238.63523864746095]},{"page":272,"text":"layers of liquid crystalline compounds forming on water different two-dimensional","rect":[53.814292907714847,259.52935791015627,385.1730790860374,250.5948028564453]},{"page":272,"text":"phases as discussed in Section 5.7.3.","rect":[53.814292907714847,271.4888916015625,201.11172147299534,262.49456787109377]},{"page":272,"text":"In liquid crystal cells the adsorption of impurities from the bulk to a liquid","rect":[65.7663345336914,283.44842529296877,385.11235896161568,274.51385498046877]},{"page":272,"text":"crystal – glass (or other solid substrates) interfaces can change conditions for","rect":[53.814292907714847,295.3511962890625,385.1233151992954,286.4166259765625]},{"page":272,"text":"alignment of the liquid crystal and often results in a misalignment of liquid crystals","rect":[53.814292907714847,307.3107604980469,385.1044045840378,298.376220703125]},{"page":272,"text":"undesirable for displays. On the other hand, using adsorption phenomena and","rect":[53.814292907714847,319.270263671875,385.14519587567818,310.3157958984375]},{"page":272,"text":"Langmuir-Blodgett technology one can prepare ultra-thin polymer films on the","rect":[53.814292907714847,331.2298278808594,385.17102850152818,322.2952880859375]},{"page":272,"text":"solid substrates. Such films can be rubbed by soft brushing or scribed by Atomic","rect":[53.814292907714847,343.1893615722656,385.1809467144188,334.25482177734377]},{"page":272,"text":"Force Microscopes or modified by polarised light for desirable alignment of liquid","rect":[53.814292907714847,355.14892578125,385.11235896161568,346.21435546875]},{"page":272,"text":"crystals. In some cases, a “negative” adsorption isobserved when foreign molecules","rect":[53.814292907714847,367.10845947265627,385.1600686465378,358.17388916015627]},{"page":272,"text":"are expelled from the surface into the bulk. Such desorption increases surface","rect":[53.814292907714847,379.0679931640625,385.0845111675438,370.1334228515625]},{"page":272,"text":"tension. Adsorption of ions at the electrodes of a liquid crystal cell may create a","rect":[53.814292907714847,390.9707336425781,385.15915716363755,382.03619384765627]},{"page":272,"text":"space charge at the interface that dramatically influences conditions for the current","rect":[53.814292907714847,402.9302978515625,385.1711259610374,393.9957275390625]},{"page":272,"text":"flow through a liquid crystal especially at low frequencies.","rect":[53.814292907714847,414.8898620605469,290.2775845101047,405.955322265625]},{"page":272,"text":"Fig. 10.2 Adsorption of","rect":[53.812843322753909,499.0357971191406,138.40832608802013,491.30596923828127]},{"page":272,"text":"surfactant molecules (black","rect":[53.812843322753909,508.60540771484377,147.72309253000737,501.332763671875]},{"page":272,"text":"spheres) at the interface","rect":[53.812843322753909,518.826904296875,135.9012999274683,511.3087158203125]},{"page":272,"text":"between water (white","rect":[53.812843322753909,528.5573120117188,126.93509814524174,521.28466796875]},{"page":272,"text":"spheres) and air. The curveC","rect":[53.812843322753909,538.7220458984375,155.34820735182147,531.203857421875]},{"page":272,"text":"qualitatively pictures the","rect":[53.812835693359378,548.7910766601563,138.40831897043706,541.1967163085938]},{"page":272,"text":"surfactant concentration asa","rect":[53.812835693359378,557.0145263671875,151.51195666574956,551.1726684570313]},{"page":272,"text":"function of the distance z","rect":[53.812835693359378,566.9591674804688,140.16821749686518,561.0919189453125]},{"page":272,"text":"from the interface shown by","rect":[53.812835693359378,578.6622314453125,150.47125763087198,571.06787109375]},{"page":272,"text":"the dash line","rect":[53.812835693359378,586.9957275390625,97.6783995368433,581.0269165039063]},{"page":272,"text":"Z","rect":[312.0286560058594,482.572509765625,316.91253992103199,476.8297424316406]},{"page":272,"text":"C","rect":[373.9764404296875,519.9589233398438,379.74757657822206,513.9202270507813]},{"page":273,"text":"260","rect":[53.815399169921878,42.55612564086914,66.50700515894815,36.68044662475586]},{"page":273,"text":"10 Liquid Crystal – Solid Interface","rect":[265.38446044921877,44.274803161621097,385.12963244699957,36.68044662475586]},{"page":273,"text":"α0=π","rect":[278.6947937011719,71.2574462890625,294.48258510818388,66.0087890625]},{"page":273,"text":"b","rect":[143.1696319580078,114.51565551757813,149.27442570748603,107.20728302001953]},{"page":273,"text":"σ23","rect":[142.12271118164063,125.96472930908203,152.93779663733455,120.54658508300781]},{"page":273,"text":"σ13","rect":[165.5336151123047,91.75489044189453,176.34856323889705,86.33625793457031]},{"page":273,"text":"σ12","rect":[163.14300537109376,115.37274169921875,173.95749573401424,110.02052307128906]},{"page":273,"text":"2","rect":[158.58724975585938,123.1640625,162.58388143767002,117.58125305175781]},{"page":273,"text":"α0","rect":[174.36483764648438,127.35752868652344,181.51276251721414,122.2408218383789]},{"page":273,"text":"σ13","rect":[171.7000732421875,137.3304443359375,182.51528076819393,132.14495849609376]},{"page":273,"text":"1","rect":[187.5364532470703,128.106201171875,191.53308492888096,122.52339172363281]},{"page":273,"text":"0<α0<π","rect":[272.6779479980469,100.1949462890625,296.76368252029325,93.39639282226563]},{"page":273,"text":"α0 =0","rect":[267.49810791015627,130.2357177734375,284.71246807119146,123.43765258789063]},{"page":273,"text":"Fig. 10.3 Surface tension in a three-phase system. Illustration of the Neumann triangle (a) and","rect":[53.812843322753909,160.65133666992188,385.1948599257938,152.71832275390626]},{"page":273,"text":"Young law (b) and the three cases of wetting phenomena: non-wetting (c), partial wetting (d) and","rect":[53.813682556152347,170.50286865234376,385.1957144179813,162.90850830078126]},{"page":273,"text":"complete wetting (e)","rect":[53.814552307128909,180.47882080078126,124.66331571204354,172.88446044921876]},{"page":273,"text":"10.1.2.3 Wetting","rect":[53.812843322753909,200.4344482421875,130.58071986249457,191.10147094726563]},{"page":273,"text":"Consider a drop of a liquid on a soft substrate, Fig. 10.3a [3]. There are three phases","rect":[53.812843322753909,224.27386474609376,385.1377297793503,215.3393096923828]},{"page":273,"text":"in contact: liquid (1), gas (2) and soft substrate (3). The soft substrate could be an","rect":[53.81282424926758,236.1766357421875,385.1507195573188,227.24208068847657]},{"page":273,"text":"elastomer or a liquid different from the liquid (1). At any point of the contact line","rect":[53.81282424926758,248.13616943359376,385.13877142145005,239.2016143798828]},{"page":273,"text":"the equilibrium condition is the vector sum of the corresponding tensions for each","rect":[53.81282424926758,260.0957336425781,385.09795466474068,251.1611785888672]},{"page":273,"text":"pair of contacting phases:","rect":[53.81282424926758,272.05523681640627,157.16142137119364,263.12066650390627]},{"page":273,"text":"~","rect":[176.90261840820313,284.44207763671877,183.9501991666009,282.4599609375]},{"page":273,"text":"~","rect":[203.12960815429688,284.4425964355469,210.17718891269466,282.4604797363281]},{"page":273,"text":"~","rect":[229.35643005371095,284.4425964355469,236.4040108121087,282.4604797363281]},{"page":273,"text":"s12 þ s23 þ s13 ¼0","rect":[177.07284545898438,291.4234313964844,261.91997614911568,283.1328125]},{"page":273,"text":"This is a so-called Neumann triangle valid for any three phases.","rect":[65.76628875732422,311.9424743652344,323.7873806526828,302.94818115234377]},{"page":273,"text":"When a substrate is solid, see Fig. 10.3b, the vertical component of its deforma-","rect":[65.76628875732422,323.8621826171875,385.10833106843605,314.9276123046875]},{"page":273,"text":"tion is negligible and we may only consider the equilibrium of horizontal projec-","rect":[53.813289642333987,335.8217468261719,385.1621030410923,326.88720703125]},{"page":273,"text":"tions of the surface tension vectors (Young’s law):","rect":[53.813289642333987,347.82110595703127,256.4830003506858,338.8268127441406]},{"page":273,"text":"s12 cosa0 ¼ s23 \u0002 s13","rect":[172.59786987304688,367.09234619140627,265.89292976936658,360.82366943359377]},{"page":273,"text":"(10.2)","rect":[361.0561828613281,367.2780456542969,385.10555396882668,358.80169677734377]},{"page":273,"text":"and the ratio ðs23 \u0002 s13Þ=s12 determines the equilibrium contact angle a0.","rect":[53.81450653076172,387.96697998046877,356.5207790901828,378.0162353515625]},{"page":273,"text":"The surface tension at a liquid-solid interface s13 may be controlled by temper-","rect":[65.7662582397461,399.53106689453127,385.10787330476418,390.5963134765625]},{"page":273,"text":"ature, composition of the liquid or adsorption. We can distinguish three different","rect":[53.8138542175293,411.4906311035156,385.1736284024436,402.55609130859377]},{"page":273,"text":"cases, shown in Fig. 10.3c–e: non-wetting (c), partial wetting (d) and complete","rect":[53.8138542175293,423.4501647949219,385.1825336284813,414.515625]},{"page":273,"text":"wetting (e). The three cases are characterized by their equilibrium contact angles:","rect":[53.814842224121097,435.4096984863281,385.15962083408427,426.47515869140627]},{"page":273,"text":"a0 ¼ p; p > a0 > 0; a0 ¼ 0, respectively.","rect":[53.814842224121097,447.36968994140627,227.20188565756565,438.43511962890627]},{"page":273,"text":"The spreading parameter S ¼ s23 \u0002 (s13 \u0002 s12) determines a wetting transi-","rect":[65.76549530029297,459.369140625,385.1603940567173,450.3946533203125]},{"page":273,"text":"tion: for S > 0 one observed complete wetting, for S < 0 the wetting is partial. The","rect":[53.813594818115237,471.2888488769531,385.14746893121568,462.35430908203127]},{"page":273,"text":"wetting transition is often observed with volatile liquids on solid substrates. The","rect":[53.81456756591797,483.2484130859375,385.14548528863755,474.3138427734375]},{"page":273,"text":"dynamics of the complete wetting is very interesting: at ��rst, a microscopically thin","rect":[53.81456756591797,495.15118408203127,385.15444270185005,486.21661376953127]},{"page":273,"text":"precursor forms that advances rather fast over the substrate followed by a macro-","rect":[53.81456756591797,507.1107177734375,385.17830787507668,498.1761474609375]},{"page":273,"text":"scopic edge of the liquid film. Afterwards all amount of liquid forms a uniform","rect":[53.81456756591797,519.0702514648438,385.1644053328093,510.1357421875]},{"page":273,"text":"layer. This has been observed in both isotropic and nematic liquids [4].","rect":[53.81456756591797,531.02978515625,341.14385648276098,522.0952758789063]},{"page":273,"text":"10.1.3 Structure of Surface Layers","rect":[53.812843322753909,566.2339477539063,233.7957981083171,555.59619140625]},{"page":273,"text":"The most interesting case is a contact of a nematic liquid crystal with a solid","rect":[53.812843322753909,593.7762451171875,385.1347283463813,584.8417358398438]},{"page":273,"text":"substrate because in most devices a nematic is sandwiched between transparent","rect":[53.812843322753909,605.7357788085938,385.14277513095927,596.80126953125]},{"page":274,"text":"10.1 General Properties","rect":[53.813716888427737,44.275779724121097,134.60593111991205,36.68142318725586]},{"page":274,"text":"261","rect":[372.49908447265627,42.55710220336914,385.1907095351688,36.68142318725586]},{"page":274,"text":"glasses, conductive or non-conductive. The interaction with a substrate causes","rect":[53.812843322753909,68.2883529663086,385.11883939849096,59.35380554199219]},{"page":274,"text":"many effects such as a change in the orientational order parameter, appearance of","rect":[53.812843322753909,80.24788665771485,385.1497739395298,71.31333923339844]},{"page":274,"text":"a short range positional order, appearance of a surface dipolar layer, etc. [5].","rect":[53.812843322753909,92.20748138427735,362.9497952034641,83.21317291259766]},{"page":274,"text":"10.1.3.1 Surface Induced Change in the Orientational Order Parameter","rect":[53.81282424926758,128.1090087890625,367.45114935113755,118.77603912353516]},{"page":274,"text":"A qualitative picture, Fig. 10.4, shows the distance dependencies of the orienta-","rect":[53.81282424926758,151.94839477539063,385.1048520645298,143.0138397216797]},{"page":274,"text":"tional order parameter for homeotropically aligned nematic liquid crystal at the","rect":[53.81281661987305,163.907958984375,385.1715473003563,154.97340393066407]},{"page":274,"text":"solid substrate. The problem is to explain such dependencies [6]. The influence of","rect":[53.81281661987305,175.86749267578126,385.1496518692173,166.9329376220703]},{"page":274,"text":"the surface on the orientational order parameter may be discussed in terms of the","rect":[53.81280517578125,187.8270263671875,385.1706012554344,178.89247131347657]},{"page":274,"text":"modified","rect":[53.81280517578125,198.0,89.25975123456488,190.7952423095703]},{"page":274,"text":"Landau–de","rect":[94.54045867919922,198.0,138.97306097223129,190.7952423095703]},{"page":274,"text":"Gennes","rect":[144.21694946289063,198.0,174.12933744536594,190.85501098632813]},{"page":274,"text":"phase","rect":[179.39312744140626,199.72979736328126,202.07877386285629,190.7952423095703]},{"page":274,"text":"transition","rect":[207.31866455078126,198.0,245.12873164227973,190.7952423095703]},{"page":274,"text":"theory.","rect":[250.36862182617188,199.72979736328126,278.4255642464328,190.7952423095703]},{"page":274,"text":"Consider","rect":[283.6186828613281,198.0,319.64299903718605,190.7952423095703]},{"page":274,"text":"a","rect":[324.9127502441406,198.0,329.36228216363755,192.0]},{"page":274,"text":"semi-infinite","rect":[334.59918212890627,198.0,385.14368475152818,190.7952423095703]},{"page":274,"text":"nematic of area A being in contact with a substrate at z ¼ 0 and uniform in the x","rect":[53.81280517578125,211.68936157226563,385.1605609722313,202.7548065185547]},{"page":274,"text":"and y directions. When writing the free energy density a surface term -Wd(z)S must","rect":[53.81280517578125,223.64886474609376,385.2072892911155,214.41549682617188]},{"page":274,"text":"be added to the standard expansion of the bulk free energy density:","rect":[53.812747955322269,235.60842895507813,325.0232072598655,226.6738739013672]},{"page":274,"text":"g ¼ g0ðSÞ þ K\u0003\u0001ddSz\u00032 \u0002 WdAðzÞS","rect":[151.6954803466797,273.67620849609377,285.6551140885688,247.85684204101563]},{"page":274,"text":"(10.3)","rect":[361.0572204589844,265.9886779785156,385.1065915664829,257.5123291015625]},{"page":274,"text":"where","rect":[53.815574645996097,293.1190185546875,78.15361822320783,286.24627685546877]},{"page":274,"text":"g0ðSÞ ¼ aðT \u0002 T\u0003ÞS2 þ bS3 þ cS4","rect":[151.35787963867188,318.7594909667969,287.10630867561658,307.4283752441406]},{"page":274,"text":"is a uniform part of the free energy density, which describes the first order N-I phase","rect":[53.812843322753909,340.2995910644531,385.0979694194969,331.36505126953127]},{"page":274,"text":"transition, T* is “virtual second order” transition temperature for the bulk, a, b, c are","rect":[53.812843322753909,352.2591552734375,385.1636127300438,343.3245849609375]},{"page":274,"text":"Landau expansion coefficients and K* is a new “gradient” elastic modulus, other","rect":[53.81185531616211,364.21868896484377,385.1357663711704,355.28411865234377]},{"page":274,"text":"than Frank moduli Kii. The surface term is chosen in the spirit of the mean field","rect":[53.81184387207031,376.1786804199219,385.11724177411568,367.24371337890627]},{"page":274,"text":"theory with a cylindrically symmetric potential of a substrate","rect":[53.8132209777832,388.1382141113281,300.27256048395005,379.20367431640627]},{"page":274,"text":"WðW;zÞ ¼ WdðzÞhP2ðcosWÞi:","rect":[160.81500244140626,410.3560485839844,278.20997559708499,400.4051513671875]},{"page":274,"text":"Here d(z) is Dirac delta function, showing that the surface potential W ¼W","rect":[65.76622772216797,431.9519958496094,385.1530630769267,422.7186279296875]},{"page":274,"text":"(z ¼ 0) is short-range, W is an angle between the longitudinal axis of a rod-like","rect":[53.81523895263672,443.9115295410156,385.1282123394188,434.67816162109377]},{"page":274,"text":"molecule and the director at the surface ns. The surface potential W may be positive","rect":[53.81523895263672,455.8718566894531,385.1248248882469,446.9365234375]},{"page":274,"text":"as in Fig. 10.4a or negative, Fig. 10.4b. In this consideration, a change of the","rect":[53.81289291381836,467.7746276855469,385.17359197809068,458.840087890625]},{"page":274,"text":"a","rect":[129.6607208251953,495.60662841796877,135.21598322242915,490.01788330078127]},{"page":274,"text":"W","rect":[143.58888244628907,507.31292724609377,151.13452306154756,501.5701599121094]},{"page":274,"text":"S0","rect":[155.58505249023438,521.4718627929688,164.47189240365166,513.8623657226563]},{"page":274,"text":"b","rect":[222.92718505859376,495.60662841796877,229.03197880807196,488.2982482910156]},{"page":274,"text":"Sbulk","rect":[289.3243713378906,510.9037780761719,306.3835848330177,503.3196716308594]},{"page":274,"text":"x","rect":[148.94886779785157,545.3893432617188,152.49547875229033,538.1981201171875]},{"page":274,"text":"0","rect":[135.236328125,557.4743041992188,138.7917318811907,552.8609008789063]},{"page":274,"text":"Sbulk","rect":[199.79660034179688,547.163330078125,216.85573754297867,539.5792846679688]},{"page":274,"text":"z","rect":[211.29661560058595,561.9513549804688,214.8935841142155,558.1865844726563]},{"page":274,"text":"z","rect":[305.21685791015627,541.1561279296875,308.8138264237858,537.391357421875]},{"page":274,"text":"Fig. 10.4 Qualitative dependencies of the orientational order parameter of the nematic phase on","rect":[53.812843322753909,583.43359375,385.20673126368447,575.7037963867188]},{"page":274,"text":"the distance z from the surface. The positive (a) and negative (b) surface potential W has the form","rect":[53.812843322753909,593.2850952148438,385.17795036668766,585.6907348632813]},{"page":274,"text":"of the d-function. The temperature is fixed","rect":[53.813716888427737,603.2610473632813,199.36949676661417,595.4127197265625]},{"page":275,"text":"262","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.68026351928711]},{"page":275,"text":"10 Liquid Crystal – Solid Interface","rect":[265.38189697265627,44.274620056152347,385.12706897043707,36.68026351928711]},{"page":275,"text":"mesophase symmetry, in particular, an appearance of the polar axis in the nematic","rect":[53.812843322753909,68.2883529663086,385.0999225444969,59.35380554199219]},{"page":275,"text":"phase is disregarded.","rect":[53.812843322753909,80.24788665771485,137.85625882651096,71.31333923339844]},{"page":275,"text":"The free energy per unit area is given by","rect":[65.76486206054688,92.20748138427735,230.35772028974066,83.27293395996094]},{"page":275,"text":"1","rect":[168.2361297607422,110.59761810302735,175.20405717056912,107.4531021118164]},{"page":275,"text":"F=A ¼ ð \"g0ðSÞ þ K\u0003\u0001ddSz\u00032#dz \u0002 WAS0","rect":[137.53370666503907,138.46070861816407,299.3415992029603,108.59919738769531]},{"page":275,"text":"0","rect":[169.48199462890626,140.9927215576172,172.9659583338197,136.19577026367188]},{"page":275,"text":"(10.4)","rect":[361.0556945800781,127.80130767822266,385.10506568757668,119.32494354248047]},{"page":275,"text":"Now we need two minimizations: (i) over function S(z) with fixed S0 (z","rect":[65.7660140991211,165.1542205810547,345.9686623965378,156.6179656982422]},{"page":275,"text":"(ii) over the boundary value S0. In our case, the standard Euler","rect":[53.81356430053711,177.51217651367188,344.42546974031105,168.57762145996095]},{"page":275,"text":"qF=qS \u0002 d=dzðqF=qS0Þ ¼ 0 reads:","rect":[53.81330490112305,193.14425659179688,182.1440568692405,179.7852783203125]},{"page":275,"text":"d2S","rect":[228.846435546875,215.5897216796875,243.05664912274848,206.989990234375]},{"page":275,"text":"g00ðSÞ \u0002 2K\u0003 dz2 ¼ 0:","rect":[174.97702026367188,229.11306762695313,263.9925073353662,214.10438537597657]},{"page":275,"text":"Its first integral: K\u0003\u0004ddSz\u00052 ¼ g0ðSÞ þ C","rect":[65.76737213134766,253.94227600097657,217.5763880890205,240.034912109375]},{"page":275,"text":"The constant C is found from dS=dzjz!1 ¼ 0;i:e: C ¼ \u0002g0ðSbulkÞ:","rect":[65.7657241821289,265.22296142578127,334.2321313588037,254.9055938720703]},{"page":275,"text":"Then we get:","rect":[65.76607513427735,276.47698974609377,118.56918199131082,267.54241943359377]},{"page":275,"text":"¼ 0) and","rect":[349.2734680175781,165.1542205810547,385.1464470963813,156.6180877685547]},{"page":275,"text":"equation","rect":[350.68768310546877,177.51229858398438,385.1760186295844,168.57774353027345]},{"page":275,"text":"(10.5)","rect":[361.0570373535156,223.8189239501953,385.10640846101418,215.2230224609375]},{"page":275,"text":"x20\u0001ddSz\u00032 ¼ g0ðSÞ \u0002aTgN0IðSbulkÞ","rect":[159.28688049316407,316.5268859863281,278.0515481387253,290.70751953125]},{"page":275,"text":"(10.10)","rect":[356.0715026855469,308.840087890625,385.1596921524204,300.3637390136719]},{"page":275,"text":"Here, we have introduced a surface correlation length, marked off in Fig. 10.4:","rect":[65.76595306396485,340.07318115234377,384.2449174649436,331.13861083984377]},{"page":275,"text":"x0 \u0004 \u0001aKTN\u0003I\u00031=2;","rect":[184.3802947998047,380.68975830078127,252.94590699356935,354.3612976074219]},{"page":275,"text":"(10.11)","rect":[356.0705261230469,373.0028381347656,385.1587155899204,364.5264892578125]},{"page":275,"text":"with the first order transition temperature TNI in the bulk. Now Eq. (10.10) for the","rect":[53.81295394897461,404.17913818359377,385.17530096246568,395.24432373046877]},{"page":275,"text":"free energy density becomes dimensionless.","rect":[53.813594818115237,416.1384582519531,230.7665981819797,407.20391845703127]},{"page":275,"text":"Next, we substitute (10.10) into (10.4) and after minimization dF/dS0 ¼ 0 find","rect":[65.76561737060547,427.7067565917969,385.17250910810005,419.1435241699219]},{"page":275,"text":"the condition 2½g0ðS0Þ \u0002 g0ðSbulkÞ\u00051=2 ¼ W=A. Using this condition the equation","rect":[53.81475067138672,440.396484375,385.17717829755318,427.87628173828127]},{"page":275,"text":"(10.10) may be integrated with the proper limits:","rect":[53.814476013183597,452.0173645019531,250.92777879306864,443.08282470703127]},{"page":275,"text":"ðaTN0IÞ1=2","rect":[140.19810485839845,490.5292663574219,176.5346076258119,477.69866943359377]},{"page":275,"text":"S0","rect":[181.7176513671875,473.06158447265627,187.71824307944065,467.2115783691406]},{"page":275,"text":"SðzÞ","rect":[178.65882873535157,504.83544921875,190.23465400590838,497.8700866699219]},{"page":275,"text":"(10.12)","rect":[356.0715026855469,489.4251708984375,385.1596921524204,480.9488220214844]},{"page":275,"text":"This equation has been solved numerically [6] for the order parameter S(z,T,S0)","rect":[65.76595306396485,529.3865966796875,385.15975318757668,520.4520874023438]},{"page":275,"text":"depending on the distance z from the boundary, the surface potential (included in","rect":[53.81393051147461,541.346435546875,385.13979426435005,532.4119262695313]},{"page":275,"text":"S0) and temperature (included in g0(Sbulk)). The found distance dependence is","rect":[53.81393051147461,553.3460083007813,385.18768705474096,544.3716430664063]},{"page":275,"text":"similar to that shown in Fig. 10.4a for the positive surface potential. The calculated","rect":[53.81406784057617,565.2656860351563,385.1498955827094,556.3311767578125]},{"page":275,"text":"thickness of the surface layer is about 10x0.","rect":[53.81406784057617,577.2252197265625,230.08883328940159,567.9719848632813]},{"page":276,"text":"10.1","rect":[53.813533782958987,43.0,68.62040466456338,36.73191833496094]},{"page":276,"text":"General","rect":[70.97681427001953,43.0,97.8237886830813,36.68111801147461]},{"page":276,"text":"Properties","rect":[100.26226806640625,44.275474548339847,134.6057480144433,36.68111801147461]},{"page":276,"text":"S0","rect":[118.1653060913086,69.3697509765625,128.05187722845253,60.61261749267578]},{"page":276,"text":"0.32","rect":[112.49728393554688,84.96244812011719,128.05217221581797,79.13969421386719]},{"page":276,"text":"0.24","rect":[112.49728393554688,115.30876159667969,128.05217221581797,109.48600769042969]},{"page":276,"text":"0.16","rect":[112.49728393554688,145.6550750732422,128.05217221581797,139.8323211669922]},{"page":276,"text":"0.08","rect":[112.49728393554688,176.00144958496095,128.05217221581797,170.17869567871095]},{"page":276,"text":"0","rect":[123.60711669921875,206.3469696044922,128.05137112939219,200.5242156982422]},{"page":276,"text":"2","rect":[164.20681762695313,90.01513671875,168.65107205712656,84.33634948730469]},{"page":276,"text":"1","rect":[150.54953002929688,103.43304443359375,154.9937844594703,97.75425720214844]},{"page":276,"text":"–0.08","rect":[144.19615173339845,214.0093536376953,164.19529202782969,208.1865997314453]},{"page":276,"text":"0","rect":[183.3799285888672,214.0093536376953,187.82418301904063,208.1865997314453]},{"page":276,"text":"132","rect":[192.76194763183595,200.08685302734376,211.85625699365,129.3091278076172]},{"page":276,"text":"4","rect":[225.04354858398438,131.5086669921875,229.4878030141578,125.82987976074219]},{"page":276,"text":"5","rect":[246.3975372314453,126.18577575683594,250.84179166161875,120.45899963378906]},{"page":276,"text":"6","rect":[266.75958251953127,118.48423767089844,271.2038369497047,112.66148376464844]},{"page":276,"text":"7","rect":[304.2535705566406,95.728271484375,308.69782498681408,90.14546203613281]},{"page":276,"text":"263","rect":[372.4989013671875,42.55679702758789,385.19052642970009,36.68111801147461]},{"page":276,"text":"Fig. 10.5 Calculated order parameter at the surface S0 as a function of temperature. The numbers","rect":[53.812843322753909,251.0572509765625,385.1725204753808,243.3274383544922]},{"page":276,"text":"at the curves corresponds to different surface potential in dimensionless units: W ¼ 0 (1), 0.0056","rect":[53.81333541870117,260.9654846191406,385.1733755507938,253.32032775878907]},{"page":276,"text":"(2), 0.008 (3), 0.01 (4), Wc ¼ 0.01078 (5), 0.012 (6), 0.017 (7). Note that at Wc the discontinuity of","rect":[53.81333541870117,270.94122314453127,385.2116708145826,263.2960510253906]},{"page":276,"text":"the first order N-Iso phase transition disappears (adapted from [7])","rect":[53.81356430053711,280.8604431152344,281.3841866837232,273.2660827636719]},{"page":276,"text":"The surface order parameter S0 shows very interesting features. Fig. 10.5 illus-","rect":[65.76496887207031,302.7769775390625,385.14022193757668,293.78265380859377]},{"page":276,"text":"trates the calculated dependence of S0 on temperature with parameters of a liquid","rect":[53.81433868408203,314.736572265625,385.11306086591255,305.802001953125]},{"page":276,"text":"crystal 5CB. The different curves correspond to the different values of the surface","rect":[53.81306076049805,326.69610595703127,385.0802387066063,317.7017822265625]},{"page":276,"text":"potential. According to the positive sign of the surface potential W, we see an","rect":[53.81306076049805,338.598876953125,385.1538933854438,329.664306640625]},{"page":276,"text":"expected increase in the orientational order in the surface nematic phase (negative","rect":[53.81307601928711,350.5584411621094,385.0991901226219,341.6239013671875]},{"page":276,"text":"values of T-TNI). Further, the increasing potential W shifts the N-I transition point to","rect":[53.81307601928711,362.5185852050781,385.1415642838813,353.58343505859377]},{"page":276,"text":"higher temperatures. At W ¼ Wc, the phase transition at the surface disappears and","rect":[53.81368637084961,374.4781188964844,385.14409724286568,365.54351806640627]},{"page":276,"text":"the surface order parameter becomes a continuous function of temperature. For high","rect":[53.81321334838867,386.4376525878906,385.12416926435005,377.50311279296877]},{"page":276,"text":"values of the surface potential, the orientational order at the interface remains finite","rect":[53.81321334838867,398.3971862792969,385.1381305523094,389.462646484375]},{"page":276,"text":"even at temperatures well above the N-I transition point in the bulk.","rect":[53.81321334838867,410.35675048828127,324.9649319222141,401.42218017578127]},{"page":276,"text":"The picture predicted by this figure has been confirmed by birefringence mea-","rect":[65.76522827148438,422.3162841796875,385.1192563614048,413.3817138671875]},{"page":276,"text":"surements on the isotropic phase [7]. Such measurements are much more precise","rect":[53.81321334838867,434.21905517578127,385.16101873590318,425.28448486328127]},{"page":276,"text":"than attempts to measure the influence of an interface on the order parameter in the","rect":[53.81321334838867,446.1786193847656,385.17002142145005,437.24407958984377]},{"page":276,"text":"nematic phase, because the isotropic phase has no background birefringence com-","rect":[53.81321334838867,458.1381530761719,385.13713966218605,449.20361328125]},{"page":276,"text":"ing from the bulk. For nematic preparations with the director homogeneously","rect":[53.81321334838867,470.0976867675781,385.1042412858344,461.16314697265627]},{"page":276,"text":"aligned along the surface of a solid substrate, the birefringence is observed at","rect":[53.81321334838867,482.0572204589844,385.1789689786155,473.1226806640625]},{"page":276,"text":"temperatures markedly exceeding the N-I transition point, Fig. 10.6. Moreover, it","rect":[53.81321334838867,494.0167541503906,385.1729570157249,485.08221435546877]},{"page":276,"text":"depends on the surface potential as predicted by theory.","rect":[53.81319808959961,505.976318359375,278.6350063851047,497.041748046875]},{"page":276,"text":"The thickness of the “quasi-nematic” layers adjacent to the substrate and shown","rect":[65.76522064208985,517.935791015625,385.1779717545844,509.00128173828127]},{"page":276,"text":"in the Inset to Fig. 10.6 can be estimated from the observed birefringence. For two","rect":[53.81319808959961,529.8385620117188,385.15007868817818,520.904052734375]},{"page":276,"text":"boundaries the phase retardation d between the ordinary and extraordinary rays is","rect":[53.81318283081055,541.798095703125,385.18588651763158,532.5647583007813]},{"page":276,"text":"given by","rect":[53.81318283081055,553.7576904296875,88.80422297528753,544.8231811523438]},{"page":276,"text":"2d \u0006 4phDnix0","rect":[187.6654510498047,582.3502197265625,249.1540381678041,568.0769653320313]},{"page":276,"text":"l","rect":[227.4869842529297,589.1607055664063,232.98170823405338,582.029052734375]},{"page":277,"text":"264","rect":[53.8127326965332,42.55643081665039,66.50433868555948,36.68075180053711]},{"page":277,"text":"10 Liquid Crystal – Solid Interface","rect":[265.38177490234377,44.275108337402347,385.1269774177027,36.68075180053711]},{"page":277,"text":"S(z)","rect":[260.6407165527344,75.66618347167969,275.9997558485224,67.57691192626953]},{"page":277,"text":"1.5","rect":[133.5574493408203,102.53861999511719,144.66808560693125,96.7718505859375]},{"page":277,"text":"1.0","rect":[133.45913696289063,132.5953826904297,144.56977322900156,126.82861328125]},{"page":277,"text":"Iso","rect":[252.9092559814453,107.03577423095703,264.32964573052586,100.7461166381836]},{"page":277,"text":"S =0","rect":[252.9092559814453,120.99896240234375,271.8176218047446,114.55633544921875]},{"page":277,"text":"0.5","rect":[133.55824279785157,165.77552795410157,144.6688790639625,160.00875854492188]},{"page":277,"text":"W3","rect":[224.82810974121095,184.224609375,235.7068480491926,176.37428283691407]},{"page":277,"text":"W1","rect":[273.1840515136719,157.40011596679688,284.0618895530988,149.6577606201172]},{"page":277,"text":"0","rect":[145.07174682617188,203.9729766845703,149.5160012563453,198.20620727539063]},{"page":277,"text":"1","rect":[176.7434539794922,203.82901000976563,181.18770840966563,198.20620727539063]},{"page":277,"text":"2","rect":[208.4151611328125,203.82901000976563,212.85941556298594,198.20620727539063]},{"page":277,"text":"T–TNI (K)","rect":[207.61959838867188,217.8831787109375,245.58261107313178,209.2790985107422]},{"page":277,"text":"4","rect":[270.5292053222656,203.82901000976563,274.97345975243908,198.20620727539063]},{"page":277,"text":"5","rect":[303.50140380859377,203.9729766845703,307.9456582387672,198.32618713378907]},{"page":277,"text":"Fig. 10.6 Temperature dependence of birefringence of thin surface layers in the isotropic phase.","rect":[53.812843322753909,239.77777099609376,385.1720301826235,232.04795837402345]},{"page":277,"text":"Surface potential W1 > W2 > W3. Inset: geometry of birefringence measurements with molecules","rect":[53.812843322753909,249.68600463867188,385.1753586101464,242.09158325195313]},{"page":277,"text":"aligned parallel to the surfaces and the gradient of the order parameter S(z) within the surface","rect":[53.81277084350586,259.6618957519531,385.12877795481207,252.06753540039063]},{"page":277,"text":"layers","rect":[53.8127326965332,269.58111572265627,74.0600708293847,261.98675537109377]},{"page":277,"text":"where l is light wavelength and <Dn> is average optical anisotropy of the surface","rect":[53.812843322753909,294.50152587890627,385.08301580621568,285.23828125]},{"page":277,"text":"layers. For typical values of Dn \u0006 0.1–0.2, x \u0006 4–10 nm.","rect":[53.812843322753909,306.404296875,287.6541714241672,297.14105224609377]},{"page":277,"text":"Some solid surfaces induce disorder in nematic liquid crystals. It means that the","rect":[65.76587677001953,318.36383056640627,385.1716693706688,309.42926025390627]},{"page":277,"text":"order parameter at the interface is lower than the bulk value. For instance, evapo-","rect":[53.81385803222656,330.3233947753906,385.08403907624855,321.38885498046877]},{"page":277,"text":"rated SiO layers of a certain thickness due to their roughness decrease the order","rect":[53.81385803222656,342.2829284667969,385.14470802156105,333.3284606933594]},{"page":277,"text":"parameter of MBBA from the bulk value Sb \u0006 0.6 down to S0 \u0006 0.1–0.2. In some","rect":[53.81385803222656,354.24249267578127,385.1481098003563,345.30792236328127]},{"page":277,"text":"cases, the surface order parameter may be equal to zero (surface melting).","rect":[53.81426239013672,366.2026672363281,352.4754604866672,357.26812744140627]},{"page":277,"text":"10.1.3.2 Surface-Induced Smectic Ordering","rect":[53.81426239013672,403.068359375,245.94246760419379,393.7353820800781]},{"page":277,"text":"Let the director of the nematic phase is perpendicular to a flat interface. Then we","rect":[53.81426239013672,426.90777587890627,385.16907537652818,417.97320556640627]},{"page":277,"text":"can anticipate two effects. First, a polar surface layer should appear due to the break","rect":[53.81426239013672,438.8105773925781,385.1471795182563,429.87603759765627]},{"page":277,"text":"of the cylindrical symmetry, n ¼6 \u0002 n. Second, due to some positional correlation","rect":[53.81426239013672,450.7701110839844,385.12819758466255,441.5068664550781]},{"page":277,"text":"of the centers of molecules in several layers adjacent to the surface, the nematic","rect":[53.81427764892578,462.7296447753906,385.10132635309068,453.79510498046877]},{"page":277,"text":"translational invariance can be broken. It means that the surface induces the short-","rect":[53.81427764892578,472.6572265625,385.11629615632668,465.754638671875]},{"page":277,"text":"range smectic A order. In the framework of the Landau theory, the smectic order","rect":[53.81427764892578,486.6487121582031,385.1441892227329,477.71417236328127]},{"page":277,"text":"decays with distance from the interface according to the exponential law","rect":[53.81427764892578,498.6082763671875,346.9548621173176,489.6737060546875]},{"page":277,"text":"r1ðzÞ ¼ r1ð0Þexpð\u0002z=lsÞ","rect":[166.82044982910157,522.8671264648438,272.2171975527878,512.9166259765625]},{"page":277,"text":"where both the smectic wave amplitude r1 and smectic correlation length ls induced","rect":[53.81388473510742,546.3910522460938,385.1270379166938,537.1378173828125]},{"page":277,"text":"by the surface increase with decreasing temperature T. More precisely, both the","rect":[53.814083099365237,558.3505859375,385.1758502788719,549.4160766601563]},{"page":277,"text":"parameters depend on the proximity (T \u0002 TNA) to the nematic ! smectic A","rect":[53.814083099365237,570.3101196289063,385.1400427813801,561.3756103515625]},{"page":277,"text":"transition because at T ¼ TNA, ls ! 1 and the smectic phase becomes stable","rect":[53.81418991088867,582.2699584960938,385.17380560113755,573.0167236328125]},{"page":277,"text":"everywhere.","rect":[53.81306076049805,594.2294921875,102.89826627280002,585.2949829101563]},{"page":278,"text":"10.1 General Properties","rect":[53.813228607177737,44.276206970214847,134.60544283866205,36.68185043334961]},{"page":278,"text":"265","rect":[372.49859619140627,42.55752944946289,385.1902212539188,36.63105392456055]},{"page":278,"text":"In the X-ray experiments on nematic 8CB, the smectic ordering was observed at","rect":[65.76496887207031,68.2883529663086,385.18171556064677,59.35380554199219]},{"page":278,"text":"the free surface (air-nematic interface). The same phenomenon has also been","rect":[53.812950134277347,80.24788665771485,385.11696711591255,71.31333923339844]},{"page":278,"text":"observed at the solid-nematic interface by the X-ray, an electrooptical technique","rect":[53.812950134277347,92.20748138427735,385.16379583551255,83.27293395996094]},{"page":278,"text":"and molecular force measurements. The principle of the latter is shown in Fig. 10.7.","rect":[53.812950134277347,104.11019134521485,385.1816677620578,95.17564392089844]},{"page":278,"text":"For two mica cylinders submerged in nematic liquid crystal, their interaction force","rect":[53.81298065185547,116.0697250366211,385.1260150737938,107.13517761230469]},{"page":278,"text":"measured with a balance oscillates with a distance between the cylinders and the","rect":[53.81298065185547,128.02932739257813,385.17078436090318,119.09477233886719]},{"page":278,"text":"period of oscillations was found to be equal to molecular length l. This clearly","rect":[53.81298065185547,139.98886108398438,385.11696711591255,131.0343780517578]},{"page":278,"text":"shows the periodicity in density characteristic of a smectic phase [8].","rect":[53.811973571777347,151.94839477539063,332.64434476401098,143.0138397216797]},{"page":278,"text":"A powerful technique for the study of molecular orientation at the surface is","rect":[65.76496887207031,163.907958984375,385.1857034121628,154.97340393066407]},{"page":278,"text":"scanning tunnel microscopy (STM): a weak tunnel electric current (of the order of","rect":[53.812950134277347,175.86749267578126,385.14690528718605,166.9329376220703]},{"page":278,"text":"0.1 nA) is measured between an extremely sharp (atomic size) tip and the conduc-","rect":[53.812950134277347,187.8270263671875,385.1547788223423,178.89247131347657]},{"page":278,"text":"tive substrate. The motion of the tip over the surface is controlled by piezoelectric","rect":[53.812950134277347,199.72976684570313,385.1229938335594,190.7952117919922]},{"page":278,"text":"drivers and a computer. As a result, we can see a current pattern correlating with the","rect":[53.812950134277347,211.6893310546875,385.1727680034813,202.75477600097657]},{"page":278,"text":"surface relief. For example, on cooling the 11th and 12th homologues of cyanobi-","rect":[53.812950134277347,223.64886474609376,385.1767514785923,214.7143096923828]},{"page":278,"text":"phenyl from the isotropic phase to a room temperature smectic phase, different","rect":[53.812950134277347,235.60842895507813,385.17170579502177,226.6738739013672]},{"page":278,"text":"types of surface layers are formed on conductive MoS2 substrates [9]: a compound","rect":[53.812950134277347,247.56985473632813,385.14727107099068,238.63340759277345]},{"page":278,"text":"11CB having intermediate nematic phase forms single-row monolayers whereas","rect":[53.81342697143555,259.5294189453125,385.14130033599096,250.59486389160157]},{"page":278,"text":"compound 12CB forms double-row ones, see Fig. 10.8. The structure of mono-","rect":[53.81342697143555,271.48895263671877,385.1492856582798,262.55438232421877]},{"page":278,"text":"layers depends on substrate properties and temperature and the latter can control the","rect":[53.81345748901367,283.448486328125,385.17221868707505,274.513916015625]},{"page":278,"text":"realignment of a liquid crystal in the bulk, i.e. cause anchoring transitions [10].","rect":[53.81345748901367,295.3910827636719,375.6380886604953,286.39678955078127]},{"page":278,"text":"10","rect":[268.9806213378906,321.25311279296877,277.8691201528297,315.486328125]},{"page":278,"text":"Fig. 10.7 Periodic force","rect":[53.812843322753909,390.4920654296875,138.99044177317144,382.7622375488281]},{"page":278,"text":"between two mica cylinders","rect":[53.812843322753909,400.4002990722656,149.5879257488183,392.8059387207031]},{"page":278,"text":"separated by nematic liquid","rect":[53.812843322753909,410.3762512207031,148.29507965235636,402.7818908691406]},{"page":278,"text":"crystal with molecules","rect":[53.812843322753909,420.29547119140627,130.72481234549799,412.70111083984377]},{"page":278,"text":"aligned perpendicular to","rect":[53.812843322753909,430.2714538574219,136.89209503321573,422.6770935058594]},{"page":278,"text":"cylinder surfaces asa function","rect":[53.812843322753909,440.2474060058594,155.33214325098917,432.6530456542969]},{"page":278,"text":"of the gap between the latter","rect":[53.812843322753909,450.1666259765625,151.59320157630138,442.572265625]},{"page":278,"text":"5","rect":[273.42486572265627,356.6199645996094,277.8691201528297,350.97314453125]},{"page":278,"text":"0","rect":[273.42486572265627,390.46795654296877,277.8691201528297,384.701171875]},{"page":278,"text":"–5","rect":[268.9814147949219,425.30133056640627,277.86991360986095,419.6545104980469]},{"page":278,"text":"0","rect":[278.7371826171875,435.21044921875,283.18143704736095,429.44366455078127]},{"page":278,"text":"Fig. 10.8 Scanning Tunnel Microscope images of smectic compounds on MoS2 substrates taken","rect":[53.812843322753909,574.0244750976563,385.18002838282509,566.0745239257813]},{"page":278,"text":"at room temperature and showing the one-row (for 11CB) or two-rows (for 12CB) molecular","rect":[53.813228607177737,583.9326782226563,385.13177579505136,576.3383178710938]},{"page":278,"text":"organization","rect":[53.813228607177737,593.9086303710938,96.26241058497354,586.3142700195313]},{"page":279,"text":"266","rect":[53.813697814941409,42.55765151977539,66.50530380396768,36.68197250366211]},{"page":279,"text":"10 Liquid Crystal – Solid Interface","rect":[265.38275146484377,44.276329040527347,385.12792346262457,36.68197250366211]},{"page":279,"text":"10.1.3.3 Polar Surface Order and Surface Polarization","rect":[53.812843322753909,66.54527282714844,293.9303013737018,59.035072326660159]},{"page":279,"text":"The interfaces in general, and particularly with solid substrates break the head-","rect":[53.812843322753909,92.20748138427735,385.15264259187355,83.27293395996094]},{"page":279,"text":"to-tail symmetry of a liquid crystal phase and induce polar orientational order.","rect":[53.812843322753909,104.11019134521485,385.1417507698703,95.17564392089844]},{"page":279,"text":"The symmetry is reduced to the conical group C1v. The latter allows a finite value","rect":[53.812843322753909,116.0697250366211,385.14145696832505,107.13517761230469]},{"page":279,"text":"of the second-order nonlinear susceptibility w2 responsible for the second optical","rect":[53.81356430053711,128.02999877929688,385.12788255283427,119.09544372558594]},{"page":279,"text":"harmonic generation [11]. This phenomenon has been observed in experiments on","rect":[53.81393051147461,139.98953247070313,385.1706475358344,131.0549774169922]},{"page":279,"text":"generation of the second harmonic in a ultrathin nematic layers on a solid substrate","rect":[53.81393051147461,151.94906616210938,385.1786578960594,143.01451110839845]},{"page":279,"text":"as shown in Fig. 10.9.","rect":[53.81393051147461,163.90863037109376,142.74287076498752,154.9740753173828]},{"page":279,"text":"The polar order parameter is the first Legendre polynomial P1 ¼ ðLhÞ ¼ cosy","rect":[65.7659683227539,176.20733642578126,385.17873469403755,166.25682067871095]},{"page":279,"text":"where L is a new polar vector parallel to n and called polar director. The polar order","rect":[53.814022064208987,187.82821655273438,385.14592872468605,178.89366149902345]},{"page":279,"text":"contributes to the elastic surface energy linearly while the quadrupolar (nematic)","rect":[53.81405258178711,199.73098754882813,385.14888892976418,190.7964324951172]},{"page":279,"text":"order contributes quadratically with a sign dependent on surface treatment:","rect":[53.81405258178711,211.6905517578125,356.3892656094749,202.75599670410157]},{"page":279,"text":"FðP1Þ ¼ \u0002spðLhÞ ¼ \u0002sp cosy","rect":[187.21241760253907,237.38502502441407,315.67592707685005,227.01768493652345]},{"page":279,"text":"FðP2Þ ¼ \u000712WðLhÞ2 ¼ \u000712WðnhÞ2 ¼ \u000712Wcos2y","rect":[123.31778717041016,254.7803955078125,315.67583552411568,241.84866333007813]},{"page":279,"text":"(10.13)","rect":[356.0710144042969,244.67735290527345,385.1592038711704,236.20098876953126]},{"page":279,"text":"where angle y is counted from the external normal to the nematic layer h [7].","rect":[53.81344223022461,278.403564453125,385.1562466194797,269.1602478027344]},{"page":279,"text":"Therefore, both contributions to the free energy vanish when both L and n perpen-","rect":[53.814476013183597,290.36309814453127,385.1154721817173,281.42852783203127]},{"page":279,"text":"dicular to h. However, when L,n || h the two contributions can compete with each","rect":[53.81444549560547,302.3226623535156,385.1025323014594,293.0096130371094]},{"page":279,"text":"other. For instance, for y ¼ 0 the linear term is negative and favors this alignment","rect":[53.81542205810547,314.2821960449219,385.1741777188499,305.03887939453127]},{"page":279,"text":"(homeotropic) but, the quadratic term with positive sign at W is unfavorable. It","rect":[53.81542205810547,326.24169921875,385.18110520908427,317.30712890625]},{"page":279,"text":"could be a reason for an oblique alignment of the director often observed at the free","rect":[53.815391540527347,338.14447021484377,385.13733709527818,329.20989990234377]},{"page":279,"text":"surface of a nematic.","rect":[53.815391540527347,348.0422058105469,138.09072537924534,341.16949462890627]},{"page":279,"text":"Note that the polar vector reflects only polar symmetry of the interfacial layer","rect":[65.76741790771485,362.0635681152344,385.12044654695168,353.1290283203125]},{"page":279,"text":"and may be associated with the conical (not rod-like) form of the molecules.","rect":[53.815391540527347,374.02313232421877,385.1811184456516,365.08856201171877]},{"page":279,"text":"However, when the electric charges are involved in the game, the same polar","rect":[53.815391540527347,385.982666015625,385.1393064102329,377.048095703125]},{"page":279,"text":"order may results in appearance of the macroscopic surface electric polarization","rect":[53.815391540527347,397.9422302246094,385.0876397233344,389.0076904296875]},{"page":279,"text":"Psurf that is the dipole moment of a unit volume [units: CGS(charge)\bcm/cm3 ¼","rect":[53.815391540527347,409.90374755859377,385.14807918760689,398.85272216796877]},{"page":279,"text":"CGSQ/cm2 ¼ StatV/cm, or C/m2 in SI system]. When an electric field is applied to","rect":[53.814231872558597,421.8633117675781,385.14031306317818,410.75567626953127]},{"page":279,"text":"a liquid crystal the surface polarization contributes to the free energy of a surface","rect":[53.81344223022461,433.7660827636719,385.08359564020005,424.83154296875]},{"page":279,"text":"layer","rect":[53.81344223022461,445.7256164550781,73.80149207917822,436.79107666015627]},{"page":279,"text":"FðPsurfÞ ¼ \u0002PsurfE","rect":[180.3583221435547,470.4626159667969,258.6438014190986,460.0328063964844]},{"page":279,"text":"(10.14)","rect":[356.07061767578127,469.24688720703127,385.1588071426548,460.7705383300781]},{"page":279,"text":"ω","rect":[152.72637939453126,497.7945556640625,158.20975806197544,494.00335693359377]},{"page":279,"text":"Glass","rect":[204.48805236816407,491.5365905761719,222.968473899877,485.809814453125]},{"page":279,"text":"2ω","rect":[267.0076599121094,494.8959655761719,276.4876816459598,489.201171875]},{"page":279,"text":"NLC","rect":[203.6274871826172,549.061279296875,221.28460142680309,543.3264770507813]},{"page":279,"text":"Polar","rect":[271.67889404296877,525.9808959960938,290.2472367172598,520.3900756835938]},{"page":279,"text":"layer","rect":[272.5941162109375,537.1864624023438,289.33201454929107,529.988037109375]},{"page":279,"text":"Fig. 10.9 Optical second harmonic generation by a polar layer at the interface between nematic","rect":[53.812843322753909,572.8909301757813,385.16442248606207,564.8563232421875]},{"page":279,"text":"liquid crystal and glass: due to non-linear interaction with surface layer the incident beam of","rect":[53.812843322753909,582.7991333007813,385.15509122473886,575.2047729492188]},{"page":279,"text":"frequency o is partially converted into the beam of frequency 2o","rect":[53.812843322753909,592.7183837890625,277.53198420729026,585.1240234375]},{"page":280,"text":"10.1 General Properties","rect":[53.812843322753909,44.274620056152347,134.60506136893549,36.68026351928711]},{"page":280,"text":"267","rect":[372.4981994628906,42.55594253540039,385.1898245254032,36.68026351928711]},{"page":280,"text":"The problem of the surface polarization have been raised [12] macroscopically","rect":[65.76496887207031,68.2883529663086,385.0950554948188,59.35380554199219]},{"page":280,"text":"in connection with the bulk flexoelectric distortion [13] discussed in the next","rect":[53.81294250488281,79.64029693603516,385.10993821689677,71.31333923339844]},{"page":280,"text":"chapter. On the microscopic level, we can distinguish between three different","rect":[53.81294250488281,92.20748138427735,385.0771623379905,83.25301361083985]},{"page":280,"text":"mechanisms of Psurf, explained with the help of Fig. 10.10.","rect":[53.81294250488281,104.11067962646485,292.11480374838598,95.17564392089844]},{"page":280,"text":"Ionic Polarization","rect":[53.8129997253418,130.55874633789063,125.49227229169378,123.68601989746094]},{"page":280,"text":"A monolayer of ionic species can be adsorbed at the interface with a solid substrate","rect":[53.8129997253418,156.53964233398438,385.17966497613755,147.60508728027345]},{"page":280,"text":"(a Helmholtz monolayer), Fig. 10.10a. A diffuse layer of ions of the opposite sign","rect":[53.8129997253418,168.49917602539063,385.11199275067818,159.5646209716797]},{"page":280,"text":"with density r(z) provides the overall electrical neutrality. This mechanism is not","rect":[53.81300735473633,180.458740234375,385.133925033303,171.52418518066407]},{"page":280,"text":"specific for liquid crystals, it takes place in the isotropic liquids as well.However, in","rect":[53.81400680541992,192.41827392578126,385.14092341474068,183.4837188720703]},{"page":280,"text":"liquid crystals the surface field E ¼ 4pPsurf can interact with the director and","rect":[53.81400680541992,204.321044921875,385.14592829755318,195.38648986816407]},{"page":280,"text":"change orientation of the latter. Qualitatively, the ionic polarization can be esti-","rect":[53.814083099365237,216.28195190429688,385.17684303132668,207.34739685058595]},{"page":280,"text":"mated as Psurf ¼ qnxD where n is the number of charges q and xD is a characteristic","rect":[53.814083099365237,228.24169921875,385.18222845270005,218.9884033203125]},{"page":280,"text":"(Debye) length for the charge distribution.","rect":[53.813533782958987,240.20126342773438,224.71238370444065,231.26670837402345]},{"page":280,"text":"Dipolar Polarization","rect":[53.813533782958987,268.7110900878906,135.4619530778266,259.77655029296877]},{"page":280,"text":"It comes in due to a polar interaction of dipoles with a substrate. A head or a tail of a","rect":[53.813533782958987,292.63018798828127,385.1614154644188,283.69561767578127]},{"page":280,"text":"molecule may have different chemical affinity to the substrate material, Fig. 10.10b.","rect":[53.813533782958987,304.532958984375,385.1822170784641,295.598388671875]},{"page":280,"text":"The molecules with electric dipole moment pe form a dipolar monolayer whose","rect":[53.813533782958987,316.4936828613281,385.13449896051255,307.5579833984375]},{"page":280,"text":"polarization Psurf ¼ pen depends on the surface density of dipoles n. The polar layer","rect":[53.81356430053711,328.45330810546877,385.1214841446079,319.5186767578125]},{"page":280,"text":"thickness is determined by the characteristic diffusion length xd ¼ (2Dt)1/2 where","rect":[53.81350326538086,340.412841796875,385.11109197809068,329.32220458984377]},{"page":280,"text":"D is a molecular diffusion coefficient and t is a characteristic time for molecular","rect":[53.81405258178711,350.3406677246094,385.15483985749855,343.43804931640627]},{"page":280,"text":"rotation. We can encounter the same mechanism in isotropic liquids, however, 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10.10 A schematic picture of the charge distribution as a function of the distance from the","rect":[53.812843322753909,574.0244750976563,385.1551756598902,566.294677734375]},{"page":280,"text":"liquid crystal–solid interface for ionic (a), dipolar (b) and quadrupolar (c) mechanisms of surface","rect":[53.812843322753909,583.9326782226563,385.1305784919214,576.3383178710938]},{"page":280,"text":"polarization Psurf","rect":[53.814552307128909,593.9086303710938,111.35025433152137,586.3142700195313]},{"page":281,"text":"268","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.68026351928711]},{"page":281,"text":"10 Liquid Crystal – Solid Interface","rect":[265.38189697265627,44.274620056152347,385.12706897043707,36.68026351928711]},{"page":281,"text":"liquid crystals the diffusion coefficients and relaxation times are different for","rect":[53.812843322753909,68.2883529663086,385.12191139070168,59.35380554199219]},{"page":281,"text":"different director orientation and different dipolar structure of the constituent","rect":[53.812843322753909,80.24788665771485,385.1477494961936,71.31333923339844]},{"page":281,"text":"molecules. For longitudinal and transverse molecular dipoles, the same character-","rect":[53.812843322753909,92.20748138427735,385.15874610749855,83.27293395996094]},{"page":281,"text":"istic times t|| and t⊥ are involved, which we met before when having discussed","rect":[53.812843322753909,104.11067962646485,385.1759881120063,95.17564392089844]},{"page":281,"text":"dielectric properties in Section 7.2.4.","rect":[53.81325149536133,116.0702133178711,205.3591274788547,107.13566589355469]},{"page":281,"text":"Ordoelectric (Quadrupolar) Polarization","rect":[53.81325149536133,151.94888305664063,214.96892634442816,142.99440002441407]},{"page":281,"text":"This polarization is related to the quadrupolar nature of any uniaxial phase,","rect":[53.81325149536133,175.86795043945313,385.1789516976047,166.9333953857422]},{"page":281,"text":"Fig. 10.10c. In the conventional nematic phase of symmetry D1h and order","rect":[53.81325149536133,187.82748413085938,385.14803443757668,178.89292907714845]},{"page":281,"text":"_","rect":[129.5475311279297,190.0,134.52463618329535,187.0]},{"page":281,"text":"parameter tensor Q, each molecule or a building block may, on average, be","rect":[53.81417465209961,199.73126220703126,385.15061224176255,190.71681213378907]},{"page":281,"text":"represented by a quadrupole, and the phase may be characterized by a tensor of","rect":[53.81276321411133,211.69082641601563,385.14265306064677,202.7164306640625]},{"page":281,"text":"density of the quadrupolar moment","rect":[53.8127555847168,223.63043212890626,195.25413377353739,214.67596435546876]},{"page":281,"text":"Q_qu ¼ \u0002qquQ_ ¼ \u0002qquSðninj \u0002 dij\u00063Þ","rect":[146.99412536621095,250.64833068847657,291.98627103911596,236.80612182617188]},{"page":281,"text":"(10.15)","rect":[356.07147216796877,248.6446075439453,385.1596616348423,240.0487060546875]},{"page":281,"text":"Here S is nematic order parameter amplitude and qquS is the modulus of tensor","rect":[65.76590728759766,272.0624084472656,385.11971412507668,262.2137451171875]},{"page":281,"text":"_","rect":[54.88914108276367,276.8572082519531,59.86624613812933,275.6121520996094]},{"page":281,"text":"Qqu (see Eq. 3.16), qqu being a scalar coefficient with dimension [charge/cm].","rect":[53.813716888427737,289.1912841796875,385.1018948128391,278.5726013183594]},{"page":281,"text":"Recall now that polarization is a gradient of charge density for any charge distribu-","rect":[53.81278610229492,300.7363586425781,385.09587989656105,291.80181884765627]},{"page":281,"text":"tion (dipolar, quadrupolar, etc.). Therefore, the gradient of the orientational order","rect":[53.81278610229492,312.6958923339844,385.1436704239048,303.7613525390625]},{"page":281,"text":"parameter creates the polarization:","rect":[53.81278610229492,324.65545654296877,192.97065599033426,315.72088623046877]},{"page":281,"text":"_","rect":[240.85520935058595,339.31951904296877,245.8323144059516,338.074462890625]},{"page":281,"text":"P ¼ \u0002qqurQ","rect":[192.08273315429688,350.8032531738281,246.9257636065754,341.0344543457031]},{"page":281,"text":"(10.16)","rect":[356.0715026855469,349.6502685546875,385.1596921524204,341.1141357421875]},{"page":281,"text":"This may be illustrated by appearance of the electric polarization in a hybrid cell,","rect":[65.76595306396485,371.3601379394531,385.2026028206516,362.42559814453127]},{"page":281,"text":"in which the quadrupolar molecules are oriented differently at the opposite inter-","rect":[53.81393051147461,383.2629089355469,385.14083228913918,374.328369140625]},{"page":281,"text":"faces, namely, homogeneously on the right plate and homeotropically on the left","rect":[53.81393051147461,395.22247314453127,385.1179338223655,386.28790283203127]},{"page":281,"text":"one Fig. 10.11. The molecular quadrupoles have an elongated form with positive","rect":[53.81393051147461,407.1820068359375,385.12488592340318,398.2474365234375]},{"page":281,"text":"charges at the apices. Therefore negative and positive charges are accumulated at","rect":[53.814903259277347,419.1415710449219,385.1826616055686,410.20703125]},{"page":281,"text":"the left and right plates, respectively, and the bulk polarization vector P(z) has its","rect":[53.814903259277347,431.1011047363281,385.15573515044408,422.16656494140627]},{"page":281,"text":"z-projection oriented from right to left. Note that in a hybrid cell the polarization","rect":[53.815940856933597,443.0606384277344,385.0960625748969,434.1260986328125]},{"page":281,"text":"_","rect":[293.13861083984377,445.0,298.1157158952094,442.0]},{"page":281,"text":"occurs due to a change of the orientational part of tensor Q i.e. the director n(r)","rect":[53.815940856933597,455.0201721191406,385.1606381973423,446.00592041015627]},{"page":281,"text":"without a change of its amplitude S. In this case we deal with a flexoelectric","rect":[53.814796447753909,466.9811706542969,385.1008685894188,457.9669189453125]},{"page":281,"text":"polarization [13], see for details Section 11.3.1. The flexoelectric mechanism","rect":[53.815834045410159,478.7743835449219,385.18057964921555,469.9294738769531]},{"page":281,"text":"may also be responsible for the surface polarization.","rect":[53.81580352783203,490.8434753417969,264.7276272347141,481.908935546875]},{"page":281,"text":"A change of the order parameter modulus S(r) can also create polarization, for","rect":[65.7678451538086,502.80303955078127,385.1825193008579,493.86846923828127]},{"page":281,"text":"_","rect":[286.56787109375,505.0,291.54497614911568,502.0]},{"page":281,"text":"example due to transformation of the ellipsoidal shape of Q tensor in space. In this","rect":[53.81584930419922,514.76318359375,385.14694608794408,505.74835205078127]},{"page":281,"text":"case we deal with the so-called ordoelectric polarization [14]. Indeed, decreasingS","rect":[53.8140983581543,526.7227172851563,385.1788262467719,517.7682495117188]},{"page":281,"text":"value results in less extended (less prolate) ellipsoid form without reorientation of its","rect":[53.81406784057617,538.6822509765625,385.1508828555222,529.7477416992188]},{"page":281,"text":"principal axes. Such a transformation may be caused by a scatter of the rigid","rect":[53.81406784057617,550.6417846679688,385.1290215592719,541.707275390625]},{"page":281,"text":"molecular quadrupoles with respect to the director axis: the stronger the scatter,","rect":[53.81406784057617,562.6013793945313,385.09820218588598,553.6668701171875]},{"page":281,"text":"_","rect":[306.4503479003906,563.7755737304688,311.4274529557563,562.530517578125]},{"page":281,"text":"the lower is the quadrupole order S and the less prolate ellipsoid Q. This is illustrated","rect":[53.81406784057617,574.504150390625,385.14421931317818,565.4899291992188]},{"page":281,"text":"by Fig. 10.12: in sketch (a) the order parameter is stronger at the surface and","rect":[53.814353942871097,586.4642944335938,385.1452569108344,577.52978515625]},{"page":282,"text":"10.1 General Properties","rect":[53.81321334838867,44.275352478027347,134.60542757987299,36.68099594116211]},{"page":282,"text":"Fig. 10.11 A hybrid cell","rect":[53.812843322753909,67.58130645751953,140.39413170065942,59.85148620605469]},{"page":282,"text":"illustrating the appearance","rect":[53.812843322753909,77.4895248413086,144.01461169504644,69.89517211914063]},{"page":282,"text":"of polarization due to the","rect":[53.812843322753909,87.4087142944336,140.22152087473394,79.81436157226563]},{"page":282,"text":"gradient of quadrupolar","rect":[53.812843322753909,97.3846664428711,134.4239968643873,89.79031372070313]},{"page":282,"text":"density of charge (the","rect":[53.812843322753909,107.36067962646485,128.36672351145269,99.76632690429688]},{"page":282,"text":"molecular quadrupoles touch","rect":[53.812843322753909,117.33663177490235,152.87421173243448,109.74227905273438]},{"page":282,"text":"the left and right surfaces by","rect":[53.812843322753909,127.28968811035156,151.94348663477823,119.62760162353516]},{"page":282,"text":"their (þ) and (\u0002) sides,","rect":[53.81369400024414,136.8931121826172,134.5204493720766,129.63742065429688]},{"page":282,"text":"respectively)","rect":[53.813690185546878,147.20773315429688,97.65895169837167,139.61337280273438]},{"page":282,"text":"+","rect":[260.65814208984377,76.34234619140625,266.2460494373112,71.49771881103516]},{"page":282,"text":"+","rect":[260.65814208984377,103.76983642578125,266.2460494373112,98.92520904541016]},{"page":282,"text":"+","rect":[260.65814208984377,131.58917236328126,266.2460494373112,126.74454498291016]},{"page":282,"text":"+","rect":[260.65814208984377,159.10601806640626,266.2460494373112,154.2613983154297]},{"page":282,"text":"+","rect":[260.65814208984377,181.43231201171876,266.2460494373112,176.5876922607422]},{"page":282,"text":"+","rect":[256.6511535644531,211.44940185546876,263.6531938284637,205.37875366210938]},{"page":282,"text":"–","rect":[277.5778503417969,68.5257797241211,282.89784432328988,67.51089477539063]},{"page":282,"text":"–","rect":[277.5778503417969,79.90779876708985,282.89784432328988,78.89291381835938]},{"page":282,"text":"–","rect":[277.5778503417969,95.95333099365235,282.89784432328988,94.93844604492188]},{"page":282,"text":"–","rect":[277.5778503417969,107.33528900146485,282.89784432328988,106.32040405273438]},{"page":282,"text":"–","rect":[277.5778503417969,123.7726058959961,282.89784432328988,122.75772094726563]},{"page":282,"text":"–","rect":[277.5778503417969,135.1546173095703,282.89784432328988,134.13973999023438]},{"page":282,"text":"–","rect":[277.5778503417969,151.2895050048828,282.89784432328988,150.27462768554688]},{"page":282,"text":"–","rect":[277.5778503417969,162.67247009277345,282.89784432328988,161.6575927734375]},{"page":282,"text":"–","rect":[277.5778503417969,173.61582946777345,282.89784432328988,172.6009521484375]},{"page":282,"text":"–","rect":[277.5778503417969,184.99876403808595,282.89784432328988,183.98388671875]},{"page":282,"text":"+","rect":[291.43853759765627,76.34234619140625,297.0264449451237,71.49771881103516]},{"page":282,"text":"+","rect":[291.43853759765627,103.76983642578125,297.0264449451237,98.92520904541016]},{"page":282,"text":"+","rect":[291.43853759765627,131.58917236328126,297.0264449451237,126.74454498291016]},{"page":282,"text":"+","rect":[291.43853759765627,159.10601806640626,297.0264449451237,154.2613983154297]},{"page":282,"text":"+","rect":[291.43853759765627,181.43231201171876,297.0264449451237,176.5876922607422]},{"page":282,"text":"269","rect":[372.4985656738281,42.62440490722656,385.1901907363407,36.68099594116211]},{"page":282,"text":"+","rect":[356.7260437011719,68.68975830078125,362.31395104863938,63.845130920410159]},{"page":282,"text":"+","rect":[356.7260437011719,100.35028076171875,362.31395104863938,95.50565338134766]},{"page":282,"text":"+","rect":[356.7260437011719,118.68963623046875,362.31395104863938,113.84500885009766]},{"page":282,"text":"+","rect":[356.7260437011719,150.35009765625,362.31395104863938,145.50547790527345]},{"page":282,"text":"+","rect":[355.92779541015627,168.02587890625,361.5157027576237,163.18125915527345]},{"page":282,"text":"+","rect":[356.061767578125,194.63302612304688,361.6496749255925,189.7884063720703]},{"page":282,"text":"–","rect":[363.8627014160156,211.0,370.52902742079285,206.0]},{"page":282,"text":"a","rect":[96.91981506347656,265.19921875,102.47507746071041,259.6104736328125]},{"page":282,"text":"+","rect":[122.2718505859375,280.9500732421875,127.67478625021163,276.26580810546877]},{"page":282,"text":"+","rect":[122.2718505859375,301.7598571777344,127.67478625021163,297.0755920410156]},{"page":282,"text":"+","rect":[122.2718505859375,316.2900695800781,127.67478625021163,311.6058044433594]},{"page":282,"text":"+","rect":[122.2718505859375,337.099853515625,127.67478625021163,332.41558837890627]},{"page":282,"text":"+","rect":[122.2718505859375,350.1810302734375,127.67478625021163,345.49676513671877]},{"page":282,"text":"+","rect":[122.2718505859375,370.9917297363281,127.67478625021163,366.3074645996094]},{"page":282,"text":"m","rect":[137.08396911621095,335.07208251953127,143.24677516956295,331.27288818359377]},{"page":282,"text":"p","rect":[143.2467803955078,338.3313903808594,146.2442541568658,334.2222595214844]},{"page":282,"text":"mh","rect":[277.1250305175781,333.7950134277344,286.285315558233,327.9962463378906]},{"page":282,"text":"–","rect":[303.9403381347656,285.0,309.08422893842387,283.0]},{"page":282,"text":"–","rect":[303.9403381347656,293.9830322265625,309.08422893842387,293.0017395019531]},{"page":282,"text":"–","rect":[303.9403381347656,308.0,309.08422893842387,306.0]},{"page":282,"text":"–","rect":[303.9403381347656,317.0,309.08422893842387,315.0]},{"page":282,"text":"S","rect":[107.3680191040039,395.4358825683594,112.19595017563117,389.7091064453125]},{"page":282,"text":"N","rect":[126.86177825927735,419.4133605957031,133.02458431262935,413.8465270996094]},{"page":282,"text":"Sbulk","rect":[193.4474639892578,416.5972595214844,209.2036659885939,408.91699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10.12 Ordoelectric surface polarization. In sketch (a) the order parameter is larger at the","rect":[53.812843322753909,458.1123046875,385.1550841071558,450.3824768066406]},{"page":282,"text":"surface and smaller in the bulk; in sketch (b) the order parameter is smaller at the surface than in","rect":[53.812843322753909,467.9638366699219,385.1678518691532,460.3694763183594]},{"page":282,"text":"the bulk. Corresponding gradient curves S(z) for a nematic liquid crystal are qualitatively pictured","rect":[53.81285858154297,477.9397888183594,385.14581817774697,470.3454284667969]},{"page":282,"text":"in the bottom sketches. Vectors mp and mh show the directions of the surface polarization","rect":[53.812828063964847,488.6633605957031,362.2259573379032,480.32098388671877]},{"page":282,"text":"decreases in the bulk due to statistical misalignment of quadrupolar molecules. On","rect":[53.812843322753909,512.779296875,385.12880793622505,503.8248596191406]},{"page":282,"text":"the contrary, in sketch (b) the order parameter is lower at the surface than in the bulk.","rect":[53.812843322753909,524.7388305664063,385.1178860237766,515.8043212890625]},{"page":282,"text":"According to Eq. (10.16) the gradient of the order parameter amplitude rS(z)","rect":[65.76486206054688,536.6983642578125,385.1596921524204,527.75390625]},{"page":282,"text":"will inevitably result in the surface ordoelectric polarization:","rect":[53.81386947631836,548.6578979492188,298.6805253750999,539.723388671875]},{"page":282,"text":"Psurf ¼ qquðrSÞ\u0001ninj \u0002 31dij\u0003","rect":[158.15390014648438,584.7406616210938,280.7958037410356,560.8554077148438]},{"page":282,"text":"(10.17)","rect":[356.07049560546877,577.0541381835938,385.1586850723423,568.5777587890625]},{"page":283,"text":"270","rect":[53.812843322753909,42.55740737915039,66.50444931178018,36.73252868652344]},{"page":283,"text":"10 Liquid Crystal – Solid Interface","rect":[265.38189697265627,44.276084899902347,385.12706897043707,36.68172836303711]},{"page":283,"text":"For a fixed orientation of n, the surface polarization is a function of rS(z) and","rect":[65.76496887207031,68.2883529663086,385.14479914716255,59.34384536743164]},{"page":283,"text":"has dimension [CGSQ/cm2 or C/m2 in the SI system]. The sign of Psurf depends on","rect":[53.812923431396487,81.16453552246094,385.17141047528755,69.19712829589844]},{"page":283,"text":"the sign of the gradient rS that is on a technique of liquid crystal alignment and,","rect":[53.813655853271487,92.20760345458985,385.1405300667453,83.26309204101563]},{"page":283,"text":"evidently, on the sign of molecular quadrupoles. In Fig. 10.12 the direction of the","rect":[53.81362533569336,104.11031341552735,385.1743549175438,95.17576599121094]},{"page":283,"text":"ordoelectric polarization in the two cases is given by vectors mp (planar alignment)","rect":[53.81362533569336,116.98374938964844,385.14678321687355,107.13529968261719]},{"page":283,"text":"and mh (homeotropic alignment).","rect":[53.81393051147461,128.02999877929688,187.35018582846409,119.09544372558594]},{"page":283,"text":"The surface polarization can be measured by different means. The most straight-","rect":[65.76547241210938,139.98953247070313,385.16228614656105,131.0549774169922]},{"page":283,"text":"forward one is based on the pyroelectric technique [15]. To measure Psurf one has to","rect":[53.813453674316409,152.86582946777345,385.1435479264594,142.95474243164063]},{"page":283,"text":"deal only with one surface of a cell with uniform director alignment, either planar or","rect":[53.81368637084961,163.908935546875,385.15059791413918,154.97438049316407]},{"page":283,"text":"homeotropic at both interfaces. The main idea is to use a spatially dependent","rect":[53.81368637084961,175.86846923828126,385.1704545743186,166.9339141845703]},{"page":283,"text":"temperature increment in order to separate the contributions to the pyroelectric","rect":[53.81368637084961,187.8280029296875,385.08985174371568,178.89344787597657]},{"page":283,"text":"response coming only from the surface under study and not from the opposite one.","rect":[53.81368637084961,199.73077392578126,385.1386379769016,190.7962188720703]},{"page":283,"text":"By definition, the pyroelectric coefficient is g ¼ dP/dT where P is macroscopic","rect":[53.81368637084961,211.69033813476563,385.1415485210594,202.73585510253907]},{"page":283,"text":"polarization of a liquid crystal and T is temperature. If we are interested only in the","rect":[53.812679290771487,223.64984130859376,385.1724323101219,214.7152862548828]},{"page":283,"text":"polarization originated from the orientational order we can subtract the “isotropic”","rect":[53.812679290771487,235.60940551757813,385.13855779840318,226.6748504638672]},{"page":283,"text":"contribution to g and calculate P in the nematic or SmA phases by integrating the","rect":[53.812679290771487,247.56893920898438,385.1743854351219,238.63438415527345]},{"page":283,"text":"pyroelectric coefficient,startingfromacertain temperatureTiinthe isotropic phase:","rect":[53.813655853271487,259.529541015625,385.207899642678,250.5939483642578]},{"page":283,"text":"T","rect":[218.6503143310547,279.3800964355469,222.49661026127917,274.77838134765627]},{"page":283,"text":"PðTÞ ¼ ð gðTÞdT","rect":[184.32394409179688,303.4861145019531,254.44366807292057,281.3638916015625]},{"page":283,"text":"Ti","rect":[217.34730529785157,310.8709716796875,222.59793785844838,305.10235595703127]},{"page":283,"text":"(10.18)","rect":[356.0721740722656,296.6537780761719,385.16036353913918,288.17742919921877]},{"page":283,"text":"In order to measure g(T) we have to change temperature by a small amount DT","rect":[65.76660919189453,335.3681945800781,385.18826646159246,326.1049499511719]},{"page":283,"text":"and record a pyroelectric response in the form of voltage Up across the load resistor","rect":[53.815589904785159,348.25787353515627,385.10247169343605,338.3931884765625]},{"page":283,"text":"R shunted by input capacitance and cell capacitance. The most convenient, dynamic","rect":[53.814369201660159,359.2876892089844,385.1801227398094,350.3531494140625]},{"page":283,"text":"regime of g measurements is based on heating the sample surface of area A by","rect":[53.814369201660159,371.1904296875,385.1711358170844,362.255859375]},{"page":283,"text":"absorbed light of a pulse laser, Fig.10.13. The light is absorbed by a semitransparent","rect":[53.813350677490237,383.1499938964844,385.1731096036155,374.2154541015625]},{"page":283,"text":"electrode or by a dye dissolved in the liquid crystal. For a very fast (in comparison","rect":[53.813350677490237,395.1095275878906,385.09047785810005,386.17498779296877]},{"page":283,"text":"with RC) jump of temperature, to the end of a laser pulse tp, the pyroelectric voltage","rect":[53.813350677490237,407.99945068359377,385.09864080621568,398.12457275390627]},{"page":283,"text":"reaches the magnitude AgDT/C and pyroelectric coefficient can be found at a given","rect":[53.81354904174805,419.0292663574219,385.15834895185005,409.7660217285156]},{"page":283,"text":"temperature. Then, on cooling the cell from the isotropic phase the temperature","rect":[53.814552307128909,430.9888000488281,385.11960638238755,422.05426025390627]},{"page":283,"text":"dependence g(T) is found and, after integrating according to (10.18), we obtain","rect":[53.814552307128909,442.9483337402344,385.1235283952094,434.0137939453125]},{"page":283,"text":"P ¼ Psurf. An example of temperature dependence of Psurf integrated over the cell","rect":[53.814537048339847,455.82476806640627,385.20796067783427,445.97369384765627]},{"page":283,"text":"Nd-YAG laser","rect":[124.61857604980469,487.0577697753906,175.511686790502,481.2110290527344]},{"page":283,"text":"Cell","rect":[222.9740753173828,488.1223449707031,236.55463262349449,482.2756042480469]},{"page":283,"text":"10k","rect":[242.4584503173828,537.5989379882813,254.89596973121113,531.7761840820313]},{"page":283,"text":"Osc","rect":[292.9926452636719,533.1255493164063,305.7338984743599,527.3987426757813]},{"page":283,"text":"Fig. 10.13 Setup for the measurements of the surface polarization by a pyroelectric technique:","rect":[53.812843322753909,572.1541137695313,385.1289339467532,564.2211303710938]},{"page":283,"text":"short pulse of a Nd-YAG laser heats the polar surface layer of a liquid crystal and the pyroelectric","rect":[53.812843322753909,582.005615234375,385.1626524665308,574.4112548828125]},{"page":283,"text":"current is detected by an oscilloscope","rect":[53.812843322753909,591.9815673828125,182.1698088386011,584.38720703125]},{"page":284,"text":"10.2","rect":[53.81307601928711,43.0,68.6199469008915,36.73167419433594]},{"page":284,"text":"Surface","rect":[70.97635650634766,43.0,96.85877368235112,36.68087387084961]},{"page":284,"text":"Energy and","rect":[99.2989501953125,44.275230407714847,138.3916220107548,36.68087387084961]},{"page":284,"text":"0.0","rect":[126.7366714477539,73.74736022949219,137.68716988250234,67.9807357788086]},{"page":284,"text":"–0.2","rect":[121.9728012084961,96.36759948730469,137.68716988250234,90.6009750366211]},{"page":284,"text":"–0.4","rect":[121.9728012084961,118.98698425292969,137.68716988250234,113.2203598022461]},{"page":284,"text":"–0.6","rect":[121.9728012084961,141.6072235107422,137.68716988250234,135.84059143066407]},{"page":284,"text":"–0.8","rect":[121.9728012084961,164.22740173339845,137.68716988250234,158.4607696533203]},{"page":284,"text":"–1.0","rect":[121.9728012084961,187.2307586669922,137.68716988250234,181.46412658691407]},{"page":284,"text":"Anchoring","rect":[140.7632598876953,44.275230407714847,176.96817535548136,36.68087387084961]},{"page":284,"text":"8OCB","rect":[149.76417541503907,76.8495864868164,174.11565133518554,70.05603790283203]},{"page":284,"text":"of","rect":[179.39566040039063,43.0,186.44372647864513,36.68087387084961]},{"page":284,"text":"Nematics","rect":[188.79844665527345,43.0,220.84053500174799,36.68087387084961]},{"page":284,"text":"1","rect":[278.3015441894531,120.74835205078125,284.0048556537728,113.53260803222656]},{"page":284,"text":"TNA","rect":[264.84600830078127,176.13412475585938,279.85922116262477,167.6740264892578]},{"page":284,"text":"2","rect":[297.5162353515625,146.20965576171876,302.516021618751,139.88400268554688]},{"page":284,"text":"TNI","rect":[311.6532287597656,95.34759521484375,323.94515477023216,86.88739776611328]},{"page":284,"text":"271","rect":[372.4984436035156,42.55655288696289,385.1900686660282,36.73167419433594]},{"page":284,"text":"Fig. 10.14 Experimental temperature dependence of surface polarization in 8OCB liquid crystal","rect":[53.812843322753909,227.6480712890625,385.15340904440947,219.9182586669922]},{"page":284,"text":"having the nematic and smectic A phase; planar (curve 1) and homeotropic (curve 2) alignment","rect":[53.812843322753909,237.55633544921876,381.4009371205813,229.9027099609375]},{"page":284,"text":"Fig. 10.15","rect":[53.812843322753909,263.13031005859377,88.76721710352823,255.40049743652345]},{"page":284,"text":"Easy direction for","rect":[94.76696014404297,263.0625915527344,155.33046048743419,255.46823120117188]},{"page":284,"text":"the equilibrium alignment of","rect":[53.812843322753909,273.0385437011719,152.05940335852794,265.4441833496094]},{"page":284,"text":"the director ðn0sÞ at the surface","rect":[53.812843322753909,283.58905029296877,155.31989428781987,274.7874755859375]},{"page":284,"text":"and definition of the zenithal","rect":[53.81243896484375,291.20635986328127,152.46513084616724,285.3391418457031]},{"page":284,"text":"ðW0sÞ and azymuthal (f0s)","rect":[53.81243896484375,303.54058837890627,138.02949613196544,294.71405029296877]},{"page":284,"text":"director angles","rect":[53.81307601928711,313.1688232421875,104.30620272635736,305.574462890625]},{"page":284,"text":"x","rect":[279.03363037109377,315.02020263671877,283.4459117478127,311.2770080566406]},{"page":284,"text":"z","rect":[294.0849304199219,262.5146179199219,297.585979773188,258.2435302734375]},{"page":284,"text":"θ0s","rect":[303.8926696777344,291.9833984375,311.8540716617486,283.0693054199219]},{"page":284,"text":"f0s","rect":[298.297607421875,318.5204772949219,306.1070284828209,309.7845458984375]},{"page":284,"text":"s","rect":[340.6269226074219,279.1152648925781,342.9349774036675,276.1938781738281]},{"page":284,"text":"n0","rect":[336.63037109375,285.10797119140627,343.62439636877988,279.24322509765627]},{"page":284,"text":"y","rect":[352.64599609375,297.7319030761719,356.64262777556066,292.373046875]},{"page":284,"text":"thickness is shown in Fig. 10.14 for two types of alignment, planar (curve 1) and","rect":[53.812843322753909,381.7333679199219,385.14272395185005,372.798828125]},{"page":284,"text":"homeotropic (curve 2). Note that the temperature behaviour of the two polarizations","rect":[53.81183624267578,393.6929016113281,385.16766752349096,384.75836181640627]},{"page":284,"text":"is quite different close to the N-SmA transition. We may guess that few dipolar","rect":[53.81183624267578,405.6524658203125,385.0919431289829,396.7178955078125]},{"page":284,"text":"smectic layers formed at the interface contribute stronger to Psurf than a not","rect":[53.81183624267578,418.52880859375,385.1363969571311,408.67742919921877]},{"page":284,"text":"stratified nematic surface layer.","rect":[53.814476013183597,429.5718688964844,180.3474087288547,420.6373291015625]},{"page":284,"text":"10.2 Surface Energy and Anchoring of Nematics","rect":[53.812843322753909,477.2400207519531,310.51161842081708,465.88507080078127]},{"page":284,"text":"10.2.1 Easy Axis","rect":[53.812843322753909,506.54888916015627,144.80293922159835,495.9828186035156]},{"page":284,"text":"Let the interface be in the xy-plane, as shown in Fig.10.15. The equilibrium position","rect":[53.812843322753909,534.0913696289063,385.14971247724068,525.0971069335938]},{"page":284,"text":"of the director n (the so-called easy direction or easy axis) is defined by the zenithal","rect":[53.81185531616211,546.0509033203125,385.206434798928,537.1163940429688]},{"page":284,"text":"ðWs0Þ and azymuthal (fs0) angles counted from the z and x axes, respectively. At the","rect":[53.812843322753909,558.7301025390625,385.17359197809068,548.3392944335938]},{"page":284,"text":"free surface of the nematic the easy direction appears spontaneously but at the","rect":[53.81284713745117,570.3098754882813,385.17258489801255,561.3753662109375]},{"page":284,"text":"nematic-solid interface it is predetermined by a specific treatment of the solid","rect":[53.81284713745117,582.2694091796875,385.1337517838813,573.3348999023438]},{"page":284,"text":"surface. We can distinguish the homeotropic (Ws0 ¼ 0), planar (Ws0 ¼ p=2) and tilted","rect":[53.81284713745117,594.94921875,385.1274041276313,584.5584106445313]},{"page":285,"text":"272","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.73106384277344]},{"page":285,"text":"10 Liquid Crystal – Solid Interface","rect":[265.38189697265627,44.274620056152347,385.12706897043707,36.68026351928711]},{"page":285,"text":"(0 < Ws0 < p=2) alignment. In turn, the planar orientation can be homogeneous","rect":[53.812843322753909,69.00798797607422,385.1577187930222,58.61709976196289]},{"page":285,"text":"with a unique angle f0s, multistable when several easy directions are possible ata","rect":[53.813880920410159,80.96746063232422,385.15979803277818,70.5766372680664]},{"page":285,"text":"crystalline or specially prepared substrate, or degenerate if all f0s-angles are equally","rect":[53.8129768371582,92.87071990966797,385.0990838151313,82.5363540649414]},{"page":285,"text":"probable and the cylindrical symmetry exists with respect to the surface normal.","rect":[53.814022064208987,104.1104965209961,385.1777309944797,95.17594909667969]},{"page":285,"text":"The same is true for the projections of the director on the xy plane in the case of the","rect":[53.814022064208987,116.0699691772461,385.1747516460594,107.13542175292969]},{"page":285,"text":"tilted orientation. Further on the angle of the director at the surface will be counted","rect":[53.81399154663086,128.02957153320313,385.12297907880318,119.09501647949219]},{"page":285,"text":"from the easy axis although other conventions may be used as, for instance, above","rect":[53.81399154663086,139.98910522460938,385.11206854059068,131.05455017089845]},{"page":285,"text":"in connection with Eq. (10.13).","rect":[53.81399154663086,151.94863891601563,179.64815946127659,143.0140838623047]},{"page":285,"text":"Surface free energy of the nematic phase FsðWs;fsÞis minimal for the easy","rect":[65.76602935791016,164.24758911132813,385.1077813248969,154.23757934570313]},{"page":285,"text":"direction ðWs ¼ W0s;fs ¼ f0sÞ. The anisotropy of the surface energy is a characteris-","rect":[53.81470489501953,176.5880584716797,385.0489438614048,166.19720458984376]},{"page":285,"text":"tic feature of liquid crystals, Fs ¼ Fisso þ Fas. Here, two terms represent the isotropic","rect":[53.81364059448242,188.55386352539063,385.17285955621568,178.89366149902345]},{"page":285,"text":"and anisotropic parts. They differ from each other by several orders of magnitude.","rect":[53.81411361694336,199.73104858398438,385.1002468636203,190.79649353027345]},{"page":285,"text":"The isotropic part, which is, in fact, the surface tension introduced earlier is of the","rect":[53.81411361694336,211.69061279296876,385.17087591363755,202.7560577392578]},{"page":285,"text":"order of 100 erg\bcm\u00022 (or 0.1 J\bm\u00022). The values for Fsa-energy are scattered over","rect":[53.81411361694336,224.34893798828126,385.1186765274204,212.5995635986328]},{"page":285,"text":"five orders of magnitude from 10\u00025 to 1 erg\bcm\u00022 (or 10\u00028 – 10\u00023 J\bm\u00022). Fsa shows","rect":[53.813655853271487,236.30859375,385.14255155669408,224.47947692871095]},{"page":285,"text":"how much energy one has to spend in order to deflect the director from the easy","rect":[53.81368637084961,247.56979370117188,385.1057976823188,238.63523864746095]},{"page":285,"text":"direction, to which it is anchored in the ground state. That is why the anisotropic","rect":[53.81368637084961,259.52935791015627,385.1654437847313,250.5948028564453]},{"page":285,"text":"part of the surface energy is usually referred to as anchoring energy.","rect":[53.81368637084961,271.5287170410156,330.80352445151098,262.534423828125]},{"page":285,"text":"In order to consider any mechanical or electro-optical effects for a liquid crystal","rect":[65.76570892333985,283.44842529296877,385.1227250821311,274.51385498046877]},{"page":285,"text":"layer placed between two solid substrates one must solve a problem of the distribu-","rect":[53.81368637084961,295.3511962890625,385.0948422989048,286.4166259765625]},{"page":285,"text":"tion of the director over the layer with allowance for the boundary conditions. The","rect":[53.81368637084961,307.3107604980469,385.14563787652818,298.376220703125]},{"page":285,"text":"standard variational procedure allows such calculations when the surface energy","rect":[53.81368637084961,319.270263671875,385.16848078778755,310.335693359375]},{"page":285,"text":"depends only on orientation of the director (angles Ws and fs) at both boundaries","rect":[53.81368637084961,331.2298278808594,385.1622659121628,321.559326171875]},{"page":285,"text":"but not on their spatial derivatives.","rect":[53.8134880065918,343.1902770996094,194.25643582846409,334.2557373046875]},{"page":285,"text":"10.2.2 Variational Problem","rect":[53.812843322753909,390.4603271484375,197.3945308849633,381.92620849609377]},{"page":285,"text":"For consistency we go back to the problem of the twisted cell discussed in","rect":[53.812843322753909,420.1062927246094,385.1417168717719,411.1717529296875]},{"page":285,"text":"Section 8.3.2, however, the director angles j at the boundaries will be not constant","rect":[53.812843322753909,432.0658264160156,385.17256028720927,423.13128662109377]},{"page":285,"text":"but can be changed due to elastic and external torques. Let a nematic layer be","rect":[53.812843322753909,444.0253601074219,385.1496967144188,435.0908203125]},{"page":285,"text":"confined by two plane surfaces with coordinates z1 ¼ \u0002d/2 and z2 ¼ þd/2 and the","rect":[53.812843322753909,455.9281311035156,385.1751178569969,446.9740905761719]},{"page":285,"text":"director is allowed to be deviated only in the xy-plane through angle j (there is no","rect":[53.814353942871097,467.8880920410156,385.1711052995063,458.95355224609377]},{"page":285,"text":"tilt, the angle W ¼ p/2 everywhere, and the azimuthal anchoring energy is finite).","rect":[53.81432342529297,479.84765625,380.9166531136203,470.6142883300781]},{"page":285,"text":"þd=2","rect":[161.43873596191407,502.10772705078127,177.49772713264785,495.1563415527344]},{"page":285,"text":"FðjÞ ¼ ð g\u0001j;qqjz\u0003dz þ Fs1ðjsÞ þ F2sðjsÞ","rect":[127.22662353515625,526.1326904296875,311.9820596133347,502.2474670410156]},{"page":285,"text":"\u0002d=2","rect":[160.92868041992188,533.8491821289063,176.98767159065566,526.8977661132813]},{"page":285,"text":"(10.19)","rect":[356.07122802734377,518.44580078125,385.1594174942173,509.96942138671877]},{"page":285,"text":"Here, g is Frank energy density, Fs12 are surface energies at opposite boundaries.","rect":[65.76566314697266,558.9864501953125,385.1571926644016,549.4159545898438]},{"page":285,"text":";","rect":[204.09237670898438,560.3320922851563,205.97371710963766,558.2543334960938]},{"page":285,"text":"Our task is to find the equilibrium alignment of the director everywhere between","rect":[65.76541900634766,570.3102416992188,385.17519465497505,561.3557739257813]},{"page":285,"text":"and at the solid surfaces. It is determined by minimization of the integral equation","rect":[53.813411712646487,582.269775390625,385.17620173505318,573.3352661132813]},{"page":285,"text":"(10.19), i.e. by solution of the correspondent differential Euler equation for the bulk","rect":[53.813411712646487,594.2293090820313,385.1243828873969,585.2947998046875]},{"page":286,"text":"10.2 Surface Energy and Anchoring of Nematics","rect":[53.812843322753909,44.274620056152347,220.84030611991205,36.68026351928711]},{"page":286,"text":"273","rect":[372.4981994628906,42.55594253540039,385.1898245254032,36.73106384277344]},{"page":286,"text":"qqjg \u0002 ddz\u0007qðqjqg=qzÞ\b ¼0","rect":[169.82276916503907,83.28526306152344,270.8155144791938,59.40004348754883]},{"page":286,"text":"(10.20)","rect":[356.07330322265627,75.5987319946289,385.1615232071079,67.12236785888672]},{"page":286,"text":"with boundary conditions","rect":[53.81670379638672,108.81502532958985,156.4386330996628,99.88047790527344]},{"page":286,"text":"\u0002 \u0007qðqjqg=qzÞ\b1 þ qqjF11 ¼ 0 and \u0007qðqjqg=qzÞ\b2 þ qqjF22 ¼0","rect":[92.10855865478516,149.1842041015625,319.98113337567818,125.03651428222656]},{"page":286,"text":"(10.21)","rect":[356.07110595703127,141.2349090576172,385.1592954239048,132.758544921875]},{"page":286,"text":"The first terms in both Eqs. (10.21) correspond to the contribution from the bulk","rect":[65.76651763916016,174.7349853515625,385.1264580827094,165.80043029785157]},{"page":286,"text":"to the surface energy. How to understand the influence of the bulk on the surface? In","rect":[53.814491271972659,186.69451904296876,385.15139094403755,177.7599639892578]},{"page":286,"text":"fact, the two equations (10.21) represent the balance of elastic and surface torques","rect":[53.814491271972659,198.65408325195313,385.10956205474096,189.7195281982422]},{"page":286,"text":"at each boundary (indices 1 and 2). One of them comes from the bulk elasticity and","rect":[53.814491271972659,210.61361694335938,385.1424187760688,201.6591339111328]},{"page":286,"text":"deflects the director from the easy axis. The other is a torque from the surface forces","rect":[53.814491271972659,222.57318115234376,385.1085244570847,213.6386260986328]},{"page":286,"text":"that tries to hold the director at its equilibrium (easy) direction. The two equations","rect":[53.814491271972659,234.53268432617188,385.1693154727097,225.59812927246095]},{"page":286,"text":"themselves are brought about from the minimization procedure.","rect":[53.814491271972659,246.49224853515626,311.0332912972141,237.5576934814453]},{"page":286,"text":"Let show it using mathematics. As was said, the boundary conditions are not","rect":[65.76651763916016,258.3949890136719,385.1354203946311,249.46043395996095]},{"page":286,"text":"fixed and the free energy depends on them. Let j(z) be a solution of the Euler","rect":[53.814491271972659,270.35455322265627,385.1423581680454,261.41998291015627]},{"page":286,"text":"equation for F(j) with fixed boundary conditions i.e. Eq. (10.20). Now we shall","rect":[53.814491271972659,282.3140869140625,385.1215043790061,273.3795166015625]},{"page":286,"text":"make variation of the boundary conditions in order to find the minimum of free","rect":[53.814491271972659,294.2736511230469,385.1363910503563,285.339111328125]},{"page":286,"text":"energy with the surface terms included.","rect":[53.814491271972659,306.2331848144531,213.08782620932346,297.29864501953127]},{"page":286,"text":"For example, we can calculate a derivative ∂F/∂j1. If we fix z1, z2 and j2 and","rect":[65.76651763916016,318.1927490234375,385.14626399091255,308.252197265625]},{"page":286,"text":"change only j1, the new solution for j(z) will get an increment dj(z). Correspond-","rect":[53.814414978027347,330.1537780761719,385.08013282624855,320.92041015625]},{"page":286,"text":"ingly the free energy will get an increment DF. Ignoring highest order terms and","rect":[53.81399154663086,342.11334228515627,385.1458367448188,332.85009765625]},{"page":286,"text":"using dj0 \u0006 (dj)0 we obtain:","rect":[53.81397247314453,354.0160827636719,172.29299790927957,344.4134521484375]},{"page":286,"text":"z2","rect":[184.0967559814453,375.34075927734377,189.24764676596409,371.29449462890627]},{"page":286,"text":"DF ¼ ð \u0001qqjg dj þ qqjg0 dj0\u0003dz","rect":[157.58665466308595,399.3943786621094,281.6684914981003,375.5091552734375]},{"page":286,"text":"z1","rect":[183.58697509765626,405.5516357421875,188.73788114096409,401.5054016113281]},{"page":286,"text":"(10.22)","rect":[356.07025146484377,391.70782470703127,385.1584409317173,383.2314758300781]},{"page":286,"text":"The second term of (10.22) can be integrated by parts:","rect":[65.76468658447266,432.12249755859377,285.56660325595927,423.18792724609377]},{"page":286,"text":"z2","rect":[209.70037841796876,453.39019775390627,214.90790982748752,449.3439636230469]},{"page":286,"text":"DF ¼ qqjg0 dj zz21þ ð \u0001qqjg \u0002 qqz qqjg0\u0003djdz","rect":[137.1373748779297,478.8965759277344,302.12579740630346,453.3116149902344]},{"page":286,"text":"z1","rect":[209.1905975341797,483.65777587890627,214.39805264975315,479.5549621582031]},{"page":286,"text":"(10.23)","rect":[356.0706481933594,469.81396484375,385.1588376602329,461.3376159667969]},{"page":286,"text":"Note that the expression under the integral vanishes because g(j) satisfies the","rect":[65.76508331298828,510.211669921875,385.17380560113755,501.2373046875]},{"page":286,"text":"Euler equation (10.20). In addition, dj (z2) ¼ 0, because j2 is fixed. Finally, the","rect":[53.81307601928711,522.1317138671875,385.17514837457505,512.8980102539063]},{"page":286,"text":"change in the bulk free energy due to variation of j1 is given by","rect":[53.81439971923828,534.0913696289063,314.4212273698188,525.15673828125]},{"page":286,"text":"DF �� \u0002qqjg0 dj z1","rect":[184.3236541748047,575.6507568359375,253.70982022787815,550.93115234375]},{"page":287,"text":"274","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.73106384277344]},{"page":287,"text":"10 Liquid Crystal – Solid Interface","rect":[265.38189697265627,44.274620056152347,385.12706897043707,36.68026351928711]},{"page":287,"text":"and the elastic torque exerted by the bulk on the j-director at the surface is given by","rect":[53.812843322753909,68.2883529663086,385.16860285810005,59.35380554199219]},{"page":287,"text":"qF","rect":[191.23402404785157,88.32222747802735,202.33098596013748,81.16064453125]},{"page":287,"text":"qg","rect":[227.5436248779297,90.43384552001953,237.50580683759223,81.16064453125]},{"page":287,"text":"¼\u0002","rect":[207.3779296875,94.0,225.52146175596625,91.0]},{"page":287,"text":":","rect":[248.9554901123047,94.99737548828125,251.64312684220216,93.95152282714844]},{"page":287,"text":"qj1","rect":[189.0251922607422,104.236328125,204.12106392464004,94.763671875]},{"page":287,"text":"(10.24)","rect":[356.07073974609377,96.7404556274414,385.1589292129673,88.26409149169922]},{"page":287,"text":"This is a contribution to be equalized by the surface torque qFs=qjjz1. The same","rect":[65.76520538330078,129.06256103515626,385.13047064020005,117.80461883544922]},{"page":287,"text":"is valid for the opposite boundary at z2, see Eqs. (10.21). Thus, two expressions","rect":[53.813533782958987,139.36611938476563,385.1335488711472,130.43150329589845]},{"page":287,"text":"(10.21) are indeed torque balance equations for the director angle j at the two","rect":[53.8136100769043,151.32565307617188,385.1524285416938,142.39109802246095]},{"page":287,"text":"boundaries.","rect":[53.81260299682617,161.19647216796876,100.12057920004611,154.2938690185547]},{"page":287,"text":"10.2.3 Surface Energy Forms","rect":[53.812843322753909,210.3917694091797,209.1317356083171,199.75399780273438]},{"page":287,"text":"In order to solve equations (10.20) and (10.21) we must know the explicit angular","rect":[53.812843322753909,237.93411254882813,385.10396705476418,228.9995574951172]},{"page":287,"text":"dependence of functions F1s and Fs2. Their simplest form is the so-called Rapini","rect":[53.812843322753909,250.52960205078126,385.1443315274436,240.95912170410157]},{"page":287,"text":"energy [16]:","rect":[53.81345748901367,261.8931579589844,103.20226152011941,252.9187774658203]},{"page":287,"text":"Fs ¼ Fsiso þ 21Wsin2djs","rect":[173.39138793945313,288.33538818359377,265.06302314653336,276.0838317871094]},{"page":287,"text":"(10.25)","rect":[356.0715026855469,286.6778869628906,385.1596921524204,278.0820007324219]},{"page":287,"text":"Here, djs ¼ js \u0002 j0s is an angle of director deflection from the equilibrium","rect":[65.76595306396485,312.56549072265627,385.08698223710618,302.6126708984375]},{"page":287,"text":"angle js0 and W is usually referred to as anchoring energy.","rect":[53.81381607055664,324.525146484375,289.7464260628391,314.870849609375]},{"page":287,"text":"When both angles Ws and jsare changed, two Rapini energies should be intro-","rect":[65.7647705078125,335.80499267578127,385.17962013093605,326.0938415527344]},{"page":287,"text":"duced: Azimuthal (for fixed Ws):","rect":[53.812923431396487,347.3263854980469,183.35461289951395,338.121337890625]},{"page":287,"text":"FjaðWsÞ ¼ 21WjðWÞsin2ðjs \u0002 js0Þ","rect":[155.77447509765626,374.2068176269531,283.2063332949753,361.955322265625]},{"page":287,"text":"Zenithal (for fixed js):","rect":[65.7655258178711,397.63775634765627,158.14745957920145,388.1138000488281]},{"page":287,"text":"(10.26a)","rect":[351.6524658203125,372.5492858886719,385.12847266999855,364.01318359375]},{"page":287,"text":"FzWðjsÞ ¼ 12WWðjÞsin2ðWs \u0002 W0sÞ","rect":[156.5667266845703,424.482666015625,282.41327299224096,411.980224609375]},{"page":287,"text":"(10.26b)","rect":[351.08636474609377,422.8252868652344,385.15166602937355,414.2891845703125]},{"page":287,"text":"The zenithal anchoring is often called polar, but this word is misleading because","rect":[65.76586151123047,448.0502014160156,385.1029743023094,439.11566162109377]},{"page":287,"text":"polar anchoring is related to polar director L as given by Eq. (10.13). Thus, the","rect":[53.813838958740237,460.0097351074219,385.17359197809068,451.0751953125]},{"page":287,"text":"Rapini form corresponds to the sine-squared shape potential well for any director","rect":[53.813838958740237,471.91253662109377,385.1069272598423,462.97796630859377]},{"page":287,"text":"deviation b (dWs or djs) from the easy direction (Ws0;j0s):","rect":[53.813838958740237,484.5917053222656,283.5598588711936,474.2007751464844]},{"page":287,"text":"Fab ¼ 21Wsin2b","rect":[190.9508514404297,510.63751220703127,248.04930466960026,498.1280822753906]},{"page":287,"text":"(10.27)","rect":[356.0713806152344,508.9800109863281,385.1595700821079,500.503662109375]},{"page":287,"text":"This function is shown by curve 1 in Fig. 10.16.","rect":[65.76583099365235,534.2049560546875,260.28106351401098,525.2704467773438]},{"page":287,"text":"The Rapini term is the first one (j ¼ 1) in an expansion of Fs in the Legendre","rect":[65.76583099365235,546.1644287109375,385.16123235895005,536.5040893554688]},{"page":287,"text":"polynomial series in terms of sin2jb:","rect":[53.81444549560547,558.067138671875,200.2056260831077,547.016357421875]},{"page":287,"text":"Fs","rect":[142.972900390625,581.8289794921875,151.77269416704119,574.5258178710938]},{"page":287,"text":"¼ Fsiso þ XWjsin2jb;","rect":[155.037841796875,586.2982177734375,244.90244996231935,572.363525390625]},{"page":287,"text":"j","rect":[198.541259765625,594.0266723632813,200.49924736778636,587.9398193359375]},{"page":287,"text":"j ¼ 1;2:3:::","rect":[249.8616180419922,583.9505004882813,295.9914087025537,575.0956420898438]},{"page":287,"text":"(10.28)","rect":[356.0720520019531,583.572021484375,385.16024146882668,575.0956420898438]},{"page":288,"text":"10.2 Surface Energy and Anchoring of Nematics","rect":[53.812843322753909,44.274986267089847,220.84030611991205,36.68062973022461]},{"page":288,"text":"Fig. 10.16 Shapes of the","rect":[53.812843322753909,67.58130645751953,141.2969298102808,59.85148620605469]},{"page":288,"text":"surface potential curves:","rect":[53.812843322753909,77.4895248413086,137.31429771628442,69.89517211914063]},{"page":288,"text":"Rapini potential (1),","rect":[53.812843322753909,87.4087142944336,123.00578567578755,79.81436157226563]},{"page":288,"text":"Legendre expansion with two","rect":[53.812843322753909,97.3846664428711,155.1451620498173,89.79031372070313]},{"page":288,"text":"terms (2) and elliptic-sine","rect":[53.812843322753909,107.36067962646485,142.05333850168706,99.76632690429688]},{"page":288,"text":"profile (3)","rect":[53.812843322753909,117.33663177490235,88.1495599137037,109.74227905273438]},{"page":288,"text":"275","rect":[372.4981994628906,42.55630874633789,385.1898245254032,36.62983322143555]},{"page":288,"text":"To improve agreement with experiment, higher order terms of expansion (10.28)","rect":[65.76496887207031,217.98248291015626,385.15581641999855,209.0479278564453]},{"page":288,"text":"are used and this change the angle dependence of the energy as shown by curve 2 in","rect":[53.812950134277347,229.9420166015625,385.14186945966255,221.00746154785157]},{"page":288,"text":"Fig. 10.16 (for both j ¼ 1 and 2 terms with ratio W2/W1 ¼ 0.5). Some experimental","rect":[53.812950134277347,241.84512329101563,385.182020736428,232.85079956054688]},{"page":288,"text":"data could be fitted better with other shapes of the surface potential, for example,","rect":[53.814292907714847,253.8046875,385.1690945198703,244.87013244628907]},{"page":288,"text":"with the elliptic sine-squared shape, 0 k 1 [1]: Fas ¼ 21Wsn2ðb;kÞ:","rect":[53.814292907714847,267.0234375,337.4614709584131,255.28204345703126]},{"page":288,"text":"Surely, this form is more general because it reduces to Rapini’s one for k ¼ 0,","rect":[65.76725006103516,277.72393798828127,385.1769680550266,268.76947021484377]},{"page":288,"text":"however, it requires numerical calculations. The corresponding angle dependence","rect":[53.815269470214847,289.6835021972656,385.08448064996568,280.74896240234377]},{"page":288,"text":"(for k close to 1) is shown by curve 3 in Fig. 10.16.","rect":[53.815269470214847,301.6430358886719,261.6979641487766,292.6885681152344]},{"page":288,"text":"When the surface energy depends not only on the director itself but also on its","rect":[65.76726531982422,313.60260009765627,385.15314115630346,304.66802978515627]},{"page":288,"text":"spatial derivatives, Fs ¼ Fsðjs;qj=qz Þ, then the so-called divergent elastic moduli","rect":[53.81523895263672,326.4949645996094,385.17243821689677,313.3251953125]},{"page":288,"text":"K13 and K24 should be taken into accosunt. In such cases, boundary conditions may","rect":[53.81370162963867,337.465576171875,385.1652764420844,327.057373046875]},{"page":288,"text":"become non-local in the sense that, for a finite cell thickness d and potential W, a","rect":[53.813438415527347,349.4251403808594,385.16126287652818,340.4706726074219]},{"page":288,"text":"situation at a boundary z1 influences the conditions at the opposite boundary z2.","rect":[53.81346130371094,361.3849182128906,374.42071195151098,352.45013427734377]},{"page":288,"text":"10.2.4 Extrapolation Length","rect":[53.812843322753909,407.89739990234377,202.7148434529221,397.3432922363281]},{"page":288,"text":"Consider again an important and fairly simple example: a twisted structure with a","rect":[53.812843322753909,435.5234069824219,385.15869939996568,426.5888671875]},{"page":288,"text":"rigid boundary condition at z ¼ 0 (easy axis y, j1 ¼ 0, W1 ! 1) and soft bound-","rect":[53.812843322753909,447.4829406738281,385.12127052156105,438.54840087890627]},{"page":288,"text":"ary condition at z ¼ d (easy axis x, j2 ¼ p/2, anchoring energy W2 ¼ W0).","rect":[53.813289642333987,459.4427185058594,385.1559109261203,450.4881286621094]},{"page":288,"text":"Fig. 10.17a clarifies the corresponding geometry. Due to the bulk elastic torque","rect":[53.81411361694336,471.4022521972656,385.13108099176255,462.46771240234377]},{"page":288,"text":"acting on the director at z ¼ d,the director deflects from the easy axis through angle","rect":[53.81411361694336,483.3050231933594,385.14197576715318,474.3505554199219]},{"page":288,"text":"jd, and forms the angle p/2 \u0002 jd with the y-axis. Our task is to find the profile of","rect":[53.81505584716797,495.2649841308594,385.15007911531105,486.3105163574219]},{"page":288,"text":"j(z) for different W0. The free energy is","rect":[53.81426239013672,507.224609375,216.15890897856907,498.28997802734377]},{"page":288,"text":"d","rect":[179.3951873779297,527.370849609375,182.87915108284316,522.476318359375]},{"page":288,"text":"F ¼ 12K22 ð \u0001ddjz\u00032dz þ 21W0ðp=2 \u0002 jdÞ2","rect":[139.743408203125,552.205810546875,298.7751929529603,526.3864135742188]},{"page":288,"text":"0","rect":[178.99871826171876,557.710205078125,182.4826819666322,552.9132690429688]},{"page":288,"text":"(10.29)","rect":[356.0715026855469,544.5189819335938,385.1596921524204,536.0426025390625]},{"page":288,"text":"Here we use Rapini surface energy (10.25) and approximation sin(p/2 \u0002 jd)","rect":[65.76595306396485,582.2701416015625,385.15975318757668,573.27587890625]},{"page":288,"text":"\u0006 (p/2 \u0002 jd).","rect":[53.81393051147461,594.1498413085938,111.95633359457736,585.2949829101563]},{"page":289,"text":"276","rect":[53.81111526489258,42.55624771118164,66.50272125391885,36.68056869506836]},{"page":289,"text":"10 Liquid Crystal – Solid Interface","rect":[265.3801574707031,44.274925231933597,385.12535998606207,36.68056869506836]},{"page":289,"text":"a","rect":[93.29457092285156,68.23336791992188,98.84983332008541,62.64461135864258]},{"page":289,"text":"φ=0","rect":[97.39020538330078,96.67709350585938,112.48149496306064,89.37464904785156]},{"page":289,"text":"d","rect":[170.69003295898438,79.06082153320313,174.68666464079502,73.3580322265625]},{"page":289,"text":"2–π – φd","rect":[207.65858459472657,111.037841796875,227.67261414221736,100.46266174316406]},{"page":289,"text":"b","rect":[233.50897216796876,68.23336791992188,239.61376591744696,60.92499542236328]},{"page":289,"text":"φ","rect":[246.50762939453126,83.71981811523438,250.67211960697794,76.4493637084961]},{"page":289,"text":"π/2","rect":[239.10276794433595,93.83048248291016,249.7330208419669,88.10369873046875]},{"page":289,"text":"φ(z)","rect":[268.72979736328127,116.68966674804688,284.38859945524816,109.4192123413086]},{"page":289,"text":"b","rect":[314.3337707519531,87.67098999023438,318.33040243376379,81.96820068359375]},{"page":289,"text":"–π2 – φd","rect":[325.7264709472656,112.03765869140625,345.74048523596738,101.46339416503906]},{"page":289,"text":"y","rect":[109.96842193603516,137.40280151367188,113.9650536178458,132.04393005371095]},{"page":289,"text":"O","rect":[249.37242126464845,144.00137329101563,255.95087701290877,138.27459716796876]},{"page":289,"text":"dz","rect":[305.5235900878906,144.60125732421876,324.870403601313,138.2818603515625]},{"page":289,"text":"z","rect":[142.3675079345703,161.81280517578126,145.86855728783645,158.2455596923828]},{"page":289,"text":"Fig. 10.17 Twisted structure with a rigid boundary condition at z ¼ 0 and soft boundary","rect":[53.812843322753909,183.43707275390626,385.16437286524697,175.70726013183595]},{"page":289,"text":"condition at z ¼ d. The geometry of the director distortion (a) and illustration of the extrapolation","rect":[53.81196975708008,193.3453369140625,385.14914459376259,185.73403930664063]},{"page":289,"text":"length b and linear dependence of the director angle f(z) (b)","rect":[53.81113052368164,203.3212890625,261.9102792130201,195.44754028320313]},{"page":289,"text":"The Euler equation for the bulk is the same as earlier, see Eq. (8.24):","rect":[65.76496887207031,228.18499755859376,344.62226731845927,219.2504425048828]},{"page":289,"text":"q2j","rect":[216.5544891357422,253.5549774169922,232.1338374298408,242.41552734375]},{"page":289,"text":"\u0002 K22 qz2 ¼0","rect":[190.8366241455078,265.1659240722656,250.36342707685004,251.50497436523438]},{"page":289,"text":"(10.30)","rect":[356.0701599121094,259.9813232421875,385.15837989656105,251.50497436523438]},{"page":289,"text":"The first boundary condition is j ¼ 0 at z ¼ 0; the second one represents the","rect":[65.76558685302735,288.66351318359377,385.17429388238755,279.72894287109377]},{"page":289,"text":"torque balance at z ¼ d according to Eq. (10.21) and (10.29):","rect":[53.81356430053711,300.5662841796875,301.792860580178,291.61181640625]},{"page":289,"text":"K22 qqjz d þ W0ðp=2 \u0002 jdÞ ¼0","rect":[159.34312438964845,339.19049072265627,279.64894953778755,315.4223937988281]},{"page":289,"text":"(10.31)","rect":[356.0704040527344,331.0590515136719,385.1585935196079,322.58270263671877]},{"page":289,"text":"The general solution of the bulk equation (10.30) is a straight line fðzÞ ¼ Az þ B","rect":[65.76583099365235,363.083984375,385.1825332013484,353.1334533691406]},{"page":289,"text":"with B ¼ 0 due to j ¼ 0 at z ¼ 0 and jðdÞ ¼ jd ¼ Ad. Now the second boundary","rect":[53.815834045410159,375.06951904296877,385.17922297528755,365.0362243652344]},{"page":289,"text":"condition reads:","rect":[53.81452178955078,384.58587646484377,118.24911363193582,377.67327880859377]},{"page":289,"text":"K22A \u0002 W0ðp=2 \u0002 AdÞ ¼ 0 that is A ¼ 2ðK22pWþ0W0dÞ:","rect":[110.45896911621094,423.5623779296875,328.5671838490381,400.90478515625]},{"page":289,"text":"Finally, we find the dependence of the twist angle on the z-coordinate in a twist","rect":[65.7660903930664,447.0860290527344,385.2086625821311,438.1514892578125]},{"page":289,"text":"cell with soft director anchoring at one boundary:","rect":[53.815086364746097,459.0455627441406,253.91960008213114,450.11102294921877]},{"page":289,"text":"pW0z","rect":[166.4801025390625,481.4742431640625,188.09906400786594,473.2862548828125]},{"page":289,"text":"jðzÞ ¼ Az ¼ 2ðK22 þ W0dÞ ¼ 2ðd þ KW202Þ ¼ 2ðd þ bÞz:","rect":[94.42923736572266,498.79461669921877,315.4274438588037,479.1906433105469]},{"page":289,"text":"(10.32)","rect":[356.073486328125,488.4042663574219,385.16167579499855,479.92791748046877]},{"page":289,"text":"Recall that for rigid director anchoring on both boundaries we had fðzÞ ¼ pz=2d,","rect":[65.7679214477539,522.639404296875,385.1866116097141,512.6889038085938]},{"page":289,"text":"see Section 8.3.2. Now, however, the situation is different and the solid line j(z) in","rect":[53.81792449951172,534.1806030273438,385.2623223405219,525.3257446289063]},{"page":289,"text":"Fig. 10.17b shows the new profile. If we extrapolate j to p/2, the profile would","rect":[53.81892776489258,546.2198486328125,385.18462458661568,537.2853393554688]},{"page":289,"text":"correspond to a virtual cell with rigid anchoring on both interfaces and enlarged","rect":[53.81991958618164,558.1793823242188,385.2692803483344,549.244873046875]},{"page":289,"text":"apparentthicknessd0 ¼ d þ b.Theadditionalthicknessb ¼ K22/W0iscalledextrap-","rect":[53.81991958618164,570.138916015625,385.15914283601418,560.480712890625]},{"page":289,"text":"olation length and it is a measure of the anchoring strength, very useful for dis-","rect":[53.81438446044922,582.1397705078125,385.25176368562355,573.1454467773438]},{"page":289,"text":"cussion of different field effects. For typical values of K22 \u0006 10\u00026 dyne (or 10\u000211N","rect":[53.81438446044922,594.0595703125,385.14025640442699,582.9688110351563]},{"page":290,"text":"10.3 Liquid Crystal Alignment","rect":[53.81367492675781,44.276084899902347,159.15316490378442,36.68172836303711]},{"page":290,"text":"277","rect":[372.49822998046877,42.55740737915039,385.1898245254032,36.73252868652344]},{"page":290,"text":"in SI system) and W0 varied in the range of 10\u00023 – 1 erg/cm2 (or 10\u00023 – 1 mJ/m2) the","rect":[53.812843322753909,68.2883529663086,385.16147649957505,57.23747253417969]},{"page":290,"text":"values of extrapolation length b ¼ 10\u00023 – 10\u00026 cm (10–0.01 mm).","rect":[53.81368637084961,80.24788665771485,314.6036343147922,69.15728759765625]},{"page":290,"text":"10.3 Liquid Crystal Alignment","rect":[53.812843322753909,130.9202117919922,219.4056439473449,119.33819580078125]},{"page":290,"text":"10.3.1 Cells","rect":[53.812843322753909,157.9915313720703,120.22364692423504,149.64865112304688]},{"page":290,"text":"In most practical applications and when examining liquid crystals, the sandwich","rect":[53.812843322753909,187.82827758789063,385.17156306317818,178.8937225341797]},{"page":290,"text":"type cells pictured in Fig. 10.18a are used. A flat capillary with a thickness of 1–100","rect":[53.812843322753909,199.73104858398438,385.1476983170844,190.79649353027345]},{"page":290,"text":"mm is made from two glass plates with transparent electrodes. The separation","rect":[53.8138427734375,211.69061279296876,385.08013239911568,202.7560577392578]},{"page":290,"text":"between the plates is fixed by means of an insulating spacer (Mylar, mica, Teflon,","rect":[53.8138427734375,223.650146484375,385.13287015463598,214.71559143066407]},{"page":290,"text":"polyethylene, etc.). To fix a very narrow gap (about 1–3 mm) glass bids or pieces of","rect":[53.8138427734375,235.60971069335938,385.15068946687355,226.67515563964845]},{"page":290,"text":"thin glass threads of proper diameter are placed between glasses. In sandwich cells","rect":[53.8138313293457,247.56924438476563,385.1865273867722,238.6346893310547]},{"page":290,"text":"light is incident along the direction of the electric field or, if required, at a specified","rect":[53.8138313293457,259.52880859375,385.09499445966255,250.59425354003907]},{"page":290,"text":"angle to it. Sometimes, e.g., when investigating the flexoelectric effect, cells with a","rect":[53.8138313293457,271.4883117675781,385.1587299175438,262.55377197265627]},{"page":290,"text":"planar arrangement of electrodes are more suitable, see Fig. 10.18b. In that case, to","rect":[53.8138313293457,283.4478454589844,385.2004326920844,274.5133056640625]},{"page":290,"text":"reduce an applied voltage, the separation between the electrodes, which are made","rect":[53.81385040283203,295.3506164550781,385.15573919488755,286.41607666015627]},{"page":290,"text":"of metallic foil or metal evaporated in vacuum, is in the range from tens of microns","rect":[53.81385040283203,307.3101806640625,385.17938627349096,298.3756103515625]},{"page":290,"text":"to few millimeters, however, even for a millimeter gap, the amount of light passing","rect":[53.81385040283203,319.26971435546877,385.17958918622505,310.33514404296877]},{"page":290,"text":"through the cell is often insufficient. A more convenient cell has interdigitated","rect":[53.81385040283203,331.2292785644531,385.18947688153755,322.29473876953127]},{"page":290,"text":"electrodes, which can be either transparent or opaque, see Fig. 10.18c. The","rect":[53.81385040283203,343.1888122558594,385.16162908746568,334.2542724609375]},{"page":290,"text":"electrodes are deposited by photolithography methods. In such structures, a large","rect":[53.81483459472656,355.14837646484377,385.16248357965318,346.21380615234377]},{"page":290,"text":"light aperture is achieved with relatively small distances (about 10 mm) between","rect":[53.81483459472656,367.10791015625,385.1805352311469,358.17333984375]},{"page":290,"text":"the electrodes and one can operate with low voltages to have quite strong field","rect":[53.81485366821289,379.06744384765627,385.1216668229438,370.13287353515627]},{"page":290,"text":"strength.","rect":[53.81485366821289,390.9701843261719,88.95122952963595,382.03564453125]},{"page":290,"text":"a","rect":[64.86354064941406,452.7554626464844,70.41880304664791,447.1667175292969]},{"page":290,"text":"b","rect":[64.29203033447266,526.1710205078125,70.39682408395086,518.8626708984375]},{"page":290,"text":"Glasses","rect":[196.98968505859376,458.1444091796875,222.46421669772858,452.29766845703127]},{"page":290,"text":"Spacers","rect":[196.73789978027345,477.04119873046877,222.7160019760489,469.7467346191406]},{"page":290,"text":"Electrodes","rect":[191.90676879882813,502.4589538574219,227.5487198715567,496.6202087402344]},{"page":290,"text":"LC","rect":[191.07546997070313,524.4622802734375,202.1541387803187,518.7274780273438]},{"page":290,"text":"Fig. 10.18","rect":[53.812843322753909,584.0003662109375,90.69380706934854,576.0504150390625]},{"page":290,"text":"Electrooptical cells of sandwich","rect":[96.69354248046875,583.9326171875,210.85201019434855,576.3382568359375]},{"page":290,"text":"in-plane interdigitated electrodes (c)","rect":[53.813682556152347,593.9085693359375,178.07803433997325,586.314208984375]},{"page":290,"text":"(a)","rect":[214.79910278320313,583.593994140625,224.64015287024669,576.3890991210938]},{"page":290,"text":"and","rect":[228.5635528564453,582.2055053710938,240.80672210841105,576.3382568359375]},{"page":290,"text":"c","rect":[249.3531951904297,452.8784484863281,254.90845758766353,447.2897033691406]},{"page":290,"text":"planar","rect":[244.7639617919922,583.9326171875,265.98431485755136,576.3382568359375]},{"page":290,"text":"type","rect":[269.91448974609377,583.9326171875,284.5352110602808,577.2018432617188]},{"page":290,"text":"(b)","rect":[288.494140625,583.593994140625,298.7887276993482,576.3890991210938]},{"page":290,"text":"and","rect":[302.7121276855469,582.2055053710938,314.95529693751259,576.3382568359375]},{"page":290,"text":"the","rect":[318.91253662109377,582.1801147460938,329.30272815012457,576.3382568359375]},{"page":290,"text":"structures","rect":[333.18719482421877,582.1801147460938,366.11768038749019,577.2018432617188]},{"page":290,"text":"with","rect":[370.1197814941406,582.1801147460938,385.18048614649697,576.3382568359375]},{"page":291,"text":"278","rect":[53.812843322753909,42.55728530883789,66.50444931178018,36.73240661621094]},{"page":291,"text":"10.3.2 Alignment","rect":[53.812843322753909,69.85308837890625,147.55626127074775,59.298980712890628]},{"page":291,"text":"10 Liquid Crystal – Solid Interface","rect":[265.38189697265627,44.275962829589847,385.12706897043707,36.68160629272461]},{"page":291,"text":"10.3.2.1 Planar Homogeneous and Tilted Alignment","rect":[53.812843322753909,97.55864715576172,282.48995338288918,88.2256851196289]},{"page":291,"text":"Most commonly, a planar alignment is produced by mechanical rubbing of the","rect":[53.812843322753909,121.3980941772461,385.1706012554344,112.46354675292969]},{"page":291,"text":"surface of the glass with paper or cloth (Chatelain’s method [17]) or using special","rect":[53.812843322753909,133.35763549804688,385.1794877774436,124.42308044433594]},{"page":291,"text":"machines with rotating brushes. The pressure under the brushes and their angular","rect":[53.81183624267578,145.31716918945313,385.10189185945168,136.3826141357422]},{"page":291,"text":"velocity is well controlled. The rubbing creates a mechanical nano-relief on the","rect":[53.81183624267578,157.27670288085938,385.17066229059068,148.34214782714845]},{"page":291,"text":"polymer coating of a glass or an electrode material in the form of ridges and","rect":[53.81183624267578,169.23623657226563,385.1437615495063,160.3016815185547]},{"page":291,"text":"troughs, Fig. 10.19, which promotes the orientation of molecules along these","rect":[53.81183624267578,181.13900756835938,385.1258014507469,172.20445251464845]},{"page":291,"text":"formations. In other words, rubbing creates an easy axis for the director n. The","rect":[53.81183624267578,193.09857177734376,385.1456683941063,184.1640167236328]},{"page":291,"text":"technique is very simple, provides sufficiently strong anchoring of the director to","rect":[53.81183624267578,205.05810546875,385.1397332291938,196.12355041503907]},{"page":291,"text":"the surface but, in the display technology, requires additional washing and drying","rect":[53.81183624267578,217.01763916015626,385.1447381120063,208.0830841064453]},{"page":291,"text":"the substrates. Another contact method is pattering the aligning layers with molec-","rect":[53.81183624267578,228.97720336914063,385.16158424226418,220.0426483154297]},{"page":291,"text":"ular size resolution by scribing a polymer coated surface by a cantilever of an","rect":[53.81183624267578,240.93673706054688,385.1507500748969,232.00218200683595]},{"page":291,"text":"atomic force microscope. The quality of alignment is very good, but the process is","rect":[53.81183624267578,252.89627075195313,385.18451322661596,243.9617156982422]},{"page":291,"text":"rather slow. Good results are obtained by evaporation of metals or oxides (e.g., SiO)","rect":[53.81183624267578,264.8558044433594,385.13179908601418,255.9013214111328]},{"page":291,"text":"onto the surface at oblique incidence, Fig. 10.20a. This method can also be applied","rect":[53.81183624267578,276.7585754394531,385.1726006608344,267.82403564453127]},{"page":291,"text":"to the orientation of various smectic mesophases.","rect":[53.81185531616211,288.7181396484375,252.5277523567844,279.7835693359375]},{"page":291,"text":"A very important technique for optical device technology is photo-alignment of","rect":[65.76387786865235,300.67767333984377,385.14775977937355,291.74310302734377]},{"page":291,"text":"photosensitive polymers illuminated by polarized light [18]. Such a technique is","rect":[53.81185531616211,312.6372375488281,385.1855508242722,303.70269775390627]},{"page":291,"text":"non-contact and allows the design of multi-pixel structures using photo-masks. In","rect":[53.810848236083987,324.5967712402344,385.14577570966255,315.6622314453125]},{"page":291,"text":"some substances (polymers included) the absorbed light causes directional destroy-","rect":[53.810848236083987,336.5563049316406,385.16866432038918,327.62176513671877]},{"page":291,"text":"ing molecules. In other materials, the light induces a molecular realignment result-","rect":[53.810848236083987,348.515869140625,385.1148923477329,339.581298828125]},{"page":291,"text":"ing in an optical anisotropy of the film promoting the alignment of the liquid crystal","rect":[53.810848236083987,360.4754333496094,385.1188798672874,351.5408935546875]},{"page":291,"text":"n","rect":[180.86355590820313,398.7919616699219,185.30781033837656,395.03277587890627]},{"page":291,"text":"Rubbing","rect":[231.82142639160157,405.0106201171875,261.58833700407629,397.8681335449219]},{"page":291,"text":"Fig. 10.19 A mechanical nano-relief obtained as a result of unidirectional rubbing","rect":[54.096107482910159,479.25421142578127,341.4645132461063,471.2196044921875]},{"page":291,"text":"surface; long polymer molecules are schematically represented by ellipsoids","rect":[54.096107482910159,489.1624450683594,315.1014755535058,481.5511474609375]},{"page":291,"text":"a","rect":[109.66497802734375,514.0713500976563,116.33130403212096,507.3648681640625]},{"page":291,"text":"LC molecules","rect":[186.93585205078126,516.4155883789063,233.99218605808015,510.5688171386719]},{"page":291,"text":"Surfactant","rect":[199.67550659179688,532.6697387695313,235.46934337000776,526.81494140625]},{"page":291,"text":"SiO","rect":[206.17002868652345,551.9152221679688,218.9112954332348,546.0364379882813]},{"page":291,"text":"Glasses","rect":[206.17002868652345,564.9044799804688,231.64456032565827,559.0576782226563]},{"page":291,"text":"b","rect":[240.2493133544922,514.0448608398438,246.3541071039704,506.73651123046877]},{"page":291,"text":"the","rect":[344.1187744140625,478.0,354.5089659430933,471.5921325683594]},{"page":291,"text":"polymer","rect":[357.1471252441406,479.1864929199219,385.4239816055982,471.5921325683594]},{"page":291,"text":"Fig. 10.20 Schemes of the planar homogeneous alignment of a nematic by an obliquely evapo-","rect":[53.812843322753909,593.9194946289063,385.14923184973886,586.189697265625]},{"page":291,"text":"rated thin film of SiO (a) and homeotropic alignment by a monolayer of surfactant molecules (b)","rect":[53.812843322753909,603.7709350585938,385.1720284805982,596.15966796875]},{"page":292,"text":"10.3 Liquid Crystal Alignment","rect":[53.812843322753909,44.274620056152347,159.15232567038599,36.68026351928711]},{"page":292,"text":"279","rect":[372.49737548828127,42.62367248535156,385.1889700332157,36.73106384277344]},{"page":292,"text":"contacting the film. Laser ablation and ion beam irradiation of polymers seem to be","rect":[53.812843322753909,68.2883529663086,385.15171087457505,59.35380554199219]},{"page":292,"text":"competing alignment techniques for displays [19].","rect":[53.812843322753909,80.24788665771485,256.3999447395969,71.31333923339844]},{"page":292,"text":"A tilted orientation of molecules at a given angle to the surface is achieved using","rect":[65.76387786865235,92.20748138427735,385.1617364030219,83.27293395996094]},{"page":292,"text":"layers of SiO produced by oblique evaporation at a very large angle (80–90 deg)","rect":[53.81186294555664,104.11019134521485,385.11287818757668,95.15572357177735]},{"page":292,"text":"between the normal to the surface and the direction to the SiO source. Tilted","rect":[53.81186294555664,114.04774475097656,385.1726006608344,107.1152572631836]},{"page":292,"text":"orientation of the nematic liquid crystal molecules can also be achieved by using","rect":[53.81186294555664,128.02932739257813,385.16762629560005,119.09477233886719]},{"page":292,"text":"photosensitive films irradiated by obliquely incident light.","rect":[53.81186294555664,139.98886108398438,287.62933011557348,131.05430603027345]},{"page":292,"text":"10.3.2.2 Homeotropic Alignment","rect":[53.81186294555664,175.94717407226563,199.98447547761573,166.61419677734376]},{"page":292,"text":"Carefully cleaned or etched glass surfaces are conducive to a homeotropic orienta-","rect":[53.81186294555664,199.78659057617188,385.1029294571079,190.85203552246095]},{"page":292,"text":"tion. Some crystalline cleavages (Al2O3, LiNbO3) also align nematics homeotropi-","rect":[53.81186294555664,211.74758911132813,385.15837989656105,202.79310607910157]},{"page":292,"text":"cally. However, the most popular technique for the homeotropic alignment is","rect":[53.81357955932617,223.65036010742188,385.12753690825658,214.71580505371095]},{"page":292,"text":"utilization of surfactants. The mechanism of homeotropic alignment by an ultrathin","rect":[53.81357955932617,235.60992431640626,385.17632380536568,226.6753692626953]},{"page":292,"text":"(even monomolecular) layer of a surfactant is demonstrated in Fig. 10.20b. An","rect":[53.81357955932617,247.5694580078125,385.13152400067818,238.63490295410157]},{"page":292,"text":"alignment layer can be obtained by withdrawing the substrate from the solution of","rect":[53.813594818115237,259.5290222167969,385.1505368789829,250.59446716308595]},{"page":292,"text":"surfactant, by polymerization of the organosilicon films directly on the substrate,","rect":[53.813594818115237,271.4885559082031,385.1773342659641,262.55401611328127]},{"page":292,"text":"and, in particular, by using a plasma discharge. Moreover, surfactant molecules can","rect":[53.813594818115237,283.44805908203127,385.12661067060005,274.51348876953127]},{"page":292,"text":"be introduced directly into the liquid crystal (e.g. lecithin or alkoxybenzoic acids)","rect":[53.813594818115237,295.4076232910156,385.09868751374855,286.47308349609377]},{"page":292,"text":"where they form the aligning layers by adsorption at the interface with a substrate.","rect":[53.813594818115237,307.3671569824219,385.1773342659641,298.4326171875]},{"page":292,"text":"10.3.2.3 Multistable Alignment","rect":[53.813594818115237,343.2687072753906,192.60816322175635,333.93572998046877]},{"page":292,"text":"When a nematic is put in contact with a crystalline substrate, the surface of which","rect":[53.813594818115237,367.1081237792969,385.1186455827094,358.173583984375]},{"page":292,"text":"possesses the N-fold rotational symmetry (e.g., N ¼ 6 for mica, N ¼ 4 for NaCl),","rect":[53.813594818115237,379.0676574707031,385.1066555550266,370.13311767578127]},{"page":292,"text":"the director is free to choose any of those N easy axes. In experiments, the","rect":[53.813594818115237,391.0272216796875,385.1175922222313,382.0926513671875]},{"page":292,"text":"orientation depends on the pre-history of the sample. A director field n(r) in a","rect":[53.81357955932617,402.9867858886719,385.16236150934068,394.05224609375]},{"page":292,"text":"nematic drop put on the surface of a crystal acquires the same N-foldsymmetry. In a","rect":[53.813594818115237,414.9463195800781,385.15647161676255,406.01177978515627]},{"page":292,"text":"sandwich cell, when crystalline axes of the opposite interfaces coincide, different","rect":[53.813594818115237,426.9058532714844,385.1723771817405,417.9713134765625]},{"page":292,"text":"domains are observed, with uniform structure or twisted through an angle 2p/N","rect":[53.813594818115237,438.8086242675781,385.1882502716377,429.87408447265627]},{"page":292,"text":"[20]. Using a properly oriented external in-plane field one can switch domains from","rect":[53.81364059448242,450.7681884765625,385.13358257890305,441.8336181640625]},{"page":292,"text":"one of the possible N orientations to another. Thus we have multistable alignment.","rect":[53.8126335144043,462.7277526855469,385.1684231331516,453.793212890625]},{"page":292,"text":"When the crystalline axes of the opposite interfaces do not coincide, many domains","rect":[53.8126335144043,474.6872863769531,385.17840971099096,465.75274658203127]},{"page":292,"text":"with different twist angles are possible.","rect":[53.8126335144043,486.6468505859375,212.4459194710422,477.7122802734375]},{"page":292,"text":"Vacuum evaporation of SiO films onto glass substrates at a grazing angle can","rect":[65.76465606689453,498.6064147949219,385.1256951432563,489.6519470214844]},{"page":292,"text":"also result in multistable alignment. Usually, the evaporation provides either the","rect":[53.8126335144043,510.5659484863281,385.17139471246568,501.63140869140627]},{"page":292,"text":"planar (⊥ to the evaporation plane) or tilted (in the evaporation plane) orientations.","rect":[53.8126335144043,522.5254516601563,385.08977933432348,513.5909423828125]},{"page":292,"text":"However, in a certain range of the incidence angle of the SiO beam and thickness of","rect":[53.81260299682617,534.42822265625,385.1474851211704,525.4737548828125]},{"page":292,"text":"a film the bistable alignment is achieved. The director is aligned at a certain polar","rect":[53.81260299682617,546.4276733398438,385.13652931062355,537.433349609375]},{"page":292,"text":"angle to a substrate and takes one of the two azimuthal angles located symmetri-","rect":[53.81260299682617,558.3472900390625,385.17034278718605,549.4127807617188]},{"page":292,"text":"cally with respect to the evaporation plane. The electric field can switch the director","rect":[53.81260299682617,570.306884765625,385.10665260163918,561.3723754882813]},{"page":292,"text":"from one stable position to the other; thus the electrically controlled surface","rect":[53.81260299682617,582.266357421875,385.07981146051255,573.3318481445313]},{"page":292,"text":"bistability has been demonstrated [21]. Multistable alignment can also be achieved","rect":[53.81260299682617,594.2259521484375,385.07976618817818,585.2914428710938]},{"page":293,"text":"280","rect":[53.81199645996094,42.55740737915039,66.50360244898721,36.73252868652344]},{"page":293,"text":"10 Liquid Crystal – Solid Interface","rect":[265.38104248046877,44.276084899902347,385.12621447824957,36.68172836303711]},{"page":293,"text":"by combination of several factors, e.g., using a microrelief in the form of a","rect":[53.812843322753909,68.2883529663086,385.1586383648094,59.35380554199219]},{"page":293,"text":"diffraction grating and treatment of the aligning film by polarized UV light.","rect":[53.812843322753909,80.24788665771485,359.3971828987766,71.31333923339844]},{"page":293,"text":"The change in the director alignment at the surface can occur spontaneously","rect":[65.76486206054688,92.20748138427735,385.1039666276313,83.27293395996094]},{"page":293,"text":"when temperature is varied (anchoring transition) due to the adsorption or desorp-","rect":[53.812843322753909,104.11019134521485,385.11687599031105,95.17564392089844]},{"page":293,"text":"tion phenomena discussed earlier. However, close to the phase transition to the","rect":[53.812843322753909,116.0697250366211,385.1706012554344,107.13517761230469]},{"page":293,"text":"isotropic phase, the order parameter and other related properties (surface tension,","rect":[53.812843322753909,128.02932739257813,385.1785854866672,119.09477233886719]},{"page":293,"text":"elasticity) are markedly changed. Due to this, close to the transition, a nematic","rect":[53.812843322753909,139.98886108398438,385.0998920269188,131.05430603027345]},{"page":293,"text":"liquid crystal aligned by a fluoropolymer with very low anchoring energy continu-","rect":[53.812843322753909,151.94839477539063,385.16170631257668,143.0138397216797]},{"page":293,"text":"ously changes the angle of the alignment at the interface from zero to p/2 demon-","rect":[53.812843322753909,163.907958984375,385.1238034805454,154.97340393066407]},{"page":293,"text":"strating a continuous anchoring transition [22].","rect":[53.812843322753909,175.86749267578126,242.97567410971409,166.9329376220703]},{"page":293,"text":"10.3.3 Berreman Model","rect":[53.812843322753909,223.87530517578126,180.02046415160712,215.3411865234375]},{"page":293,"text":"The macroscopic theory of elasticity can explain why longitudinal ridges and","rect":[53.812843322753909,253.52127075195313,385.14077082685005,244.5867156982422]},{"page":293,"text":"troughs on the surface of a glass are conducive to the planar homogeneous align-","rect":[53.812843322753909,265.4808349609375,385.09896217195168,256.5462646484375]},{"page":293,"text":"ment of nematic liquid crystals [23]. For simplicity, a sinusoidal shape is chosen for","rect":[53.812843322753909,277.44036865234377,385.17754493562355,268.50579833984377]},{"page":293,"text":"the cross-section of a surface relief with the wavevector q directed along x, see","rect":[53.812843322753909,289.3999328613281,385.11484564020005,280.46539306640627]},{"page":293,"text":"Fig. 10.21a:","rect":[53.81185531616211,301.3594665527344,102.09474809238503,292.4249267578125]},{"page":293,"text":"aðxÞ ¼ A sin qx","rect":[187.3804473876953,325.6171875,251.58412969781723,315.6666564941406]},{"page":293,"text":"(10.33)","rect":[356.0704650878906,324.8800964355469,385.15865455476418,316.40374755859377]},{"page":293,"text":"The amplitude A is assumed to be small and the components of the director n at","rect":[65.76490783691406,349.1408386230469,385.18257005283427,340.206298828125]},{"page":293,"text":"any distance from the surface remain in the figure plane at an angle y(x,z) with","rect":[53.81386947631836,361.10040283203127,385.11879817060005,351.8570861816406]},{"page":293,"text":"respect to the x-axis: nx ¼ cosW \u0006 1, ny ¼ 0, nz ¼ sinW \u0006 W. With a distancez","rect":[53.81385040283203,373.97772216796877,385.16864408599096,363.82781982421877]},{"page":293,"text":"from the surface the amplitude of the relief decreases and deeply in the bulk the","rect":[53.814857482910159,385.020751953125,385.17466009332505,376.086181640625]},{"page":293,"text":"director is parallel to the x-axis. From Fig. 10.21a, we can see that such a director","rect":[53.814857482910159,396.98028564453127,385.11089454499855,388.04571533203127]},{"page":293,"text":"field requires some energy due to elastic bend distortion. If the director were","rect":[53.814842224121097,408.9398498535156,385.1178668804344,400.00531005859377]},{"page":293,"text":"parallel to the grooves n ¼ ny everywhere as in Fig. 10.21b the director field","rect":[53.814842224121097,421.8161926269531,385.1192254166938,411.96484375]},{"page":293,"text":"would be uniform with zero elastic energy. Therefore Berreman has calculated","rect":[53.814231872558597,432.8592834472656,385.14711848310005,423.92474365234377]},{"page":293,"text":"the extra energy for the geometry (a) with respect to the case (b).","rect":[53.814231872558597,444.7620544433594,317.1459621956516,435.8275146484375]},{"page":293,"text":"Fig. 10.21 Berreman model","rect":[53.812843322753909,528.3964233398438,152.19225792136255,520.6666259765625]},{"page":293,"text":"[23] illustrating an elastic free","rect":[53.812843322753909,538.2478637695313,155.32198474192144,530.6535034179688]},{"page":293,"text":"energy difference between the","rect":[53.81199645996094,548.223876953125,155.34398791574956,540.6295166015625]},{"page":293,"text":"two configurations of the","rect":[53.81199645996094,558.1998291015625,139.53957507395269,550.60546875]},{"page":293,"text":"director, perpendicular (a)","rect":[53.81199645996094,568.17578125,143.6930856095045,560.5814208984375]},{"page":293,"text":"and parallel (b) to grooves of","rect":[53.81199645996094,578.094970703125,154.26773160559825,570.5006103515625]},{"page":293,"text":"the surface relief","rect":[53.81199645996094,586.3184204101563,111.4175042496412,580.4765625]},{"page":293,"text":"a","rect":[210.66964721679688,484.5536804199219,216.2249096140307,478.9649353027344]},{"page":293,"text":"2π","rect":[211.13864135742188,527.3534545898438,220.57169649551583,520.9468383789063]},{"page":293,"text":"q","rect":[213.37326049804688,538.0377807617188,218.3370770798599,531.8650512695313]},{"page":293,"text":"A","rect":[221.79196166992188,562.4000854492188,227.78990670627929,555.939453125]},{"page":293,"text":"x","rect":[234.39364624023438,489.3587646484375,238.8059276169533,485.6155700683594]},{"page":293,"text":"θ","rect":[259.3838806152344,485.9110107421875,264.0689321353876,479.5403747558594]},{"page":293,"text":"z","rect":[296.2510986328125,585.1743774414063,299.7521479860786,580.9032592773438]},{"page":293,"text":"bx","rect":[312.5827331542969,487.7543029785156,325.86455188453149,478.91094970703127]},{"page":294,"text":"10.3 Liquid Crystal Alignment","rect":[53.812843322753909,44.274620056152347,159.15232567038599,36.68026351928711]},{"page":294,"text":"281","rect":[372.49737548828127,42.55594253540039,385.1889700332157,36.73106384277344]},{"page":294,"text":"At the surface (z ¼ 0) the z-component of the director is assumed to be tangen-","rect":[65.76496887207031,68.2883529663086,385.10698829499855,59.35380554199219]},{"page":294,"text":"tial to the relief (strong anchoring boundary condition):","rect":[53.81393814086914,80.24788665771485,277.35468919345927,71.31333923339844]},{"page":294,"text":"Wðz ¼ 0Þ ¼ nzðxÞ ¼ qa=qx ¼ Aqcosqx","rect":[142.74684143066407,105.68626403808594,296.2235492534813,92.49664306640625]},{"page":294,"text":"(10.34)","rect":[356.0732116699219,104.3356704711914,385.1614011367954,95.85930633544922]},{"page":294,"text":"In the one constant approximation (Kii ¼ K), the Frank distortion energy (8.15)","rect":[65.76766204833985,127.1800765991211,385.10396705476418,118.1855239868164]},{"page":294,"text":"can be written in terms of the angle W:","rect":[53.813899993896487,139.13937377929688,208.47646958896707,129.906005859375]},{"page":294,"text":"gd ¼ K2 \"\u0001qqWz\u00032 þ \u0001qqWx\u00032#","rect":[163.13900756835938,181.19801330566407,275.83385334935397,151.3365020751953]},{"page":294,"text":"(10.35)","rect":[356.07073974609377,170.53932189941407,385.1589292129673,161.94342041015626]},{"page":294,"text":"The minimization gives us the Laplace equation:","rect":[65.76519012451172,202.73556518554688,262.7570024258811,193.80101013183595]},{"page":294,"text":"q2W þ q2W ¼ r2W ¼0","rect":[175.20346069335938,231.62144470214845,265.4317254166938,214.98199462890626]},{"page":294,"text":"qx2","rect":[175.7698974609375,237.7523956298828,188.71365425667129,230.05902099609376]},{"page":294,"text":"qz2","rect":[202.67691040039063,237.7324676513672,214.9970099695619,230.05902099609376]},{"page":294,"text":"with a solution","rect":[53.81418991088867,257.1839904785156,113.75943842938909,250.31126403808595]},{"page":294,"text":"(10.36)","rect":[356.0707702636719,232.5477752685547,385.1589597305454,224.0116424560547]},{"page":294,"text":"Wðx;zÞ ¼ AqcosðqxÞexpð\u0002qzÞ","rect":[158.21095275878907,281.46368408203127,280.7741738711472,271.5131530761719]},{"page":294,"text":"(10.37)","rect":[356.07476806640627,280.7265930175781,385.1629575332798,272.250244140625]},{"page":294,"text":"that satisfies the boundary condition (10.34) and the second boundary condition of","rect":[53.817176818847659,303.0599670410156,385.15105567781105,294.12542724609377]},{"page":294,"text":"W ¼ 0 at z ! 1.","rect":[53.817176818847659,312.99755859375,124.44925351645236,305.7861633300781]},{"page":294,"text":"Then, we find derivatives","rect":[65.7691879272461,325.0,168.207965985405,318.04449462890627]},{"page":294,"text":"qqWx ¼ \u0002Aq2 sinqx \b expð\u0002qzÞ; qqWz ¼ \u0002Aq2 cosqx \b expð\u0002qzÞ;","rect":[97.59963989257813,360.0127868652344,341.3697961537256,339.228271484375]},{"page":294,"text":"and substitute them into (10.35) to obtain the energy density","rect":[53.814552307128909,381.50634765625,297.5314873795844,372.51202392578127]},{"page":294,"text":"gd ¼ K2 ðAq2Þ2 expð\u00022qzÞ:","rect":[166.53802490234376,413.93994140625,272.43250977677249,393.76300048828127]},{"page":294,"text":"(10.38)","rect":[356.0717468261719,408.8250732421875,385.1599362930454,400.3487243652344]},{"page":294,"text":"Hence, the major part of the elastic energy is concentrated within a layer of p/q","rect":[65.76619720458985,435.46636962890627,385.17888728192818,426.53179931640627]},{"page":294,"text":"thickness. Integrating over z we find for the total elastic energy per unit area","rect":[53.814144134521487,447.4259338378906,362.08408392145005,438.49139404296877]},{"page":294,"text":"1","rect":[189.47801208496095,465.87310791015627,196.44593949478787,462.72857666015627]},{"page":294,"text":"Fd ¼ ð gdðzÞdz ¼ 14KA2q3","rect":[165.6885223388672,489.9098205566406,272.77500984749158,467.78759765625]},{"page":294,"text":"0","rect":[190.72459411621095,496.2682189941406,194.2085578211244,491.47125244140627]},{"page":294,"text":"(10.39)","rect":[356.0715026855469,483.0769348144531,385.1596921524204,474.6005859375]},{"page":294,"text":"Thus, the orientation of the director perpendicular to the grooves costs an excess","rect":[65.76595306396485,520.828125,385.1200295840378,511.89361572265627]},{"page":294,"text":"elastic energy quadratically dependent on the relief depth A and inversely propor-","rect":[53.81393051147461,532.7308959960938,385.1308835586704,523.79638671875]},{"page":294,"text":"tional to the cube of its period L ¼ 2p/q. For typical values of A ¼ 1 nm, L ¼ 20","rect":[53.81393051147461,544.6904296875,385.1726922135688,535.4570922851563]},{"page":294,"text":"nm and modulus K ¼10\u00026 dyn, Fd ¼ 8.10\u00022 erg/cm2 (or 8.10\u00022 mJ/m2), which is","rect":[53.81493377685547,556.6503295898438,385.1880227481003,545.5595092773438]},{"page":294,"text":"close to experimental data. If a relief is two-dimensional in the x- and y-direction","rect":[53.81432342529297,568.60986328125,385.1591424088813,559.6753540039063]},{"page":294,"text":"(e.g. at an etched surface) the director acquires the most profitable, homeotropic","rect":[53.815330505371097,580.5694580078125,385.16712225152818,571.6349487304688]},{"page":294,"text":"alignment along the z-axis.","rect":[53.815330505371097,592.5289916992188,162.48247952963596,583.594482421875]},{"page":295,"text":"282","rect":[53.807762145996097,42.56283950805664,66.49936813502237,36.73796081542969]},{"page":295,"text":"References","rect":[53.812843322753909,68.09864807128906,109.59614448282879,59.31352233886719]},{"page":295,"text":"10 Liquid Crystal – Solid Interface","rect":[265.3768005371094,44.281517028808597,385.1220030524683,36.68716049194336]},{"page":295,"text":"1.","rect":[58.06126022338867,94.0,64.40706131055318,87.80046081542969]},{"page":295,"text":"2.","rect":[58.06126022338867,114.0,64.40706131055318,107.75236511230469]},{"page":295,"text":"3.","rect":[58.06126022338867,124.0,64.40706131055318,117.67155456542969]},{"page":295,"text":"4.","rect":[58.06126022338867,144.0,64.40706131055318,137.62351989746095]},{"page":295,"text":"5.","rect":[58.06126022338867,163.34353637695313,64.40706131055318,157.4170684814453]},{"page":295,"text":"6.","rect":[58.059593200683597,184.0,64.4053942878481,177.4197998046875]},{"page":295,"text":"7.","rect":[58.060428619384769,203.19064331054688,64.40622970654927,197.4842987060547]},{"page":295,"text":"8.","rect":[58.059566497802737,223.14254760742188,64.40536758496724,217.3176727294922]},{"page":295,"text":"9.","rect":[58.05957794189453,243.10545349121095,64.40537902905904,237.2128448486328]},{"page":295,"text":"10.","rect":[53.81127166748047,273.0,64.38761016919576,267.083984375]},{"page":295,"text":"11.","rect":[53.810420989990237,293.0,64.38675949170552,287.035888671875]},{"page":295,"text":"12.","rect":[53.80955505371094,313.0,64.38589355542622,306.9311218261719]},{"page":295,"text":"13.","rect":[53.80955505371094,333.0,64.38589355542622,326.883056640625]},{"page":295,"text":"14.","rect":[53.809539794921878,343.0,64.38587829663716,336.8590087890625]},{"page":295,"text":"15.","rect":[53.80949020385742,363.0,64.3858287055727,356.652587890625]},{"page":295,"text":"16.","rect":[53.808616638183597,393.0,64.38495513989888,386.5745544433594]},{"page":295,"text":"17.","rect":[53.808616638183597,413.0,64.38495513989888,406.5772399902344]},{"page":295,"text":"18.","rect":[53.80862045288086,433.0,64.38495895459615,426.4723815917969]},{"page":295,"text":"19.","rect":[53.80862045288086,452.3169250488281,64.38495895459615,446.4242858886719]},{"page":295,"text":"20.","rect":[53.807769775390628,472.14434814453127,64.38410827710591,466.3194580078125]},{"page":295,"text":"21.","rect":[53.807769775390628,492.0,64.38410827710591,486.2146301269531]},{"page":295,"text":"22.","rect":[53.807769775390628,522.0,64.38410827710591,516.1425170898438]},{"page":295,"text":"23.","rect":[53.80860900878906,542.0,64.38494751050435,536.0377197265625]},{"page":295,"text":"Blinov, L.M., Chigrinov, V.G.: Electrooptic Effects in Liquid Crystal Materials. 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(eds.)","rect":[68.5926513671875,453.9678649902344,385.1694650040357,446.04412841796877]},{"page":295,"text":"Surface and Interfaces of Liquid Crystals, pp. 3–15. Springer, Berlin (2004)","rect":[68.591796875,463.8871154785156,328.26461881262949,456.241943359375]},{"page":295,"text":"Blinov, L.M., Sonin, A.A.: The interaction of nematic liquid crystals with anisotropic sub-","rect":[68.591796875,473.863037109375,385.16098111731699,466.2686767578125]},{"page":295,"text":"strates. Mol. Cryst. Liq. Cryst. 179, 13–25 (1990)","rect":[68.591796875,483.8390197753906,238.59585660559825,475.87213134765627]},{"page":295,"text":"Barberi, R., Durand, G.: Controlled textural bistability in nematic liquid crystals. In: Collings,","rect":[68.591796875,493.7582092285156,385.1431605537172,486.1638488769531]},{"page":295,"text":"P., Patel, J. (eds.) Handbook of Liquid Crystal Research, pp. 567–589. Oxford University","rect":[68.591796875,503.7341613769531,385.13983673243447,496.0889892578125]},{"page":295,"text":"Press, New York (1997)","rect":[68.591796875,513.3714599609375,151.5331277237623,506.1157531738281]},{"page":295,"text":"Patel, J.S., Yokoyama, H.: Continuous anchoring transition in liquid crystals. Nature 362,","rect":[68.591796875,523.68603515625,385.2007777412172,515.82080078125]},{"page":295,"text":"525–527 (1993)","rect":[68.59263610839844,533.2666625976563,123.15469449622323,525.9601440429688]},{"page":295,"text":"Berreman, D.: Solid surface shape and the alignment of the adjacent liquid crystal. Phys. Rev.","rect":[68.59263610839844,543.5812377929688,385.1441371162172,535.9868774414063]},{"page":295,"text":"Lett. 28, 1683–1686 (1972)","rect":[68.59263610839844,553.2185668945313,162.80580228430919,545.6749877929688]},{"page":296,"text":"Part III","rect":[332.1671447753906,70.58036041259766,385.1733819380572,59.488277435302737]},{"page":296,"text":"Electro-Optics","rect":[273.4261779785156,94.0221939086914,385.1072071029615,77.90404510498047]},{"page":297,"text":"Chapter 11","rect":[53.812843322753909,72.10812377929688,121.10908599090695,59.571903228759769]},{"page":297,"text":"Optics and Electric Field Effects in Nematic","rect":[53.812843322753909,90.43364715576172,355.14616933591989,76.0426254272461]},{"page":297,"text":"and Smectic A Liquid Crystals","rect":[53.812843322753909,109.14990997314453,264.2419549361041,94.07360076904297]},{"page":297,"text":"11.1 Optical Properties of Uniaxial Phases","rect":[53.812843322753909,212.0922088623047,278.9205844852702,201.29905700683595]},{"page":297,"text":"11.1.1 Dielectric Ellipsoid, Birefringence and Light Transmission","rect":[53.812843322753909,241.87918090820313,385.13946503495336,231.3250732421875]},{"page":297,"text":"11.1.1.1 Dielectric Ellipsoid","rect":[53.812843322753909,269.435546875,177.86920518229557,260.49102783203127]},{"page":297,"text":"We begin with the electric displacement vector Dj ¼ eijEi where i, j ¼ x0, y0, z0 are","rect":[53.812843322753909,294.24188232421877,385.1655963726219,283.7650146484375]},{"page":297,"text":"Cartesian coordinates and the summation over repeated indices is inferred. The","rect":[53.813838958740237,305.3273620605469,385.14475286676255,296.392822265625]},{"page":297,"text":"tensor of dielectric permittivity is symmetric eij ¼ ejiand generally (even for biaxial","rect":[53.813838958740237,318.1612854003906,385.126356673928,308.35235595703127]},{"page":297,"text":"medium) has six independent components. If an insulator is placed in the electric","rect":[53.81339645385742,329.2468566894531,385.15219915582505,320.31231689453127]},{"page":297,"text":"field, the stored electric energy density is given by","rect":[53.81339645385742,341.2064208984375,257.6059502702094,332.2718505859375]},{"page":297,"text":"1","rect":[197.46498107910157,362.13409423828127,202.44208613446723,355.40081787109377]},{"page":297,"text":"gelectr ¼ 8pE \u0002 D ¼ 8pEieijEj","rect":[159.8526153564453,375.8566589355469,278.669565971302,362.1520080566406]},{"page":297,"text":"(11.1)","rect":[361.0561828613281,370.6789245605469,385.10555396882668,362.20257568359377]},{"page":297,"text":"or","rect":[53.81450653076172,397.3560791015625,62.10636268464697,392.7145080566406]},{"page":297,"text":"8pgelect ¼ ex0x0Ex20 þ ey0y0E2y0 þ ez0z0Ez20 þ 2ey0z0Ey0Ez0 þ 2ex0z0Ex0Ez0 þ 2ex0y0Ex0Ey0","rect":[65.37035369873047,426.70526123046877,372.65277832588625,413.648193359375]},{"page":297,"text":"This is an equation of an ellipsoid arbitrary oriented with respect to any Carte-","rect":[65.76496887207031,450.2604675292969,385.1000913223423,441.325927734375]},{"page":297,"text":"sian frame [1]. The frame may be chosen in such a way that the ellipsoid will be","rect":[53.812950134277347,462.2200012207031,385.1517719097313,453.28546142578127]},{"page":297,"text":"oriented with its principal axes along the co-ordinate axes. In the new frame x, y, z,","rect":[53.81294250488281,474.1227722167969,385.1816372444797,465.188232421875]},{"page":297,"text":"the tensor is diagonal that is all the off-diagonal terms vanish:","rect":[53.81294250488281,486.0823059082031,304.04292161533427,477.14776611328127]},{"page":297,"text":"8pgelect ¼ exxEx2 þ eyyEy2 þ ezzE2z","rect":[154.69786071777345,513.4267578125,283.76423714241346,500.313232421875]},{"page":297,"text":"(11.2)","rect":[361.0561828613281,510.9071960449219,385.10555396882668,502.43084716796877]},{"page":297,"text":"The same energy may be expressed in terms of the electric displacement vector","rect":[65.76653289794922,536.9249267578125,385.11361060945168,527.9904174804688]},{"page":297,"text":"components:","rect":[53.81450653076172,548.884521484375,104.89056260654519,540.9659423828125]},{"page":297,"text":"8pgelect ¼ Dx2 þ Dy2 þ Dz2","rect":[168.5775909423828,581.7421264648438,268.2434241541322,563.1156005859375]},{"page":297,"text":"exx","rect":[211.116455078125,587.9566040039063,221.707528542427,581.6720581054688]},{"page":297,"text":"eyy","rect":[234.3409881591797,589.3789672851563,244.93204636469265,581.6720581054688]},{"page":297,"text":"ezz","rect":[258.0753173828125,587.8938598632813,267.80231147661149,581.6720581054688]},{"page":297,"text":"(11.3)","rect":[361.0561828613281,581.3048095703125,385.10555396882668,572.8284301757813]},{"page":297,"text":"L.M. Blinov, Structure and Properties of Liquid Crystals,","rect":[53.812843322753909,623.7623291015625,251.63688919141254,616.1510620117188]},{"page":297,"text":"DOI 10.1007/978-90-481-8829-1_11, # Springer ScienceþBusiness Media B.V. 2011","rect":[53.812843322753909,633.698486328125,351.58160919337197,625.4920043945313]},{"page":297,"text":"285","rect":[372.4981994628906,622.0606079101563,385.18979400782509,616.1340942382813]},{"page":298,"text":"286","rect":[53.812843322753909,42.55630874633789,66.50444931178018,36.68062973022461]},{"page":298,"text":"11 Optics and","rect":[114.25279998779297,44.274986267089847,161.84293121485636,36.6636962890625]},{"page":298,"text":"Fig. 11.1 Dielectric ellipsoid","rect":[53.812843322753909,67.58130645751953,155.3329824843876,59.85148620605469]},{"page":298,"text":"for a biaxial medium","rect":[53.812843322753909,75.76238250732422,125.6067223393439,69.89517211914063]},{"page":298,"text":"Electric","rect":[164.21456909179688,43.0,190.70617053293706,36.68062973022461]},{"page":298,"text":"Field","rect":[193.10403442382813,43.0,210.54230255274698,36.68062973022461]},{"page":298,"text":"Effects","rect":[212.93002319335938,43.0,236.95947726249018,36.68062973022461]},{"page":298,"text":"in","rect":[239.38360595703126,43.0,245.9916738906376,36.68062973022461]},{"page":298,"text":"Nematic","rect":[248.33370971679688,43.0,277.11827990793707,36.68062973022461]},{"page":298,"text":"and","rect":[279.544921875,43.0,291.7880911269657,36.68062973022461]},{"page":298,"text":"y","rect":[281.1170349121094,125.10384368896485,285.11366659392,119.74498748779297]},{"page":298,"text":"Smectic","rect":[294.15972900390627,43.0,321.5397581794214,36.68062973022461]},{"page":298,"text":"Öe1","rect":[308.2989501953125,71.38873291015625,319.21115784338925,62.08525466918945]},{"page":298,"text":"Öe2","rect":[300.4643859863281,104.33905029296875,311.37644104651425,95.03557586669922]},{"page":298,"text":"A Liquid","rect":[323.9554138183594,44.274986267089847,355.03708404688759,36.68062973022461]},{"page":298,"text":"x","rect":[327.9566955566406,66.30203247070313,332.3689769333596,62.55883026123047]},{"page":298,"text":"Crystals","rect":[357.4324035644531,44.274986267089847,385.1677597331933,36.68062973022461]},{"page":298,"text":"and now the constant energy curves (8pgelect) form ellipsoids in the space Dx,","rect":[53.812843322753909,163.8354949951172,385.1832241585422,154.86109924316407]},{"page":298,"text":"Dy, Dz,.","rect":[53.814537048339847,176.67176818847657,84.62018247153049,167.039794921875]},{"page":298,"text":"ffiffiffiffiffiffiffiffiffiffiffiffiffiffi","rect":[306.6189270019531,179.0,335.71506798818839,177.0]},{"page":298,"text":"Finally we go back to the x ,y, z space replacing vector D=p8pgelect by vector r.","rect":[65.7649154663086,188.0435028076172,385.18255277182348,177.90988159179688]},{"page":298,"text":"Then we obtain the dielectric ellipsoid shown in Fig. 11.1 with three semi-axes","rect":[53.8138542175293,199.67453002929688,385.0720254336472,190.73997497558595]},{"page":298,"text":"equal to pe3, pe3, pe3 and satisfying to equation","rect":[53.813838958740237,211.94996643066407,255.97046748212348,201.99929809570313]},{"page":298,"text":"2","rect":[190.95086669921876,230.52120971679688,194.4348304041322,225.80792236328126]},{"page":298,"text":"2","rect":[211.51304626464845,230.52120971679688,214.9970099695619,225.80792236328126]},{"page":298,"text":"2","rect":[231.7352752685547,230.52120971679688,235.21923897346816,225.80792236328126]},{"page":298,"text":"x","rect":[186.47584533691407,234.23851013183595,190.92537725641098,229.7064971923828]},{"page":298,"text":"y","rect":[207.03799438476563,236.27044677734376,211.48752630426254,229.6367645263672]},{"page":298,"text":"z","rect":[227.88345336914063,234.14886474609376,231.71582426177219,229.82601928710938]},{"page":298,"text":"þ þ ¼1","rect":[197.1251678466797,241.7672576904297,254.15952387860785,234.21722412109376]},{"page":298,"text":"e1 e2 e3","rect":[186.53250122070313,249.3662872314453,235.44587776741347,243.06089782714845]},{"page":298,"text":"(11.4)","rect":[361.0557861328125,242.69358825683595,385.10515724031105,234.21722412109376]},{"page":298,"text":"From this ellipsoid we can find pe for any direction specified by radius-vector r,","rect":[65.7661361694336,273.22149658203127,385.18377347494848,263.2709655761719]},{"page":298,"text":"see the figure. For example such an ellipsoid corresponds to the biaxial phase of the","rect":[53.815101623535159,284.8652648925781,385.2325824566063,275.93072509765627]},{"page":298,"text":"SmC liquid crystal. In this case all the three semi-axes are different e1 ¼ exx 6¼ e2 ¼ eyy","rect":[53.815101623535159,297.741943359375,385.2338106300847,287.5622253417969]},{"page":298,"text":"6¼ e3 ¼ ezz. For a uniaxial phase (nematic, smectic A) the ellipsoid degenerates into an","rect":[53.812843322753909,308.72845458984377,385.2090996842719,299.4652099609375]},{"page":298,"text":"ellipsoid of revolution that is invariant for rotation about, e.g., the z-axis. For an","rect":[53.81351852416992,320.68798828125,385.2100762467719,311.75341796875]},{"page":298,"text":"isotropic liquid or a cubic crystal the ellipsoid degenerates into a sphere of radius pe.","rect":[53.813507080078128,333.02008056640627,385.1831936409641,323.0695495605469]},{"page":298,"text":"At optical frequencies e ¼ n2 and the same ellipsoid becomes the so-called","rect":[65.76654052734375,344.6073913574219,385.1758965592719,333.55645751953127]},{"page":298,"text":"“optical indicatrix” with its semi-axes exactly equal to refraction indices n1, n2 and n3.","rect":[53.81313705444336,356.56695556640627,385.1832241585422,347.63238525390627]},{"page":298,"text":"2","rect":[190.3843231201172,375.5108337402344,193.86828682503066,370.79754638671877]},{"page":298,"text":"2","rect":[211.4563751220703,375.5108337402344,214.94033882698379,370.79754638671877]},{"page":298,"text":"2","rect":[232.24514770507813,375.5108337402344,235.7291114099916,370.79754638671877]},{"page":298,"text":"x","rect":[185.90940856933595,379.2279968261719,190.35894048883285,374.69598388671877]},{"page":298,"text":"y","rect":[206.98147583007813,381.25994873046877,211.43100774957504,374.62628173828127]},{"page":298,"text":"z","rect":[228.393310546875,379.13836669921877,232.22568143950657,374.8155212402344]},{"page":298,"text":"2þ 2þ 2¼1","rect":[190.66761779785157,389.4542541503906,255.00930110028754,379.1501159667969]},{"page":298,"text":"n1 n2 n3","rect":[185.6261444091797,395.7725524902344,236.2955481775697,388.4127502441406]},{"page":298,"text":"(11.5)","rect":[361.05645751953127,387.62646484375,385.1058286270298,379.03057861328127]},{"page":298,"text":"Therefore, electromagnetic waves with polarization vectors along x, y or z axes","rect":[65.76581573486328,419.3127746582031,385.10886015044408,410.37823486328127]},{"page":298,"text":"propagate with three different velocities c/n1, c/n2 and c/n3. In addition, two waves","rect":[53.81379318237305,431.27227783203127,385.1181680117722,422.33770751953127]},{"page":298,"text":"with the same wave normal h but orthogonal polarizations s propagate in different","rect":[53.814205169677737,443.2319641113281,385.174940658303,434.29742431640627]},{"page":298,"text":"directions; the wavevector of the ordinary ray is parallel to the normal, ko||h, but","rect":[53.814205169677737,455.1914978027344,385.13679368564677,446.2569580078125]},{"page":298,"text":"wavevector ke for the extraordinary ray forms an angle with h. It means that the","rect":[53.81386947631836,467.09466552734377,385.17408025934068,458.16009521484377]},{"page":298,"text":"Snell law is not fulfilled for the extraordinary index of biaxial crystals. This results","rect":[53.81330871582031,479.0542297363281,385.12631620513158,470.0997619628906]},{"page":298,"text":"in a double refraction phenomenon. For biaxial crystals the double refraction occurs","rect":[53.81330871582031,491.0137634277344,385.1273538027878,482.0792236328125]},{"page":298,"text":"even at normal light incidence onto their surface; for uniaxial ones only at oblique","rect":[53.81330871582031,502.9732971191406,385.11438787652818,494.03875732421877]},{"page":298,"text":"incidence.","rect":[53.81330871582031,512.90087890625,94.71217008139377,505.998291015625]},{"page":298,"text":"11.1.1.2 Extraordinary Index of a Birefringent Layer","rect":[53.81330871582031,557.1612548828125,288.1433795757469,547.6987915039063]},{"page":298,"text":"The most interesting for applications are uniaxial phases in which n1 ¼ n2 ¼ n⊥ and","rect":[53.81330871582031,580.681884765625,385.14626399091255,571.7274169921875]},{"page":298,"text":"n3 ¼ n||. For k || n, z (n is the director) light of any polarization propagates along the","rect":[53.814414978027347,592.642578125,385.17163885309068,583.7080688476563]},{"page":299,"text":"11.1 Optical Properties of Uniaxial Phases","rect":[53.81251525878906,44.275840759277347,199.20247347831049,36.66455078125]},{"page":299,"text":"287","rect":[372.4970703125,42.55716323852539,385.18866485743447,36.73228454589844]},{"page":299,"text":"optical axis with the same velocity c/n⊥ (no extraordinary ray). For example, this","rect":[53.812843322753909,68.2883529663086,385.1440469180222,59.35380554199219]},{"page":299,"text":"corresponds to the normal incidence of light onto homeotropically aligned nematic","rect":[53.813167572021487,80.24788665771485,385.09925115777818,71.31333923339844]},{"page":299,"text":"layer (case a in Fig. 11.2). For an arbitrary angle between k and n, the beam of","rect":[53.813167572021487,92.20748138427735,385.15203224031105,83.27293395996094]},{"page":299,"text":"unpolarised light can always be decomposed into two beams. The ordinary ray with","rect":[53.81415939331055,104.11019134521485,385.11513606122505,95.17564392089844]},{"page":299,"text":"electric polarization vector e ⊥ n propagates with velocity c/no independent of the","rect":[53.81415939331055,116.07039642333985,385.17469061090318,107.13517761230469]},{"page":299,"text":"incidence angle. The extraordinary ray propagates with velocity c/ne; Index ne","rect":[53.81393051147461,128.02999877929688,385.1820833351628,119.09544372558594]},{"page":299,"text":"depends on the incident angle and can be found from the optical indicatrix. For","rect":[53.812843322753909,139.98959350585938,385.1467221817173,131.05503845214845]},{"page":299,"text":"example, in Fig. 11.2b, a nematic liquid crystal has a tilted orientation with an angle","rect":[53.812843322753909,151.94912719726563,385.13974798395005,143.0145721435547]},{"page":299,"text":"a between the light vector e and the director n. Then the refraction index for the","rect":[53.812835693359378,163.90869140625,385.1715778179344,154.97413635253907]},{"page":299,"text":"extraordinary ray as a function of the tilt angle W ¼ p/2 \u0003 a between n and z is given","rect":[53.81280517578125,175.86822509765626,385.1575860123969,166.63485717773438]},{"page":299,"text":"by:","rect":[53.81181716918945,187.8277587890625,66.56316239902566,178.89320373535157]},{"page":299,"text":"neðWÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffinffiffijffijffinffiffi?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi","rect":[158.09510803222657,216.28054809570313,279.234507685454,201.9895782470703]},{"page":299,"text":"qnj2jcos2W þ n?2 sin2W","rect":[193.10409545898438,231.03750610351563,279.26077622721746,213.1186065673828]},{"page":299,"text":"(11.6)","rect":[361.0565185546875,215.5439910888672,385.10588966218605,207.0078582763672]},{"page":299,"text":"Here n|| and n⊥ are principal refractive indices of the nematic (semi-axes of the","rect":[65.7668685913086,254.54159545898438,385.1736530132469,245.60704040527345]},{"page":299,"text":"ellipsoid).","rect":[53.81289291381836,266.4443664550781,94.10454984213595,257.50982666015627]},{"page":299,"text":"This result came about from the consideration of the ellipsoid cross-section in","rect":[65.7649154663086,278.4039306640625,385.1398858170844,269.4693603515625]},{"page":299,"text":"plane (n, k) and the position of point P on the indicatrix, see Fig. 11.3. The","rect":[53.81289291381836,290.36346435546877,385.1467670269188,281.40899658203127]},{"page":299,"text":"projections of the segment ne(W) on the semi-axes of the ellipse are","rect":[53.81290054321289,302.3235168457031,324.55432928277818,293.09014892578127]},{"page":299,"text":"X ¼ neðWÞcosW; Z ¼ neðWÞsinW and Y ¼0","rect":[65.76575469970703,314.621826171875,241.86745539716254,304.6712951660156]},{"page":299,"text":"The point P is situated on the indicatrix, therefore, from (11.5) we obtain the","rect":[65.76653289794922,326.2427062988281,385.1762164898094,317.2483825683594]},{"page":299,"text":"expression","rect":[53.81451416015625,338.2022705078125,96.50911799481878,329.2677001953125]},{"page":299,"text":"Fig. 11.2 Normal light","rect":[53.812843322753909,407.4963073730469,134.77512840476099,399.7664794921875]},{"page":299,"text":"incidence on a planar layer of","rect":[53.812843322753909,417.4045715332031,155.34399503333263,409.8102111816406]},{"page":299,"text":"a nematic liquid crystal with","rect":[53.812843322753909,427.3237609863281,151.57544464259073,419.7294006347656]},{"page":299,"text":"homeotropic (a) and tilted (b)","rect":[53.812843322753909,437.2997131347656,155.3625954971998,429.7053527832031]},{"page":299,"text":"alignment (k is light","rect":[53.812843322753909,447.2756652832031,123.67251304831568,439.6813049316406]},{"page":299,"text":"propagation vector, e is light","rect":[53.81284713745117,457.25164794921877,152.22187523093286,449.65728759765627]},{"page":299,"text":"polarization vector, n is the","rect":[53.81285095214844,467.1708679199219,147.9811796882105,459.5765075683594]},{"page":299,"text":"director)","rect":[53.81285858154297,476.80816650390627,83.04585355384042,469.5524597167969]},{"page":299,"text":"Fig. 11.3 The geometry for","rect":[53.812843322753909,516.3234252929688,150.79872220618419,508.3904113769531]},{"page":299,"text":"calculation of the","rect":[53.812843322753909,524.4791259765625,113.37116381906987,518.6372680664063]},{"page":299,"text":"extraordinary refraction index","rect":[53.812843322753909,536.2075805664063,155.3093008437626,528.6132202148438]},{"page":299,"text":"ne(W) for the tilted nematic","rect":[53.812843322753909,545.841796875,145.44167468332769,538.2781982421875]},{"page":299,"text":"shown in Fig. 11.2b. Z andX","rect":[53.81251525878906,556.1024780273438,154.7691705380103,548.5081176757813]},{"page":299,"text":"are projections of segment OP","rect":[53.81251525878906,566.0784301757813,155.3605893473342,558.4671630859375]},{"page":299,"text":"on the long and short ellipsoid","rect":[53.81251525878906,575.9976196289063,155.33435577540323,568.4032592773438]},{"page":299,"text":"axes","rect":[53.81251525878906,584.2210693359375,68.8647583293847,580.2756958007813]},{"page":299,"text":"a","rect":[187.66543579101563,385.6458740234375,193.22069818824947,380.05712890625]},{"page":299,"text":"e","rect":[226.05177307128907,389.83428955078127,229.5528224245552,385.9311218261719]},{"page":299,"text":"k","rect":[259.0882568359375,397.3121643066406,263.50053821265649,391.81732177734377]},{"page":299,"text":"b","rect":[294.8265380859375,385.6458740234375,300.9313318354157,378.3374938964844]},{"page":299,"text":"O","rect":[317.3996887207031,541.214111328125,323.56249477405518,535.4873046875]},{"page":299,"text":"k","rect":[317.03997802734377,576.4175415039063,321.28440087342667,570.5947875976563]},{"page":299,"text":"ϑ","rect":[330.317626953125,529.0829467773438,335.36137613557,523.4201049804688]},{"page":299,"text":"J","rect":[339.9493103027344,423.8899841308594,344.9930594851794,418.2272033691406]},{"page":299,"text":"Z","rect":[342.5430908203125,533.6843872070313,348.2902471787562,528.2855224609375]},{"page":299,"text":"X","rect":[338.2067565917969,559.7044677734375,343.9539129502406,554.3056030273438]},{"page":299,"text":"k","rect":[356.2414855957031,390.10321044921877,360.6537669724221,384.6083679199219]},{"page":299,"text":"a","rect":[350.10162353515627,434.5739440917969,355.1453727176013,430.7747497558594]},{"page":299,"text":"e","rect":[360.3929748535156,442.5219421386719,363.89402420678177,438.6187744140625]},{"page":299,"text":"n","rect":[352.6657409667969,511.20599365234377,356.6623726486075,507.40679931640627]},{"page":299,"text":"P","rect":[352.1870422363281,550.65869140625,357.01497330795538,545.1638793945313]},{"page":299,"text":"x","rect":[356.0998840332031,558.6917114257813,360.5121654099221,554.9485473632813]},{"page":299,"text":"z","rect":[364.43048095703127,516.7471313476563,367.9315303102974,513.1798706054688]},{"page":299,"text":"n","rect":[376.92779541015627,397.92974853515627,380.9244270919669,394.13055419921877]},{"page":300,"text":"288","rect":[53.81200408935547,42.55801773071289,66.50361007838174,36.73313903808594]},{"page":300,"text":"11 Optics and Electric Field Effects in Nematic and Smectic A Liquid Crystals","rect":[114.25196075439453,44.276695251464847,385.1669357585839,36.6654052734375]},{"page":300,"text":"n2eðWÞ2cos2W þ n2eðWÞ2sin2W ¼1","rect":[161.0421905517578,78.56005859375,279.59319392255318,59.44980239868164]},{"page":300,"text":"n?","rect":[178.0920867919922,84.73822021484375,188.44240676669956,77.57508087158203]},{"page":300,"text":"njj","rect":[235.13442993164063,87.09064483642578,243.88152350612203,77.57508087158203]},{"page":300,"text":"and then arrive at Eq. (11.6):","rect":[53.81380844116211,110.62862396240235,170.61054856845926,101.69407653808594]},{"page":300,"text":"neðWÞ ¼  cons?22W þ sinnj22jW!\u00031=2 ¼ qnffiffiffij2ffijfficffiffioffiffisffiffi2ffinffiWffijffijffinffiþffi?ffiffiffiffinffiffi2?ffiffiffisffiffiiffinffiffiffi2ffiffiWffiffi","rect":[107.51380157470703,162.05709838867188,329.8448216373737,124.91726684570313]},{"page":300,"text":"For example, for the homeotropic orientation, W ¼","rect":[65.7666244506836,185.50408935546876,283.3009722540131,176.27072143554688]},{"page":300,"text":"homogeneous planar alignment W ¼ p/2 and ne(W) ¼ n||.","rect":[53.81444549560547,197.46383666992188,279.03020902182348,188.23046875]},{"page":300,"text":"0","rect":[288.1546325683594,183.4821014404297,293.13173762372505,176.62930297851563]},{"page":300,"text":"and","rect":[297.9545593261719,183.47213745117188,312.35830012372505,176.5695343017578]},{"page":300,"text":"ne(W)","rect":[317.2139587402344,185.2027587890625,337.69085059968605,176.27093505859376]},{"page":300,"text":"¼","rect":[342.53253173828127,182.03805541992188,350.19727352354439,179.70730590820313]},{"page":300,"text":"n⊥;","rect":[355.0509948730469,185.06332397460938,368.7248368985374,178.80088806152345]},{"page":300,"text":"for","rect":[373.57452392578127,183.44247436523438,385.1811460098423,176.5697479248047]},{"page":300,"text":"11.1.1.3 Light Ellipticity","rect":[53.81421661376953,239.6923828125,164.51499262860785,230.04067993164063]},{"page":300,"text":"Consider the normal incidence of unpolarised monochromatic light of wavelength l","rect":[53.81421661376953,263.21307373046877,385.18692368815496,253.95977783203126]},{"page":300,"text":"onto a homogeneously aligned (n || x) nematic or SmA layer of thickness d. The","rect":[53.81421661376953,275.172607421875,385.14807928277818,266.2181396484375]},{"page":300,"text":"layer is between the polariser (P) and an analyser (A) with an arbitrary anglew","rect":[53.81421661376953,287.1321716308594,385.18692368815496,278.1976318359375]},{"page":300,"text":"between them, Fig. 11.4. We are interested in the light polarization and intensity of","rect":[53.81421661376953,299.0916748046875,385.1490720352329,290.1571044921875]},{"page":300,"text":"the transmitted beam [2]. The reflected light intensity is negligible (few percents of","rect":[53.81421661376953,311.0512390136719,385.1511472305454,302.11669921875]},{"page":300,"text":"the incident intensity), the absorption in a liquid crystal, is absent and both","rect":[53.815208435058597,322.9539794921875,385.12917414716255,314.0194091796875]},{"page":300,"text":"polarizers and analyser considered to be ideal.","rect":[53.815208435058597,334.9135437011719,240.3272671272922,325.97900390625]},{"page":300,"text":"The amplitude E of the linearly polarised light beam after a polarizer can be","rect":[65.7672348022461,346.8730773925781,385.15210760309068,337.93853759765627]},{"page":300,"text":"projected onto the two principal directions of the nematic, parallel and perpendicu-","rect":[53.815208435058597,358.8326416015625,385.1291745742954,349.8980712890625]},{"page":300,"text":"lar to the optical axis x:","rect":[53.815208435058597,370.79217529296877,149.1138292081077,361.85760498046877]},{"page":300,"text":"Ejj ¼ a ¼ Ecosj;E? ¼ b ¼ Esinj;","rect":[143.93765258789063,396.0579528808594,293.39146363419436,385.77862548828127]},{"page":300,"text":"(11.7)","rect":[361.05718994140627,394.31475830078127,385.1065610489048,385.8384094238281]},{"page":300,"text":"The ordinary and extraordinary rays passing the layer acquire an additional","rect":[65.76753997802735,419.5964660644531,385.1733537442405,410.66192626953127]},{"page":300,"text":"phase shifts equal, respectively, to 2pnod/l and 2pned/l, therefore their phase","rect":[53.81549835205078,431.5559997558594,385.1010211773094,422.3024597167969]},{"page":300,"text":"difference is d ¼ \u00032p=l\u0004ðne \u0003 noÞd.","rect":[53.81394577026367,447.8311767578125,198.9907650520969,429.9222412109375]},{"page":300,"text":"Generally, the interference of two fields results in an elliptic polarization. It can","rect":[65.7656021118164,455.4182434082031,385.1285332780219,446.4637756347656]},{"page":300,"text":"be shown if we consider time dependencies of the fields. After the liquid crystal","rect":[53.813594818115237,467.3778076171875,385.1215959317405,458.4432373046875]},{"page":300,"text":"layer, the output fields are:","rect":[53.813594818115237,479.33734130859377,161.94717271396707,470.40277099609377]},{"page":300,"text":"Fig. 11.4 Transmission of","rect":[53.812843322753909,525.5624389648438,145.20510953528575,517.8326416015625]},{"page":300,"text":"unpolarised light through a","rect":[53.812843322753909,535.4706420898438,144.4876036384058,527.8762817382813]},{"page":300,"text":"homogeneously aligned","rect":[53.812843322753909,545.389892578125,133.91632599024698,537.7955322265625]},{"page":300,"text":"nematic (or smectic A) layer;","rect":[53.812843322753909,555.3658447265625,152.46047691550317,547.771484375]},{"page":300,"text":"geometry of experiment with","rect":[53.812843322753909,565.341796875,151.74634308009073,557.7474365234375]},{"page":300,"text":"polarizer P and analyser A (a)","rect":[53.812843322753909,575.3177490234375,154.79571622473888,567.723388671875]},{"page":300,"text":"anddefinition of characteristic","rect":[53.81199645996094,583.5098876953125,155.44551989817144,577.6426391601563]},{"page":300,"text":"angles j and w (b)","rect":[53.81199645996094,595.2129516601563,116.39007657630136,587.6185913085938]},{"page":300,"text":"a","rect":[183.53082275390626,522.4165649414063,189.08608515114009,516.8278198242188]},{"page":300,"text":"z","rect":[189.75257873535157,544.3043212890625,193.2536280886177,540.737060546875]},{"page":300,"text":"y","rect":[184.8814697265625,571.6160888671875,188.87810140837315,566.2572631835938]},{"page":300,"text":"x","rect":[209.03311157226563,569.760498046875,213.44539294898457,566.017333984375]},{"page":300,"text":"P","rect":[260.5600891113281,531.3630981445313,265.38802018295538,525.8682861328125]},{"page":300,"text":"A","rect":[263.2698059082031,566.8219604492188,269.43261196155518,561.087158203125]},{"page":300,"text":"b","rect":[284.3828125,522.4165649414063,290.4876062494782,515.1082153320313]},{"page":300,"text":"A-axis","rect":[285.2954406738281,555.9425048828125,307.3568558578848,550.0797119140625]},{"page":300,"text":"P-axis","rect":[297.7134704589844,567.7849731445313,318.46397255710357,561.9221801757813]},{"page":300,"text":"χ","rect":[351.57379150390627,542.6253051757813,355.9620930905343,537.2584838867188]},{"page":300,"text":"x","rect":[380.76806640625,557.572998046875,385.18034778296899,553.829833984375]},{"page":301,"text":"11.1 Optical Properties of Uniaxial Phases","rect":[53.812843322753909,44.274620056152347,199.2027939128808,36.663330078125]},{"page":301,"text":"289","rect":[372.49737548828127,42.62367248535156,385.1889700332157,36.73106384277344]},{"page":301,"text":"x ¼ acosot;y ¼ bcosðot \u0003 dÞ ¼ bðcosotcosd þ sinotsindÞ","rect":[81.62570190429688,69.47722625732422,333.2850686465378,59.52671432495117]},{"page":301,"text":"(11.8)","rect":[361.0602722167969,68.74015045166016,385.1096433242954,60.26378631591797]},{"page":301,"text":"From these two equations for we obtain y=b \u0003 cosotcosd ¼ sinotsind and","rect":[65.7706298828125,93.38639831542969,374.7293633561469,83.45580291748047]},{"page":301,"text":"sinot \u0002 sind ¼ y \u0003 x cosd","rect":[166.9969482421875,118.75570678710938,273.7010125260688,107.26293182373047]},{"page":301,"text":"ba","rect":[228.33612060546876,125.52739715576172,250.42009821942816,118.58494567871094]},{"page":301,"text":"or","rect":[53.8134880065918,146.99624633789063,62.105344160477049,142.3546600341797]},{"page":301,"text":"sin2ot \u0002 sin2d ¼ by22 \u0003 2bxay cosd þ ax22 cos2d:","rect":[133.5128631591797,185.38272094726563,305.5128930775537,163.28909301757813]},{"page":301,"text":"In addition, using cosot ¼ x/a we may write cos2ot \u0002 sin2d ¼ ax22 sin2d.","rect":[65.7667465209961,210.23562622070313,347.1745876839328,197.2271728515625]},{"page":301,"text":"Now we make a sum of the last two equations and exclude the time dependence.","rect":[65.76607513427735,220.81637573242188,385.0822720101047,211.88182067871095]},{"page":301,"text":"Then we arrive at the equation for the output field (but before an analyser) in the","rect":[53.81405258178711,232.77590942382813,385.17185247613755,223.8413543701172]},{"page":301,"text":"form of the equation for an ellipse:","rect":[53.81405258178711,244.7354736328125,195.03642137119364,235.80091857910157]},{"page":301,"text":"x2 þ y2 \u0003 2xy cosd ¼ sin2d","rect":[163.98770141601563,274.8689270019531,276.9305047135688,258.9662780761719]},{"page":301,"text":"a2 b2","rect":[163.7611846923828,281.05987548828127,193.3018958338197,273.3065185546875]},{"page":301,"text":"ab","rect":[207.88804626464845,281.0595397949219,217.84226313642035,274.1170654296875]},{"page":301,"text":"(11.9)","rect":[361.0555725097656,275.79522705078127,385.10494361726418,267.3188781738281]},{"page":301,"text":"The orientation of the ellipse axes depends on polarizer angle j (through a and","rect":[65.76590728759766,304.5342102050781,385.1457451920844,295.59967041015627]},{"page":301,"text":"b) and phase retardation d. Consider few interesting consequences of (11.9).","rect":[53.81386947631836,316.4937744140625,361.25164456869848,307.2604064941406]},{"page":301,"text":"Case 1 Corresponds to the So-Called l/4 Plate","rect":[53.8138542175293,353.7897644042969,241.47063482477035,344.5364685058594]},{"page":301,"text":"The layer thickness satisfies the condition ðne \u0003 noÞd ¼ l=4; i.e., d ¼ p/2, and the","rect":[53.81386947631836,378.0476989746094,385.1762164898094,368.09698486328127]},{"page":301,"text":"ellipse becomes oriented along the principal axes of the nematic layer and the ratio","rect":[53.81549835205078,389.6685791015625,385.1115349870063,380.7340087890625]},{"page":301,"text":"of its semi-axes depends only on polarizer angle j:","rect":[53.81549835205078,401.6281433105469,261.1576982266624,392.693603515625]},{"page":301,"text":"2","rect":[200.9203643798828,420.572021484375,204.40432808479629,415.8587341308594]},{"page":301,"text":"2","rect":[221.99241638183595,420.572021484375,225.4763800867494,415.8587341308594]},{"page":301,"text":"x þy ¼1","rect":[196.44544982910157,431.7613525390625,244.4722832780219,419.68756103515627]},{"page":301,"text":"a2 b2","rect":[196.2188262939453,437.95245361328127,225.75952217659316,430.198974609375]},{"page":301,"text":"(11.10)","rect":[356.0709228515625,432.6876525878906,385.15911231843605,424.2113037109375]},{"page":301,"text":"Particularly, for j ¼ \u0004p/4 i.e. a ¼ \u0004b the equation for an ellipse degenerates","rect":[65.76534271240235,461.4266662597656,385.1063577090378,452.49212646484377]},{"page":301,"text":"into the equation for a circumference:","rect":[53.81333541870117,473.3861999511719,206.2670274502952,464.45166015625]},{"page":301,"text":"2","rect":[198.37130737304688,492.33001708984377,201.85527107796035,487.6167297363281]},{"page":301,"text":"2","rect":[218.93357849121095,492.33001708984377,222.4175421961244,487.6167297363281]},{"page":301,"text":"2","rect":[241.1384735107422,492.33001708984377,244.62243721565566,487.6167297363281]},{"page":301,"text":"x þy ¼a","rect":[193.89593505859376,498.5892333984375,241.13118830731879,490.66070556640627]},{"page":301,"text":"(11.11)","rect":[356.0715026855469,498.2107238769531,385.1596921524204,489.734375]},{"page":301,"text":"The light beam transmitted through the layer becomes left or right circularly","rect":[65.76595306396485,522.5281982421875,385.14492121747505,513.5936889648438]},{"page":301,"text":"polarised. Thus a l/4 plate converts the linear polarization to one of the possible","rect":[53.81393051147461,534.4309692382813,385.1746295757469,525.177734375]},{"page":301,"text":"circular polarizations, left or right dependent on a sign of parameter b as illustrated","rect":[53.814903259277347,546.3905029296875,385.1447686295844,537.4559936523438]},{"page":301,"text":"by Fig. 11.5. Note that, in the figure, the right polarization corresponds to the","rect":[53.81487274169922,558.35009765625,385.17371404840318,549.3558349609375]},{"page":301,"text":"electric vector of light e rotating clockwise for an observer looking at the incoming","rect":[53.814903259277347,570.3096313476563,385.1497735123969,561.3751220703125]},{"page":301,"text":"beam (according to the convention used in many classical books, for instance","rect":[53.814903259277347,582.2691650390625,385.1577533550438,573.3346557617188]},{"page":301,"text":"in [2,3]). However, more recently, another convention is often used according to","rect":[53.814903259277347,594.2286987304688,385.14382258466255,585.294189453125]},{"page":302,"text":"290","rect":[53.812843322753909,42.62403869628906,66.50444931178018,36.73143005371094]},{"page":302,"text":"11 Optics and Electric Field Effects in Nematic and Smectic A Liquid Crystals","rect":[114.25279998779297,44.274986267089847,385.1677597331933,36.6636962890625]},{"page":302,"text":"Fig. 11.5 Right and left","rect":[53.812843322753909,67.58130645751953,138.12742332663599,59.85148620605469]},{"page":302,"text":"circular polarization of light","rect":[53.812843322753909,77.4895248413086,150.18273643698755,69.89517211914063]},{"page":302,"text":"according to the classical","rect":[53.812843322753909,87.4087142944336,140.19698814597192,79.81436157226563]},{"page":302,"text":"convention (according to the","rect":[53.812843322753909,97.3846664428711,151.946877930398,89.79031372070313]},{"page":302,"text":"modern connection the","rect":[53.812843322753909,105.63353729248047,132.12088916086675,99.76632690429688]},{"page":302,"text":"handedness is reversed)","rect":[53.812843322753909,116.99797821044922,134.78189176184825,109.74227905273438]},{"page":302,"text":"which the right polarization of light follows the right screw law [4]. Personally,I","rect":[53.812843322753909,217.47225952148438,385.15862403718605,208.53770446777345]},{"page":302,"text":"like more the second one, however, throughout the book we follow the traditional","rect":[53.8138313293457,229.43182373046876,385.16865403720927,220.4972686767578]},{"page":302,"text":"convention.","rect":[53.8138313293457,239.32952880859376,100.73997159506564,232.45680236816407]},{"page":302,"text":"Case 2 Corresponds to the l Plate","rect":[53.8138313293457,279.76409912109377,190.88730657770004,270.51080322265627]},{"page":302,"text":"The layer thickness satisfies the condition ðne \u0003 noÞd ¼ l; i.e., d ¼ 2p. Then,","rect":[53.8138427734375,304.0224609375,385.13714261557348,294.0713195800781]},{"page":302,"text":"according to (11.9), the ellipse degenerates into the straight line x=a \u0003 y=b ¼ 0 or","rect":[53.81519317626953,316.6553649902344,385.15203224031105,305.98565673828127]},{"page":302,"text":"y ¼ xb/a ¼ x\u0002tanj. This means that after the cell the light is linearly polarised. The","rect":[53.814231872558597,327.6029052734375,385.1471332378563,318.6683349609375]},{"page":302,"text":"same equationis valid for d ¼ 2kp where k is any integer. The angle j is determined","rect":[53.814247131347659,339.56243896484377,385.16799250653755,330.3290710449219]},{"page":302,"text":"by the angular position of polarizer; for j ¼ p/4 the cell transmit light without","rect":[53.814231872558597,351.5220031738281,385.12519700595927,342.58746337890627]},{"page":302,"text":"change of polarization, y ¼ x.","rect":[53.81519317626953,363.48150634765627,173.92966886069065,354.54693603515627]},{"page":302,"text":"11.1.1.4 Light Transmission (Cell Between Polarizers)","rect":[53.81519317626953,401.9339294433594,291.42522559968605,392.7503662109375]},{"page":302,"text":"The transmitted light intensity is calculated as follows. The analyser can only","rect":[53.81519317626953,425.7733459472656,385.1252068620063,416.83880615234377]},{"page":302,"text":"transmit the field components parallel to its axis, that is projections of E|| and E⊥","rect":[53.81519317626953,437.7328796386719,385.20126792144796,428.79833984375]},{"page":302,"text":"on the analyser direction A, see again Fig. 11.4:","rect":[53.812843322753909,449.6935729980469,247.50385911533426,440.759033203125]},{"page":302,"text":"EjAj ¼ Ecosjcosðj \u0003 wÞ; EA? ¼ Esinjsinðj \u0003 wÞ;","rect":[96.80706787109375,477.2249755859375,311.40565430802249,463.8952941894531]},{"page":302,"text":"(11.12)","rect":[356.07318115234377,474.4614562988281,385.1613706192173,465.985107421875]},{"page":302,"text":"The total light intensity I after an analyser is a result of interference of the two","rect":[65.76761627197266,500.7063293457031,385.15444270185005,491.77178955078127]},{"page":302,"text":"rays","rect":[53.81658172607422,512.6658935546875,70.39033903228,505.9624938964844]},{"page":302,"text":"I ¼ IjAj þ I?A þ 2ðIjAjI?AÞ1=2 cosd:","rect":[157.36325073242188,541.444580078125,281.60853516739749,526.9542236328125]},{"page":302,"text":"(11.13)","rect":[356.07098388671877,538.6808471679688,385.1591733535923,530.2044677734375]},{"page":302,"text":"Then","rect":[65.76541900634766,562.920654296875,86.28104487470159,556.0479125976563]},{"page":302,"text":"I ¼ E2f½cosjcosðj \u0003 wÞ\u00052 þ ½sinjsinðj \u0003 wÞ\u00052 þ 21sin2jsin2ðj \u0003 wÞcosdg","rect":[61.68717575073242,592.0878295898438,377.30461970380318,579.2129516601563]},{"page":303,"text":"11.1 Optical Properties of Uniaxial Phases","rect":[53.812843322753909,44.274620056152347,199.2027939128808,36.663330078125]},{"page":303,"text":"291","rect":[372.49737548828127,42.62367248535156,385.1889700332157,36.73106384277344]},{"page":303,"text":"and using cos2ð2j \u0003 wÞ ¼ 12½1 þ cos2ð2j \u0003 wÞ\u0005 and cosd ¼ (1 - 2sin22dÞ we find","rect":[53.812843322753909,69.5474853515625,385.1713799577094,57.55512237548828]},{"page":303,"text":"the transmitted intensity:","rect":[53.81362533569336,80.24788665771485,154.03164537021707,71.31333923339844]},{"page":303,"text":"I ¼ E2½cos2w \u0003 sin2jsin2ðj \u0003 wÞsin2d=2\u0005","rect":[134.3630828857422,107.58062744140625,304.66236817520999,94.4805908203125]},{"page":303,"text":"(11.14)","rect":[356.0708923339844,106.3195571899414,385.1590818008579,97.84319305419922]},{"page":303,"text":"Consider again two important particular cases:","rect":[65.76532745361328,131.14712524414063,253.713759017678,122.21257019042969]},{"page":303,"text":"Case a, parallel polarizers, A || P. In this case w ¼ 0 and","rect":[65.76532745361328,143.10659790039063,292.1917656998969,134.1720428466797]},{"page":303,"text":"Ijj ¼ E2ð1 \u0003 sin22j \u0002 sin2d=2Þ","rect":[160.81610107421876,170.9784393310547,278.16525663481908,157.33956909179688]},{"page":303,"text":"(11.15)","rect":[356.0718688964844,169.17860412597657,385.1600583633579,160.58270263671876]},{"page":303,"text":"There are maxima of transmission for j ¼ 0, p/2, p...etc, and I|| (max) ¼ E2,","rect":[65.76628875732422,194.3795928955078,385.1832241585422,183.4088592529297]},{"page":303,"text":"when the incoming polarization coincides with the principal axes x or y. Between","rect":[53.814537048339847,206.41943359375,385.08071223310005,197.48487854003907]},{"page":303,"text":"them there are minima of transmission corresponding to j ¼ p/4, 3p/4, 5p/4....","rect":[53.81450653076172,218.37896728515626,385.1842007210422,209.3846435546875]},{"page":303,"text":"Their intensities I|| (min) ¼ E2 cos2(d/2) do not show full darkness (except a special","rect":[53.81554412841797,230.33883666992188,385.126112533303,219.2879180908203]},{"page":303,"text":"value for phase retardation when cos2(d/2) ¼ 0).","rect":[53.81415939331055,242.29840087890626,249.36922879721409,231.2476043701172]},{"page":303,"text":"Case b, crossed polarizers, A ⊥ P. Now w ¼ p/2 and","rect":[65.76482391357422,254.258056640625,278.48232356122505,245.32350158691407]},{"page":303,"text":"I? ¼ E2sin22j \u0002 sin2d=2","rect":[172.42715454101563,281.5906677246094,266.5644463639594,268.49066162109377]},{"page":303,"text":"(11.16)","rect":[356.0707092285156,280.3296813964844,385.15889869538918,271.7935791015625]},{"page":303,"text":"Now minima for j ¼ 0, p/2, p... correspond to complete darkness and the","rect":[53.81313705444336,305.1571960449219,385.1181110210594,296.22265625]},{"page":303,"text":"maximum intensity is observed at j ¼ p/4, 3p/4, 5p/4... This case is the most","rect":[53.81412887573242,317.1167297363281,385.1390214688499,308.1224060058594]},{"page":303,"text":"interesting because provides a high contrast for a cell under a microscope. Let us","rect":[53.81513214111328,329.0762939453125,385.15597929106908,320.1417236328125]},{"page":303,"text":"select the angle j ¼ p/4 between the director and polarizer. Then we have","rect":[53.81513214111328,340.97906494140627,385.1221393413719,332.04449462890627]},{"page":303,"text":"maximum light intensity after analyser","rect":[53.816139221191409,352.9385986328125,210.09325538484229,344.0040283203125]},{"page":303,"text":"I?ðmaxÞ ¼ E2sin2 d ¼ E2sin2 pðne \u0003 noÞd","rect":[134.42234802246095,383.94244384765627,302.64785090497505,367.2469482421875]},{"page":303,"text":"2","rect":[209.13394165039063,388.3203430175781,214.11104670575629,381.5870666503906]},{"page":303,"text":"(11.17)","rect":[356.07208251953127,383.2046203613281,385.1602719864048,374.728271484375]},{"page":303,"text":"The light intensity has an oscillatory character as a function of cell thickness d,","rect":[65.76653289794922,411.8868103027344,385.1842007210422,402.9323425292969]},{"page":303,"text":"optical anisotropy Dn ¼ ne-no and wavelength l. This can be used for measure-","rect":[53.81549835205078,423.8471984863281,385.09795509187355,414.5831298828125]},{"page":303,"text":"ments of Dn.","rect":[53.81386184692383,433.8445129394531,105.69720120932345,426.5434875488281]},{"page":303,"text":"11.1.1.5 Measurements of Birefringence of Nematics","rect":[53.81386184692383,485.6519775390625,284.3335000430222,476.50823974609377]},{"page":303,"text":"For example, we can use a wedge form cell, in which thickness d(x) changes along","rect":[53.81386184692383,509.4913635253906,385.1596612077094,500.5368957519531]},{"page":303,"text":"the optical axis of a nematic, see Fig. 11.6. The nematic director is parallel to x, the","rect":[53.81486511230469,521.4508666992188,385.17661321832505,512.516357421875]},{"page":303,"text":"polarizer P is installed at an angle of 45\u0006 to the x-axis and analyser A is crossed with","rect":[53.81486511230469,533.4110717773438,385.1173028092719,524.4161376953125]},{"page":303,"text":"polarizer. If such a cell is illuminated by a filtered light of wavelength l, then, a","rect":[53.814308166503909,545.3706665039063,385.16111028863755,536.117431640625]},{"page":303,"text":"series of contrast interference stripes is seen which are parallel to the y-direction.","rect":[53.814292907714847,557.3302001953125,385.15508695151098,548.3956909179688]},{"page":303,"text":"The dark stripes correspond to dDn/l ¼ 0, 1, 2 .. k and the distance between them is","rect":[53.815269470214847,569.2329711914063,385.18893827544408,559.9697265625]},{"page":303,"text":"constant, l ¼ lsina/Dn where a is the angle of the wedge. The latter can be found","rect":[53.815269470214847,581.1925048828125,385.12822810224068,571.9292602539063]},{"page":303,"text":"from the “stripes of equal thickness” in a part of the wedge not filled with the","rect":[53.81529998779297,593.1520385742188,385.17404974176255,584.217529296875]},{"page":304,"text":"292","rect":[53.813289642333987,42.62403869628906,66.50489563136026,36.73143005371094]},{"page":304,"text":"11 Optics and Electric Field Effects in Nematic and Smectic A Liquid Crystals","rect":[114.25325012207031,44.274986267089847,385.16821749686519,36.6636962890625]},{"page":304,"text":"z","rect":[153.05450439453126,75.2484130859375,156.55555374779738,71.6811752319336]},{"page":304,"text":"y","rect":[169.6109619140625,76.7123031616211,173.60759359587315,71.35344696044922]},{"page":304,"text":"x","rect":[172.30548095703126,89.74551391601563,176.7177623337502,86.00231170654297]},{"page":304,"text":"M","rect":[198.17086791992188,66.2615966796875,206.08419864990695,60.76676940917969]},{"page":304,"text":"A","rect":[197.87173461914063,80.25140380859375,204.03454067249263,74.5166244506836]},{"page":304,"text":"k k+1","rect":[238.67605590820313,71.32284545898438,267.1672249435294,63.488548278808597]},{"page":304,"text":"α","rect":[276.7134094238281,90.8013687133789,281.75715860627317,87.00218200683594]},{"page":304,"text":"light","rect":[197.57217407226563,140.12985229492188,212.9751874862187,132.6674346923828]},{"page":304,"text":"P","rect":[267.95477294921877,122.51953125,272.782704020846,117.02470397949219]},{"page":304,"text":"Fig. 11.6 Scheme of simple measurements of optical anisotropy Dn ¼ n|| -n⊥ by observation of","rect":[53.812843322753909,160.08453369140626,385.15499967200449,152.14306640625]},{"page":304,"text":"interference lines in a wedge-form nematic cell placed between cross polarisers (a: wedge angle;","rect":[53.813594818115237,169.93606567382813,385.17953209128447,162.34170532226563]},{"page":304,"text":"P: polarizer; A: analyzer; M; microscope; k: order of interference). The same scheme can easily be","rect":[53.813533782958987,179.9119873046875,385.1566710212183,172.30068969726563]},{"page":304,"text":"modified to measure refraction indices n|| and n⊥ separately as explained in the text","rect":[53.813533782958987,189.8876953125,339.3836946889407,182.2933349609375]},{"page":304,"text":"Fig. 11.7 Temperature","rect":[53.812843322753909,229.6319580078125,134.3207792975855,221.9021453857422]},{"page":304,"text":"dependencies of the","rect":[53.812843322753909,239.48345947265626,121.75438830637455,231.88909912109376]},{"page":304,"text":"transmitted light intensity","rect":[53.812843322753909,249.45941162109376,141.62605804591105,241.86505126953126]},{"page":304,"text":"and phase retardation d","rect":[53.812843322753909,259.4349365234375,133.5508016982548,251.58657836914063]},{"page":304,"text":"(above) and principal","rect":[53.812843322753909,269.35418701171877,126.89870933737818,261.75982666015627]},{"page":304,"text":"refraction indices (below)","rect":[53.81199645996094,278.991455078125,141.59643643958263,271.71881103515627]},{"page":304,"text":"Itr","rect":[144.0484619140625,249.15750122070313,151.10710813643426,242.0513153076172]},{"page":304,"text":"Itr","rect":[274.576904296875,243.44766235351563,282.3440076766484,236.03054809570313]},{"page":304,"text":"Δn","rect":[284.3931884765625,297.3861999511719,294.7278848012672,291.9080810546875]},{"page":304,"text":" ","rect":[289.28448486328127,298.0,290.2836427837339,297.0]},{"page":304,"text":"δ(T)","rect":[315.6044921875,230.32443237304688,329.7605421004765,223.05398559570313]},{"page":304,"text":"n⊥","rect":[313.6755065917969,308.0716857910156,322.0646095519784,302.37689208984377]},{"page":304,"text":"...32","rect":[336.9635009765625,235.35787963867188,354.94768148649816,227.5227813720703]},{"page":304,"text":"1=K","rect":[356.380859375,239.19149780273438,372.26397126314478,233.6886749267578]},{"page":304,"text":" ","rect":[360.3776550292969,240.0,361.37681294974956,239.0]},{"page":304,"text":"n||","rect":[342.4515075683594,284.07879638671877,349.29380761977759,277.0943908691406]},{"page":304,"text":"TNI","rect":[356.0872497558594,330.88037109375,367.6322665199303,323.4279479980469]},{"page":304,"text":"T","rect":[376.29278564453127,268.958740234375,381.17666955970386,263.6078796386719]},{"page":304,"text":"nematic. With known wedge angle one can also use the “stripes of equal thickness”","rect":[53.812843322753909,367.7898254394531,385.08994329645005,358.85528564453127]},{"page":304,"text":"for measurements of n|| and n⊥ separately: to this effect the electric vector of a","rect":[53.812843322753909,379.6927795410156,385.15830267145005,370.758056640625]},{"page":304,"text":"linearly polarised light should be installed parallel or perpendicular to the director","rect":[53.813472747802737,391.65234375,385.1056150039829,382.7177734375]},{"page":304,"text":"and analyser is not used.","rect":[53.813472747802737,403.61187744140627,152.87678189780002,394.67730712890627]},{"page":304,"text":"Thetemperaturedependenceof Dn ¼ n||-n⊥ where n||andn⊥ areprincipal indices","rect":[65.76549530029297,415.5717468261719,385.19711698638158,406.30816650390627]},{"page":304,"text":"of a nematic (lower plot in Fig. 11.7) can simply be measured using geometry of","rect":[53.81351852416992,427.5312805175781,385.20806251374855,418.59674072265627]},{"page":304,"text":"Fig. 11.4. A cell with fixed thickness d about 30 mm and well-aligned nematic (or a","rect":[53.81351852416992,439.4908447265625,385.1612933941063,430.536376953125]},{"page":304,"text":"smectic with intermediate nematic phase) is heated up to the isotropic phase. Due to a","rect":[53.81351852416992,451.45037841796877,385.15833318902818,442.51580810546877]},{"page":304,"text":"decrease in Dn with increasing temperature, the transmitted intensity oscillates and","rect":[53.81351852416992,463.409912109375,385.2588738541938,454.14666748046877]},{"page":304,"text":"these oscillations can be numbered as 1,2,3..k counted from the isotropic phase. This","rect":[53.81351852416992,475.31268310546877,385.1662637148972,466.35821533203127]},{"page":304,"text":"way the phase retardation d(T) ¼ 2pdDn(T)/l ¼ 2p, 4p, 6p...2kp can be plotted as a","rect":[53.81350326538086,487.2722473144531,385.1642535991844,478.0090026855469]},{"page":304,"text":"function of temperature and Dn(T) found from Eq. (11.17), see upper plot in","rect":[53.81749725341797,499.2317810058594,385.20513239911568,489.9685363769531]},{"page":304,"text":"Fig. 11.7. The absolute valuesof nisoand n⊥ caneasilybe found with a refractometer","rect":[53.818504333496097,511.1922912597656,385.1595395645298,502.25677490234377]},{"page":304,"text":"although one can meet some problems with refractometry of n||. We can also change","rect":[53.81374740600586,523.15185546875,385.2584613628563,514.21728515625]},{"page":304,"text":"thedirection of thedirector by external factors (electricandmagnetic fields,acoustic","rect":[53.814144134521487,535.1114501953125,385.2375873394188,526.1769409179688]},{"page":304,"text":"vibrations, flow of a liquid crystal) and follow these changes by birefringence","rect":[53.814144134521487,547.0709838867188,385.1918109722313,538.136474609375]},{"page":304,"text":"measurements with high accuracy. For a comprehensive review of experimental","rect":[53.814144134521487,559.030517578125,385.1828447110374,550.0960083007813]},{"page":304,"text":"data on optical properties of liquid crystal see [5].","rect":[53.814144134521487,570.9332885742188,251.7008480599094,561.9390258789063]},{"page":305,"text":"11.1 Optical Properties of Uniaxial Phases","rect":[53.812843322753909,44.274620056152347,199.2027939128808,36.663330078125]},{"page":305,"text":"293","rect":[372.49737548828127,42.62367248535156,385.1889700332157,36.73106384277344]},{"page":305,"text":"11.1.1.6 Twist Structure","rect":[53.812843322753909,66.76439666748047,163.7172778911766,59.26416015625]},{"page":305,"text":"Consider one more particular case related to elliptic polarization of light that passes","rect":[53.812843322753909,92.20748138427735,385.12179960356908,83.27293395996094]},{"page":305,"text":"a birefringent layer. Let the layer be very thin, d!0 and the principal axis of the","rect":[53.812843322753909,104.11019134521485,385.17356146051255,95.15572357177735]},{"page":305,"text":"layer forms an angle j with the input linear polarization. Then we go back to the","rect":[53.81282424926758,116.0697250366211,385.1725543804344,107.13517761230469]},{"page":305,"text":"general form of the ellipse (11.9) and using smallness of phase shift d (cosd \u0007 1 \u0003d","rect":[53.813812255859378,128.02932739257813,385.17852107099068,118.79595947265625]},{"page":305,"text":"\u0007 1 and sin2d \u0007 d2) get","rect":[53.81379318237305,139.98959350585938,149.98498399326395,128.93870544433595]},{"page":305,"text":"ax22 þ by22 \u0003 2axbyð1 \u0003 dÞ ¼ d2:","rect":[164.1009979248047,181.30169677734376,276.5103296009912,159.20803833007813]},{"page":305,"text":"(11.18)","rect":[356.0713195800781,176.0370330810547,385.15950904695168,167.5606689453125]},{"page":305,"text":"Neglecting d << 1 in parentheses we obtain","rect":[65.76578521728516,209.76321411132813,245.6831444840766,200.52984619140626]},{"page":305,"text":"x \u0003 y ¼ d:","rect":[199.10769653320313,240.33934020996095,241.7869409291162,229.01329040527345]},{"page":305,"text":"ab","rect":[198.82400512695313,247.27783203125,220.90796748212348,240.3353729248047]},{"page":305,"text":"(11.19)","rect":[356.07122802734377,242.01271057128907,385.1594174942173,233.53634643554688]},{"page":305,"text":"This equation describes a straight line","rect":[65.7656478881836,275.7388916015625,218.65833318902816,266.8043212890625]},{"page":305,"text":"y \u0007 bx \u0003 bd ¼ Esinf x \u0003 bd ¼ xtanf \u0003 bd","rect":[127.62211608886719,310.9740905761719,311.31536189130318,294.9874572753906]},{"page":305,"text":"a","rect":[145.35150146484376,315.8018493652344,150.3286065202094,311.26983642578127]},{"page":305,"text":"E cos f","rect":[191.744384765625,317.7640686035156,221.14512893374707,308.5904846191406]},{"page":305,"text":"showing that the outgoing beam is linearly polarised and its electric vector forms a","rect":[53.81488800048828,346.1932678222656,385.1597369976219,337.25872802734377]},{"page":305,"text":"small angle bd with respect to the vector of the linearly polarised incident beam.","rect":[53.81488800048828,358.15283203125,385.1387905647922,348.9194641113281]},{"page":305,"text":"So, there is no ellipticity!","rect":[53.814857482910159,370.11236572265627,156.4168943008579,361.17779541015627]},{"page":305,"text":"Now imagine a stack of very thin plates or layers, in which the director turns by a","rect":[65.76688385009766,382.0718994140625,385.16172064020005,373.1373291015625]},{"page":305,"text":"small angle upon proceeding from one layer to the next one. Then, after each","rect":[53.814857482910159,394.03143310546877,385.10097590497505,385.09686279296877]},{"page":305,"text":"passage of a successive plate, the electric vector of the beam rotates through a small","rect":[53.814857482910159,405.9342346191406,385.13874681064677,396.99969482421877]},{"page":305,"text":"angle and such a stack of plates “guides” the light polarisation. We can prepare such","rect":[53.814857482910159,417.89373779296877,385.1248406510688,408.95916748046877]},{"page":305,"text":"a “stack” using different boundary conditions for alignment of the nematic. For","rect":[53.814857482910159,429.8533020019531,385.1487973770298,420.91876220703127]},{"page":305,"text":"instance, if the directors at the top and bottom glasses are strictly perpendicular to","rect":[53.814857482910159,441.8128356933594,385.1437615495063,432.8782958984375]},{"page":305,"text":"each other, the nematic is twisted through angle p/2, as discussed in Section 8.3.2. It","rect":[53.814857482910159,453.7723693847656,385.18159349033427,444.83782958984377]},{"page":305,"text":"is of great importance that the light polarization follows the p/2-twisted structure at","rect":[53.81488800048828,465.7319030761719,385.1826005704124,456.79736328125]},{"page":305,"text":"any wavelength. This is so-called “waveguide” or Mauguin regime [6]. When light","rect":[53.81488800048828,477.69146728515627,385.2094560391624,468.75689697265627]},{"page":305,"text":"leaves the entire nematic cell its electric vector is turned through p/2 with respect to","rect":[53.81586456298828,489.6510314941406,385.20244685224068,480.71649169921877]},{"page":305,"text":"the electric vector of the incident beam. It is evident that, a planar cell is non-","rect":[53.81586456298828,501.5538024902344,385.1347898086704,492.6192626953125]},{"page":305,"text":"transparent when installed between crossed polarizers with polarizer P parallel (or","rect":[53.81586456298828,513.5133056640625,385.18062721101418,504.57879638671877]},{"page":305,"text":"perpendicular) to the director, because an analyser absorbs light almost completely,","rect":[53.81586456298828,525.4728393554688,385.12286038901098,516.538330078125]},{"page":305,"text":"Fig. 11.8a. A homeotropic cell is also non-transparent when observed through","rect":[53.81586456298828,537.432373046875,385.1547783952094,528.4978637695313]},{"page":305,"text":"crossed polarizers. On the contrary, a twist cell completely transmits light under","rect":[53.814857482910159,549.3919677734375,385.10994850007668,540.4375]},{"page":305,"text":"the same conditions, Fig. 11.8b.","rect":[53.814857482910159,561.3515014648438,182.6790737679172,552.4169921875]},{"page":306,"text":"294","rect":[53.81196975708008,42.62409973144531,66.50357574610635,36.73149108886719]},{"page":306,"text":"11 Optics and Electric Field Effects in Nematic and Smectic A Liquid Crystals","rect":[114.25192260742188,44.275047302246097,385.1669052410058,36.66375732421875]},{"page":306,"text":"a","rect":[145.12522888183595,68.23306274414063,150.68049127906978,62.64430618286133]},{"page":306,"text":"P","rect":[145.68072509765626,80.89288330078125,150.12497952782969,75.59801483154297]},{"page":306,"text":"b","rect":[220.06031799316407,68.23306274414063,226.16511174264228,60.92469024658203]},{"page":306,"text":"A","rect":[148.6566162109375,120.25885009765625,154.42775235947208,114.86799621582031]},{"page":306,"text":"z","rect":[272.0374450683594,101.34444427490235,275.14682451680809,97.13734436035156]},{"page":306,"text":"x","rect":[289.17657470703127,122.69515991210938,293.1732063888419,118.90396881103516]},{"page":306,"text":"Fig. 11.8 A planar cell installed between crossed polarizers is non-transparent (a) whereas a twist","rect":[53.812843322753909,150.618896484375,385.1677828237063,142.66896057128907]},{"page":306,"text":"cell rotates the linear polarization through p/2 and transmits light (b). Polarizer P is parallel to the","rect":[53.81196975708008,160.52713012695313,385.15429065012457,152.93276977539063]},{"page":306,"text":"director at the input plate","rect":[53.81196975708008,170.50308227539063,140.31879565500737,162.90872192382813]},{"page":306,"text":"11.1.2 Light Absorption and Linear Dichroism","rect":[53.812843322753909,202.03268432617188,293.5974727794945,191.47857666015626]},{"page":306,"text":"11.1.2.1 Extinction Index, Absorption Coefficient, Optical Density","rect":[53.812843322753909,230.05706787109376,341.63091364911568,220.50497436523438]},{"page":306,"text":"An electromagnetic wave propagating with velocity v in a medium is described by","rect":[53.812843322753909,253.52096557617188,385.16960993817818,244.58641052246095]},{"page":306,"text":"the wave equation:","rect":[53.812862396240237,265.48052978515627,130.01333482334207,256.54595947265627]},{"page":306,"text":"1 q2E","rect":[209.81365966796876,287.8953552246094,235.41926833804394,278.74786376953127]},{"page":306,"text":"DE \u0003 v2 qt2 ¼ 0:","rect":[182.6233367919922,301.6079406738281,256.3443597523584,287.4483642578125]},{"page":306,"text":"Assuming E(r, t) ¼ E(r)exp(\u0003iot) we exclude the time dependence and get the","rect":[65.76602935791016,324.0884704589844,385.17371404840318,315.1539306640625]},{"page":306,"text":"Helmholtz equation:","rect":[53.81301498413086,336.0480041503906,136.18708665439676,327.11346435546877]},{"page":306,"text":"2","rect":[226.07086181640626,354.0287170410156,229.5548255213197,349.28057861328127]},{"page":306,"text":"o","rect":[219.38670349121095,357.73602294921877,226.016207424958,352.9350891113281]},{"page":306,"text":"DE þ em c2 E ¼0","rect":[182.45327758789063,371.4091796875,256.53838435224068,357.3355407714844]},{"page":306,"text":"where e and m are dielectric and magnetic permeability and c light velocity in","rect":[53.813961029052737,393.9191589355469,385.14382258466255,384.984619140625]},{"page":306,"text":"vacuum.","rect":[53.814918518066409,403.8168640136719,87.90809293295627,399.17529296875]},{"page":306,"text":"For the plane wave E(r) / expikr and we obtain the dispersion relation:","rect":[65.76696014404297,417.8382263183594,385.15778977939677,408.7343444824219]},{"page":306,"text":"k2 ¼ emo2=c2: For the absorbing medium the wavevector amplitude k becomes","rect":[53.814918518066409,430.9209289550781,385.08920683013158,420.1098327636719]},{"page":306,"text":"complex:","rect":[53.81308364868164,442.5517883300781,91.09857041904519,433.61724853515627]},{"page":306,"text":"k\b ¼ ðemÞ1=2 o \u0007 ðn þ ikÞo","rect":[161.4947509765625,469.7568359375,275.80732986148146,455.75408935546877]},{"page":306,"text":"c","rect":[215.6479034423828,474.2277526855469,220.09743536187973,469.5562744140625]},{"page":306,"text":"(11.20)","rect":[356.07110595703127,469.0194091796875,385.1592954239048,460.5430603027344]},{"page":306,"text":"The absorbing medium can be described in terms of a complex refraction index","rect":[65.76555633544922,496.7374572753906,385.15837946942818,487.80291748046877]},{"page":306,"text":"n* ¼ n + ik where n is real refraction index and k is real extinction index. For non-","rect":[53.813533782958987,506.7646484375,385.1384214004673,499.7026672363281]},{"page":306,"text":"magnetic medium m \u0007 1 and ðn þ ikÞ2 ¼ n2 \u0003 k2 þ 2ink ¼ e\b ¼ e0 þ e00 we find","rect":[53.81549835205078,520.9951782226563,385.17189875653755,509.0420837402344]},{"page":306,"text":"the relations between the components of e* and n*:","rect":[53.814144134521487,532.6175537109375,261.659071517678,523.623291015625]},{"page":306,"text":"n2 \u0003 k2 ¼ e0 and 2nk ¼ e00","rect":[165.0086669921875,555.8783569335938,273.4502796096376,546.84814453125]},{"page":306,"text":"(11.21)","rect":[356.0715026855469,557.4421997070313,385.1596921524204,548.9658203125]},{"page":306,"text":"From (11.21) it is seen how any kind of energy dissipation contributes to real part of","rect":[53.81393051147461,581.702880859375,385.1488278946079,572.7683715820313]},{"page":306,"text":"the dielectric permittivity e0 at optical frequencies.","rect":[53.81393051147461,593.662841796875,257.48069425131566,584.0032958984375]},{"page":307,"text":"11.1 Optical Properties of Uniaxial Phases","rect":[53.81282043457031,44.276329040527347,199.20277865409174,36.6650390625]},{"page":307,"text":"295","rect":[372.49737548828127,42.62538146972656,385.1889700332157,36.63117599487305]},{"page":307,"text":"The dimensionless extinction index k can be related to the absorption coefficient","rect":[65.76496887207031,68.2883529663086,385.2065263516624,59.35380554199219]},{"page":307,"text":"aabs (cm\u00031) by the well-known Buger law for the light intensity I transmitted","rect":[53.81296920776367,80.28784942626953,385.15126887372505,69.19712829589844]},{"page":307,"text":"through an absorbing layer of thickness z (the reflection is ignored):","rect":[53.81241226196289,92.20760345458985,327.7104936368186,83.27305603027344]},{"page":307,"text":"I ¼ I0 expð\u0003aabszÞ","rect":[181.9430389404297,116.40904998779297,257.09323515044408,106.45853424072266]},{"page":307,"text":"(11.22)","rect":[356.0709228515625,115.6719741821289,385.15911231843605,107.19561004638672]},{"page":307,"text":"Indeed, comparing (11.22) with a general form for imaginary part of Imk* ¼","rect":[65.76537322998047,139.98953247070313,385.14820125791939,131.01513671875]},{"page":307,"text":"iko/c, we find","rect":[53.81433868408203,150.86337280273438,111.24515620282659,143.01451110839845]},{"page":307,"text":"I / ðEexpikzÞ2 ¼ E2 expð\u00032kcozÞ ¼ E2 expð\u00034lpkzÞ","rect":[109.21349334716797,186.62936401367188,329.7693215762253,166.14364624023438]},{"page":307,"text":"and then obtain","rect":[53.814720153808597,208.12802124023438,116.06138697675238,201.2254180908203]},{"page":307,"text":"2ko 4pk","rect":[209.92698669433595,231.18771362304688,256.2633091862018,224.35482788085938]},{"page":307,"text":"aabs ¼ c ¼l","rect":[181.0394744873047,244.84054565429688,250.7606133561469,233.12088012695313]},{"page":307,"text":"(11.23)","rect":[356.0711669921875,239.57542419433595,385.15935645906105,231.09906005859376]},{"page":307,"text":"In experiment, a spectrometer usually measures the so-called absorbance or","rect":[65.76558685302735,268.3711853027344,385.1583596621628,259.4366455078125]},{"page":307,"text":"optical density D of a sample with thickness d:","rect":[53.81357955932617,280.2739562988281,243.3686204678733,271.3194885253906]},{"page":307,"text":"D ¼ log10I0=I ¼ log101=T ¼ aabsdlog10e ¼ 0:434aabsd","rect":[97.37320709228516,307.2104187011719,312.22048274091255,294.468994140625]},{"page":307,"text":"(11.24)","rect":[356.0717468261719,305.94940185546877,385.1599362930454,297.4730529785156]},{"page":307,"text":"Note that here, the losses due to reflection from and scattering in the sample are","rect":[65.76619720458985,330.77691650390627,385.16403997613755,321.84234619140627]},{"page":307,"text":"disregarded.","rect":[53.81415939331055,342.7364501953125,102.91031308676486,333.8018798828125]},{"page":307,"text":"This relationship may be used for calculation of the absorption coefficient aabs","rect":[65.76619720458985,354.6960144042969,385.1607501541408,345.761474609375]},{"page":307,"text":"and extinction coefficient k from measured values of D. A typical absorption","rect":[53.812843322753909,366.6561584472656,385.1476983170844,357.72161865234377]},{"page":307,"text":"spectrum of a liquid crystalline substance in the isotropic phase is shown in","rect":[53.81183624267578,378.61572265625,385.1387871842719,369.68115234375]},{"page":307,"text":"Fig. 11.9a. In the UV part of the spectrum, the absorption originates from molecular","rect":[53.81183624267578,390.57525634765627,385.14971290437355,381.64068603515627]},{"page":307,"text":"electronic transitions (with vibronic structure). Except for dyes, the long-wave edge","rect":[53.81084442138672,402.5347900390625,385.1159137554344,393.6002197265625]},{"page":307,"text":"of organic compounds is situated at about 250–350 nm depending on particular","rect":[53.81084442138672,414.4375915527344,385.1417172989048,405.4432678222656]},{"page":307,"text":"molecular structure. As a rule, liquid crystalline materials are transparent in the","rect":[53.81084442138672,426.3971252441406,385.1686481304344,417.46258544921877]},{"page":307,"text":"Fig. 11.9 Typical absorbance spectrum of a mesogenic compound (in the isotropic phase) with","rect":[53.812843322753909,564.1054077148438,385.1804556289188,556.07080078125]},{"page":307,"text":"absorption bands in the visible and infrared spectra (a), typical molecular moieties responsible for","rect":[53.812843322753909,573.9569091796875,385.1974191055982,566.362548828125]},{"page":307,"text":"the UV and IR absorption (b) and characteristic polarization absorption spectra (dichroism) in the","rect":[53.812828063964847,583.932861328125,385.15514514231207,576.3385009765625]},{"page":307,"text":"nematic phase (c)","rect":[53.81282043457031,593.9088134765625,114.23839658362557,586.314453125]},{"page":308,"text":"296","rect":[53.812843322753909,42.62367248535156,66.50444931178018,36.68026351928711]},{"page":308,"text":"11 Optics and Electric Field Effects in Nematic and Smectic A Liquid Crystals","rect":[114.25279998779297,44.274620056152347,385.1677597331933,36.663330078125]},{"page":308,"text":"visible and near IR range (400–2,500 nm) although thick layers can strongly scatter","rect":[53.812843322753909,68.2883529663086,385.1028989395298,59.294044494628909]},{"page":308,"text":"light. Some IR absorption bands are very characteristic; they are caused by vibra-","rect":[53.812843322753909,80.24788665771485,385.1079343399204,71.31333923339844]},{"page":308,"text":"tions related to particular molecular bonds, e.g., -C ¼ O, -C-F, -C","rect":[53.812843322753909,92.20748138427735,332.8193843048408,83.25301361083985]},{"page":308,"text":"N, -C-H,","rect":[348.5370788574219,90.18550109863281,385.13970609213598,83.33269500732422]},{"page":308,"text":"stretch vibration of benzene ring, etc. Such bands can be used for identification of","rect":[53.8138313293457,104.11019134521485,385.14974342195168,95.17564392089844]},{"page":308,"text":"substances by IR spectroscopy.","rect":[53.8138313293457,116.0697250366211,179.4468960335422,107.13517761230469]},{"page":308,"text":"Each particular electronic or vibrational band originates from a quantum transi-","rect":[65.76585388183594,128.02932739257813,385.1198667129673,119.09477233886719]},{"page":308,"text":"tion between two energy states i and j characterised by a transition dipole moment","rect":[53.8138313293457,139.98886108398438,385.1735673672874,131.05430603027345]},{"page":308,"text":"ptr that is a vector. The molecule only absorbs light when (eptr) ¼6 0, i.e., when there","rect":[53.81382369995117,151.94937133789063,385.1226276226219,142.68612670898438]},{"page":308,"text":"is a non-zero projection of the light electric vector e onto ptr. The absorption cross-","rect":[53.814598083496097,163.908935546875,385.1162351211704,154.97438049316407]},{"page":308,"text":"section of a molecule sabs in (cm2) units is proportional to |ptr|2 and can be related","rect":[53.81421661376953,175.86868286132813,385.11797419599068,164.81764221191407]},{"page":308,"text":"to the absorption coefficient and a number of molecules in a unit volume nv [cm\u00033]","rect":[53.813961029052737,187.82821655273438,385.15975318757668,176.7772979736328]},{"page":308,"text":"as sabs ¼ aabs=nv.","rect":[53.81393051147461,200.05995178222657,126.71136136557345,190.1293487548828]},{"page":308,"text":"11.1.2.2 Linear Dichroism","rect":[53.81270217895508,244.06932067871095,172.19015719313763,236.7981719970703]},{"page":308,"text":"Generally, the transition moment can be oriented at an arbitrary angle to the","rect":[53.81270217895508,269.67474365234377,385.11374700738755,260.74017333984377]},{"page":308,"text":"molecular frame but the symmetry imposes some constraints. Consider a cyanobi-","rect":[53.81270217895508,281.63427734375,385.17550025788918,272.69970703125]},{"page":308,"text":"phenyl compound as an example, see Inset to Fig. 11.9b. This molecule, has two","rect":[53.81270217895508,293.5938415527344,385.15252009442818,284.6593017578125]},{"page":308,"text":"transitions especially interesting for us.","rect":[53.81269454956055,305.5533447265625,211.93235440756565,296.6187744140625]},{"page":308,"text":"1. The electronic transition between the p and p* states of a p-electron delocalized","rect":[53.81269454956055,321.4323425292969,385.11870661786568,314.469970703125]},{"page":308,"text":"over the biphenyl moiety due to a chain of conjugated single and double bonds.","rect":[66.27434539794922,335.4237976074219,385.1306118538547,326.4892578125]},{"page":308,"text":"2. The vibration transition of the triple bond C N in the cyano-group.","rect":[53.81267166137695,347.38336181640627,339.9195828011203,338.44879150390627]},{"page":308,"text":"For both transitions the dipole moment ptr is directed exactly along the longitu-","rect":[65.7646713256836,365.2958679199219,385.13460670320168,356.3597412109375]},{"page":308,"text":"dinal molecular axis. Thus, if such molecules form the nematic phase, the absorp-","rect":[53.81368637084961,377.2554016113281,385.11773048249855,368.32086181640627]},{"page":308,"text":"tion coefficient would depend on the average angle between the light polarization","rect":[53.81368637084961,389.2149658203125,385.0878838639594,380.2803955078125]},{"page":308,"text":"vector e and the longitudinal molecular axes, i.e., between e and the director n.","rect":[53.81368637084961,401.17449951171877,373.45830960776098,392.23992919921877]},{"page":308,"text":"As a result, the absorption acquires properties of a second rank tensor with two","rect":[65.7667007446289,413.134033203125,385.15358820966255,404.199462890625]},{"page":308,"text":"principal components aabs|| and aabs⊥ (|| and ⊥ to n). The qualitative absorption","rect":[53.814674377441409,425.0941467285156,385.1492852311469,415.9902648925781]},{"page":308,"text":"spectra for two polarizations are shown in Fig. 11.9c. For the isotropic phase, due to","rect":[53.81344223022461,437.0536804199219,385.1413506608344,428.119140625]},{"page":308,"text":"complete averaging over oscillator directions the tensor degenerates into a scalar","rect":[53.81248092651367,449.0132141113281,385.07967506257668,440.07867431640627]},{"page":308,"text":"aabs ¼ (aabs||+2aabs⊥)/3.","rect":[53.81248092651367,460.6147766113281,151.05538602377659,451.9818115234375]},{"page":308,"text":"We can introduce a dichroic ratio K ¼ aabs||/aabs⊥ or, in terms of the optical","rect":[65.76571655273438,472.8760070800781,385.1277910000999,463.9214782714844]},{"page":308,"text":"density, K ¼ D||/D⊥. In the isotropic phase K ¼ 1. In the nematic phase, if the","rect":[53.8138542175293,484.835693359375,385.17420232965318,475.9010009765625]},{"page":308,"text":"alignment were ideal (S ¼ 1) the dichroic ratio would be infinite, K!1. Thus we","rect":[53.81349563598633,496.7952575683594,385.1712116069969,487.8607177734375]},{"page":308,"text":"can use a factor similar to ratio S ¼ ea/eamax (see Section 3.5.2)","rect":[53.814476013183597,508.3565368652344,309.8786557754673,499.23150634765627]},{"page":308,"text":"Sabs ¼ aaaabbssjjjjþ\u00032aaaabbss?? ¼ KK þ\u0003 12","rect":[157.0218048095703,546.8289184570313,280.32912531903755,522.9494018554688]},{"page":308,"text":"(11.25)","rect":[356.0706787109375,538.226806640625,385.15886817781105,529.6309204101563]},{"page":308,"text":"as a measure of the orientational order parameter of the nematic or SmA phase.","rect":[53.813106536865237,570.3094482421875,374.9539455940891,561.3749389648438]},{"page":308,"text":"Experiments [7] show that, for a properly selected absorption bands with ptr || n or","rect":[65.76512908935547,582.2689819335938,385.26531349031105,573.33447265625]},{"page":308,"text":"ptr ⊥ n one can obtain Sabs values very close to S measured by other techniques such","rect":[53.813961029052737,594.2296142578125,385.1239861588813,585.2951049804688]},{"page":309,"text":"11.1 Optical Properties of Uniaxial Phases","rect":[53.812843322753909,44.274620056152347,199.2027939128808,36.663330078125]},{"page":309,"text":"297","rect":[372.49737548828127,42.62367248535156,385.1889700332157,36.73106384277344]},{"page":309,"text":"as NMR, magnetic anisotropy or birefringence. Moreover, if the angle b between","rect":[53.812843322753909,68.2883529663086,385.17754450849068,59.04502868652344]},{"page":309,"text":"the longitudinal molecular axis l and selected moment ptr is known the parameter","rect":[53.812843322753909,80.24800872802735,385.1544431289829,71.29341888427735]},{"page":309,"text":"Sabs can still be found using a correction factor dependent on b:","rect":[53.81362533569336,92.20772552490235,310.606093002053,82.96440124511719]},{"page":309,"text":"Sabs \u0002 \u00051 \u0003 32sin2b\u0006 ¼ KK \u0003þ 12:","rect":[157.8129425048828,130.27342224121095,281.2138208119287,106.38819885253906]},{"page":309,"text":"(11.26)","rect":[356.0734558105469,122.5861587524414,385.1616452774204,114.05003356933594]},{"page":309,"text":"When pij forms with l a “magic” angle bm \u0007 54.7\u0006 the dichroism is not observed","rect":[65.76790618896485,154.7494659423828,385.11492243817818,144.5196990966797]},{"page":309,"text":"at all (K ¼ 1), for b < bm the dichroic ratio K > 1; for b > bm K < 1. For small","rect":[53.81393051147461,165.6331024169922,385.1420121915061,156.47923278808595]},{"page":309,"text":"angles b formula (11.26) works quite well. Note that the order parameter S is the","rect":[53.814144134521487,177.68228149414063,385.17090643121568,168.43894958496095]},{"page":309,"text":"fundamental characteristic of a liquid crystal microscopically related to a more or","rect":[53.813106536865237,189.641845703125,385.14898048249855,180.70729064941407]},{"page":309,"text":"less rigid molecular skeletons. Therefore, Sabs S should be considered as a value","rect":[53.813106536865237,201.60137939453126,385.1416400737938,192.6668243408203]},{"page":309,"text":"independent of the electronic oscillator angle b.","rect":[53.813777923583987,213.56121826171876,246.62978787924534,204.31788635253907]},{"page":309,"text":"Sometimes it is difficult to perform measurements in the UV or IR range on a","rect":[65.76677703857422,225.520751953125,385.1596149273094,216.58619689941407]},{"page":309,"text":"pure liquid crystal. Then, genuine orientational order parameter of the mesophaseS","rect":[53.814735412597659,237.48031616210938,385.17949763349068,228.54576110839845]},{"page":309,"text":"can be estimated using dichroism of guest dye molecules dissolved in a liquid","rect":[53.814735412597659,249.383056640625,385.11476985028755,240.44850158691407]},{"page":309,"text":"crystal. Then the dichroism measurements can be made in the visible range where","rect":[53.814735412597659,261.3426208496094,385.1088336773094,252.40806579589845]},{"page":309,"text":"good quality polarisers and fast spectrometers are available. However, the dye","rect":[53.814735412597659,273.3021545410156,385.14267767145005,264.36761474609377]},{"page":309,"text":"molecules should have molecular structure similar to that of the liquid crystal","rect":[53.814735412597659,285.2616882324219,385.14655931064677,276.3271484375]},{"page":309,"text":"molecules; only in this case Sabs(dye) \u0007 S.","rect":[53.814735412597659,297.2221984863281,225.2178158333469,288.28668212890627]},{"page":309,"text":"11.1.2.3 Kramers – Kronig Relations","rect":[53.81426239013672,341.62603759765627,218.3045998965378,332.2930603027344]},{"page":309,"text":"The real and imaginary parts of the complex refraction index n\bðoÞ ¼ nðoÞ þ ikðoÞ","rect":[53.81426239013672,365.8045654296875,385.1687356387253,355.8540344238281]},{"page":309,"text":"are related to each other through the Kramers-Kronig relation:","rect":[53.81494903564453,377.4653015136719,305.5688920743186,368.49090576171877]},{"page":309,"text":"1","rect":[222.84210205078126,395.8155517578125,229.81002946060819,392.6710205078125]},{"page":309,"text":"nðoÞ \u0003 1 ¼ p2 ð uu2k\u0003ðuoÞ2 du","rect":[164.15936279296876,419.8515625,274.8269890885688,397.0610046386719]},{"page":309,"text":"0","rect":[224.0879669189453,426.26715087890627,227.57193062385879,421.4701843261719]},{"page":309,"text":"(11.27)","rect":[356.0712585449219,413.01947021484377,385.1594480117954,404.5431213378906]},{"page":309,"text":"Here o and u are the same angular frequencies lettered differently in order to","rect":[65.76570892333985,450.77069091796877,385.1415948014594,441.83612060546877]},{"page":309,"text":"perform a proper integration [8,9].","rect":[53.81269454956055,462.730224609375,192.73267789389377,453.795654296875]},{"page":309,"text":"Mathematically, integral Kramers-Kronig relations have very general character. They","rect":[65.76570892333985,474.6897888183594,385.05727473310005,465.7552490234375]},{"page":309,"text":"represent the Hilbert transform of any complex function eðoÞ ¼ e0ðoÞ þ ie00ðoÞ","rect":[53.81368637084961,486.9881591796875,385.1680032168503,476.9898681640625]},{"page":309,"text":"satisfying the condition e \b ðoÞ ¼ eð\u0003oÞ(here the star means complex conjugate).","rect":[53.814205169677737,498.94769287109377,385.1112331917453,488.9971618652344]},{"page":309,"text":"In our particular example, this condition is applied to function n(o) related to","rect":[53.81523895263672,510.568603515625,385.1451348405219,501.634033203125]},{"page":309,"text":"dielectric permittivity e(o). The latter is Fourier transform of the time dependent","rect":[53.815269470214847,522.528076171875,385.1760392911155,513.5935668945313]},{"page":309,"text":"dielectric function e(t), which takes into account a time lag (and never advance) in","rect":[53.815269470214847,534.430908203125,385.14214411786568,525.4963989257813]},{"page":309,"text":"the response of a substance to the external, e.g. optical, electric field. Therefore the","rect":[53.81426239013672,546.390380859375,385.15808904840318,537.4558715820313]},{"page":309,"text":"Kramers-Kronig relations follow directly from the causality principle.","rect":[53.81426239013672,558.3499755859375,329.86846585776098,549.4154663085938]},{"page":309,"text":"For practical purpose, the Eq. (11.27) is very useful because, using a spectro-","rect":[65.76628875732422,570.3095092773438,385.17998634187355,561.375]},{"page":309,"text":"scopic technique, it is much easier to measure the frequency dependence of the","rect":[53.81426239013672,582.26904296875,385.1720660991844,573.3345336914063]},{"page":309,"text":"extinction index k(o) than the frequency dependence of the real value of refraction","rect":[53.81426239013672,594.2285766601563,385.1580742936469,585.2940673828125]},{"page":310,"text":"298","rect":[53.812252044677737,42.62544250488281,66.503858033704,36.73283386230469]},{"page":310,"text":"11 Optics and Electric Field Effects in Nematic and Smectic A Liquid Crystals","rect":[114.25221252441406,44.276390075683597,385.1671798992089,36.66510009765625]},{"page":310,"text":"index n(o). In the ideal case, having k(o) at any wavelength (l ¼ 2pc/o) from 0 to","rect":[53.812843322753909,68.2883529663086,385.1436699967719,59.035072326660159]},{"page":310,"text":"1, we would obtain exact n(o) values over the whole spectrum from the UV to","rect":[53.814796447753909,80.24788665771485,385.1426934342719,71.31333923339844]},{"page":310,"text":"microwave range. In practice, however, the accuracy is limited by an experimen-","rect":[53.8148078918457,92.20748138427735,385.06707130281105,83.27293395996094]},{"page":310,"text":"tally available range of k(o). On the other hand, very often the frequency range of","rect":[53.8148078918457,104.11019134521485,385.15068946687355,95.15572357177735]},{"page":310,"text":"the desirable values of n(o) isalsolimited and the calculation technique may still be","rect":[53.8148078918457,116.0697250366211,385.15064275934068,107.13517761230469]},{"page":310,"text":"applied.","rect":[53.8148078918457,128.02932739257813,85.81460996176486,119.09477233886719]},{"page":310,"text":"Consider a typical absorption spectrum aabs(l) measured in the range from ls to","rect":[65.76683044433594,139.98959350585938,385.1442498307563,130.73629760742188]},{"page":310,"text":"ll shown in Fig. 11.10a. It corresponds to a liquid crystal (in the isotropic phase)","rect":[53.81438446044922,151.94937133789063,385.1280454239048,142.69583129882813]},{"page":310,"text":"with some amount of a dye dissolved in it. The liquid crystal has a strong absorption","rect":[53.813072204589847,163.908935546875,385.14797297528755,154.97438049316407]},{"page":310,"text":"in the UV whereas the dye has UV absorption similar to that of the liquid crystal,","rect":[53.813072204589847,175.86846923828126,385.11907620932348,166.9339141845703]},{"page":310,"text":"but additionally absorbs in the visible range. We meet such a situation in the display","rect":[53.813072204589847,187.8280029296875,385.1718377213813,178.89344787597657]},{"page":310,"text":"technology (guest-host effect) or in the technology of non-linear optical materials.","rect":[53.813072204589847,199.73077392578126,385.1817898323703,190.7962188720703]},{"page":310,"text":"The spectrum of n(l) qualitatively corresponding to aabs(l) is shown below. Such a","rect":[53.813072204589847,211.69033813476563,385.15988958551255,202.43704223632813]},{"page":310,"text":"picture follows from the light dispersion theory and from Eq. (11.27). The back-","rect":[53.814083099365237,223.65048217773438,385.09319434968605,214.71592712402345]},{"page":310,"text":"ground value nb is provided by all short-wave absorption bands not included in the","rect":[53.81406784057617,235.61026000976563,385.17212713434068,226.6754913330078]},{"page":310,"text":"spectrum (l < ls). That part of the whole spectrum is unknown.","rect":[53.81338119506836,247.56985473632813,313.6581997444797,238.31649780273438]},{"page":310,"text":"Going from the left to the right along the wavelength axis (i.e. o!0) we","rect":[65.76551055908203,259.5294189453125,385.1702350444969,250.59486389160157]},{"page":310,"text":"subsequently meet regions of anomalous and normal dispersion located on the left","rect":[53.81349563598633,271.48895263671877,385.1185136563499,262.55438232421877]},{"page":310,"text":"and right slopes of each absorption band. It is very important, that the structure of the","rect":[53.81349563598633,283.448486328125,385.1732562847313,274.513916015625]},{"page":310,"text":"n(l) curve in the vicinity of each absorption band aabs(l) is determined exclusively","rect":[53.81349563598633,295.3517761230469,385.16689387372505,286.09796142578127]},{"page":310,"text":"a","rect":[55.96542739868164,338.7710876464844,61.52068979591549,333.1823425292969]},{"page":310,"text":"UV","rect":[89.21221160888672,352.5549621582031,100.7536877232416,347.0041198730469]},{"page":310,"text":"Vis","rect":[151.68116760253907,353.1122131347656,162.78221575215964,347.4494323730469]},{"page":310,"text":"dye","rect":[177.34274291992188,385.38055419921877,189.3326393045416,378.342041015625]},{"page":310,"text":"band","rect":[177.34274291992188,393.4508361816406,194.67214444482188,387.94000244140627]},{"page":310,"text":"b 0.15","rect":[218.71661376953126,338.7710876464844,249.97452339812734,331.46270751953127]},{"page":310,"text":"0.10","rect":[234.7557373046875,359.5220031738281,249.97452339812734,353.69940185546877]},{"page":310,"text":"e","rect":[329.9557189941406,352.0982360839844,334.39997342431408,347.627197265625]},{"page":310,"text":"o","rect":[320.3182373046875,387.6538391113281,325.2021212198601,383.18280029296877]},{"page":310,"text":"nb","rect":[177.15159606933595,476.5666198730469,184.59232239417049,470.7998352050781]},{"page":310,"text":"4","rect":[245.53054809570313,428.33135986328127,249.97469124480703,422.6527099609375]},{"page":310,"text":"2","rect":[245.53054809570313,445.03375244140627,249.97469124480703,439.3551025390625]},{"page":310,"text":"0","rect":[245.53054809570313,461.88092041015627,249.97469124480703,456.0583190917969]},{"page":310,"text":"e","rect":[343.7752685546875,430.4143371582031,348.21952298486095,425.94329833984377]},{"page":310,"text":"o","rect":[337.330322265625,451.1546936035156,342.2142061807976,446.68365478515627]},{"page":310,"text":"l","rect":[70.1194076538086,497.1031188964844,74.50770924043668,490.4085388183594]},{"page":310,"text":"s","rect":[75.50646209716797,498.9107360839844,77.83849668350448,496.06732177734377]},{"page":310,"text":"ll","rect":[200.260009765625,498.8867492675781,207.31306452752598,490.4085388183594]},{"page":310,"text":"450 500 550 600 650","rect":[261.3542175292969,499.6475524902344,376.68611702605707,493.824951171875]},{"page":310,"text":"l (nm)","rect":[307.46917724609377,513.7400512695313,332.9443675156687,505.224853515625]},{"page":310,"text":"Fig. 11.10 Kramers-Kronig relations: (a) Qualitative spectra of absorbance of an isotropic liquid","rect":[53.812843322753909,534.1779174804688,385.1855520644657,526.4481201171875]},{"page":310,"text":"with admixture of a dye in the UV and visible range (above) and corresponding spectrum of the","rect":[53.812843322753909,544.0861206054688,385.15258166574957,536.4917602539063]},{"page":310,"text":"refraction index (below); (b) Experimental polarization spectra of absorption coefficient aabs fora","rect":[53.81200408935547,554.0620727539063,385.17394397043707,546.4508056640625]},{"page":310,"text":"nematic liquid crystal E7 doped with a small amount of dye Chromene (upper plot) and","rect":[53.81393051147461,564.0377807617188,385.1384329238407,556.426513671875]},{"page":310,"text":"corresponding spectra of the increment of refraction indices d for two polarizations calculated","rect":[53.81393051147461,573.9569702148438,385.1375784316532,566.108642578125]},{"page":310,"text":"with Eq. (11.29) (lower plot). For both plots symbols (e) and (o) mean linear polarizations parallel","rect":[53.81307601928711,583.9329223632813,385.16634849753447,576.3216552734375]},{"page":310,"text":"and perpendicular to the director","rect":[53.812252044677737,593.9088745117188,165.70567411048106,586.3145141601563]},{"page":311,"text":"11.1 Optical Properties of Uniaxial Phases","rect":[53.812843322753909,44.274620056152347,199.2027939128808,36.663330078125]},{"page":311,"text":"299","rect":[372.49737548828127,42.62367248535156,385.1889700332157,36.73106384277344]},{"page":311,"text":"by the spectral features of that particular band and, with increasing l, the value of","rect":[53.812843322753909,68.2883529663086,385.1506589492954,59.035072326660159]},{"page":311,"text":"n(l) is systematically growing upon crossing each new band encountered.","rect":[53.812843322753909,80.24788665771485,350.5369534065891,70.9946060180664]},{"page":311,"text":"Therefore, for the practical purpose, using aabs ¼ 2 ko/c, we can rewrite","rect":[65.76486206054688,92.20748138427735,385.1602252788719,83.23309326171875]},{"page":311,"text":"Eq. (11.27) for a limited spectral range u1–u2 related to our experiment as follows [10]:","rect":[53.81439971923828,104.11067962646485,385.23713548252177,95.17594909667969]},{"page":311,"text":"u2","rect":[225.27783203125,123.50787353515625,231.2783169319797,119.37802124023438]},{"page":311,"text":"nðoÞ ¼ nb þ pc ð u2a\u0003ðuoÞ 2 du","rect":[162.3454132080078,146.76345825195313,276.9228142838813,123.9161148071289]},{"page":311,"text":"u1","rect":[224.76795959472657,153.71875,230.7685513069797,149.58883666992188]},{"page":311,"text":"(11.28)","rect":[356.07073974609377,139.93125915527345,385.1589292129673,131.45489501953126]},{"page":311,"text":"or in terms of the wavelength:","rect":[53.81315231323242,178.30596923828126,175.5610948331077,169.3714141845703]},{"page":311,"text":"ls","rect":[220.06643676757813,199.67813110351563,225.49464613967516,193.52532958984376]},{"page":311,"text":"nðlÞ ¼ nb þ 21p2 lðl 1 \u0003a\u0003ðll00=Þl\u00042 dl0","rect":[150.05245971679688,233.0109100341797,288.44921149440327,199.31065368652345]},{"page":311,"text":"(11.29)","rect":[356.0715026855469,216.0536651611328,385.1596921524204,207.57730102539063]},{"page":311,"text":"It is easy to calculate numerically a spectral dependence of n(l)–nb. The only","rect":[65.76595306396485,256.4686584472656,385.1279229264594,247.21502685546876]},{"page":311,"text":"problem is to find nb. A practical way is to measure nb at a convenient wavelength,","rect":[53.81399154663086,268.42840576171877,385.1741604378391,259.49365234375]},{"page":311,"text":"using, say, a laser and after this, to pin the whole spectrum to this particular point.","rect":[53.81344223022461,280.3879699707031,385.1423306038547,271.45343017578127]},{"page":311,"text":"The technique described can also be applied to measuring the dispersion of each","rect":[65.7654800415039,292.3475036621094,385.1005486588813,283.4129638671875]},{"page":311,"text":"principal components n|| and n⊥ of the refraction index for nematic or SmA liquid","rect":[53.81344223022461,304.2507019042969,385.11333552411568,295.31573486328127]},{"page":311,"text":"crystals using spectral data on principal absorption coefficients. An example of our","rect":[53.81332015991211,316.21026611328127,385.1402829727329,307.27569580078127]},{"page":311,"text":"measurements and calculations is shown in Fig. 11.10b. On the upper plot are","rect":[53.81332015991211,328.1697998046875,385.1641010112938,319.21533203125]},{"page":311,"text":"presented the experimental polarization absorption spectra aabs(l) (|| and ⊥ to the","rect":[53.81232833862305,340.12933349609377,385.1745075054344,330.8763427734375]},{"page":311,"text":"director) of a homogeneously oriented, 10 mm thick cell filled with nematic liquid","rect":[53.81379318237305,352.0892028808594,385.11379328778755,343.1546630859375]},{"page":311,"text":"crystal mixture E7 doped with 0.5% of lasing dye Chromene. On the lower plot, the","rect":[53.81476593017578,364.0487365722656,385.1715778179344,355.0544128417969]},{"page":311,"text":"spectra of the refraction index increment d ¼ n \u0003 nb calculated for each polarization","rect":[53.81476593017578,376.00848388671877,385.08220759442818,366.7749328613281]},{"page":311,"text":"are given. We see that a small amount of dye substantially changes the refraction","rect":[53.81405258178711,387.968017578125,385.1549004655219,379.033447265625]},{"page":311,"text":"index of the mixture in the vicinity of its absorption bands (by 5\u000210\u00033 in the","rect":[53.81405258178711,399.87078857421877,385.1192096538719,388.82025146484377]},{"page":311,"text":"maximum). Such an effect can influence the performance of the liquid crystal dye","rect":[53.814231872558597,411.830810546875,385.14417303277818,402.896240234375]},{"page":311,"text":"lasers. Note that, for solid anisotropic films of dyes, aabs||(l) may reach values as","rect":[53.814231872558597,423.7904968261719,385.14056791411596,414.5372009277344]},{"page":311,"text":"high","rect":[53.81368637084961,435.7500305175781,71.54213801435003,426.81549072265627]},{"page":311,"text":"as","rect":[76.5849380493164,434.0,84.86684049712375,428.0]},{"page":311,"text":"10","rect":[89.8967056274414,433.7280578613281,99.91561213544378,426.875244140625]},{"page":311,"text":"\u00031","rect":[118.21847534179688,429.1875,127.728779128469,424.6991271972656]},{"page":311,"text":"mm","rect":[104.90766143798828,435.6603698730469,118.17661236405928,429.0167541503906]},{"page":311,"text":"and","rect":[132.7196502685547,434.0,147.12339106610785,426.81561279296877]},{"page":311,"text":"the","rect":[152.20501708984376,434.0,164.4287799663719,426.81561279296877]},{"page":311,"text":"corresponding","rect":[169.48153686523438,435.7501525878906,226.5699395280219,426.81561279296877]},{"page":311,"text":"increment","rect":[231.61972045898438,434.0,271.60578782627177,426.81561279296877]},{"page":311,"text":"d","rect":[276.7113037109375,433.6783752441406,281.68840876630318,426.51678466796877]},{"page":311,"text":"(at","rect":[286.7371520996094,435.3517150878906,297.2985673672874,426.8753662109375]},{"page":311,"text":"absorption","rect":[302.3712463378906,435.7501525878906,344.44768611005318,426.81561279296877]},{"page":311,"text":"maxima)","rect":[349.5561828613281,435.3517150878906,385.09270606843605,426.81561279296877]},{"page":311,"text":"approaches 0.4 – 0.5 (compare with nb \u0007 2).","rect":[53.81362533569336,447.709716796875,233.7927670052219,438.71539306640627]},{"page":311,"text":"11.1.3 Light Scattering in Nematics and SmecticA","rect":[53.812843322753909,494.84576416015627,314.0091333536742,484.2916564941406]},{"page":311,"text":"Light scattering in nematics is very strong. A thick (hundreds of micrometer)","rect":[53.812843322753909,522.5284423828125,385.1238034805454,513.5939331054688]},{"page":311,"text":"homeotropically oriented preparation between crossed polarisers does not look","rect":[53.812843322753909,534.43115234375,385.1248406510688,525.4966430664063]},{"page":311,"text":"black under a microscope but rather sparking at random. In the beginning of the","rect":[53.812843322753909,546.3907470703125,385.1715778179344,537.4562377929688]},{"page":311,"text":"liquid crystal history it was taken as a strong argument in favour of the so-called","rect":[53.812843322753909,558.3502807617188,385.0880059342719,549.415771484375]},{"page":311,"text":"“swarm” model. Later Chatelain [11] made a series of careful experiments using","rect":[53.812843322753909,570.309814453125,385.1726006608344,561.3753051757813]},{"page":311,"text":"polarised light. He observed strong anisotropy of light scattering in nematics. When","rect":[53.81285095214844,582.2693481445313,385.1079033952094,573.3348388671875]},{"page":311,"text":"the electric vector of the scattered light s was perpendicular to the electric vector of","rect":[53.81285095214844,594.2288818359375,385.1486753067173,585.2943725585938]},{"page":312,"text":"300","rect":[53.81200408935547,42.55795669555664,66.50361007838174,36.73307800292969]},{"page":312,"text":"11 Optics and Electric Field Effects in Nematic and Smectic A Liquid Crystals","rect":[114.25196075439453,44.276634216308597,385.1669357585839,36.66534423828125]},{"page":312,"text":"incident light f, light scattering was 106 times stronger than in the isotropic liquid.","rect":[53.812843322753909,68.2883529663086,385.1805080940891,57.1976318359375]},{"page":312,"text":"For s || f, the scattering was much weaker but still considerably stronger than in the","rect":[53.81379318237305,80.24788665771485,385.17554510309068,71.31333923339844]},{"page":312,"text":"isotropic phase.","rect":[53.814781188964847,92.20748138427735,116.85678525473361,83.27293395996094]},{"page":312,"text":"Where does such a strong scattering anisotropy originate from? It is evident that","rect":[65.76680755615235,104.11019134521485,385.1377397305686,95.17564392089844]},{"page":312,"text":"the optical anisotropy of nematic liquid crystals plays the crucial role. In fact, the","rect":[53.814781188964847,116.0697250366211,385.1745990581688,107.13517761230469]},{"page":312,"text":"scattering is caused by fluctuations of the director n, i.e. the local orientation of the","rect":[53.814781188964847,128.02932739257813,385.17453802301255,119.09477233886719]},{"page":312,"text":"order parameter tensor. The local changes in orientation of n imply local changes in","rect":[53.81476593017578,139.98886108398438,385.20037165692818,131.05430603027345]},{"page":312,"text":"orientation of the optical indicatrix.","rect":[53.81476593017578,151.94839477539063,197.22707791830784,143.0138397216797]},{"page":312,"text":"11.1.3.1 Isotropic Phase","rect":[53.81476593017578,193.70867919921876,162.13053167535629,184.52511596679688]},{"page":312,"text":"Let us recall the reason for the light scattering in gas or in isotropic liquid. In that","rect":[53.81476593017578,217.697509765625,385.13972337314677,208.76295471191407]},{"page":312,"text":"case, we deal with fluctuations of the mass density. They can be represented bya","rect":[53.81476593017578,229.65704345703126,385.1596149273094,220.7224884033203]},{"page":312,"text":"sum of normal elastic vibration modes (Fourier harmonics) with wavevector q and","rect":[53.81476593017578,241.546875,385.14663020185005,232.6820526123047]},{"page":312,"text":"frequency O. When such a particular mode interacts with light of frequency o and","rect":[53.81476593017578,253.51934814453126,385.14663020185005,244.40550231933595]},{"page":312,"text":"wavevector k the conservation laws for energy and momentum read:","rect":[53.81476593017578,265.4789123535156,331.0833574063499,256.54437255859377]},{"page":312,"text":"O ¼ o0 \u0003 o and \u0004 q ¼ k0 \u0003 k","rect":[156.68157958984376,289.8406982421875,282.3190648014362,279.74066162109377]},{"page":312,"text":"(11.30)","rect":[356.07080078125,289.5119934082031,385.15899024812355,281.03564453125]},{"page":312,"text":"where o0 and k0 are frequency and wavevector of the scattered light. When O <<o","rect":[53.813228607177737,313.7730712890625,385.1882502716377,304.1700439453125]},{"page":312,"text":"we have a case of quasi-elastic scattering with k0 \u0007 k. Then, we have the same","rect":[53.81364822387695,325.73260498046877,385.1282123394188,316.1297302246094]},{"page":312,"text":"Eq. (5.10) for the wavevector of scattering,","rect":[53.81426239013672,337.6922607421875,228.03384061362034,328.69793701171877]},{"page":312,"text":"q ¼ 2ksin2y ¼ 2\u0007on=c\bsin2y;","rect":[156.002197265625,363.8624267578125,281.3257287709131,351.9297790527344]},{"page":312,"text":"(11.31)","rect":[356.0718688964844,362.1203308105469,385.1600583633579,353.64398193359377]},{"page":312,"text":"where n is refraction index, c velocity of light and 2y is the angle between k0 and k,","rect":[53.814308166503909,387.4010009765625,385.18328519369848,377.7417907714844]},{"page":312,"text":"Fig. 11.11a. Note that, in this case, normal vibrations are nothing more than sound","rect":[53.814598083496097,399.30413818359377,385.13359919599068,390.36956787109377]},{"page":312,"text":"waves with velocity vs and simple dispersion law O ¼ vsq. The frequency shift","rect":[53.81462860107422,411.2640686035156,372.06562669345927,402.1502380371094]},{"page":312,"text":"Do ¼ O ¼ \u0004qvs ¼ \u00042on\u0007vs=c\bsin2y","rect":[143.3129119873047,437.43475341796877,295.6796502213813,425.5021057128906]},{"page":312,"text":"(11.32)","rect":[356.0708312988281,435.7484436035156,385.15902076570168,427.2720947265625]},{"page":312,"text":"due to interaction of the incident light wave with the sound wave results in the","rect":[53.81327438354492,460.9733581542969,385.17102850152818,452.038818359375]},{"page":312,"text":"appearance of two satellites on both sides of the main frequency (Rayleigh line),","rect":[53.81327438354492,472.97271728515627,385.15508695151098,463.9784240722656]},{"page":312,"text":"namely, o þ O and o \u0003 O called Mandelshtam-Brillouin doublet.","rect":[53.813289642333987,484.8924255371094,321.45611234213598,475.7785949707031]},{"page":312,"text":"Fig. 11.11 Geometry of","rect":[53.812843322753909,525.5624389648438,138.4650124893873,517.8326416015625]},{"page":312,"text":"quasi-elastic scattering in","rect":[53.812843322753909,535.4706420898438,140.6877188125126,527.8762817382813]},{"page":312,"text":"general (a) and scattering on","rect":[53.812843322753909,545.389892578125,151.94348663477823,537.7955322265625]},{"page":312,"text":"the director fluctuations with","rect":[53.812843322753909,553.6387329101563,152.42493194727823,547.771484375]},{"page":312,"text":"director n||z, and vectors of","rect":[53.812843322753909,563.6146850585938,146.96160977942638,557.7474365234375]},{"page":312,"text":"incident (f) and scattered (s)","rect":[53.81284713745117,574.9791259765625,150.5474099503248,567.6133422851563]},{"page":312,"text":"light polarizations","rect":[53.81200408935547,585.2369995117188,115.62266238211908,577.6426391601563]},{"page":312,"text":"a","rect":[232.07586669921876,522.4165649414063,237.63112909645259,516.8278198242188]},{"page":312,"text":"y","rect":[303.3052673339844,582.2714233398438,307.301899015795,576.91259765625]},{"page":312,"text":"z","rect":[334.1446838378906,530.6939697265625,337.64573319115677,527.126708984375]},{"page":312,"text":"δn","rect":[365.4560852050781,521.4134521484375,373.4013863204825,515.7506103515625]},{"page":312,"text":"n0","rect":[356.34857177734377,538.9299926757813,363.34287171057675,533.0653076171875]},{"page":312,"text":"n","rect":[375.6307067871094,539.6063232421875,379.62733846892,535.80712890625]},{"page":312,"text":"s","rect":[351.9586486816406,570.2796020507813,355.0360550766348,566.3843994140625]},{"page":312,"text":"x","rect":[373.3843078613281,557.16650390625,379.05717777800569,552.3538818359375]},{"page":313,"text":"11.1 Optical Properties of Uniaxial Phases","rect":[53.812843322753909,44.274620056152347,199.2027939128808,36.663330078125]},{"page":313,"text":"301","rect":[372.49737548828127,42.55594253540039,385.1889700332157,36.73106384277344]},{"page":313,"text":"It follows from the theory of elastic scattering (o ¼ o0) that, for small modula-","rect":[65.76496887207031,68.2883529663086,385.15545020906105,58.62885665893555]},{"page":313,"text":"tion or fluctuation of dielectric permittivity (de << e⊥), the differential cross","rect":[53.81367111206055,80.24788665771485,385.09527982817846,71.0145263671875]},{"page":313,"text":"section for the scattered light per unit solid angle around k0 is given by","rect":[53.81418991088867,92.20772552490235,339.2317742448188,82.54804992675781]},{"page":313,"text":"s ¼ \u00054opc22\u00062hf_eðqÞsi2","rect":[171.12506103515626,129.9337615966797,267.39378426155408,103.43405151367188]},{"page":313,"text":"(11.33)","rect":[356.0715026855469,122.24686431884766,385.1596921524204,113.77050018310547]},{"page":313,"text":"_","rect":[360.37646484375,141.84356689453126,365.35356989911568,140.59849548339845]},{"page":313,"text":"where f and s are polarisation vectors of the incident and scattered light and eðqÞis","rect":[53.81393051147461,150.757568359375,385.1900674258347,140.8070526123047]},{"page":313,"text":"the spatial Fourier component of the dielectric permittivity, which scatters light","rect":[53.815452575683597,162.37847900390626,385.12348802158427,153.4439239501953]},{"page":313,"text":"with wavevector q. This expression is independent of the physical mechanism of","rect":[53.815452575683597,174.33804321289063,385.15227638093605,165.4034881591797]},{"page":313,"text":"scattering and can be found in textbooks on optics. For scalar e (mass density","rect":[53.816429138183597,186.29757690429688,385.10747614911568,177.36302185058595]},{"page":313,"text":"fluctuations) and f⊥s, s ¼ 0 because the incident field Ei has no projection on the","rect":[53.81639862060547,198.25750732421876,385.1749957866844,189.19309997558595]},{"page":313,"text":"induction vector Ds ¼ eEs ¼ 0. In this type of scattering, maximum scattering","rect":[53.81426239013672,210.21713256835938,385.16338435224068,201.28248596191407]},{"page":313,"text":"intensity is observed when polarizations of the incident and scattered light coincide","rect":[53.812625885009769,222.17666625976563,385.14551580621568,213.2421112060547]},{"page":313,"text":"with each other, f||s.","rect":[53.812625885009769,232.22677612304688,134.96033902670627,225.01539611816407]},{"page":313,"text":"11.1.3.2 Nematic Phase","rect":[53.81162643432617,269.29168701171877,159.39595831109848,261.781494140625]},{"page":313,"text":"The strong light scattering by nematics is due to specific properties of their","rect":[53.81162643432617,294.8970947265625,385.10860572663918,285.9625244140625]},{"page":313,"text":"dielectric tensor (ni and nj are components of the director, see Section 7.2.2):","rect":[53.81162643432617,307.7743835449219,364.674696517678,297.922119140625]},{"page":313,"text":"_","rect":[179.05528259277345,324.46917724609377,184.0323876481391,323.22412109375]},{"page":313,"text":"e ¼ e?dij þ eaðninjÞ","rect":[178.6016082763672,333.9184265136719,259.92511381255346,323.4325866699219]},{"page":313,"text":"(11.34)","rect":[356.07086181640627,332.6460266113281,385.1590512832798,324.169677734375]},{"page":313,"text":"Consider a slab of a nematic with the director n ¼ n0 þ dn having small","rect":[65.7652816772461,357.4172668457031,385.1732621915061,348.18389892578127]},{"page":313,"text":"fluctuating components nx, ny << nz \u0007 1, see Fig. 11.15b. The slab is illuminated","rect":[53.81450653076172,370.29339599609377,385.1408623795844,360.3825988769531]},{"page":313,"text":"byalightbeamalongthex-directionwithlinearpolarizationinthey,zplane,f¼ fyjþ","rect":[53.81296920776367,382.2530212402344,385.14780452940377,372.2724609375]},{"page":313,"text":"fzk. The scattered beam propagates in the same direction and has polarization","rect":[53.813961029052737,393.2962341308594,385.0978936295844,384.32171630859377]},{"page":313,"text":"s ¼ syj þ szk. We would like to find the “polarization structure” of the scattering","rect":[53.81278991699219,406.1723327636719,385.1645135026313,396.2415771484375]},{"page":313,"text":"cross-section sfs. To this effect, we only take the anisotropic part of the dielectric","rect":[53.81368637084961,418.1319885253906,385.18064153863755,408.22406005859377]},{"page":313,"text":"tensor responsible for fluctuations and s f scattering. Using Eq. (11.33) we have","rect":[53.81294631958008,429.1183776855469,385.1209186382469,420.14398193359377]},{"page":313,"text":"sfs / s\u0002","rect":[99.18621826171875,470.23583984375,132.91525208634278,462.5090637207031]},{"page":313,"text":"Then, neglecting","rect":[65.7651596069336,555.4032592773438,133.14919367841254,546.46875]},{"page":313,"text":"nz2 \u0007 1 we obtain:","rect":[53.813289642333987,566.865234375,128.65761430332254,556.3123168945313]},{"page":313,"text":"2","rect":[233.0948486328125,448.062255859375,236.57881233772597,443.3489685058594]},{"page":313,"text":"0 nx","rect":[213.60887145996095,462.1114196777344,236.20951218500515,444.1028747558594]},{"page":313,"text":"nxny nxnz 1 0 0 1","rect":[251.1646270751953,462.1114196777344,339.80057105392117,444.1028747558594]},{"page":313,"text":"eaðnanbÞ \u0002 f ¼ eas \u0002 Bnynx","rect":[135.15594482421876,470.1870422363281,240.68444199457546,459.75]},{"page":313,"text":"2","rect":[260.284423828125,463.0259704589844,263.76838753303846,458.31268310546877]},{"page":313,"text":"C \u0002 Bfy C","rect":[296.76336669921877,470.23583984375,339.80057105392117,460.3872375488281]},{"page":313,"text":"@ nznx nzny","rect":[213.60887145996095,486.1064758300781,267.6475004662551,467.34564208984377]},{"page":313,"text":"A @fz A","rect":[296.76336669921877,485.35418701171877,339.80057105392117,467.34564208984377]},{"page":313,"text":"¼ eas \u0002 B0nnxny2fyyfyþþnnynxnzfzzfz C1","rect":[183.70062255859376,516.7139892578125,292.78485450118679,490.3544006347656]},{"page":313,"text":"@ nznyfy þ n2zfz A","rect":[213.49559020996095,531.9046020507813,292.78485450118679,513.5404052734375]},{"page":313,"text":"second order terms nxny and ny2 and non-fluctuating","rect":[135.77810668945313,556.31982421875,343.81926814130318,544.3526611328125]},{"page":313,"text":"term","rect":[346.4402160644531,554.0,364.7758555281218,547.4847412109375]},{"page":313,"text":"with","rect":[367.3988037109375,554.0,385.1173028092719,546.4688110351563]},{"page":313,"text":"sfs / eaðsyj þ szkÞ \u0002 ðnxfzi þ nyfzj þ nyfykÞ ¼ eanyðsyfz þ szfyÞ","rect":[80.77687072753906,592.099609375,329.08884061919408,581.6704711914063]},{"page":313,"text":"(11.35)","rect":[356.07171630859377,590.8839111328125,385.1599057754673,582.2880249023438]},{"page":314,"text":"302","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.73106384277344]},{"page":314,"text":"11 Optics and Electric Field Effects in Nematic and Smectic A Liquid Crystals","rect":[114.25279998779297,44.274620056152347,385.1677597331933,36.663330078125]},{"page":314,"text":"Now we see that only fluctuations of the director in the y-direction are essential,","rect":[65.76496887207031,68.2883529663086,385.1557888558078,59.35380554199219]},{"page":314,"text":"dn ¼ ny, because nz – fluctuations are not related to the director realignment and nx–","rect":[53.813968658447269,81.16453552246094,385.17937556317818,71.0145263671875]},{"page":314,"text":"fluctuations are not seen by the transverse light wave incident from the x-direction.","rect":[53.81462860107422,92.20760345458985,385.15545316244848,83.27305603027344]},{"page":314,"text":"Considering the term syfz + szfy we can distinguish two important particular","rect":[65.76766204833985,105.08378601074219,385.1468442520298,95.13629150390625]},{"page":314,"text":"cases:","rect":[53.81394577026367,114.04823303222656,77.62441880771707,109.36681365966797]},{"page":314,"text":"Case A: f || n0||z and f ⊥ dn","rect":[53.81394577026367,157.76133728027345,166.98959702799869,148.66741943359376]},{"page":314,"text":"Then only the first term remains in Eq. 11.35. It means that there is no scattering","rect":[53.813201904296878,181.81988525390626,385.16497126630318,172.8255615234375]},{"page":314,"text":"with polarization s || f and the maximum scattering corresponds to sy that is s ⊥f","rect":[53.81318283081055,194.6962127685547,381.4213803848423,184.8448944091797]},{"page":314,"text":"Case B: f ⊥ n0","rect":[53.814353942871097,241.47850036621095,115.32370405522681,232.59375]},{"page":314,"text":"Then we have only the second term with fy and again there is no scattering with","rect":[53.812843322753909,266.4541015625,385.1177605729438,256.50653076171877]},{"page":314,"text":"polarization s || f. The maximum scattering corresponds to sz that is s ⊥ f. Indeed,","rect":[53.81374740600586,277.4404602050781,385.17486234213598,268.50592041015627]},{"page":314,"text":"maximum scattering by the director fluctuation is always takes place when s ⊥ f in","rect":[53.81315231323242,289.400146484375,385.1997918229438,280.465576171875]},{"page":314,"text":"agreement with experiments on liquid crystals and in contrast to the case of","rect":[53.81321334838867,301.35968017578127,385.14812599031105,292.42510986328127]},{"page":314,"text":"scattering by the density fluctuations in liquids and gases.","rect":[53.81321334838867,313.3192443847656,286.3923306038547,304.38470458984377]},{"page":314,"text":"As to the large amplitude of light scattering in nematics it is explained by very","rect":[65.7652359008789,325.2787780761719,385.1162957291938,316.34423828125]},{"page":314,"text":"low elasticity of the nematic phase with respect to the director distortion that is","rect":[53.81321334838867,337.2383117675781,385.18686308013158,328.30377197265627]},{"page":314,"text":"small Frank moduli Kii discussed below in Section 8.2. The strict theory of light","rect":[53.81321334838867,349.1418151855469,385.20588548252177,340.20654296875]},{"page":314,"text":"scattering in the static and dynamic regime has been developed by de Gennes","rect":[53.813289642333987,361.10137939453127,385.10837186919408,352.16680908203127]},{"page":314,"text":"[12]. His expression for the mean square amplitude of the director fluctuations has","rect":[53.813289642333987,373.0609130859375,385.1302529727097,364.1263427734375]},{"page":314,"text":"been discussed earlier, see Eq. (8.33). Using that equation and the tensor of","rect":[53.81229019165039,385.0204772949219,385.14617286531105,376.0859375]},{"page":314,"text":"dielectric anisotropy (11.34), de Gennes found the amplitude of the dielectric","rect":[53.81229019165039,396.9800109863281,385.1790241069969,388.04547119140627]},{"page":314,"text":"tensor","rect":[53.81229019165039,406.8777160644531,78.16028724519384,401.0209655761719]},{"page":314,"text":"fluctuations.","rect":[85.24967956542969,407.0,134.35679288412815,400.0050048828125]},{"page":314,"text":"After","rect":[141.38446044921876,406.8777160644531,162.40775428620948,400.0050048828125]},{"page":314,"text":"substitution","rect":[169.480224609375,406.8777160644531,216.10474482587348,400.0050048828125]},{"page":314,"text":"of","rect":[223.12246704101563,406.8777160644531,231.4143308242954,400.0050048828125]},{"page":314,"text":"the","rect":[238.41612243652345,406.8777160644531,250.6398853130516,400.0050048828125]},{"page":314,"text":"dielectric","rect":[257.675537109375,406.9075927734375,295.05164373590318,400.0050048828125]},{"page":314,"text":"tensor","rect":[302.0852966308594,406.8777160644531,326.43328224031105,401.0209655761719]},{"page":314,"text":"(11.34)","rect":[333.5226745605469,408.5411071777344,362.55709205476418,400.06475830078127]},{"page":314,"text":"into","rect":[369.608642578125,406.8777160644531,385.15712824872505,400.0050048828125]},{"page":314,"text":"Eq. (11.33) the differential cross-section for the light scattering by a nematic","rect":[53.813289642333987,420.8990783691406,385.10141790582505,411.96453857421877]},{"page":314,"text":"was given by","rect":[53.81427764892578,432.8586120605469,108.46090022138128,423.924072265625]},{"page":314,"text":"sa ¼ V\u0005e4apoc22\u00062 K33q2jjkBþTKaq?2 ðsyfz þ szfyÞ","rect":[131.5875244140625,476.20440673828127,307.3938332949753,447.0331726074219]},{"page":314,"text":"(11.36)","rect":[356.0708923339844,465.8460388183594,385.1590818008579,457.3099365234375]},{"page":314,"text":"We can see that large optical anisotropy (factor e2a in the first multiplier) and","rect":[65.76534271240235,500.38470458984377,385.1445244889594,489.2604064941406]},{"page":314,"text":"small energy terms (Kq2 in denominator of the second multiplier) are responsible","rect":[53.81362533569336,511.6457824707031,385.17139471246568,500.5084228515625]},{"page":314,"text":"for the high intensity of scattering. In addition, due to the factor of q2, especially","rect":[53.8136100769043,523.6054077148438,385.16228571942818,512.4680786132813]},{"page":314,"text":"strong scattering is observed in small solid angles around the incident beam.","rect":[53.81450653076172,535.56494140625,385.1374477913547,526.6304321289063]},{"page":314,"text":"Finally, the polarization factor (the third multiplier) makes the scattering extremely","rect":[53.81450653076172,547.5244750976563,385.17327204755318,538.5899658203125]},{"page":314,"text":"anisotropic. Eq. (11.36) is useful for the determination of elastic moduli from the","rect":[53.81450653076172,559.4840087890625,385.1732868023094,550.5494995117188]},{"page":314,"text":"intensity of scattering in different geometries and the viscosity coefficients from the","rect":[53.81450653076172,571.443603515625,385.1723102398094,562.5090942382813]},{"page":314,"text":"optical frequency spectra of scattering [13].","rect":[53.81450653076172,583.4031372070313,229.7790951302219,574.4686279296875]},{"page":315,"text":"11.1 Optical Properties of Uniaxial Phases","rect":[53.812984466552737,44.276084899902347,199.20294650077143,36.664794921875]},{"page":315,"text":"303","rect":[372.4975280761719,42.55740737915039,385.1891226211063,36.73252868652344]},{"page":315,"text":"11.1.3.3 Smectic A Phase","rect":[53.812843322753909,66.54527282714844,167.15747869684066,59.035072326660159]},{"page":315,"text":"The SmA phase has the same symmetry and the same dielectric ellipsoid as in","rect":[53.812843322753909,92.20748138427735,385.13979426435005,83.27293395996094]},{"page":315,"text":"nematics, therefore, everything said above about the birefringence and dichroism is","rect":[53.812843322753909,104.11019134521485,385.1865579043503,95.17564392089844]},{"page":315,"text":"valid for the SmA phase. However, due to specific elastic properties of the layered","rect":[53.812843322753909,116.0697250366211,385.08400813153755,107.13517761230469]},{"page":315,"text":"structure, the director fluctuations are strongly quenched, and the SmA preparations","rect":[53.812843322753909,128.02932739257813,385.16766752349096,119.09477233886719]},{"page":315,"text":"are much more transparent than the nematic ones. This is related to specific elastic","rect":[53.812843322753909,139.98886108398438,385.09494817926255,131.05430603027345]},{"page":315,"text":"properties of the lamellar SmA phase [14].","rect":[53.812843322753909,151.94839477539063,226.3780483772922,143.0138397216797]},{"page":315,"text":"According to Fig. 8.23 in Section 8.5.1, a fluctuation component of the layer","rect":[65.76385498046875,163.907958984375,385.1187985977329,154.91363525390626]},{"page":315,"text":"displacement uz ¼ u along the x-direction (within the smectic layer) is described by","rect":[53.812843322753909,175.86868286132813,385.1708916764594,166.9329376220703]},{"page":315,"text":"uðxÞ ¼ ucosq?x. Since the director angle Wf \u0007 qu=qx ¼ q?u, the free energy","rect":[53.814144134521487,188.64547729492188,385.17080012372505,178.2163543701172]},{"page":315,"text":"density in Eq. (8.49) can be rewritten in terms of the Wf-angle fluctuations:","rect":[53.81403732299805,200.7044219970703,353.8090959317405,190.4976806640625]},{"page":315,"text":"gdistðqÞ ¼ 21\u0005Bqq?2z2 þ K11q2?\u0006 \u0002 W2f","rect":[151.97952270507813,236.81277465820313,286.5398108728822,211.9779052734375]},{"page":315,"text":"(11.37)","rect":[356.0715026855469,228.8635711669922,385.1596921524204,220.38720703125]},{"page":315,"text":"For a cell of thickness d the z-component of the fluctuation wavevector is qz ¼","rect":[65.76595306396485,258.3953857421875,385.14807918760689,249.44090270996095]},{"page":315,"text":"mp/d, see Fig. 11.12a where the corresponding harmonics are marked off by the","rect":[53.814231872558597,270.3554382324219,385.17395818902818,261.4009704589844]},{"page":315,"text":"numbers 1, 2, 3, etc. For fluctuations along the x-axis, there is a critical vector","rect":[53.814231872558597,282.2582092285156,385.11422096101418,273.32366943359377]},{"page":315,"text":"equals qc ¼ (p/dls)1/2 (Eq. 8.48b), and, according to Eqs. (8.50) and (11.37), the","rect":[53.814231872558597,294.2181396484375,385.1754840679344,283.12738037109377]},{"page":315,"text":"intensity of the scattered light is given by","rect":[53.813724517822269,306.1777038574219,221.2385644059516,297.2431640625]},{"page":315,"text":"I / hW2i ¼ K11\u0003q2? þkBlTs\u00032q2z q2?\u0004 ¼ K11\u0007q?2 þkBmT2qc4=q?2 \b","rect":[90.1227035522461,348.7536315917969,318.09323919488755,318.4220275878906]},{"page":315,"text":"(11.38)","rect":[356.0705261230469,333.7232360839844,385.1587155899204,325.24688720703127]},{"page":315,"text":"This formula predicts that, in the SmA","rect":[65.76496124267578,370.2273864746094,239.57442998841135,361.2928466796875]},{"page":315,"text":"Fig. 11.12b, scattering vanishes in two limits:","rect":[53.81293869018555,382.18695068359377,237.95287950107645,373.25238037109377]},{"page":315,"text":"phase,","rect":[245.38470458984376,370.2273864746094,270.5589260628391,361.2928466796875]},{"page":315,"text":"1. For large scattering angles 2y and wavevectors q⊥","rect":[53.81290817260742,400.097900390625,269.47519370269796,390.8545837402344]},{"page":315,"text":"in","rect":[276.36920166015627,369.0,284.14345637372505,361.2928466796875]},{"page":315,"text":"a","rect":[289.9636535644531,369.0,294.41318548395005,363.0]},{"page":315,"text":"typical","rect":[300.2165222167969,370.2273864746094,327.4611955411155,361.2928466796875]},{"page":315,"text":"geometry","rect":[333.2973327636719,370.2273864746094,371.1004571061469,362.3088073730469]},{"page":315,"text":"of","rect":[376.8569641113281,369.0,385.1488278946079,361.2928466796875]},{"page":315,"text":"q?>>qc; I / 1 q?2 !0","rect":[171.29493713378907,429.8108825683594,267.6975945573188,414.292236328125]},{"page":315,"text":"a","rect":[76.0744857788086,451.1692199707031,81.62974817604244,445.5804748535156]},{"page":315,"text":"z","rect":[81.09564208984375,461.5238952636719,84.59669144310988,457.81268310546877]},{"page":315,"text":"m=1","rect":[108.32469177246094,461.3559265136719,123.81563192595127,455.7731018066406]},{"page":315,"text":"m=2","rect":[139.63189697265626,461.3559265136719,155.12285238493565,455.7731018066406]},{"page":315,"text":"m=3","rect":[169.4403839111328,461.44390869140627,184.93132406462315,455.7731018066406]},{"page":315,"text":"b","rect":[247.68515014648438,451.1692199707031,253.78994389596259,443.86083984375]},{"page":315,"text":"d","rect":[210.0238037109375,500.9363098144531,214.02043539274815,495.2335205078125]},{"page":315,"text":"x","rect":[226.7265167236328,531.5961303710938,231.13879810035176,527.8529663085938]},{"page":315,"text":"K0","rect":[284.5926208496094,527.9087524414063,293.33716492346738,520.3484497070313]},{"page":315,"text":"K","rect":[328.0547180175781,521.01123046875,333.80187437602185,515.5164184570313]},{"page":315,"text":"2q","rect":[307.4073181152344,529.75732421875,315.56845139408736,523.87060546875]},{"page":315,"text":"q^","rect":[318.1183166503906,552.4212036132813,326.059391046119,546.53857421875]},{"page":315,"text":"Fig. 11.12 Light scattering by the smectic A phase. Fluctuating elastic modes in the z andx","rect":[53.812843322753909,574.0244750976563,385.17369982981207,566.294677734375]},{"page":315,"text":"directions in a planar cell with director n0 || z (a) and typical geometry of scattering on fluctuating","rect":[53.813682556152347,583.9326782226563,385.16794342188759,576.3382568359375]},{"page":315,"text":"smectic layers (b)","rect":[53.812984466552737,593.9085693359375,115.14473813635995,586.314208984375]},{"page":316,"text":"304","rect":[53.812843322753909,42.55752944946289,66.50444931178018,36.73265075683594]},{"page":316,"text":"11 Optics and Electric Field Effects in Nematic and Smectic A Liquid Crystals","rect":[114.25279998779297,44.276206970214847,385.1677597331933,36.6649169921875]},{"page":316,"text":"2. For small scattering angles 2y!0:","rect":[53.812843322753909,68.2883529663086,205.18154771396707,59.04502868652344]},{"page":316,"text":"2","rect":[242.61123657226563,87.23223876953125,246.0952002771791,82.51895141601563]},{"page":316,"text":"q? ! 0and I / mq2?q4c !0","rect":[165.40252685546876,107.28829956054688,273.58855525067818,86.22805786132813]},{"page":316,"text":"The maximum scattering occurs at some resonance values of the wavevector","rect":[65.76554107666016,136.81521606445313,385.17134986726418,127.88066101074219]},{"page":316,"text":"q? ¼ m1=2qc. In fact, light propagating along z probes different qz modes with the","rect":[53.81350326538086,150.13546752929688,385.1751178569969,139.14413452148438]},{"page":316,"text":"number m. However, in comparison with nematics, well aligned, defect-free SmA","rect":[53.814353942871097,162.09503173828126,385.1512732501301,153.1604766845703]},{"page":316,"text":"liquid crystals are weakly scattering.","rect":[53.814353942871097,174.05459594726563,201.46712918783909,165.1200408935547]},{"page":316,"text":"11.2 Frederiks Transition and Related Phenomena","rect":[53.812843322753909,222.08248901367188,321.9418462743718,213.48861694335938]},{"page":316,"text":"11.2.1 Field Free Energy and Torques","rect":[53.812843322753909,254.0926055908203,253.1694858524577,243.454833984375]},{"page":316,"text":"Consider a nematic liquid crystal with director n¼ (1, 0, 0) aligned along the x-axis,","rect":[53.812843322753909,281.6350402832031,385.1247829964328,272.70050048828127]},{"page":316,"text":"Fig. 11.13. The liquid crystal is placed in the magnetic field oriented at an anglea","rect":[53.81385040283203,293.5946044921875,385.1875035221393,284.6600341796875]},{"page":316,"text":"with respect to the director, H ¼ (Hcosa, 0, Hsina), the diamagnetic anisotropy","rect":[53.814815521240237,305.4973449707031,385.1486443620063,296.56280517578127]},{"page":316,"text":"wa ¼ w|| – w⊥ being positive. We are interested in the excess free energy of the","rect":[53.81477737426758,317.457275390625,385.17160833551255,308.522705078125]},{"page":316,"text":"nematic due to the magnetic field. First we find the magnetization vector:","rect":[53.8128547668457,329.41680908203127,350.1059709317405,320.48223876953127]},{"page":316,"text":"M ¼ ^wH ¼ ðwjjH cosaÞi þ ðw?H sinaÞk","rect":[122.80648803710938,357.3450012207031,284.8695103580768,345.0]},{"page":316,"text":"¼ w?½ðH sinaÞk þ ðH cosaÞi\u0005 þ waðH cosaÞi","rect":[134.9869384765625,374.19342041015627,316.2158036954124,364.2428894042969]},{"page":316,"text":"or, on account of (Hn) ¼ cosa and n ¼ i, we obtain the magnetisation vector in the","rect":[53.814857482910159,398.7370300292969,385.17554510309068,389.7227783203125]},{"page":316,"text":"form","rect":[53.815818786621097,408.6347351074219,73.1966624128874,401.76202392578127]},{"page":316,"text":"M ¼ w?H þ waðHnÞn","rect":[175.0362091064453,434.95513916015627,263.9675328190143,425.0046081542969]},{"page":316,"text":"(11.39)","rect":[356.07183837890627,434.2180480957031,385.1600278457798,425.74169921875]},{"page":316,"text":"When, instead of the magnetic field, an electric field is applied at some angle to","rect":[65.7662582397461,458.5356140136719,385.19793025067818,449.60107421875]},{"page":316,"text":"the director of a nematic liquid crystal, in analogy to (11.39), the electric polariza-","rect":[53.814231872558597,470.4383544921875,385.16597877351418,461.5037841796875]},{"page":316,"text":"tion P is given by","rect":[53.81517791748047,482.3979187011719,125.51137629560003,473.46337890625]},{"page":316,"text":"P ¼ w?EE þ waEðEnÞn","rect":[177.47036743164063,508.2069396972656,261.5318333561237,496.6838073730469]},{"page":316,"text":"(11.40)","rect":[356.0719299316406,507.1662902832031,385.16011939851418,498.68994140625]},{"page":316,"text":"Fig. 11.13 Vector diagram","rect":[53.812843322753909,545.7408447265625,148.42622612352359,537.807861328125]},{"page":316,"text":"for calculations of","rect":[53.812843322753909,553.8965454101563,116.09054654700448,548.0546875]},{"page":316,"text":"magnetisation and free energy","rect":[53.812843322753909,565.625,155.33975738673136,558.0306396484375]},{"page":316,"text":"of a nematic liquid crystal ina","rect":[53.812843322753909,575.6009521484375,155.36515185617925,568.006591796875]},{"page":316,"text":"magnetic field","rect":[53.812843322753909,585.5202026367188,102.46399444727823,577.9258422851563]},{"page":316,"text":"z","rect":[290.5890808105469,541.5883178710938,294.09056785644256,537.3167114257813]},{"page":316,"text":"y","rect":[308.39947509765627,554.2744750976563,312.39660642858737,548.9149169921875]},{"page":316,"text":"H","rect":[339.5547180175781,539.0374145507813,346.1339961882908,533.5418701171875]},{"page":316,"text":"a","rect":[347.1690979003906,557.223876953125,352.2134776400257,552.7122802734375]},{"page":316,"text":"n","rect":[350.9161682128906,572.5696411132813,354.9132995438217,568.7699584960938]},{"page":316,"text":"x","rect":[370.8394775390625,573.3255615234375,375.2523105284105,569.5818481445313]},{"page":317,"text":"11.2 Frederiks Transition and Related Phenomena","rect":[53.812843322753909,42.54747772216797,225.17406604074956,36.68026351928711]},{"page":317,"text":"The magnetic field exerts a torque on the magnetisation:","rect":[65.76496887207031,68.32819366455078,294.2917314297874,59.333885192871097]},{"page":317,"text":"305","rect":[372.4990539550781,42.55594253540039,385.19064850001259,36.62946701049805]},{"page":317,"text":"GH ¼ M","rect":[157.7558135986328,93.52374267578125,192.73068360572194,85.21722412109375]},{"page":317,"text":"H ¼ waðHnÞn","rect":[204.88577270507813,94.53020477294922,261.3618962223346,84.5796890258789]},{"page":317,"text":"H","rect":[273.4831237792969,92.050048828125,281.25736187577805,85.2669677734375]},{"page":317,"text":"(11.41)","rect":[356.07122802734377,93.79312896728516,385.1594174942173,85.31676483154297]},{"page":317,"text":"that directed along the y-axis in Fig. 11.13. Correspondingly, the electric fieldE","rect":[53.81364059448242,120.0947494506836,385.18928786929396,111.16020202636719]},{"page":317,"text":"exerts the torque on the polarization P:","rect":[53.814659118652347,132.05435180664063,211.30918748447489,123.11979675292969]},{"page":317,"text":"GE ¼P","rect":[126.43362426757813,161.25732421875,157.1848983136531,152.89361572265626]},{"page":317,"text":"E ¼ wEaðEnÞn","rect":[169.31309509277345,162.56732177734376,224.31601304362369,151.04417419433595]},{"page":317,"text":"E ¼ ea ðEnÞn","rect":[236.437255859375,162.2076416015625,293.81985825846746,148.29054260253907]},{"page":317,"text":"4p","rect":[256.3193664550781,166.6549835205078,266.7991673404987,159.85198974609376]},{"page":317,"text":"E","rect":[305.9411315917969,159.72747802734376,312.57063552554396,152.94439697265626]},{"page":317,"text":"(11.42)","rect":[356.0715026855469,161.47056579589845,385.1596921524204,152.99420166015626]},{"page":317,"text":"E","rect":[248.9554901123047,183.13613891601563,253.23379754193844,178.53440856933595]},{"page":317,"text":"Note that, from the tensor form _e ¼ 1 þ 4p_w , follows ea ¼ 4pwaE because the","rect":[65.76595306396485,192.8343505859375,385.1747211284813,181.76512145996095]},{"page":317,"text":"_","rect":[243.970703125,195.52029418945313,248.94780818036566,194.2752227783203]},{"page":317,"text":"unity is included in the isotropic part of tensor e.","rect":[53.81399154663086,204.09555053710938,251.44404263998752,195.16099548339845]},{"page":317,"text":"In each particular situation, these torques may be balanced by the elastic, surface","rect":[65.76616668701172,216.05517578125,385.0853961773094,207.12062072753907]},{"page":317,"text":"or viscous torques. The magnetic and electric field torques may be obtained","rect":[53.814144134521487,228.01473999023438,385.17693415692818,219.08018493652345]},{"page":317,"text":"differently. Using minimisation of the free energy with respect to the director one","rect":[53.814144134521487,239.91751098632813,385.1441119976219,230.9829559326172]},{"page":317,"text":"obtains the “molecular field” introduced earlier, see Eq. (8.27) and then finds the","rect":[53.814144134521487,251.87704467773438,385.17087591363755,242.94248962402345]},{"page":317,"text":"torques as vector products with the director. Let us show it. The magnetic free","rect":[53.814144134521487,263.8365783691406,385.13608587457505,254.9020233154297]},{"page":317,"text":"energy density is given by","rect":[53.814144134521487,275.796142578125,159.89418879560004,266.861572265625]},{"page":317,"text":"H","rect":[98.27936553955078,297.6313171386719,103.28233741980651,293.02960205078127]},{"page":317,"text":"H","rect":[150.5628204345703,297.6313171386719,155.56579231482605,293.02960205078127]},{"page":317,"text":"gH ¼ \u0003ð MdH ¼ \u0003 ð ½w?H þ waðHnÞn\u0005dH ¼ \u000321½w?H2 þ waðHnÞ2\u0005 (11.43)","rect":[64.74685668945313,321.794677734375,385.1593259414829,299.6724548339844]},{"page":317,"text":"0","rect":[98.67584228515625,328.15264892578127,102.15980599006972,323.3556823730469]},{"page":317,"text":"0","rect":[150.90284729003907,328.15264892578127,154.38681099495254,323.3556823730469]},{"page":317,"text":"The first term in (11.43) is independent of the director; the absolute value of the","rect":[65.76555633544922,354.6961364746094,385.17331731988755,345.7615966796875]},{"page":317,"text":"second one is maximal for H || n that correspond to the minimum of magnetic free","rect":[53.81356430053711,366.5989074707031,385.13648260309068,357.66436767578127]},{"page":317,"text":"energy. Minimisation of (11.43) results in a vector of the “molecular field”","rect":[53.813594818115237,378.5584411621094,356.2025836773094,369.6239013671875]},{"page":317,"text":"hH ¼ qqgnH ¼ \u0003waðnHÞH","rect":[170.10462951660157,415.706787109375,268.9094004499968,394.83221435546877]},{"page":317,"text":"directed along H and coinciding with (8.27). The torque exerted by the “molecular","rect":[53.814842224121097,441.1910705566406,385.11690650788918,432.25653076171877]},{"page":317,"text":"field” on the director will be","rect":[53.814842224121097,451.11865234375,168.37387884332504,444.216064453125]},{"page":317,"text":"GH ¼ hH","rect":[157.98464965820313,478.386474609375,194.08449959021668,470.02264404296877]},{"page":317,"text":"n ¼ waðHnÞn","rect":[206.92428588867188,479.3363037109375,261.0785710270221,469.3857727050781]},{"page":317,"text":"H","rect":[273.2008056640625,476.85614013671877,280.97504376054368,470.07305908203127]},{"page":317,"text":"(11.44)","rect":[356.0726013183594,478.5992126464844,385.1607907852329,470.12286376953127]},{"page":317,"text":"that coincides with (11.41).","rect":[53.815025329589847,504.5024108886719,164.41128201987034,495.96630859375]},{"page":317,"text":"For the fixed electric field applied to the sample from the electrodes, instead of","rect":[65.76703643798828,516.8603515625,385.15291725007668,507.92584228515627]},{"page":317,"text":"Helmholtz free energy one should minimise the thermodynamic potential density:","rect":[53.81501007080078,528.8198852539063,384.243055892678,519.8853759765625]},{"page":317,"text":"H","rect":[179.56515502929688,550.6546020507813,184.5681269095526,546.0528564453125]},{"page":317,"text":"gE ¼ \u000341p Z DdE ¼ \u0003 81p½e?E2 þ eaðEnÞ2\u0005","rect":[128.52931213378907,574.8169555664063,310.44078004044436,552.6947631835938]},{"page":317,"text":"0","rect":[175.65684509277345,581.1759033203125,179.1408087976869,576.3789672851563]},{"page":317,"text":"(11.45)","rect":[356.07086181640627,567.9281616210938,385.1590512832798,559.332275390625]},{"page":318,"text":"306","rect":[53.813358306884769,42.55722427368164,66.50496429591104,36.68154525756836]},{"page":318,"text":"11 Optics and Electric Field Effects in Nematic and Smectic A Liquid Crystals","rect":[114.25331115722656,44.275901794433597,385.1682785320214,36.66461181640625]},{"page":318,"text":"By analogy, the “molecular field” coming from minimisation of (11.45) is given","rect":[65.76496887207031,68.2883529663086,385.15874568036568,59.294044494628909]},{"page":318,"text":"by (8.27) and the corresponding torque exerted by hE on the director is equal","rect":[53.81294250488281,80.24800872802735,385.0856767422874,71.31333923339844]},{"page":318,"text":"to (11.42):","rect":[53.81356430053711,91.8091812133789,96.29114396640847,83.33281707763672]},{"page":318,"text":"GE ¼ hE","rect":[158.49302673339845,119.42694091796875,192.90672539838375,111.063720703125]},{"page":318,"text":"n ¼ ea ðEnÞn","rect":[205.5648193359375,120.37627410888672,261.75870103190496,106.4601058959961]},{"page":318,"text":"4p","rect":[224.2581787109375,124.82466888427735,234.73797959635807,118.02165985107422]},{"page":318,"text":"E","rect":[273.8799133300781,117.8961181640625,280.5094172638252,111.113037109375]},{"page":318,"text":"(11.46)","rect":[356.07183837890627,119.63919830322266,385.1600278457798,111.10307312011719]},{"page":318,"text":"Example: Let the electric field 1 V/mm is applied at an angle of a ¼ p/3 to the","rect":[53.81426239013672,148.32144165039063,385.1749957866844,139.3868865966797]},{"page":318,"text":"director of liquid crystal 5CB (e⊥ ¼ 6.7, e|| ¼ þ19.7). What are the values of the free","rect":[53.81427764892578,160.28097534179688,385.24683416559068,151.28665161132813]},{"page":318,"text":"energydensityandtheelectrictorque?WeusetheGausssystem:E¼ 1/(300\u000210\u00034)¼","rect":[53.814430236816409,172.24093627929688,385.14780452940377,161.1334686279297]},{"page":318,"text":"33.3 statV/cm, (En) ¼ Ecos(p/3); the energy(11.45) and torque (11.46) are g ¼ (6.7","rect":[53.813961029052737,184.18360900878907,385.1796807389594,175.14944458007813]},{"page":318,"text":"þ 13\u00020.25)E2/8p ¼ 439 erg/cm3 (or 43.9 J/m3 in the SI system) and GE ¼ (13/4p)\u0002","rect":[53.813961029052737,196.10366821289063,385.15635621231936,185.05271911621095]},{"page":318,"text":"E2cos(p/3)\u0002sin(p/3) ¼ 493 erg/cm3 (49.3 J/m3 in the SI system).","rect":[53.81456756591797,208.06326293945313,307.0978664925266,197.0123748779297]},{"page":318,"text":"11.2.2 Experiments on Field Alignment of a Nematic","rect":[53.812843322753909,258.146728515625,325.91474708051887,247.59262084960938]},{"page":318,"text":"We shall discuss a very important macroscopic effect used in almost all the types of","rect":[53.812843322753909,285.7727355957031,385.1487973770298,276.83819580078127]},{"page":318,"text":"modern displays. In his original experiment Frederiks [15] used a liquid crystal p,p0-","rect":[53.812843322753909,297.7322998046875,385.15975318757668,288.0726623535156]},{"page":318,"text":"azoxyanisol (PAA), the grandfather of all other nematics, Fig. 11.14c. It was","rect":[53.81393051147461,309.69195556640627,385.09503568755346,300.75738525390627]},{"page":318,"text":"oriented homeotropically in a wedge-form gap between a flat and convex glasses","rect":[53.81393051147461,321.5947265625,385.08807767974096,312.66015625]},{"page":318,"text":"as shown in Fig. 11.14a, b. The cell with PAA was placed between crossed","rect":[53.81393051147461,333.55426025390627,385.13384333661568,324.61968994140627]},{"page":318,"text":"polarizers, heated up to about 120\u0006C and observed with an optical system. All","rect":[53.81393051147461,345.5142822265625,385.12397630283427,336.57928466796877]},{"page":318,"text":"this construction was installed between the poles of a magnet. In the figure,a","rect":[53.8129997253418,357.47381591796877,385.15586126520005,348.53924560546877]},{"page":318,"text":"magnetic field H was oriented horizontally. In the absence of the field the cell","rect":[53.8129997253418,369.4333801269531,385.1200395352561,360.49884033203127]},{"page":318,"text":"looked black. With increasing field H the PAA realignment began very sharply ata","rect":[53.8129997253418,381.3929138183594,385.15979803277818,372.4583740234375]},{"page":318,"text":"certain critical field strength Hc depending on the gap thickness d. Therefore, the","rect":[53.812984466552737,393.3529052734375,385.17530096246568,384.3984375]},{"page":318,"text":"birefringence appeared and the optical pattern resembled the Newton rings, but the","rect":[53.814598083496097,405.31243896484377,385.1724017925438,396.37786865234377]},{"page":318,"text":"contrast was much higher. It was shown that product Hcd ¼ const.","rect":[53.814598083496097,417.2152099609375,321.8155483772922,408.2610168457031]},{"page":318,"text":"Fig. 11.14 Experiment by Frederiks. Homeotropically aligned nematic PAA in a weak (a) anda","rect":[53.812843322753909,564.1054077148438,385.1728148200464,556.3756103515625]},{"page":318,"text":"strong (b) magnetic field and the correspondent optical pictures seen between crossed polarizers;","rect":[53.812843322753909,573.9569091796875,385.16442589011259,566.362548828125]},{"page":318,"text":"above a certain field a distortion occurs that causes the interference pattern. (c) The chemical","rect":[53.813690185546878,583.932861328125,385.15084556784697,576.3385009765625]},{"page":318,"text":"formula of PAA (p,p0-azoxyanisol)","rect":[53.81365203857422,593.9083862304688,173.37165921790294,585.7205810546875]},{"page":319,"text":"11.2 Frederiks Transition and Related Phenomena","rect":[53.81368637084961,42.54930877685547,225.174905274148,36.68209457397461]},{"page":319,"text":"307","rect":[372.4999084472656,42.55777359008789,385.19150299220009,36.73289489746094]},{"page":319,"text":"What is the physics of the observed phenomenon? In the absence of an external","rect":[65.76496887207031,68.2883529663086,385.109022689553,59.35380554199219]},{"page":319,"text":"field, the elastic free energy is minimal for the vertical alignment of the director. We","rect":[53.812950134277347,80.24788665771485,385.1249469585594,71.31333923339844]},{"page":319,"text":"know that PAA has positive anisotropy of magnetic susceptibility and the magnetic","rect":[53.812950134277347,92.20748138427735,385.17871893121568,83.27293395996094]},{"page":319,"text":"field should align the director horizontally. However, such a rotation of the director","rect":[53.812950134277347,104.11019134521485,385.1040280899204,95.17564392089844]},{"page":319,"text":"seems impossible as the magnetic torque GH is zero because, according to (11.44),","rect":[53.812950134277347,116.07039642333985,385.1540798714328,106.91604614257813]},{"page":319,"text":"the molecular field equals hH ¼ \u0003waðHnÞH and, in the experiment, H ⊥ n results in","rect":[53.81327438354492,128.36865234375,385.14373103192818,118.41812896728516]},{"page":319,"text":"(Hn) ¼ 0. Therefore, the system must accumulate some threshold amount of the","rect":[53.813838958740237,139.98953247070313,385.1736224956688,131.0549774169922]},{"page":319,"text":"magnetic field energy and then, accompanied by a small thermal fluctuation of the","rect":[53.813838958740237,151.94906616210938,385.17163885309068,143.01451110839845]},{"page":319,"text":"director, abruptly change its state so that the elastic and magnetic field forces will","rect":[53.813838958740237,163.90863037109376,385.1447587735374,154.9740753173828]},{"page":319,"text":"be in balance satisfying the minimum of free energy. This abrupt field-induced","rect":[53.813838958740237,175.8681640625,385.12587824872505,166.93360900878907]},{"page":319,"text":"change of the director alignment is called Frederiks transition. The threshold field","rect":[53.813838958740237,187.82766723632813,385.11782160810005,178.87318420410157]},{"page":319,"text":"is proportional to the inverse thickness of the nematic layer and this will be","rect":[53.81382369995117,199.73043823242188,385.15073431207505,190.79588317871095]},{"page":319,"text":"discussed below.","rect":[53.81382369995117,209.65805053710938,121.73637052084689,202.7554473876953]},{"page":319,"text":"In practice we meet numerous situations, but there are three basic geometries","rect":[65.7658462524414,223.6495361328125,385.13080228911596,214.71498107910157]},{"page":319,"text":"shown in Fig. 11.15. All upper sketches correspond to H < Hc, lower ones to","rect":[53.81382369995117,235.60910034179688,385.14360896161568,226.61477661132813]},{"page":319,"text":"H > Hc. In case (a), initially nx ¼ 1, ny ¼ 0, nz ¼ 0 and just above the threshold the z-","rect":[53.81374740600586,248.48634338378907,385.15938697663918,238.6352996826172]},{"page":319,"text":"component of the director appears. The distortion ∂nz/∂z ¼ cosW corresponds to the","rect":[53.813533782958987,259.529541015625,385.1734088726219,249.58885192871095]},{"page":319,"text":"splay term in the Frank energy (modulus K11). In case (b), initially nz ¼ 1, ny ¼ 0,","rect":[53.81370162963867,272.4056396484375,385.1757778694797,262.55450439453127]},{"page":319,"text":"nx ¼ 0 and above the threshold, the component nx appears. The term ∂nx/∂z ¼ sinW","rect":[53.814083099365237,283.44879150390627,385.1871067936237,273.50823974609377]},{"page":319,"text":"corresponds to the bend term (modulus K33). In both the cases, with further increase","rect":[53.814414978027347,295.3515625,385.1474994487938,286.4169921875]},{"page":319,"text":"of H > Hc the distortion becomes of the mixed type. In case (c), initially nx ¼ 1,","rect":[53.81364059448242,307.31146240234377,385.1755642464328,298.37677001953127]},{"page":319,"text":"nz ¼ 0, ny ¼ 0 but above the threshold, ny appears. The term ∂ny/∂z corresponds to","rect":[53.81386947631836,320.2442321777344,385.14251032880318,309.3305358886719]},{"page":319,"text":"the twist term (angle j, modulus K22). In this simple geometry, the twist distortion","rect":[53.81264877319336,331.2307434082031,385.1563653092719,322.29608154296877]},{"page":319,"text":"is “pure”; it does not mixed with bend or splay.","rect":[53.81354904174805,343.1902770996094,245.7964291145969,334.2557373046875]},{"page":319,"text":"11.2.3 Theory of Frederiks Transition","rect":[53.812843322753909,393.3575134277344,250.47708100175024,382.7197265625]},{"page":319,"text":"Our task now is to find the threshold field strength for the distortion and the","rect":[53.812843322753909,420.89971923828127,385.17160833551255,411.94525146484377]},{"page":319,"text":"distribution of the director n(z) over the cell thickness above the threshold. This","rect":[53.812843322753909,432.4608459472656,385.18552030669408,423.92474365234377]},{"page":319,"text":"time we shall take an initial homeotropic alignment, case (b) in Fig. 11.15. In its","rect":[53.812835693359378,444.7620544433594,385.15265287505346,435.7677307128906]},{"page":319,"text":"modern form, the theory was developed by Saupe [16].","rect":[53.81282424926758,456.7215881347656,276.5094265511203,447.78704833984377]},{"page":319,"text":"a","rect":[257.0557861328125,484.72515869140627,262.61104853004636,479.13641357421877]},{"page":319,"text":"nH","rect":[257.47137451171877,499.0061950683594,279.8789013049009,493.1994323730469]},{"page":319,"text":"b","rect":[305.5404357910156,484.6111755371094,311.6452295404938,477.30279541015627]},{"page":319,"text":"n","rect":[309.09429931640627,494.2152099609375,313.0909309982169,490.416015625]},{"page":319,"text":"H","rect":[328.21258544921877,496.1947937011719,334.79104119747907,490.699951171875]},{"page":319,"text":"c","rect":[352.74517822265627,484.6741638183594,358.3004406198901,479.0854187011719]},{"page":319,"text":"n","rect":[353.4305114746094,496.9234313964844,357.42714315642,493.1242370605469]},{"page":319,"text":"H","rect":[378.6284484863281,496.6107177734375,385.2069042345884,491.1158752441406]},{"page":319,"text":"Fig. 11.15 Three basic","rect":[53.812843322753909,538.8823852539063,134.74213549875737,531.152587890625]},{"page":319,"text":"configurations of the director","rect":[53.812843322753909,548.73388671875,153.29979032141856,541.1395263671875]},{"page":319,"text":"and the magnetic field for the","rect":[53.812843322753909,558.7098388671875,154.83548877024175,551.115478515625]},{"page":319,"text":"Frederiks transition onset,","rect":[53.812843322753909,567.0,142.6303889472719,561.0914306640625]},{"page":319,"text":"namely, splay (a), bend","rect":[53.812843322753909,578.6617431640625,134.40875763087198,571.0673828125]},{"page":319,"text":"(b) and twist (c)","rect":[53.812843322753909,588.2423706054688,109.42406552893807,580.9866333007813]},{"page":319,"text":"q","rect":[288.5075378417969,580.4942626953125,292.6720280542436,574.7994384765625]},{"page":319,"text":"d","rect":[320.3998718261719,579.9655151367188,324.3965035079825,574.1427612304688]},{"page":319,"text":"x","rect":[340.578857421875,566.53076171875,344.99113879859399,562.78759765625]},{"page":319,"text":"z","rect":[361.3301696777344,585.032470703125,364.8312190310005,581.4652099609375]},{"page":320,"text":"308","rect":[53.813777923583987,42.55722427368164,66.50538391261026,36.73234558105469]},{"page":320,"text":"11 Optics and Electric Field Effects in Nematic and Smectic A Liquid Crystals","rect":[114.25373840332031,44.275901794433597,385.16870577811519,36.66461181640625]},{"page":320,"text":"11.2.3.1 Simplest Model","rect":[53.812843322753909,68.2186279296875,162.69599778720926,59.035072326660159]},{"page":320,"text":"The geometry of the problem is illustrated in more details by Fig. 11.16. A plane","rect":[53.812843322753909,92.20748138427735,385.14170110895005,83.27293395996094]},{"page":320,"text":"nematic layer with normal z and thickness d is confined between two infinitely wide","rect":[53.812843322753909,104.11019134521485,385.1258014507469,95.15572357177735]},{"page":320,"text":"plates. The magnetic field is applied along x. The molecules are rigidly fixed","rect":[53.81282424926758,116.0697250366211,385.14373103192818,107.13517761230469]},{"page":320,"text":"(anchoring energy W!1) at the opposite boundaries (W0,d ¼ 0). When the field","rect":[53.812862396240237,128.02932739257813,385.11785212567818,118.79595947265625]},{"page":320,"text":"exceeds the criticalone (Hc), the director turns through an angle W(z) in the direction","rect":[53.813838958740237,139.98959350585938,385.09294978192818,130.7562255859375]},{"page":320,"text":"of H. Due to the up-and-down symmetry of our cell, the deflection angle must be","rect":[53.812870025634769,151.94912719726563,385.14881170465318,143.0145721435547]},{"page":320,"text":"symmetric with respect to the middle of the cell and the maximum deflection","rect":[53.812870025634769,163.90869140625,385.1676873307563,154.97413635253907]},{"page":320,"text":"Wm ¼ W(d/2) occurs at z ¼ d/2 as shown schematically by the dot curve.","rect":[53.812870025634769,175.86868286132813,343.1153225472141,166.63485717773438]},{"page":320,"text":"Our task isto find ananalytical expressionfor W(z) atdifferent fields. The scheme","rect":[65.76615142822266,187.82821655273438,385.09717596246568,178.5948486328125]},{"page":320,"text":"is as follows. First we shall write a proper integral equation for the free energy.","rect":[53.81412887573242,199.73098754882813,385.16692777182348,190.7964324951172]},{"page":320,"text":"Then, following the variational procedure discussed in Section 8.3, we compose the","rect":[53.81412887573242,211.6905517578125,385.1748737163719,202.75599670410157]},{"page":320,"text":"Euler equationcorrespondingto the free energy minimum andsolvethisdifferential","rect":[53.814151763916019,223.65008544921876,385.1330705411155,214.7155303955078]},{"page":320,"text":"equation for W(z). To simplify the problem we use the one-constant approximation","rect":[53.814151763916019,235.60964965820313,385.1042412858344,226.37628173828126]},{"page":320,"text":"K11 ¼ K22 ¼ K33 ¼ K. In our geometry, W \u0007 nx and only one derivative, namely the","rect":[53.81514358520508,247.56985473632813,385.17371404840318,238.33648681640626]},{"page":320,"text":"bend term with ∂nx/∂z, is essential in the Frank free energy form (8.15):","rect":[53.81394577026367,259.529541015625,344.42011888095927,249.58885192871095]},{"page":320,"text":"d","rect":[169.9921417236328,277.69189453125,173.47610542854629,272.7973327636719]},{"page":320,"text":"F ¼ 21 ð \"K\u0005ddWz\u00062 \u0003 waH2sin2WðzÞ#dz","rect":[141.27261352539063,305.4420166015625,297.70065702544408,275.58050537109377]},{"page":320,"text":"0","rect":[169.59567260742188,308.0312194824219,173.07963631233535,303.2342529296875]},{"page":320,"text":"(11.47)","rect":[356.0721130371094,294.78326416015627,385.1603025039829,286.3069152832031]},{"page":320,"text":"The magnetic part of the energy is given by Eq. (11.43).","rect":[65.76654815673828,330.55035400390627,293.3914761116672,321.61578369140627]},{"page":320,"text":"Thus we have an integral equation for F ¼ Fðz;W;W0Þ. The Euler equation of the","rect":[65.76654815673828,342.848876953125,385.1753619976219,332.1136779785156]},{"page":320,"text":"general form","rect":[53.814613342285159,354.4697570800781,105.4829401838835,345.53521728515627]},{"page":320,"text":"qF q qF","rect":[187.09901428222657,375.8638916015625,234.4488143292781,368.7023010253906]},{"page":320,"text":"\u0003","rect":[200.41078186035157,381.0,208.07552364561469,379.0]},{"page":320,"text":"¼0","rect":[238.13619995117188,382.7154846191406,253.53635493329535,375.8626708984375]},{"page":320,"text":"qW qz qW0","rect":[187.38270568847657,389.7607421875,234.8628925246767,381.42596435546877]},{"page":320,"text":"gives us the expression","rect":[53.8140983581543,413.2483215332031,147.01935664227973,404.31378173828127]},{"page":320,"text":"\u0003 2waH2 sinWcosW \u0003 K qqz2qqWz ¼0","rect":[148.35421752929688,448.30169677734377,292.8492364030219,427.5370788574219]},{"page":320,"text":"d","rect":[147.5369415283203,497.90289306640627,151.53222437774893,492.2022399902344]},{"page":320,"text":"d/2","rect":[141.8396759033203,522.6923217773438,152.07558985626455,516.9757080078125]},{"page":320,"text":"ϑ","rect":[172.06776428222657,477.95928955078127,177.1115134646716,472.2965087890625]},{"page":320,"text":"m","rect":[177.11131286621095,479.9128723144531,181.73341740622494,477.10546875]},{"page":320,"text":"z","rect":[253.5526885986328,481.7727966308594,257.05255637473229,477.5033264160156]},{"page":320,"text":"ϑ","rect":[264.7752380371094,491.906005859375,269.8189872195544,486.24322509765627]},{"page":320,"text":"n","rect":[281.4980773925781,486.0330505371094,285.4933602358569,482.2351379394531]},{"page":320,"text":"H","rect":[277.4294738769531,519.5623168945313,284.0079296252134,514.1634521484375]},{"page":320,"text":"x","rect":[284.0079650878906,521.6278686523438,287.31717612042987,518.8204956054688]},{"page":320,"text":"0","rect":[148.08030700683595,552.51318359375,152.07558985626455,546.8445434570313]},{"page":320,"text":"Fig. 11.16 Magnetic field induced Frederiks transition in a homeotropically aligned nematic","rect":[53.812843322753909,574.0244750976563,385.16442248606207,566.294677734375]},{"page":320,"text":"liquid crystal. Below the threshold the director is parallel to z; magnetic field is in the x-direction.","rect":[53.812843322753909,583.9326782226563,385.14831802442037,576.3383178710938]},{"page":320,"text":"Dot line shows the distortion above the threshold with maximum angle Wm in the middle of the cell","rect":[53.812843322753909,593.9086303710938,385.15856651511259,586.060302734375]},{"page":321,"text":"11.2 Frederiks Transition and Related Phenomena","rect":[53.812843322753909,42.54747772216797,225.17406604074956,36.68026351928711]},{"page":321,"text":"309","rect":[372.4990539550781,42.62367248535156,385.19064850001259,36.73106384277344]},{"page":321,"text":"Therefore we obtain the differential equation, which express","rect":[65.76496887207031,68.2883529663086,310.1100198184128,59.35380554199219]},{"page":321,"text":"elastic and magnetic torques:","rect":[53.812950134277347,80.24788665771485,171.18504197666239,71.31333923339844]},{"page":321,"text":"x2 qq2zW2 þ sinWcosW ¼0","rect":[171.29454040527345,117.22896575927735,267.6402520280219,94.47854614257813]},{"page":321,"text":"where x is the so-called magnetic field coherence length.","rect":[53.8128776550293,140.70960998535157,283.16485257651098,131.41647338867188]},{"page":321,"text":"ffiffiffiffi","rect":[233.66091918945313,156.0,241.94282067373525,155.0]},{"page":321,"text":"x ¼ H1 swKa","rect":[196.38900756835938,185.28900146484376,242.13003609260879,155.4175262451172]},{"page":321,"text":"the","rect":[313.0743713378906,67.0,325.2981647319969,59.35380554199219]},{"page":321,"text":"balance of the","rect":[328.19879150390627,67.0,385.17176092340318,59.35380554199219]},{"page":321,"text":"(11.48)","rect":[356.0704650878906,112.04434967041016,385.1586850723423,103.56798553466797]},{"page":321,"text":"(11.49)","rect":[356.07073974609377,175.1302947998047,385.1589292129673,166.6539306640625]},{"page":321,"text":"We can easily check by differentiation that the first integral of (11.48) is","rect":[65.7651596069336,208.85650634765626,358.6219827090378,199.9219512939453]},{"page":321,"text":"\u0005qqWz\u00062 ¼ \u0003x12 ðsin2WðzÞ \u0003 CÞ:","rect":[157.4743194580078,249.4077911376953,281.49607789200686,223.08731079101563]},{"page":321,"text":"As in the middle of the cell qW=qzjd=2 ¼ 0, the arbitrary constant C is easily","rect":[65.76642608642578,275.2710876464844,385.09490290692818,263.3042907714844]},{"page":321,"text":"found:","rect":[53.81379318237305,282.84375,79.83409745761941,275.93115234375]},{"page":321,"text":"C ¼ sin2Wm:","rect":[194.91671752929688,309.4897766113281,244.10928284806153,299.0963134765625]},{"page":321,"text":"Hence,","rect":[65.7656478881836,332.0,93.72702451254611,325.2159118652344]},{"page":321,"text":"ddWz ¼ 1xðsin2Wm \u0003 sin2WÞ1=2","rect":[165.74371337890626,370.9306335449219,274.4176184900697,348.18389892578127]},{"page":321,"text":"(11.50)","rect":[356.0715026855469,363.8205871582031,385.1596921524204,355.2247009277344]},{"page":321,"text":"The next step is to integrate (11.50). We can do it for the lower half of the cell","rect":[65.76595306396485,394.4301452636719,385.123915267678,385.4358215332031]},{"page":321,"text":"0- d/2 (dashed area in Fig. 11.16):","rect":[53.81493377685547,406.38970947265627,191.5702501065452,397.43524169921877]},{"page":321,"text":"Wm","rect":[166.65003967285157,427.7467346191406,174.0754535195963,421.615966796875]},{"page":321,"text":"2dx ¼ ð pffisffiiffiffinffiffi2ffiffiWffiffidffimffiWffiffi\u0003ffiffiffiffiffisffiffiiffinffiffiffi2ffiffiWffiffi ¼ FðsinWmÞ","rect":[143.19903564453126,453.59832763671877,297.4242288027878,428.50054931640627]},{"page":321,"text":"0","rect":[168.3494110107422,457.271484375,171.83337471565566,452.4745178222656]},{"page":321,"text":"(11.51)","rect":[356.0714416503906,444.08056640625,385.15963111726418,435.48468017578127]},{"page":321,"text":"First order elliptic integrals F(k) are tabulated and our problem is solved. In","rect":[65.76587677001953,481.831787109375,385.1507195573188,472.8773193359375]},{"page":321,"text":"Fig. 11.17a the distribution of the director is qualitatively shown for increasing field","rect":[53.81386947631836,493.79132080078127,385.1169365983344,484.85675048828127]},{"page":321,"text":"from H ¼ 0 to H4.","rect":[53.813899993896487,505.253173828125,128.1841244270969,496.81634521484377]},{"page":321,"text":"11.2.3.2 Threshold Condition","rect":[53.813236236572269,544.56787109375,185.23968858073307,537.1373901367188]},{"page":321,"text":"However, to make the result more transparent we shall look more carefully at a","rect":[53.813236236572269,570.3097534179688,385.15708196832505,561.375244140625]},{"page":321,"text":"simpler case of small distortions. From Fig. 11.17a one can see that, for a small","rect":[53.813236236572269,582.269287109375,385.1420732266624,573.3347778320313]},{"page":321,"text":"distortion, the director profile has a sine form. Consider, at first, a very severe","rect":[53.81321334838867,594.2288208007813,385.1122516460594,585.2943115234375]},{"page":322,"text":"310","rect":[53.813053131103519,42.55636978149414,66.50465912012979,36.73149108886719]},{"page":322,"text":"11 Optics and Electric Field Effects in Nematic and Smectic A Liquid Crystals","rect":[114.25300598144531,44.275047302246097,385.16797335624019,36.66375732421875]},{"page":322,"text":"a","rect":[115.44280242919922,68.23446655273438,120.99806482643307,62.64570999145508]},{"page":322,"text":"0","rect":[131.2101287841797,162.53204345703126,135.20676046599034,156.96524047851563]},{"page":322,"text":"H4","rect":[169.34486389160157,77.27886962890625,178.56120972327205,69.92843627929688]},{"page":322,"text":"H3","rect":[168.8452911376953,95.35294342041016,178.0616369693658,87.92446899414063]},{"page":322,"text":"H2","rect":[169.34486389160157,115.103515625,178.56120972327205,107.75308227539063]},{"page":322,"text":"1","rect":[264.6007080078125,161.76107788085938,268.59733968962316,156.2582550048828]},{"page":322,"text":"4","rect":[286.2469482421875,99.77496337890625,290.24357992399816,94.27213287353516]},{"page":322,"text":"3","rect":[287.57891845703127,114.20804595947266,291.5755501388419,108.60123443603516]},{"page":322,"text":"2","rect":[287.079345703125,130.4342041015625,291.07597738493566,124.9313735961914]},{"page":322,"text":"1","rect":[287.9138488769531,145.43017578125,291.91048055876379,139.92735290527345]},{"page":322,"text":"Fig. 11.17 (a) Qualitative picture of the director distribution over the cell thickness with","rect":[53.812843322753909,186.83795166015626,385.17877716212197,179.10813903808595]},{"page":322,"text":"increasing magnetic field from H ¼ 0 to H4 in the form of elliptic-sine functions (homeotropic","rect":[53.81199264526367,196.74618530273438,385.13378283762457,189.15164184570313]},{"page":322,"text":"alignment, W0 ¼ 0). (b) Absence of the threshold field for the initial tilted director alignment;","rect":[53.81350326538086,206.72195434570313,385.13976768698759,198.87359619140626]},{"page":322,"text":"calculation of the maximum distortion angle Wm changing with variation of the initial uniform","rect":[53.81355285644531,216.64102172851563,385.1972069584845,208.79266357421876]},{"page":322,"text":"director tilt W0: no tilt (curve 1 showing threshold at H/Hc ¼ 1), tilt angle W0 ¼ 1.7\u0006 (curve 2), 10\u0006","rect":[53.81344223022461,226.61685180664063,384.69291324909366,218.76849365234376]},{"page":322,"text":"(curve 3), 50\u0006 (curve 4)","rect":[53.812843322753909,236.25392150878907,134.78295987708263,228.9474334716797]},{"page":322,"text":"approximation sinW \u0007 W, and sinWm \u0007 Wm. Then the torque balance equation (11.48)","rect":[53.812843322753909,270.4122009277344,385.1593259414829,261.1788330078125]},{"page":322,"text":"reads:","rect":[53.813472747802737,280.34979248046877,77.63389451572488,273.43719482421877]},{"page":322,"text":"x2 qq2zW2 þ W ¼ 0:","rect":[187.66574096679688,321.3366394042969,251.30409180802247,298.5861511230469]},{"page":322,"text":"(11.52)","rect":[356.0721435546875,316.15191650390627,385.16033302156105,307.5560302734375]},{"page":322,"text":"Evidently the general solution is a harmonic function, e.g. W ¼ Wmsinqz þ B with","rect":[65.76656341552735,346.8182678222656,385.1189812760688,337.58489990234377]},{"page":322,"text":"wavevector q ¼ p/d. From the boundary condition at z ¼ 0 or z ¼ d we immediately","rect":[53.81399154663086,358.7774658203125,385.12993708661568,349.822998046875]},{"page":322,"text":"find B ¼ 0. Substituting W ¼ Wmsinqz into the approximate equation we get","rect":[53.814964294433597,370.73724365234377,356.3990922696311,361.503662109375]},{"page":322,"text":"\u0003 x2q2 þ 1 ¼ 0:","rect":[187.666259765625,398.0807189941406,253.5124505726709,386.9515075683594]},{"page":322,"text":"As we are interested in the extremely small W-angles, this result gives us the","rect":[65.7660903930664,424.0740051269531,385.1748431987938,414.84063720703127]},{"page":322,"text":"threshold condition,","rect":[53.814083099365237,434.0015869140625,134.0201382210422,427.0989990234375]},{"page":322,"text":"d","rect":[226.92054748535157,459.23760986328127,231.89765254071723,452.245361328125]},{"page":322,"text":"xc ¼","rect":[204.9996337890625,468.2319030761719,224.04926326963813,458.6916198730469]},{"page":322,"text":"p","rect":[226.8070831298828,472.7310791015625,232.3018071110065,468.22894287109377]},{"page":322,"text":"or, according to (11.49), the threshold magnetic field is given by","rect":[53.813838958740237,498.2109680175781,314.1950615983344,489.27642822265627]},{"page":322,"text":"ffiffiffiffi","rect":[235.5877685546875,516.0,243.86967003896963,514.0]},{"page":322,"text":"Hc ¼ pd swKa","rect":[194.46385192871095,544.7587890625,244.05590889534316,514.8873291015625]},{"page":322,"text":"(11.53a)","rect":[351.65277099609377,467.5456237792969,385.1287778457798,458.9497375488281]},{"page":322,"text":"(11.53b)","rect":[351.08648681640627,534.6001586914063,385.15178809968605,526.0042724609375]},{"page":322,"text":"Thus we have obtained a nice result in complete agreement with the Frederiks","rect":[65.7659683227539,570.3103637695313,385.14987577544408,561.3758544921875]},{"page":322,"text":"measurements, Hcd ¼ const, but we have not found yet the amplitude Wm of the sine-","rect":[53.81394577026367,582.2699584960938,385.1285642227329,573.03662109375]},{"page":322,"text":"form solution.","rect":[53.813655853271487,592.167724609375,110.70893521811252,585.2949829101563]},{"page":323,"text":"11.2 Frederiks Transition and Related Phenomena","rect":[53.812843322753909,42.54747772216797,225.17406604074956,36.68026351928711]},{"page":323,"text":"311","rect":[372.4990539550781,42.55594253540039,385.19064850001259,36.73106384277344]},{"page":323,"text":"We can find Wm going back to the strict equation (11.51), and using the second","rect":[65.76496887207031,68.2883529663086,385.1789483170844,59.05499267578125]},{"page":323,"text":"approximation, take sinW \u0007 W \u0003 W3/3!þ..., (and same for sinWm). Then, neglecting","rect":[53.81422424316406,80.24800872802735,385.0874871354438,69.19712829589844]},{"page":323,"text":"terms of the order of W6 and higher,","rect":[53.814369201660159,92.20772552490235,197.26372952963596,81.11688232421875]},{"page":323,"text":"ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi","rect":[127.6220703125,106.0,187.52651513174306,104.0]},{"page":323,"text":"ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi","rect":[210.7198028564453,106.0,243.32181847646963,105.0]},{"page":323,"text":"ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi","rect":[260.4537658691406,107.0,320.6010848338915,105.0]},{"page":323,"text":"qsin2Wm \u0003 sin2W \u0007 qW2m \u0003 W2 \u0002 q1 \u0003 13ðWm2 þ W2Þ","rect":[117.65192413330078,123.62704467773438,321.3284951602097,104.91523742675781]},{"page":323,"text":"and using expansion (1\u0003x)\u00031/2 \u0007 1 þ x/2þ...we obtain","rect":[53.81405258178711,145.09085083007813,280.16164485028755,134.00006103515626]},{"page":323,"text":"1","rect":[177.18606567382813,168.96585083007813,182.16317072919379,162.23257446289063]},{"page":323,"text":"1 þ 61ðWm2 þ W2Þ","rect":[227.82676696777345,171.88331604003907,288.75754179106908,159.3214111328125]},{"page":323,"text":"\u0007","rect":[217.40390014648438,175.2303466796875,225.0686419317475,170.96726989746095]},{"page":323,"text":"\u0002","rect":[290.9860534667969,173.7859344482422,293.67369019669436,172.7500457763672]},{"page":323,"text":"ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi","rect":[154.75425720214845,176.0,214.60294640615713,175.0]},{"page":323,"text":"ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi","rect":[246.97276306152345,176.0,279.57477868154776,175.0]},{"page":323,"text":"psin2Wm \u0003 sin2W","rect":[144.78494262695313,187.02853393554688,214.6286656315143,175.08592224121095]},{"page":323,"text":"qWm2 \u0003 W2","rect":[237.00335693359376,193.00466918945313,279.11924812874158,175.0857696533203]},{"page":323,"text":"Now the equation (11.51) takes the approximate form:","rect":[65.76604461669922,216.45260620117188,285.5211625821311,207.45828247070313]},{"page":323,"text":"2dx ¼ Wð0m pffisffiiffiffinffiffi2ffiffiWffiffidffimffiWffiffi\u0003ffiffiffiffiffisffiffiiffinffiffiffi2ffiffiWffiffi \u0007 Wð0m qWffiffiffi2mdffiffiffiWffi\u0003ffiffiffiffiffiWffiffi2ffiffiþ61W2m Wð0m qWffiffiffi2mdffiffiffiWffi\u0003ffiffiffiffiffiWffiffi2ffiffiþ16 Wð0m qWffiWffiffi2mffi2ffiffidffi\u0003ffiWffiffiffiffiWffiffi2ffiffi","rect":[59.307552337646487,269.6938171386719,381.309947138579,231.7352294921875]},{"page":323,"text":"Finally, using standard integrals","rect":[65.7656478881836,293.1407775878906,194.86676420317844,284.20623779296877]},{"page":323,"text":"Wð0m qWffiffiffi2mdffiffiffiWffi\u0003ffiffiffiffiffiWffiffi2ffiffi ¼ arcsinWWm W0m¼ p2","rect":[156.51058959960938,346.3829345703125,284.41543931315496,308.42425537109377]},{"page":323,"text":"and","rect":[53.81418991088867,367.79779052734377,68.21793452313909,360.89520263671877]},{"page":323,"text":"Wð0m qWffiWffiffim2ffi2ffiffidffi\u0003ffiWffiffiffiffiWffiffi2ffiffi ¼ \u0003W2 qffiWffiffi2mffiffiffiffi\u0003ffiffiffiffiffiWffiffi2ffiffi 0WmþW22m arcsinWWm 0Wm ¼ 0 þ W2m2 \u0002 p2","rect":[90.5755615234375,418.9343566894531,348.7077671940143,382.677490234375]},{"page":323,"text":"we arrive at the expression","rect":[53.81432342529297,442.3811340332031,162.1061944108344,433.44659423828127]},{"page":323,"text":"2dx \u0007 p2 \u00051 þ W62m þ W12m2 þ :::\u0006 ¼ p2 \u00051 þ W42m þ ::\u0006","rect":[121.84375762939453,481.9210205078125,318.8045622859575,456.6122741699219]},{"page":323,"text":"and finally to the form:","rect":[53.81356430053711,505.4097900390625,147.19302232334207,496.4752197265625]},{"page":323,"text":"pdx ¼ xxc ¼ HHc \u0007 \u00051 þ W4m2 þ ::\u0006","rect":[155.49075317382813,544.9500732421875,285.1572539363481,519.641357421875]},{"page":323,"text":"(11.54)","rect":[356.0700378417969,537.2635498046875,385.1582273086704,528.6676635742188]},{"page":323,"text":"The result contains two terms, the first one presents the same threshold condition","rect":[65.76447296142578,568.4398193359375,385.15334406903755,559.5053100585938]},{"page":323,"text":"d ¼ xcp, already found above and the sum of the first and the second term allows us","rect":[53.81245040893555,580.3995971679688,385.1562844668503,571.1461181640625]},{"page":323,"text":"to find the amplitude of the low-field sine-form distortion:","rect":[53.81342697143555,592.359130859375,288.3137651211936,583.4246215820313]},{"page":324,"text":"312","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.73106384277344]},{"page":324,"text":"11 Optics and Electric Field Effects in Nematic and Smectic A Liquid Crystals","rect":[114.25279998779297,44.274620056152347,385.1677597331933,36.663330078125]},{"page":324,"text":"ffiffiffiffiffi","rect":[176.3930206298828,61.0,186.74540246084463,60.0]},{"page":324,"text":"ffiffiffiffiffiffiffiffiffiffiffiffi","rect":[215.0261688232422,61.0,239.9475203807665,59.0]},{"page":324,"text":"ffiffiffiffiffiffiffiffiffiffiffiffiffi","rect":[268.951904296875,64.0,295.8680831249071,63.0]},{"page":324,"text":"Wm \u0007 sx4pd ¼ 2sxxc \u0003 1 ¼ 2rHHc \u00031","rect":[142.1793670654297,89.95144653320313,296.8124932389594,59.96675491333008]},{"page":324,"text":"(11.55)","rect":[356.0709228515625,80.07633209228516,385.15911231843605,71.4804458618164]},{"page":324,"text":"This form shows that the distortion develops at H > Hc smoothly, without a jump","rect":[65.7653579711914,113.51961517333985,385.14885798505318,104.58506774902344]},{"page":324,"text":"as in the case of second order thermodynamic transitions.","rect":[53.81400680541992,125.4791488647461,285.8337063362766,116.54460144042969]},{"page":324,"text":"11.2.4 Generalizations of the Simplest Model","rect":[53.812843322753909,175.50604248046876,285.154710977779,164.95193481445313]},{"page":324,"text":"Equations (11.53a) and (11.55) have been derived using several assumptions:","rect":[53.812843322753909,203.13201904296876,365.92011888095927,194.1376953125]},{"page":324,"text":"1.","rect":[53.812835693359378,218.95822143554688,61.27849240805392,212.22494506835938]},{"page":324,"text":"2.","rect":[53.81511688232422,291.0,61.280773597018768,283.92572021484377]},{"page":324,"text":"3.","rect":[53.81511688232422,303.0,61.280773597018768,295.8852844238281]},{"page":324,"text":"4.","rect":[53.81511688232422,327.0,61.280773597018768,319.8043212890625]},{"page":324,"text":"5.","rect":[53.81511688232422,339.0,61.280773597018768,331.64434814453127]},{"page":324,"text":"The electric field case has not been considered and that situation is more difficult","rect":[66.2745132446289,219.06777954101563,385.1227861172874,212.16517639160157]},{"page":324,"text":"for two reasons. First the dielectric anisotropy ea can be comparable with","rect":[66.2745132446289,233.05953979492188,385.1181267838813,224.1247100830078]},{"page":324,"text":"average <e> and therefore the field in a distorted nematic is no longer uniform","rect":[66.27579498291016,245.01910400390626,385.1519846785124,236.0845489501953]},{"page":324,"text":"in the z-direction. In addition, a liquid crystal can be conductive, and this can","rect":[66.27677154541016,256.921875,385.12612238935005,247.98731994628907]},{"page":324,"text":"result in some specific features, for example, there could be a flow of mass even","rect":[66.27680206298828,268.88140869140627,385.12114802411568,259.94683837890627]},{"page":324,"text":"in the steady-state regime.","rect":[66.27680206298828,280.8409729003906,172.28615994955784,271.90643310546877]},{"page":324,"text":"A difference in Frank elastic constants was ignored.","rect":[66.27679443359375,292.8005065917969,275.99309964682348,283.865966796875]},{"page":324,"text":"The field direction was selected along one of the principal axis of the liquid","rect":[66.27679443359375,304.76007080078127,385.11415949872505,295.82550048828127]},{"page":324,"text":"crystal.","rect":[66.27680206298828,316.7195739746094,95.4625439217258,307.7850341796875]},{"page":324,"text":"The infinitely strong anchoring was assumed.","rect":[66.27679443359375,328.6791076660156,249.23321195151096,319.74456787109377]},{"page":324,"text":"A steady-state situation was only considered. For example, a transient flow ofa","rect":[66.27679443359375,340.638671875,385.16098821832505,331.7041015625]},{"page":324,"text":"nematic (backflow) that occurs even in the case of the magnetic field was","rect":[66.27680206298828,352.54144287109377,385.1500283633347,343.60687255859377]},{"page":324,"text":"disregarded.","rect":[66.27680206298828,364.5009765625,115.37196012045627,355.56640625]},{"page":324,"text":"Below we shall consider qualitatively other situations (all of them are easily","rect":[65.76712799072266,382.46868896484377,385.09426203778755,373.53411865234377]},{"page":324,"text":"modeled numerically).","rect":[53.815101623535159,394.42822265625,144.7338833382297,385.49365234375]},{"page":324,"text":"11.2.4.1 Electric Field Case","rect":[53.815101623535159,434.51513671875,177.37075079156723,427.2340393066406]},{"page":324,"text":"Now the free energy density has a form (11.45) wherein, due to a large ea the field E","rect":[53.815101623535159,460.1205749511719,385.1890132110908,451.1262512207031]},{"page":324,"text":"becomes dependent on coordinates. In this case, one should operate with electric","rect":[53.814353942871097,472.0823974609375,385.1542438335594,463.1478271484375]},{"page":324,"text":"displacement D. For example, in the case of the Frederiks transition and the splay","rect":[53.814353942871097,484.0419616699219,385.14727107099068,475.107421875]},{"page":324,"text":"geometry of Fig. 11.15a the field strength is:","rect":[53.814353942871097,496.0014953613281,233.79844529697489,487.0071716308594]},{"page":324,"text":"EðzÞ ¼ 4peDz z ¼ e?sin2Wðz4ÞpþDezjj cosWðzÞ","rect":[137.53524780273438,534.8125610351563,299.80414213286596,510.19610595703127]},{"page":324,"text":"Evidently, that the correction does not influence the threshold condition:","rect":[65.76671600341797,558.3501586914063,357.9626603848655,549.4156494140625]},{"page":324,"text":"ffiffiffiffiffiffiffiffi","rect":[230.77281188964845,574.0,247.33661950186025,573.0]},{"page":324,"text":"Ec ¼ pd r4epaK","rect":[190.66917419433595,596.93701171875,247.881685940583,573.0418090820313]},{"page":324,"text":"(11.56)","rect":[356.070556640625,590.2037353515625,385.15874610749855,581.6078491210938]},{"page":325,"text":"11.2 Frederiks Transition and Related Phenomena","rect":[53.812843322753909,42.54747772216797,225.17406604074956,36.68026351928711]},{"page":325,"text":"313","rect":[372.4990539550781,42.55594253540039,385.19064850001259,36.73106384277344]},{"page":325,"text":"Thus, in full analogy with (11.53b), the threshold voltage Uc ¼ Ecd is indepen-","rect":[65.76496887207031,68.2883529663086,385.0992368301548,59.294044494628909]},{"page":325,"text":"dent of thickness! However, already at relatively small amplitude of the director","rect":[53.81417465209961,80.24788665771485,385.10625587312355,71.31333923339844]},{"page":325,"text":"deflection from its initial orientation, Wm depends on ez and one should correct","rect":[53.81417465209961,92.20772552490235,385.15742356845927,82.97412109375]},{"page":325,"text":"Eq. (11.54) for dielectric anisotropy ea/e⊥:","rect":[53.8136100769043,104.1104965209961,225.63923509189676,95.1161880493164]},{"page":325,"text":"EEc ¼ UUc \u0007 1 þ 14\u00051 þ ee?a\u0006Wm2","rect":[158.4931182861328,142.23350524902345,281.65787910681828,118.34828186035156]},{"page":325,"text":"(11.57)","rect":[356.0715026855469,134.54661560058595,385.1596921524204,125.95072174072266]},{"page":325,"text":"11.2.4.2 Anisotropy of Elastic Properties","rect":[53.81393051147461,196.04861450195313,233.72136320220188,186.58616638183595]},{"page":325,"text":"The Saupe solution (11.51) is not valid for different K11 and K33. To take into","rect":[53.81393051147461,219.51248168945313,385.15761652997505,210.51815795898438]},{"page":325,"text":"account a ratio of k ¼ K33 \u0003 K11/ K11 one more term should be added to the","rect":[53.81380844116211,231.05441284179688,385.17481268121568,222.53785705566407]},{"page":325,"text":"approximate form (11.57). Then we arrive at an even more correct form [17]:","rect":[53.813106536865237,243.43203735351563,366.82722337314677,234.43771362304688]},{"page":325,"text":"EEc \u0007 1 þ 41\u00051 þ ee?a þ k\u0006W2m","rect":[161.72190856933595,281.49786376953127,278.4291498587714,257.6126403808594]},{"page":325,"text":"(11.58)","rect":[356.0715026855469,273.81134033203127,385.1596921524204,265.2154541015625]},{"page":325,"text":"For positive k the initial slope of the W(z) curves in Fig. 11.17a would be steeper,","rect":[65.76595306396485,305.04443359375,385.1527370979953,295.8110656738281]},{"page":325,"text":"for k < 0 smoother.","rect":[53.81493377685547,315.240966796875,134.42705960776096,308.06939697265627]},{"page":325,"text":"11.2.4.3 Oblique Field or Tilted Alignment","rect":[53.814903259277347,358.9136657714844,243.2495359024204,349.5806884765625]},{"page":325,"text":"If the electric (or magnetic) field is applied at a certain angle to the director in the","rect":[53.814903259277347,382.7530822753906,385.1737445659813,373.81854248046877]},{"page":325,"text":"initial state, it creates a finite torque on the director and the Frederiks transition","rect":[53.814903259277347,394.6558532714844,385.15679255536568,385.7213134765625]},{"page":325,"text":"becomes “thresholdless”. The same situation occurs if the field is applied along the","rect":[53.814903259277347,406.6153564453125,385.1716693706688,397.6807861328125]},{"page":325,"text":"cell normal z but the initial alignment of a nematic is tilted at an angle 0 < W0 < p/2.","rect":[53.814903259277347,418.5749206542969,385.1408352425266,409.341552734375]},{"page":325,"text":"With increasing magnetic field the director deflection angle Wm in the middle of the","rect":[53.81493377685547,430.5354919433594,385.17432439996568,421.3021240234375]},{"page":325,"text":"cell is growing without threshold as shown in Fig. 11.17b (results of calculations,","rect":[53.813594818115237,442.4950256347656,385.0956997444797,433.56048583984377]},{"page":325,"text":"MBBA, d ¼ 24 mm [18]).","rect":[53.813594818115237,454.3648986816406,157.92564054037815,445.5000915527344]},{"page":325,"text":"11.2.4.4 Weak Anchoring","rect":[53.81356430053711,496.3642578125,169.30031672528754,487.27032470703127]},{"page":325,"text":"When the anchoring energy Ws is finite and the field is applied, the director at the","rect":[53.81356430053711,520.2045288085938,385.1741412944969,510.6010437011719]},{"page":325,"text":"surface has a certain freedom to turn under the action of the elastic torque from the","rect":[53.81345748901367,532.1072998046875,385.17221868707505,523.1329345703125]},{"page":325,"text":"bulk. Then, the profile of W(z) changes. The sine-form is still can be taken as an","rect":[53.81345748901367,544.0668334960938,385.15129939130318,534.83349609375]},{"page":325,"text":"approximation but its half-period is no longer equal to cell thickness d. Instead we","rect":[53.81344223022461,556.0263671875,385.1702045269188,547.0718994140625]},{"page":325,"text":"have d + 2b where b ¼ Kii/Ws is the surface extrapolation length already discussed","rect":[53.81345748901367,567.9864501953125,385.17705622724068,558.3829345703125]},{"page":325,"text":"in Section 10.2.4. Figure 11.18 clarifies the situation. Therefore, the threshold field","rect":[53.81333541870117,579.9459838867188,385.1153496842719,571.011474609375]},{"page":325,"text":"for the weak anchoring conditions is reduced according to formula:","rect":[53.81332778930664,591.905517578125,325.5563188321311,582.9710083007813]},{"page":326,"text":"314","rect":[53.812843322753909,42.55630874633789,66.50444931178018,36.73143005371094]},{"page":326,"text":"11 Optics and Electric Field Effects in Nematic and Smectic A Liquid Crystals","rect":[114.25279998779297,44.274986267089847,385.1677597331933,36.6636962890625]},{"page":326,"text":"Fig. 11.18 Profiles of the","rect":[53.812843322753909,67.58130645751953,143.10929248117925,59.6313591003418]},{"page":326,"text":"director angle W(z) in the same","rect":[53.812843322753909,77.4895248413086,155.30422351145269,69.64117431640625]},{"page":326,"text":"magnetic field for rigid (curve","rect":[53.812843322753909,87.4087142944336,155.3431486823511,79.81436157226563]},{"page":326,"text":"1) and weak (curve 2)","rect":[53.812843322753909,97.04601287841797,129.2314385758131,89.79031372070313]},{"page":326,"text":"anchoring in the cell","rect":[53.812843322753909,107.36067962646485,124.19371513571802,99.76632690429688]},{"page":326,"text":"geometry shown on the left","rect":[53.812843322753909,117.33663177490235,147.64272789206567,109.74227905273438]},{"page":326,"text":"side","rect":[53.812843322753909,125.52867889404297,67.46054980539799,119.66146850585938]},{"page":326,"text":"H","rect":[217.5954132080078,66.1893310546875,223.81417210490518,60.838470458984378]},{"page":326,"text":"b","rect":[263.96661376953127,116.13345336914063,267.9632454513419,110.44667053222656]},{"page":326,"text":"1","rect":[304.1344909667969,101.4322509765625,308.1311226486075,95.9294204711914]},{"page":326,"text":"b","rect":[364.6313781738281,115.13363647460938,368.62800985563879,109.44685363769531]},{"page":326,"text":"z","rect":[382.0614929199219,115.65352630615235,385.17087236837059,111.44642639160156]},{"page":326,"text":"ffiffiffiffi","rect":[246.5184783935547,170.0,254.8003798778368,169.0]},{"page":326,"text":"Hc ¼ d þp2bswKa","rect":[183.53033447265626,199.4024658203125,254.98848030647597,169.53099060058595]},{"page":326,"text":"(11.59)","rect":[356.0713195800781,189.24378967285157,385.15950904695168,180.64788818359376]},{"page":326,"text":"In principle, the measurements of this threshold allow us to find the value ofb","rect":[65.7657699584961,222.969970703125,385.1784600358344,214.03541564941407]},{"page":326,"text":"and then Ws. However, very thin cells have to be used to have d comparable withb","rect":[53.813716888427737,234.93008422851563,385.1790703873969,225.32647705078126]},{"page":326,"text":"(less than 1 mm). Note that, at the ordinate axes, curve 2 cuts angles W0 and Wd off.","rect":[53.81434631347656,246.83285522460938,385.1776089241672,237.5994873046875]},{"page":326,"text":"They depend on the anchoring strength that can be different at either interfaces. The","rect":[53.813899993896487,258.7926025390625,385.1448444194969,249.85804748535157]},{"page":326,"text":"general solution of the same problem for arbitrary fields (up to saturation of","rect":[53.813899993896487,270.75213623046877,385.1487973770298,261.81756591796877]},{"page":326,"text":"distortion, i.e. break of anchoring) is also known [19] and we discuss it below.","rect":[53.813899993896487,282.7117004394531,370.8604702522922,273.77716064453127]},{"page":326,"text":"11.2.4.5 Break of Anchoring","rect":[53.81289291381836,324.6213684082031,181.53438654950629,315.4776611328125]},{"page":326,"text":"Equation (11.53a) states that, for the infinitely strong anchoring, the threshold for","rect":[53.81289291381836,348.4607849121094,385.1745847305454,339.4664611816406]},{"page":326,"text":"the director field distortion is determined by equality of the field coherence length x","rect":[53.812862396240237,360.4203186035156,385.1776055436469,351.1670227050781]},{"page":326,"text":"(magnetic or electric) to the characteristic length of the cell d or, more precisely, to","rect":[53.812862396240237,372.3798522949219,385.14275446942818,363.4253845214844]},{"page":326,"text":"the reciprocal wavevector of the weak distortion d/p. Equation (11.59) points out","rect":[53.812862396240237,384.33941650390627,385.1357866055686,375.3450927734375]},{"page":326,"text":"that, for a weak anchoring, one reaches the threshold with increased characteristic","rect":[53.81386947631836,396.2421569824219,385.18064153863755,387.3076171875]},{"page":326,"text":"length (d + 2b) and, consequently, field coherence length xc ¼ (d + 2b)/p. For","rect":[53.81386947631836,408.20172119140627,385.1528256973423,398.94842529296877]},{"page":326,"text":"infinitely weak anchoring, xc !1 and the distortion is thresholdless.","rect":[53.81499481201172,420.1628112792969,333.1485256722141,410.9095153808594]},{"page":326,"text":"Now the question arises, how strong should be the field in order to force the","rect":[65.76545715332031,432.1224060058594,385.1721881694969,423.1878662109375]},{"page":326,"text":"director to be parallel to the field everywhere in the cell, including near-surface","rect":[53.81343460083008,444.0819396972656,385.13440740777818,435.14739990234377]},{"page":326,"text":"regions. Surely, for infinitely strong anchoring (b ¼ 0), such a field is infinite and,","rect":[53.81343460083008,456.0414733886719,385.1423306038547,447.10693359375]},{"page":326,"text":"for b!1, the threshold tends to zero. Therefore, for b >> d the value of p/b can be","rect":[53.814422607421878,466.91534423828127,385.1542438335594,459.04656982421877]},{"page":326,"text":"taken as a wavevector of the uniform distortion throughout the cell and, by analogy","rect":[53.814414978027347,479.9606018066406,385.1244744401313,471.02606201171877]},{"page":326,"text":"with Eq. (11.53a), we may write the threshold condition for the director saturation","rect":[53.814414978027347,491.8633728027344,385.17022028974068,482.8690490722656]},{"page":326,"text":"b","rect":[212.1361083984375,513.054443359375,217.11321345380316,506.11199951171877]},{"page":326,"text":"K","rect":[235.58724975585938,512.905029296875,242.21675368960644,506.33111572265627]},{"page":326,"text":"xb ¼ p ¼ pWs","rect":[189.705322265625,528.1853637695313,247.10655647172869,512.5941162109375]},{"page":326,"text":"(11.60)","rect":[356.0715026855469,521.4498901367188,385.1596921524204,512.9137573242188]},{"page":326,"text":"or the break-of-anchoring field. The formulas for the magnetic and electric fields","rect":[53.81393051147461,551.7586059570313,385.1527444277878,542.744384765625]},{"page":326,"text":"sufficient to overcome surface energy Ws and to break anchoring are given by:","rect":[53.81393051147461,563.6787719726563,370.7545610196311,554.7437744140625]},{"page":327,"text":"11.2 Frederiks Transition and Related Phenomena","rect":[53.812843322753909,42.54747772216797,225.17406604074956,36.68026351928711]},{"page":327,"text":"ffiffiffiffiffiffiffiffi","rect":[193.78306579589845,61.0,209.78843224600088,59.0]},{"page":327,"text":"ffiffiffiffiffiffiffi","rect":[275.6919250488281,63.0,290.18525231436026,62.0]},{"page":327,"text":"Hb ¼ Wsswa1K and Eb ¼ Wsre4apK","rect":[147.9004669189453,89.83889770507813,290.6483332794502,59.96742630004883]},{"page":327,"text":"315","rect":[372.4990539550781,42.55594253540039,385.19064850001259,36.62946701049805]},{"page":327,"text":"(11.61)","rect":[356.0713806152344,79.67923736572266,385.1595700821079,71.14311218261719]},{"page":327,"text":"Here, for the threshold estimation, we used an isotropic approximation with","rect":[65.76581573486328,113.3496322631836,385.11586848310005,104.41508483886719]},{"page":327,"text":"elastic modulus K. A precise value of the saturation field can be obtained without","rect":[53.81379318237305,125.30916595458985,385.1227861172874,116.37461853027344]},{"page":327,"text":"such an approximation. For a homogeneous planar cell it can be found from the","rect":[53.81379318237305,137.26870727539063,385.1725543804344,128.3341522216797]},{"page":327,"text":"equation [20]:","rect":[53.81379318237305,149.22830200195313,110.50898606845925,140.2937469482422]},{"page":327,"text":"cth\"p2 EEsFat \u0005KK1331\u00061=2# ¼ pWKs3d3 \u0002 EEsFat \u0005KK1313\u00061=2","rect":[127.16835021972656,193.2710418701172,311.29365608772596,163.40953063964845]},{"page":327,"text":"(11.62)","rect":[356.0715026855469,182.6118927001953,385.1596921524204,174.0757598876953]},{"page":327,"text":"Here Esat/EF represents the ratio of the saturation field Esat ¼ Eb to the Frederiks","rect":[65.76595306396485,216.848876953125,385.1540872012253,207.91432189941407]},{"page":327,"text":"transition field (electric or magnetic). As Esat/EF \u0007 d/b \u000B 1, the left part of","rect":[53.81426239013672,228.8084716796875,385.15203224031105,219.85398864746095]},{"page":327,"text":"Eq. (11.62) is close to 1. Therefore, assuming K11 ¼ K33 ¼ K, we turn back to","rect":[53.81418991088867,240.76803588867188,385.14370051435005,231.83348083496095]},{"page":327,"text":"Eqs. (11.60) and (11.61). In the next chapter we shall meet the break of anchoring","rect":[53.813838958740237,252.6708984375,385.1785821061469,243.73634338378907]},{"page":327,"text":"effect when discussing bistable devices.","rect":[53.81482696533203,264.6304626464844,214.59222074057346,255.69590759277345]},{"page":327,"text":"11.2.5 Dynamics of Frederiks Transition","rect":[53.812843322753909,302.10137939453127,264.3811032185471,291.4635925292969]},{"page":327,"text":"It is simpler to examine the dynamics of the Frederiks effect for the experimental","rect":[53.812843322753909,329.6436767578125,385.1775346524436,320.7091064453125]},{"page":327,"text":"geometry of Fig. 11.15c, since a pure twist distortion is not accompanied by the","rect":[53.812843322753909,341.60321044921877,385.17160833551255,332.60888671875]},{"page":327,"text":"backflow effect (see the next Section). For a twist distortion we operate with","rect":[53.81184387207031,353.5627746582031,385.1148614030219,344.62823486328127]},{"page":327,"text":"the azimuthal director angle j(z) (sinj \u0007 ny) and the equation for rotation of the","rect":[53.81184387207031,366.4391174316406,385.17258489801255,356.5877685546875]},{"page":327,"text":"director that expresses the balance of elastic, magnetic field and viscous torques is","rect":[53.813838958740237,377.4822082519531,385.1855508242722,368.54766845703127]},{"page":327,"text":"given by","rect":[53.813838958740237,389.38494873046877,88.80487910321722,380.45037841796877]},{"page":327,"text":"K22 qq2zj2 þ waH2 sinjcosj ¼ g1 qqjt :","rect":[145.1825408935547,426.4757385253906,293.8439477650537,403.6156311035156]},{"page":327,"text":"(11.63)","rect":[356.0716857910156,421.23748779296877,385.15987525788918,412.70135498046877]},{"page":327,"text":"Here, the first two terms came from minimisation procedure of the free energy,","rect":[65.76612091064453,449.9764709472656,385.1360745003391,441.04193115234377]},{"page":327,"text":"see Eqs. (8.15) and (11.43) and the viscous term was discussed earlier, see","rect":[53.814083099365237,461.9360046386719,385.11512029840318,452.9416809082031]},{"page":327,"text":"Eqs. (9.31) and (9.32) [21]. In terms of the phase transition theory, Eq. (11.63)","rect":[53.814083099365237,473.8955383300781,385.15889869538918,464.96099853515627]},{"page":327,"text":"may be regarded as the Landau-Khalatnikov equation discussed in Section 6.5.1. It","rect":[53.81306076049805,485.8551025390625,385.18073899814677,476.86077880859377]},{"page":327,"text":"describes the director rotation in magnetic field H with rotational viscosity g1 ¼","rect":[53.81306076049805,497.81463623046877,385.14807918760689,488.88006591796877]},{"page":327,"text":"a2 \u0003 a3 and without the director inertia term. In the limit of small j-angles, it","rect":[53.814231872558597,509.7185974121094,385.1721025235374,500.7840576171875]},{"page":327,"text":"reduces to the linear form:","rect":[53.81334686279297,519.6561279296875,160.56130845615457,512.7435913085938]},{"page":327,"text":"K22 qq2zj2 þ waH2j ¼ g1 qqjt","rect":[165.62998962402345,558.768798828125,271.66453811343458,535.9087524414063]},{"page":327,"text":"(11.64)","rect":[356.0712585449219,553.5305786132813,385.1594480117954,544.9944458007813]},{"page":327,"text":"with general solution j ¼ jm expðt=tRÞsinðpz=dÞ. By substitution this into (11.64)","rect":[53.81368637084961,582.6912841796875,385.1601804336704,572.6581420898438]},{"page":327,"text":"we find the characteristic time for the director reaction to the field:","rect":[53.81432342529297,592.20751953125,323.1683183438499,585.2949829101563]},{"page":328,"text":"316","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.68026351928711]},{"page":328,"text":"11 Optics and Electric Field Effects in Nematic and Smectic A Liquid Crystals","rect":[114.25279998779297,44.274620056152347,385.1677597331933,36.663330078125]},{"page":328,"text":"g1","rect":[187.43881225585938,66.487060546875,195.90759346565566,59.49162292480469]},{"page":328,"text":"tR ¼ waH2 \u0003 K22p2=d2 ¼ K22ðx\u00032 \u0003 xc\u00032Þ","rect":[134.36236572265626,81.72531127929688,302.9754678164597,66.53231048583985]},{"page":328,"text":"(11.65)","rect":[356.0711975097656,72.70767974853516,385.15938697663918,64.1117935180664]},{"page":328,"text":"where the field coherence length (11.49) includes the twist modulus K22.","rect":[53.81362533569336,108.2477035522461,385.1832241585422,99.31315612792969]},{"page":328,"text":"At the threshold (x ¼ xc) there is a singularity, tR!1. Above the Frederiks","rect":[53.814537048339847,120.2080307006836,385.1824990664597,110.95462799072266]},{"page":328,"text":"transition threshold H > Hc and tR is positive and distortion rises. When","rect":[53.813777923583987,132.16763305664063,385.14409724286568,123.23301696777344]},{"page":328,"text":"the applied field is switched off, the field induced distortion relaxes with the","rect":[53.813228607177737,144.12716674804688,385.07663763238755,135.19261169433595]},{"page":328,"text":"decay time","rect":[53.813228607177737,156.08676147460938,98.61712682916486,147.15220642089845]},{"page":328,"text":"g1","rect":[225.73095703125,179.33843994140626,234.19970772346816,172.3430938720703]},{"page":328,"text":"tD ¼ K22q2","rect":[195.59503173828126,192.64991760253907,241.2803351648744,179.3832550048828]},{"page":328,"text":"(11.66)","rect":[356.0715026855469,185.5593719482422,385.1596921524204,177.0232391357422]},{"page":328,"text":"as for a typical hydrodynamic mode.","rect":[53.81393051147461,218.15206909179688,201.9923061897922,209.21751403808595]},{"page":328,"text":"When the distortion is weak it is described only by the single Fourier harmonic","rect":[65.76595306396485,230.11160278320313,385.1647418804344,221.1770477294922]},{"page":328,"text":"with wavevector q ¼ 2p/L where L ¼ 2d. With increasing field, the distortion","rect":[53.81393051147461,242.07113647460938,385.1577080827094,232.8377685546875]},{"page":328,"text":"is characterised by elliptic-sine functions, Fig. 11.17a, with higher harmonics.","rect":[53.815879821777347,253.97390747070313,385.1397976448703,245.0393524169922]},{"page":328,"text":"Therefore we have multiple odd sine harmonics with wavevectors qm ¼ mp/d","rect":[53.815879821777347,265.9334716796875,385.1794060807563,256.97998046875]},{"page":328,"text":"(m ¼ 1, 2, 3 ...). Then, according to (11.66), each harmonic decays with its own","rect":[53.81468963623047,277.89398193359377,385.1236504655219,268.95941162109377]},{"page":328,"text":"time, the higher the number m the faster is the decay (in analogy with string of a","rect":[53.81468963623047,289.8535461425781,385.16150701715318,280.91900634765627]},{"page":328,"text":"guitar):","rect":[53.81468963623047,301.8130798339844,83.3209596402366,292.8785400390625]},{"page":328,"text":"tmD ¼ K22gm12q2 ;m ¼ 1;2;3:::","rect":[161.72328186035157,338.3763427734375,277.2422631947412,318.0694885253906]},{"page":328,"text":"(11.67)","rect":[356.0726318359375,331.28570556640627,385.16082130281105,322.74957275390627]},{"page":328,"text":"11.2.6 Backflow Effect","rect":[53.812843322753909,399.2252197265625,174.73240873168525,388.6711120605469]},{"page":328,"text":"We know that the shear-induced flow of a nematic liquid strongly influences the","rect":[53.812843322753909,426.9079284667969,385.1715778179344,417.973388671875]},{"page":328,"text":"alignment of the director (Section 9.3.2), i.e. there is a coupling of the two vector","rect":[53.812843322753909,438.8106994628906,385.11089454499855,429.87615966796877]},{"page":328,"text":"fields, the director n(r) and velocity of the liquid v(r). It is quite natural to think that","rect":[53.812843322753909,450.7702331542969,385.1397538907249,441.835693359375]},{"page":328,"text":"the opposite effect should also exist. Indeed, one observes a strange, not monotonic","rect":[53.81483459472656,462.72979736328127,385.1317828960594,453.79522705078127]},{"page":328,"text":"director rotation during its relaxation from the field-induced quasi-homeotropic","rect":[53.81483459472656,474.6893615722656,385.1666950054344,465.75482177734377]},{"page":328,"text":"alignment to the initial, field-off planar one. Normally, the elastic force should","rect":[53.81483459472656,486.6488952636719,385.17864314130318,477.71435546875]},{"page":328,"text":"smoothly rotate the director from y ¼ 0 (parallel to the cell normal) to 90\u0006, but, in","rect":[53.81483459472656,498.6084289550781,385.14394465497505,489.3651123046875]},{"page":328,"text":"experiment, the director angle may exceed 90\u0006 during the relaxation. As a result, in","rect":[53.814083099365237,510.5688171386719,385.14348689130318,501.6341552734375]},{"page":328,"text":"the optical transmission one observes a characteristic bump.","rect":[53.8136100769043,522.5283203125,295.9637417366672,513.5938110351563]},{"page":328,"text":"The reason for this is a flow of the nematic, which is lunched by the director","rect":[65.76563262939453,534.4310913085938,385.1067136367954,525.49658203125]},{"page":328,"text":"rotation. The flow arises in the beginningof the director relaxation process when the","rect":[53.8136100769043,546.390625,385.17139471246568,537.4561157226563]},{"page":328,"text":"elastic torque exerted on the director is very high near both interfaces due toa","rect":[53.8136100769043,558.3501586914063,385.1584857769188,549.4156494140625]},{"page":328,"text":"strong curvature of the director field. However, the curvature at the two interfaces","rect":[53.8136100769043,570.3096923828125,385.1455117617722,561.3751831054688]},{"page":328,"text":"has different sign, see Fig. 11.19a, where dW(z) ¼ (p/2) \u0003 W. Therefore, the flow of","rect":[53.8136100769043,582.269287109375,385.15044532624855,573.0359497070313]},{"page":328,"text":"nematic fluid coupled to the director rotation (backflow) at the two interfaces is","rect":[53.81362533569336,594.228759765625,385.1843301211472,585.2942504882813]},{"page":329,"text":"11.2 Frederiks Transition and Related Phenomena","rect":[53.81200408935547,42.54833221435547,225.17321154856206,36.68111801147461]},{"page":329,"text":"317","rect":[372.49822998046877,42.55679702758789,385.1898245254032,36.73191833496094]},{"page":329,"text":"a","rect":[119.86117553710938,68.35824584960938,125.41643793434322,62.76948928833008]},{"page":329,"text":"d","rect":[124.70509338378906,76.44161987304688,128.7017250655997,70.75483703613281]},{"page":329,"text":"torque","rect":[135.86050415039063,95.639892578125,157.1785407937994,89.25725555419922]},{"page":329,"text":"0","rect":[130.36671447753907,132.43045043945313,134.3633461593497,126.8316421508789]},{"page":329,"text":"dq (z)","rect":[169.1826171875,105.06505584716797,186.7290021834843,97.49066925048828]},{"page":329,"text":"Vx(z)","rect":[239.06968688964845,110.21199798583985,255.83088511805463,102.66160583496094]},{"page":329,"text":"Vx(z)","rect":[298.7459411621094,89.71639251708985,315.3874341903203,82.16600036621094]},{"page":329,"text":"n","rect":[287.5165100097656,120.34188079833985,291.96076443993908,116.58267974853516]},{"page":329,"text":"s","rect":[291.9601745605469,122.34944915771485,294.2922091468834,119.50605773925781]},{"page":329,"text":"Fig. 11.19","rect":[53.812843322753909,154.35980224609376,88.4846167006962,146.3251953125]},{"page":329,"text":"Backflow effect. The profile ofthe director in the field-on regime with steep parts close","rect":[94.48435974121094,154.29208374023438,385.1703428962183,146.69772338867188]},{"page":329,"text":"to interfaces at z ¼ 0 and z ¼ d (a). The direction of the torques is shown by small arrows in the","rect":[53.812843322753909,164.2113037109375,385.15514514231207,156.60000610351563]},{"page":329,"text":"right part of sketch (b) and a profile of the velocity is shown by thin arrows in the left part of the","rect":[53.812835693359378,174.18722534179688,385.15429065012457,166.59286499023438]},{"page":329,"text":"sketch. The strongest gradient of velocity is in the middle of the cell (dash arrows)","rect":[53.812835693359378,184.1632080078125,338.49566739661386,176.55191040039063]},{"page":329,"text":"opposite. It is shown in Fig. 11.19b by velocity arrows. In the middle of the cell the","rect":[53.812843322753909,209.02685546875,385.17261541559068,200.09230041503907]},{"page":329,"text":"gradient of velocity (shear) is very strong (shown by dash arrows). No wander that,","rect":[53.812843322753909,220.98641967773438,385.13274808432348,212.05186462402345]},{"page":329,"text":"in the middle of the cell, where the elastic torque is weak, the flow-induced torque","rect":[53.812843322753909,232.94595336914063,385.1288226909813,224.0113983154297]},{"page":329,"text":"prevails and rotates the director to the angles y exceeding 90\u0006.","rect":[53.812843322753909,244.905517578125,305.5967678597141,235.6621856689453]},{"page":329,"text":"It is interesting that, although the same effect is also observed during the director","rect":[65.76598358154297,256.86541748046877,385.1060727676548,247.9308624267578]},{"page":329,"text":"relaxation from the field-on planar texture to the field-off homeotropic one, it is","rect":[53.813961029052737,268.7681884765625,385.1866494570847,259.8336181640625]},{"page":329,"text":"much weaker. Note that, in the first case, we deal with the bend distortion near","rect":[53.813961029052737,278.70574951171877,385.1557859024204,271.79315185546877]},{"page":329,"text":"surfaces (torque MB), but, in the second case, with the splay one (torque MS). In","rect":[53.813961029052737,292.6877746582031,385.1521844010688,283.75274658203127]},{"page":329,"text":"both cases, a strong elastic torque rotates the director, let say, with the same angular","rect":[53.81432342529297,304.6473083496094,385.1064694961704,295.7127685546875]},{"page":329,"text":"rate N. However, due to friction a viscous torque appears, which is exerted by","rect":[53.81432342529297,316.60687255859377,385.16912165692818,307.67230224609377]},{"page":329,"text":"rotating molecules onto adjacent parts of the liquid crystal. The absolute value of","rect":[53.81432342529297,328.56640625,385.1491635879673,319.6318359375]},{"page":329,"text":"the viscous torque related to the bend distortion Mb ¼ 21ðg1 \u0003 g2ÞN ¼ \u0003a2N con-","rect":[53.81432342529297,341.7854309082031,385.11910377351418,330.0440368652344]},{"page":329,"text":"siderably exceeds the torque related to the splay Ms ¼ 12ðg1 þ g2ÞN ¼ a3N because","rect":[53.81411361694336,353.6886291503906,385.1071246929344,342.0036926269531]},{"page":329,"text":"|a2|>>|a3|. Therefore, the backflow is especially important for the initial home-","rect":[53.814083099365237,364.38897705078127,385.12871681062355,355.4542236328125]},{"page":329,"text":"otropic orientation. Note that there is no backflow for the twist distortion that does","rect":[53.8127555847168,376.3485107421875,385.1207314883347,367.4139404296875]},{"page":329,"text":"not change the position of the centers of gravity of the molecules.","rect":[53.8127555847168,388.3080749511719,319.3343166878391,379.37353515625]},{"page":329,"text":"If we are going to discuss the problem of the director relaxation with allowance","rect":[65.76477813720703,400.2676086425781,385.06308782770005,391.33306884765627]},{"page":329,"text":"for the backflow, we should write two equations of motion, one for the director and","rect":[53.8127555847168,412.2271423339844,385.1426629166938,403.2926025390625]},{"page":329,"text":"the other for the mass of liquid [22]. Each of the equations should include coupling","rect":[53.8127555847168,424.1866760253906,385.1048211198188,415.25213623046877]},{"page":329,"text":"terms describing influence of the director motion on the flow and vice versa.","rect":[53.81175994873047,436.1462097167969,385.1804470589328,427.211669921875]},{"page":329,"text":"According to Fig. 11.19, the splay and bend distortions take place in the xz plane","rect":[53.81175994873047,448.10577392578127,385.14063299371568,439.17120361328127]},{"page":329,"text":"and the vector of flow velocity is assumed to have only one component, v ¼ vx(z)","rect":[53.812747955322269,460.008544921875,385.16060767976418,451.073974609375]},{"page":329,"text":"parallel to the substrates because the y -component is forbidden by symmetry","rect":[53.81475067138672,471.9691162109375,385.17348567060005,463.0345458984375]},{"page":329,"text":"and the z-component should vanish according to the mass continuity equation","rect":[53.81475067138672,483.9286804199219,385.1774529557563,474.994140625]},{"page":329,"text":"(divv ¼ 0). Therefore, in the absence of an external force and neglecting the","rect":[53.81475067138672,495.8882141113281,385.1177448101219,486.95367431640627]},{"page":329,"text":"convective term and pressure in the tensor of momentum density flux (9.10), the","rect":[53.81475067138672,507.8477478027344,385.1725238628563,498.9132080078125]},{"page":329,"text":"equation for momentum conservation (9.8) is given by:","rect":[53.81475067138672,519.8072509765625,276.9433732754905,510.87274169921877]},{"page":329,"text":"rqqvtx ¼ qsqz0xz ¼ qqz\u0005AqqWt þ Bqqvzx\u0006","rect":[147.84519958496095,557.8741455078125,291.1625029597856,533.9888916015625]},{"page":329,"text":"(11.68)","rect":[356.0718994140625,550.1876220703125,385.16008888093605,541.6514892578125]},{"page":329,"text":"Here, director n ¼ (sinW \u0007 W, 0, cosW \u0007 1), the term with friction coefficient B is","rect":[65.76631927490235,581.0222778320313,385.18893827544408,572.1873168945313]},{"page":329,"text":"the standard Navier-Stokes terms (9.15) of the type r∂v/∂t ¼ \u0002∂2v/∂z2 (B is","rect":[53.81432342529297,593.3801879882813,385.1318398867722,582.2718505859375]},{"page":330,"text":"318","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.73106384277344]},{"page":330,"text":"11 Optics and Electric Field Effects in Nematic and Smectic A Liquid Crystals","rect":[114.25279998779297,44.274620056152347,385.1677597331933,36.663330078125]},{"page":330,"text":"combination of Leslie coefficients coming from the viscous stress tensor for","rect":[53.812843322753909,68.2883529663086,385.1208737930454,59.35380554199219]},{"page":330,"text":"nematics s0xz (9.20)). The coupling term with friction coefficient A takes into","rect":[53.812843322753909,80.24800872802735,385.1569451432563,70.58845520019531]},{"page":330,"text":"account the influence of the director rotation with angular velocity N ¼ ∂W/∂t on","rect":[53.813106536865237,92.20760345458985,385.16985407880318,82.26704406738281]},{"page":330,"text":"the flow acceleration controlled by Leslie coefficients a2 and a3 (backflow effect).","rect":[53.81209945678711,104.11031341552735,385.1021389534641,95.17576599121094]},{"page":330,"text":"The second equation describes the relaxation of the director","rect":[65.76607513427735,116.0702133178711,307.1188901504673,107.13566589355469]},{"page":330,"text":"qW","rect":[184.04010009765626,137.47433471679688,194.51193589518619,130.30279541015626]},{"page":330,"text":"q2W","rect":[216.44119262695313,137.47433471679688,230.9424321110065,128.31692504882813]},{"page":330,"text":"qvx","rect":[250.82479858398438,138.93109130859376,263.3990873803176,130.30279541015626]},{"page":330,"text":"g1 qt ¼ K qz2 \u0003 A qz","rect":[173.3909912109375,151.17697143554688,261.79474271880346,137.5028533935547]},{"page":330,"text":"(11.69)","rect":[356.071044921875,145.9392547607422,385.15923438874855,137.4031219482422]},{"page":330,"text":"It is the same general equation (9.22) for director motion adapted to our simple","rect":[65.76549530029297,172.69406127929688,385.13144720270005,163.75950622558595]},{"page":330,"text":"situation: we have the familiarform (9.32) for the elastic and viscous torques and, in","rect":[53.81345748901367,184.65362548828126,385.1393975358344,175.7190704345703]},{"page":330,"text":"addition, the coupling term with the same coefficient A describing the torque","rect":[53.81345748901367,196.55636596679688,385.1333698101219,187.62181091308595]},{"page":330,"text":"exerted on the director by shear ∂vx/∂z.","rect":[53.814491271972659,208.51593017578126,214.34124417807346,198.5753631591797]},{"page":330,"text":"Numerical solution of the two equations results in the time variation of the","rect":[65.76567840576172,220.476318359375,385.11475408746568,211.54176330566407]},{"page":330,"text":"velocity vx(z) and director W(z) along the cell thickness as shown in the same","rect":[53.813655853271487,232.43609619140626,385.1287006206688,223.20272827148438]},{"page":330,"text":"Fig. 11.19 for a particular time t. At a certain moment, the angle W in the middle","rect":[53.81275177001953,244.3956298828125,385.1655658550438,235.16226196289063]},{"page":330,"text":"of the cell may cross zero (vertical line) and change sign. In the figure the profile of","rect":[53.811790466308597,256.3551940917969,385.1476987442173,247.42063903808595]},{"page":330,"text":"velocity is antisymmetric. The question arises how this symmetry is consistent with","rect":[53.811790466308597,268.3147277832031,385.1138543229438,259.38018798828127]},{"page":330,"text":"initial symmetry of the cell. The symmetry is indeed broken locally on the scale of","rect":[53.811790466308597,280.2742919921875,385.1457456192173,271.3397216796875]},{"page":330,"text":"one vortex. But in the neighbor area the direction of the director rotation is different","rect":[53.811790466308597,292.1770324707031,385.170546127053,283.24249267578127]},{"page":330,"text":"and the flow velocity has an opposite sign. Therefore the total dynamic symmetry of","rect":[53.811790466308597,304.1365966796875,385.14669166413918,295.2020263671875]},{"page":330,"text":"the whole cell is consistent with boundary conditions. The backflow effect consid-","rect":[53.811790466308597,316.09613037109377,385.10787330476418,307.16156005859377]},{"page":330,"text":"erably influences the dynamic of the director relaxation and this phenomenon is used","rect":[53.811790466308597,328.0556640625,385.12172785810005,319.12109375]},{"page":330,"text":"in bistable displays (Chapter 12). By controlling the velocity of flow using a special","rect":[53.811790466308597,340.01519775390627,385.18049485752177,331.08062744140627]},{"page":330,"text":"form of the applied voltage one can select one of the two final stable field-off states.","rect":[53.81178283691406,351.9747619628906,385.1296658089328,343.04022216796877]},{"page":330,"text":"11.2.7 Electrooptical Response","rect":[53.812843322753909,385.67852783203127,214.7208078715345,375.1244201660156]},{"page":330,"text":"If a nematic liquid crystal has negligible conductivity the results of Sections","rect":[53.812843322753909,413.3045349121094,385.2153359805222,404.3699951171875]},{"page":330,"text":"11.2.1–11.2.5 for the Frederiks transition induced by a magneticfield may be directly","rect":[53.812843322753909,425.2640686035156,385.2542351823188,416.2697448730469]},{"page":330,"text":"applied to the electric field case. To this effect, it suffices to substitute H by E and all","rect":[53.812843322753909,437.2236022949219,385.207411361428,428.2890625]},{"page":330,"text":"components of magnetic susceptibility tensor wij by correspondent components of","rect":[53.81282424926758,450.0999755859375,385.26531349031105,440.24859619140627]},{"page":330,"text":"dielectric permittivity tensor eij. From the practical point of view the electrooptical","rect":[53.8139762878418,462.05963134765627,385.23652513095927,452.20849609375]},{"page":330,"text":"effects are much more important and further on we discuss the optical response of","rect":[53.81309127807617,473.1026916503906,385.2555173477329,464.16815185546877]},{"page":330,"text":"nematics to the electric field.","rect":[53.81309127807617,483.0302734375,168.11129422690159,476.127685546875]},{"page":330,"text":"11.2.7.1 Splay-Bend Distortions","rect":[53.81309127807617,522.9259643554688,195.25249113188938,513.5033569335938]},{"page":330,"text":"We discuss the splay-bend distortion induced by an electric voltage applied toa","rect":[53.81309127807617,546.3898315429688,385.1579974956688,537.455322265625]},{"page":330,"text":"cell similar to that shown in Fig. 11.16 using two transparent electrodes at z ¼0","rect":[53.81309127807617,558.349365234375,385.1788262467719,549.4148559570313]},{"page":330,"text":"and z ¼ d. The distortion is easy to observe optically for the cell birefringence.","rect":[53.8140754699707,570.3089599609375,385.1051907112766,561.3544921875]},{"page":330,"text":"The splay-bend cell behaves like a birefringent plate discussed in Section 11.1.1","rect":[53.8150634765625,582.2684936523438,385.14992610028755,573.333984375]},{"page":330,"text":"but now the plate birefringence is controlled by the field. The optical anisotropy","rect":[53.8150634765625,594.22802734375,385.14995661786568,585.2935180664063]},{"page":331,"text":"11.2 Frederiks Transition and Related Phenomena","rect":[53.812843322753909,42.54875946044922,225.17406604074956,36.68154525756836]},{"page":331,"text":"319","rect":[372.4990539550781,42.62495422363281,385.19064850001259,36.73234558105469]},{"page":331,"text":"Dn(z) ¼ ne \u0003 no depends on the angle of the director W(z), which has a certain","rect":[53.812843322753909,68.2883529663086,385.1176690202094,59.025108337402347]},{"page":331,"text":"distribution over the cell thickness. In the absence of the twist, the splay-bend","rect":[53.81364059448242,80.24788665771485,385.1684197526313,71.31333923339844]},{"page":331,"text":"distortion is limited by the x,z-plane, the ordinary refraction index is independent of","rect":[53.81364059448242,92.20748138427735,385.14949928132668,83.27293395996094]},{"page":331,"text":"W, no ¼ n⊥, but the extraordinary index ne(W) given by Eq. (11.6) becomes a","rect":[53.81364822387695,104.11067962646485,385.16111028863755,94.8768310546875]},{"page":331,"text":"function of position z of the dielectric ellipsoid within the cell thickness.","rect":[53.81330490112305,116.0702133178711,346.6791042854953,107.13566589355469]},{"page":331,"text":"neðzÞ ¼ hn2jjcos2WðzÞnþjjnn??2 sin2WðzÞi1=2","rect":[142.23655700683595,161.36647033691407,294.6397864588197,130.2882843017578]},{"page":331,"text":"(11.70)","rect":[356.0715026855469,143.84226989746095,385.1596921524204,135.36590576171876]},{"page":331,"text":"The corresponding phase retardation is obtained by integrating (11.70) over the","rect":[65.76595306396485,184.82363891601563,385.17566717340318,175.8890838623047]},{"page":331,"text":"cell thickness:","rect":[53.81393051147461,194.76121520996095,111.12031419345925,187.84864807128907]},{"page":331,"text":"d","rect":[180.64144897460938,216.92994689941407,184.12541267952285,212.03538513183595]},{"page":331,"text":"d ¼ 2lp ð ½neðzÞ \u0003 noÞdz ¼ 2pdlhDni","rect":[147.5616912841797,240.910400390625,291.4209329043503,218.07301330566407]},{"page":331,"text":"0","rect":[180.18853759765626,247.26927185058595,183.6725013025697,242.47232055664063]},{"page":331,"text":"(11.71)","rect":[356.072509765625,234.07752990722657,385.16069923249855,225.60116577148438]},{"page":331,"text":"The averaged value of <Dn> and, consequently, d are voltage dependent.","rect":[65.76692962646485,271.8287353515625,385.1696743538547,262.56549072265627]},{"page":331,"text":"Usually a liquid crystal cell equipped with thin tin dioxide (SnO2) or indium-tin-oxide","rect":[53.813899993896487,283.7314758300781,385.1770709819969,274.7770080566406]},{"page":331,"text":"(ITO) electrodes is placed between two crossed polarizers and illuminated by","rect":[53.81432342529297,295.69171142578127,385.16909113935005,286.73724365234377]},{"page":331,"text":"filtered white light or laser light (for example, of a He-Ne laser, l ¼ 632.8 nm)","rect":[53.81432342529297,307.6512451171875,385.16509376374855,298.39794921875]},{"page":331,"text":"and the transmitted intensity is recorded using a photodetector, Fig. 11.20. The","rect":[53.81432342529297,319.6108093261719,385.14820135309068,310.67626953125]},{"page":331,"text":"output light intensity depends on the angle f between the axis of the polarizer and","rect":[53.81433868408203,331.5703430175781,385.1462029557563,322.3070983886719]},{"page":331,"text":"the projection of the director on the cell plane and on the phase retardation d(U)","rect":[53.813350677490237,343.5298767089844,385.15816627351418,334.2965087890625]},{"page":331,"text":"controlled by voltage:","rect":[53.81232833862305,355.4894104003906,142.22561509677957,346.55487060546877]},{"page":331,"text":"I ¼ I0sin22j \u0002 sin2 dðUÞ","rect":[170.67076110839845,386.0747375488281,266.6660500918503,369.74090576171877]},{"page":331,"text":"2","rect":[254.11080932617188,390.8143005371094,259.08791438153755,384.0810241699219]},{"page":331,"text":"The oscillations of I (U) are well seen in the experimental plot, Fig. 11.21. The","rect":[65.76622772216797,414.38079833984377,385.14804876520005,405.44622802734377]},{"page":331,"text":"measurements were made at 27\u0006C on 55 mm thick cell filled with a mixture having","rect":[53.814205169677737,426.3411560058594,385.1505059342719,417.3468322753906]},{"page":331,"text":"ea ¼ 22. From the I (U) curve, the field dependence of the phase retardation d(U)","rect":[53.813655853271487,438.3008117675781,385.1573422989048,429.06744384765627]},{"page":331,"text":"and the Frederiks transition threshold Uc were obtained. In turn, from Ec ¼ Uc/d and","rect":[53.81153106689453,449.8622741699219,385.1467217545844,441.3059997558594]},{"page":331,"text":"Eq. (11.56) the splay elastic constant K11 was found. The bend modulus K33 was","rect":[53.81487274169922,462.2200012207031,385.1518899356003,453.2256774902344]},{"page":331,"text":"calculated from the derivative dd/dU. The same material parameters may be found","rect":[53.81405258178711,474.1796569824219,385.12896052411568,464.9462890625]},{"page":331,"text":"for the whole temperature range of the nematic phase.","rect":[53.814022064208987,486.1391906738281,272.0889858772922,477.20465087890627]},{"page":331,"text":"He-Ne laser","rect":[91.01420593261719,556.0416870117188,132.29836598372686,550.2050170898438]},{"page":331,"text":"Polarizer","rect":[162.01144409179688,524.0,192.72487965560186,518.2002563476563]},{"page":331,"text":"LC cell","rect":[204.61074829101563,524.0689086914063,229.67446662616207,518.2242431640625]},{"page":331,"text":"Analyzer","rect":[243.27899169921876,525.6599731445313,274.0802873704456,518.2242431640625]},{"page":331,"text":"Photodiode or","rect":[301.3660583496094,550.6536254882813,351.1275590989612,544.79296875]},{"page":331,"text":"photomultiplier","rect":[299.4484558105469,561.8231201171875,353.0451616380237,554.3873901367188]},{"page":331,"text":"Fig. 11.20 A typical set-up for electrooptic measurements of liquid crystal physical parameters","rect":[53.812843322753909,593.9761352539063,382.48987277030269,586.246337890625]},{"page":332,"text":"320","rect":[53.8120002746582,42.55765151977539,66.50360626368448,36.73277282714844]},{"page":332,"text":"11 Optics and Electric Field Effects in Nematic and Smectic A Liquid Crystals","rect":[114.251953125,44.276329040527347,385.1669357585839,36.6650390625]},{"page":332,"text":"0","rect":[127.4522933959961,77.67446899414063,131.44892507780674,72.00367736816406]},{"page":332,"text":"U","rect":[113.91010284423828,87.33085632324219,119.68123899277285,81.97200012207031]},{"page":332,"text":"c","rect":[119.68163299560547,89.27225494384766,122.30742001055507,86.34487915039063]},{"page":332,"text":"2","rect":[155.27684020996095,77.58648681640625,159.27347189177159,72.00367736816406]},{"page":332,"text":"4","rect":[181.362060546875,77.58648681640625,185.35869222868565,72.00367736816406]},{"page":332,"text":"d/2p","rect":[217.36959838867188,66.59312438964844,232.7478291745187,60.12250518798828]},{"page":332,"text":"6","rect":[207.2570343017578,77.67446899414063,211.25366598356846,72.00367736816406]},{"page":332,"text":"8","rect":[232.63644409179688,77.67446899414063,236.63307577360752,72.00367736816406]},{"page":332,"text":"10","rect":[255.95199584960938,77.67446899414063,263.94527059782629,72.00367736816406]},{"page":332,"text":"12","rect":[281.8013916015625,77.58648681640625,289.7946663497794,72.00367736816406]},{"page":332,"text":"14","rect":[307.81146240234377,77.58648681640625,315.80473715056066,72.00367736816406]},{"page":332,"text":"1.0","rect":[112.14759063720703,113.96597290039063,122.30702596892002,108.29518127441406]},{"page":332,"text":"103","rect":[328.4365234375,117.72036743164063,339.427161261358,110.4459457397461]},{"page":332,"text":"1.5","rect":[112.14787292480469,145.95700073242188,122.30730825651767,140.2862091064453]},{"page":332,"text":"d","rect":[271.2480163574219,148.04942321777345,275.1966884590508,141.60279846191407]},{"page":332,"text":"102","rect":[328.4365234375,149.81155395507813,339.427161261358,142.53713989257813]},{"page":332,"text":"2.0","rect":[112.14787292480469,179.08673095703126,122.30730825651767,173.4159393310547]},{"page":332,"text":"d = 55 mm","rect":[138.60397338867188,199.4327850341797,175.89254266232283,191.92239379882813]},{"page":332,"text":"2.5","rect":[112.14787292480469,212.59237670898438,122.30730825651767,206.9215850830078]},{"page":332,"text":"0.6 0.8 1.0","rect":[125.99060821533203,218.71246337890626,165.84741598356846,213.0416717529297]},{"page":332,"text":"2","rect":[199.45989990234376,218.62448120117188,203.4565315841544,213.0416717529297]},{"page":332,"text":"3","rect":[221.19517517089845,218.71246337890626,225.19180685270909,213.0416717529297]},{"page":332,"text":"4 5 6 8 10","rect":[239.0249481201172,218.71246337890626,297.19990926970129,213.0416717529297]},{"page":332,"text":"scale U for I (V)","rect":[196.5911102294922,231.88665771484376,253.52552623133588,224.69619750976563]},{"page":332,"text":"101","rect":[328.4375915527344,182.86013793945313,339.427161261358,175.58587646484376]},{"page":332,"text":"1","rect":[328.4375915527344,212.27304077148438,332.434223234545,206.6902313232422]},{"page":332,"text":"15","rect":[313.2168273925781,218.71246337890626,321.2100716232169,213.0416717529297]},{"page":332,"text":"Fig. 11.21 The oscillating experimental curve I(U) (right axis) is voltage dependent intensity of","rect":[53.812843322753909,251.96417236328126,385.2101144180982,244.23435974121095]},{"page":332,"text":"the light transmitted by the 50 mm thick planar nematic cell placed between crossed polarizers (the","rect":[53.81200408935547,261.8724060058594,385.1965880134058,254.2272491455078]},{"page":332,"text":"logarithmic voltage scale for I(U) is the bottom axis). The pointed curve is the voltage dependence","rect":[53.81200408935547,271.8483581542969,385.14415881418707,264.2539978027344]},{"page":332,"text":"of phase retardation d calculated from curve I(U) with a Frederiks transition threshold at Uc (the","rect":[53.81200408935547,281.767578125,385.1964354255152,273.9192199707031]},{"page":332,"text":"scale for d(U) is on the top axis and its argument i.e. voltage is on the left axis)","rect":[53.81356430053711,291.74322509765627,326.08660977942636,283.8948669433594]},{"page":332,"text":"Fig. 11.22 Dynamic of the","rect":[53.812843322753909,382.1600341796875,148.4338469245386,374.4302062988281]},{"page":332,"text":"electrooptical response ofa","rect":[53.812843322753909,392.0682678222656,147.71718737864019,384.4739074707031]},{"page":332,"text":"planar nematic cell: the","rect":[53.812843322753909,402.04425048828127,134.27254626535894,394.44989013671877]},{"page":332,"text":"voltage pulse U (upper plot)","rect":[53.812843322753909,411.9634704589844,150.887543617317,404.3521728515625]},{"page":332,"text":"and the transient intensity of","rect":[53.812843322753909,421.9394226074219,151.55004972083263,414.3450622558594]},{"page":332,"text":"transmitted monochromatic","rect":[53.812843322753909,430.188232421875,147.71549365305425,424.3210144042969]},{"page":332,"text":"light I (lower plot)","rect":[53.812843322753909,441.8345947265625,117.52300351721932,434.2232971191406]},{"page":332,"text":"U","rect":[218.25375366210938,335.89947509765627,224.02488981064395,330.41265869140627]},{"page":332,"text":"I","rect":[220.96026611328126,381.2010498046875,224.06964556172995,375.8501892089844]},{"page":332,"text":"ton","rect":[253.10118103027345,437.6144714355469,262.94058218005196,431.41912841796877]},{"page":332,"text":"toff","rect":[332.205078125,437.6134948730469,342.70426430240817,431.4168395996094]},{"page":332,"text":"t","rect":[370.06982421875,364.4862060546875,372.29195143383677,359.6712341308594]},{"page":332,"text":"t","rect":[379.581787109375,423.96392822265627,381.80391432446177,419.1489562988281]},{"page":332,"text":"Note that the voltage necessary to modulate monochromatic light by 2p (between","rect":[65.76496887207031,490.6171569824219,385.18462458661568,481.6826171875]},{"page":332,"text":"two spikes) is less than 0.1 V. The modulation by p or 2p can also be obtained in","rect":[53.81393051147461,502.5766906738281,385.1428155045844,493.64215087890627]},{"page":332,"text":"dynamics, during switching the field on and off. The oscillograms are shown in","rect":[53.81393051147461,514.5362548828125,385.1408623795844,505.60174560546877]},{"page":332,"text":"Fig. 11.22. By proper selection of the voltage shape and using the dual frequency","rect":[53.81393051147461,526.4957275390625,385.1787041764594,517.5612182617188]},{"page":332,"text":"mode of addressing (for materials with frequency inversion of sign of dielectric","rect":[53.81492614746094,538.455322265625,385.1817096538719,529.5208129882813]},{"page":332,"text":"anisotropy), one can modulate the optical transmission as fast as 1 ms. Of course,","rect":[53.81492614746094,550.4148559570313,385.1278957894016,541.4603881835938]},{"page":332,"text":"solid state modulators are much faster, but we should not forget that a liquid crystal","rect":[53.815921783447269,562.3743896484375,385.1269365079124,553.4398803710938]},{"page":332,"text":"cell may consist of thousands pixels and be controlled by low voltages. Such","rect":[53.815921783447269,574.3339233398438,385.14479914716255,565.3994140625]},{"page":332,"text":"regimes are used for image processing in adaptive optics and other applications.","rect":[53.815921783447269,586.2366943359375,377.0610317757297,577.3021850585938]},{"page":333,"text":"11.2 Frederiks Transition and Related Phenomena","rect":[53.812843322753909,42.54747772216797,225.17406604074956,36.68026351928711]},{"page":333,"text":"321","rect":[372.4990539550781,42.55594253540039,385.19064850001259,36.73106384277344]},{"page":333,"text":"11.2.7.2 Twist and Supertwist Effects","rect":[53.812843322753909,68.2186279296875,219.57532133208469,59.22431945800781]},{"page":333,"text":"Let the director at the two opposite electrodes be aligned along x and y, respec-","rect":[53.812843322753909,92.20748138427735,385.16460548249855,83.27293395996094]},{"page":333,"text":"tively. As discussed in Sections 8.3.2, for strong anchoring at the boundaries, the","rect":[53.81385040283203,104.11019134521485,385.17356146051255,95.17564392089844]},{"page":333,"text":"azimuthal angle j in the bulk changes linearly with z. Then, under condition Dnd/l","rect":[53.81385803222656,116.0697250366211,385.1875035221393,106.80648040771485]},{"page":333,"text":"> 1 such a twisted layer rotates the polarization vector of light of any wavelength l","rect":[53.814815521240237,128.02932739257813,385.1875035221393,118.77603912353516]},{"page":333,"text":"through the angle p/2. This is the waveguide regime already discussed in Sec-","rect":[53.814815521240237,139.98886108398438,385.10982642976418,131.05430603027345]},{"page":333,"text":"tion 11.1.1. With typical values of Dn \u0007 0.15, l \u0007 0.5 mm, this regime takes place","rect":[53.814796447753909,151.94839477539063,385.1237262554344,142.68515014648438]},{"page":333,"text":"for cell thickness d > 3 mm. Therefore, a typical twist cell of thickness about 5 mm","rect":[53.814796447753909,163.907958984375,385.14362286210618,154.91363525390626]},{"page":333,"text":"placed between crossed polarizer and analyser oriented, respectively along x and y,","rect":[53.81375503540039,175.86749267578126,385.1834377815891,166.9329376220703]},{"page":333,"text":"is completely transparent (with some attenuation due to non-ideal polarizers).","rect":[53.81477737426758,187.8270263671875,385.1138577034641,178.89247131347657]},{"page":333,"text":"However, upon application of a voltage, the director is realigned along the field,","rect":[53.81477737426758,199.72976684570313,385.11285062338598,190.7952117919922]},{"page":333,"text":"the twist cell no longer rotates the light polarization, and the outgoing light is","rect":[53.81477737426758,211.6893310546875,385.18649686919408,202.75477600097657]},{"page":333,"text":"completely absorbed by the analyser. For parallel polarizers, on the contrary, the","rect":[53.81477737426758,223.64886474609376,385.17258489801255,214.7143096923828]},{"page":333,"text":"OFF-state is dark and the ON-state bright. It is important that the so-called twist","rect":[53.81477737426758,235.60842895507813,385.12678392002177,226.65394592285157]},{"page":333,"text":"effect is almost insensitive to light wavelength [23].","rect":[53.81477737426758,247.56796264648438,263.7659878304172,238.63340759277345]},{"page":333,"text":"Twist cells are widely used in modern technology of simple, low-informative","rect":[65.76680755615235,259.52752685546877,385.0989154644188,250.5929718017578]},{"page":333,"text":"displays (watches, calculators, telephones, dashboards, etc). Their advantages are","rect":[53.81478500366211,271.4870910644531,385.1646198101219,262.55255126953127]},{"page":333,"text":"high contrast, simplicity and stability. But for high information displays their","rect":[53.81478500366211,283.4466247558594,385.10982642976418,274.5120849609375]},{"page":333,"text":"contrast characteristics are not steep enough. This hampers the application of twist","rect":[53.81478500366211,295.3493957519531,385.2432695157249,286.41485595703127]},{"page":333,"text":"cells to multiplexing schemes. Multiplexed displays use electrodes in the form of","rect":[53.81478500366211,307.3089599609375,385.1506589492954,298.3743896484375]},{"page":333,"text":"the x,y matrix and each pixel is situated on an intersection of the x and y bars. When","rect":[53.81478500366211,319.26849365234377,385.11379328778755,310.33392333984377]},{"page":333,"text":"a selected pixel is activated by voltage U, other pixels along the same x and y bars","rect":[53.81475830078125,331.2280578613281,385.16351713286596,322.29351806640627]},{"page":333,"text":"inevitably activated by voltage U/2 (the so-called cross-talk effect). Therefore, to","rect":[53.813716888427737,343.1875915527344,385.14263239911568,334.2530517578125]},{"page":333,"text":"activate solely one selected pixel, the contrast curve must be steep and the larger the","rect":[53.813697814941409,355.1471862792969,385.1704486675438,346.212646484375]},{"page":333,"text":"number of bars in a matrix the steeper should be the contrast curve. For this reason,","rect":[53.813697814941409,367.1067199707031,385.1236538460422,358.17218017578127]},{"page":333,"text":"the cells with an initial director twist angle larger than p/2, the so-called supertwist","rect":[53.813697814941409,379.0662536621094,385.188368392678,370.1317138671875]},{"page":333,"text":"cells are more preferable for high information content displays. In addition they","rect":[53.813697814941409,390.9690246582031,385.1625603776313,382.0145568847656]},{"page":333,"text":"show better angular characteristics but, unfortunately, they are more sensitive to the","rect":[53.813697814941409,402.9285583496094,385.17246282770005,393.9940185546875]},{"page":333,"text":"cell gap and have longer response times.","rect":[53.813697814941409,414.8880920410156,217.55050321127659,405.95355224609377]},{"page":333,"text":"11.2.7.3 Guest-Host Effect","rect":[53.813697814941409,449.0236511230469,172.69384132234229,441.5134582519531]},{"page":333,"text":"This effect is a version of the splay-bend Frederiks transition, but it is observed in","rect":[53.813697814941409,474.6858215332031,385.1406182389594,465.75128173828127]},{"page":333,"text":"liquid crystals doped with dyes. The liquid crystalline matrix (the host) is subjected","rect":[53.813697814941409,486.6453552246094,385.1207207780219,477.7108154296875]},{"page":333,"text":"to the influence of a field; the function of the dye (the guest) is to enable the effect to","rect":[53.813697814941409,498.6048889160156,385.1993340592719,489.67034912109377]},{"page":333,"text":"be seen with only one polarizer or even without any.","rect":[53.813697814941409,510.5644226074219,266.35796781088598,501.6298828125]},{"page":333,"text":"In the Fig. 11.23a a typical electro-optical cell is shown with a homogeneous","rect":[65.76571655273438,522.52392578125,385.15753568755346,513.5894165039063]},{"page":333,"text":"alignment of a nematic and ea > 0. A small amount (few percents) of a proper dye is","rect":[53.813697814941409,534.431396484375,385.1875344668503,525.4921875]},{"page":333,"text":"dissolved in the liquid crystal. The dye molecules are elongated in shape, and the","rect":[53.813838958740237,546.3909301757813,385.1736530132469,537.4564208984375]},{"page":333,"text":"dipole moment of their long-wave optical transition is parallel to the long molecular","rect":[53.813838958740237,558.3504638671875,385.1537107071079,549.4159545898438]},{"page":333,"text":"axis. In the absence of a field, the optical density of the cell varies with the linear","rect":[53.813838958740237,570.31005859375,385.16365943757668,561.3755493164063]},{"page":333,"text":"polarization of the light e from D|| (e || n) to D⊥ (e ⊥ n). When a voltage exceeding","rect":[53.813838958740237,582.2699584960938,385.17815486005318,573.3350830078125]},{"page":333,"text":"the threshold for the Frederiks transition is applied to the cell, the liquid crystal is","rect":[53.814476013183597,594.2294921875,385.1892129336472,585.2949829101563]},{"page":334,"text":"322","rect":[53.812843322753909,42.55716323852539,66.50444931178018,36.73228454589844]},{"page":334,"text":"11 Optics and","rect":[114.25279998779297,44.275840759277347,161.84293121485636,36.66455078125]},{"page":334,"text":"Fig. 11.23 Guest-host effect.","rect":[53.812843322753909,67.58130645751953,155.3262355048891,59.648292541503909]},{"page":334,"text":"Field-off (a) and field-on (b)","rect":[53.812843322753909,77.15087127685547,152.2472390518873,69.89517211914063]},{"page":334,"text":"cell configurations and","rect":[53.812843322753909,87.4087142944336,131.5954565444462,79.81436157226563]},{"page":334,"text":"absorbance spectra fora","rect":[53.812843322753909,97.3846664428711,136.78548571848394,89.79031372070313]},{"page":334,"text":"nematic liquid crystal doped","rect":[53.812843322753909,107.36067962646485,151.10159820704386,99.76632690429688]},{"page":334,"text":"with a dye having elongated","rect":[53.812843322753909,117.33663177490235,150.5059561172001,109.74227905273438]},{"page":334,"text":"molecules shown by small","rect":[53.812843322753909,127.25582122802735,144.2684450551516,119.66146850585938]},{"page":334,"text":"black spherocylinders","rect":[53.812843322753909,137.23178100585938,128.23641665458,129.63742065429688]},{"page":334,"text":"Electric Field","rect":[164.21456909179688,43.0,210.54230255274698,36.68148422241211]},{"page":334,"text":"a","rect":[177.41258239746095,68.23361206054688,182.96784479469478,62.64485549926758]},{"page":334,"text":"e","rect":[188.6890869140625,91.49161529541016,192.19013626732863,87.58844757080078]},{"page":334,"text":"D","rect":[182.968505859375,145.90606689453126,189.131311912727,140.41123962402345]},{"page":334,"text":"Effects in Nematic","rect":[212.93002319335938,43.0,277.11827990793707,36.68148422241211]},{"page":334,"text":"n0","rect":[229.9393310546875,74.06775665283203,236.93302063635799,68.20343780517578]},{"page":334,"text":"D||(0)","rect":[239.08718872070313,156.5325164794922,257.1955396590703,147.612548828125]},{"page":334,"text":"D^(0)","rect":[220.5942840576172,203.37030029296876,240.0216047225468,195.65594482421876]},{"page":334,"text":"l","rect":[269.7375793457031,222.6124267578125,274.1258809323312,215.9178466796875]},{"page":334,"text":"and Smectic","rect":[279.544921875,43.0,321.5397581794214,36.68148422241211]},{"page":334,"text":"b","rect":[285.54534912109377,68.23361206054688,291.6501428705719,60.92523956298828]},{"page":334,"text":"e","rect":[295.5989685058594,91.49076080322266,299.1000178591255,87.58759307861328]},{"page":334,"text":"D","rect":[290.90313720703127,136.43927001953126,297.0659432603833,130.94444274902345]},{"page":334,"text":"A Liquid Crystals","rect":[323.9554138183594,44.275840759277347,385.1677597331933,36.68148422241211]},{"page":334,"text":"DD","rect":[342.0280456542969,160.795654296875,353.082727928352,154.87692260742188]},{"page":334,"text":"D||(E)","rect":[344.81048583984377,186.7759246826172,364.2522108016484,177.8558349609375]},{"page":334,"text":"D^(E)","rect":[326.228271484375,203.36932373046876,346.9908582625859,195.65496826171876]},{"page":334,"text":"l","rect":[378.1366271972656,222.6124267578125,382.5249287838937,215.9178466796875]},{"page":334,"text":"realigned along the field and so the dye molecules are reoriented. If the field is","rect":[53.812843322753909,263.4971618652344,385.18552030669408,254.56260681152345]},{"page":334,"text":"strong enough, the optical densities for light of both polarizations become the same,","rect":[53.812843322753909,275.3999328613281,385.1258205940891,266.46539306640627]},{"page":334,"text":"Fig. 11.23b. Therefore, for the light polarised along the initial alignment of the","rect":[53.812843322753909,287.3594665527344,385.17063177301255,278.4249267578125]},{"page":334,"text":"director, the field induced decrement of density DD(E) ¼ D||(0) – D⊥(0) is very","rect":[53.812843322753909,299.3194580078125,385.11846247724068,290.0557861328125]},{"page":334,"text":"large. The ratio of the corresponding transmitted light intensities for the field-on","rect":[53.814476013183597,311.27899169921877,385.1304558854438,302.34442138671877]},{"page":334,"text":"and field-off states can be as high as 100. This effect is interesting for colour","rect":[53.814476013183597,323.238525390625,385.1314939102329,314.303955078125]},{"page":334,"text":"displays. For more detailed information about various electrooptical effects and","rect":[53.814476013183597,335.1980895996094,385.1424187760688,326.2635498046875]},{"page":334,"text":"liquid crystal displays and other devices see [24].","rect":[53.814476013183597,347.1576232910156,253.05994840170627,338.22308349609377]},{"page":334,"text":"11.3 Flexoelectricity","rect":[53.812843322753909,395.9030456542969,165.4571050634343,384.3210144042969]},{"page":334,"text":"We know that the quadratic-in-field coupling of an electric field to the dielectric","rect":[53.812843322753909,422.88360595703127,385.17954290582505,413.94903564453127]},{"page":334,"text":"tensor contributes to the free energy density with the term gE ¼ \u0003eaE2=8p. When","rect":[53.812843322753909,435.17193603515627,385.11379328778755,424.3609313964844]},{"page":334,"text":"liquid crystals possess macroscopic electric polarization P (spontaneous or induced","rect":[53.814781188964847,446.7460021972656,385.1267022233344,437.81146240234377]},{"page":334,"text":"by some external, other than electric field factors), then an additional, linear-in-field","rect":[53.81475067138672,458.70556640625,385.13967219403755,449.77099609375]},{"page":334,"text":"term gE ¼ \u0003PE is added to the free energy density. One of such a source of the","rect":[53.81475067138672,470.7049560546875,385.17270696832505,461.7110290527344]},{"page":334,"text":"macroscopic electric polarization is orientational distortion of a liquid crystal.","rect":[53.81291580200195,482.62506103515627,367.95293851401098,473.69049072265627]},{"page":334,"text":"11.3.1 Flexoelectric Polarization","rect":[53.812843322753909,528.9916381835938,224.2493893513596,520.170654296875]},{"page":334,"text":"11.3.1.1 Dipolar and Quadrupolar Flexoelectricity","rect":[53.812843322753909,558.7491455078125,275.4802178483344,549.0974731445313]},{"page":334,"text":"Let us look at Fig. 11.24. In the upper two sketches, we can see undistorted nematic","rect":[53.812843322753909,582.269775390625,385.1008991069969,573.3352661132813]},{"page":334,"text":"liquid crystals with pear- and banana-shape molecules. Such nematics in the bulk","rect":[53.812843322753909,594.2293090820313,385.12380305341255,585.2947998046875]},{"page":335,"text":"11.3 Flexoelectricity","rect":[53.81284713745117,44.275840759277347,124.3316397109501,36.68148422241211]},{"page":335,"text":"Fig. 11.24 Dipolar","rect":[53.812843322753909,67.58130645751953,120.99797147376229,59.85148620605469]},{"page":335,"text":"flexoelectric polarization.","rect":[53.812843322753909,77.4895248413086,140.93393203809223,69.89517211914063]},{"page":335,"text":"Pear-shape and banana-shape","rect":[53.812843322753909,87.4087142944336,154.16112658762456,79.81436157226563]},{"page":335,"text":"molecules in undistorted","rect":[53.812843322753909,95.65752410888672,137.7813467422001,89.79031372070313]},{"page":335,"text":"nematic liquid crystals","rect":[53.812843322753909,107.36067962646485,131.24263461112299,99.76632690429688]},{"page":335,"text":"without any polar axes (a) and","rect":[53.812843322753909,117.33663177490235,155.32877105860636,109.74227905273438]},{"page":335,"text":"appearance of polar axes and","rect":[53.81285095214844,127.25582122802735,153.3463339248173,119.66146850585938]},{"page":335,"text":"flexoelectric polarization","rect":[53.81285095214844,137.23178100585938,138.81866974024698,129.63742065429688]},{"page":335,"text":"along the z-direction in the","rect":[53.81285095214844,147.20773315429688,145.88622424387456,139.61337280273438]},{"page":335,"text":"same nematics due,","rect":[53.81285095214844,155.45655822753907,120.6248423774477,149.58935546875]},{"page":335,"text":"correspondingly, splay and","rect":[53.81285095214844,167.10293579101563,145.92596954493448,159.50857543945313]},{"page":335,"text":"bend distortion (b)","rect":[53.81285095214844,176.74021911621095,117.86399167639901,169.48452758789063]},{"page":335,"text":"a","rect":[221.82244873046876,68.23348999023438,227.37771112770259,62.64473342895508]},{"page":335,"text":"b","rect":[221.98431396484376,149.67636108398438,228.08910771432196,142.36798095703126]},{"page":335,"text":"–","rect":[235.8422088623047,73.0,239.339261583889,71.0]},{"page":335,"text":"+","rect":[235.59742736816407,88.18267822265625,239.58406747077019,84.64143371582031]},{"page":335,"text":"–","rect":[327.111083984375,70.0,330.6081367059593,68.0]},{"page":335,"text":"+","rect":[326.8663024902344,83.0814208984375,330.85294259284049,79.54017639160156]},{"page":335,"text":"Bend","rect":[333.14990234375,134.25869750976563,350.9189248403297,128.7478790283203]},{"page":335,"text":"z","rect":[348.3970642089844,149.67141723632813,351.50644365743309,145.4643096923828]},{"page":335,"text":"Polar axis","rect":[283.1768493652344,219.36373901367188,317.1482283254018,213.73294067382813]},{"page":335,"text":"x","rect":[380.06719970703127,225.1072998046875,384.0638313888419,221.3161163330078]},{"page":335,"text":"are non-polar due to free or partially hindered rotation of molecules (even polar)","rect":[53.812843322753909,258.225830078125,385.18355689851418,249.29127502441407]},{"page":335,"text":"about all their axes. Imagine now that, in the absence of the electric field, the same","rect":[53.812843322753909,270.1285705566406,385.12683904840318,261.19403076171877]},{"page":335,"text":"nematics are subjected to the splay (left) or bend (right) distortions, respectively.","rect":[53.812843322753909,282.088134765625,385.0969204476047,273.153564453125]},{"page":335,"text":"For example, such distortion arises spontaneously in wedge form cells with rigid","rect":[53.812843322753909,294.04766845703127,385.1267937760688,285.11309814453127]},{"page":335,"text":"boundary conditions for the director. For a moment we may forget that molecules","rect":[53.812843322753909,306.0072326660156,385.16171659575658,297.07269287109377]},{"page":335,"text":"have dipoles. Nevertheless, due to the change in symmetry from cylindrical D1h to","rect":[53.812843322753909,317.9667663574219,385.1442498307563,309.0322265625]},{"page":335,"text":"conical C1v (for splay) or to C2v (for bend), in both cases the corresponding polar","rect":[53.81438446044922,329.927001953125,385.1373227676548,320.992431640625]},{"page":335,"text":"axes appear. Their directions are shown by long vertical arrows. It is not surprising","rect":[53.813411712646487,341.88653564453127,385.1035088639594,332.95196533203127]},{"page":335,"text":"because the splay (ndivn) and bend (n curln) distortions are vectors.","rect":[53.813411712646487,353.8460998535156,340.5474514534641,344.91156005859377]},{"page":335,"text":"The new polar symmetry allows for the existence of macroscopic polarization,","rect":[65.76544189453125,365.74884033203127,385.0826687386203,356.81427001953127]},{"page":335,"text":"large or small, depending on the magnitude of the strain and molecular dipole","rect":[53.81342697143555,377.7084045410156,385.12143743707505,368.77386474609377]},{"page":335,"text":"moments shown by small arrows. Due to the distortions, the densest packing of our","rect":[53.81342697143555,389.6679382324219,385.14238868562355,380.7333984375]},{"page":335,"text":"pears and bananas results in some preferable alignment of molecular skeletons in","rect":[53.81342697143555,401.6274719238281,385.13933650067818,392.69293212890627]},{"page":335,"text":"such a way that molecular dipoles look more up than down.By definition, the dipole","rect":[53.81342697143555,413.5870056152344,385.12143743707505,404.6524658203125]},{"page":335,"text":"moment of the unit volume is electric polarization. These simple arguments brought","rect":[53.81342697143555,425.5465393066406,385.13838059970927,416.61199951171877]},{"page":335,"text":"R. Meyer to the brilliant idea of piezoelectric polarization [25]:","rect":[53.81342697143555,437.506103515625,309.5022416836936,428.51177978515627]},{"page":335,"text":"Pf ¼ e1ndivn \u0003 e3ðn curlnÞ","rect":[157.64280700683595,462.2438659667969,281.3376504336472,451.7580261230469]},{"page":335,"text":"(11.72)","rect":[356.0718688964844,460.9714660644531,385.1600583633579,452.4951171875]},{"page":335,"text":"The term piezoelectric was borrowed from the physics of solids by analogy to","rect":[65.76630401611328,485.7421569824219,385.1431817155219,476.8076171875]},{"page":335,"text":"the piezoelectric effect in crystals without center of symmetry. As a rule, the","rect":[53.81427764892578,497.70172119140627,385.1162799663719,488.76715087890627]},{"page":335,"text":"piezoelectric polarization manifests itself as a charge on the surfaces of a crystal","rect":[53.81427764892578,509.6612548828125,385.1233049161155,500.7266845703125]},{"page":335,"text":"due to a translational deformation, e.g. compression or extension. Piezo-effects are","rect":[53.81427764892578,521.6207885742188,385.1641010112938,512.686279296875]},{"page":335,"text":"also characteristic of polar liquid crystalline phases, e.g., of the chiral smectic C*","rect":[53.81427764892578,533.580322265625,385.18007746747505,524.6458129882813]},{"page":335,"text":"phase. The polarization, we are interested now, is caused by the mechanical","rect":[53.81427764892578,545.5398559570313,385.04365403720927,536.6053466796875]},{"page":335,"text":"curvature (or flexion) of the director field, and, following De Gennes, we call it","rect":[53.81427764892578,557.442626953125,385.1700273282249,548.5081176757813]},{"page":335,"text":"flexoelectric.","rect":[53.81427764892578,569.3822631835938,105.3571743538547,560.387939453125]},{"page":335,"text":"In Eq. (11.72) there are two terms related, respectively to the splay and bend","rect":[65.76628875732422,581.3616943359375,385.1381768327094,572.4271850585938]},{"page":335,"text":"distortions with corresponding flexoelectric coefficients e1 and e3. Indeed, the divn","rect":[53.81426239013672,593.3212280273438,385.1869847233112,584.38671875]},{"page":336,"text":"324","rect":[53.812828063964847,42.55850601196289,66.50443405299112,36.73362731933594]},{"page":336,"text":"11 Optics and Electric Field Effects in Nematic and Smectic A Liquid Crystals","rect":[114.2527847290039,44.277183532714847,385.1677597331933,36.6658935546875]},{"page":336,"text":"is scalar and vector Pf (splay) coincides with the director n. In the case of bend, with","rect":[53.812843322753909,69.20487976074219,385.1182793717719,59.35380554199219]},{"page":336,"text":"director components (nx¼1, 0, 0), the curvature vector curln ¼ (∂nx/∂z)j (along y)","rect":[53.813297271728519,80.33765411376953,385.1603025039829,70.30744934082031]},{"page":336,"text":"and n curln ¼ nx(∂nx/∂z)k, therefore vector Pf (bend) is also directed along z, as","rect":[53.814476013183597,93.12413024902344,385.14203275786596,82.26716613769531]},{"page":336,"text":"shown in the same figure. Note, that the twist distortion corresponding to scalar","rect":[53.81417465209961,104.1104965209961,385.08333717195168,95.17594909667969]},{"page":336,"text":"product n\u0002curln cannot create polarization.","rect":[53.81417465209961,116.0699691772461,225.81197782065159,107.13542175292969]},{"page":336,"text":"Now, what would happen if molecules have no dipole moments? Would flex-","rect":[65.76618194580078,128.02957153320313,385.1241696914829,119.09501647949219]},{"page":336,"text":"oelectricity be observed? Generally yes, because in addition to the dipolar mecha-","rect":[53.81415939331055,139.98910522460938,385.0882810196079,131.05455017089845]},{"page":336,"text":"nism, there is, at least, one more, namely, the quadrupolar one. An example is","rect":[53.81415939331055,151.94863891601563,385.18588651763158,143.0140838623047]},{"page":336,"text":"shown in Fig. 11.25: a splay distortion creates additional positive charges at the","rect":[53.81415939331055,163.908203125,385.17493475152818,154.91387939453126]},{"page":336,"text":"bottom of the ensemble of quadrupoles due to an enhanced packing density. The","rect":[53.813167572021487,175.86773681640626,385.1440814800438,166.9331817626953]},{"page":336,"text":"upper part is less positively charged, therefore polarization Pf is directed down [26].","rect":[53.813167572021487,188.74476623535157,385.15520902182348,178.89271545410157]},{"page":336,"text":"A similar difference in the negative charge density will be seen for the bend","rect":[53.81344223022461,199.73104858398438,385.13637629560005,190.79649353027345]},{"page":336,"text":"distortion. Now, if we introduce the density of the quadrupole moment, as a sum","rect":[53.81344223022461,211.69061279296876,385.17121075273118,202.7560577392578]},{"page":336,"text":"of molecular quadrupole moments in a unit volume that is a tensor of quadrupole","rect":[53.81344223022461,223.650146484375,385.1652301616844,214.71559143066407]},{"page":336,"text":"density, see Eq. (10.17), then its gradient is the vector of flexoelectric polarization.","rect":[53.81344223022461,235.60971069335938,385.18023343588598,226.67515563964845]},{"page":336,"text":"_","rect":[335.3392639160156,236.84063720703126,340.3163689713813,235.59556579589845]},{"page":336,"text":"Since this tensor is proportional to the orientational order parameter Q, the quad-","rect":[53.81344223022461,247.56985473632813,385.11129127351418,238.5550079345703]},{"page":336,"text":"rupolar contribution to the flexoelectric polarization (for e ¼ e1 + e3 in the first","rect":[53.81426239013672,259.5294189453125,385.1616044766624,250.59486389160157]},{"page":336,"text":"approximation) is given by:","rect":[53.813777923583987,271.48907470703127,165.52290208408426,262.55450439453127]},{"page":336,"text":"PfQ ¼ erQ_","rect":[197.1255645751953,302.0638427734375,241.82848882630197,287.0]},{"page":336,"text":"(11.73)","rect":[356.0715026855469,299.4311218261719,385.1596921524204,290.95477294921877]},{"page":336,"text":"We already discussed this case in relation to the surface polarization (Section 10.1.3).","rect":[65.76595306396485,325.56243896484377,385.2124905159641,316.62786865234377]},{"page":336,"text":"Generally","rect":[53.81393051147461,337.5220031738281,93.27341548017034,328.58746337890627]},{"page":336,"text":"both","rect":[98.50634002685547,336.0,116.23478785565863,328.58746337890627]},{"page":336,"text":"dipolar","rect":[121.44779968261719,337.5220031738281,149.73766456452979,328.58746337890627]},{"page":336,"text":"and","rect":[154.98155212402345,336.0,169.3852929215766,328.58746337890627]},{"page":336,"text":"quadrupolar","rect":[174.5803985595703,337.5220031738281,222.92699561921729,328.58746337890627]},{"page":336,"text":"mechanisms","rect":[228.10916137695313,336.0,277.5347939883347,328.58746337890627]},{"page":336,"text":"contribute","rect":[282.7139892578125,336.0,323.3032916851219,328.58746337890627]},{"page":336,"text":"to","rect":[328.5392150878906,336.0,336.3134698014594,329.6034240722656]},{"page":336,"text":"Pf","rect":[341.5134582519531,338.4955139160156,349.7290943932709,328.597412109375]},{"page":336,"text":"but","rect":[354.9385681152344,335.4604797363281,367.6899247891624,328.5877685546875]},{"page":336,"text":"the","rect":[372.89495849609377,335.4604797363281,385.1187213726219,328.5877685546875]},{"page":336,"text":"temperature dependence of the corresponding coefficients is different, eq / S(T)","rect":[53.81374740600586,350.39532470703127,385.1246274551548,340.54730224609377]},{"page":336,"text":"for the quadrupolar mechanism, but ed / S2(T) for the dipolar one. The flexoelectric","rect":[53.81368637084961,361.44158935546877,385.12412298395005,350.3906555175781]},{"page":336,"text":"coefficients have the dimension of (CGSQ/cm or C/m) and the order of magnitude,","rect":[53.81315231323242,373.401123046875,385.0992703011203,364.466552734375]},{"page":336,"text":"e ~ 10\u00034 CGS units (or ~3 pC/m). The flexoelectricity is also observed in the SmA","rect":[53.81315231323242,385.36090087890627,385.1491980548176,374.3099365234375]},{"page":336,"text":"phase [27].","rect":[53.81332778930664,397.3204650878906,98.13444180990939,388.38592529296877]},{"page":336,"text":"Fig. 11.25 Quadrupolar","rect":[53.812843322753909,470.241943359375,137.91165250403575,462.5121154785156]},{"page":336,"text":"flexoelectric polarization.","rect":[53.812843322753909,480.09344482421877,140.93393203809223,472.49908447265627]},{"page":336,"text":"Undistorted nematic phase","rect":[53.812843322753909,490.06939697265627,144.80826709055425,482.47503662109377]},{"page":336,"text":"consisted solely of molecular","rect":[53.812843322753909,500.04534912109377,153.79222196204356,492.45098876953127]},{"page":336,"text":"quadrupoles (a) and","rect":[53.812843322753909,510.0213317871094,122.07759613184854,502.4269714355469]},{"page":336,"text":"appearance of a polar axis and","rect":[53.812843322753909,519.9404907226563,155.32877105860636,512.3461303710938]},{"page":336,"text":"flexoelectric polarization due","rect":[53.812843322753909,529.9164428710938,153.45971057199956,522.3220825195313]},{"page":336,"text":"to splay distortion (b). Note","rect":[53.812843322753909,539.8923950195313,149.2410216071558,532.2980346679688]},{"page":336,"text":"that in the lower part of (b)","rect":[53.812835693359378,549.8116455078125,147.71548551184825,542.21728515625]},{"page":336,"text":"the densityofpositive charges","rect":[53.812828063964847,559.78759765625,155.31689913749018,552.1932373046875]},{"page":336,"text":"is larger than in the upper part","rect":[53.812828063964847,569.7635498046875,155.33044151511255,562.169189453125]},{"page":336,"text":"whereas in sketch (a) these","rect":[53.812828063964847,579.40087890625,146.63668200754644,572.1451416015625]},{"page":336,"text":"densities are equal","rect":[53.812828063964847,589.6587524414063,117.07877830710474,582.0643920898438]},{"page":336,"text":"a","rect":[239.3258056640625,467.0965881347656,244.88106806129634,461.5078430175781]},{"page":336,"text":"+++","rect":[252.9275360107422,491.4978942871094,281.33273865591738,485.6942443847656]},{"page":336,"text":"----","rect":[264.02386474609377,498.3741455078125,283.8266431746953,497.10784912109377]},{"page":336,"text":"--","rect":[252.0306854248047,499.9032897949219,260.0894910750859,498.863525390625]},{"page":336,"text":"+++","rect":[252.9275360107422,509.4780578613281,281.33273865591738,503.6752014160156]},{"page":336,"text":"++++","rect":[248.18443298339845,521.4074096679688,287.4360711754486,515.603759765625]},{"page":336,"text":"------","rect":[258.8628234863281,528.0563354492188,289.9299756942265,527.0165405273438]},{"page":336,"text":"--","rect":[247.28758239746095,529.8136596679688,255.34638804774213,528.7738647460938]},{"page":336,"text":"++++","rect":[248.18443298339845,539.3883666992188,287.4360711754486,533.5839233398438]},{"page":336,"text":"+++","rect":[254.62818908691407,548.4854736328125,281.7235772789642,544.3816528320313]},{"page":336,"text":"------","rect":[253.73133850097657,556.8916625976563,284.217451280164,555.79443359375]},{"page":336,"text":"+++","rect":[254.62818908691407,566.4664306640625,281.7235772789642,562.36181640625]},{"page":337,"text":"11.3 Flexoelectricity","rect":[53.81287384033203,44.275596618652347,124.33167022852823,36.68124008178711]},{"page":337,"text":"325","rect":[372.49822998046877,42.55691909790039,385.1898550429813,36.63044357299805]},{"page":337,"text":"11.3.1.2 A Hybrid Cell","rect":[53.812843322753909,68.68677520751953,157.21618516513895,59.035072326660159]},{"page":337,"text":"The director at one of the boundaries of a hybrid cell is aligned homeotropically, at","rect":[53.812843322753909,92.20748138427735,385.1785722500999,83.27293395996094]},{"page":337,"text":"the opposite boundary homogeneously as was shown earlier in Fig. 10.11. There-","rect":[53.812843322753909,104.11019134521485,385.1208432754673,95.17564392089844]},{"page":337,"text":"fore, a hybrid cell has intrinsic bend-splay distortion and must have a projection","rect":[53.812843322753909,116.0697250366211,385.1776055436469,107.13517761230469]},{"page":337,"text":"of the macroscopic polarization along the cell normal. We can clearly see in","rect":[53.812843322753909,128.02932739257813,385.1418084245063,119.09477233886719]},{"page":337,"text":"Figs. 10.11 and 11.25 how the quadrupolar polarization emerges. The molecules","rect":[53.812843322753909,139.98886108398438,385.15964140044408,130.99453735351563]},{"page":337,"text":"may have positive (e0 > 0) quadrupoles shown in Fig. 11.25 or negative ones","rect":[53.812835693359378,151.94937133789063,385.1257668887253,142.95504760742188]},{"page":337,"text":"(e0 < 0) seen in Inset to Fig. 11.26b.","rect":[53.813846588134769,163.908935546875,202.10593839194065,154.97438049316407]},{"page":337,"text":"For a hybrid cell the flexoelectric polarization can easily be calculated. In","rect":[65.7651138305664,175.86846923828126,385.15004817060005,166.9339141845703]},{"page":337,"text":"Fig. 11.26a, the profile of the director is n(z) with boundary conditions W(0) ¼ p/2 and","rect":[53.8130989074707,187.8280029296875,385.14495173505318,178.59463500976563]},{"page":337,"text":"W(d) ¼ 0. These angles are rigidly fixed. The components of the director are nx ¼","rect":[53.8130989074707,199.73077392578126,385.14807918760689,190.49740600585938]},{"page":337,"text":"sinW, ny ¼ 0, nz ¼ cosW. To calculate the polarization we have to find distribution","rect":[53.814231872558597,212.6639862060547,385.17607966474068,202.45745849609376]},{"page":337,"text":"Pf(z) using Meyer’s equation (11.72), and after integrating over z, to obtain total","rect":[53.8133430480957,224.62376403808595,385.15998704502177,214.7160186767578]},{"page":337,"text":"polarization of the cell. In the considered geometry:","rect":[53.81319046020508,235.61013793945313,263.3274064786155,226.6755828857422]},{"page":337,"text":"dW","rect":[259.6613464355469,257.17047119140627,270.4238316471393,249.89932250976563]},{"page":337,"text":"dW","rect":[334.65966796875,257.17047119140627,345.42215318034246,249.89932250976563]},{"page":337,"text":"n ¼ isinWðzÞ þ kcosWðzÞ; divn ¼ \u0003sinW ; curln ¼ cosW j;","rect":[85.87471008300781,266.2166748046875,353.09486329239749,256.2661437988281]},{"page":337,"text":"dz","rect":[260.6239013671875,270.77349853515627,269.4333840762253,263.78125]},{"page":337,"text":"dz","rect":[335.6222229003906,270.77349853515627,344.43170560942846,263.78125]},{"page":337,"text":"dW","rect":[239.9488983154297,281.88336181640627,250.71138352702213,274.6122131347656]},{"page":337,"text":"n curln ¼ ð\u0003icosW þ ksinWÞcosW","rect":[85.87509155273438,290.98529052734377,238.30588120768619,281.0347595214844]},{"page":337,"text":"dz","rect":[240.9114532470703,295.48638916015627,249.72093595610813,288.494140625]},{"page":337,"text":"Then splay and bend polarization contributions are:","rect":[65.7651596069336,318.98712158203127,273.7076249844749,310.05255126953127]},{"page":337,"text":"Psfplay ¼ \u0003e1 sinWddWz \u0002 ðisinW þ kcosWÞ","rect":[138.78028869628907,354.0943603515625,300.1979104434128,333.2201843261719]},{"page":337,"text":"Pbfend ¼ \u0003e3 cosWddWz ð\u0003icosW þ ksinWÞ","rect":[137.30722045898438,388.8963317871094,301.6723672305222,368.02215576171877]},{"page":337,"text":"a","rect":[96.97647094726563,412.1725158691406,102.53173334449947,406.5837707519531]},{"page":337,"text":"x","rect":[118.3582534790039,420.6127014160156,122.35488516081455,416.89349365234377]},{"page":337,"text":"P","rect":[128.43614196777345,452.9700927734375,133.76764863130885,447.4752502441406]},{"page":337,"text":"z","rect":[133.76756286621095,454.96966552734377,136.39334988116054,452.29425048828127]},{"page":337,"text":"RL","rect":[193.50982666015626,462.885009765625,204.58849518892604,457.3901672363281]},{"page":337,"text":"b","rect":[229.53042602539063,412.1725158691406,235.63521977486884,404.8641357421875]},{"page":337,"text":"10","rect":[245.8082275390625,450.95294189453127,252.80233531435776,446.0539855957031]},{"page":337,"text":"5CB","rect":[309.6958923339844,437.40594482421877,322.9077441118435,432.4440002441406]},{"page":337,"text":"n(z)","rect":[125.94757080078125,495.9331359863281,141.43851095427159,488.90264892578127]},{"page":337,"text":"J","rect":[141.88961791992188,506.4729919433594,146.9333671023669,500.8102111816406]},{"page":337,"text":"5","rect":[249.30528259277345,492.5324401855469,252.80233531435776,487.7524719238281]},{"page":337,"text":"0","rect":[123.29158782958985,532.874267578125,127.28821951140049,527.2034912109375]},{"page":337,"text":"d","rect":[157.02476501464845,532.88232421875,161.02139669645909,527.3634643554688]},{"page":337,"text":"z","rect":[179.03500366210938,532.6143798828125,182.14438311055808,528.4312744140625]},{"page":337,"text":"0","rect":[249.30528259277345,532.5569458007813,252.80233531435776,527.657958984375]},{"page":337,"text":"28 30 32 34 36 38","rect":[262.13177490234377,541.4947509765625,338.2689033075218,536.5887451171875]},{"page":337,"text":"T(°C)","rect":[320.2607727050781,551.898193359375,339.8594800887578,544.3958129882813]},{"page":337,"text":"Fig. 11.26 A scheme of a hybrid cell that supports the splay-bend distortion and manifests the","rect":[53.812843322753909,572.6641845703125,385.1550841071558,564.9343872070313]},{"page":337,"text":"flexoelectric polarization (a) and an experimental temperature dependence of the sum of flexo-","rect":[53.812843322753909,582.5723876953125,385.1500558243482,574.97802734375]},{"page":337,"text":"electric coefficients in the nematic phase of liquid crystal 5CB (b)","rect":[53.81285095214844,592.4916381835938,280.43499845130136,584.8464965820313]},{"page":338,"text":"326","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.68026351928711]},{"page":338,"text":"11 Optics and Electric Field Effects in Nematic and Smectic A Liquid Crystals","rect":[114.25279998779297,44.274620056152347,385.1677597331933,36.663330078125]},{"page":338,"text":"Combining the x and z components we obtain projections","rect":[65.76496887207031,68.2883529663086,307.3914834414597,59.35380554199219]},{"page":338,"text":"electric polarization on the x- and z- axes with Pfy ¼ 0:","rect":[53.81295394897461,81.16453552246094,276.15937669345927,71.31333923339844]},{"page":338,"text":"Pfx ¼ \u0007\u0003e1sin2W þ e3cos2W\bddWz","rect":[155.20875549316407,115.35470581054688,282.1495396549518,94.4805908203125]},{"page":338,"text":"of","rect":[311.37615966796877,67.0,319.73272071687355,59.35380554199219]},{"page":338,"text":"the","rect":[323.725341796875,67.0,335.9491351909813,59.35380554199219]},{"page":338,"text":"total flexo-","rect":[340.039306640625,67.0,385.11699806062355,59.35380554199219]},{"page":338,"text":"(11.74a)","rect":[351.65283203125,110.1165542602539,385.12883888093605,101.64019012451172]},{"page":338,"text":"dW","rect":[263.0034484863281,136.55368041992188,273.82266587565496,129.28253173828126]},{"page":338,"text":"Pfz ¼ \u000321ðe1 þ e3Þsin2W dz","rect":[163.5352020263672,150.15664672851563,272.83221830474096,134.83529663085938]},{"page":338,"text":"(11.74b)","rect":[351.08624267578127,144.9185028076172,385.15154395906105,136.3823699951172]},{"page":338,"text":"From (11.74) we see, that the z-component depends on the sum e ¼ e1 + e3 and,","rect":[65.7657241821289,173.65744018554688,385.1424221565891,164.72288513183595]},{"page":338,"text":"for negative e, Pfz should be directed from the homeotropic to planar interface.","rect":[53.814537048339847,186.5341339111328,385.1817593147922,176.6830596923828]},{"page":338,"text":"After integrating over cell thickness the average cell polarization along z is given by","rect":[53.81300735473633,197.57720947265626,385.1707696061469,188.6426544189453]},{"page":338,"text":"d","rect":[124.33611297607422,215.6830596923828,127.82007668098768,210.7884979248047]},{"page":338,"text":"\u000BPfz\f ¼ d1 ð Pfzdz ¼ e14þde3 ðcos2Wd \u0003 cos2W0Þ ¼ \u0003e12þde3 :","rect":[80.83267974853516,239.66326904296876,329.07752930802249,217.54104614257813]},{"page":338,"text":"0","rect":[123.93963623046875,246.07884216308595,127.42359993538222,241.28189086914063]},{"page":338,"text":"(11.75)","rect":[356.07232666015627,232.8307342529297,385.1605161270298,224.23483276367188]},{"page":338,"text":"Therefore, if we could measure the z-component of the polarization of a hybrid","rect":[65.76676177978516,268.59783935546877,385.14959040692818,259.66326904296877]},{"page":338,"text":"cell we find e ¼ e1 + e3. The main problem is screening the polarization by free","rect":[53.81574249267578,280.55804443359377,385.13575018121568,271.622802734375]},{"page":338,"text":"charges. What we do measure is a change in polarization, induced by some external","rect":[53.81283187866211,292.517578125,385.1089006192405,283.5830078125]},{"page":338,"text":"factors, but not polarization itself.","rect":[53.81283187866211,304.4771423339844,190.59861417319065,295.5426025390625]},{"page":338,"text":"11.3.1.3 Measurements of Pf","rect":[53.81283187866211,345.81005859375,181.37143594558087,337.0538330078125]},{"page":338,"text":"It is not difficult to measure the temperature derivative dPf /dT that is pyroelectric","rect":[53.812843322753909,371.1436462402344,385.08664739801255,361.2726745605469]},{"page":338,"text":"coefficient within the temperature range of the nematic phase. Then, integrating it","rect":[53.813472747802737,382.1299133300781,385.16926438877177,373.19537353515627]},{"page":338,"text":"over temperature from the temperature point where Pf ¼ 0 we can find Pf(T). As a","rect":[53.813472747802737,395.06292724609377,385.1614459819969,385.1549072265625]},{"page":338,"text":"zero point, any temperature Ti within the isotropic phase may be taken.","rect":[53.81362533569336,406.04937744140627,341.2171902229953,397.11468505859377]},{"page":338,"text":"We measure pyroelectric coefficient g ¼ dPf/dT, using heating the hybrid cell by","rect":[65.76530456542969,418.925537109375,385.16997614911568,409.0544738769531]},{"page":338,"text":"short (~10 ns) laser pulses, as shown in Fig. 10.13. The only difference from the","rect":[53.813228607177737,429.9685974121094,385.1739887066063,421.0340576171875]},{"page":338,"text":"surface polarization measurements is using a hybrid cell instead of uniform (planar","rect":[53.813228607177737,441.9281311035156,385.15609107820168,432.99359130859377]},{"page":338,"text":"or homeotropic) cells [28]. The laser pulse produces a temperature increment about","rect":[53.813228607177737,453.8876647949219,385.1560502774436,444.953125]},{"page":338,"text":"DT \u0007 0.05 K and the flexoelectric polarization changes. To compensate this change,","rect":[53.81222152709961,465.8471984863281,385.17596097494848,456.5839538574219]},{"page":338,"text":"a charge passes through the external circuit and the current i ¼ dq/dt is measured by","rect":[53.81222152709961,477.7499694824219,385.16899958661568,468.7955017089844]},{"page":338,"text":"an oscilloscope. From the identity (A is cell area)","rect":[53.81222152709961,489.7095031738281,252.67444740144385,480.77496337890627]},{"page":338,"text":"qq","rect":[163.9310302734375,511.139404296875,173.89321223310004,502.0155944824219]},{"page":338,"text":"qP","rect":[194.91603088378907,509.17718505859377,206.01299279607498,502.0155944824219]},{"page":338,"text":"qP qT","rect":[227.0337677001953,509.17718505859377,251.95808762858463,502.0155944824219]},{"page":338,"text":"qT","rect":[278.4676513671875,509.17718505859377,288.94742173502996,502.0155944824219]},{"page":338,"text":"i ¼ qt ¼ A qt ¼ AqT qt ¼ Ag qt","rect":[147.8994140625,522.8897094726563,287.8365312344749,509.1582336425781]},{"page":338,"text":"the polarization is given by","rect":[53.81505584716797,544.349609375,163.97332087567816,535.4151000976563]},{"page":338,"text":"T","rect":[218.99014282226563,564.201171875,222.8364387524901,559.5994262695313]},{"page":338,"text":"PfðTÞ ¼ ð gðTÞdT:","rect":[181.2090301513672,588.3077392578125,258.04539429825686,566.185546875]},{"page":338,"text":"Ti","rect":[217.74427795410157,595.6920776367188,222.99439171587026,589.9234619140625]},{"page":338,"text":"(11.76)","rect":[356.07244873046877,581.4749145507813,385.1606381973423,572.9387817382813]},{"page":339,"text":"11.3 Flexoelectricity","rect":[53.812843322753909,44.274620056152347,124.3316397109501,36.68026351928711]},{"page":339,"text":"327","rect":[372.4981994628906,42.55594253540039,385.1898245254032,36.73106384277344]},{"page":339,"text":"With a short laser pulse, the derivative ∂T/∂t is just a jump, therefore, pyroelec-","rect":[65.76496887207031,68.2883529663086,385.1279233535923,58.34779739379883]},{"page":339,"text":"tric coefficient g(T) can be easily calculated at any temperature of the nematic","rect":[53.811973571777347,80.24788665771485,385.10108221246568,71.31333923339844]},{"page":339,"text":"phase. An example of the Pf(T) dependence is shown in Fig. 11.26b. The maximum","rect":[53.81296920776367,93.12413024902344,385.0944895613249,83.27293395996094]},{"page":339,"text":"value of e for 5CB is \u00033.6\u000210\u00034 CGS (or –12 pC/m). It means that the molecular","rect":[53.81240463256836,104.11067962646485,385.15410743562355,93.05973815917969]},{"page":339,"text":"quadrupole has the form shown in the Inset to the same figure. There are some other","rect":[53.81327438354492,116.0702133178711,385.13726173249855,107.13566589355469]},{"page":339,"text":"methods to find the sum of the flexoelectric coefficients based, e.g., on the electro-","rect":[53.81327438354492,128.02981567382813,385.17803321687355,119.09526062011719]},{"page":339,"text":"optical techniques but they are not as straightforward as the pyroelectric technique.","rect":[53.81327438354492,139.98934936523438,385.1611294319797,131.05479431152345]},{"page":339,"text":"For conventional nematics the order of magnitude \u000710\u00034 CGS of the flexo-","rect":[65.76528930664063,151.94888305664063,385.11846290437355,140.8982391357422]},{"page":339,"text":"electric coefficient is quite reasonable. There are, however, nematics composed of","rect":[53.813472747802737,163.908935546875,385.1473630508579,154.97438049316407]},{"page":339,"text":"bent-shape (banana-like) molecules with transverse dipole moments for which three","rect":[53.813472747802737,175.86846923828126,385.11850774957505,166.9339141845703]},{"page":339,"text":"orders of magnitude larger flexoelectric coefficient has been reported [29]. If such a","rect":[53.813472747802737,187.8280029296875,385.16123235895005,178.89344787597657]},{"page":339,"text":"material is placed between two flexible polymer sheets covered with electrodes and","rect":[53.813472747802737,199.73077392578126,385.14043513349068,190.7962188720703]},{"page":339,"text":"subjected to periodic bending, the current in the range of few nA is observed. The","rect":[53.813472747802737,211.69033813476563,385.14438665582505,202.7557830810547]},{"page":339,"text":"reason for such a giant effect is probably related to the formation of big polar","rect":[53.813472747802737,223.64984130859376,385.13738380281105,214.7152862548828]},{"page":339,"text":"clusters in the nematic phase, that is to a short-range order effect related to the break","rect":[53.813472747802737,235.60940551757813,385.14736262372505,226.6748504638672]},{"page":339,"text":"of quadrupolar symmetry similar to the break of mirror symmetry that discussed in","rect":[53.813472747802737,247.56893920898438,385.14238825849068,238.63438415527345]},{"page":339,"text":"Section 4.11. Whatever mechanism is, the effect may be useful for micro-devices","rect":[53.813472747802737,259.52850341796877,385.13547147856908,250.5939483642578]},{"page":339,"text":"converting mechanical energy in the electric one.","rect":[53.813472747802737,271.488037109375,252.81806607748752,262.553466796875]},{"page":339,"text":"11.3.2 Converse Flexoelectric Effect","rect":[53.812843322753909,327.5240478515625,242.82029935668525,316.9699401855469]},{"page":339,"text":"11.3.2.1 Uniform Distortion","rect":[53.812843322753909,353.3272705078125,178.73224229167057,345.8967590332031]},{"page":339,"text":"As has been shown, the splay and bend distortions of a nematic create electric","rect":[53.812843322753909,379.0691223144531,385.1496967144188,370.13458251953127]},{"page":339,"text":"polarization. There is also a converse effect; the external electric field causesa","rect":[53.812843322753909,390.9718933105469,385.1566852398094,382.037353515625]},{"page":339,"text":"distortion due to the flexoelectric mechanism. For example, if the banana-shape","rect":[53.812843322753909,402.93145751953127,385.1267780132469,393.99688720703127]},{"page":339,"text":"molecules with transverse dipoles are placed in the electric field, the dipoles are","rect":[53.812843322753909,414.8909606933594,385.1626666851219,405.9564208984375]},{"page":339,"text":"partially aligned along the field and their banana shape induces some bend. This","rect":[53.812843322753909,426.85052490234377,385.18353666411596,417.91595458984377]},{"page":339,"text":"effect takes place even in nematics with zero dielectric anisotropy.","rect":[53.812843322753909,438.81005859375,322.8920864632297,429.87548828125]},{"page":339,"text":"Let the director of a nematic liquid crystal be aligned homeotropically (n || z) and","rect":[65.76486206054688,450.7696228027344,385.1447381120063,441.8350830078125]},{"page":339,"text":"the uniform field E || x as shown in Fig. 11.27a. For negative ea, in the absence of","rect":[53.812862396240237,462.7291564941406,385.1511167129673,453.4161071777344]},{"page":339,"text":"the flexoelectric effect, such a situation is stable at any field strength. However, in","rect":[53.814292907714847,474.6897277832031,385.1412285905219,465.75518798828127]},{"page":339,"text":"experiment [30] the bend distortion is observed and its magnitude calculated. For","rect":[53.814292907714847,486.5924987792969,385.15020118562355,477.657958984375]},{"page":339,"text":"zero","rect":[53.814292907714847,497.0,71.00521174481878,491.0]},{"page":339,"text":"anchoring","rect":[76.41532897949219,498.5520324707031,116.4123467545844,489.61749267578127]},{"page":339,"text":"energy","rect":[121.7876205444336,498.5520324707031,149.01836482099066,491.0]},{"page":339,"text":"and","rect":[154.41453552246095,497.0,168.81826106122504,489.61749267578127]},{"page":339,"text":"small","rect":[174.24032592773438,497.0,195.89073045322489,489.61749267578127]},{"page":339,"text":"distortions,","rect":[201.3167724609375,497.0,245.95745511557346,489.61749267578127]},{"page":339,"text":"the","rect":[251.33370971679688,497.0,263.55750311090318,489.61749267578127]},{"page":339,"text":"components","rect":[269.00640869140627,498.5520324707031,317.30426420317846,490.6334533691406]},{"page":339,"text":"of","rect":[322.648681640625,497.0,331.0052426895298,489.61749267578127]},{"page":339,"text":"the","rect":[336.3576354980469,497.0,348.58139837457505,489.61749267578127]},{"page":339,"text":"director","rect":[353.974609375,497.0,385.12130103913918,489.61749267578127]},{"page":339,"text":"are:nz ¼ cosyðzÞ \u0007 1;nx ¼ sinyðzÞ \u0007 yðzÞ and ny ¼ 0, hence,","rect":[53.814292907714847,511.48529052734377,303.4447292854953,500.90032958984377]},{"page":339,"text":"divn ¼ qnz=qz ¼ \u0003sinWqW=qz;","rect":[82.8166275024414,537.970947265625,202.53184449356935,524.8012084960938]},{"page":339,"text":"and","rect":[207.4910125732422,535.0067138671875,221.89475337079535,528.1041259765625]},{"page":339,"text":"ndivn \u0007 \u0003sinWqW=qzk;","rect":[225.27720642089845,537.970947265625,320.74998414200686,524.8012084960938]},{"page":339,"text":"curln ¼ qnx=qzj ¼ qW=qzj; and n curln ¼ k qW=qzj ¼ \u0003qW=qzi:","rect":[90.23637390136719,555.4852905273438,356.1515649525537,542.3156127929688]},{"page":339,"text":"Therefore, for a small distortion, we can leave only the x-component of the total","rect":[65.76435852050781,579.435302734375,385.15913255283427,570.5007934570313]},{"page":339,"text":"flexoelectric polarization (11.72):","rect":[53.81235122680664,591.3948364257813,188.7347093350608,582.4603271484375]},{"page":340,"text":"328","rect":[53.81313705444336,42.55667495727539,66.50474304346963,36.73179626464844]},{"page":340,"text":"11 Optics and Electric Field Effects in Nematic and Smectic A Liquid Crystals","rect":[114.25309753417969,44.275352478027347,385.16806490897457,36.6640625]},{"page":340,"text":"a","rect":[111.25103759765625,68.23361206054688,116.8062999948901,62.64485549926758]},{"page":340,"text":"b","rect":[197.4125518798828,68.23361206054688,203.51734562936103,60.92523956298828]},{"page":340,"text":"ne(z)","rect":[230.642822265625,84.09358978271485,246.1824933944218,76.59918975830078]},{"page":340,"text":"Light","rect":[117.0098876953125,124.56228637695313,135.66616649012495,117.41980743408203]},{"page":340,"text":"z","rect":[181.0990753173828,118.11888122558594,184.2084547658315,113.9357681274414]},{"page":340,"text":"Jm","rect":[316.9520568847656,108.13053131103516,327.7669247227533,102.46773529052735]},{"page":340,"text":"z","rect":[318.0807189941406,126.3770751953125,321.6297279275885,122.79383850097656]},{"page":340,"text":"d/2","rect":[304.2739562988281,163.44915771484376,314.48054891813879,157.75437927246095]},{"page":340,"text":"Fig. 11.27 Converse flexoelectric effect: (a) Structure of the electrooptical cell. (b) Distribution","rect":[53.812843322753909,185.64761352539063,385.1356253066532,177.9178009033203]},{"page":340,"text":"of the director angle over the cell thickness pictured by lower straight lines for zero (solid line) and","rect":[53.813682556152347,195.55584716796876,385.19489044337197,187.94454956054688]},{"page":340,"text":"finite (dot line) anchoring energies, respectively. The upper curves show spatial dependence of two","rect":[53.813716888427737,205.53179931640626,385.1839651504032,197.92050170898438]},{"page":340,"text":"principal refraction indices no (dash line) and ne (z) (solid line)","rect":[53.81455612182617,215.45101928710938,269.7861642227857,207.83932495117188]},{"page":340,"text":"qy","rect":[231.2255096435547,252.80982971191407,241.1797265153266,245.63829040527345]},{"page":340,"text":"Pfx \u0007 e3 qz","rect":[196.1621856689453,266.4128723144531,240.60934843169407,252.96792602539063]},{"page":340,"text":"For ea ¼ 0, the free energy density in the bulk includes only the flexoelectric and","rect":[65.76563262939453,289.9104309082031,385.14202204755318,280.97589111328127]},{"page":340,"text":"the elastic (bend) terms:","rect":[53.8121223449707,301.4715576171875,151.2061143643577,292.9354248046875]},{"page":340,"text":"g ¼ \u0003PfxE þ K233 \u0002 \u0005qqyz\u00062 ¼ K233 \u0002 \u0005qqyz\u00062 \u0003 e3EqqWz","rect":[114.93197631835938,343.90386962890627,322.3600804264362,318.0843811035156]},{"page":340,"text":"(11.77)","rect":[356.07196044921877,336.2163391113281,385.1601499160923,327.739990234375]},{"page":340,"text":"Here we ignore the surface energy (Ws ¼ 0) and the director is free to deflect at","rect":[65.76641082763672,369.37579345703127,385.18195970127177,360.44122314453127]},{"page":340,"text":"both boundaries perpendicular to z. According to Euler equation (8.22), the mini-","rect":[53.814308166503909,381.33648681640627,385.15117774812355,372.40191650390627]},{"page":340,"text":"misation over ∂W/∂z results in the torque balance:","rect":[53.81529998779297,393.2960205078125,255.71653611728739,383.35546875]},{"page":340,"text":"0 \u0003 ddz\u0005K33 qqWz \u0003 e3E\u0006 ¼ 0 or \u0005K33 qqWz \u0003 e3E\u0006 ¼ const","rect":[78.28573608398438,433.4024963378906,331.5612931973655,409.51727294921877]},{"page":340,"text":"(11.78)","rect":[356.070556640625,425.7159729003906,385.15874610749855,417.2396240234375]},{"page":340,"text":"Hence","rect":[65.76500701904297,456.8135681152344,91.23783148004377,450.1400451660156]},{"page":340,"text":"qy e3E","rect":[188.7416229248047,481.81439208984377,226.46157067693435,473.15509033203127]},{"page":340,"text":"¼","rect":[201.4302520751953,485.6764221191406,209.09499386045844,483.3456726074219]},{"page":340,"text":"þ C1;","rect":[228.6764678955078,488.91436767578127,251.9262689320459,480.2686767578125]},{"page":340,"text":"qz","rect":[189.3080291748047,493.9296569824219,198.125446454155,486.76806640625]},{"page":340,"text":"K33","rect":[212.136474609375,495.4177551269531,225.73153755745254,487.2860107421875]},{"page":340,"text":"In zero field ∂W/∂z ¼ 0 everywhere, therefore C1 ¼ 0,","rect":[65.7656021118164,518.9571533203125,285.4805874397922,509.0166320800781]},{"page":340,"text":"y ¼ eK33E3 z and Pf ¼ eK323E3 ¼ const","rect":[143.99343872070313,556.689697265625,294.9129472500999,533.1880493164063]},{"page":340,"text":"(11.79)","rect":[356.0716247558594,550.0170288085938,385.1598142227329,541.5406494140625]},{"page":340,"text":"Here z ¼ 0 is taken in the middle of the layer. The resulting distortion angle is","rect":[65.76605987548828,580.2290649414063,385.1877480898972,571.2945556640625]},{"page":340,"text":"shown in Fig. 11.27b. In the middle of the cell the director keeps its equilibrium","rect":[53.81403732299805,592.1885986328125,385.08917950273118,583.2540893554688]},{"page":341,"text":"11.3 Flexoelectricity","rect":[53.81288528442383,44.276206970214847,124.33168548731729,36.68185043334961]},{"page":341,"text":"329","rect":[372.4982604980469,42.62525939941406,385.18988556055947,36.73265075683594]},{"page":341,"text":"orientation and the maximal deflection angles of the director occur at the restricting","rect":[53.812843322753909,68.2883529663086,385.17559138349068,59.35380554199219]},{"page":341,"text":"surfaces,","rect":[53.812843322753909,78.18606567382813,89.02187772299533,71.31333923339844]},{"page":341,"text":"ym ¼ \u0004e3Ed:","rect":[190.83653259277345,109.63325500488281,248.13108765763185,94.4759750366211]},{"page":341,"text":"2K33","rect":[224.31446838378907,116.44976043701172,242.8950049646791,108.15868377685547]},{"page":341,"text":"Such an antisymmetric distortion differs from the symmetric distortion charac-","rect":[65.76598358154297,139.98916625976563,385.1638425430454,131.0546112060547]},{"page":341,"text":"teristic of the Frederiks transition. It is instructive to compare these two cases. In","rect":[53.81394577026367,151.94869995117188,385.1498955827094,143.01414489746095]},{"page":341,"text":"Fig. 11.28 the space distributions of the director n and its x-projection nx ¼ sinW \u0007 W","rect":[53.81394577026367,163.90823364257813,385.18768662760808,154.67556762695313]},{"page":341,"text":"are pictured for the Fredericks transition (a) and flexoelectric effect (b); the anchor-","rect":[53.81499481201172,175.86846923828126,385.11699806062355,166.9339141845703]},{"page":341,"text":"ing energy at both surfaces is infinitely strong in case (a) and finite in case (b).","rect":[53.816001892089847,187.8280029296875,368.9573635628391,178.89344787597657]},{"page":341,"text":"Note that, in the free energy density expansion (11.77), the flexoelectric term is","rect":[65.76802825927735,199.78756713867188,385.1916848574753,190.85301208496095]},{"page":341,"text":"proportional to the first derivative ∂W/∂z. Therefore, upon integration over the cell","rect":[53.81700897216797,211.69033813476563,385.2086015469749,201.74977111816407]},{"page":341,"text":"thickness, it gives only surface energy terms W(W\u0004d/2). Correspondingly, the torque","rect":[53.816001892089847,223.65057373046876,385.1314166851219,214.41650390625]},{"page":341,"text":"balance (11.78) shows the absence of the flexoelectric torque in the bulk of a cell for","rect":[53.8134880065918,235.61013793945313,385.1801999649204,226.6755828857422]},{"page":341,"text":"the uniform field Ex:","rect":[53.814476013183597,247.13186645507813,136.70628221103739,238.63511657714845]},{"page":341,"text":"ddz\u0005K33 qqWz \u0003 e3Ex\u0006 ¼ K33 dd2zW2","rect":[156.7938690185547,286.3161315917969,282.2063023502643,261.8002624511719]},{"page":341,"text":"(11.80)","rect":[356.0715026855469,278.62921142578127,385.1596921524204,270.1528625488281]},{"page":341,"text":"Evidently, the distortion comes in from the boundaries. It means that weak","rect":[65.76595306396485,309.8623046875,385.1757439713813,300.927734375]},{"page":341,"text":"anchoring of a nematic liquid crystal at the surfaces is a necessary condition for","rect":[53.81393051147461,321.82183837890627,385.1787351211704,312.88726806640627]},{"page":341,"text":"the one-dimensional distortion considered. It is interesting that, for a finite, but","rect":[53.81393051147461,333.724609375,385.13386399814677,324.7900390625]},{"page":341,"text":"weak anchoring, the linear profile of W(z) remains, although the maximum values","rect":[53.81393051147461,345.6841735839844,385.1208840762253,336.4508056640625]},{"page":341,"text":"Wm at the glass surfaces (z ¼ –d/2 and +d/2) reduces. The higher the anchoring","rect":[53.813899993896487,357.6440124511719,385.17815486005318,348.41033935546877]},{"page":341,"text":"energy the smaller is Wm. In experiment this may look like a decrease in an effective","rect":[53.81446838378906,369.6036682128906,385.17102850152818,360.3702087402344]},{"page":341,"text":"flexoelectric coefficient. The profiles of the director angles and refraction indices","rect":[53.813236236572269,381.5632019042969,385.0843850527878,372.628662109375]},{"page":341,"text":"are shown in Fig. 11.27b, the solid and dotted lines for W correspond to Ws ¼ 0 and","rect":[53.813236236572269,393.5227355957031,385.14653864911568,384.28936767578127]},{"page":341,"text":"Ws > 0, respectively.","rect":[53.81368637084961,405.48260498046877,139.4018978645969,396.54803466796877]},{"page":341,"text":"Fig. 11.28 Comparison of the distortion profile (molecular picture below and angle W(z) above)","rect":[53.812843322753909,574.0244750976563,385.17120450598886,566.0745239257813]},{"page":341,"text":"for the Frederiks transition with infinite anchoring energies (a) and flexoelectric effect with finite","rect":[53.812034606933597,583.9326782226563,385.17797229074957,576.3383178710938]},{"page":341,"text":"anchoring energies (b) (homeotropic initial director alignment in both cases)","rect":[53.812034606933597,593.9086303710938,316.19744962317636,586.3142700195313]},{"page":342,"text":"330","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.73106384277344]},{"page":342,"text":"11 Optics and Electric Field Effects in Nematic and Smectic A Liquid Crystals","rect":[114.25279998779297,44.274620056152347,385.1677597331933,36.663330078125]},{"page":342,"text":"11.3.2.2 Electrooptical Properties","rect":[53.812843322753909,68.2186279296875,203.50424589263157,59.035072326660159]},{"page":342,"text":"Due to the linear profile of W(z) it is very easy to calculate the phase retardation of","rect":[53.812843322753909,92.20748138427735,385.1496823867954,82.97412109375]},{"page":342,"text":"the initially homeotropic cell for the normal light incidence, k||z. Without electric","rect":[53.81285095214844,104.11019134521485,385.1506732769188,95.17564392089844]},{"page":342,"text":"field, the longest axis of the dielectric ellipsoid coincides with the director axis z.","rect":[53.8138313293457,116.0697250366211,385.1834988167453,107.13517761230469]},{"page":342,"text":"Therefore, refraction index for any polarization is no ¼ n⊥. With increasing field Ex,","rect":[53.814815521240237,128.02999877929688,385.1832241585422,119.09477233886719]},{"page":342,"text":"due to deflection of the director within plane xz, the y- and x-components of the","rect":[53.814537048339847,139.98953247070313,385.17527044488755,131.0549774169922]},{"page":342,"text":"refraction index will correspond to the ordinary and extraordinary rays, ny ¼ no ¼","rect":[53.814537048339847,152.86582946777345,385.14807918760689,143.01451110839845]},{"page":342,"text":"n⊥, nx(z) ¼ ne(z). Integration provides us with the average extraordinary index:","rect":[53.814231872558597,163.908935546875,373.0204301602561,154.97438049316407]},{"page":342,"text":"d=2","rect":[181.88760375976563,186.1688995361328,192.5652472742494,179.21749877929688]},{"page":342,"text":"<ne> ¼ d1 \u0003dð=2 \u0003n2?sin2Wnþ?nnj2jjjcos2W\u00041=2 dz","rect":[132.88900756835938,219.9744415283203,306.3664590273972,187.2169189453125]},{"page":342,"text":"For small distortions, expanding sinW \u0007 W, cosW \u0007 1\u0003W2/2!, cos2W \u0007 1\u0003W2+.. we","rect":[65.76622772216797,243.48846435546876,385.1708148784813,232.38111877441407]},{"page":342,"text":"obtain","rect":[53.81405258178711,253.32974243164063,78.78916255048284,246.45701599121095]},{"page":342,"text":"<ne>","rect":[127.28209686279297,293.8668212890625,151.26021602842719,286.86767578125]},{"page":342,"text":"¼","rect":[154.01852416992188,290.9219665527344,161.683265955185,288.5912170410156]},{"page":342,"text":"\u0007","rect":[154.0185546875,345.3858337402344,161.68329647276313,341.12274169921877]},{"page":342,"text":"1","rect":[164.61085510253907,285.4450378417969,169.58796015790473,278.7117614746094]},{"page":342,"text":"d","rect":[164.49737548828126,299.28411865234377,169.4744805436469,292.2918701171875]},{"page":342,"text":"1","rect":[164.61085510253907,339.1783752441406,169.58796015790473,332.4450988769531]},{"page":342,"text":"d","rect":[164.49737548828126,352.960693359375,169.4744805436469,345.96844482421877]},{"page":342,"text":"d=2","rect":[176.27972412109376,277.65167236328127,186.9573828943666,270.7002868652344]},{"page":342,"text":"ð","rect":[178.71498107910157,300.822021484375,183.48304772214187,278.6997985839844]},{"page":342,"text":"\u0003d=2","rect":[173.10723876953126,309.4496154785156,189.10977242073379,302.49822998046877]},{"page":342,"text":"d=2","rect":[176.27972412109376,331.3851318359375,186.9573828943666,324.4337463378906]},{"page":342,"text":"ð","rect":[178.71498107910157,354.55535888671877,183.48304772214187,332.4331359863281]},{"page":342,"text":"\u0003d=2","rect":[173.10723876953126,363.1265869140625,189.10977242073379,356.1752014160156]},{"page":342,"text":"n?","rect":[238.0795440673828,286.97540283203127,248.4296504190433,281.0026550292969]},{"page":342,"text":"dz","rect":[297.3870544433594,292.4259948730469,306.1965371523972,285.4337463378906]},{"page":342,"text":"1=2","rect":[284.7552185058594,298.113525390625,295.20640633186658,291.1621398925781]},{"page":342,"text":"1 \u0003 \u00051 \u0003 n2?.n2jj\u0006W2\u000E","rect":[196.50209045410157,317.4906311035156,284.7198904294553,293.60540771484377]},{"page":342,"text":"n? þ n2? \u00051 \u0003 n2?.nj2j\u0006W2\u000Edz","rect":[196.50209045410157,355.4100341796875,311.9735452090378,331.5248107910156]},{"page":342,"text":"Substituting y ¼ e3Ez=K33 from (11.79) (case of zero anchoring energy) and","rect":[65.76510620117188,387.956787109375,385.14348689130318,378.0262145996094]},{"page":342,"text":"integrating we obtain","rect":[53.81361770629883,399.587646484375,139.4705742936469,390.653076171875]},{"page":342,"text":"2 d=2","rect":[229.07302856445313,424.709716796875,248.58755562385879,414.8963623046875]},{"page":342,"text":"<ne> \u0007 n? þ n2?d \u00051 \u0003 n?2.n2jj\u0006\u0005eK33E3\u0006 ð z2dz with","rect":[70.80744934082031,445.81597900390627,297.31926814130318,421.9307556152344]},{"page":342,"text":"d=2","rect":[312.1715087890625,421.8477478027344,322.8491370447572,414.8963623046875]},{"page":342,"text":"ð z2dz ¼ 1d23","rect":[314.60772705078127,444.9619140625,367.0452508073188,421.3000793457031]},{"page":342,"text":"\u0003d=2","rect":[234.73741149902345,453.5892333984375,250.79638741096816,446.6378479003906]},{"page":342,"text":"\u0003d=2","rect":[308.94256591796877,453.5892333984375,325.00154182991346,446.6378479003906]},{"page":342,"text":"Finally the phase retardation d ¼ 2pd(ne\u0003no)/ l:","rect":[65.7661361694336,478.0901184082031,262.74177415439677,468.8374328613281]},{"page":342,"text":"d ¼ 2lp\u0005Ke333\u00062 1 \u0003 nn?2j2j!n?24d3 E2","rect":[149.4300079345703,522.1331176757813,289.08885262092908,492.2716369628906]},{"page":342,"text":"(11.81)","rect":[356.0715026855469,511.4739685058594,385.1596921524204,502.99761962890627]},{"page":342,"text":"The dependencies d / d3 and d / E2 agree well with experiment [30]. Therefore,","rect":[65.76595306396485,545.7111206054688,385.1513332894016,534.6599731445313]},{"page":342,"text":"in principle, we can find e3 from the measured value of the cell retardation because","rect":[53.813472747802737,557.67041015625,385.10315740777818,548.7359008789063]},{"page":342,"text":"usually K33 is known from the Frederiks transition threshold. However, in a real","rect":[53.8131217956543,569.5733032226563,385.1288285977561,560.6387939453125]},{"page":342,"text":"experiment it is almost impossible to have zero anchoring energy. For the finite","rect":[53.81285095214844,581.5330200195313,385.1367877788719,572.5985107421875]},{"page":342,"text":"anchoring energy, we can only find ratio e3/Ws and the accuracy of determination","rect":[53.81285095214844,593.4927978515625,385.1304864030219,584.5580444335938]},{"page":343,"text":"11.3 Flexoelectricity","rect":[53.813350677490237,44.275047302246097,124.33214325098916,36.68069076538086]},{"page":343,"text":"a","rect":[98.27936553955078,68.23306274414063,103.83462793678463,62.64430618286133]},{"page":343,"text":"z=d/2","rect":[131.62896728515626,74.75005340576172,153.89819723356846,68.431396484375]},{"page":343,"text":"W1Ƶ","rect":[183.7761993408203,75.71234130859375,209.0752038583401,68.21781921386719]},{"page":343,"text":"b","rect":[233.90469360351563,68.23306274414063,240.00948735299384,60.92469024658203]},{"page":343,"text":"z=d/2","rect":[250.91482543945313,74.75005340576172,273.1840706466544,68.431396484375]},{"page":343,"text":"W1Ƶ","rect":[301.53094482421877,75.71234130859375,326.8303613290432,68.21733093261719]},{"page":343,"text":"331","rect":[372.49871826171877,42.55636978149414,385.1903433242313,36.73149108886719]},{"page":343,"text":"E=0","rect":[143.5402374267578,127.30520629882813,159.79854208220127,121.63441467285156]},{"page":343,"text":"E","rect":[212.82025146484376,124.7952880859375,218.15175812837917,119.30046081542969]},{"page":343,"text":"E=0","rect":[272.3301086425781,129.82785034179688,288.5883980392325,124.15705871582031]},{"page":343,"text":"E","rect":[330.6793212890625,124.7952880859375,336.0108279525979,119.30046081542969]},{"page":343,"text":"q","rect":[208.52706909179688,153.80116271972657,212.69155930424356,147.9144287109375]},{"page":343,"text":"s","rect":[212.69134521484376,155.62481689453126,214.9994000110894,152.70343017578126]},{"page":343,"text":"q","rect":[314.4090270996094,152.8491973876953,318.5735173120561,146.96246337890626]},{"page":343,"text":"s","rect":[318.572509765625,154.67282104492188,320.88056456187067,151.75143432617188]},{"page":343,"text":"z","rect":[103.52415466308594,174.03936767578126,107.02520401635207,169.7682647705078]},{"page":343,"text":"x","rect":[124.29703521728516,195.0421142578125,128.7093165940041,191.2989044189453]},{"page":343,"text":"W2Æ0","rect":[184.32806396484376,185.03082275390626,208.52405172575596,177.4484405517578]},{"page":343,"text":"z=–d/2","rect":[243.4954833984375,184.0699462890625,270.5606575607169,177.75128173828126]},{"page":343,"text":"W2Æ0","rect":[302.08441162109377,185.03082275390626,326.2780953048575,177.4484405517578]},{"page":343,"text":"Fig. 11.29 Conversed flexoelectric effect in cells with homeotropic (a) and homogeneous (b)","rect":[53.812843322753909,216.59536743164063,385.1711739884107,208.56076049804688]},{"page":343,"text":"director alignment and electric field applied along the cell normal. Weak anchoring energy at the","rect":[53.81200408935547,226.44686889648438,385.1551146247339,218.85250854492188]},{"page":343,"text":"bottom plate allows the flexoelectric deflection of the director Ws at the surface propagating up in","rect":[53.81200408935547,236.42282104492188,385.16721100001259,228.574462890625]},{"page":343,"text":"the vertical direction (ea ¼ 0)","rect":[53.813106536865237,246.0598602294922,155.1185769425123,238.80416870117188]},{"page":343,"text":"of e3 depends on the value of Ws, which varies within several orders of magnitude","rect":[53.812843322753909,265.31097412109377,385.10340154840318,256.37640380859377]},{"page":343,"text":"and is difficult to measure with sufficient accuracy. Another factor that may","rect":[53.81332015991211,277.2137451171875,385.1073845963813,268.2791748046875]},{"page":343,"text":"influence the estimation of flexoelectric coefficients is the surface polarization","rect":[53.81332015991211,289.17327880859377,385.09746638349068,280.23870849609377]},{"page":343,"text":"discussed above. When the dielectric anisotropy is finite the distortion has a more","rect":[53.81332015991211,301.1328430175781,385.1382526226219,292.19830322265627]},{"page":343,"text":"complicated character due to a competition of the dielectric (as for Frederiks","rect":[53.81332015991211,313.0923767089844,385.15216459380346,304.1578369140625]},{"page":343,"text":"transition) and flexoelectric torques [31].","rect":[53.81332015991211,325.05194091796877,218.6182064583469,316.11737060546877]},{"page":343,"text":"The flexoelectric effect can also be observed in other geometries. For example,","rect":[65.76434326171875,337.011474609375,385.1671413948703,328.076904296875]},{"page":343,"text":"the field can be applied along the normal of an electrooptical cell. For a home-","rect":[53.81232833862305,348.97100830078127,385.1283200821079,340.03643798828127]},{"page":343,"text":"otropic cell, the splay flexoelectric distortion shown in Fig. 11.29a is observed for","rect":[53.81232833862305,360.9305419921875,385.17900977937355,351.9959716796875]},{"page":343,"text":"ea \f 0 and weak anchoring energy, at least, at one interface. Another interesting","rect":[53.812313079833987,372.83447265625,385.11199275067818,363.89990234375]},{"page":343,"text":"geometry where the splay flexoelectric distortion is also possible is a planar","rect":[53.81292724609375,384.79400634765627,385.1109555801548,375.85943603515627]},{"page":343,"text":"homogeneous alignment of the director with asymmetric anchoring: it is strong","rect":[53.81292724609375,396.7535705566406,385.1348809342719,387.81903076171877]},{"page":343,"text":"on the top and weak at the bottom, see Fig. 11.29b. In both cases, the surface","rect":[53.81292724609375,408.7131042480469,385.08408392145005,399.778564453125]},{"page":343,"text":"flexoelectric torque is equal to (e1 þ e3)EWs [31].","rect":[53.812957763671878,420.6726379394531,251.75571866537815,411.439697265625]},{"page":343,"text":"11.3.2.3 Dynamics of the Flexoelectric Effect","rect":[53.81325912475586,462.334716796875,251.24906287995948,452.6830139160156]},{"page":343,"text":"Consider the same bend distortions caused by field Ex and shown in Fig. 11.28 and","rect":[53.81325912475586,485.8559875488281,385.14531794599068,476.9208984375]},{"page":343,"text":"assume that distortions are small. What happens if we switched the field off? In the","rect":[53.813472747802737,497.8155212402344,385.17127264215318,488.8809814453125]},{"page":343,"text":"torque balance equation for “Frederiks” distortion (a), we shall have two contribu-","rect":[53.813472747802737,509.7182922363281,385.11849342195168,500.78375244140627]},{"page":343,"text":"tions, elastic and viscous:","rect":[53.81446075439453,519.6558227539063,156.86344773838114,512.7432861328125]},{"page":343,"text":"q2W","rect":[202.16651916503907,545.0662231445313,216.61117905924869,535.9087524414063]},{"page":343,"text":"qW","rect":[240.4586639404297,545.0662231445313,250.93049973795963,537.8946533203125]},{"page":343,"text":"K33 qz2 ¼ g1 qt","rect":[186.36373901367188,558.768798828125,249.6005998379905,545.2134399414063]},{"page":343,"text":"(11.82)","rect":[356.0718078613281,553.5305786132813,385.15999732820168,545.05419921875]},{"page":343,"text":"with general solution W ¼ Wm expt=tsinqz where q ¼ p=d. In this case, we have a","rect":[53.814247131347659,583.2022705078125,385.1609577007469,571.6062622070313]},{"page":343,"text":"relaxation process with a single spatial Fourier harmonic and the characteristic bulk","rect":[53.8141975402832,594.2294921875,385.12722102216255,585.2949829101563]},{"page":344,"text":"332","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.73106384277344]},{"page":344,"text":"11 Optics and Electric Field Effects in Nematic and Smectic A Liquid Crystals","rect":[114.25279998779297,44.274620056152347,385.1677597331933,36.663330078125]},{"page":344,"text":"relaxation","rect":[53.812843322753909,67.0,93.90142909101019,59.35380554199219]},{"page":344,"text":"time","rect":[99.0716552734375,67.0,116.88968694879377,59.35380554199219]},{"page":344,"text":"is","rect":[122.06886291503906,67.0,128.69837583647922,59.35380554199219]},{"page":344,"text":"basically","rect":[133.90740966796876,68.2883529663086,169.43398371747504,59.35380554199219]},{"page":344,"text":"controlled","rect":[174.69180297851563,67.0,215.29799738935004,59.35380554199219]},{"page":344,"text":"by","rect":[220.46026611328126,68.2883529663086,230.41448298505316,59.35380554199219]},{"page":344,"text":"the","rect":[235.58468627929688,67.0,247.80847967340316,59.35380554199219]},{"page":344,"text":"cell","rect":[252.97470092773438,67.0,267.4680314786155,59.35380554199219]},{"page":344,"text":"thickness:","rect":[272.6870422363281,67.0,312.6601701504905,59.35380554199219]},{"page":344,"text":"tbulk","rect":[317.8353271484375,67.74604797363281,334.23822922602076,61.714439392089847]},{"page":344,"text":"¼ g1=K33q2","rect":[337.6618347167969,68.61705017089844,384.70588753303846,57.80612564086914]},{"page":344,"text":"¼ g1d2=p2K33. For the finite anchoring the situation is similar; we should just","rect":[53.812843322753909,81.37022399902344,385.142469955178,70.55923461914063]},{"page":344,"text":"substitute thickness d by d + 2b where b is surface extrapolation length given by","rect":[53.81352615356445,93.00106048583985,385.16826716474068,84.04659271240235]},{"page":344,"text":"b ¼ K33/Ws. For the anchoring energies W s\u0007 10\u00033 – 10\u00032 erg/cm2 (homeotropic","rect":[53.81253433227539,104.90413665771485,385.13999212457505,93.85325622558594]},{"page":344,"text":"alignment), b \u0007 10\u00033 – 10\u00034cm, d þ 2b \u0007 30 mm (3\u000210\u00033 cm), g1 \u0007 1 P, and K33 \u0007","rect":[53.814083099365237,116.86373138427735,385.14807918760689,105.81291198730469]},{"page":344,"text":"10\u00036 dyn, the relaxation time is tbulk ¼ g1/K33q2 \u0007 1 s.","rect":[53.814231872558597,128.82345581054688,276.6319851448703,117.73272705078125]},{"page":344,"text":"Equation (11.82) is valid for the flexoelectric distortion (b) as well, but it does","rect":[65.7659683227539,140.78298950195313,385.12689603911596,131.8484344482422]},{"page":344,"text":"not have a sine-form profile and we cannot expect a simple relaxation process.","rect":[53.81394577026367,152.74252319335938,385.1070217659641,143.80796813964845]},{"page":344,"text":"Moreover,","rect":[53.81394577026367,163.0,95.66642423178439,155.96673583984376]},{"page":344,"text":"this","rect":[100.99889373779297,163.0,115.40263761626437,155.7675323486328]},{"page":344,"text":"distortion","rect":[120.71122741699219,163.0,159.03095332196723,155.7675323486328]},{"page":344,"text":"is","rect":[164.3843231201172,163.0,171.0138284121628,155.7675323486328]},{"page":344,"text":"controlled","rect":[176.3363494873047,163.0,216.94255915692816,155.7675323486328]},{"page":344,"text":"by","rect":[222.2183074951172,164.70208740234376,232.17250910810004,155.7675323486328]},{"page":344,"text":"boundary","rect":[237.51194763183595,164.70208740234376,275.2294549088813,155.7675323486328]},{"page":344,"text":"conditions","rect":[280.5619201660156,163.0,322.1207314883347,155.7675323486328]},{"page":344,"text":"that","rect":[327.5199279785156,163.0,342.5408159024436,155.7675323486328]},{"page":344,"text":"generally","rect":[347.8553771972656,164.70208740234376,385.13784113935005,155.7675323486328]},{"page":344,"text":"include the Rapini-type surface torque, elastic and flexoelectric torques:","rect":[53.81394577026367,176.66162109375,343.90754563877177,167.72706604003907]},{"page":344,"text":"qy","rect":[160.7589111328125,196.63917541503907,170.7131280045844,189.46763610839845]},{"page":344,"text":"K33 qz z¼\u0004d=2 \u0004 Wisy z¼\u0004d=2 \u0003 e3E ¼0","rect":[144.9556884765625,214.9060516357422,294.0375908952094,196.0679931640625]},{"page":344,"text":"(11.83)","rect":[356.07122802734377,205.11424255371095,385.1594174942173,196.63787841796876]},{"page":344,"text":"For free relaxation, the term e3E ¼ 0 but the Rapini terms Ws at each surface","rect":[65.7666244506836,236.40377807617188,385.08545721246568,226.82672119140626]},{"page":344,"text":"dramatically influence the relaxation process. The relaxation time of the director at","rect":[53.81432342529297,248.36334228515626,385.17903001377177,239.4287872314453]},{"page":344,"text":"the surface ts is controlled by the wavevector qs \u0007 p/b. For the same parameters as","rect":[53.81432342529297,260.266357421875,385.1396829043503,251.33152770996095]},{"page":344,"text":"above ts ¼ g1/K33qs2 \u0007 1 – 100 ms that is ts << tbulk and relaxation process starting","rect":[53.81380844116211,272.2260437011719,385.1052178483344,261.17486572265627]},{"page":344,"text":"from the surface propagates into the bulk. When an oscillating field is applied to a","rect":[53.81417465209961,284.1855773925781,385.1580280132469,275.25103759765627]},{"page":344,"text":"cell the waves of the director realignment spread from the boundaries into the bulk","rect":[53.81417465209961,296.1451416015625,385.19680110028755,287.2105712890625]},{"page":344,"text":"[32]. It is very convenient to observe the near-surface director oscillations using","rect":[53.81417465209961,308.10467529296877,385.16997614911568,299.17010498046877]},{"page":344,"text":"total internal reflection technique [33]. With such a technique the flexoelectric","rect":[53.813167572021487,320.0642395019531,385.15198553277818,311.12969970703127]},{"page":344,"text":"effect is observed at frequencies as high as 10 kHz.","rect":[53.812191009521487,332.02374267578127,261.2738918831516,323.08917236328127]},{"page":344,"text":"11.3.3 Flexoelectric Domains","rect":[53.812843322753909,377.08355712890627,208.18212012980147,368.5494384765625]},{"page":344,"text":"There is a very interesting example of the flexoelectric torque acting on the director","rect":[53.812843322753909,406.72955322265627,385.10491309968605,397.79498291015627]},{"page":344,"text":"in the bulk. In a typical planar nematic cell the director is strongly anchored at both","rect":[53.812843322753909,418.6891174316406,385.1248406510688,409.75457763671877]},{"page":344,"text":"interfaces, ns ¼ (1, 0, 0) and the electric field is directed along z. The conductivity is","rect":[53.812843322753909,430.5922546386719,385.1877480898972,421.6573486328125]},{"page":344,"text":"low and the dielectric anisotropy is either zero or small negative, such that the","rect":[53.814048767089847,442.5517883300781,385.17383611871568,433.61724853515627]},{"page":344,"text":"dielectric torque may only weakly stabilize the initial planar structure. Upon the dc","rect":[53.814048767089847,454.5113525390625,385.15097845270005,445.5767822265625]},{"page":344,"text":"field application, a pattern in the form of stripes parallel to the initial director","rect":[53.814048767089847,466.47088623046877,385.10515724031105,457.53631591796877]},{"page":344,"text":"orientation in the bulk n0||x is observed in the polarization microscope. The most","rect":[53.814048767089847,478.43084716796877,385.13703782627177,469.49591064453127]},{"page":344,"text":"interesting feature of these domains is substantial field dependence of their spatial","rect":[53.81313705444336,490.390380859375,385.11018235752177,481.455810546875]},{"page":344,"text":"period as shown in Fig. 11.30 [34].","rect":[53.81313705444336,502.3499450683594,195.50735135580784,493.4154052734375]},{"page":344,"text":"Theperiodofthe stripes andthe threshold voltage fortheir appearancehavebeen","rect":[65.76515197753906,514.3094482421875,385.11614314130318,505.37493896484377]},{"page":344,"text":"found [35] by minimising the free energy of the nematic in an electric field, taking","rect":[53.81313705444336,526.2122192382813,385.1211785416938,517.2179565429688]},{"page":344,"text":"into account the flexoelectric (PfE) and dielectric ea(En)2/4p terms. The solution of","rect":[53.812137603759769,539.1455078125,385.15081153718605,527.1213989257813]},{"page":344,"text":"the torque balance equations for angles j(counted from x within the xy plane) andW","rect":[53.81394577026367,550.1318359375,385.1866185123737,540.8984985351563]},{"page":344,"text":"(counted from x within the xy plane) has been found in the form of equations","rect":[53.81393051147461,562.0913696289063,362.3307229434128,553.1568603515625]},{"page":344,"text":"j ¼ j0 sinðqyÞcosðpz=dÞ;W ¼ W0 cosðqyÞcosðpz=dÞ","rect":[114.42411804199219,586.3904418945313,324.5588418398972,576.399169921875]},{"page":345,"text":"11.3 Flexoelectricity","rect":[53.812843322753909,44.274742126464847,124.3316397109501,36.68038558959961]},{"page":345,"text":"333","rect":[372.4981994628906,42.55606460571289,385.1898245254032,36.73118591308594]},{"page":345,"text":"Fig. 11.30 Flexoelectric instability. Photos of flexoelectric domains with a period variable by","rect":[53.812843322753909,166.14935302734376,385.20847076563759,158.21633911132813]},{"page":345,"text":"electric field (nematic cell thickness 12 mm)","rect":[53.812843322753909,175.98138427734376,204.45457547766856,168.46322631835938]},{"page":345,"text":"The emergence","rect":[65.76496887207031,204.09564208984376,127.18244970514142,195.1610870361328]},{"page":345,"text":"the stripe width at","rect":[53.812950134277347,216.05517578125,126.65387590000222,207.12062072753907]},{"page":345,"text":"and","rect":[53.81421661376953,270.6466369628906,68.21795741132269,263.7440185546875]},{"page":345,"text":"where","rect":[53.81393051147461,330.755859375,78.15197027398908,323.88311767578127]},{"page":345,"text":"and","rect":[53.812843322753909,388.99700927734377,68.21658412030706,382.09442138671877]},{"page":345,"text":"of the pattern has a threshold character.","rect":[129.83026123046876,204.09564208984376,287.5567287972141,195.1610870361328]},{"page":345,"text":"the threshold are given by:","rect":[129.48983764648438,216.05517578125,237.35563523838114,207.12062072753907]},{"page":345,"text":"2pK","rect":[221.8791046142578,235.0688018798828,239.0451380890205,228.26580810546876]},{"page":345,"text":"Uc ¼ jejð1 \u0003 mÞ","rect":[186.5881805419922,251.082275390625,250.7493020449753,235.22708129882813]},{"page":345,"text":"wc ¼ pd \u000511 \u0003þ mm\u00061=2","rect":[180.1321563720703,311.31231689453127,258.3870246179994,284.9839782714844]},{"page":345,"text":"eaK","rect":[220.3496856689453,355.2251281738281,235.30597182925488,347.0580749511719]},{"page":345,"text":"m ¼ 4pe2","rect":[199.8441925048828,367.4441833496094,236.9752814539369,355.84130859375]},{"page":345,"text":"The","rect":[290.1348876953125,203.0,305.6734088726219,195.1610870361328]},{"page":345,"text":"critical","rect":[308.26153564453127,203.0,336.2089982754905,195.1610870361328]},{"page":345,"text":"voltage and","rect":[338.736328125,204.09564208984376,385.1439141373969,195.1610870361328]},{"page":345,"text":"(11.84)","rect":[356.0715026855469,303.62548828125,385.1596921524204,295.1491394042969]},{"page":345,"text":"e ¼ e1 þ e3","rect":[196.1620330810547,414.3641357421875,242.3565528162416,406.9998474121094]},{"page":345,"text":"Therefore, in a nematic with compensated anisotropy, ea \u0007 0, the threshold","rect":[65.76496887207031,438.8108825683594,385.13872614911568,429.8763427734375]},{"page":345,"text":"voltage is controlled exclusively by ratio K/e, therefore the flexoelectric coefficient","rect":[53.813838958740237,450.7704162597656,385.0859514004905,441.83587646484377]},{"page":345,"text":"is easily found from the bulk effect, e \u0007 2\u000210\u00034 CGSE at room temperature [34].","rect":[53.814842224121097,462.7301940917969,381.0765652229953,451.6792297363281]},{"page":345,"text":"It is very peculiar that the spatial distribution of the director field of the","rect":[65.7649917602539,474.6897277832031,385.1169818706688,465.75518798828127]},{"page":345,"text":"modulated structure forms a chiral structure. This became evident much later [36]","rect":[53.81296920776367,486.04168701171877,385.15780006257668,477.7147216796875]},{"page":345,"text":"when the numerical calculations had been made in the same geometry with director","rect":[53.81196212768555,498.60882568359377,385.10503516999855,489.67425537109377]},{"page":345,"text":"components nx \u0007 1, n0y ¼ cosqy, n0z ¼ sinqy. Thus the projections of the director on","rect":[53.81196212768555,511.4850769042969,385.17022028974068,499.2522888183594]},{"page":345,"text":"the zy plane, i.e. n⊥ ¼ ( ny, nz) rotate about the x-axis upon translation along the","rect":[53.814491271972659,523.4449462890625,385.1734088726219,513.5370483398438]},{"page":345,"text":"y-axis. The corresponding picture is demonstrated in Fig. 11.31. The calculations","rect":[53.81369400024414,534.431396484375,385.10275663481908,525.4968872070313]},{"page":345,"text":"show that the chirality changes its handedness when the sign of the electric field","rect":[53.81468200683594,546.3909301757813,385.11968318036568,537.4564208984375]},{"page":345,"text":"applied in the z-direction inverses. Therefore, we again see the field induced break","rect":[53.81468200683594,558.3504638671875,385.1495293717719,549.4159545898438]},{"page":345,"text":"of the mirror symmetry.","rect":[53.81468200683594,570.3099975585938,150.77167935873752,561.37548828125]},{"page":345,"text":"As shown both in experiments and calculations, the domain period w decreases","rect":[65.7667007446289,582.26953125,385.12863554106908,573.3350219726563]},{"page":345,"text":"with increasing voltage approximately as w ~ U –1. This is a very rare or even","rect":[53.81468200683594,594.2296142578125,385.1195000748969,583.1787719726563]},{"page":346,"text":"334","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.73106384277344]},{"page":346,"text":"11 Optics and Electric Field Effects in Nematic and Smectic A Liquid Crystals","rect":[114.25279998779297,44.274620056152347,385.1677597331933,36.663330078125]},{"page":346,"text":"Fig. 11.31 Flexoelectric","rect":[53.812843322753909,67.58130645751953,139.47695300364019,59.648292541503909]},{"page":346,"text":"instability. Periodic structure","rect":[53.812843322753909,77.4895248413086,152.64998767160894,69.89517211914063]},{"page":346,"text":"of the field induced director","rect":[53.812843322753909,85.68157196044922,149.10818570716075,79.81436157226563]},{"page":346,"text":"distribution along the y-axis","rect":[53.812843322753909,97.3846664428711,149.58115084647455,89.79031372070313]},{"page":346,"text":"represented by projections nz","rect":[53.812843322753909,107.36067962646485,152.94924070479363,99.76632690429688]},{"page":346,"text":"and ny","rect":[53.812843322753909,118.0783920288086,75.51225733201545,109.74191284179688]},{"page":346,"text":"unique case: in fact, the “flexoelectric cells” discussed represent diffraction gratings","rect":[53.812843322753909,272.1693115234375,385.1367532168503,263.2347412109375]},{"page":346,"text":"with period controlled by the electric field. Such gratings have been used for","rect":[53.812843322753909,284.07208251953127,385.12383399812355,275.13751220703127]},{"page":346,"text":"processing of optical information.","rect":[53.812843322753909,296.0316467285156,190.19051785971409,287.09710693359377]},{"page":346,"text":"11.4 Electrohydrodynamic Instability","rect":[53.812843322753909,346.4206848144531,254.0713323583562,335.1255187988281]},{"page":346,"text":"In this Section, we shall briefly discuss the electrohydrodynamic (EHD) instabilities","rect":[53.812843322753909,373.4012756347656,385.1437112246628,364.46673583984377]},{"page":346,"text":"of nematics, which are caused by an electric field induced flow of the substance.","rect":[53.812843322753909,385.3608093261719,385.15667386557348,376.42626953125]},{"page":346,"text":"There are many interesting critical phenomena of this sort discussed in detail","rect":[53.812843322753909,397.32037353515627,385.13374192783427,388.38580322265627]},{"page":346,"text":"elsewhere [7,37,38], but here we shall consider in more detail only one but very","rect":[53.812843322753909,409.2799072265625,385.11589900067818,400.3453369140625]},{"page":346,"text":"representative example of the EHD instability owed to the anisotropy of electric","rect":[53.81285095214844,421.2394714355469,385.1516803569969,412.304931640625]},{"page":346,"text":"conductivity.","rect":[53.81285095214844,433.1990051269531,106.3332485726047,424.26446533203127]},{"page":346,"text":"11.4.1 The Reasons for Instabilities","rect":[53.812843322753909,482.94281005859377,238.4555881473796,472.3887023925781]},{"page":346,"text":"Let us take a small volume of a liquid and consider two forces, the gravity force that","rect":[53.812843322753909,510.5688171386719,385.1367631680686,501.63427734375]},{"page":346,"text":"push that volume down and the buoyancy force that push it up. Such a situation","rect":[53.812843322753909,522.5283203125,385.17665949872505,513.5938110351563]},{"page":346,"text":"happens when a liquid is heated from below in a shallow pan: then, with increasing","rect":[53.812843322753909,534.4310913085938,385.0820244889594,525.49658203125]},{"page":346,"text":"temperature, warm bottom layers of the liquid tend to rise but the upper cool layers","rect":[53.812843322753909,546.390625,385.09894193755346,537.4561157226563]},{"page":346,"text":"tend to sink, Fig. 11.32a. Evidently, the two vertical forces (both along the z-axis)","rect":[53.812843322753909,558.3501586914063,385.0998776992954,549.4156494140625]},{"page":346,"text":"counteract and we are tempted to conclude that warm liquid would penetrate","rect":[53.812843322753909,570.3096923828125,385.08994329645005,561.3751831054688]},{"page":346,"text":"through the cold one. In reality, however, a nice steady-state periodic pattern of","rect":[53.812843322753909,582.269287109375,385.14974342195168,573.3347778320313]},{"page":346,"text":"flow is observed in the horizontal plane xy due to up and down vertical streams.","rect":[53.812843322753909,594.228759765625,385.0989040901828,585.2942504882813]},{"page":347,"text":"11.4 Electrohydrodynamic Instability","rect":[53.811988830566409,44.275413513183597,181.42947906641886,36.68105697631836]},{"page":347,"text":"a","rect":[83.47145080566406,68.2331771850586,89.02671322493952,62.64448547363281]},{"page":347,"text":"T","rect":[148.13294982910157,72.648193359375,153.01683374427416,66.9054183959961]},{"page":347,"text":"g","rect":[172.6155242919922,70.94376373291016,177.05977872216563,64.88905334472656]},{"page":347,"text":"b","rect":[226.8331756591797,68.2331771850586,232.93796943287988,60.92488479614258]},{"page":347,"text":"–","rect":[289.1034240722656,70.0,293.10005575407629,68.0]},{"page":347,"text":"335","rect":[372.4973449707031,42.55673599243164,385.1889700332157,36.6302604675293]},{"page":347,"text":"z","rect":[82.19622039794922,113.3910140991211,85.3055998463979,109.18391418457031]},{"page":347,"text":"x","rect":[100.82131958007813,133.91146850585938,104.81795126188877,130.1202850341797]},{"page":347,"text":"d","rect":[148.88560485839845,125.94674682617188,152.88223654020909,120.25996398925781]},{"page":347,"text":"d","rect":[208.6648406982422,102.80130004882813,212.66147238005284,97.11451721191406]},{"page":347,"text":"+","rect":[289.6031188964844,138.24383544921876,294.1592790137485,134.1967010498047]},{"page":347,"text":"Fig. 11.32 A convective instability caused by a temperature gradient","rect":[53.812843322753909,161.04818725585938,299.21520714011259,153.11517333984376]},{"page":347,"text":"namic instability caused by unipolar charge injection (b) in an isotropic","rect":[53.811988830566409,170.9564208984375,299.2540831305933,163.362060546875]},{"page":347,"text":"(a) and","rect":[302.4287414550781,160.6417999267578,327.75522369532509,153.3861083984375]},{"page":347,"text":"liquid","rect":[301.6341857910156,170.9564208984375,321.45847076563759,163.362060546875]},{"page":347,"text":"electrohydrody-","rect":[330.9763488769531,160.98046875,385.14413541419199,153.3861083984375]},{"page":347,"text":"Such a pattern occurs at a critical value of the vertical temperature gradient rTc and","rect":[53.812843322753909,199.78799438476563,385.14626399091255,190.84347534179688]},{"page":347,"text":"has a form of a two-dimensional hexagonal lattice. This is another example of a","rect":[53.814414978027347,211.74752807617188,385.1582721538719,202.81297302246095]},{"page":347,"text":"break of the symmetry of the system caused by convective hydrodynamic instabil-","rect":[53.814414978027347,223.70709228515626,385.1781247696079,214.7725372314453]},{"page":347,"text":"ity, the so-called Benard instability.","rect":[53.814414978027347,235.60986328125,198.48093839194065,226.65538024902345]},{"page":347,"text":"Imagine now that there is a capacitor filled with an insulating liquid and the","rect":[65.76543426513672,247.56939697265626,385.1711810894188,238.6348419189453]},{"page":347,"text":"electric field E is applied along the normal to the capacitor plates. Assume that the","rect":[53.813411712646487,259.5289611816406,385.1721881694969,250.5944061279297]},{"page":347,"text":"lower electrode injects positive charges Q into the liquid, Fig. 11.32b, and there","rect":[53.81240463256836,271.48846435546877,385.1213764019188,262.4742126464844]},{"page":347,"text":"appears the space charge discussed in Section 7.3.3. Then, under the action of the","rect":[53.812435150146487,283.447998046875,385.17322576715318,274.513427734375]},{"page":347,"text":"electric field, the charged liquid layers will be pushed up against the counteracting","rect":[53.81246566772461,295.4075622558594,385.10649958661568,286.4730224609375]},{"page":347,"text":"gravity force like in the previous example. To reduce the energy, the charge layer","rect":[53.81246566772461,307.3670959472656,385.1195310196079,298.43255615234377]},{"page":347,"text":"will not move as an entire block but will be broken into vortices almost cylin-","rect":[53.81246566772461,319.32666015625,385.1015561660923,310.39208984375]},{"page":347,"text":"drically symmetric about the z-axis. That results in a periodic distribution of the","rect":[53.81246566772461,331.2294006347656,385.1721881694969,322.29486083984377]},{"page":347,"text":"space charge within the xy plane. Therefore, one again observes an appearance of","rect":[53.812435150146487,343.1889343261719,385.14824806062355,334.25439453125]},{"page":347,"text":"the convective instability, this time electrohydrodynamic one.","rect":[53.81142807006836,355.14849853515627,303.1146511604953,346.21392822265627]},{"page":347,"text":"In both the cases considered, an optical contrast of the patterns observed in","rect":[65.7634506225586,367.1080322265625,385.13738337567818,358.1734619140625]},{"page":347,"text":"isotropic liquids is very small. Certainly, the anisotropy of liquid crystals brings","rect":[53.81142807006836,379.06756591796877,385.13437284575658,370.13299560546877]},{"page":347,"text":"new features in. For instance, the anisotropy of dielectric or diamagnetic suscepti-","rect":[53.81142807006836,391.0271301269531,385.08956275788918,382.09259033203127]},{"page":347,"text":"bility causes the Fredericks transition in nematics and wave like instabilities in","rect":[53.81142807006836,402.9866943359375,385.1373223405219,394.0521240234375]},{"page":347,"text":"cholesterics (see next Section), and the flexoelectric polarization results in the field-","rect":[53.81142807006836,414.9462585449219,385.14129005281105,406.01171875]},{"page":347,"text":"controllable domain patterns. In turn, the anisotropy of electric conductivity is","rect":[53.81142807006836,426.8490295410156,385.12546171294408,417.91448974609377]},{"page":347,"text":"responsible for instability in the form of rolls to be discussed below. All these","rect":[53.81142807006836,438.8085632324219,385.1254047222313,429.8740234375]},{"page":347,"text":"instabilities are not observed in the isotropic liquids and have an electric field","rect":[53.81142807006836,450.7680969238281,385.11245051435005,441.83355712890627]},{"page":347,"text":"threshold controlled by the corresponding parameters of anisotropy. In addition,","rect":[53.81142807006836,462.7276611328125,385.1473354866672,453.7930908203125]},{"page":347,"text":"due to the optical anisotropy, the contrast of the patterns that are driven by","rect":[53.81142807006836,474.6872253417969,385.1672295670844,465.752685546875]},{"page":347,"text":"“isotropic mechanisms”, i.e. only indirectly dependent on anisotropy parameters,","rect":[53.81142807006836,486.64678955078127,385.1164822151828,477.71221923828127]},{"page":347,"text":"increases dramatically. Thanks to this, one can easily study specific features and","rect":[53.81142807006836,498.6063537597656,385.14040461591255,489.67181396484377]},{"page":347,"text":"mechanisms of different instability modes, both isotropic and anisotropic. The","rect":[53.81142807006836,510.5658874511719,385.14136541559068,501.63134765625]},{"page":347,"text":"characteristic pattern formation is a special branch of physics dealing witha","rect":[53.81142807006836,522.4686279296875,385.1572650737938,513.5341186523438]},{"page":347,"text":"nonlinear response of dissipative media to external fields, and liquid crystals are","rect":[53.81142807006836,534.4281005859375,385.1602252788719,525.4935913085938]},{"page":347,"text":"suitable model objects for investigation of the relevant phenomena [39].","rect":[53.81142807006836,546.3876953125,344.93523831869848,537.4531860351563]},{"page":347,"text":"Assume that our capacitor is filled by a nematic mixture with ea \u0007 0 well aligned","rect":[65.76342010498047,558.3507080078125,385.1000908952094,549.4126586914063]},{"page":347,"text":"along the x-axis and let the same charge injection mechanism works. Then, in a dc","rect":[53.814083099365237,570.3102416992188,385.1529010601219,561.375732421875]},{"page":347,"text":"regime, the periodic flow will inevitably interact with the director. The maximum","rect":[53.81505584716797,582.269775390625,385.0942149031218,573.3352661132813]},{"page":347,"text":"realignment, i.e. the deflection of the director angle W in the z-direction, will be","rect":[53.81505584716797,594.2293090820313,385.15589178277818,584.9959716796875]},{"page":348,"text":"336","rect":[53.812835693359378,42.55752944946289,66.50444168238565,36.68185043334961]},{"page":348,"text":"11 Optics and Electric Field Effects in Nematic and Smectic A Liquid Crystals","rect":[114.25279235839844,44.276206970214847,385.1677597331933,36.6649169921875]},{"page":348,"text":"observed where the shear rate has maximum, namely, in the middle of the vortices","rect":[53.812843322753909,68.2883529663086,385.11288847075658,59.35380554199219]},{"page":348,"text":"as shown by thick lines in Fig. 11.33a. On the contrary, the velocity is maximal","rect":[53.812843322753909,80.24788665771485,385.15962083408427,71.29341888427735]},{"page":348,"text":"where the space charge is accumulated. Such a mechanismof the director alignment","rect":[53.81285095214844,92.20748138427735,385.17060716220927,83.27293395996094]},{"page":348,"text":"is especially strong when the anisotropy of electric conductivity sa ¼ sjj \u0003 s? is","rect":[53.81285095214844,105.51209259033203,385.1891213809128,95.17564392089844]},{"page":348,"text":"high. The conductivity induced torque My may even exceed the dielectric torque if","rect":[53.81450653076172,117.0567855834961,385.1795896133579,107.13566589355469]},{"page":348,"text":"dielectric anisotropy ea is not very strong. It is the torque, which is responsible, for","rect":[53.81386947631836,128.02999877929688,385.17916236726418,119.09544372558594]},{"page":348,"text":"example, for alignment of the director in the nematic phase and smectic layers in","rect":[53.813419342041019,139.98953247070313,385.1393670182563,131.0549774169922]},{"page":348,"text":"the SmA phase (in both cases along the flow lines) shown earlier in Fig. 5.16. The","rect":[53.813419342041019,151.94906616210938,385.14426458551255,142.95474243164063]},{"page":348,"text":"same torque described by Carr et al [40] is responsible for the Carr-Helfrich","rect":[53.813419342041019,163.90863037109376,385.11541071942818,154.9740753173828]},{"page":348,"text":"instability. The latter is also driven by the space charge, however, accumulated","rect":[53.81342697143555,175.8681640625,385.14531794599068,166.93360900878907]},{"page":348,"text":"due to anisotropy of conductivity in the bulk of the nematic without any injection.","rect":[53.81342697143555,187.82766723632813,385.0686306526828,178.87318420410157]},{"page":348,"text":"11.4.2 Carr-Helfrich Mode","rect":[53.812843322753909,237.85494995117188,195.92292579145639,227.30084228515626]},{"page":348,"text":"This mode is observed at the ac current at frequencies not exceeding the inverse of","rect":[53.812843322753909,265.4809265136719,385.1467221817173,256.54638671875]},{"page":348,"text":"the space charge relaxation time oq ¼ 1/tq ¼ 4ps/e. When a sine-form electric field","rect":[53.812843322753909,278.29730224609377,385.1183709245063,268.50592041015627]},{"page":348,"text":"is applied to homogeneously oriented fairly conductive nematics with negative","rect":[53.81438446044922,289.400146484375,385.1721881694969,280.465576171875]},{"page":348,"text":"dielectric anisotropy, a very regular vortex motion is often observed. In fact, such","rect":[53.81438446044922,301.35968017578127,385.12542048505318,292.42510986328127]},{"page":348,"text":"vortices have a form of long rolls perpendicular to the initial alignment of the","rect":[53.81438446044922,313.3192443847656,385.17417181207505,304.38470458984377]},{"page":348,"text":"director. They are usually called Williams domains [41], see photo in Fig. 11.33b.","rect":[53.81438446044922,325.2787780761719,385.1830715706516,316.34423828125]},{"page":348,"text":"The instability appears in thin cells (d ¼ 10 – 100 mm) and has a well-defined","rect":[53.81438446044922,337.2383117675781,385.1531914811469,328.2838439941406]},{"page":348,"text":"voltage threshold independent of thickness. Upon illumination, the rolls behave like","rect":[53.814369201660159,349.1410827636719,385.1372760601219,340.20654296875]},{"page":348,"text":"lenses: they form a diffraction grating and focus light onto the screen, Fig. 11.34.","rect":[53.814369201660159,361.10064697265627,382.29534574057348,352.16607666015627]},{"page":348,"text":"a","rect":[62.649497985839847,423.16827392578127,68.20481604615574,417.57952880859377]},{"page":348,"text":"b","rect":[265.2217712402344,423.16827392578127,271.3266261590348,415.8598937988281]},{"page":348,"text":"–v","rect":[74.67745971679688,461.8158264160156,82.22309645786192,458.0246276855469]},{"page":348,"text":"z","rect":[82.22269439697266,464.18133544921877,84.55472898330916,461.0260009765625]},{"page":348,"text":"v","rect":[116.08087158203125,486.5486755371094,119.6298805154791,482.7574768066406]},{"page":348,"text":"z","rect":[119.62947082519531,488.9141845703125,121.96150541153182,485.75885009765627]},{"page":348,"text":"E","rect":[229.960693359375,457.5869140625,235.2922000229104,452.2360534667969]},{"page":348,"text":"z","rect":[235.29208374023438,460.04840087890627,237.62411832657089,456.89306640625]},{"page":348,"text":"q (x)","rect":[230.7310791015625,485.35894775390627,246.214033311414,478.29644775390627]},{"page":348,"text":"z","rect":[64.78612518310547,509.0650634765625,67.89550463155415,504.8579406738281]},{"page":348,"text":"y","rect":[264.9757995605469,523.1780395507813,268.5248084939947,517.8511962890625]},{"page":348,"text":"x","rect":[100.86372375488281,541.6381225585938,104.86035543669346,537.846923828125]},{"page":348,"text":"x","rect":[303.4433898925781,543.8880615234375,307.44002157438879,540.0968627929688]},{"page":348,"text":"20 mm","rect":[346.0341491699219,554.1177978515625,366.8475430765849,546.9913330078125]},{"page":348,"text":"Fig. 11.33 Carr-Helfrich EHD instability in nematic liquid crystals: (a) onset of the instability","rect":[53.812843322753909,574.0244750976563,385.13986725001259,566.0914916992188]},{"page":348,"text":"showing a competition of the elastic and hydrodynamic torques; (b) photo of Williams domains","rect":[53.812828063964847,583.9326782226563,385.1847275066308,576.3383178710938]},{"page":348,"text":"observed at a voltage 7.5 V in a 20 mm thick cell filled with liquid crystal MBBA","rect":[53.812843322753909,593.9086303710938,333.32346130949466,586.2634887695313]},{"page":349,"text":"11.4 Electrohydrodynamic Instability","rect":[53.812843322753909,44.274986267089847,181.43033355860636,36.68062973022461]},{"page":349,"text":"Fig. 11.34 Roll-type vortex","rect":[53.812843322753909,67.58130645751953,151.27252716212198,59.648292541503909]},{"page":349,"text":"motion of a liquid crystal and","rect":[53.812843322753909,77.4895248413086,155.2720693984501,69.89517211914063]},{"page":349,"text":"the pattern of black and white","rect":[53.812843322753909,87.4087142944336,155.30085131906987,79.81436157226563]},{"page":349,"text":"stripes in the screen plane due","rect":[53.812843322753909,97.3846664428711,155.32875964426519,89.79031372070313]},{"page":349,"text":"to diffraction on the roll","rect":[53.812843322753909,105.63353729248047,136.4224977895266,99.76632690429688]},{"page":349,"text":"structure","rect":[53.812843322753909,115.58409118652344,83.48581836008549,110.60585021972656]},{"page":349,"text":"337","rect":[372.4981994628906,42.55630874633789,385.1898245254032,36.73143005371094]},{"page":349,"text":"11.4.2.1 The Instability Threshold in the Simplest Model","rect":[53.812843322753909,224.7291259765625,302.3237138516624,215.3065185546875]},{"page":349,"text":"The physical mechanism of the instability is related to several coupled phenomena","rect":[53.812843322753909,248.24981689453126,385.0840228862938,239.3152618408203]},{"page":349,"text":"discussed by Helfrich [42]. His elegant calculation of the instability threshold is","rect":[53.812843322753909,260.2093811035156,385.1835061465378,251.2748260498047]},{"page":349,"text":"reproduced here for the simplest steady state one-dimensional model shown in","rect":[53.81184387207031,272.1689147949219,385.1387871842719,263.234375]},{"page":349,"text":"Fig. 11.33a. A homogeneously aligned nematic liquid crystal layer of thicknessd","rect":[53.81184387207031,284.0716857910156,385.17751399091255,275.1172180175781]},{"page":349,"text":"is stabilised by the rubbed surfaces of the limiting glasses. The dielectric torque is","rect":[53.81280517578125,296.03125,385.18451322661596,287.0966796875]},{"page":349,"text":"considered negligible (ea ¼ 0). At first, a small director fluctuation W(x) with a","rect":[53.81280517578125,307.9907531738281,385.1621784038719,298.7582092285156]},{"page":349,"text":"period wx \u0007 d is postulated:","rect":[53.81437683105469,319.95123291015627,166.54588182041239,310.99676513671877]},{"page":349,"text":"px","rect":[242.9510498046875,338.70123291015627,252.89530218316879,334.1692199707031]},{"page":349,"text":"WðxÞ ¼ Wm cos","rect":[184.266357421875,347.83660888671877,241.0857736025925,337.8860778808594]},{"page":349,"text":"wx","rect":[242.78082275390626,353.75146484375,252.5799345482864,347.7622985839844]},{"page":349,"text":"(11.85)","rect":[356.0715026855469,347.099609375,385.1596921524204,338.50372314453127]},{"page":349,"text":"With the field applied, this fluctuation causes a slight periodic deflection of the","rect":[65.76595306396485,377.2549743652344,385.1737445659813,368.3204345703125]},{"page":349,"text":"electric current lines along the director proportional to the anisotropy of conductiv-","rect":[53.81393051147461,389.21453857421877,385.13485084382668,380.27996826171877]},{"page":349,"text":"ity sa ¼ sjj \u0003 s?>0. This creates the x-component of the current that, in turn,","rect":[53.81393051147461,402.5195617675781,385.1607327034641,392.239501953125]},{"page":349,"text":"results in the accumulation of a space charge Q(x) close to the points where angle","rect":[53.81389236450195,413.1343688964844,385.1417926616844,404.1201171875]},{"page":349,"text":"W ¼ 0. Therefore, the x-component of the field (Ex) emerges. The electric current","rect":[53.81391525268555,425.0941467285156,385.1743608243186,415.8605651855469]},{"page":349,"text":"densityisJi¼sijEjwherethetensoroftheelectricconductivity hasastandardform:","rect":[53.81364059448242,437.97027587890627,385.1552568204124,428.06231689453127]},{"page":349,"text":"sij ¼ s?dij þ saninj","rect":[177.58209228515626,461.79046630859377,260.93961602013015,451.6830139160156]},{"page":349,"text":"According to our geometry, E ¼ (Ex, 0, Ez), n ¼ (cosW, 0, sinW) and the","rect":[65.76496887207031,485.2891845703125,385.1184772319969,476.0558166503906]},{"page":349,"text":"conductivity is given by","rect":[53.8134880065918,497.2487487792969,151.1218651872016,488.314208984375]},{"page":349,"text":"s_xz ¼ s?\u000501","rect":[115.04682159423828,535.3145141601563,167.66199580243598,511.4292907714844]},{"page":349,"text":"01\u0006 þ sa\u0005cocsoWss2iWnW","rect":[177.63909912109376,535.31494140625,264.1365696842487,511.4292907714844]},{"page":349,"text":"cossiWn2sWin W\u0006","rect":[274.105712890625,535.31494140625,323.9591948543169,511.4297180175781]},{"page":349,"text":"Then the x-component of the current for small W is given by","rect":[65.76590728759766,558.8604736328125,308.1339959245063,549.6271362304688]},{"page":349,"text":"Jx ¼ s?Ex þ saExcos2W þ saEz sinWcosW \u0007 sjjEx þ saEzW","rect":[96.97733306884766,585.4285888671875,341.96695298502996,573.0913696289063]},{"page":350,"text":"338","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.73106384277344]},{"page":350,"text":"11 Optics and Electric Field Effects in Nematic and Smectic A Liquid Crystals","rect":[114.25279998779297,44.274620056152347,385.1677597331933,36.663330078125]},{"page":350,"text":"where","rect":[53.812843322753909,67.0,78.15088689996564,59.35380554199219]},{"page":350,"text":"the","rect":[83.26834106445313,67.0,95.49211919977033,59.35380554199219]},{"page":350,"text":"new","rect":[100.54486846923828,67.0,117.11862707093086,61.0]},{"page":350,"text":"component","rect":[122.24006652832031,68.2883529663086,166.59602220127176,60.369773864746097]},{"page":350,"text":"of","rect":[171.74732971191407,67.0,180.03917823640479,59.35380554199219]},{"page":350,"text":"electric","rect":[185.05911254882813,67.0,214.65296209527816,59.35380554199219]},{"page":350,"text":"field","rect":[219.72561645507813,67.0,237.44411555341254,59.35380554199219]},{"page":350,"text":"distribution Q(x) by the Poisson equation:","rect":[53.81370162963867,80.24788665771485,222.62817246982645,71.23365783691406]},{"page":350,"text":"E","rect":[242.4968719482422,66.1468505859375,248.60875695623123,59.57293701171875]},{"page":350,"text":"x","rect":[248.67222595214845,67.85040283203125,252.00930806660814,64.82939147949219]},{"page":350,"text":"is","rect":[257.0557861328125,67.0,263.68527616606908,59.35380554199219]},{"page":350,"text":"related","rect":[268.72509765625,67.0,295.9598321061469,59.35380554199219]},{"page":350,"text":"to","rect":[301.0693359375,67.0,308.84356013349068,60.369773864746097]},{"page":350,"text":"the","rect":[313.92718505859377,67.0,326.1509479351219,59.35380554199219]},{"page":350,"text":"space","rect":[331.2037048339844,68.2883529663086,353.3617633648094,61.0]},{"page":350,"text":"charge","rect":[358.4493408203125,68.2883529663086,385.1525043316063,59.35380554199219]},{"page":350,"text":"qE","rect":[201.77003479003907,98.6380844116211,212.86699670232498,91.47650146484375]},{"page":350,"text":"1","rect":[231.5654296875,98.568359375,236.54253474286566,91.83507537841797]},{"page":350,"text":"divE ¼ x ¼ 4pQðxÞ","rect":[169.1422119140625,107.85021209716797,269.83920683013158,96.98910522460938]},{"page":350,"text":"qx","rect":[204.43240356445313,112.26127624511719,213.8670123882469,105.07977294921875]},{"page":350,"text":"ejj","rect":[229.7527618408203,115.71460723876953,237.8771747512392,107.47994232177735]},{"page":350,"text":"(11.86)","rect":[356.072509765625,107.1131362915039,385.16069923249855,98.57701110839844]},{"page":350,"text":"and J obeys the current conservation law:","rect":[53.81493377685547,136.24856567382813,222.133680892678,127.31401062011719]},{"page":350,"text":"divJ ¼ qqJxx ¼ sjj qqExx þ saEz qqWx ¼0","rect":[145.9759979248047,168.26210021972657,293.01824275067818,147.4771728515625]},{"page":350,"text":"(11.87)","rect":[356.07220458984377,163.1131134033203,385.1603940567173,154.63674926757813]},{"page":350,"text":"Combining (11.85–11.87), we get the periodic space charge distribution over x:","rect":[65.76660919189453,188.73446655273438,385.20918138095927,179.74014282226563]},{"page":350,"text":"QðxÞ ¼ saejjEzWm sinpx","rect":[169.1440887451172,216.95944213867188,268.0195697612938,200.02029418945313]},{"page":350,"text":"4psjjwx","rect":[205.67857360839845,224.82518005371095,236.8325743187942,214.54904174804688]},{"page":350,"text":"wx","rect":[257.90509033203127,222.87542724609376,267.70417160883326,216.82962036132813]},{"page":350,"text":"(11..88)","rect":[353.5790710449219,216.22361755371095,385.1588071426548,207.74725341796876]},{"page":350,"text":"Due to the space charge and corresponding force \u0003Q(x)E the nematic liquid","rect":[65.7660140991211,245.35903930664063,385.1140069108344,236.3448028564453]},{"page":350,"text":"begins to move with a velocity vz determined by reduced form of the Navier-Stokes","rect":[53.81399154663086,257.26220703125,385.0882913027878,248.3272247314453]},{"page":350,"text":"equation (7.16):","rect":[53.81318283081055,269.2217712402344,118.01186997959207,260.2872314453125]},{"page":350,"text":"\u0002 qq2xv2z ¼ \u0003QðxÞEz","rect":[183.0198211669922,301.2351989746094,255.43333869829119,278.4644470214844]},{"page":350,"text":"(11.89)","rect":[356.0715026855469,296.0868835449219,385.1596921524204,287.61053466796877]},{"page":350,"text":"where Z ¼ (1/2) (a4 + a5 \u0003 a2) is a combination of Leslie’s viscosity coefficients ai.","rect":[53.81393051147461,319.7244873046875,385.1832241585422,310.78955078125]},{"page":350,"text":"At a certain critical voltage the destabilising shear-induced torque My ¼ a2(∂vz/","rect":[65.76656341552735,332.60064697265627,385.208632064553,321.74359130859377]},{"page":350,"text":"∂x), which comes from the interaction of a field driven charged volume of a liquid","rect":[53.814083099365237,343.6437072753906,385.1141289811469,333.7031555175781]},{"page":350,"text":"with the director, becomes large enough to equalise the stabilising elastic torque.","rect":[53.814083099365237,355.6032409667969,385.1280483772922,346.668701171875]},{"page":350,"text":"This balance of the elastic and hydrodynamic torques is the condition for the onset","rect":[53.814083099365237,367.56280517578127,385.1459794766624,358.62823486328127]},{"page":350,"text":"of instability:","rect":[53.814083099365237,379.5223388671875,107.79576737949441,370.5877685546875]},{"page":350,"text":"q2W","rect":[200.80706787109376,400.8701171875,215.30838364909244,391.7126770019531]},{"page":350,"text":"K33 qx2 ¼ a2 qx","rect":[185.00360107421876,414.48345947265627,250.6862720318016,401.0180358886719]},{"page":350,"text":"(11.90)","rect":[356.07147216796877,409.33489990234377,385.1596616348423,400.8585510253906]},{"page":350,"text":"Integrating once Eq. (11.89) on account of (11.88) we obtain the shear rate","rect":[65.76592254638672,435.9762268066406,368.3062213726219,427.04168701171877]},{"page":350,"text":"qvz","rect":[172.0879669189453,458.60491943359377,184.23033576860369,450.0392761230469]},{"page":350,"text":"Ez2saejjWm","rect":[197.91819763183595,460.05059814453127,237.87121223670105,448.2235412597656]},{"page":350,"text":"¼","rect":[187.49545288085938,462.5509948730469,195.1601946661225,460.2202453613281]},{"page":350,"text":"px","rect":[256.7724609375,457.22076416015627,266.77344549371568,452.6887512207031]},{"page":350,"text":"(11.91)","rect":[356.0715026855469,465.6192626953125,385.1596921524204,457.1429138183594]},{"page":350,"text":"and finally, combining (11.90) and (11.91) and using Eq. (11.85), we have an","rect":[53.81393051147461,495.7746276855469,385.2652520280219,486.7803039550781]},{"page":350,"text":"equation for W equivalent to (11.52). It solution results in the threshold voltage for","rect":[53.81295394897461,507.7341613769531,385.15377174226418,498.50079345703127]},{"page":350,"text":"the instability:","rect":[53.811946868896487,519.6369018554688,110.36381394931863,510.702392578125]},{"page":350,"text":"Ucrit ¼ Ecritd ¼ 4ð\u0003p3aK23Þ3esjjsjj\u0002a\u000E1=2 \u0002 wdx","rect":[144.2170867919922,560.489013671875,292.5714201439895,531.942138671875]},{"page":350,"text":"(11.91)","rect":[356.0715026855469,551.830810546875,385.1596921524204,543.3544311523438]},{"page":350,"text":"Above the threshold a periodic pattern of vortices forms with a period of wx \u0007d","rect":[65.76595306396485,581.986083984375,385.1791924577094,573.0321655273438]},{"page":350,"text":"along the x-axis. The entire process is governed by the anisotropy sa in the","rect":[53.81444549560547,593.9461669921875,385.1188434429344,585.0116577148438]},{"page":351,"text":"11.4 Electrohydrodynamic Instability","rect":[53.812843322753909,44.274620056152347,181.43033355860636,36.68026351928711]},{"page":351,"text":"339","rect":[372.4981994628906,42.62367248535156,385.1898245254032,36.73106384277344]},{"page":351,"text":"denominator. The threshold is diverged when anisotropy sa vanishes, e.g., in","rect":[53.812843322753909,68.2883529663086,385.1437615495063,59.35380554199219]},{"page":351,"text":"nematics with a short-range smectic order close to the N-SmA phase transition.","rect":[53.81386947631836,80.24788665771485,385.1527065804172,71.31333923339844]},{"page":351,"text":"The threshold is proportional to the ratio Z/a2 of the two viscosities. Roughly","rect":[53.81386947631836,92.20772552490235,385.11379328778755,83.27293395996094]},{"page":351,"text":"speaking, they are proportional to each other, thus the threshold weakly depends on","rect":[53.81376266479492,104.1104965209961,385.1705254655219,95.17594909667969]},{"page":351,"text":"viscosity and the instability may easily be observed in very viscous, e.g., polymer","rect":[53.81376266479492,116.0699691772461,385.0869077285923,107.13542175292969]},{"page":351,"text":"liquid crystals. The reason is a compensation for the two effects: on the one hand, in","rect":[53.81376266479492,128.02957153320313,385.14067927411568,119.09501647949219]},{"page":351,"text":"very viscous media the velocity of vortex motion is low (low Z) but, on the other","rect":[53.81376266479492,139.98910522460938,385.1396726211704,131.05455017089845]},{"page":351,"text":"hand, the coupling between the flow and the director is strong (high a2). A more","rect":[53.81379318237305,151.94863891601563,385.14087713434068,143.0140838623047]},{"page":351,"text":"precise expression for the threshold voltage derived by Helfrich [42] includes also a","rect":[53.81399154663086,163.908935546875,385.15976751520005,154.97438049316407]},{"page":351,"text":"finite value of dielectric anisotropy. The dependencies predicted by the simplest","rect":[53.814022064208987,175.86846923828126,385.1598344571311,166.9339141845703]},{"page":351,"text":"theory have been confirmed qualitatively by many experiments [7].","rect":[53.814022064208987,187.8280029296875,325.4524807503391,178.89344787597657]},{"page":351,"text":"Going back to Fig. 11.33a we may see that, for the same structure of the director","rect":[65.76705169677735,199.73077392578126,385.1110776504673,190.7962188720703]},{"page":351,"text":"fluctuation W(z), when the field direction changes sign, the space charge sign is also","rect":[53.815025329589847,211.69033813476563,385.1558465104438,202.45697021484376]},{"page":351,"text":"reversed. However, their product (the electric force Q(x)E) keeps its direction. It","rect":[53.81499481201172,223.64984130859376,385.1816850430686,214.63560485839845]},{"page":351,"text":"means that the Carr-Helfrich instability may be observed at the ac voltage. Indeed,","rect":[53.814022064208987,235.60940551757813,385.17278714682348,226.6748504638672]},{"page":351,"text":"in experiment the instability is observed up to the frequency oq ¼ 4ps||/e|| corre-","rect":[53.814022064208987,248.4265899658203,385.10320411531105,238.63438415527345]},{"page":351,"text":"spondent to the space charge oscillation along the x-axis. The theory of the ac","rect":[53.81417465209961,259.5294189453125,385.13510931207505,250.59486389160157]},{"page":351,"text":"regime of the same instability requires the consideration of a set of two coupled","rect":[53.81417465209961,271.48895263671877,385.12612238935005,262.55438232421877]},{"page":351,"text":"linear equations for the space charge Q(x) and curvature c(x) ¼ ∂W/∂x dependent","rect":[53.81417465209961,283.448486328125,385.1758867032249,273.5079345703125]},{"page":351,"text":"on time and the problem of the threshold has been solved for frequencies below and","rect":[53.81417465209961,295.35125732421877,385.1451043229438,286.41668701171877]},{"page":351,"text":"above oq [43].","rect":[53.81417465209961,308.22479248046877,112.74946256186252,298.37628173828127]},{"page":351,"text":"11.4.2.2 Behaviour Above the Threshold","rect":[53.81356430053711,341.7557678222656,233.006274884444,334.17584228515627]},{"page":351,"text":"At voltages higher than the threshold, the one-dimensional roll structure subse-","rect":[53.81356430053711,367.1091613769531,385.1335385879673,358.17462158203127]},{"page":351,"text":"quently transforms in more complex hydrodynamic patterns. One can distinguish","rect":[53.81356430053711,379.0686950683594,385.1613701920844,370.1142272949219]},{"page":351,"text":"the zigzag, fluctuating and other domain structures, which, in turn, are substituted","rect":[53.81356430053711,391.02825927734377,385.1743096452094,382.09368896484377]},{"page":351,"text":"by a turbulent motion of a liquid crystal. To calculate the wavevectors and ampli-","rect":[53.81356430053711,402.98779296875,385.12362037507668,394.05322265625]},{"page":351,"text":"tudes of the distortions a set of nonlinear equations must be solved. More generally,","rect":[53.81356430053711,414.9473571777344,385.0967068245578,406.0128173828125]},{"page":351,"text":"the problem for describing a transition from a regular electrohydrodynamic vortex","rect":[53.81356430053711,426.9068908691406,385.1295098405219,417.97235107421877]},{"page":351,"text":"motion to turbulence is a part of the classical problem concerning the transition","rect":[53.81356430053711,438.8096618652344,385.15541926435005,429.8751220703125]},{"page":351,"text":"from the laminar to turbulent flow of a liquid. Some progress has been achieved in","rect":[53.81356430053711,450.7691955566406,385.14052668622505,441.83465576171877]},{"page":351,"text":"understanding the nonlinear behaviour of nematics in terms of bifurcation mechan-","rect":[53.81356430053711,462.7287292480469,385.0787595352329,453.794189453125]},{"page":351,"text":"isms, phase transitions and dynamic chaos theory [44].","rect":[53.81356430053711,474.6882629394531,274.86773343588598,465.75372314453127]},{"page":351,"text":"As known from general theory of dissipative dynamic systems, after a finite","rect":[65.76558685302735,486.6478271484375,385.1365436382469,477.7132568359375]},{"page":351,"text":"number of bifurcations the system undergoes to the dynamic chaos. This scenario is","rect":[53.81356430053711,498.6073913574219,385.1853372012253,489.6728515625]},{"page":351,"text":"also observed in the electrohydrodynamic convective motion. With increasing","rect":[53.81356430053711,510.5669250488281,385.07878962567818,501.63238525390627]},{"page":351,"text":"voltage the velocity of vortices increases rapidly and the periodic flow of a liquid","rect":[53.81356430053711,522.5264282226563,385.11162653974068,513.5919189453125]},{"page":351,"text":"transforms to turbulence. 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The DSL effect have been initially","rect":[53.81356430053711,558.38818359375,385.1036309342719,549.3739624023438]},{"page":351,"text":"proposed for manufacturing field-controllable shutters and displays, and the semi-","rect":[53.81356430053711,570.307861328125,385.1783383926548,561.3733520507813]},{"page":351,"text":"nal paper [45] was the starting point for development of the modern technology of","rect":[53.81356430053711,582.2674560546875,385.1484311660923,573.273193359375]},{"page":351,"text":"liquid crystal materials and displays.","rect":[53.81356430053711,594.2269897460938,201.46238370444065,585.29248046875]},{"page":352,"text":"340","rect":[53.806209564208987,42.56271743774414,66.49781555323526,36.73783874511719]},{"page":352,"text":"11 Optics and Electric Field Effects in Nematic and Smectic A Liquid Crystals","rect":[114.24617004394531,44.281394958496097,385.16113741874019,36.67010498046875]},{"page":352,"text":"Reference","rect":[53.812843322753909,68.09864807128906,104.99731238813605,59.31352233886719]},{"page":352,"text":"1.","rect":[58.06126022338867,94.0,64.40706131055318,87.80046081542969]},{"page":352,"text":"2.","rect":[58.06126022338867,104.0,64.40706131055318,97.77641296386719]},{"page":352,"text":"3.","rect":[58.06126022338867,114.0,64.40706131055318,107.75236511230469]},{"page":352,"text":"4.","rect":[58.06126022338867,134.0,64.40706131055318,127.64750671386719]},{"page":352,"text":"5.","rect":[58.06126022338867,144.0,64.40706131055318,137.5219268798828]},{"page":352,"text":"6.","rect":[58.06126022338867,164.0,64.40706131055318,157.46786499023438]},{"page":352,"text":"7.","rect":[58.06124496459961,174.0,64.40704605176411,167.61317443847657]},{"page":352,"text":"8.","rect":[58.06124496459961,193.27142333984376,64.40704605176411,187.44654846191407]},{"page":352,"text":"9.","rect":[58.06124496459961,203.2583770751953,64.40704605176411,197.3657684326172]},{"page":352,"text":"10.","rect":[53.81294250488281,223.14254760742188,64.3892810065981,217.3176727294922]},{"page":352,"text":"11.","rect":[53.81294250488281,243.0,64.3892810065981,237.2128448486328]},{"page":352,"text":"12.","rect":[53.812103271484378,263.0,64.38844177319966,257.1647644042969]},{"page":352,"text":"13.","rect":[53.811279296875,283.0,64.38761779859029,277.0599365234375]},{"page":352,"text":"14.","rect":[53.811279296875,303.0,64.38761779859029,297.0118713378906]},{"page":352,"text":"15.","rect":[53.811279296875,323.0,64.38761779859029,316.8055114746094]},{"page":352,"text":"16.","rect":[53.81044006347656,343.0,64.38677856519185,336.8082275390625]},{"page":352,"text":"17.","rect":[53.81044006347656,363.0,64.38677856519185,356.7541809082031]},{"page":352,"text":"18.","rect":[53.81044006347656,393.0,64.38677856519185,386.6253356933594]},{"page":352,"text":"19.","rect":[53.81044006347656,413.0,64.38677856519185,406.5772399902344]},{"page":352,"text":"20.","rect":[53.80960464477539,433.0,64.38594314649068,426.4723815917969]},{"page":352,"text":"21.","rect":[53.808746337890628,452.1475830078125,64.38508483960591,446.4242858886719]},{"page":352,"text":"22.","rect":[53.807891845703128,472.0427551269531,64.38423034741841,466.3194580078125]},{"page":352,"text":"23.","rect":[53.807891845703128,492.0395202636719,64.38423034741841,486.2146301269531]},{"page":352,"text":"24.","rect":[53.80706024169922,512.0,64.3833987434145,506.1665344238281]},{"page":352,"text":"25.","rect":[53.80706024169922,532.0,64.3833987434145,525.9601440429688]},{"page":352,"text":"26.","rect":[53.80705261230469,542.0,64.38339111401997,535.9868774414063]},{"page":352,"text":"27.","rect":[53.80620574951172,562.0,64.382544251227,555.9895629882813]},{"page":352,"text":"28.","rect":[53.80620574951172,582.0,64.382544251227,575.884765625]},{"page":352,"text":"Yariv, A.: Quantum Electronics, 3rd edn. 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Physique (Paris) 39, 417–422 (1978)","rect":[68.59516906738281,176.51223754882813,232.59524625403575,168.54534912109376]},{"page":353,"text":"Bobylev, YuP, Pikin, S.A.: Threshold piezoelectric instability in liquid crystals. Zh. Exp.","rect":[68.59432983398438,186.48822021484376,385.1525599677797,178.89385986328126]},{"page":353,"text":"Teor. Fiz. 72, 369–374 (1977)","rect":[68.59432983398438,196.12550354003907,171.98437589270763,188.86981201171876]},{"page":353,"text":"Palto, S.P., Mottram, N.J., Osipov, M.A.: Flexoelectric instability and spontaneous chiral-","rect":[68.59349060058594,206.44012451171876,385.16525357825449,198.82882690429688]},{"page":353,"text":"symmetry breaking in a nematic liquid crystal cell with asymmetric boundary conditions.","rect":[68.59349060058594,216.35934448242188,385.182162018561,208.76498413085938]},{"page":353,"text":"Phys. Rev. E, 75, 061707(1–8) (2007).","rect":[68.59349060058594,226.33529663085938,201.06169387891254,218.6901397705078]},{"page":353,"text":"Blinov, L.M.: Behavior of liquid crystal in electric and magnetic fields. In: Demus, D.,","rect":[68.59349060058594,236.31124877929688,385.180453034186,228.71688842773438]},{"page":353,"text":"Goodby, J., Gray, G.W., Spiess, H.-W., Vill, V. (eds.) Physical Properties of Liquid Crystals,","rect":[68.59349060058594,246.23046875,385.1263148505922,238.6361083984375]},{"page":353,"text":"pp. 375–432. Wiley-VCH, Weinheim (1999).","rect":[68.59349060058594,256.2064208984375,224.05974075391254,248.56126403808595]},{"page":353,"text":"Dubois-Violette, E., Durand, G., Guyon, E., Manneville, P., Pieranski, P.: Instabilities in","rect":[68.59349060058594,266.1824035644531,385.1643118300907,258.5880432128906]},{"page":353,"text":"nematic liquid crystals. In: Liebert, L. (ed.) Liquid Crystals, pp. 147–208. (Ehrenreich, H.,","rect":[68.59349060058594,276.1583557128906,385.182162018561,268.5639953613281]},{"page":353,"text":"Seitz, F., Turnbull, D. (eds.) series Solid State Physics), Academic Press, New-York, (1978).","rect":[68.59349060058594,286.07763671875,385.1585414130922,278.4832763671875]},{"page":353,"text":"Kramer, L., Pesch, W.: Electrodynamic convection in nematics. In: Dunmur, D., Fukuda, A.,","rect":[68.59349060058594,296.0535888671875,385.1796595771547,288.459228515625]},{"page":353,"text":"Luckhurst, G. (eds.) Physical Properties of Liquid crystals: Nematics, pp. 441–454. INSPEC,","rect":[68.59349060058594,306.029541015625,385.1525599677797,298.3843688964844]},{"page":353,"text":"London (2001).","rect":[68.59349060058594,315.6668701171875,122.04207107617818,308.4111633300781]},{"page":353,"text":"Carr, E.F., Flint, W.T., Parker, J.H.: Effect of dielectric and conductivity anisotropies on","rect":[68.59349060058594,325.9247131347656,385.2049917617313,318.3303527832031]},{"page":353,"text":"molecular alignment in a liquid crystal. Phys. Rev. A11, 1732–1736 (1975)","rect":[68.59349060058594,335.90069580078127,326.96493619544199,328.23858642578127]},{"page":353,"text":"Williams, R.: Domains in liquid crystals. J. Chem. Phys. 39, 384–388 (1963)","rect":[68.59351348876953,345.87664794921877,332.2328194962232,337.9097595214844]},{"page":353,"text":"Helfrich, W.: Conduction induced alignment in nematic liquid crystals. Basic models and","rect":[68.59351348876953,355.8525695800781,385.1922964492313,348.2582092285156]},{"page":353,"text":"stability consideration. J. Chem. Phys. 51, 4092–4105 (1969)","rect":[68.59351348876953,365.7718200683594,277.9660653458326,358.1097106933594]},{"page":353,"text":"Dubois-Violette, E., de Gennes, P.G., Parodi, O.: Hydrodynamic instabilities of nematic liquid","rect":[68.59265899658203,375.7477722167969,385.1829885879032,368.136474609375]},{"page":353,"text":"crystals under a.c. electric field. J. Physique (Paris) 32, 305–317 (1971)","rect":[68.59265899658203,385.7237243652344,313.36883634192636,377.8584289550781]},{"page":353,"text":"Buka, A., Kramer, L. (eds.): Pattern Formation in Liquid Crystals. Springer, New York (1995)","rect":[68.5926284790039,395.6429443359375,385.1965340958326,387.9977722167969]},{"page":353,"text":"Heilmeier, G., Zanoni, L.A., Barton, L.A.: Dynamic scattering: a new electrooptic effecting","rect":[68.5926284790039,405.618896484375,385.1330618300907,398.0245361328125]},{"page":353,"text":"certain classes of nematic liquid crystals. Proc. I.E.E.E 56, 1162–1171 (1968)","rect":[68.5926284790039,415.5948486328125,334.55365079505136,407.94122314453127]},{"page":354,"text":"Chapter 12","rect":[53.812843322753909,72.10812377929688,121.10908599090695,59.571903228759769]},{"page":354,"text":"Electro-Optical Effects in Cholesteric Phase","rect":[53.812843322753909,90.43364715576172,354.200154931623,76.0426254272461]},{"page":354,"text":"12.1 Cholesteric as One-Dimensional Photonic Crystal","rect":[53.812843322753909,212.6539764404297,340.5665456945759,201.34686279296876]},{"page":354,"text":"A cholesteric forms a helical structure and its optical properties are characterised","rect":[53.812843322753909,239.63455200195313,385.07207575849068,230.6999969482422]},{"page":354,"text":"by the tensor of dielectric permittivity rotating in space. We are already familiar","rect":[53.812843322753909,251.5941162109375,385.09993873445168,242.65956115722657]},{"page":354,"text":"with the form of the cholesteric tensor (see Section 4.7). It was Oseen [1] who","rect":[53.812843322753909,263.09844970703127,385.1237420182563,254.5424041748047]},{"page":354,"text":"suggested the first quantitative model of the helical cholesteric phase as a periodic","rect":[53.812843322753909,275.4564208984375,385.16663397027818,266.5218505859375]},{"page":354,"text":"medium with local anisotropy and very specific optical properties. First we shall","rect":[53.812843322753909,287.4159851074219,385.1158891446311,278.4814453125]},{"page":354,"text":"discuss more carefully the Bragg reflection from the so-called “cholesteric planes”.","rect":[53.812843322753909,299.41534423828127,385.09197659994848,290.36126708984377]},{"page":354,"text":"12.1.1 Bragg Reflection","rect":[53.812843322753909,345.4918212890625,179.33721894120337,334.9377136230469]},{"page":354,"text":"12.1.1.1 Experimental Data","rect":[53.812843322753909,373.0480651855469,177.74276057294379,364.1035461425781]},{"page":354,"text":"The most characteristic features of cholesteric liquid crystals are as follows:","rect":[53.812843322753909,396.9801025390625,360.79377610752177,388.0455322265625]},{"page":354,"text":"1.","rect":[53.812843322753909,413.0,61.27850003744845,406.0730285644531]},{"page":354,"text":"2.","rect":[53.814022064208987,461.0,61.27967877890353,453.8551940917969]},{"page":354,"text":"3.","rect":[53.814022064208987,545.0,61.27967877890353,537.5151977539063]},{"page":354,"text":"There is a strong rotation of the plane of polarisation of linearly polarised light","rect":[66.27452087402344,414.94781494140627,385.2064042813499,405.9734191894531]},{"page":354,"text":"(C \u0001 10–100 full revolutions per mm to be compared, e.g., with 24\u0003/mm in","rect":[66.2745132446289,426.9073486328125,385.14418879560005,417.86322021484377]},{"page":354,"text":"quartz). The sign of the optical rotation changes at a certain wavelength l0 of the","rect":[66.27599334716797,438.8106994628906,385.17575872613755,429.5574035644531]},{"page":354,"text":"incident light as shown by curve OR in Fig. 12.1.","rect":[66.27571868896485,450.7704162597656,265.88848538901098,441.8159484863281]},{"page":354,"text":"The regions of rotation with different handedness are not separated by an","rect":[66.27569580078125,462.72998046875,385.1498955827094,453.79541015625]},{"page":354,"text":"absorption band as in typical gyrotropic materials. Instead, there is a band of a","rect":[66.27571868896485,474.68951416015627,385.15894354059068,465.75494384765627]},{"page":354,"text":"selective reflection of the beam with a particular circular polarization, curve R in","rect":[66.27571868896485,486.6490783691406,385.1419610123969,477.63482666015627]},{"page":354,"text":"Fig. 12.1. The beam with the opposite circular polarization is transmitted","rect":[66.27571868896485,498.6086120605469,385.14992610028755,489.674072265625]},{"page":354,"text":"without any change, therefore the reflection is negligible and not shown in the","rect":[66.27571868896485,510.5681457519531,385.1727985210594,501.63360595703127]},{"page":354,"text":"plot. Only one band is observed in the wavelength spectrum without higher","rect":[66.27571868896485,522.5276489257813,385.13301978913918,513.5731811523438]},{"page":354,"text":"diffraction orders.","rect":[66.27571868896485,532.3984985351563,138.7194790413547,525.4959106445313]},{"page":354,"text":"The electric vectors of the circularly polarised incident and reflected light are","rect":[66.27569580078125,546.3899536132813,385.16284979059068,537.4554443359375]},{"page":354,"text":"rotated in the same direction when viewed against the wavevectors of each","rect":[66.27571868896485,558.3494873046875,385.0992058854438,549.4149780273438]},{"page":354,"text":"beam. In contrast, upon reflection from a conventional mirror the beam changes","rect":[66.27571868896485,570.30908203125,385.1618997012253,561.3745727539063]},{"page":354,"text":"the sign of rotation. An example is shown in Fig. 12.2. Note that, in this figure,","rect":[66.27571868896485,582.2686157226563,385.1140408089328,573.3341064453125]},{"page":354,"text":"the circular light handedness is defined not conventionally: for the right circular","rect":[66.27574920654297,594.2281494140625,385.08724342195168,585.2936401367188]},{"page":354,"text":"L.M. Blinov, Structure and Properties of Liquid Crystals,","rect":[53.812843322753909,623.7623291015625,251.63688919141254,616.1510620117188]},{"page":354,"text":"DOI 10.1007/978-90-481-8829-1_12, # Springer ScienceþBusiness Media B.V. 2011","rect":[53.812843322753909,633.698486328125,351.58160919337197,625.4920043945313]},{"page":354,"text":"343","rect":[372.4981994628906,622.0606079101563,385.18979400782509,616.2357177734375]},{"page":355,"text":"344","rect":[53.812843322753909,42.55630874633789,66.50444931178018,36.73143005371094]},{"page":355,"text":"Fig. 12.1 Spectra of optical","rect":[53.812843322753909,67.58130645751953,151.28945640769067,59.85148620605469]},{"page":355,"text":"rotatory power (OR) and","rect":[53.812843322753909,77.4895248413086,138.56146759180948,69.87823486328125]},{"page":355,"text":"selective reflection (R) ofa","rect":[53.812843322753909,87.07006072998047,147.2636809089136,79.81436157226563]},{"page":355,"text":"planar cholesteric texture for","rect":[53.812843322753909,97.3846664428711,152.49768155677013,89.79031372070313]},{"page":355,"text":"light propagation parallel to","rect":[53.812843322753909,107.36067962646485,149.2410177627079,99.76632690429688]},{"page":355,"text":"the helical axis.","rect":[53.812843322753909,115.58409118652344,107.47125503613911,109.74227905273438]},{"page":355,"text":"12 Electro-Optical Effects in Cholesteric Phase","rect":[223.29483032226563,44.274986267089847,385.1406798102808,36.6636962890625]},{"page":355,"text":"OR","rect":[260.474853515625,66.52562713623047,272.46009238642929,60.489253997802737]},{"page":355,"text":"deg","rect":[259.5479736328125,76.56759643554688,272.8755766583427,69.07610321044922]},{"page":355,"text":"–20","rect":[259.5479736328125,87.73284149169922,272.8755766583427,81.96830749511719]},{"page":355,"text":"Linear","rect":[130.41639709472657,242.88369750976563,152.6296827254765,237.02894592285157]},{"page":355,"text":"R.H","rect":[174.1443328857422,261.9356689453125,187.9087399083002,256.1929016113281]},{"page":355,"text":"L.H","rect":[173.75106811523438,285.76336669921877,186.18858609970645,280.0205993652344]},{"page":355,"text":"R-screw helix","rect":[217.3743133544922,228.88027954101563,268.9068703969703,222.9775390625]},{"page":355,"text":"R.H","rect":[269.2314147949219,263.2681579589844,282.9958065586908,257.525390625]},{"page":355,"text":"Fig. 12.2 Transmission and reflection of linearly polarized light through the planar cholesteric","rect":[53.812843322753909,331.6007080078125,385.10851428293707,323.8708801269531]},{"page":355,"text":"structure. The linear light is decomposed into two circularly polarized components, left-handed (L.","rect":[53.812843322753909,341.50897216796877,385.1762110908266,333.91461181640627]},{"page":355,"text":"H.) and right handed (R.H). In this particular case, the handedness is defined according to the","rect":[53.812843322753909,351.48492431640627,385.1542601325464,343.89056396484377]},{"page":355,"text":"modern convention, see the text","rect":[53.812843322753909,359.73370361328127,163.0469332143313,353.8664855957031]},{"page":355,"text":"4.","rect":[53.8141975402832,545.0,61.27985425497775,537.1746826171875]},{"page":355,"text":"polarization, the observer looking at the light source sees the counterclockwise","rect":[66.27484893798828,390.728759765625,385.0962604351219,381.7344665527344]},{"page":355,"text":"rotation of the light electric vector. This “new definition” (as discussed in","rect":[66.27486419677735,402.64849853515627,385.14214411786568,393.71392822265627]},{"page":355,"text":"Section 11.1.1) is used here deliberately because I may suggest a mnemonic","rect":[66.27486419677735,414.5512390136719,385.18186224176255,405.61669921875]},{"page":355,"text":"rule: a right-hand circular beam goes as easily (i.e. transmitted) through the","rect":[66.27587127685547,426.51080322265627,385.17295110895005,417.55633544921877]},{"page":355,"text":"right-hand helix as a right-hand screw goes into a right-hand female screw. And","rect":[66.27587127685547,438.4703674316406,385.12224665692818,429.53582763671877]},{"page":355,"text":"this may be explained as follows: the right-hand circularly polarized light going","rect":[66.27587127685547,450.4299011230469,385.1182488541938,441.495361328125]},{"page":355,"text":"along the right helix does not see periodicity of the helix and, therefore, does not","rect":[66.27587127685547,462.3894348144531,385.20783860752177,453.45489501953127]},{"page":355,"text":"diffract. In the figure we see the right-screw helical cholesteric structure that","rect":[66.27587127685547,474.3489685058594,385.1362138516624,465.4144287109375]},{"page":355,"text":"transmit the right-hand (R.H.) circularly polarised light and completely reflects","rect":[66.27587127685547,486.3085021972656,385.1849404727097,477.37396240234377]},{"page":355,"text":"the left circular polarized light (L.H.) without change of its handedness. By the","rect":[66.27587127685547,498.2680358886719,385.17404974176255,489.33349609375]},{"page":355,"text":"way, direct modelling of the light transmission or reflection results in exactly","rect":[66.27587127685547,510.1708068847656,385.1750115495063,501.23626708984377]},{"page":355,"text":"that situation, which corresponds to the non-conventional case. Nevertheless,","rect":[66.27587127685547,522.1303100585938,385.17998929526098,513.19580078125]},{"page":355,"text":"further on we follow the old convention.","rect":[66.27587127685547,532.0579223632813,229.89022489096409,525.1553344726563]},{"page":355,"text":"The wavelength of selected reflection l0 (in vacuum) depends on the angle of","rect":[66.27587127685547,546.051025390625,385.14946876374855,536.7962036132813]},{"page":355,"text":"light incidence i measured from the layer normal, namely, l0 ¼ 2(P0/2) <n>","rect":[66.27532196044922,558.0105590820313,385.14820125791939,548.75732421875]},{"page":355,"text":"cosi. It is the same Bragg condition discussed in Section 5.2.2, l0 ¼ 2dsinY.","rect":[66.2760238647461,570.0100708007813,385.1842007210422,560.7169799804688]},{"page":355,"text":"However, in the case of the X-ray diffraction on a stack of the layers in vacuum,","rect":[66.27717590332031,581.9298706054688,385.1225246956516,572.995361328125]},{"page":355,"text":"we used refraction index n ¼ 1, sliding angle Y ¼ (p/2) \u0004 i, and interlayer","rect":[66.27717590332031,593.889404296875,385.15731178132668,584.7855224609375]},{"page":356,"text":"12.1 Cholesteric as One-Dimensional Photonic Crystal","rect":[53.81256866455078,44.275901794433597,241.14489464011255,36.66461181640625]},{"page":356,"text":"345","rect":[372.4979248046875,42.55722427368164,385.18954986720009,36.6307487487793]},{"page":356,"text":"distance d instead of half-pitch P0/2. The factor ½ appeared in a cholesteric","rect":[66.27484893798828,68.2883529663086,385.10437811090318,59.31396484375]},{"page":356,"text":"because, due to the head-to-tail symmetry n ¼ \u0004n, the period of its optical","rect":[66.2750015258789,80.24788665771485,385.1262651211936,71.31333923339844]},{"page":356,"text":"properties is doubled.","rect":[66.27498626708985,92.20748138427735,153.18719144369846,83.27293395996094]},{"page":356,"text":"We see that the optical properties of cholesterics are quite peculiar. How to","rect":[65.76532745361328,110.11837005615235,385.14125910810005,101.18382263183594]},{"page":356,"text":"explain them on the quantitative basis?","rect":[53.81330490112305,122.0779037475586,211.26700628473129,113.14335632324219]},{"page":356,"text":"12.1.1.2 The Simplest Model","rect":[53.81330490112305,163.83822631835938,182.12507493564676,154.88375854492188]},{"page":356,"text":"Consider the optical properties of a cholesteric helix shown in Fig. 12.3a under the","rect":[53.81330490112305,187.8270263671875,385.1749957866844,178.89247131347657]},{"page":356,"text":"following assumptions:","rect":[53.813289642333987,199.72976684570313,147.568007064553,190.7952117919922]},{"page":356,"text":"1.","rect":[53.813289642333987,216.0,61.27894635702853,208.82272338867188]},{"page":356,"text":"2.","rect":[53.814308166503909,240.0,61.27996488119845,232.7418212890625]},{"page":356,"text":"3.","rect":[53.814308166503909,276.0,61.27996488119845,268.5636901855469]},{"page":356,"text":"4.","rect":[53.81350326538086,288.0,61.279159980075409,280.5253601074219]},{"page":356,"text":"The light propagates along the helical axis z, and the helix is regarded as ideal,","rect":[66.27496337890625,217.697509765625,385.1103176644016,208.76295471191407]},{"page":356,"text":"corresponding to the sinusoidal form for the variation of the director.","rect":[66.2759780883789,229.65704345703126,345.0417141487766,220.7224884033203]},{"page":356,"text":"The semi-infinite structure is assumed, bordered at the front plane by a dielectric","rect":[66.27598571777344,241.61660766601563,385.1810687847313,232.6820526123047]},{"page":356,"text":"of the same refractive index as the average refractive index of the cholesteric","rect":[66.2759780883789,253.51934814453126,385.10544622613755,244.5847930908203]},{"page":356,"text":"<n>. In such a case, we neglect the reflection from the front boundary.","rect":[66.2759780883789,265.4789123535156,355.38311429526098,256.54437255859377]},{"page":356,"text":"The optical anisotropy is small, i.e. n|| \u0001 n⊥ \u0001 <n> and Dn ¼ n||-n⊥ \u0005 <n>.","rect":[66.27598571777344,277.4384765625,384.3331265022922,268.1773376464844]},{"page":356,"text":"The wavevectors of the incident light and the cholesteric helix have the same","rect":[66.27517700195313,289.400146484375,385.1274799175438,280.465576171875]},{"page":356,"text":"amplitude, ki ¼ q0.","rect":[66.2751693725586,301.35968017578127,143.7615932991672,292.40521240234377]},{"page":356,"text":"Now","rect":[65.76537322998047,318.0,85.03672552063789,310.5354919433594]},{"page":356,"text":"we","rect":[90.34928894042969,318.0,101.94593847467267,312.0]},{"page":356,"text":"would","rect":[107.22962951660156,318.0,132.10520258954535,310.3363037109375]},{"page":356,"text":"like","rect":[137.42074584960938,318.0,152.51233709527816,310.3363037109375]},{"page":356,"text":"to","rect":[157.75621032714845,318.0,165.53044978192816,311.3522644042969]},{"page":356,"text":"understand","rect":[170.8410186767578,318.0,214.6455010514594,310.3363037109375]},{"page":356,"text":"why","rect":[219.95208740234376,319.2708435058594,237.05342951825629,310.3363037109375]},{"page":356,"text":"only","rect":[242.32717895507813,319.2708435058594,260.0556267838813,310.3363037109375]},{"page":356,"text":"one","rect":[265.3811340332031,318.0,279.78488958551255,312.0]},{"page":356,"text":"diffraction","rect":[285.0934753417969,318.0,327.35610285810005,310.3363037109375]},{"page":356,"text":"maximum","rect":[332.6178894042969,318.0,373.2171397078093,310.3363037109375]},{"page":356,"text":"is","rect":[378.4998474121094,318.0,385.12936796294408,310.3363037109375]},{"page":356,"text":"observed in the normal reflection from the cholesteric helix and why the reflected","rect":[53.813350677490237,331.23040771484377,385.1720818620063,322.29583740234377]},{"page":356,"text":"light is circularly polarized. Therefore, at first, we write the Bragg condition on","rect":[53.813350677490237,343.18994140625,385.1691521745063,334.25537109375]},{"page":356,"text":"account of possible higher diffraction orders:","rect":[53.813350677490237,355.1495056152344,235.5314622647483,346.2149658203125]},{"page":356,"text":"ml0 ¼ P0<n>cosi","rect":[181.54776000976563,378.4851989746094,257.473768783303,369.82525634765627]},{"page":356,"text":"(12.1)","rect":[361.0563049316406,378.6708984375,385.10567603913918,370.1945495605469]},{"page":356,"text":"where m is the order of diffraction (i.e. reflection). By analogy with crystals,","rect":[53.81462860107422,402.9316711425781,385.0997585823703,393.99713134765627]},{"page":356,"text":"the values of m ¼ 2, 3... seem to allow the presence of higher order reflections.","rect":[53.81462860107422,414.8912048339844,385.1355862190891,405.9566650390625]},{"page":356,"text":"However, the latter are not observed in experiment on the cholesteric structure for","rect":[53.81562042236328,426.8507385253906,385.1804135879673,417.91619873046877]},{"page":356,"text":"the normal incidence of light (i ¼ 0). This is a result of some selection rules: the","rect":[53.81562042236328,438.8102722167969,385.1753619976219,429.875732421875]},{"page":356,"text":"reflections with m ¼ 2, 3,.. are forbidden due to a specific form of the dielectric","rect":[53.816627502441409,450.76983642578127,385.18430364801255,441.83526611328127]},{"page":356,"text":"permittivity tensor of a cholesteric.","rect":[53.816612243652347,462.7293701171875,195.45504422690159,453.7947998046875]},{"page":356,"text":"a","rect":[147.4473419189453,502.96136474609377,153.00260431617915,497.37261962890627]},{"page":356,"text":"k","rect":[161.96530151367188,512.5489501953125,165.96408118666387,506.8031005859375]},{"page":356,"text":"r","rect":[165.96383666992188,514.5494995117188,167.9612187061769,511.3204650878906]},{"page":356,"text":"ki","rect":[160.70506286621095,539.4161987304688,166.0357811394044,531.6697998046875]},{"page":356,"text":"q0","rect":[221.29995727539063,520.7724609375,229.08159601229529,514.3577880859375]},{"page":356,"text":"x","rect":[228.09788513183595,537.191650390625,232.09666480482793,533.0062866210938]},{"page":356,"text":"y","rect":[214.08375549316407,552.755859375,218.08253516615606,546.865966796875]},{"page":356,"text":"z","rect":[233.98568725585938,552.41015625,237.98446692885137,548.2247924804688]},{"page":356,"text":"b","rect":[248.55361938476563,502.8050842285156,254.65841313424384,495.4967041015625]},{"page":356,"text":"k","rect":[253.60536193847657,516.8966674804688,257.60414161146857,511.15081787109377]},{"page":356,"text":"r","rect":[257.6038818359375,518.8971557617188,259.60126387219256,515.6681518554688]},{"page":356,"text":"q=2q0","rect":[256.3499450683594,542.1056518554688,279.29441034334999,534.3712768554688]},{"page":356,"text":"Fig. 12.3 The geometry for discussion of the Bragg diffraction in a cholesteric (a) and illustration","rect":[53.812843322753909,574.0244750976563,385.18134063868447,566.0914916992188]},{"page":356,"text":"of the wavevector conservation law (b). ki and kr are wavevectors of the incident and reflected","rect":[53.81287384033203,583.6309204101563,385.12628692774697,576.3382568359375]},{"page":356,"text":"beams, q0 is the helix wavevector","rect":[53.81364059448242,593.8493041992188,169.43054288489513,586.3140258789063]},{"page":357,"text":"346","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.68026351928711]},{"page":357,"text":"12 Electro-Optical Effects in Cholesteric Phase","rect":[223.29483032226563,44.274620056152347,385.1406798102808,36.663330078125]},{"page":357,"text":"To show this, it is necessary to insert the Fourier components e(q) of the","rect":[65.76496887207031,68.2883529663086,385.11695135309068,59.35380554199219]},{"page":357,"text":"_","rect":[173.84397888183595,71.67272186279297,178.8210839372016,70.42766571044922]},{"page":357,"text":"dielectric permittivity tensor eðqÞ of the cholesteric into the general formula for","rect":[53.812923431396487,80.58666229248047,385.18047462312355,70.63614654541016]},{"page":357,"text":"the scattering cross section s / ðr \u0006 _eðqÞ \u0006 fÞ2 as already discussed for nematics in","rect":[53.81474304199219,92.54637908935547,385.1435174088813,80.53506469726563]},{"page":357,"text":"Section 11.1.3. Here f and r are polarization vectors for the incident and reflected","rect":[53.81362533569336,104.1104965209961,385.17534724286568,95.04646301269531]},{"page":357,"text":"light, q is the wavevector of scattering coinciding in this simple geometry with the","rect":[53.81462860107422,116.0699691772461,385.1743854351219,107.13542175292969]},{"page":357,"text":"wavevector of the reflected wave [2].","rect":[53.81462860107422,127.4219741821289,204.11922879721409,119.09501647949219]},{"page":357,"text":"_","rect":[95.61701202392578,131.414306640625,100.59411707929144,130.1692352294922]},{"page":357,"text":"_","rect":[279.6571044921875,131.414306640625,284.63420954755318,130.1692352294922]},{"page":357,"text":"Tensor eðqÞ is Fourier transform of cholesteric tensor eðrÞ. The latter is obtained","rect":[65.76663970947266,140.3282470703125,385.1674737077094,130.3777313232422]},{"page":357,"text":"_","rect":[290.193115234375,143.9974365234375,295.17022028974068,142.7523651123047]},{"page":357,"text":"_","rect":[158.32315063476563,145.86785888671876,163.30025569013129,144.62278747558595]},{"page":357,"text":"from the nematic tensor eNðrÞ using the rotation matix R as explained in Sec-","rect":[53.81468963623047,154.781982421875,385.10845313874855,144.8314666748047]},{"page":357,"text":"\u00041","rect":[138.38406372070313,157.79986572265626,147.24935982307754,153.08657836914063]},{"page":357,"text":"_","rect":[116.29248809814453,158.4510498046875,121.26959315351019,157.2059783935547]},{"page":357,"text":"_","rect":[132.83282470703126,158.4510498046875,137.8099297623969,157.2059783935547]},{"page":357,"text":"_","rect":[97.48634338378906,160.32156372070313,102.46344843915472,159.0764923095703]},{"page":357,"text":"_","rect":[121.84370422363281,160.32156372070313,126.82080927899847,159.0764923095703]},{"page":357,"text":"tion 4.7.3: e ¼ ReNR . The rotation matrix and local nematic tensor can be taken","rect":[53.81339645385742,168.2987518310547,385.1385430436469,159.9623260498047]},{"page":357,"text":"in their simplest, “plane” form because now the director has only two non-zero","rect":[53.81364059448242,180.85641479492188,385.1046990495063,171.92185974121095]},{"page":357,"text":"components (nx ¼ sinf(z), ny ¼ cosf(z), nz ¼ 0), and in our geometry only com-","rect":[53.81364059448242,193.7325897216797,385.1391538223423,183.49624633789063]},{"page":357,"text":"ponents of the optical field Ex and Ey are of interest. Therefore","rect":[53.81427764892578,205.70558166503907,307.38062322809068,195.7844696044922]},{"page":357,"text":"_","rect":[77.26398468017578,225.5611572265625,82.24108973554144,224.3160858154297]},{"page":357,"text":"eðzÞ","rect":[77.60460662841797,234.47509765625,93.78571714507297,224.5245819091797]},{"page":357,"text":"¼","rect":[96.58085632324219,233.0,104.24559810850529,226.0]},{"page":357,"text":"¼","rect":[96.58069610595703,259.9744873046875,104.24543789122014,257.64373779296877]},{"page":357,"text":"¼","rect":[96.5802993774414,292.0,104.24504116270451,285.0]},{"page":357,"text":"\u0001 csoins qq00zz","rect":[107.0031967163086,241.5087127685547,141.65001310454563,217.5398712158203]},{"page":357,"text":"\u0004cosisnqq00zz\u0003 \u0006 \u0001e0jj","rect":[151.5823211669922,241.50877380371095,216.0121082229189,217.5397186279297]},{"page":357,"text":"e0? \u0003 \u0006 \u0001\u0004cosisnqq00zz","rect":[226.46693420410157,241.5087127685547,293.0058633242722,217.5397186279297]},{"page":357,"text":"\u0001<e2D>e2a sþine22aqc0ozs2q0z","rect":[107.00335693359375,271.728271484375,195.17969144927219,245.9986572265625]},{"page":357,"text":"<e2D>e2a s\u0004ine22aqc0ozs2q0z\u0003","rect":[205.16812133789063,271.728271484375,293.3302419919553,245.9986572265625]},{"page":357,"text":"he2Di\u0001 01","rect":[107.00334930419922,300.0892333984375,139.90566340497504,276.2040100097656]},{"page":357,"text":"10\u0003 þ e2a \u0001csoins22qq00zz","rect":[149.8827667236328,300.17333984375,223.44566740142063,276.2040100097656]},{"page":357,"text":"\u0004sicno2s q20qz0z \u0003","rect":[233.43511962890626,300.17340087890627,282.34104521461156,276.2044982910156]},{"page":357,"text":"csoins qq00zz \u0003","rect":[302.9949035644531,241.50877380371095,337.5701101560178,217.5398712158203]},{"page":357,"text":"(12.2)","rect":[361.1126708984375,263.0426940917969,385.16204200593605,254.56634521484376]},{"page":357,"text":"Here we introduced ea ¼ ejj \u0004 e?, a two-dimensional average he2Di¼12ðejj þe?Þ","rect":[65.76627349853516,323.56317138671877,385.1676064883347,311.679443359375]},{"page":357,"text":"and also used the expression cos2a ¼ (1/2)(1 þ cos2a). Note that the wavevector","rect":[53.81380844116211,334.121337890625,385.17339454499855,323.0704650878906]},{"page":357,"text":"of","rect":[53.814659118652347,344.01922607421877,62.10651527253759,337.146484375]},{"page":357,"text":"a","rect":[68.82560729980469,344.01922607421877,73.27513921930158,339.3776550292969]},{"page":357,"text":"cholesteric","rect":[79.98428344726563,344.01922607421877,123.35079229547346,337.146484375]},{"page":357,"text":"q0 ¼ 2p/P0 >0","rect":[130.1724090576172,345.97149658203127,195.7017754655219,337.14654541015627]},{"page":357,"text":"(right-handed","rect":[202.3939971923828,346.0810852050781,256.7778252702094,337.14654541015627]},{"page":357,"text":"helix).","rect":[263.5705871582031,345.6826477050781,289.37188382651098,337.14654541015627]},{"page":357,"text":"Left-handed","rect":[296.1407470703125,345.0,345.0328301530219,337.14654541015627]},{"page":357,"text":"helix","rect":[351.7648620605469,345.0,371.76287165692818,337.14654541015627]},{"page":357,"text":"is","rect":[378.5576171875,345.0,385.1871377383347,337.14654541015627]},{"page":357,"text":"described by q0 < 0.","rect":[53.8134651184082,358.0406188964844,138.2029995491672,349.1060791015625]},{"page":357,"text":"As an example, we can apply the Fourier transform to a single component exx (z)","rect":[65.76520538330078,370.0002746582031,385.15960059968605,361.06573486328127]},{"page":357,"text":"from tensor (12.2):","rect":[53.81374740600586,381.56146240234377,130.10878617832254,373.02532958984377]},{"page":357,"text":"exxðqÞ ¼ ðea=2Þð cosð2q0zÞexpðiqzÞdz","rect":[142.40621948242188,415.37451171875,296.56857694731908,393.2522888183594]},{"page":357,"text":"V","rect":[206.64154052734376,421.7178039550781,210.9198479569775,416.9906005859375]},{"page":357,"text":"(12.3)","rect":[361.1146545410156,408.5416259765625,385.16402564851418,400.0652770996094]},{"page":357,"text":"Here, scattering vector q ¼ ki \u0004 kr and the integral can only be non-zero (equal","rect":[65.76825714111328,443.2320556640625,385.20637376377177,434.29742431640627]},{"page":357,"text":"to ½) if q \u0007 2q0 ¼ 0. Since ki ¼ q0 we have two possibilities: either |kr| ¼ q0 þ","rect":[53.81380844116211,455.19171142578127,385.14807918760689,446.257080078125]},{"page":357,"text":"2q0 ¼3q0 or |kr| ¼ q0 \u0004 2q0 ¼\u0004q0. Only the second case satisfies the conservation","rect":[53.814231872558597,467.1512451171875,385.17839900067818,458.19677734375]},{"page":357,"text":"of energy i.e. frequency o ¼ c|kr|/n ¼ cq0/n. Therefore, our scattering vector is","rect":[53.81368637084961,479.0938415527344,385.18793119536596,470.0796203613281]},{"page":357,"text":"q ¼ 2q0 as shown in Fig.12.3b.","rect":[53.81327438354492,491.0139465332031,181.82760282065159,482.07940673828127]},{"page":357,"text":"For q ¼ 4q0 (m ¼ 2) the integral is zero and the second order reflection is absent","rect":[65.76651000976563,502.9735412597656,385.20673997470927,494.03900146484377]},{"page":357,"text":"(the same is true for all integers m \b 2). Thus, only the first order reflection with","rect":[53.814109802246097,514.9330444335938,385.1160821061469,505.99853515625]},{"page":357,"text":"q ¼ þ2q0 is possible and exxð2q0Þ ¼ eaV=4:","rect":[53.81410217285156,527.2315673828125,232.04446351212403,517.2810668945313]},{"page":357,"text":"_","rect":[180.64144897460938,530.2771606445313,185.61855402997504,529.0321044921875]},{"page":357,"text":"From the structure of tensor e (12.2) it is seen that eyy¼\u0004exx and exy ¼ eyx. The","rect":[65.76636505126953,539.76904296875,385.1477741069969,529.9179077148438]},{"page":357,"text":"latter two are imaginary due to the Euler expansion of sin(2q0z). Therefore, for the","rect":[53.813899993896487,550.8120727539063,385.17481268121568,541.8775634765625]},{"page":357,"text":"anisotropic","rect":[53.81405258178711,562.771728515625,98.26059759332502,553.8372192382813]},{"page":357,"text":"part","rect":[103.1511001586914,562.771728515625,118.68961961338113,554.8531494140625]},{"page":357,"text":"of","rect":[123.60003662109375,561.0,131.89190040437354,553.8372192382813]},{"page":357,"text":"the","rect":[136.74159240722657,561.0,148.96535528375473,553.8372192382813]},{"page":357,"text":"_","rect":[153.90487670898438,555.0,158.88198176435004,552.0]},{"page":357,"text":"eðqÞtensor","rect":[154.18931579589845,563.1105346679688,196.46782813874854,553.1600341796875]},{"page":357,"text":"in","rect":[201.31753540039063,561.0,209.09175959638129,553.8373413085938]},{"page":357,"text":"the","rect":[214.00616455078126,561.0,226.2299274273094,553.8373413085938]},{"page":357,"text":"wavevector","rect":[231.1124725341797,561.0,277.2074216446079,554.853271484375]},{"page":357,"text":"space,","rect":[282.0919494628906,562.7718505859375,306.8420681526828,555.0]},{"page":357,"text":"we","rect":[311.6609191894531,561.0,323.25757635309068,555.0]},{"page":357,"text":"may","rect":[328.2008361816406,562.7718505859375,345.4017266373969,555.0]},{"page":357,"text":"write","rect":[350.292236328125,561.0,370.7979205913719,553.8373413085938]},{"page":357,"text":"an","rect":[375.72528076171877,561.0,385.15190974286568,555.0]},{"page":357,"text":"expression","rect":[53.81405258178711,574.6746215820313,96.5086602311469,565.7401123046875]},{"page":357,"text":"_","rect":[261.6439514160156,590.3587646484375,266.6210564713813,589.1137084960938]},{"page":357,"text":"_eðqÞ ¼ _eð2q0Þ ¼ 14eaVM","rect":[170.61520385742188,602.0076293945313,268.1174185456767,590.0]},{"page":358,"text":"12.1 Cholesteric as One-Dimensional Photonic Crystal","rect":[53.812843322753909,44.274620056152347,241.14516929831567,36.663330078125]},{"page":358,"text":"where","rect":[53.812843322753909,66.22653198242188,78.15088689996564,59.35380554199219]},{"page":358,"text":"347","rect":[372.4981994628906,42.55594253540039,385.1898245254032,36.73106384277344]},{"page":358,"text":"M_ ¼ \u00041i \u0004i1\u0005","rect":[185.9091796875,106.35423278808594,253.09658499103564,82.46900939941406]},{"page":358,"text":"and V is the sample volume. This form allows us to find the polarisation of the","rect":[53.814308166503909,129.89981079101563,385.17408025934068,120.96525573730469]},{"page":358,"text":"reflected beam r. Indeed, the polarization vector of the reflected beam is given by","rect":[53.81529998779297,141.85934448242188,383.6968621354438,132.92478942871095]},{"page":358,"text":"r ¼ \u0004rryx \u0005 ¼ M_ \u0006 \u0004ffxy \u0005 ¼ \u0004ifxfxþ\u0004iffyy \u0005 ¼ ðfx þ ifyÞi þ ðifx \u0004 fyÞj ¼ rxi þ ryj","rect":[67.58097839355469,179.9262237548828,371.3949216446079,156.04100036621095]},{"page":358,"text":"(12.4)","rect":[361.0564880371094,193.2679901123047,385.1058591446079,184.7916259765625]},{"page":358,"text":"From Eq. (12.4) we see that ry ¼ irx. It means that the reflected light is circularly","rect":[65.76680755615235,217.6519317626953,385.14544001630318,207.8008575439453]},{"page":358,"text":"polarised in agreement with experiment.","rect":[53.81357955932617,228.69497680664063,216.5489925911594,219.7604217529297]},{"page":358,"text":"Therefore, the simplest model predicts the existence of one maximum of selec-","rect":[65.7656021118164,240.59774780273438,385.16143165437355,231.66319274902345]},{"page":358,"text":"tive reflection centered at the wavelength l0 ¼ hniP0 and the circular polarisation","rect":[53.81357955932617,252.88636779785157,385.10317317060005,242.9557647705078]},{"page":358,"text":"of the reflected beam. However, the spectral dependence of the selective reflection","rect":[53.81412887573242,264.5172424316406,385.15505305341255,255.5826873779297]},{"page":358,"text":"and the magnitude of the angle of the light polarisation rotation by the cholesteric","rect":[53.81412887573242,276.4767761230469,385.10422552301255,267.542236328125]},{"page":358,"text":"structure can only be discussed by analysing the Maxwell equations for the optical","rect":[53.81412887573242,288.43634033203127,385.12507493564677,279.50177001953127]},{"page":358,"text":"waves propagating in the periodic medium.","rect":[53.81412887573242,300.3958740234375,228.64291043783909,291.4613037109375]},{"page":358,"text":"12.1.2 Waves in Layered Medium and Photonic Crystals","rect":[53.812843322753909,348.01275634765627,341.64037208292646,337.3749694824219]},{"page":358,"text":"There are several examples of waves in periodic media:","rect":[53.812843322753909,375.6117858886719,279.0298906094749,366.67724609375]},{"page":358,"text":"1.","rect":[53.812843322753909,392.0,61.27850003744845,384.64788818359377]},{"page":358,"text":"2.","rect":[53.812843322753909,404.0,61.27850003744845,396.6074523925781]},{"page":358,"text":"3.","rect":[53.812843322753909,416.0,61.27850003744845,408.5670166015625]},{"page":358,"text":"4.","rect":[53.812843322753909,428.0,61.27850003744845,420.46978759765627]},{"page":358,"text":"5.","rect":[53.81285858154297,464.0,61.278515296237518,456.2288513183594]},{"page":358,"text":"Electron or neutron waves (C-functions) in crystals","rect":[66.27452087402344,393.5226745605469,274.44195951567846,384.4785461425781]},{"page":358,"text":"X-ray (electromagnetic) waves in crystals","rect":[66.27452087402344,405.48223876953127,234.67486967192844,396.54766845703127]},{"page":358,"text":"Light waves in the natural media such as opal, mother-of-pearl,beetleshells, etc.","rect":[66.27452087402344,417.4418029785156,385.15069242026098,408.50726318359377]},{"page":358,"text":"Light waves interacting with artificial diffraction gratings, one or two dimen-","rect":[66.27452087402344,429.3445739746094,385.09295020906105,420.4100341796875]},{"page":358,"text":"sional and three dimensional photonic bandgap crystals, in particular, artificial","rect":[66.27452087402344,441.34393310546877,385.14668138095927,432.3496398925781]},{"page":358,"text":"opals","rect":[66.2745361328125,453.2636413574219,87.38343443022922,444.3291015625]},{"page":358,"text":"Acoustic waves between periodically arranged columns in a theatre","rect":[66.2745361328125,465.2231750488281,338.6665729351219,456.28863525390627]},{"page":358,"text":"A common feature of all these media is a spatial periodicity with a period","rect":[65.76487731933594,483.1341247558594,385.13079157880318,474.1995849609375]},{"page":358,"text":"comparable to that of the external wave of any sort. In the three dimensional","rect":[53.81285858154297,495.0936584472656,385.1308427579124,486.15911865234377]},{"page":358,"text":"case, the diffraction may result in light localisation and trapping like electrons","rect":[53.81285858154297,507.05322265625,385.08807767974096,498.11865234375]},{"page":358,"text":"may be completely localised in a disordered metal (metal–insulator transition).","rect":[53.81285858154297,519.0126953125,371.8678860237766,510.07818603515627]},{"page":358,"text":"12.1.2.1 Hill and Mathieu Equations","rect":[53.81285858154297,558.2789306640625,215.91022123442844,549.3344116210938]},{"page":358,"text":"Theoretically one should solve a wave equation with dielectric permittivityperiodic","rect":[53.81285858154297,582.2677001953125,385.13083685113755,573.3331909179688]},{"page":358,"text":"in one, two or three dimensions but, for simplicity, consider a medium with periodic","rect":[53.81285858154297,594.227294921875,385.1656879253563,585.2927856445313]},{"page":359,"text":"348","rect":[53.812843322753909,42.55630874633789,66.50444931178018,36.73143005371094]},{"page":359,"text":"Fig. 12.4 Periodic medium","rect":[53.812843322753909,67.58130645751953,149.39755485887515,59.85148620605469]},{"page":359,"text":"with modulation of scalar","rect":[53.812843322753909,75.76238250732422,141.52874845130138,69.89517211914063]},{"page":359,"text":"dielectric permittivity","rect":[53.812843322753909,87.4087142944336,128.0781988540165,79.81436157226563]},{"page":359,"text":"12 Electro-Optical Effects in Cholesteric Phase","rect":[223.29483032226563,44.274986267089847,385.1406798102808,36.6636962890625]},{"page":359,"text":"modulation of dielectric permittivity in one direction, Fig. 12.4. A wave equation","rect":[53.812843322753909,180.11965942382813,385.17751399091255,171.1851043701172]},{"page":359,"text":"for the electric field in the one-dimensional case is called Hill equation [3]:","rect":[53.8138313293457,192.0792236328125,357.6521440274436,183.14466857910157]},{"page":359,"text":"d2E","rect":[176.7329559326172,214.82989501953126,192.13462060369216,206.309814453125]},{"page":359,"text":"dz2 þ eðzÞm c2 E ¼0","rect":[178.0354766845703,228.43292236328126,263.9023064713813,213.9253387451172]},{"page":359,"text":"(12.5)","rect":[361.056396484375,223.13877868652345,385.10576759187355,214.54287719726563]},{"page":359,"text":"where m \u0001 1 is magnetic susceptibility. The analysis of the Hill equation is a large","rect":[53.8137321472168,251.87774658203126,385.1217731304344,242.9431915283203]},{"page":359,"text":"area of mathematics. In the simplest case of the cosine (or sine) form of the spatial","rect":[53.8137321472168,263.8372802734375,385.1147599942405,254.90272521972657]},{"page":359,"text":"modulation of the scalar parameter eðzÞ ¼ e0 þ e1 cosð2pz=lÞ shown in the figure,","rect":[53.8137321472168,276.1355285644531,385.1168484261203,266.1849670410156]},{"page":359,"text":"we obtain a standard form of the Mathieu wave equation","rect":[53.815834045410159,287.75640869140627,282.8204278092719,278.82183837890627]},{"page":359,"text":"dd2zE2 þ ðZ þ gcos2xÞE ¼0","rect":[167.55638122558595,324.1101379394531,273.07949152997505,301.98699951171877]},{"page":359,"text":"(12.6)","rect":[361.0567932128906,318.81597900390627,385.10616432038918,310.27984619140627]},{"page":359,"text":"where l is period of the structure, x ¼ pz/l is the normalised coordinate along the","rect":[53.81511688232422,347.5549621582031,385.1748431987938,338.3016662597656]},{"page":359,"text":"z-axis and dimensionless parameters g and Z are related to the amplitude of the","rect":[53.815101623535159,359.5144958496094,385.1768268413719,350.5799560546875]},{"page":359,"text":"dielectric permittivity modulation e1 and the average value of permittivity e0 as","rect":[53.816062927246097,371.4740295410156,385.14215482817846,362.53948974609377]},{"page":359,"text":"shown in Fig. 12.4,","rect":[53.81432342529297,383.43359375,131.69704862143284,374.4990234375]},{"page":359,"text":"g ¼ \u0004pocl\u00052e1","rect":[192.25448608398438,423.48370361328127,246.26513741096816,397.6642150878906]},{"page":359,"text":"and","rect":[53.812843322753909,444.9410095214844,68.21658412030706,438.03839111328127]},{"page":359,"text":"2","rect":[219.83982849121095,465.9166564941406,223.3237921961244,461.203369140625]},{"page":359,"text":"\u0002 ¼ \u0004opcl\u0005 ðe0 \u0004 e1Þ:","rect":[177.01510620117188,487.0227355957031,261.9529565541162,463.13751220703127]},{"page":359,"text":"The general solution of Eq. (12.6) is given by the Floquet-Bloch theorem as a","rect":[65.76642608642578,510.5691223144531,385.1612018413719,501.6146545410156]},{"page":359,"text":"sum of products of a spatially periodic amplitudes A(x) and B(x) with oscillating","rect":[53.813411712646487,522.4718627929688,385.0875176530219,513.2186279296875]},{"page":359,"text":"exponential functions","rect":[53.813411712646487,534.431396484375,140.47676481353,525.4968872070313]},{"page":359,"text":"EðxÞ ¼ D1AðxÞexpibx þ D2BðxÞexpð\u0004ibxÞ","rect":[129.5480194091797,558.6893920898438,309.4344521914597,548.7387084960938]},{"page":359,"text":"(12.7)","rect":[361.0570068359375,557.9523315429688,385.10637794343605,549.4759521484375]},{"page":359,"text":"where D1 and D2 are arbitrary constants. The solution describes two waves","rect":[53.815330505371097,582.2699584960938,385.13888944731908,573.3352661132813]},{"page":359,"text":"with dimensionless wavevectors \u0007 b ¼ \u0007ðl=pÞk ¼ \u0007ðl=pÞðohni=cÞ propagating in","rect":[53.812992095947269,594.5681762695313,385.18361750653755,584.61767578125]},{"page":360,"text":"12.1 Cholesteric as One-Dimensional Photonic Crystal","rect":[53.812843322753909,44.274620056152347,241.14516929831567,36.663330078125]},{"page":360,"text":"349","rect":[372.4981994628906,42.62367248535156,385.1898245254032,36.73106384277344]},{"page":360,"text":"opposite directions. Due to periodicity of A and B, the electromagnetic waves can","rect":[53.812843322753909,68.2883529663086,385.12575617841255,59.35380554199219]},{"page":360,"text":"be presented as a discrete sum of infinite numbers of spatial Fourier harmonics","rect":[53.81282424926758,80.24788665771485,372.2231789981003,71.31333923339844]},{"page":360,"text":"m¼þ1","rect":[175.76995849609376,100.16023254394531,198.45334000748319,95.52364349365235]},{"page":360,"text":"E ¼ X As exp½iðk þ mq0Þz ;","rect":[156.39694213867188,115.50703430175781,280.9289087025537,101.57232666015625]},{"page":360,"text":"m¼\u00041","rect":[175.76995849609376,121.59315490722656,198.45334000748319,118.35800170898438]},{"page":360,"text":"(12.8)","rect":[361.0563049316406,112.78116607666016,385.10567603913918,104.30480194091797]},{"page":360,"text":"where q0 is the vector of one-dimensional reciprocal lattice, q0 ¼ 2p/l, as discussed","rect":[53.81464385986328,146.05453491210938,385.1780633073188,137.1000518798828]},{"page":360,"text":"in Section 5.3.1. Usually the periodicity of E on the wavevector axis allows one to","rect":[53.81432342529297,158.01406860351563,385.14321223310005,149.01974487304688]},{"page":360,"text":"consider only the waves with wavevectors in the range \u0004 p/l k p/l, i.e. in the","rect":[53.81433868408203,169.97360229492188,385.1770709819969,161.0191192626953]},{"page":360,"text":"first Brillouin zone.","rect":[53.815330505371097,180.00082397460938,131.81153531576877,172.9786834716797]},{"page":360,"text":"The Mathieu equation has no analytical solution despite e is scalar and its","rect":[65.76734161376953,193.8927001953125,385.15414823638158,184.95814514160157]},{"page":360,"text":"solutions can only be found numerically. The crucial parameter is the depth of","rect":[53.81531524658203,205.85220336914063,385.15023170320168,196.9176483154297]},{"page":360,"text":"the e-modulation.","rect":[53.81531524658203,215.77981567382813,124.61961026694064,208.87721252441407]},{"page":360,"text":"1.","rect":[53.81531524658203,233.58120727539063,61.28097196127658,226.84793090820313]},{"page":360,"text":"2.","rect":[53.81400680541992,258.0,61.27966352011447,250.76791381835938]},{"page":360,"text":"3.","rect":[53.8136100769043,281.5399475097656,61.27926679159884,274.6871337890625]},{"page":360,"text":"4.","rect":[53.814537048339847,342.0,61.28019376303439,334.4283447265625]},{"page":360,"text":"When e1 (i.e. g) is zero we have ordinary Maxwell equation for uniform","rect":[66.27699279785156,235.72360229492188,385.16278790116868,226.7881622314453]},{"page":360,"text":"medium.","rect":[66.27568817138672,245.65118408203126,101.58924527670627,238.7485809326172]},{"page":360,"text":"For very shallow e-relief, e1 \u0005 e0, g is small, Z > 0 and the waves are propa-","rect":[66.27568054199219,259.642822265625,385.1404965957798,250.70814514160157]},{"page":360,"text":"gating although with velocities depending on the z-coordinate.","rect":[66.27527618408203,271.60235595703127,317.8589748909641,262.66778564453127]},{"page":360,"text":"In the intermediate case e1 < e0 and g \u0001 Z, we observe a photon energy (or","rect":[66.27528381347656,283.56219482421877,385.17925391999855,274.62738037109377]},{"page":360,"text":"frequency) bands either allowed or forbidden for the wave propagation. There-","rect":[66.2762222290039,295.4649353027344,385.12261329499855,286.5303955078125]},{"page":360,"text":"fore, there are some selection rules for the Bragg diffraction of electromagnetic","rect":[66.2762222290039,307.42449951171877,385.17832220270005,298.48992919921877]},{"page":360,"text":"waves on the periodical structure. Here, we see a deep analogy with the Bloch -","rect":[66.2762222290039,319.384033203125,385.15642677156105,310.449462890625]},{"page":360,"text":"de Broglie waves in crystals. For this reason we speak of photonic crystals.","rect":[66.2762222290039,331.3435974121094,369.9981350472141,322.4090576171875]},{"page":360,"text":"For a very deep relief, e1 > e0/2, Z < 0 and g > Z, the waves cannot propagate","rect":[66.27621459960938,343.30377197265627,385.14841497613755,334.36859130859377]},{"page":360,"text":"at all. In such a structure one may observe only evanescent waves.","rect":[66.27526092529297,355.2633361816406,334.7204250862766,346.32879638671877]},{"page":360,"text":"From the analysis of the Mathieu equation, we can make the following general","rect":[65.7656021118164,373.17425537109377,385.18134934970927,364.23968505859377]},{"page":360,"text":"conclusions which are useful for further discussion of cholesteric liquid crystals:","rect":[53.81357955932617,385.1337890625,378.7220292813499,376.19921875]},{"page":360,"text":"1.","rect":[53.81357955932617,400.9032287597656,61.27923627402072,394.1699523925781]},{"page":360,"text":"2.","rect":[53.81357955932617,425.0,61.27923627402072,418.08905029296877]},{"page":360,"text":"3.","rect":[53.81357955932617,437.0,61.27923627402072,430.048583984375]},{"page":360,"text":"4.","rect":[53.81458282470703,473.0,61.28023953940158,465.92718505859377]},{"page":360,"text":"For a scalar e there isno general analyticalsolution even for the one-dimensional","rect":[66.27525329589844,403.04473876953127,385.130507064553,394.11016845703127]},{"page":360,"text":"problem.","rect":[66.27526092529297,415.0043029785156,102.1273388436008,406.06976318359377]},{"page":360,"text":"The wave characteristics are independent of the wave polarisation.","rect":[66.27525329589844,426.9638366699219,335.1395229866672,418.029296875]},{"page":360,"text":"A monochromatic wave is superposition of infinite number of plane waves: one-","rect":[66.27525329589844,438.9233703613281,385.11565528718605,429.98883056640627]},{"page":360,"text":"or two-waves approximations can only be used for waves with k \u0005 p/l (far from","rect":[66.27526092529297,450.8829040527344,385.1385264265593,441.9284362792969]},{"page":360,"text":"the gaps).","rect":[66.2762680053711,462.8424377441406,105.38337369467502,453.90789794921877]},{"page":360,"text":"There exists an infinite number of forbidden zones with the frequency gap Do","rect":[66.27626037597656,474.8019714355469,385.13450881655958,465.5387268066406]},{"page":360,"text":"decreasing with increasing the zone number.","rect":[66.2762680053711,486.7047424316406,246.12295194174534,477.77020263671877]},{"page":360,"text":"12.1.2.2 One Dimensional Photonic Band-Gap Structure (Modelling)","rect":[53.81458282470703,534.509033203125,354.00374732820168,525.325439453125]},{"page":360,"text":"An example of the numerical solution of the Maxwell equations for a one-dimen-","rect":[53.81458282470703,558.3484497070313,385.0837644180454,549.4139404296875]},{"page":360,"text":"sional photonic crystal is shown in Fig. 12.5a and b. I have modelled a stack of five","rect":[53.81458282470703,570.3079833984375,385.1524127788719,561.313720703125]},{"page":360,"text":"alternating layers, each with optical thickness of l/4: dielectric layers with thick-","rect":[53.81456756591797,582.267578125,385.1056150039829,573.0143432617188]},{"page":360,"text":"ness dd ¼ 0.075 mm and refraction index nd ¼ 2 and air gaps between them with","rect":[53.81458282470703,594.2296142578125,385.11660090497505,585.2353515625]},{"page":361,"text":"350","rect":[53.81285095214844,42.55923843383789,66.50445694117471,36.63276290893555]},{"page":361,"text":"12 Electro-Optical Effects in Cholesteric Phase","rect":[223.2948455810547,44.277915954589847,385.1406798102808,36.6666259765625]},{"page":361,"text":"thickness dair ¼ 0.15 mm and refraction index nair ¼ 1. In fact, I have verified the","rect":[53.812843322753909,68.1987075805664,385.1754535503563,59.294044494628909]},{"page":361,"text":"analytical result of a seminal paper [4] using the Palto’s software [5] based on the","rect":[53.813716888427737,80.24788665771485,385.17542303277818,71.25357818603516]},{"page":361,"text":"direct numerical solution of the Maxwell equations. In Fig. 12.5a the several band-","rect":[53.813716888427737,92.20748138427735,385.1097348770298,83.21317291259766]},{"page":361,"text":"gap (or stop-bands) for the unpolarised light transmission are seen in the curve","rect":[53.813716888427737,104.11019134521485,385.1466449566063,95.17564392089844]},{"page":361,"text":"marked off by the “stack”, which are separated by equal frequency intervals. The","rect":[53.813716888427737,116.0697250366211,385.1426471538719,107.13517761230469]},{"page":361,"text":"width of the stop-bands is determined by the difference of the refraction indices","rect":[53.813716888427737,128.02932739257813,385.0848733340378,119.09477233886719]},{"page":361,"text":"nd \u0004 nair ¼ 1: the larger the difference the wider the stop-band. Note that, within","rect":[53.813716888427737,139.98959350585938,385.1282586198188,131.05503845214845]},{"page":361,"text":"the stop band, the transmission of the unpolarized beam completely vanishes.","rect":[53.81327819824219,151.94912719726563,366.1894192269016,143.0145721435547]},{"page":361,"text":"For the stack considered here ddnd ¼ dairnair ¼ 0.15 mm ¼ l/4 and the position","rect":[65.76529693603516,163.908935546875,385.15361872724068,154.6556396484375]},{"page":361,"text":"of the first Bragg transmittance minimum (m ¼ 1) is expected at lB(m ¼ 1) ¼2","rect":[53.814796447753909,175.86846923828126,385.1794365983344,166.61517333984376]},{"page":361,"text":"<n> (dd þ dair) ¼ 2(ddnd þ dairnair) ¼ 0.6 mm. It is instructive to compare the","rect":[57.15635681152344,187.82827758789063,385.1741412944969,178.8737335205078]},{"page":361,"text":"results obtained for such a stack with the transmission of a slab of a cholesteric","rect":[53.81439971923828,197.6990966796875,385.1044391460594,190.79649353027345]},{"page":361,"text":"liquid crystal whose Bragg diffraction wavelength is approximately located at the","rect":[53.81439971923828,211.69061279296876,385.1731647319969,202.7560577392578]},{"page":361,"text":"same wavelength lB as for the stack stop-band at m ¼ 1. The cholesteric slab has","rect":[53.81439971923828,223.65057373046876,385.1329690371628,214.3968505859375]},{"page":361,"text":"the following parameters: the pitch is P ¼ 0.4 mm, slab thickness d ¼ 20 mm (50","rect":[53.81306076049805,235.61013793945313,385.1439141373969,226.61581420898438]},{"page":361,"text":"fullhelicalturns),no ¼ 1.5,ne ¼ 1.7,<n> ¼1.6,therefore,lB ¼ P<n> ¼ 0.64mm.","rect":[53.81406784057617,247.48020935058595,385.1840481331516,238.31655883789063]},{"page":361,"text":"The slab is bordered by infinitely thick glasses with refraction index ng ¼ 1.5. We","rect":[53.815330505371097,260.4591979980469,385.12830389215318,250.53521728515626]},{"page":361,"text":"see in Fig. 12.5a that in the cholesteric spectrum marked off by “CLC” there is a","rect":[53.81432342529297,271.48907470703127,385.1581500835594,262.4947509765625]},{"page":361,"text":"stop-band corresponding to m ¼ 1 and period P/2, as discussed above. Therefore a","rect":[53.81432342529297,283.4486083984375,385.16208685113755,274.5140380859375]},{"page":361,"text":"cholesteric liquid crystal may be regarded as a one-dimensional photonic crystal.","rect":[53.81529998779297,295.35137939453127,380.55535550619848,286.41680908203127]},{"page":361,"text":"Fig. 12.5 Comparison of the","rect":[53.812843322753909,394.346435546875,154.49535510324956,386.6166076660156]},{"page":361,"text":"non-polarized light","rect":[53.812843322753909,404.1979675292969,118.9148531362063,396.6036071777344]},{"page":361,"text":"transmission by a stack of","rect":[53.812843322753909,414.1739196777344,142.88336270911388,406.5795593261719]},{"page":361,"text":"dielectric layers anda","rect":[53.812843322753909,424.14984130859377,128.96830127024175,416.55548095703127]},{"page":361,"text":"cholesteric liquid crystal","rect":[53.812843322753909,434.1258239746094,137.84142784323755,426.5314636230469]},{"page":361,"text":"(CLC). The two materials","rect":[53.812843322753909,443.7063903808594,142.24371798514643,436.45068359375]},{"page":361,"text":"have the same Bragg","rect":[53.812843322753909,454.0209655761719,125.4891104140751,446.4266052246094]},{"page":361,"text":"reflection frequency","rect":[53.812843322753909,463.9969482421875,122.15629333643838,456.402587890625]},{"page":361,"text":"(numerical calculations, for","rect":[53.812843322753909,473.5775146484375,148.0234841690748,466.3218078613281]},{"page":361,"text":"parameters see the text). (a)","rect":[53.812843322753909,483.8921203613281,149.4144753800123,476.2977600097656]},{"page":361,"text":"Transmission spectra on the","rect":[53.81200408935547,493.8680725097656,149.51009509348394,486.2737121582031]},{"page":361,"text":"frequency scale showing the","rect":[53.81200408935547,503.8440246582031,150.92647692942144,496.2496643066406]},{"page":361,"text":"absence of high harmonics in","rect":[53.81200408935547,513.7632446289063,154.2804464736454,506.1689147949219]},{"page":361,"text":"the case of CLC; (b) blown","rect":[53.81200408935547,523.4005737304688,148.0852560439579,516.1448364257813]},{"page":361,"text":"transmission spectra at the","rect":[53.81285095214844,533.7151489257813,144.4131407477808,526.1207885742188]},{"page":361,"text":"wavelength scale showing the","rect":[53.81285095214844,543.6911010742188,155.34484240793706,536.0967407226563]},{"page":361,"text":"flat form of the CLC Bragg","rect":[53.81285095214844,553.6103515625,147.63765472559855,546.0159912109375]},{"page":361,"text":"band and oscillations of","rect":[53.81285095214844,561.8591918945313,135.406342445442,555.991943359375]},{"page":361,"text":"transmission at the edges of","rect":[53.81285095214844,573.562255859375,149.0574197159498,565.9678955078125]},{"page":361,"text":"the band","rect":[53.81285095214844,581.75439453125,83.0314688125126,575.8871459960938]},{"page":361,"text":"a","rect":[194.42063903808595,343.1327819824219,199.97445710106829,337.54547119140627]},{"page":361,"text":"b","rect":[194.08743286132813,466.95281982421877,200.19063940176378,459.6463317871094]},{"page":361,"text":"1.0","rect":[206.61619567871095,356.8321228027344,217.7233525982567,351.0671691894531]},{"page":361,"text":"0.8","rect":[206.61619567871095,372.5286865234375,217.7233525982567,366.76373291015627]},{"page":361,"text":"0.6","rect":[206.61619567871095,388.2252502441406,217.7233525982567,382.4602966308594]},{"page":361,"text":"0.4","rect":[206.61619567871095,403.9217834472656,217.7233525982567,398.1568298339844]},{"page":361,"text":"0.2","rect":[206.61619567871095,419.6183166503906,217.7233525982567,413.8533630371094]},{"page":361,"text":"0.0","rect":[206.61619567871095,435.31488037109377,217.7233525982567,429.5499267578125]},{"page":361,"text":"0","rect":[220.43621826171876,445.06256103515627,224.87908379942858,439.297607421875]},{"page":361,"text":"1.0","rect":[206.511474609375,480.08380126953127,217.61863152892077,474.31884765625]},{"page":361,"text":"0.8","rect":[206.511474609375,494.683349609375,217.61863152892077,488.91839599609377]},{"page":361,"text":"0.6","rect":[206.511474609375,509.28204345703127,217.61863152892077,503.51708984375]},{"page":361,"text":"0.4","rect":[206.511474609375,523.8807983398438,217.61863152892077,518.1158447265625]},{"page":361,"text":"0.2","rect":[206.511474609375,538.4794921875,217.61863152892077,532.7145385742188]},{"page":361,"text":"0.0","rect":[206.511474609375,553.0790405273438,217.61863152892077,547.3140869140625]},{"page":361,"text":"400","rect":[215.98297119140626,568.5231323242188,229.26823503153015,562.7769165039063]},{"page":361,"text":"1","rect":[245.2754669189453,444.91864013671877,249.71833245665514,439.297607421875]},{"page":361,"text":"CLC","rect":[259.1548767089844,406.2779846191406,275.14140592392519,400.2392578125]},{"page":361,"text":"CLC","rect":[276.883056640625,518.0787963867188,292.8695858555658,512.0401000976563]},{"page":361,"text":"600","rect":[287.8898620605469,568.5231323242188,301.17512590067079,562.7769165039063]},{"page":361,"text":"Wavelength, nm","rect":[274.72125244140627,581.66650390625,332.2667479893789,574.1988525390625]},{"page":362,"text":"12.1 Cholesteric as One-Dimensional Photonic Crystal","rect":[53.812843322753909,44.274620056152347,241.14516929831567,36.663330078125]},{"page":362,"text":"351","rect":[372.4981994628906,42.55594253540039,385.1898245254032,36.62946701049805]},{"page":362,"text":"As discussed in the previous section, indeed, there is only one stop-band in the","rect":[65.76496887207031,68.2883529663086,385.17176092340318,59.35380554199219]},{"page":362,"text":"cholesteric transmission spectrum. This is only valid for the light propagating along","rect":[53.812950134277347,80.24788665771485,385.15679255536568,71.31333923339844]},{"page":362,"text":"the helical axis. The minimum transmission of an unpolarized light is 0.5, because","rect":[53.812950134277347,92.20748138427735,385.10300481988755,83.21317291259766]},{"page":362,"text":"one circular polarization is totally reflected. As in the case of a stack, the width of","rect":[53.812950134277347,104.11019134521485,385.14791236726418,95.17564392089844]},{"page":362,"text":"the band is determined by the difference Dn ¼ ne \u0004 no. In the case of our","rect":[53.812950134277347,116.0697250366211,385.14238868562355,106.80648040771485]},{"page":362,"text":"cholesteric, Dn ¼ 0.2, that is much less than for the virtual stack modeled and,","rect":[53.81350326538086,126.06777954101563,385.1374172737766,118.7667465209961]},{"page":362,"text":"for this reason, in Fig. 12.5 the cholesteric spectrum is muchnarrower than the stack","rect":[53.81351852416992,139.98953247070313,385.1295098405219,130.99520874023438]},{"page":362,"text":"spectrum. However, anisotropy Dn ¼ 0.2 is quite large for both liquid crystals and","rect":[53.81450653076172,151.94906616210938,385.1453789811469,142.68582153320313]},{"page":362,"text":"real stacks made of alternating dielectric films. The structure of the cholesteric stop-","rect":[53.81450653076172,163.90863037109376,385.1444333633579,154.9740753173828]},{"page":362,"text":"band on the wavelength scale is well seen in Fig. 12.5b: there is a wide plateau","rect":[53.81450653076172,175.86813354492188,385.1791924577094,166.87380981445313]},{"page":362,"text":"between the two wavelengths corresponding to lo ¼ Pno and le ¼ Pne. A number","rect":[53.81549835205078,187.82766723632813,385.12569557038918,178.57437133789063]},{"page":362,"text":"of fringes on both sides of the stop-band increases with increasing slab thickness","rect":[53.81374740600586,199.73104858398438,385.1466409121628,190.79649353027345]},{"page":362,"text":"and their amplitude is determined by the reflection coefficient between the chole-","rect":[53.81374740600586,211.69061279296876,385.1118100723423,202.7560577392578]},{"page":362,"text":"steric and surrounding media. In our case, the fringes are not well seen because the","rect":[53.81374740600586,223.650146484375,385.1725238628563,214.71559143066407]},{"page":362,"text":"surrounding glasses have refraction index close to the average index of the chole-","rect":[53.81374740600586,235.60971069335938,385.1137631973423,226.67515563964845]},{"page":362,"text":"steric.","rect":[53.81374740600586,245.50741577148438,77.94275327231174,238.6346893310547]},{"page":362,"text":"12.1.3 Simple Analytical Solution for Light Incident Parallel","rect":[53.812843322753909,297.7369079589844,363.92268583129467,287.09912109375]},{"page":362,"text":"to the Helical Axis","rect":[95.61687469482422,309.57708740234377,188.30675391886397,301.04296875]},{"page":362,"text":"Our task is to find the spectrum of eigenmodes propagating along the helical axis of","rect":[53.812843322753909,339.2227783203125,385.14571510163918,330.268310546875]},{"page":362,"text":"a cholesteric liquid crystal and discuss some consequences of that. It is very rare","rect":[53.812843322753909,351.18231201171877,385.13278997613755,342.24774169921877]},{"page":362,"text":"and even unique case when, despite chirality and anisotropy of a medium, there is","rect":[53.812843322753909,363.0850830078125,385.18649686919408,354.1505126953125]},{"page":362,"text":"an analytical solution found many years ago by De Vries [6]. Here, we follow a","rect":[53.812843322753909,375.0446472167969,385.1586383648094,366.110107421875]},{"page":362,"text":"rather simple and very elegant analytic solution of this problem given by Kats [7].","rect":[53.812843322753909,387.0041809082031,385.1556362679172,378.06964111328127]},{"page":362,"text":"12.1.3.1 Wave Equations","rect":[53.81386947631836,422.8130798339844,166.5164376650925,413.6295166015625]},{"page":362,"text":"We again consider an electromagnetic wave propagating parallel to the helical axis","rect":[53.81386947631836,446.8018798828125,385.1537209902878,437.8673095703125]},{"page":362,"text":"of an infinite cholesteric medium (k || q0 || z) in the geometry corresponding to","rect":[53.81386947631836,458.7058410644531,385.14260188153755,449.77008056640627]},{"page":362,"text":"Fig. 12.3. Therefore non-zero components of the electric field Ex and Ey depend","rect":[53.81374740600586,471.638671875,385.10549250653755,461.7308349609375]},{"page":362,"text":"only on z. The Helmholtz wave equation","rect":[53.81444549560547,482.62506103515627,218.07585993817816,473.69049072265627]},{"page":362,"text":"]2E","rect":[189.4214630126953,502.2721252441406,204.49104568179394,493.8515930175781]},{"page":362,"text":"¼e","rect":[207.26461791992188,509.08380126953127,232.5166920888974,504.30279541015627]},{"page":362,"text":"]z2","rect":[190.78111267089845,515.8751831054688,202.5916602381166,508.1917419433594]},{"page":362,"text":"c2 ]t2","rect":[217.74417114257813,515.9747314453125,247.1713721521791,508.1917419433594]},{"page":362,"text":"(12.9)","rect":[361.0561828613281,510.73724365234377,385.10555396882668,502.2608947753906]},{"page":362,"text":"is writtenusing the dielectric permittivity tensor (12.2) with dimensionless dielectric","rect":[53.81450653076172,536.4721069335938,385.15930975152818,527.53759765625]},{"page":362,"text":"anisotropy d ¼ (e|| \u0004 e⊥)/(e|| þ e⊥) ¼ ea/2 <e2D> where <e2D> ¼ (e|| þ e⊥)/2","rect":[53.81350326538086,548.431640625,378.9484795670844,539.1983032226563]},{"page":362,"text":"_eðzÞ ¼ \u0001eexyxx eeyxyy \u0003 ¼ he2Di\u0001ð1 þdsdinco2sq20zq0zÞ ð1 \u0004dsdinco2sq20zq0zÞ\u0003","rect":[88.53638458251953,586.497802734375,350.4856680173459,562.6121826171875]},{"page":363,"text":"352","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.62946701049805]},{"page":363,"text":"12 Electro-Optical Effects in Cholesteric Phase","rect":[223.29483032226563,44.274620056152347,385.1406798102808,36.663330078125]},{"page":363,"text":"For the field components we write","rect":[65.76496887207031,68.2883529663086,203.97820318414535,59.35380554199219]},{"page":363,"text":"qq2zE2x ¼ c12 \u0004exx qq2tE2x þ exy qq2tE2y\u0005","rect":[67.29448699951172,107.82829284667969,198.99966116291064,82.51895141601563]},{"page":363,"text":"and","rect":[212.30545043945313,98.50830841064453,226.70919123700629,91.60569763183594]},{"page":363,"text":"qq2zE2y ¼ c12 \u0004eyx qq2tE2x þ eyy qq2tE2y\u0005","rect":[241.64825439453126,107.82829284667969,373.3543364558794,82.51895141601563]},{"page":363,"text":"Using Exðz;tÞ ¼ FxðzÞexpiot;Eyðz;tÞ ¼ FyðzÞexpiot we exclude the time","rect":[65.76624298095703,132.19140625,385.1061481304344,121.7623062133789]},{"page":363,"text":"dependence and, on account of k2 ¼ o2 <e2D>/c2, obtain the set of equations for","rect":[53.815086364746097,143.33377075195313,385.17827735749855,132.22618103027345]},{"page":363,"text":"the field amplitudes, which contains all optical properties of the cholesteric:","rect":[53.81356430053711,155.23660278320313,360.0310502774436,146.3020477294922]},{"page":363,"text":"qd2zF2x þ k2½Fx þ Fxdcos2q0z þ Fydsin2q0z","rect":[122.6367416381836,192.32769775390626,295.44159330474096,169.46728515625]},{"page":363,"text":"qd2zF2y þ k2½Fy \u0004 Fydcos2q0z þ Fxdsin2q0z","rect":[122.6367416381836,219.08099365234376,295.44159330474096,196.16384887695313]},{"page":363,"text":"¼0","rect":[300.9562072753906,186.0,316.35540095380318,178.6134033203125]},{"page":363,"text":"¼0","rect":[300.9562072753906,213.0,316.35540095380318,205.36669921875]},{"page":363,"text":"(12.10)","rect":[356.0706787109375,198.53977966308595,385.15886817781105,190.06341552734376]},{"page":363,"text":"These equations become simpler if one introduces circular field components:","rect":[65.7651138305664,242.58203125,376.0090471036155,233.64747619628907]},{"page":363,"text":"Eþ ¼ Fx þ iFy and E\u0004 ¼ Fx \u0004 iFy","rect":[142.68927001953126,267.31854248046877,295.8002104271926,257.56671142578127]},{"page":363,"text":"Then, Fx ¼ 21ðEþ þ E\u0004Þ and Fy ¼ 21iðEþ \u0004 E\u0004Þ","rect":[65.76595306396485,292.2591552734375,254.14731229888157,280.3916015625]},{"page":363,"text":"For","rect":[65.76586151123047,301.0,79.55244575349463,294.09814453125]},{"page":363,"text":"the","rect":[87.85723876953125,301.0,100.08100927545392,293.89892578125]},{"page":363,"text":"circular","rect":[108.3629150390625,301.0,139.01192603913916,293.89892578125]},{"page":363,"text":"components","rect":[147.2779083251953,302.83349609375,195.54188932524876,294.9149169921875]},{"page":363,"text":"on","rect":[203.8656005859375,301.0,213.8198174577094,295.0]},{"page":363,"text":"account","rect":[222.16143798828126,301.0,253.23850877353739,294.9149169921875]},{"page":363,"text":"Eqs. (12.10) read:","rect":[53.814857482910159,314.73626708984377,126.00279862949441,305.80169677734377]},{"page":363,"text":"of","rect":[261.5860900878906,301.0,269.8779538711704,293.89892578125]},{"page":363,"text":"exp(\u0007ia)","rect":[278.1827392578125,302.83349609375,315.25920997468605,293.89892578125]},{"page":363,"text":"qq2Fz2þ þ k2½Fþ þ F\u0004dexpð2iq0zÞ","rect":[141.0464324951172,351.7178649902344,271.02801908599096,328.96710205078127]},{"page":363,"text":"¼0","rect":[276.48492431640627,346.0,291.88506403974068,338.1131896972656]},{"page":363,"text":"qq2Fz2\u0004 þ k2½F\u0004 þ Fþdexpð\u00042iq0zÞ","rect":[141.0464324951172,378.35784912109377,278.6750832949753,355.6070556640625]},{"page":363,"text":"¼0","rect":[284.188720703125,372.0,299.5888604264594,364.696533203125]},{"page":363,"text":"¼","rect":[318.57196044921877,299.36724853515627,326.23670223448189,297.0364990234375]},{"page":363,"text":"cosa","rect":[329.5614318847656,300.7816162109375,348.3679234440143,295.9806823730469]},{"page":363,"text":"(12.11)","rect":[356.0715026855469,266.1028137207031,385.1596921524204,257.62646484375]},{"page":363,"text":"\u0007 isina,","rect":[351.70953369140627,302.0964050292969,385.1835598519016,293.89892578125]},{"page":363,"text":"(12.12)","rect":[356.071044921875,357.8695983886719,385.15923438874855,349.39324951171877]},{"page":363,"text":"12.1.3.2 Dispersion Relation","rect":[53.813472747802737,431.65679931640627,180.42905015299869,422.4732360839844]},{"page":363,"text":"Now we shall look for a solution of Eq. (12.12) in a form compatible with the","rect":[53.813472747802737,455.5888671875,385.17420232965318,446.654296875]},{"page":363,"text":"Floquet-Bloch theorem:","rect":[53.813472747802737,467.4388732910156,150.01695115634989,458.5939636230469]},{"page":363,"text":"Fþ ¼ Aþ expiðb þ q0Þz","rect":[171.5220184326172,492.8262939453125,267.45929350005346,482.8757629394531]},{"page":363,"text":"F\u0004 ¼ A\u0004 expiðb \u0004 q0Þz","rect":[171.52261352539063,507.73333740234377,267.45929350005346,497.7828063964844]},{"page":363,"text":"(12.13)","rect":[356.0716552734375,499.513916015625,385.15984474031105,491.0375671386719]},{"page":363,"text":"Here b is a wavevector of an electromagnetic wave, which can propagates in our","rect":[65.76610565185547,532.3336791992188,385.1409848770298,523.0903930664063]},{"page":363,"text":"periodic structure (an eigenmode). Equations (12.13) state that, due toperiodicity of","rect":[53.814083099365237,544.2364501953125,385.1509030899204,535.3019409179688]},{"page":363,"text":"the medium, the difference in wavevectors of possible modes should be equal to the","rect":[53.815040588378909,556.1959838867188,385.1737750835594,547.261474609375]},{"page":363,"text":"“lattice vector” of the structure (b þ q0) \u0004 (b \u0004 q0) ¼ 2q0 (remember, that, ina","rect":[53.815040588378909,568.06689453125,385.15992010309068,558.9122314453125]},{"page":363,"text":"cholesteric, the period of the e-modulation is P0/2). Substituting Fþ and F\u0004 into","rect":[53.814083099365237,580.162841796875,385.1580437760688,571.1815185546875]},{"page":363,"text":"Eq. (12.12) we find","rect":[53.814231872558597,592.07568359375,131.9697198014594,583.1411743164063]},{"page":364,"text":"12.1 Cholesteric as One-Dimensional Photonic Crystal","rect":[53.812843322753909,44.274620056152347,241.14516929831567,36.663330078125]},{"page":364,"text":"½k2 \u0004 ðb þ q0Þ2 Aþ þ dk2A\u0004 ¼0","rect":[151.92236328125,71.40447235107422,284.35149470380318,59.44980239868164]},{"page":364,"text":"dk2Aþ þ ½k2 \u0004 ðb \u0004 q0Þ2 A\u0004 ¼ 0:","rect":[151.9236602783203,88.29540252685547,287.04710328263186,76.34072875976563]},{"page":364,"text":"353","rect":[372.4981994628906,42.55594253540039,385.1898245254032,36.62946701049805]},{"page":364,"text":"(12.14)","rect":[356.0715637207031,78.6596450805664,385.15975318757668,70.18328094482422]},{"page":364,"text":"The two equations have a non-trivial solution only if the corresponding determi-","rect":[65.76599884033203,112.8967514038086,385.16475807038918,103.96220397949219]},{"page":364,"text":"nant (a11a22–a12a21) ¼ 0. From this condition we have a biquadratic equation for","rect":[53.813961029052737,124.8558578491211,385.17904029695168,115.92131042480469]},{"page":364,"text":"determination of the wave vector b as a function of k or frequency o. This is a","rect":[53.81332778930664,136.75863647460938,385.16111028863755,127.51530456542969]},{"page":364,"text":"dispersion relation","rect":[53.81433868408203,148.60853576660157,129.77192774823676,139.7636260986328]},{"page":364,"text":"b4 \u0004 2ðk2 þ q02Þb2 þ ½ðk2 \u0004 q20Þ2 \u0004 k4d2 ¼0","rect":[129.32200622558595,175.2847442626953,309.6708916764594,162.94894409179688]},{"page":364,"text":"(12.15)","rect":[356.0704345703125,174.16651916503907,385.15862403718605,165.57061767578126]},{"page":364,"text":"Its solution for the wave vectors of the two propagating modes is given by","rect":[65.7658462524414,198.82470703125,366.2496270280219,189.89015197753907]},{"page":364,"text":"ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi","rect":[237.22972106933595,214.0,280.79929650381339,213.0]},{"page":364,"text":"b2 ¼ k2 þ q20 \u0007 kq4q20 þ k2d2","rect":[158.04039001464845,231.490966796875,280.42219612678846,213.5720672607422]},{"page":364,"text":"(12.16)","rect":[356.0708312988281,228.23985290527345,385.15902076570168,219.70372009277345]},{"page":364,"text":"Note that the dispersion relation of the type b(o) is the key equation for","rect":[65.76526641845703,254.93792724609376,385.1212400039829,245.69459533691407]},{"page":364,"text":"discussion of the spectrum of the photonic band-gap. Indeed the derivative","rect":[53.81327438354492,266.8974609375,385.1600726909813,257.962890625]},{"page":364,"text":"vg ¼ do/db is nothing else as the group velocity of light in the considered sample.","rect":[53.81327438354492,279.81390380859377,385.1697048714328,269.6142883300781]},{"page":364,"text":"When this derivative tends to zero the velocity decreases and eventually the light","rect":[53.812896728515628,290.8171691894531,385.1189409024436,281.88262939453127]},{"page":364,"text":"does not propagate. We say the light stops. The inverse ratio db/do defines the","rect":[53.812896728515628,302.7767028808594,385.17456854059068,293.53338623046877]},{"page":364,"text":"density of the possible wavevectors db for the unit frequency interval do. It is so-","rect":[53.81387710571289,314.73626708984377,385.13179908601418,305.4929504394531]},{"page":364,"text":"called density of optical b-modes (DOM) or density of photonic states (DOS), the","rect":[53.81385803222656,326.69580078125,385.1746295757469,317.4524841308594]},{"page":364,"text":"concept playing the principal role in calculation of properties of photonic crystals,","rect":[53.81385803222656,338.59857177734377,385.0999416878391,329.66400146484377]},{"page":364,"text":"see, for instance [4,8]. Due to great importance of the dispersion relation, it is useful","rect":[53.81385803222656,350.5581359863281,385.2085099942405,341.62359619140627]},{"page":364,"text":"to present it in a more familiar form of the frequency (i.e., photon energy) depend-","rect":[53.81386184692383,362.51763916015627,385.15374122468605,353.58306884765627]},{"page":364,"text":"ing on wavevector. Since the wavevector of the incident light wave k ¼o","rect":[53.81386184692383,374.4771728515625,385.13276931460646,365.522705078125]},{"page":364,"text":"<e2D> 1/2/c is proportional to light frequency, Eq. (12.15) may be rewritten as","rect":[53.814918518066409,386.4378662109375,372.1112100039597,375.3470153808594]},{"page":364,"text":"k4ð1 \u0004 d2Þ \u0004 2k2ðb2 þ q20Þ þ ðb2 \u0004 q02Þ2 ¼0","rect":[129.99993896484376,413.00408935546877,308.99199763349068,400.66827392578127]},{"page":364,"text":"(12.17)","rect":[356.0714416503906,411.8858337402344,385.15963111726418,403.40948486328127]},{"page":364,"text":"After substituting the values of b into the field equations (12.13) we find the","rect":[65.76685333251953,436.54400634765627,385.1755756206688,427.3006896972656]},{"page":364,"text":"analytical solution of the original wave equation (12.9). Usually d is small, about","rect":[53.814857482910159,448.5035705566406,385.1029496915061,439.27020263671877]},{"page":364,"text":"0.01–01 but it is important for our consideration and cannot be ignored. However,","rect":[53.81386947631836,460.4631042480469,385.1507229378391,451.528564453125]},{"page":364,"text":"for a moment, consider a limit of infinitely small anisotropy, d ! 0. Then from","rect":[53.81386947631836,472.4226379394531,385.13776348710618,463.18927001953127]},{"page":364,"text":"Eq. (12.15) we have","rect":[53.813899993896487,484.3821716308594,135.3737262554344,475.3878479003906]},{"page":364,"text":"b2 ¼ k2 þ q20 \u0007 2kq0","rect":[177.97972106933595,510.49505615234377,260.4830786951478,498.6128234863281]},{"page":364,"text":"that is four solutions for b: b1 ¼ \u0007ðkþq0Þ and b2 ¼ \u0007ðk\u0004q0Þ. Equivalently, from","rect":[53.812843322753909,534.3733520507813,385.13751934648118,524.4228515625]},{"page":364,"text":"Eq. (12.17) we get four solutions for k: k1 ¼ \u0007ðbþq0Þ and k2 ¼ \u0007ðb\u0004q0Þ. Four","rect":[53.814598083496097,546.3330078125,385.1445554336704,536.3825073242188]},{"page":364,"text":"solutions mean that, at any frequency o, we have four circularly polarised eigen-","rect":[53.814659118652347,557.953857421875,385.11467872468605,549.0193481445313]},{"page":364,"text":"waves shown in Fig. 12.6a. The four waves differ by their polarisation and direction","rect":[53.814659118652347,569.9133911132813,385.09279719403755,560.9788818359375]},{"page":364,"text":"of propagation. The curve numbers in the figure corresponds to the following","rect":[53.814659118652347,581.8729858398438,385.16741267255318,572.9384765625]},{"page":364,"text":"wavevectors: \u0004(b \u0004 q0) (curve 1), \u0004(b þ q0) (curve 2), (b \u0004 q0) (curve 3) and","rect":[53.814659118652347,593.7433471679688,385.1453789811469,584.5892333984375]},{"page":365,"text":"354","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.62946701049805]},{"page":365,"text":"12 Electro-Optical Effects in Cholesteric Phase","rect":[223.29483032226563,44.274620056152347,385.1406798102808,36.663330078125]},{"page":365,"text":"Fig. 12.6 Dispersion relation for a cholesteric that has helical structure with wavevector q0. The","rect":[53.812843322753909,156.91046142578126,385.1964964606714,149.18064880371095]},{"page":365,"text":"abscissa corresponds to the wavevectors of the eigenmodes propagating in the medium and the","rect":[53.813655853271487,166.761962890625,385.15587756418707,159.1676025390625]},{"page":365,"text":"ordinate is the frequency of the incident light proportional to its wavevector, k ¼ oe1/2/c. (a) Very","rect":[53.813655853271487,176.7379150390625,385.1555532851688,167.3275146484375]},{"page":365,"text":"small","rect":[53.81332015991211,185.0,72.2161455556399,179.11929321289063]},{"page":365,"text":"optical","rect":[76.5845947265625,186.71365356445313,99.74254326071802,179.11929321289063]},{"page":365,"text":"anisotropy","rect":[104.1143798828125,186.71365356445313,139.8793234267704,179.11929321289063]},{"page":365,"text":"d ! 0:","rect":[144.27569580078126,185.0,169.23753074851099,178.86529541015626]},{"page":365,"text":"two","rect":[173.56198120117188,185.0,186.2451376357548,179.9828643798828]},{"page":365,"text":"pairs","rect":[190.6118927001953,186.71365356445313,207.0771377849511,179.11929321289063]},{"page":365,"text":"of","rect":[211.45742797851563,185.0,218.50550931555919,179.11929321289063]},{"page":365,"text":"circularly","rect":[222.843505859375,186.71365356445313,255.41861480860636,179.11929321289063]},{"page":365,"text":"polarized","rect":[259.832763671875,186.71365356445313,291.44330352930947,179.11929321289063]},{"page":365,"text":"eigenmodes","rect":[295.80242919921877,186.71365356445313,336.31406100272457,179.11929321289063]},{"page":365,"text":"propagate","rect":[340.7222900390625,186.71365356445313,374.1858153083277,179.9828643798828]},{"page":365,"text":"in","rect":[378.56103515625,185.0,385.16913360743447,179.11929321289063]},{"page":365,"text":"opposite direction without diffraction and each pair consists of a right- and left polarized beams.","rect":[53.81417465209961,196.68960571289063,385.17248794629537,189.09524536132813]},{"page":365,"text":"Note the absence of a stop-band on the o-scale at frequency oB. (b) Finite optical anisotropy d:","rect":[53.81417465209961,206.60882568359376,385.18514732565947,198.76004028320313]},{"page":365,"text":"modes 2 and 3 propagating in opposite directions suffer diffraction on the periodic structure anda","rect":[53.81327438354492,216.5843505859375,385.17492053293707,208.989990234375]},{"page":365,"text":"stop-band appears with a frequency gap Do ¼ oo \u0004 oe centered at Bragg frequency oB","rect":[53.81327438354492,226.56033325195313,355.43605598117679,218.68658447265626]},{"page":365,"text":"(b þ q0) (curve 4). The dashed (1, 2) and solid (3, 4) lines correspond to back and","rect":[53.812843322753909,262.25018310546877,385.1427849870063,253.00685119628907]},{"page":365,"text":"forward propagating waves, respectively, because their slopes corresponding to","rect":[53.812862396240237,274.1529235839844,385.14080134442818,265.2183837890625]},{"page":365,"text":"group velocities do/db have different signs. The crossover of lines 2 and 3 at","rect":[53.812862396240237,286.11248779296877,385.1258378750999,276.8691711425781]},{"page":365,"text":"b ¼ 0 and k ¼ q0 determines the Bragg frequency oB <e2D> 1/2/c ¼ q0 ¼ 2p/P0.","rect":[53.8138542175293,298.072509765625,385.1832241585422,286.98162841796877]},{"page":365,"text":"The situation changes when the optical anisotropy is finite. Consider a particular","rect":[65.76656341552735,310.0320739746094,385.1415036758579,301.0975341796875]},{"page":365,"text":"case of small wavevectors b \u0001 0. Then from Eq. (12.17) we have","rect":[53.814537048339847,321.9916076660156,319.4137653179344,312.748291015625]},{"page":365,"text":"2","rect":[250.88134765625,340.9355163574219,254.36531136116347,336.22222900390627]},{"page":365,"text":"ðk2 \u0004 q20Þ2 \u0004 k4d2 ¼ 0 or k2 ¼ ð1q\u00070 dÞ \u0001 q20ð1 \u000B dÞ","rect":[92.73049926757813,360.6466979980469,317.19357694731908,339.9315185546875]},{"page":365,"text":"(12.18)","rect":[356.07171630859377,353.0511474609375,385.1599057754673,344.5747985839844]},{"page":365,"text":"Since both q0 and d ¼ ea/2 <e2D> are fixed material parameters and k2 ¼ o2","rect":[65.76616668701172,384.1705627441406,385.18130140630105,373.1196594238281]},{"page":365,"text":"<e2D>/c2, the allowed frequency at b ¼ 0 takes two values","rect":[53.812843322753909,396.1302795410156,295.58441557036596,385.0792541503906]},{"page":365,"text":"ffiffiffiffiffiffiffiffiffiffi","rect":[111.59095001220703,412.0,132.37136376455556,410.0]},{"page":365,"text":"ffiffiffiffiffiffiffiffiffiffi","rect":[220.91612243652345,412.0,241.7104598338915,411.0]},{"page":365,"text":"oe ¼ cq0rhee2jDj i and oo ¼ cq0rhee2?Di with pffieffiffijffij ¼ njj;pffieffiffi?ffiffi ¼ n?","rect":[64.68854522705078,435.056640625,373.7854090860355,410.8775939941406]},{"page":365,"text":"Between the two frequencies there is a gap. Above we have qualitatively","rect":[65.76496887207031,458.57562255859377,385.1298760514594,449.60125732421877]},{"page":365,"text":"discussed an appearance of the forbidden frequency bands. Now, in Fig. 12.6b we","rect":[53.81294250488281,470.4953308105469,385.1706928081688,461.560791015625]},{"page":365,"text":"see the frequency gap formed by the corresponding dispersion curves. The width of","rect":[53.813968658447269,482.4548645019531,385.15081153718605,473.52032470703127]},{"page":365,"text":"the gap","rect":[53.813968658447269,494.4143981933594,83.28540125897894,485.4798583984375]},{"page":365,"text":"Do ¼ cq0Dn=hni","rect":[184.89002990722657,518.662841796875,254.09198392974094,508.73223876953127]},{"page":365,"text":"(12.19)","rect":[356.0718688964844,517.9357299804688,385.1600583633579,509.4593505859375]},{"page":365,"text":"determines the spectral interval of the Bragg diffraction where only two waves","rect":[53.814292907714847,542.1964111328125,385.11630643950658,533.2619018554688]},{"page":365,"text":"(no. 1 and 4 in the figure) can propagate at any o. The gap in Fig. 12.6b corresponds","rect":[53.814292907714847,554.1559448242188,385.1521340762253,545.221435546875]},{"page":365,"text":"to the minimum in the optical transmission at l0 ¼ 600–670nm in Fig. 12.5. The","rect":[53.814292907714847,566.1160278320313,385.1487506694969,556.8622436523438]},{"page":365,"text":"other two waves (nos. 2 and 3) cannot propagate within the gap: due to the","rect":[53.81493377685547,578.0755615234375,385.1179889507469,569.1410522460938]},{"page":365,"text":"diffraction they are completely reflected.","rect":[53.81493377685547,590.0350952148438,218.29729886557346,581.1005859375]},{"page":366,"text":"12.1 Cholesteric as One-Dimensional Photonic Crystal","rect":[53.812843322753909,44.274620056152347,241.14516929831567,36.663330078125]},{"page":366,"text":"355","rect":[372.4981994628906,42.55594253540039,385.1898245254032,36.62946701049805]},{"page":366,"text":"12.1.3.3 Rotation of Linearly Polarised Light","rect":[53.812843322753909,68.68677520751953,252.62230811921729,59.035072326660159]},{"page":366,"text":"Giant optical rotatory power is observed at wavelengths corresponding to the","rect":[53.812843322753909,92.20748138427735,385.1725543804344,83.24304962158203]},{"page":366,"text":"slopes of the selective reflection curve. In some cases, its magnitude reaches 3\u0003/mm.","rect":[53.812843322753909,104.11019134521485,385.1836208870578,95.17564392089844]},{"page":366,"text":"A sign of rotation changes at some wavelength within the reflection band, see","rect":[53.81493377685547,116.0702133178711,385.1149371929344,107.13566589355469]},{"page":366,"text":"Fig. 12.1. In that figure, the reflection band (stop band for transmission) has no flat","rect":[53.81493377685547,128.02981567382813,385.2055803067405,119.09526062011719]},{"page":366,"text":"top, the band is narrow because the particular cholesteric material has low optical","rect":[53.814964294433597,139.98934936523438,385.1299272305686,131.05479431152345]},{"page":366,"text":"anisotropy Dn \u0001 0.01.","rect":[53.814964294433597,151.94888305664063,144.64316983725315,142.68563842773438]},{"page":366,"text":"The magnitude of the rotation angle per unit length can be found from the same","rect":[65.7669906616211,163.90841674804688,385.1289752788719,154.97386169433595]},{"page":366,"text":"theory. To this effect, we consider the incident wave whose wavevector is far from","rect":[53.814964294433597,175.86795043945313,385.13495586991868,166.9333953857422]},{"page":366,"text":"the Bragg resonance on both sides of the latter jk \u0004 q0j \f q0d. Then, the disper-","rect":[53.814964294433597,188.15696716308595,385.1633237442173,178.22557067871095]},{"page":366,"text":"sion relation (12.16) for propagating waves becomes simpler if we use an expansion","rect":[53.814537048339847,199.73104858398438,385.19219294599068,190.79649353027345]},{"page":366,"text":"(1 þ x)1/2 \u0001 1 þ x/2:","rect":[53.81554412841797,211.2921905517578,142.55552537509989,200.60006713867188]},{"page":366,"text":"ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi","rect":[198.42843627929688,224.0,233.12681115225088,223.0]},{"page":366,"text":"b2 ¼ k2 þ q20 \u0007 2q0ks1 þ k42qd022 \u0001 ðk \u0007 q0Þ2 þ k43qd02 þ :::","rect":[105.2481460571289,253.305908203125,333.7154077259912,223.43443298339845]},{"page":366,"text":"Applying the same expansion for the second time, we find moduli of wavevec-","rect":[65.7660140991211,273.8697814941406,385.1309751114048,264.93524169921877]},{"page":366,"text":"tors |b1| and |b2|:","rect":[53.81399154663086,285.7397766113281,120.76734025547097,276.5860290527344]},{"page":366,"text":"k3d2","rect":[248.33241271972657,307.3130798339844,265.46788094124158,298.0760498046875]},{"page":366,"text":"jb1j ¼ ðk þ q0Þ þ 8q0ðk þ q0Þ","rect":[159.79580688476563,323.2372741699219,280.82721342192846,306.4283752441406]},{"page":366,"text":"k3d2","rect":[246.68968200683595,336.390380859375,263.82502815803846,327.1532287597656]},{"page":366,"text":"jb2j ¼ ðk \u0004 q0Þ \u0004 8q0ðk \u0004 q0Þ","rect":[158.15353393554688,352.3145751953125,279.18451322661596,335.505615234375]},{"page":366,"text":"(12.20)","rect":[356.1275634765625,329.4721984863281,385.1540464004673,320.995849609375]},{"page":366,"text":"We can see that there are two modes with wavevectors b1 \u0004 q0 and b2 þ q0","rect":[65.7652359008789,373.82220458984377,385.18130140630105,364.6680908203125]},{"page":366,"text":"compatible with dispersion relation (12.16). Further, according to Eqs. (12.14), the","rect":[53.812843322753909,385.8144226074219,385.17359197809068,376.8798828125]},{"page":366,"text":"field amplitude ratios for the two modes are dramatically different:","rect":[53.81185531616211,397.77398681640627,323.5969072110374,388.83941650390627]},{"page":366,"text":"Aþ","rect":[125.69557189941406,420.57696533203127,137.178597562598,411.8381042480469]},{"page":366,"text":"¼","rect":[140.47999572753907,423.9517822265625,148.14473751280219,421.62103271484377]},{"page":366,"text":"A\u0004 ðb þ q0Þ2 \u0004 k2","rect":[125.69557189941406,436.08880615234377,209.4457862146791,424.1341552734375]},{"page":366,"text":"and","rect":[224.8812713623047,425.3860778808594,239.28501215985785,418.48345947265627]},{"page":366,"text":"ðb \u0004 q0Þ2 \u0004 k2","rect":[254.2800750732422,421.01171875,312.8231360681947,409.0003967285156]},{"page":366,"text":"dk2","rect":[276.938232421875,432.2943115234375,290.10847542366346,424.5308837890625]},{"page":366,"text":"(12.21)","rect":[356.1281433105469,427.01959228515627,385.15462623445168,418.5432434082031]},{"page":366,"text":"Indeed Aþ/A\u0004 \f 1 for the first wave and Aþ/A\u0004 \u0005 1 for the second wave.","rect":[65.76583099365235,456.7120361328125,368.4809841683078,447.73046875]},{"page":366,"text":"We see that the two waves are nearly circular and polarised in opposite direc-","rect":[65.7664566040039,468.62493896484377,385.1632321914829,459.69036865234377]},{"page":366,"text":"tions. The optical rotation C of linearly polarised light per unit length is defined as a","rect":[53.814430236816409,480.58447265625,385.1592487163719,471.54034423828127]},{"page":366,"text":"half of the wavevector difference between the two circular waves with refraction","rect":[53.814430236816409,490.5120544433594,385.15630427411568,483.60943603515627]},{"page":366,"text":"indices n1and n2","rect":[53.814430236816409,504.0062255859375,119.62877302495338,495.56903076171877]},{"page":366,"text":"C ¼ c=d ¼ Dk=2","rect":[181.4344024658203,525.8046875,257.55922785810005,515.8740844726563]},{"page":366,"text":"Therefore, from Eq. (12.20) we get","rect":[65.76747131347656,546.334228515625,207.65078599521707,537.3997192382813]},{"page":366,"text":"k3d2","rect":[216.384521484375,567.818115234375,233.51998970589004,558.5810546875]},{"page":366,"text":"k3d2","rect":[275.9185485839844,567.818115234375,293.0540015467103,558.5810546875]},{"page":366,"text":"k4d2","rect":[338.5115051269531,567.818115234375,355.64683601936658,558.5810546875]},{"page":366,"text":"Dk ¼ ðjb1j \u0004 q0Þ \u0004 ðjb2j þ q0Þ ¼ 8q0ðk þ q0Þ þ 8q0ðk \u0004 q0Þ ¼ 4q0ðk2 \u0004 q02Þ","rect":[64.29423522949219,584.29296875,373.04547514067846,566.9334716796875]},{"page":366,"text":"(12.22)","rect":[356.1283264160156,597.6290893554688,385.1548093399204,589.1527099609375]},{"page":367,"text":"356","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.62946701049805]},{"page":367,"text":"12 Electro-Optical Effects in Cholesteric Phase","rect":[223.29483032226563,44.274620056152347,385.1406798102808,36.663330078125]},{"page":367,"text":"Remembering that k2 ¼ <e2D> (2p/l0)2 and d ¼ ea=2he2Di ¼ ðn2jj \u0004 n2?Þ=","rect":[65.76496887207031,71.04999542236328,385.1798028092719,57.23747253417969]},{"page":367,"text":"ðn2jj þ n?2 Þ, for the optical rotatory power we finally find","rect":[53.815086364746097,83.80316925048828,279.01589289716255,70.55923461914063]},{"page":367,"text":"k2d2","rect":[161.94851684570313,108.13678741455078,179.08395454964004,98.89956665039063]},{"page":367,"text":"C¼","rect":[117.3127212524414,114.71189880371094,135.9669115850678,107.81925201416016]},{"page":367,"text":"8q0\b1 \u0004 l02\u0007lB2","rect":[138.7806396484375,132.1195831298828,196.36200188764156,113.57998657226563]},{"page":367,"text":"pP0ðnj2j \u0004 n?2 Þ","rect":[240.23211669921876,110.89655303955078,295.7815286074753,97.65267944335938]},{"page":367,"text":"¼","rect":[205.56527709960938,113.39791107177735,213.2300188848725,111.0671615600586]},{"page":367,"text":"8l02ðn2jj þ n?2 Þ\b1 \u0004 l20\u0007lB2","rect":[216.0453338623047,132.1195831298828,313.5608696854931,113.57998657226563]},{"page":367,"text":"(12.23)","rect":[356.0709228515625,116.46573638916016,385.15911231843605,107.98937225341797]},{"page":367,"text":"Here l0 is the wavelength of the linearly polarised incident light (in vacuum) and","rect":[65.7653579711914,155.57675170898438,385.14247981122505,146.32321166992188]},{"page":367,"text":"lB ¼ ¼ P0 <e2D>1/2 is the wavelength of the Bragg reflection maximum. In this","rect":[53.81357192993164,167.5364990234375,385.14230741606908,156.44577026367188]},{"page":367,"text":"approximation, the value of c diverges at l0 ! lB, however, the formula describes","rect":[53.81338882446289,179.49606323242188,385.1550637637253,170.24276733398438]},{"page":367,"text":"both the spectral shape and the magnitude of the optical rotation on both sides of the","rect":[53.81325912475586,191.45571899414063,385.1730426616844,182.5211639404297]},{"page":367,"text":"Bragg reflection maximum (except the top of the reflection band) in agreement with","rect":[53.81325912475586,203.35848999023438,385.11428156903755,194.42393493652345]},{"page":367,"text":"experimental data shown in Fig. 12.1.","rect":[53.81325912475586,215.31802368164063,206.35458035971409,206.3834686279297]},{"page":367,"text":"12.1.3.4 Waveguide Regime","rect":[53.81325912475586,241.47024536132813,178.84510076715316,232.13726806640626]},{"page":367,"text":"This case corresponds to a large pitch of a cholesteric with respect to the wave-","rect":[53.81325912475586,265.3096618652344,385.1063474258579,256.3751220703125]},{"page":367,"text":"length P0 \f l0 that is a small wavevector of the helical structure and rather high","rect":[53.81325912475586,277.27056884765627,385.1272820573188,268.01727294921877]},{"page":367,"text":"frequency of the incident light satisfying a condition of kd \f q0. Then, from","rect":[53.813297271728519,289.2301025390625,385.13825176835618,279.9967346191406]},{"page":367,"text":"Eq. (12.16)","rect":[53.814353942871097,301.1329650878906,99.27324806062353,292.19842529296877]},{"page":367,"text":"b2 ¼ k2 þ q20 \u0007 k2d \u0001 k2 \u0007 k2d","rect":[156.5686798095703,327.302734375,282.42507258466255,315.3636779785156]},{"page":367,"text":"and, according to Eqs. (12.14) for q0 ! 0 the ratio Aþ/A\u0004 ¼ k2d/\u0007 k2d ¼ \u00071. This","rect":[53.81370162963867,350.94561767578127,385.18881620513158,339.79132080078127]},{"page":367,"text":"corresponds to the linearly polarised waves with polarisation vector rotating in","rect":[53.814144134521487,362.80181884765627,385.14006892255318,353.86724853515627]},{"page":367,"text":"space following the helical structure. We meet such a waveguide (or Mauguin)","rect":[53.814144134521487,374.80120849609377,385.17388282624855,365.8069152832031]},{"page":367,"text":"regime [9] in cholesterics with very long pitch, l0 \u0005 P0(n|| \u0004 n⊥) and also in the","rect":[53.814144134521487,386.7607421875,385.17578924371568,377.4676208496094]},{"page":367,"text":"case of the twisted nematic cell, already discussed in Section 11.1.1.","rect":[53.81405258178711,398.6807556152344,330.1807522347141,389.7462158203125]},{"page":367,"text":"12.1.4 Other Important Cases","rect":[53.812843322753909,434.93414306640627,209.1078403446452,424.3800354003906]},{"page":367,"text":"12.1.4.1 Cholesteric Slab of Finite Thickness","rect":[53.812843322753909,461.12579345703127,250.4065286074753,453.49609375]},{"page":367,"text":"In thin cells there is an additional effect of the interference from the parallel","rect":[53.812843322753909,486.4792175292969,385.0800004727561,477.544677734375]},{"page":367,"text":"boundaries resulting in the spectral oscillations observed on both sides of the","rect":[53.812843322753909,498.4387512207031,385.11484564020005,489.50421142578127]},{"page":367,"text":"Bragg maximum. Such fringes are well seen in Fig. 12.7, the corresponding","rect":[53.812843322753909,510.3982849121094,385.11882868817818,501.4637451171875]},{"page":367,"text":"numerical calculations being made for the cholesteric slab of thickness 4 mm. Of","rect":[53.812862396240237,522.3577880859375,385.1675962051548,513.4033203125]},{"page":367,"text":"course, in this case, the theory is more difficult [3].","rect":[53.81288146972656,534.3173828125,260.25323148276098,525.3828735351563]},{"page":367,"text":"12.1.4.2 Oblique Incidence of Light","rect":[53.81386947631836,558.4296875,211.5005124649204,549.2859497070313]},{"page":367,"text":"When light impinges on a cholesteric at some angle i with respect to the helical axis,","rect":[53.81386947631836,582.2691040039063,385.1516995003391,573.3345947265625]},{"page":367,"text":"the following new features should be mentioned:","rect":[53.81386947631836,594.2286376953125,250.99186570713114,585.2941284179688]},{"page":368,"text":"12.1 Cholesteric as One-Dimensional","rect":[53.8123779296875,43.0,181.7369966908938,36.66363525390625]},{"page":368,"text":"Fig. 12.7 Calculated","rect":[53.812843322753909,67.58130645751953,127.27271026759073,59.85148620605469]},{"page":368,"text":"transmission spectra ofa","rect":[53.812843322753909,77.4895248413086,138.65453479074956,69.89517211914063]},{"page":368,"text":"cholesteric for non-polarized","rect":[53.812843322753909,87.4087142944336,152.36652130274698,79.81436157226563]},{"page":368,"text":"light and different angles of","rect":[53.812843322753909,97.3846664428711,149.17164701087169,89.79031372070313]},{"page":368,"text":"incoming light incidence: 5\u0003,","rect":[53.812843322753909,107.36067962646485,153.4144465644594,99.71552276611328]},{"page":368,"text":"45\u0003 and 60\u0003 with respect to","rect":[53.81326675415039,117.33626556396485,147.4847921767704,109.69116973876953]},{"page":368,"text":"the helical axis. Both","rect":[53.81312942504883,125.50291442871094,126.30841583399698,119.66110229492188]},{"page":368,"text":"materials have helical pitch","rect":[53.81312942504883,137.23141479492188,147.90530914454386,129.63705444335938]},{"page":368,"text":"0.25 mm, refraction indices","rect":[53.81312942504883,147.13116455078126,146.22238619803705,139.5622100830078]},{"page":368,"text":"n|| ¼ 1.73 and n⊥ ¼ 1.51,","rect":[53.81312942504883,156.722900390625,142.84398147656879,149.5376434326172]},{"page":368,"text":"cell thickness 4 mm","rect":[53.8123779296875,167.02584838867188,120.35107658739078,159.5076904296875]},{"page":368,"text":"Photonic","rect":[184.1526336669922,43.0,214.28249499582769,36.68056869506836]},{"page":368,"text":"Crystal","rect":[216.6668243408203,44.274925231933597,241.1447115346438,36.68056869506836]},{"page":368,"text":"°","rect":[284.1072692871094,120.9405746459961,286.504728727663,119.1893310546875]},{"page":368,"text":"40","rect":[275.21923828125,126.54788970947266,284.1069819317803,120.78162384033203]},{"page":368,"text":"400","rect":[301.3110656738281,173.00985717773438,314.6426569806084,167.24359130859376]},{"page":368,"text":"Wavelength, nm","rect":[273.2253112792969,186.11094665527345,330.9715873895275,178.6171875]},{"page":368,"text":"357","rect":[372.49774169921877,42.55624771118164,385.1893667617313,36.6297721862793]},{"page":368,"text":"500","rect":[360.2290344238281,173.00985717773438,373.5606257306084,167.24359130859376]},{"page":368,"text":"1.","rect":[53.812843322753909,228.0,61.27850003744845,220.78387451171876]},{"page":368,"text":"2.","rect":[53.81278610229492,252.0,61.27844281698947,244.70297241210938]},{"page":368,"text":"3.","rect":[53.813411712646487,312.0,61.27906842734103,304.44427490234377]},{"page":368,"text":"4.","rect":[53.814414978027347,348.0,61.28007169272189,340.3228759765625]},{"page":368,"text":"The spectral maximum of the Bragg reflection lB1 is displaced to the short-wave","rect":[66.27452087402344,229.65866088867188,385.15955389215318,220.40536499023438]},{"page":368,"text":"side with the increase of i, see Fig. 12.7.","rect":[66.27544403076172,241.61822509765626,229.63499112631565,232.6836700439453]},{"page":368,"text":"An infinite number of reflections of higher orders emerge. The correspondent","rect":[66.27445983886719,253.5777587890625,385.171522689553,244.64320373535157]},{"page":368,"text":"minima of the transmission lB2, lB3 etc. are not seen in Fig. 12.7 because they","rect":[66.27445220947266,265.53765869140627,385.16619196942818,256.2840270996094]},{"page":368,"text":"are deeply in the UV region. The frequency range of the reflection (energy gap)","rect":[66.27509307861328,277.4971923828125,385.1164792617954,268.5626220703125]},{"page":368,"text":"is reduced with the order of reflection m and the peaks of reflections (or pits of","rect":[66.27509307861328,289.39996337890627,385.1512693008579,280.46539306640627]},{"page":368,"text":"transmission) becomes sharper.","rect":[66.27509307861328,301.3594970703125,192.50442166830784,292.4249267578125]},{"page":368,"text":"Higher orders of reflection have a complicated spectral and angular dependence.","rect":[66.27508544921875,313.3190612792969,385.07964749838598,304.384521484375]},{"page":368,"text":"There is a fine structure in the form of spectral satellites separated from the main","rect":[66.27509307861328,325.2785949707031,385.12944880536568,316.34405517578127]},{"page":368,"text":"harmonic by a distance dependent on the incidence angle [3].","rect":[66.27509307861328,337.2381286621094,314.0662502815891,328.3035888671875]},{"page":368,"text":"In the applied electric field the high order reflections may appear even","rect":[66.27609252929688,349.1976623535156,385.1164177995063,340.26312255859377]},{"page":368,"text":"for the incident light propagating parallel to the helical axis as discussed in","rect":[66.2761001586914,361.1572265625,385.14138117841255,352.22265625]},{"page":368,"text":"Section 12.2.2.","rect":[66.2761001586914,372.0,126.54287381674533,364.18218994140627]},{"page":368,"text":"12.1.4.3 Diffraction and Scattering","rect":[53.814414978027347,415.0264587402344,208.78554621747504,405.6934814453125]},{"page":368,"text":"Diffraction on the one-dimensional helical structure. Such a structure can be","rect":[53.814414978027347,437.06597900390627,385.1532672710594,429.74505615234377]},{"page":368,"text":"obtained from the initial quasi-planar texture with a small tilt (e.g. in the x-","rect":[53.81439971923828,450.7686462402344,385.16024146882668,441.8341064453125]},{"page":368,"text":"direction) of the helical axis with respect to the cell normal z. A cholesteric should","rect":[53.81439971923828,462.7281799316406,385.17913142255318,453.79364013671877]},{"page":368,"text":"have positive dielectric anisotropy. Then upon application of the strong electric","rect":[53.814414978027347,474.687744140625,385.15027654840318,465.753173828125]},{"page":368,"text":"field Ez the helix unwinds. After switching the field off, a one-dimensional helical","rect":[53.814414978027347,486.6495056152344,385.1089921719749,477.7127685546875]},{"page":368,"text":"structure appears with the axis parallel to the cell boundaries. When white light is","rect":[53.812896728515628,498.6090393066406,385.1846047793503,489.67449951171877]},{"page":368,"text":"incident onto that structure as shown in Fig. 4.29a the helix behaves as a diffraction","rect":[53.812896728515628,510.568603515625,385.0910576920844,501.634033203125]},{"page":368,"text":"grating and iridescent colours are observed [10]. The spectral positions of the","rect":[53.812896728515628,522.4713134765625,385.1158832378563,513.5368041992188]},{"page":368,"text":"diffraction maxima depend on the light incidence angle as expected from the","rect":[53.8118896484375,534.430908203125,385.1159137554344,525.4963989257813]},{"page":368,"text":"theory. For a monochromatic light the diffraction spots are located at angles \u00072y,","rect":[53.8118896484375,546.390380859375,385.1815456917453,537.1470947265625]},{"page":368,"text":"which are symmetric with respect to the incident beam direction and satisfy the","rect":[53.8128776550293,558.3499755859375,385.16968572809068,549.4154663085938]},{"page":368,"text":"conditions q ¼ 2mk0 sinW ¼ q0 ¼ 4p=P0. Therefore P0 ¼ l0=msinW, where l0 is","rect":[53.8128776550293,570.6390380859375,385.1891213809128,560.7084350585938]},{"page":368,"text":"light wavelength in air. Note that the refraction index of the medium is not included","rect":[53.81450653076172,582.2698974609375,385.17534724286568,573.3353881835938]},{"page":368,"text":"in the formula. In fact, it appears twice, once in the wavevector conservation law","rect":[53.81450653076172,594.2294311523438,385.1324744220051,585.294921875]},{"page":369,"text":"358","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.62946701049805]},{"page":369,"text":"12 Electro-Optical Effects in Cholesteric Phase","rect":[223.29483032226563,44.274620056152347,385.1406798102808,36.663330078125]},{"page":369,"text":"within the medium and then in the Snell law for the diffracted beam leaving the","rect":[53.812843322753909,68.2883529663086,385.17160833551255,59.35380554199219]},{"page":369,"text":"medium for air. Therefore, n is compensated, at least, for small diffraction angles","rect":[53.812843322753909,80.24788665771485,385.12082304106908,71.31333923339844]},{"page":369,"text":"2y. The period of the optical properties in this geometry is close to P0/2 although","rect":[53.8138313293457,92.20772552490235,385.10976496747505,82.96415710449219]},{"page":369,"text":"may depend on cell thickness. The first order diffraction (m ¼ 1) is very intense, the","rect":[53.813716888427737,104.1104965209961,385.1754840679344,95.17594909667969]},{"page":369,"text":"highest orders are much weaker (even should absent in the ideal case). By measur-","rect":[53.81475067138672,116.0699691772461,385.1585935196079,107.13542175292969]},{"page":369,"text":"ing scattering angle 2y one can find the pitch of the helix.","rect":[53.81475067138672,128.02957153320313,287.14043851401098,118.78623962402344]},{"page":369,"text":"Scattering by inhomogeneous focal-conic structure. When a white light beam","rect":[65.76677703857422,140.38751220703126,385.14264629960618,130.92506408691407]},{"page":369,"text":"is incident at an angle a on a non-aligned layer of a short-pitch cholesteric liquid","rect":[53.814720153808597,151.94863891601563,385.11480036786568,143.0140838623047]},{"page":369,"text":"crystal the scattered light shows iridescent colours. The reason lies in a light","rect":[53.814720153808597,163.908203125,385.1217485196311,154.97364807128907]},{"page":369,"text":"diffraction","rect":[53.814720153808597,174.0,96.07730952313909,166.9331817626953]},{"page":369,"text":"from","rect":[101.5650634765625,174.0,120.94591473222335,166.9331817626953]},{"page":369,"text":"randomly","rect":[126.43167877197266,175.86773681640626,164.77230158856879,166.9331817626953]},{"page":369,"text":"oriented","rect":[170.21826171875,174.0,202.95766535810004,166.9331817626953]},{"page":369,"text":"cholesteric","rect":[208.453369140625,174.0,251.8198932476219,166.9331817626953]},{"page":369,"text":"planes.","rect":[257.3375244140625,175.86773681640626,285.3845181038547,166.9331817626953]},{"page":369,"text":"By","rect":[290.8713073730469,175.86773681640626,302.4778985123969,167.13238525390626]},{"page":369,"text":"averaging","rect":[307.9208679199219,175.86773681640626,347.37542048505318,166.9331817626953]},{"page":369,"text":"variable","rect":[352.84027099609377,174.0,385.1476520366844,166.9331817626953]},{"page":369,"text":"Bragg conditions a formula has been derived [11] that relates the scattered wave-","rect":[53.814720153808597,187.8272705078125,385.1097348770298,178.89271545410157]},{"page":369,"text":"length to the observation angle b:","rect":[53.813716888427737,199.73004150390626,189.4995713956077,190.48670959472657]},{"page":369,"text":"l ¼ P0mhni cos\u00041=2acosshinnia þ 1=2acosshinnib\u0005","rect":[127.50872802734375,237.8539276123047,311.49804373127,213.9687042236328]},{"page":369,"text":"(12.24)","rect":[356.0719909667969,230.1674346923828,385.1601804336704,221.69107055664063]},{"page":369,"text":"With this equation and knowing the angles of incidence a, and of reflection b of","rect":[65.76642608642578,261.3437194824219,385.1532224258579,252.1003875732422]},{"page":369,"text":"monochromatic light, one can also determine the helical pitch of the cholesteric","rect":[53.815406799316409,273.3032531738281,385.10450018121568,264.36871337890627]},{"page":369,"text":"liquid crystal.","rect":[53.815406799316409,285.2628173828125,109.09113736655002,276.3282470703125]},{"page":369,"text":"When the pitch changes with variation of temperature the colours also change.","rect":[65.7674331665039,297.22235107421877,385.1801724007297,288.28778076171877]},{"page":369,"text":"This phenomenon is used in thermography, a sensitive technique for measurements","rect":[53.815406799316409,309.1819152832031,385.1801797305222,300.24737548828127]},{"page":369,"text":"of the distribution of temperature over various objects, for example, in medicine for","rect":[53.815406799316409,321.14141845703127,385.1801694473423,312.20684814453127]},{"page":369,"text":"making temperature maps of human skin. In technics, cholesterics are used for the","rect":[53.815406799316409,333.1009826660156,385.17417181207505,324.16644287109377]},{"page":369,"text":"estimation of the temperature distribution over plane electronic circuits and other","rect":[53.815406799316409,345.00372314453127,385.13836036531105,336.06915283203127]},{"page":369,"text":"objects with a relatively flat surface.","rect":[53.815406799316409,356.9632873535156,200.40707059408909,348.02874755859377]},{"page":369,"text":"12.2 Dielectric Instability of Cholesterics","rect":[53.812843322753909,406.7857360839844,271.0177829715983,395.4307861328125]},{"page":369,"text":"In this chapter we consider the most characteristic phenomena related to the electric","rect":[53.812843322753909,433.7662658691406,385.1477435894188,424.83172607421877]},{"page":369,"text":"field interaction with chiral, quasi-layered structure of cholesteric liquid crystals.","rect":[53.812843322753909,445.725830078125,379.65899320151098,436.791259765625]},{"page":369,"text":"12.2.1 Untwisting of the Cholesteric Helix","rect":[53.812843322753909,494.84576416015627,271.32471980952809,484.2916564941406]},{"page":369,"text":"12.2.1.1 De Gennes–Meyer Model for Field Induced Cholesteric–Nematic","rect":[53.812843322753909,522.8702392578125,374.26877630426255,513.4077758789063]},{"page":369,"text":"Transition","rect":[96.18294525146485,532.5787963867188,141.0266606266315,525.4171752929688]},{"page":369,"text":"In the simplest case, this transition is observed in a cholesteric with positive","rect":[53.812843322753909,558.3504638671875,385.13779485895005,549.4159545898438]},{"page":369,"text":"dielectric","rect":[53.812843322753909,569.0,91.18891180230939,561.37548828125]},{"page":369,"text":"or","rect":[96.46662902832031,569.0,104.75849281160009,563.0]},{"page":369,"text":"diamagnetic","rect":[110.00436401367188,570.3099975585938,158.98604620172348,561.37548828125]},{"page":369,"text":"anisotropy","rect":[164.2697296142578,570.3099975585938,206.43377009442816,561.37548828125]},{"page":369,"text":"in","rect":[211.681640625,569.0,219.45586482099066,561.37548828125]},{"page":369,"text":"the","rect":[224.76644897460938,569.0,236.99021185113754,561.37548828125]},{"page":369,"text":"electric","rect":[242.26992797851563,569.0,271.7742389507469,561.37548828125]},{"page":369,"text":"or","rect":[277.1067199707031,569.0,285.3985532364048,563.0]},{"page":369,"text":"magnetic","rect":[290.6444396972656,570.3099975585938,327.3993610210594,561.37548828125]},{"page":369,"text":"field","rect":[332.67510986328127,569.0,350.39360896161568,561.37548828125]},{"page":369,"text":"applied","rect":[355.6723327636719,570.3099975585938,385.09694758466255,561.37548828125]},{"page":369,"text":"perpendicular to the helical axis. Let a helix has a pitch P0 in the absence of a","rect":[53.812843322753909,582.26953125,385.16184271051255,573.3350219726563]},{"page":369,"text":"field and the thickness of a sample is much larger than P0. Therefore, the boundary","rect":[53.81405258178711,594.2296142578125,385.17876521161568,585.2949829101563]},{"page":370,"text":"12.2 Dielectric Instability of Cholesterics","rect":[53.81438446044922,44.276206970214847,195.13373263358393,36.68185043334961]},{"page":370,"text":"359","rect":[372.4989318847656,42.62525939941406,385.19052642970009,36.63105392456055]},{"page":370,"text":"conditions can be neglected. Such a model was investigated by Meyer and de","rect":[53.812843322753909,68.2883529663086,385.1496967144188,59.35380554199219]},{"page":370,"text":"Gennes [12,13] for a cholesteric having anisotropy of magnetic susceptibility wa","rect":[53.812843322753909,80.24788665771485,385.18130140630105,71.31333923339844]},{"page":370,"text":"and placed in magnetic field H perpendicular to the helical axis h. Inevitably, in the","rect":[53.812843322753909,92.20772552490235,385.1725543804344,83.27317810058594]},{"page":370,"text":"initial state, in certain parts (A) of the helix, the molecules are arranged favourably","rect":[53.81185531616211,104.1104965209961,385.13384333661568,95.17594909667969]},{"page":370,"text":"relative to the field, but in other parts (B) they are arranged unfavourably,","rect":[53.81185531616211,116.0699691772461,385.1169094612766,107.13542175292969]},{"page":370,"text":"Fig. 12.8a. Due to wa > 0, the latter would tend to realign themselves along the","rect":[53.81185531616211,128.02999877929688,385.1726764507469,119.09501647949219]},{"page":370,"text":"field.","rect":[53.81392288208008,137.95758056640626,74.02096982504611,131.0549774169922]},{"page":370,"text":"With the field applied, regions A will increase in size and regions B decreased.A","rect":[65.76593780517578,151.94906616210938,385.1339392657551,143.01451110839845]},{"page":370,"text":"decrease in the dimensions of the B regions would cost very large elastic energy","rect":[53.81392288208008,163.90863037109376,385.16875544599068,154.9740753173828]},{"page":370,"text":"K22(∂j/∂z)2. In a strong field, regions B transforms into thin two-dimensional","rect":[53.81392288208008,175.86868286132813,385.0787187344749,164.81764221191407]},{"page":370,"text":"defects (walls) perpendicular to z with the director turned by angle p across the","rect":[53.81352233886719,187.82821655273438,385.1752399273094,178.89366149902345]},{"page":370,"text":"wall. If a number of regions B were reduced by increasing period (pitch) of the","rect":[53.81452178955078,199.73098754882813,385.1732868023094,190.7964324951172]},{"page":370,"text":"structure, the total elastic energy would be lower. Therefore, our structure becomes","rect":[53.81452178955078,211.6905517578125,385.08771146880346,202.75599670410157]},{"page":370,"text":"unstable: a strong field tries to expel all the walls from the helical structure. As a","rect":[53.81452178955078,223.65008544921876,385.15735662652818,214.7155303955078]},{"page":370,"text":"result of such instability, at a certain critical field Hu, the helical structure trans-","rect":[53.81452178955078,235.60964965820313,385.10186134187355,226.6750946044922]},{"page":370,"text":"forms into a uniform (nematic) structure. We can say, that there occurs a cholesteric","rect":[53.813777923583987,247.56979370117188,385.10288274957505,238.63523864746095]},{"page":370,"text":"to nematic phase transition with a threshold field Hu.","rect":[53.813777923583987,259.52935791015627,266.8515286019016,250.5948028564453]},{"page":370,"text":"The threshold field can be calculated thermodynamically by comparison of the","rect":[65.76555633544922,271.48907470703127,385.1703265972313,262.55450439453127]},{"page":370,"text":"free energy of the helical and uniform structures in the presence of the field. In our","rect":[53.813533782958987,283.4486083984375,385.14348731843605,274.5140380859375]},{"page":370,"text":"geometry, the free energy density of a cholesteric in a magnetic field is","rect":[53.813533782958987,295.35137939453127,341.11603178130346,286.41680908203127]},{"page":370,"text":"2","rect":[236.8900604248047,315.3724060058594,240.37402412971816,310.65911865234377]},{"page":370,"text":"gCh ¼ 12\"K22\u0004ddjz \u0004 q0\u0005 \u0004 waH2cos2j#","rect":[135.94871520996095,339.450439453125,303.02419148411959,309.5889587402344]},{"page":370,"text":"(12.25)","rect":[356.0711669921875,328.791748046875,385.15935645906105,320.19586181640627]},{"page":370,"text":"where j is an angle between the field and the director. For the unwound, nematic-","rect":[53.813594818115237,362.9721374511719,385.1484616836704,354.03759765625]},{"page":370,"text":"like cholesteric, ∂j/∂z ¼ 0, j ¼ 0 or p, and the free energy density is given by","rect":[53.813594818115237,374.9316711425781,379.1659783463813,364.9911193847656]},{"page":370,"text":"gN ¼ 21hK22ð\u0004q0Þ2 \u0004 waH2i","rect":[162.17613220214845,409.46221923828127,276.78149513650978,389.12591552734377]},{"page":370,"text":"(12.26)","rect":[356.07073974609377,404.4044494628906,385.1589292129673,395.86834716796877]},{"page":370,"text":"a","rect":[57.22526931762695,439.6304626464844,62.7805317148608,434.0417175292969]},{"page":370,"text":"H","rect":[132.5040740966797,448.63592529296877,138.27513799264467,442.8932189941406]},{"page":370,"text":"b","rect":[230.4555206298828,439.6054992675781,236.56031437936103,432.297119140625]},{"page":370,"text":"1.5","rect":[246.54014587402345,471.91790771484377,257.6506502057061,466.15118408203127]},{"page":370,"text":"H = O","rect":[78.78369903564453,542.8549194335938,99.88565033956677,536.8162841796875]},{"page":370,"text":"1","rect":[253.20724487304688,520.3341674804688,257.65144366273736,514.71142578125]},{"page":370,"text":"0","rect":[258.8616027832031,538.4868774414063,263.3058015728936,532.7201538085938]},{"page":370,"text":"Fig. 12.8 Influence of a magnetic field H on the planar cholesteric texture having wa > 0. The","rect":[53.812843322753909,574.0244750976563,385.1964354255152,566.0745239257813]},{"page":370,"text":"helical axis is parallel to z. Horizontal lines show the projections of the director parallel to H. Helix","rect":[53.813533782958987,583.9326782226563,385.1676382461063,576.3214111328125]},{"page":370,"text":"unwinding according to de Gennes (a) and Meyer experiment (b)","rect":[53.81438446044922,593.9086303710938,277.71713346106699,586.3142700195313]},{"page":371,"text":"360","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.68026351928711]},{"page":371,"text":"12 Electro-Optical Effects in Cholesteric Phase","rect":[223.29483032226563,44.274620056152347,385.1406798102808,36.663330078125]},{"page":371,"text":"The difference between the two densities (12.25) and (12.26) reads","rect":[65.76496887207031,67.88993072509766,336.33932890044408,59.294044494628909]},{"page":371,"text":"Dg","rect":[83.04249572753906,102.11011505126953,94.64711848310003,92.8070297241211]},{"page":371,"text":"¼","rect":[97.43031311035156,98.6040267944336,105.09505489561467,96.27327728271485]},{"page":371,"text":"¼","rect":[97.43087005615235,132.38485717773438,105.09561184141545,130.05410766601563]},{"page":371,"text":"gch \u0004 gN ¼ 12\"K22\u0004ddjz \u0004 q0\u00052 \u0004 K22q20 \u0004","rect":[107.91010284423828,112.33045959472656,278.8817828008881,82.46894836425781]},{"page":371,"text":"21K22\"\u0004ddjz\u00052 \u0004 2q0 ddjz# þ 12waH2sin2j","rect":[107.9090576171875,146.11256408691407,271.67222854312208,116.25105285644531]},{"page":371,"text":"waH2ðcos2j","rect":[281.1294250488281,102.40880584716797,329.28035232241896,91.13436889648438]},{"page":371,"text":"\u0004","rect":[331.5439453125,98.0,339.2086870977631,97.0]},{"page":371,"text":"1Þ#","rect":[341.400634765625,112.33045959472656,355.93178425755709,82.46894836425781]},{"page":371,"text":"Introducing the field coherence length x2 ¼ K22=waH2 and integrating over one","rect":[65.7654037475586,169.96238708496095,385.1442340679344,158.41452026367188]},{"page":371,"text":"period of the helix P0 along the z-axis we find the free energy","rect":[53.81438446044922,181.59335327148438,303.1761101823188,172.65867614746095]},{"page":371,"text":"wDaHF2 ¼ wa1H2 Pð0 Dgdz ¼ Pð0 \"12x2\u0004ddjz\u00052 \u0004 q0x2 ddjz þ 12sin2j#dz","rect":[75.56437683105469,229.6030731201172,335.9924355898972,196.84230041503907]},{"page":371,"text":"0","rect":[134.5888671875,232.13539123535157,138.07283089241347,227.33843994140626]},{"page":371,"text":"(12.27)","rect":[356.0720520019531,218.9443817138672,385.16024146882668,210.468017578125]},{"page":371,"text":"The","rect":[65.76648712158203,255.0,81.30500829889142,247.7610321044922]},{"page":371,"text":"(12.27)","rect":[53.814476013183597,268.2567138671875,82.90264259187353,259.7803649902344]},{"page":371,"text":"Euler equation (8.22) corresponding to the minimal free energy","rect":[85.47881317138672,256.6955871582031,352.1172722916938,247.7610321044922]},{"page":371,"text":"within a period of the structure reads:","rect":[85.70574951171875,268.6551513671875,237.4586778409202,259.7205810546875]},{"page":371,"text":"x2 dd2zj2 ¼ sinjcosj or 21x2 ddz\u0004ddjz\u00052 ¼ sinjcosjddjz","rect":[104.73820495605469,308.70501708984377,332.56550857241896,282.8855895996094]},{"page":371,"text":"density","rect":[356.2960510253906,256.6955871582031,385.10353938153755,247.7610321044922]},{"page":371,"text":"This equation is easily integrated:","rect":[65.76580047607422,332.19378662109377,202.03694017002176,323.25921630859377]},{"page":371,"text":"x2\u0004ddjz\u00052 ¼ 2ð sinjcosjddjzdz ¼ 2ð sinjcosjdj ¼ sin2j þC","rect":[70.92007446289063,372.2441101074219,338.9658381622627,346.4247741699219]},{"page":371,"text":"z","rect":[131.2469024658203,377.3812255859375,133.92955451860369,374.3552551269531]},{"page":371,"text":"(12.28)","rect":[356.07012939453127,364.55731201171877,385.1583188614048,356.0809631347656]},{"page":371,"text":"For the particular periodic structure shown in Fig. 12.8a with angle j counted","rect":[65.76457977294922,403.3852844238281,385.12551203778755,394.45074462890627]},{"page":371,"text":"from the field H direction, the derivative dj/dz ¼ 0 at any values of z where","rect":[53.812557220458987,415.3448181152344,385.11060369684068,406.3903503417969]},{"page":371,"text":"j ¼ 0 or p (middle points of regions A). Therefore, C ¼ 0 and x(dj/dz) ¼ \u0007sinj,","rect":[53.81356430053711,427.2475891113281,385.1832241585422,417.9942932128906]},{"page":371,"text":"where for the right-handed helix the sign at the right side is either positive (if j","rect":[53.81456756591797,439.2071228027344,385.18919631655958,430.2725830078125]},{"page":371,"text":"belongs to an interval from 0 to p) or negative (if p < j < 2p). Then, substituting","rect":[53.81456756591797,451.1666564941406,385.10463801435005,442.23211669921877]},{"page":371,"text":"Eq. (12.28) into Eq. (12.27) we find","rect":[53.81456756591797,463.1261901855469,198.64832392743598,454.191650390625]},{"page":371,"text":"wDaHF2 ¼ Pð0 x2\"\u0004ddjz\u00052 \u0004 q0 ddjz#dz ¼ x2 2ðp\u0004ddjz \u0004 q0\u0005dj","rect":[105.41632080078125,511.136962890625,335.22798415835646,478.3758850097656]},{"page":371,"text":"0","rect":[140.7066192626953,513.6690063476563,144.19058296760879,508.8720397949219]},{"page":371,"text":"0","rect":[268.157958984375,513.669189453125,271.64192268928846,508.8722229003906]},{"page":371,"text":"p","rect":[151.6390838623047,526.3306274414063,155.48537979252917,523.1791381835938]},{"page":371,"text":"2p","rect":[221.4825439453125,526.3306274414063,228.84068069584948,521.5685424804688]},{"page":371,"text":"¼ 2xð sinjdj \u0004 q0x2 ð dj ¼ 2xð2 \u0004 pq0xÞ","rect":[128.58554077148438,550.3883056640625,309.8864785586472,528.265625]},{"page":371,"text":"0","rect":[151.35618591308595,556.7466430664063,154.8401496179994,551.94970703125]},{"page":371,"text":"0","rect":[222.89840698242188,556.7466430664063,226.38237068733535,551.94970703125]},{"page":371,"text":"Therefore DF ¼ 2K22 ð2 \u0004 pq0xÞ and the threshold condition (DF ¼","rect":[65.76642608642578,581.6451416015625,344.30637386534127,565.6866455078125]},{"page":371,"text":"helix unwinding readsx","rect":[53.81364059448242,593.2660522460938,149.25229731610785,578.91162109375]},{"page":371,"text":"0)","rect":[347.6878356933594,580.9080810546875,355.97969947663918,572.4317016601563]},{"page":371,"text":"for","rect":[358.62054443359377,579.2446899414063,370.2271665176548,572.3719482421875]},{"page":371,"text":"the","rect":[372.9516296386719,579.2446899414063,385.17539251520005,572.3719482421875]},{"page":372,"text":"12.2 Dielectric Instability of Cholesterics","rect":[53.812843322753909,44.274620056152347,195.13217623709955,36.68026351928711]},{"page":372,"text":"2","rect":[178.37557983398438,71.134765625,183.35268488935004,64.40148162841797]},{"page":372,"text":"xu¼","rect":[151.41249084472657,80.26648712158203,170.85939815733344,70.76810455322266]},{"page":372,"text":"pq0","rect":[173.61746215820313,86.76966857910156,187.58071968635879,80.10610961914063]},{"page":372,"text":"ffiffiffiffiffiffiffi","rect":[270.7638854980469,61.0,284.84311974600089,59.0]},{"page":372,"text":"Hu ¼ Pp20 sKw2a2:","rect":[224.5979461669922,89.83847045898438,287.5561975209131,59.96699905395508]},{"page":372,"text":"361","rect":[372.49737548828127,42.55594253540039,385.1889700332157,36.68026351928711]},{"page":372,"text":"(12.29)","rect":[356.0710144042969,79.6230239868164,385.1592038711704,71.14665985107422]},{"page":372,"text":"This result is in good agreement with experiment. It tells us that, at H > Hu","rect":[65.76546478271485,113.34920501708985,385.18130140630105,104.41465759277344]},{"page":372,"text":"cholesteric should become uniform. Moreover, for each value of H < Hu, de","rect":[53.812843322753909,124.871337890625,385.15430486871568,116.37474060058594]},{"page":372,"text":"Gennes has estimated a stationary value of the pitch [12]:","rect":[53.814476013183597,137.26882934570313,286.2224260098655,128.3342742919922]},{"page":372,"text":"PðHÞ ¼ P0\u00011 þ 32ðw2a2pP4Þ04K222 H4 þ :::\u0003","rect":[143.65322875976563,176.39083862304688,295.31260283179906,151.49951171875]},{"page":372,"text":"(12.30)","rect":[356.0710754394531,168.32835388183595,385.15926490632668,159.85198974609376]},{"page":372,"text":"The results (12.29) and (12.30) are in very good agreement with experiments","rect":[65.76551055908203,199.95785522460938,385.13339628325658,191.02330017089845]},{"page":372,"text":"made under thermodynamic equilibrium. Figure 12.8b shows the results obtained","rect":[53.81345748901367,211.91738891601563,385.1662224870063,202.9828338623047]},{"page":372,"text":"by R. Meyer [14] on rather a thick cell (d ¼ 130 mm) filled with a cholesteric","rect":[53.81345748901367,223.82015991210938,385.1055072612938,214.8656768798828]},{"page":372,"text":"mixture based on p-azoxyanisol (PAA). The mixture was not oriented by bound-","rect":[53.81350326538086,235.77969360351563,385.12053809968605,226.8451385498047]},{"page":372,"text":"aries and contained a number of defects. Meyer mentioned that, in order to reach the","rect":[53.81350326538086,247.7392578125,385.17331731988755,238.80470275878907]},{"page":372,"text":"equilibrium state for each value of magnetic field, “the tendency to hysteresis was","rect":[53.81350326538086,259.69879150390627,385.1463967715378,250.7642364501953]},{"page":372,"text":"overcome by cycling the field while observing the cell”. This comment is very","rect":[53.81350326538086,271.6583251953125,385.1155633073188,262.7237548828125]},{"page":372,"text":"important, because the hysteresis is a fingerprint of the topological constraints","rect":[53.81350326538086,283.61785888671877,385.14239896880346,274.68328857421877]},{"page":372,"text":"discussed below.","rect":[53.81350326538086,293.54547119140627,121.73605008627658,286.64288330078127]},{"page":372,"text":"For the electric field, in Eqs. (12.29) and (12.30) we should substitute ea/4p for","rect":[65.7655258178711,307.5369567871094,385.18120704499855,298.6024169921875]},{"page":372,"text":"wa. Therefore, if we apply magnetic field (or electric field more convenient for","rect":[53.814537048339847,319.4411315917969,385.1793149551548,310.506591796875]},{"page":372,"text":"practical purposes) to a cholesteric sample for a long enough time, we should","rect":[53.812538146972659,331.4006652832031,385.1752556901313,322.46612548828127]},{"page":372,"text":"change the helical pitch of the sample according to Eq. (12.30). Such a field-","rect":[53.812538146972659,343.3602294921875,385.1463864883579,334.4256591796875]},{"page":372,"text":"induced pitch tuning would be very promising for applicable to fast displays,","rect":[53.81252670288086,355.31976318359377,385.1594204476047,346.38519287109377]},{"page":372,"text":"tunable photonic filters, diffraction gratings and lasers. Unfortunately, pitch tuning","rect":[53.81252670288086,367.2793273925781,385.13945857099068,358.34478759765627]},{"page":372,"text":"may be realized only via an intermediate, very slow stage of the defect formation.","rect":[53.81252670288086,379.23883056640627,384.88164945151098,370.30426025390627]},{"page":372,"text":"12.2.1.2 Topological Limitation","rect":[53.81252670288086,426.98736572265627,194.44760484049869,417.8934326171875]},{"page":372,"text":"What is a reason for such a disappointing situation with tuning the helical pitch by","rect":[53.81252670288086,450.7699890136719,385.1673516373969,441.83544921875]},{"page":372,"text":"electric or magnetic field? It is very simple: despite the fact that field unwinding of","rect":[53.81252670288086,462.72955322265627,385.1494382461704,453.79498291015627]},{"page":372,"text":"the cholesteric helix is thermodynamically profitable there is a strong topological","rect":[53.81252670288086,474.6890869140625,385.15437181064677,465.7545166015625]},{"page":372,"text":"limitation on the unwinding process. It can be understood as follows. In Fig. 12.9","rect":[53.81252670288086,486.6486511230469,385.1583184342719,477.714111328125]},{"page":372,"text":"there is a helical structure of the director field n (shown by arrows) with vertical","rect":[53.81252670288086,498.6081848144531,385.18025071689677,489.67364501953127]},{"page":372,"text":"helical axis h. We assume that the helix is either infinite or limited by two","rect":[53.81254959106445,510.5677490234375,385.1503838639594,501.6331787109375]},{"page":372,"text":"boundaries with infinitely weak azimuthal anchoring at least at one of the bound-","rect":[53.81354904174805,522.5272216796875,385.11858497468605,513.5927124023438]},{"page":372,"text":"aries. It means that there is no confinement, which would prevent a free rotation of","rect":[53.81354904174805,534.4300537109375,385.14837013093605,525.4955444335938]},{"page":372,"text":"the non-anchored director at that boundary. Therefore unwinding the helix due, for","rect":[53.81354904174805,546.3895874023438,385.1762937149204,537.455078125]},{"page":372,"text":"instance, to a heating process is possible.","rect":[53.81354904174805,558.34912109375,219.41181607748752,549.4146118164063]},{"page":372,"text":"Now, imagine that dielectric anisotropy is positive and we apply a certain","rect":[65.76557159423828,570.3087158203125,385.17632380536568,561.3742065429688]},{"page":372,"text":"electric field E⊥h to structure (a) with equilibrium pitch P0 trying to increase the","rect":[53.81354904174805,582.2699584960938,385.17481268121568,573.333740234375]},{"page":372,"text":"pitch twice, PE ! 2P0, as shown in sketch (b). To do this we must turn the director","rect":[53.814083099365237,594.2294921875,385.1053708633579,585.2949829101563]},{"page":373,"text":"362","rect":[53.812843322753909,42.55752944946289,66.50444931178018,36.68185043334961]},{"page":373,"text":"Fig. 12.9 Field behaviour of","rect":[53.812843322753909,67.58130645751953,154.0426949845045,59.546695709228519]},{"page":373,"text":"a cholesteric helix (ea > 0).","rect":[53.812843322753909,77.15099334716797,149.5863673408266,69.89517211914063]},{"page":373,"text":"(a) Zero-field structure,","rect":[53.812992095947269,87.07018280029297,134.14578506543598,79.81448364257813]},{"page":373,"text":"(b) unfavorable structure with","rect":[53.8121452331543,97.04613494873047,155.31368011622355,89.79043579101563]},{"page":373,"text":"a larger field induced pitch,","rect":[53.812137603759769,107.36080169677735,147.98045608594379,99.76644897460938]},{"page":373,"text":"(c) favorable wall structure","rect":[53.812137603759769,116.99810028076172,146.75698230051519,109.74240112304688]},{"page":373,"text":"with unchanged pitch","rect":[53.812129974365237,127.25594329833985,127.3989004531376,119.66159057617188]},{"page":373,"text":"Fig. 12.10 Helical structure","rect":[53.812843322753909,169.83367919921876,151.7429747321558,162.10386657714845]},{"page":373,"text":"of a cholesteric liquid crystal","rect":[53.812843322753909,179.741943359375,153.41825584616724,172.1475830078125]},{"page":373,"text":"between two glass plates. On","rect":[53.812843322753909,189.7178955078125,153.44788116602823,182.10659790039063]},{"page":373,"text":"the bottom plate, an array of","rect":[53.812843322753909,199.63711547851563,151.32330411536388,192.04275512695313]},{"page":373,"text":"metal interdigitated","rect":[53.812843322753909,209.61306762695313,120.4632315078251,202.01870727539063]},{"page":373,"text":"electrodes is deposited. The","rect":[53.812843322753909,219.58901977539063,148.87296435617925,211.99465942382813]},{"page":373,"text":"array is covered bya","rect":[53.812843322753909,229.56500244140626,125.51280352854252,221.97064208984376]},{"page":373,"text":"polyimide layers and rubbed","rect":[53.812843322753909,239.48419189453126,151.42651886134073,231.88983154296876]},{"page":373,"text":"to align the molecules in the","rect":[53.812843322753909,249.46017456054688,151.15322253489019,241.86581420898438]},{"page":373,"text":"plane of the substrate. The","rect":[53.812843322753909,259.4361267089844,144.62381884836675,251.84176635742188]},{"page":373,"text":"upper glass is also covered by","rect":[53.812843322753909,269.3553771972656,155.34229034571573,261.7610168457031]},{"page":373,"text":"polyimide but not rubbed","rect":[53.812843322753909,279.331298828125,140.60733551173136,271.7369384765625]},{"page":373,"text":"12 Electro-Optical","rect":[223.29483032226563,44.276206970214847,286.55321983542509,36.6649169921875]},{"page":373,"text":"E","rect":[237.9285888671875,231.98452758789063,243.2600955307229,226.2417449951172]},{"page":373,"text":"stripe","rect":[211.05880737304688,245.58352661132813,232.82309663071497,238.1375732421875]},{"page":373,"text":"electrodes","rect":[211.05880737304688,253.63729858398438,251.0386117674337,247.79090881347657]},{"page":373,"text":"U","rect":[205.73492431640626,290.23291015625,211.50606046494083,284.3461608886719]},{"page":373,"text":"Effects","rect":[288.9468688964844,43.0,312.97632296561519,36.68185043334961]},{"page":373,"text":"in","rect":[315.4004211425781,43.0,322.00851959376259,36.68185043334961]},{"page":373,"text":"Cholesteric","rect":[324.35052490234377,43.0,363.0006346442652,36.68185043334961]},{"page":373,"text":"Phase","rect":[365.41796875,43.0,385.1406798102808,36.68185043334961]},{"page":373,"text":"from the central favourable position A’( n||E, ea > 0) to the unfavourable position","rect":[53.812843322753909,343.1903991699219,385.1516045670844,334.255859375]},{"page":373,"text":"B, where n⊥E, and this situation takes place within each period. Moreover, the","rect":[53.812740325927737,355.14996337890627,385.17346990777818,346.21539306640627]},{"page":373,"text":"director must make a p/2-turn against the field to change its initial A position (at the","rect":[53.81374740600586,367.1094970703125,385.1745075054344,358.1749267578125]},{"page":373,"text":"bottom) to new position A’. In other words, the director should overcome a high","rect":[53.814735412597659,379.06903076171877,385.1266717057563,370.13446044921877]},{"page":373,"text":"potential barrier. Therefore, a very serious topological problem exists for the ideal","rect":[53.814735412597659,391.0285949707031,385.1127763516624,382.09405517578127]},{"page":373,"text":"cholesteric helix. In reality, the structure (c) very often forms with favourable","rect":[53.814735412597659,402.98809814453127,385.12073553277818,394.05352783203127]},{"page":373,"text":"orientation of the director everywhere. The positions of the walls W separating","rect":[53.814735412597659,414.9476623535156,385.08391657880318,406.01312255859377]},{"page":373,"text":"areas where n differs by p are fixed and the energy of the structure (c) with the same","rect":[53.814735412597659,426.8504333496094,385.13071478082505,417.9158935546875]},{"page":373,"text":"initial pitch P0 is, of course, larger than the more profitable stationary structure with","rect":[53.814735412597659,438.8108825683594,385.1162957291938,429.87542724609377]},{"page":373,"text":"an enhanced pitch.","rect":[53.81328582763672,450.7704162597656,129.14973874594456,441.83587646484377]},{"page":373,"text":"The numerical modelling with software [5] and experiment [15] confirm this","rect":[65.76531219482422,462.72998046875,385.1451760684128,453.73565673828127]},{"page":373,"text":"picture. In the experiment, a cell was used pictured in Fig. 12.10. The dielectric","rect":[53.81329345703125,474.68951416015627,385.1819537944969,465.75494384765627]},{"page":373,"text":"anisotropy of the material is ea ¼þ7.8 and the electric voltage is applied between","rect":[53.81429672241211,486.6495056152344,385.17681208661568,477.71453857421877]},{"page":373,"text":"the in-plane interdigitated electrodes with a gap 20 mm. In calculations, both","rect":[53.8130989074707,498.6090393066406,385.1270684342719,489.67449951171877]},{"page":373,"text":"the zenithal and azimuthal anchoring strengths at the bottom substrate is strong,","rect":[53.8130989074707,510.568603515625,385.12908597494848,501.634033203125]},{"page":373,"text":"Wz1 ¼ Wa1 ¼ 0.1 erg/cm2. At the upper substrate the zenithal anchoring energy","rect":[53.8130989074707,522.4718017578125,385.1682976823188,511.4208068847656]},{"page":373,"text":"Wz2 is also that strong, therefore the director is always confined within the plane of","rect":[53.813533782958987,534.431396484375,385.1487973770298,525.4968872070313]},{"page":373,"text":"substrates perpendicular to the helical axis. However, the azimuthal anchoring","rect":[53.81290817260742,546.3909301757813,385.1756524186469,537.4564208984375]},{"page":373,"text":"energy at the second substrate is negligibly small Wa2 ¼ 0.001 erg/cm2 and pro-","rect":[53.81290817260742,558.3507080078125,385.17342506257668,547.2998046875]},{"page":373,"text":"vides easy rotation (sliding) of the director in the substrate plane.","rect":[53.813716888427737,570.3102416992188,317.04888577963598,561.375732421875]},{"page":373,"text":"Figure 12.11 shows the calculated distribution of the azimuthal angle j for the","rect":[65.76573944091797,582.269775390625,385.17542303277818,573.3352661132813]},{"page":373,"text":"planar cholesteric structure of thickness d ¼ 25P0, however, only two periods are","rect":[53.813716888427737,594.2296142578125,385.1639789409813,585.2350463867188]},{"page":374,"text":"12.2","rect":[53.81332015991211,43.0,68.6201910415165,36.73161315917969]},{"page":374,"text":"Dielectric","rect":[70.97660064697266,43.0,104.50782153391362,36.68081283569336]},{"page":374,"text":"Instability","rect":[106.94630432128906,44.275169372558597,141.3828634658329,36.68081283569336]},{"page":374,"text":"360","rect":[121.38967895507813,87.49942779541016,134.42848692002145,81.73539733886719]},{"page":374,"text":"270","rect":[121.38967895507813,114.7974624633789,134.42848692002145,109.03343200683594]},{"page":374,"text":"180","rect":[121.38967895507813,142.09544372558595,134.42848692002145,136.33140563964845]},{"page":374,"text":"90","rect":[125.6880111694336,169.39427185058595,134.42848692002145,163.63023376464845]},{"page":374,"text":"0","rect":[129.98634338378907,196.6923065185547,134.42848692002145,190.9282684326172]},{"page":374,"text":"of","rect":[143.7655029296875,43.0,150.813569007942,36.68081283569336]},{"page":374,"text":"Cholesterics","rect":[153.2249755859375,43.0,195.13266451834955,36.68081283569336]},{"page":374,"text":"2nd boundary","rect":[260.5509338378906,68.08588409423828,308.64753490956437,60.643009185791019]},{"page":374,"text":"500 V","rect":[244.8312530517578,132.18833923339845,265.0797976791604,126.30159759521485]},{"page":374,"text":"222 V","rect":[266.8945617675781,166.81793212890626,287.39394627394867,161.077880859375]},{"page":374,"text":"U = 0","rect":[256.9508972167969,184.94468688964845,275.598301983498,179.06072998046876]},{"page":374,"text":"363","rect":[372.49786376953127,42.55649185180664,385.1894583144657,36.68081283569336]},{"page":374,"text":"Fig. 12.11 Calculated director azimuth j for the last two periods of the helix adjacent to the top","rect":[53.812843322753909,242.49850463867188,385.2642874160282,234.76869201660157]},{"page":374,"text":"boundary of the cell (see Fig. 12.10). It repeatedly increases from 0\u0003 to 360\u0003 within each period P0.","rect":[53.812843322753909,252.35000610351563,385.2064845283266,244.7554931640625]},{"page":374,"text":"Without field the dependence j (z) is linear. With increasing voltage the director is progressively","rect":[53.81344223022461,262.3258361816406,385.1684622207157,254.73147583007813]},{"page":374,"text":"reorientedbuttheperiodremainsunchanged.Cellparameters:thicknessd ¼ 4mm,pitchP0 ¼ 2.5mm,","rect":[53.812618255615237,272.3017883300781,385.206087799811,264.6563415527344]},{"page":374,"text":"ea ¼ 7.8, twist elastic modulus K22 ¼ 9 10\u00047 dyn","rect":[53.81307601928711,282.220703125,232.53143829493448,272.9234313964844]},{"page":374,"text":"shown for clarity. It is seen that, with increasing voltage, the director is progres-","rect":[53.812843322753909,313.0928955078125,385.1666196426548,304.1583251953125]},{"page":374,"text":"sively realigned along the field direction but the period of the distorted helix","rect":[53.812843322753909,325.0524597167969,385.13176814130318,316.117919921875]},{"page":374,"text":"remains unchanged (some total shift of the curves along the z-coordinate is due","rect":[53.812843322753909,337.0119934082031,385.14377630426255,328.07745361328127]},{"page":374,"text":"to boundary effects). Note that within each period there are horizontal parts of the","rect":[53.812843322753909,348.9715576171875,385.17063177301255,340.0369873046875]},{"page":374,"text":"curves that correspond to very narrow ranges of angle j. These are field induced","rect":[53.812843322753909,360.93109130859377,385.12575617841255,351.99652099609377]},{"page":374,"text":"defects (walls) having half-pitch periodicity which called p-solitons observed also","rect":[53.81282424926758,372.8338623046875,385.1546563248969,363.8992919921875]},{"page":374,"text":"in SmC* materials (see Section 13.4.2).","rect":[53.81282424926758,384.39495849609377,214.0875668099094,375.85882568359377]},{"page":374,"text":"The calculated field induced transmission of the same planar cholesteric texture","rect":[65.76583862304688,396.7529602050781,385.1587299175438,387.81842041015627]},{"page":374,"text":"in the non-polarized light is shown in Fig. 12.12. It is clearly seen that, with","rect":[53.813819885253909,408.71246337890627,385.11782160810005,399.77789306640627]},{"page":374,"text":"increasing field, the Bragg minimum is only slightly shifted to shorter wavelengths","rect":[53.8138313293457,420.6720275878906,385.1645852481003,411.7175598144531]},{"page":374,"text":"due to a distortion of the helix seen in the previous figure and then disappeared ata","rect":[53.813812255859378,432.6315612792969,385.1576618023094,423.697021484375]},{"page":374,"text":"field of about 25V/mm. Therefore, in the absence of defects the field cannot increase","rect":[53.813812255859378,444.5014343261719,385.14868963434068,435.5967712402344]},{"page":374,"text":"the period of the helix. An essential increase of the cell thickness does not influence","rect":[53.814815521240237,456.5506286621094,385.10788763238755,447.6160888671875]},{"page":374,"text":"the result. The measurements of the field dependence of the transmission spectra of","rect":[53.814815521240237,468.4533996582031,385.14974342195168,459.51885986328127]},{"page":374,"text":"a cholesteric with the same parameters have confirmed the absence of the red shift","rect":[53.814815521240237,480.4129638671875,385.207411361428,471.4783935546875]},{"page":374,"text":"of the Bragg minimum [15].","rect":[53.814815521240237,492.3725280761719,167.9796566536594,483.3782043457031]},{"page":374,"text":"The characteristic field, at which the Bragg band disappears, is considerably","rect":[65.7668228149414,504.3320617675781,385.16362849286568,495.39752197265627]},{"page":374,"text":"higher than the critical field (Eu ¼ 7 V/mm) calculated from the thermodynamic","rect":[53.81480407714844,516.2935791015625,385.18128240777818,507.35711669921877]},{"page":374,"text":"approach, see Eq. (12.29). However, the periodic structure with very thin defect","rect":[53.8136100769043,528.2530517578125,385.179335189553,519.3185424804688]},{"page":374,"text":"walls separating area of opposite director orientation (j ¼ 0 or p) may still exist","rect":[53.8136100769043,540.212646484375,385.1196428067405,531.2781372070313]},{"page":374,"text":"but","rect":[53.81364059448242,551.0,66.56498582431863,543.2376708984375]},{"page":374,"text":"not","rect":[72.67587280273438,551.0,85.42721421787332,544.2536010742188]},{"page":374,"text":"seen","rect":[91.48136901855469,551.0,109.18990412763128,545.0]},{"page":374,"text":"optically.","rect":[115.32867431640625,552.1721801757813,152.9287075569797,543.2376708984375]},{"page":374,"text":"Metastable,","rect":[159.0007781982422,551.0,205.40829129721409,543.2376708984375]},{"page":374,"text":"non-unwound","rect":[211.51022338867188,551.0,266.90139094403755,543.2376708984375]},{"page":374,"text":"helical","rect":[272.970458984375,551.0,299.6876054532249,543.2376708984375]},{"page":374,"text":"structures","rect":[305.8243713378906,551.0,344.6627236758347,544.2536010742188]},{"page":374,"text":"are","rect":[350.7427673339844,551.0,362.9565814800438,545.0]},{"page":374,"text":"also","rect":[369.0386047363281,551.0,385.17434016278755,543.2376708984375]},{"page":374,"text":"observed at field strengths E > Eu in experiments with short voltage pulses when","rect":[53.81364059448242,564.075439453125,385.17629328778755,555.1404418945313]},{"page":374,"text":"the defects have not enough time to form.","rect":[53.81354904174805,576.0350341796875,222.5622982552219,567.1005249023438]},{"page":375,"text":"364","rect":[53.81279373168945,42.55667495727539,66.50439972071573,36.68099594116211]},{"page":375,"text":"12 Electro-Optical Effects in Cholesteric Phase","rect":[223.29478454589845,44.275352478027347,385.1406492927027,36.6640625]},{"page":375,"text":"0.9","rect":[118.13693237304688,81.73577117919922,129.1089022101629,75.96626281738281]},{"page":375,"text":"0.8","rect":[118.13693237304688,107.50896453857422,129.1089022101629,101.73945617675781]},{"page":375,"text":"0.7","rect":[118.13693237304688,133.280517578125,129.1089022101629,127.51100158691406]},{"page":375,"text":"0.6","rect":[118.13693237304688,159.05364990234376,129.1089022101629,153.2841339111328]},{"page":375,"text":"500 V","rect":[169.88217163085938,101.35002899169922,190.7784973026131,95.46048736572266]},{"page":375,"text":"444 V","rect":[170.23483276367188,114.51983642578125,190.77927550085529,108.77433013916016]},{"page":375,"text":"389 V","rect":[170.5930938720703,127.05596160888672,191.13753660925372,121.16641998291016]},{"page":375,"text":"278 V","rect":[170.38919067382813,140.65234375,190.93363341101154,134.76280212402345]},{"page":375,"text":"U = 0","rect":[170.9337921142578,155.40496826171876,189.88682579414729,149.5154266357422]},{"page":375,"text":"500","rect":[127.87223052978516,187.04354858398438,141.06738304512386,181.27403259277345]},{"page":375,"text":"550","rect":[176.03932189941407,187.04354858398438,189.37841820137386,181.27403259277345]},{"page":375,"text":"600","rect":[224.34954833984376,187.04354858398438,237.40075706856136,181.27403259277345]},{"page":375,"text":"Wavelength (nm)","rect":[200.37432861328126,199.68553161621095,261.0162620839912,192.06753540039063]},{"page":375,"text":"650","rect":[272.3719177246094,187.04354858398438,285.4231111945379,181.27403259277345]},{"page":375,"text":"700","rect":[320.3942565917969,187.04354858398438,333.44548057930356,181.27403259277345]},{"page":375,"text":"Fig. 12.12","rect":[53.812843322753909,219.54269409179688,88.54130310206338,211.81288146972657]},{"page":375,"text":"Calculated optical transmission spectra of the planar cholesteric texture as functions of","rect":[94.54020690917969,219.4749755859375,385.21102994544199,211.880615234375]},{"page":375,"text":"the electric voltage applied (unpolarized light). Principal refraction indices used are n|| ¼ 1.550,","rect":[53.812843322753909,229.450927734375,385.1540248115297,221.80567932128907]},{"page":375,"text":"n⊥ ¼ 1.474, for other parameters see Fig. 12.11","rect":[53.813472747802737,239.4266357421875,218.48553985743448,231.832275390625]},{"page":375,"text":"12.2.2 Field Induced Anharmonicity and Dynamics of the Helix","rect":[53.812843322753909,278.4654235839844,380.3673833837468,267.82763671875]},{"page":375,"text":"Let us come back to Fig. 12.11. It is evident that a sufficiently strong electric field","rect":[53.812843322753909,306.0076904296875,385.11489192060005,297.0731201171875]},{"page":375,"text":"perpendicular to the helical axis causes a snake-like picture of the director field with","rect":[53.81285095214844,317.96722412109377,385.1984795670844,309.03265380859377]},{"page":375,"text":"the pitch of the helical structure remaining unchanged. It means that the distribution","rect":[53.81285095214844,329.9267883300781,385.1776055436469,320.99224853515627]},{"page":375,"text":"of the x- and y- components of the director is no longer described by a simple sine","rect":[53.81285095214844,341.8863220214844,385.1547321148094,332.9517822265625]},{"page":375,"text":"law but contains a contribution of higher harmonics. The amplitudes of the field","rect":[53.813846588134769,353.84588623046877,385.11589900067818,344.91131591796877]},{"page":375,"text":"induced harmonics characterize a degree of the field-induced anharmonicity of the","rect":[53.813846588134769,365.8053894042969,385.17160833551255,356.870849609375]},{"page":375,"text":"helical structure. The higher harmonics of the helix had been observed long ago","rect":[53.813846588134769,377.70819091796877,385.1437615495063,368.77362060546877]},{"page":375,"text":"[16], but only recently understood as very promising issue for applications. Indeed,","rect":[53.813846588134769,389.667724609375,385.1716579964328,380.733154296875]},{"page":375,"text":"with an experimental cell of the type shown in Fig. 12.10 one can detect several","rect":[53.81285095214844,401.62725830078127,385.1815019375999,392.69268798828127]},{"page":375,"text":"spectacular effects.","rect":[53.812843322753909,413.5868225097656,130.8823208382297,404.65228271484377]},{"page":375,"text":"Let us simulate an appearance of the higher harmonics and optical properties of","rect":[65.76486206054688,425.5463562011719,385.14974342195168,416.61181640625]},{"page":375,"text":"the cholesteric structure with the following parameters typical of chiral materials","rect":[53.812843322753909,437.5058898925781,385.18454374419408,428.57135009765627]},{"page":375,"text":"based on the well-known nematic mixture E7: helical pitch 0.4 mm, elastic modulus","rect":[53.812843322753909,449.4654235839844,385.17953886138158,440.5308837890625]},{"page":375,"text":"K22 ¼ 5","rect":[53.812862396240237,460.9288330078125,87.22620478681097,452.4322509765625]},{"page":375,"text":"10\u00047 dyn (or 5 pN); principal dielectric permittivity values e|| ¼ 20,","rect":[102.35759735107422,461.42657470703127,385.1683010628391,450.4118347167969]},{"page":375,"text":"e⊥ ¼ 8; refraction indices n|| ¼ 1.7, n⊥ ¼ 1.5. Cell thickness is d ¼ 10 mm,","rect":[53.814537048339847,473.2398681640625,385.1402554085422,464.335205078125]},{"page":375,"text":"zenithal and azimuthal anchoring energies is strong (Wz,a ¼ 0.1 erg/cm2) at both","rect":[53.814353942871097,485.2891845703125,385.1287774186469,474.2381591796875]},{"page":375,"text":"boundaries. The electric voltage is applied across the in-plane electrodes separated","rect":[53.813838958740237,497.2487487792969,385.1476983170844,488.314208984375]},{"page":375,"text":"by a distance of l ¼20 mm, see Fig. 12.10. The helix is confined by two glasses with","rect":[53.813838958740237,509.2082824707031,385.11980525067818,500.2538146972656]},{"page":375,"text":"refractive index ng ¼ 1.5.","rect":[53.814842224121097,522.1244506835938,158.0739712288547,512.173828125]},{"page":375,"text":"Theinset to Fig.12.13 shows the calculatedspace dependenceofthe x-component","rect":[65.76558685302735,533.127685546875,385.25691087314677,524.1931762695313]},{"page":375,"text":"of the director nx(z) within one period of the cholesteric structure. The voltage","rect":[53.81357192993164,545.08740234375,385.09583318902818,536.1527099609375]},{"page":375,"text":"applied to the in-plane electrodes is either 0 or 200 V (E ¼ 10 V/mm). As expected,","rect":[53.81370162963867,557.0469360351563,385.1714138558078,548.1124267578125]},{"page":375,"text":"at the field applied, the apices of the curve nx(z) for U ¼ 200 V become very flat.","rect":[53.81468963623047,568.9500122070313,385.1217922737766,560.0151977539063]},{"page":375,"text":"The main plot of Fig. 12.13 represents the Fourier transform of the director","rect":[53.81379318237305,580.9095458984375,385.10784278718605,571.9750366210938]},{"page":375,"text":"component nx(q/2p). In zero field, on the wavevector axis, the helix is represented","rect":[53.81379318237305,592.8693237304688,385.1652459245063,583.934814453125]},{"page":376,"text":"12.2 Dielectric Instability of Cholesterics","rect":[53.811649322509769,44.275474548339847,195.13098605155268,36.68111801147461]},{"page":376,"text":"Fig. 12.13 Inset: calculated","rect":[53.812843322753909,67.58130645751953,150.85370391993448,59.648292541503909]},{"page":376,"text":"space dependence of the x-","rect":[53.81284713745117,77.4895248413086,146.29995816809825,69.89517211914063]},{"page":376,"text":"component of the director","rect":[53.813690185546878,87.4087142944336,142.53814786536388,79.81436157226563]},{"page":376,"text":"nx(z) within one period of the","rect":[53.813690185546878,97.3844223022461,154.721780274148,89.79006958007813]},{"page":376,"text":"cholesteric structure. Main","rect":[53.81251525878906,105.60789489746094,145.10323089747355,99.76608276367188]},{"page":376,"text":"plot: Fourier transform nx(q)","rect":[53.81251525878906,117.33638763427735,151.39740079505138,109.74203491210938]},{"page":376,"text":"showing appearance of the","rect":[53.812503814697269,127.25545501708985,145.26229235910894,119.66110229492188]},{"page":376,"text":"third harmonic of the helix in","rect":[53.812503814697269,135.50425720214845,154.84699005274698,129.63705444335938]},{"page":376,"text":"a strong field. In both plots","rect":[53.812503814697269,147.20736694335938,146.4163711833886,139.61300659179688]},{"page":376,"text":"solid lines correspond to zero","rect":[53.812503814697269,157.183349609375,154.6955313125126,149.57205200195313]},{"page":376,"text":"voltage, dot (or dash) curves","rect":[53.812503814697269,167.10260009765626,152.03537447928705,159.49130249023438]},{"page":376,"text":"to U ¼ 200 V.For parameters","rect":[53.812503814697269,177.07852172851563,155.28528292655268,169.5349578857422]},{"page":376,"text":"see the text","rect":[53.811649322509769,185.3019561767578,92.5794268056399,179.46014404296876]},{"page":376,"text":"1.0","rect":[189.527587890625,89.42747497558594,200.63822415673594,83.66070556640625]},{"page":376,"text":"0.5","rect":[189.527587890625,144.5853729248047,200.63822415673594,138.818603515625]},{"page":376,"text":"0.0","rect":[189.527587890625,199.74317932128907,200.63822415673594,193.97640991210938]},{"page":376,"text":"0.0","rect":[200.07308959960938,210.54408264160157,211.1837258657203,204.77731323242188]},{"page":376,"text":"Fig. 12.14 Calculated","rect":[53.812843322753909,259.5026550292969,131.46431488184855,251.77284240722657]},{"page":376,"text":"transmission spectra ofa","rect":[53.812843322753909,269.35418701171877,138.65453479074956,261.75982666015627]},{"page":376,"text":"planar cholesteric texture in","rect":[53.812843322753909,279.3301086425781,149.24017852930948,271.7357482910156]},{"page":376,"text":"zero field and in field","rect":[53.812843322753909,287.57891845703127,126.93510955958291,281.7117004394531]},{"page":376,"text":"E ¼ 5.7 V/mm applied","rect":[53.812843322753909,299.28204345703127,131.2773184218876,291.6368713378906]},{"page":376,"text":"perpendicular to the helical","rect":[53.812843322753909,309.2012634277344,147.38549522605005,301.6069030761719]},{"page":376,"text":"axis. Note appearance of the","rect":[53.812843322753909,319.1772155761719,151.26659533762456,311.5828552246094]},{"page":376,"text":"strong second order photonic","rect":[53.812843322753909,329.1531677246094,153.17543933176519,321.5588073730469]},{"page":376,"text":"stop-band even for non-","rect":[53.812843322753909,339.1291198730469,135.7667703018873,331.5347595214844]},{"page":376,"text":"polarized light","rect":[53.812843322753909,349.04833984375,103.39387230124537,341.4539794921875]},{"page":376,"text":"1.0","rect":[200.88040161132813,266.6135559082031,211.80719998681406,260.8467712402344]},{"page":376,"text":"0.8","rect":[200.88040161132813,291.2210998535156,211.80719998681406,285.4543151855469]},{"page":376,"text":"0.6","rect":[200.88040161132813,315.8270568847656,211.80719998681406,310.0602722167969]},{"page":376,"text":"0.4","rect":[200.88040161132813,340.4338073730469,211.80719998681406,334.6670227050781]},{"page":376,"text":"0.2","rect":[200.88040161132813,365.0397644042969,211.80719998681406,359.2729797363281]},{"page":376,"text":"1st","rect":[221.8136444091797,176.07174682617188,232.47665907543829,170.33697509765626]},{"page":376,"text":"365","rect":[372.4961853027344,42.55679702758789,385.1877798476688,36.63032150268555]},{"page":376,"text":"by a single harmonic at q/2p ¼ q0/2p ¼ 1/P0 ¼ 2.5 mm\u00041. At U ¼ 200 V, a strong","rect":[53.812843322753909,450.4305114746094,385.13771906903755,439.3794860839844]},{"page":376,"text":"third harmonic of the distorted helix appears at q/2p ¼ 3q0/2p ¼3P0 ¼7.5 mm\u00041.","rect":[53.81380844116211,462.39007568359377,385.1832241585422,451.3391418457031]},{"page":376,"text":"The amplitude of the field induced third harmonic reaches the value as high as 27%","rect":[53.814537048339847,474.3497314453125,385.13554598708296,465.4151611328125]},{"page":376,"text":"of the first harmonic amplitude at zero field. Note that characteristic relaxation time","rect":[53.814537048339847,486.3092956542969,385.1583637066063,477.374755859375]},{"page":376,"text":"of any elastic distortion mode is described by universal (hydrodynamic) formula","rect":[53.814537048339847,498.268798828125,385.12448919488755,489.334228515625]},{"page":376,"text":"t ¼ g/Kq2, where g is a rotational viscosity. Therefore, the higher the harmonic of","rect":[53.814537048339847,510.1720886230469,385.1480954727329,499.1210632324219]},{"page":376,"text":"distortion the shorter is its relaxation time. This fact is of principal importance for","rect":[53.812225341796878,522.0220336914063,385.1799253067173,513.1771240234375]},{"page":376,"text":"the fast devices based on the helix anharmonicity [15].","rect":[53.81324768066406,534.0911254882813,274.8106655647922,525.0968627929688]},{"page":376,"text":"Figure 12.14 shows the calculated transmission spectra of a cholesteric mixture","rect":[65.76527404785156,546.0506591796875,385.11826360895005,537.1161499023438]},{"page":376,"text":"in zero field and at E ¼ 5.7 V/mm. In this case, the pitch is 0.4 mm and the cell","rect":[53.81325149536133,558.0101928710938,385.122206283303,549.0159301757813]},{"page":376,"text":"thickness d ¼10 mm. The incident light is circularly polarised. Upon application of","rect":[53.81324768066406,569.9697875976563,385.1511167129673,561.0153198242188]},{"page":376,"text":"the field, a strong second Bragg reflection band emerges. The transmission is almost","rect":[53.814239501953128,581.9293212890625,385.1321855313499,572.9948120117188]},{"page":377,"text":"366","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.68026351928711]},{"page":377,"text":"12 Electro-Optical Effects in Cholesteric Phase","rect":[223.29483032226563,44.274620056152347,385.1406798102808,36.663330078125]},{"page":377,"text":"completely suppressed within a narrow spectral band. Within this range, using","rect":[53.812843322753909,68.2883529663086,385.17559138349068,59.35380554199219]},{"page":377,"text":"electric field one can directly modulate light without polarizers.","rect":[53.812843322753909,80.24788665771485,310.6513943245578,71.31333923339844]},{"page":377,"text":"The appearance of the high harmonics in the director distribution results in","rect":[65.76486206054688,92.20748138427735,385.13979426435005,83.27293395996094]},{"page":377,"text":"considerably","rect":[53.812843322753909,104.11019134521485,104.88690272382269,95.17564392089844]},{"page":377,"text":"faster","rect":[110.00436401367188,103.0,132.16243873445166,95.17564392089844]},{"page":377,"text":"electro-optical","rect":[137.36351013183595,104.11019134521485,195.26017625400614,95.17564392089844]},{"page":377,"text":"switching.","rect":[200.40847778320313,104.11019134521485,241.72842069174534,95.17564392089844]},{"page":377,"text":"The","rect":[246.85684204101563,103.0,262.39536321832505,95.17564392089844]},{"page":377,"text":"dynamics","rect":[267.53173828125,104.11019134521485,305.8564492617722,95.17564392089844]},{"page":377,"text":"of","rect":[310.9779052734375,103.0,319.26973853913918,95.17564392089844]},{"page":377,"text":"the","rect":[324.40313720703127,103.0,336.6269000835594,95.17564392089844]},{"page":377,"text":"cholesteric","rect":[341.7364196777344,103.0,385.1029437847313,95.17564392089844]},{"page":377,"text":"helix in the electric field is described by the balance of viscous, elastic and electric","rect":[53.812843322753909,116.0697250366211,385.1487506694969,107.13517761230469]},{"page":377,"text":"torques in the infinitely thick sample is given by","rect":[53.812843322753909,128.02932739257813,248.93825617841254,119.09477233886719]},{"page":377,"text":"g1 qqjt ¼ K22 qq2zj2 þ 4epa E2 sinjcosj","rect":[146.4268035888672,163.13665771484376,292.5741145294502,140.27664184570313]},{"page":377,"text":"(12.31)","rect":[356.0710144042969,157.8424835205078,385.1592038711704,149.36611938476563]},{"page":377,"text":"A low field only slightly changes the angle j keeping only the first harmonic of","rect":[65.76544952392578,184.65411376953126,385.1502622207798,175.7195587158203]},{"page":377,"text":"the structure with wavevector 2p/P0. Then, as soon as the field is switched off, the","rect":[53.81342697143555,196.22222900390626,385.17530096246568,187.62229919433595]},{"page":377,"text":"helical structure jE(z) would relax to the field-off structure according to the same","rect":[53.81357955932617,208.51678466796876,385.1295856304344,199.58216857910157]},{"page":377,"text":"equation (12.31) without the field term. With ∂j/∂z ffi q0 we find solution","rect":[53.814613342285159,220.476318359375,355.30523005536568,210.53575134277345]},{"page":377,"text":"j ¼ jE sinq0zexp\u0004\u0004K2g21q20 t\u0005","rect":[156.17127990722657,257.2385559082031,282.7782866511919,232.72308349609376]},{"page":377,"text":"and the field-free relaxation time","rect":[53.81354904174805,276.7680358886719,186.28020513727035,269.86541748046877]},{"page":377,"text":"g1","rect":[224.9945831298828,298.0847473144531,233.46334908089004,291.08935546875]},{"page":377,"text":"t1 ¼ K22q20","rect":[196.33197021484376,312.3951721191406,240.4873206385072,298.0724792480469]},{"page":377,"text":"(12.32)","rect":[356.0715026855469,304.2489318847656,385.1596921524204,295.7725830078125]},{"page":377,"text":"Note that in contrast to nematics t1 is controlled by the helical pitch P0 ¼ 2p/q0","rect":[65.76595306396485,333.95135498046877,385.18130140630105,325.01654052734377]},{"page":377,"text":"and not by cell thickness d. At a strong field, the distortion involves several harmonics","rect":[53.812843322753909,345.9109802246094,385.1656838809128,336.9565124511719]},{"page":377,"text":"with number m and wavevectors qm ¼ 2pm/P0 and each harmonic relaxes with its","rect":[53.8138427734375,357.760986328125,385.2666360293503,348.93597412109377]},{"page":377,"text":"own time","rect":[53.81330490112305,367.7117004394531,91.0599597759422,360.8389892578125]},{"page":377,"text":"g1","rect":[224.37149047851563,389.1142272949219,232.84016487678847,382.1187438964844]},{"page":377,"text":"tm ¼ K22m2q02 :","rect":[188.62814331054688,403.424560546875,250.34031617325685,389.10284423828127]},{"page":377,"text":"(12.33)","rect":[356.0709533691406,395.2783508300781,385.15914283601418,386.802001953125]},{"page":377,"text":"For instance, the third harmonic of the distorted helix relaxes nine-times faster","rect":[65.76538848876953,423.0,385.1353696426548,416.04595947265627]},{"page":377,"text":"than the first one and this agrees with experimental data showing submillisecond","rect":[53.8133659362793,436.9400634765625,385.1353386979438,428.0054931640625]},{"page":377,"text":"response times of the cholesteric helix in the external electric field.","rect":[53.8133659362793,448.8996276855469,324.5012173226047,439.965087890625]},{"page":377,"text":"It is very spectacular that the electrooptical cell shown in Fig. 12.10 can provide","rect":[65.76538848876953,460.8023986816406,385.1253131694969,451.86785888671877]},{"page":377,"text":"very high and spectrally tunable optical contrast between the field -off and -on","rect":[53.814353942871097,472.7619323730469,385.1422967057563,463.827392578125]},{"page":377,"text":"states. To this effect, we install the cell between two polarizers and each of them","rect":[53.814353942871097,484.7214660644531,385.1303477156218,475.78692626953127]},{"page":377,"text":"should precisely be oriented at particular angles. Using variable optical anisotropy","rect":[53.814353942871097,496.6809997558594,385.1483086686469,487.7464599609375]},{"page":377,"text":"the spectral band of high contrast may be done either very narrow and tunable (for","rect":[53.814353942871097,508.64056396484377,385.1512693008579,499.70599365234377]},{"page":377,"text":"large Dn) or very wide for white light applications (small Dn).","rect":[53.814353942871097,520.6000366210938,305.34429593588598,511.3368225097656]},{"page":377,"text":"12.2.3 Instability of the Planar Cholesteric Texture","rect":[53.812843322753909,554.7277221679688,315.87019141645637,544.0899658203125]},{"page":377,"text":"For unwinding the helical structure, Eq. (12.29) relates the threshold coherence length","rect":[53.812843322753909,582.2699584960938,385.2521905045844,573.33544921875]},{"page":377,"text":"to a characteristic size of the system, namely, the pitch of the helix xu ¼ P0/p2.","rect":[53.812843322753909,594.2294921875,385.1832241585422,583.1787719726563]},{"page":378,"text":"12.2 Dielectric Instability of Cholesterics","rect":[53.81254196166992,44.275901794433597,195.13188632010736,36.68154525756836]},{"page":378,"text":"367","rect":[372.4970703125,42.55722427368164,385.18866485743447,36.68154525756836]},{"page":378,"text":"In Chapter 11 we have found that, for the Frederiks transition in nematics,","rect":[53.812843322753909,68.2883529663086,385.1816372444797,59.35380554199219]},{"page":378,"text":"the threshold field coherence length is determined by the cell thickness, xF ¼ d/p,","rect":[53.812843322753909,80.24788665771485,385.1833462288547,70.9946060180664]},{"page":378,"text":"see Eq.(11.53). Now we shall briefly discuss another type of instability with a","rect":[53.814659118652347,92.20760345458985,385.1615680523094,83.21329498291016]},{"page":378,"text":"threshold determined by the geometrical average of the two parameters mentioned","rect":[53.814659118652347,104.11031341552735,385.17641535810005,95.17576599121094]},{"page":378,"text":"(P0d)1/2 [17].","rect":[53.814659118652347,115.73550415039063,105.49942441488986,104.97955322265625]},{"page":378,"text":"Let both the helical axis and the electric field are parallel to the normal z of a","rect":[65.76619720458985,128.02999877929688,385.1620258159813,119.09544372558594]},{"page":378,"text":"cholesteric liquid crystal layer of thickness d and ea >0. In the case of a very weak","rect":[53.81420135498047,139.98959350585938,385.13662043622505,131.03504943847657]},{"page":378,"text":"field the elastic forces tend to preserve the original stack-like arrangement of the","rect":[53.81370162963867,151.94912719726563,385.17444647027818,143.0145721435547]},{"page":378,"text":"cholesteric quasi-layers as shown in Fig. 12.15a. On the contrary, in a very strong","rect":[53.81370162963867,163.90869140625,385.13762751630318,154.91436767578126]},{"page":378,"text":"field, the dielectric torque causes the local directors to be parallel to the cell normal,","rect":[53.81370162963867,175.86822509765626,385.1773953011203,166.9336700439453]},{"page":378,"text":"as shown in Fig. 12.15c. At intermediate fields, due to competition of the elastic and","rect":[53.81370162963867,187.8277587890625,385.14360896161568,178.83343505859376]},{"page":378,"text":"electric forces an undulation pattern appears pictured in Fig. 12.15b. Such a","rect":[53.81370162963867,199.73052978515626,385.16049993707505,190.7362060546875]},{"page":378,"text":"structure has two wavevectors, one along the z-axis (p/d) and the other along the","rect":[53.81368637084961,211.69009399414063,385.17441595270005,202.73561096191407]},{"page":378,"text":"arbitrary direction x within the xy-plane. The periodicity of the director pattern","rect":[53.814674377441409,223.64959716796876,385.17839900067818,214.7150421142578]},{"page":378,"text":"results in periodicity in the distribution of the refractive index. Hence, a diffraction","rect":[53.815650939941409,235.60916137695313,385.0938347916938,226.6746063232422]},{"page":378,"text":"grating forms. Let us find a threshold field for this instability.","rect":[53.815650939941409,247.56869506835938,301.6108059456516,238.63414001464845]},{"page":378,"text":"In the absence of the field, the director components are n ¼ (cosq0z, sinq0z, 0)","rect":[65.76769256591797,259.52825927734377,385.15252052156105,250.5937042236328]},{"page":378,"text":"and q0 ¼ ∂j/∂z. For a small field perturbation, both the conical distortion appears","rect":[53.814720153808597,271.4891662597656,385.1026955996628,261.5486145019531]},{"page":378,"text":"(angle W) and the azimuthal angle j slightly changes. The new components of the","rect":[53.812538146972659,283.4486999511719,385.1732868023094,274.21533203125]},{"page":378,"text":"director are:","rect":[53.813541412353519,293.3294982910156,102.7544084317405,286.41693115234377]},{"page":378,"text":"nx ¼ cosðq0z þ jÞ \u0001 cosq0z \u0004 jsinq0z","rect":[138.5547332763672,320.6300354003906,300.4263955508347,310.67950439453127]},{"page":378,"text":"ny ¼ sinðq0z þ jÞ \u0001 sinq0z þ jcosq0z","rect":[138.5550079345703,336.0723876953125,298.78366483794408,325.5865173339844]},{"page":378,"text":"nz ¼ Wcosq0z","rect":[138.95089721679688,350.0525207519531,195.29301847563938,340.9287109375]},{"page":378,"text":"(12.34)","rect":[356.0704345703125,334.8000183105469,385.15862403718605,326.32366943359377]},{"page":378,"text":"If we intend to calculate precisely the threshold field for the two-dimensional","rect":[65.76487731933594,375.0453796386719,385.17158372470927,366.11083984375]},{"page":378,"text":"distortion we should write the Frank free energy with the director components","rect":[53.812862396240237,387.0049133300781,385.1278420840378,378.07037353515627]},{"page":378,"text":"(12.34) and the field term (ea/4p)(En)2 and then make minimization of the free","rect":[53.812862396240237,398.56597900390627,385.1367572612938,387.8801574707031]},{"page":378,"text":"energy with respect to the two variables j and W [18]. For a qualitative estimation of","rect":[53.8138542175293,410.923828125,385.14971290437355,401.6904602050781]},{"page":378,"text":"the threshold we prefer to follow the simple arguments by Helfrich [17]. We","rect":[53.8138542175293,422.8833923339844,385.12781561090318,413.9488525390625]},{"page":378,"text":"consider a one-dimensional (in layer plane xy) periodic distortion of a cholesteric","rect":[53.81284713745117,434.8429260253906,385.1039203472313,425.90838623046877]},{"page":378,"text":"Fig. 12.15 A planar cholesteric structure in the electric field parallel to the helical axis (ea > 0).","rect":[53.812843322753909,564.1054077148438,385.1734950263735,556.3756103515625]},{"page":378,"text":"The local director orientation is shown by solid lines: field-off planar alignment (a), undulated","rect":[53.813472747802737,573.9569091796875,385.18198150782509,566.3456420898438]},{"page":378,"text":"structure ina weak field Eu > E > Eth (b), and the homeotropic structure in the field exceeding the","rect":[53.813472747802737,583.9326171875,385.1546873786402,576.3382568359375]},{"page":378,"text":"threshold for helix unwinding E > Eu (c)","rect":[53.81325912475586,593.9085693359375,195.5236062393873,586.314208984375]},{"page":379,"text":"368","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.68026351928711]},{"page":379,"text":"12 Electro-Optical Effects in Cholesteric Phase","rect":[223.29483032226563,44.274620056152347,385.1406798102808,36.663330078125]},{"page":379,"text":"with helical axis and electric field E parallel to axis z and layer thickness d \f P0.","rect":[53.812843322753909,68.2883529663086,385.1832241585422,59.333885192871097]},{"page":379,"text":"Further, the deformation is assumed to be sinusoidal along both the x- axis (period","rect":[53.814537048339847,80.24788665771485,385.10659113935005,71.31333923339844]},{"page":379,"text":"equals w) and the z-axis (half-period equals d). Hence, we have two variables: one","rect":[53.814537048339847,92.20748138427735,385.1464008159813,83.25301361083985]},{"page":379,"text":"of them is the x- and z-dependent tilt angle W of the helical axis with respect to the","rect":[53.81554412841797,104.11019134521485,385.17624700738755,94.8768310546875]},{"page":379,"text":"cell normal, see Fig. 12.15b,","rect":[53.814537048339847,116.0697250366211,168.85635037924534,107.0754165649414]},{"page":379,"text":"y ¼ \u0004ym sinðpz=dÞsinð2px=wÞ","rect":[157.19102478027345,140.3282470703125,281.7917519961472,130.3777313232422]},{"page":379,"text":"(12.35a)","rect":[351.6543884277344,139.59117126464845,385.1303952774204,130.99526977539063]},{"page":379,"text":"and the other is","rect":[53.815467834472659,161.87673950195313,120.26181425200656,154.97413635253907]},{"page":379,"text":"distorted and the","rect":[53.815467834472659,174.0,121.04122198297346,166.9336700439453]},{"page":379,"text":"x- and z-dependent difference between the wavevectors","rect":[124.4515380859375,163.90869140625,356.18158353911596,154.97413635253907]},{"page":379,"text":"equilibrium helix, Dq ¼ q\u0004q0 (q0 ¼ 2p/P0):","rect":[123.88513946533203,175.86822509765626,304.5752397305686,166.60498046875]},{"page":379,"text":"of","rect":[360.4330139160156,162.0,368.7248776992954,154.97413635253907]},{"page":379,"text":"the","rect":[372.9514465332031,162.0,385.1752094097313,154.97413635253907]},{"page":379,"text":"Dq ¼ Dqm cosðpz=dÞcosð2px=wÞ","rect":[152.71578979492188,200.06991577148438,286.2663613711472,190.11940002441407]},{"page":379,"text":"(12.35b)","rect":[351.08819580078127,199.3328399658203,385.15349708406105,190.7369384765625]},{"page":379,"text":"The splay and bend distortions are described by angle W while the twist distortion","rect":[65.7676773071289,223.65036010742188,385.15850153974068,214.4169921875]},{"page":379,"text":"is related to a slight change of the period of the helical structure. The maximum","rect":[53.81566619873047,235.60992431640626,385.0968088972624,226.6753692626953]},{"page":379,"text":"values of two variables Wm and Dqm are coupled to each other by equation","rect":[53.81566619873047,247.56985473632813,352.50847712567818,238.30661010742188]},{"page":379,"text":"y ¼ 2d \u0006 Dqm","rect":[189.53517150878907,275.7659912109375,247.27428840857605,261.7724304199219]},{"page":379,"text":"m","rect":[194.51947021484376,277.2950744628906,199.52244209509949,274.0599365234375]},{"page":379,"text":"w q0","rect":[215.08169555664063,284.5295104980469,243.20622322639785,277.8659362792969]},{"page":379,"text":"(12.36)","rect":[356.0715026855469,277.43902587890627,385.1596921524204,268.90289306640627]},{"page":379,"text":"that can be understood with the help of Fig. 12.15b. Indeed, due to strong anchoring","rect":[53.81393051147461,308.048583984375,385.17763606122505,299.05426025390627]},{"page":379,"text":"the number of helical turns in the cell is fixed, but for the helical axis tilted through","rect":[53.81393051147461,320.00811767578127,385.1398552995063,311.07354736328127]},{"page":379,"text":"angle W the helical pitch becomes larger (\u0001P0/cosW) and the wavevector q smaller","rect":[53.81393051147461,331.9676513671875,385.08879981843605,322.7341613769531]},{"page":379,"text":"by Dqm. In addition, for fixed cell thickness d and q0, with decreasing period of","rect":[53.81370162963867,343.92724609375,385.15035377351418,334.6638488769531]},{"page":379,"text":"distortion w, the tilt angle Wm will be larger because the sin(2p/w) function in","rect":[53.81350326538086,355.8868713378906,385.14333430341255,346.6534118652344]},{"page":379,"text":"Eq. (12.35a) becomes sharper. Using Eq. (12.36) we have only one independent","rect":[53.81345748901367,367.7896423339844,385.126356673928,358.7953186035156]},{"page":379,"text":"variable.","rect":[53.8134880065918,377.6873779296875,88.62037320639377,370.81463623046877]},{"page":379,"text":"Now we are looking fora difference betweenthe elastic energies of structures (a)","rect":[65.76549530029297,391.708740234375,385.1583493789829,382.774169921875]},{"page":379,"text":"and (b) in Fig. 12.15 irrespective of a source of the distortion. For a small distortion","rect":[53.8134880065918,403.6683044433594,385.1573113541938,394.6739807128906]},{"page":379,"text":"and director compounds nx ¼ cosW \u0001 1, ny and nz ¼ sinW \u0001 W the highest order","rect":[53.8134880065918,416.544921875,385.14864478913918,406.39508056640627]},{"page":379,"text":"terms for splay, bend and twist are divn ¼ ∂W/∂z, n curln¼\u0004∂W/∂x and","rect":[53.81481170654297,427.5879821777344,385.14568415692818,417.6474304199219]},{"page":379,"text":"ncurln¼\u0004∂ny/∂z ¼ Dqm. Then, using Eqs. (12.35) we can write the Frank energy","rect":[53.81382369995117,440.4642333984375,385.16933527997505,429.60699462890627]},{"page":379,"text":"density:","rect":[53.813541412353519,451.5072937011719,85.49978501865457,442.57275390625]},{"page":379,"text":"gelast ¼ 18\"K11\bpd","rect":[112.78128814697266,495.5498046875,182.62413377116276,465.6883239746094]},{"page":379,"text":"2 þ K33\u00042wp\u00052#W2m þ 14K22ðDqmÞ2","rect":[188.51513671875,495.5498046875,325.73823616585096,465.6883239746094]},{"page":379,"text":"Here, the average values of <cos2W> ¼ <sin2W> ¼ 1/2 are used. As the cell","rect":[65.76496887207031,519.127685546875,385.1235185391624,508.0199279785156]},{"page":379,"text":"thickness is assumed to be large, d \f w Helfrich discarded the splay term and the","rect":[53.814537048339847,531.0303955078125,385.1742633648094,522.075927734375]},{"page":379,"text":"elastic free energy density is reduced to the form","rect":[53.814537048339847,542.989990234375,251.2284006941374,534.0554809570313]},{"page":379,"text":"gelast ¼ 81\"K33\u00042wp\u00052#W2m þ 41K22ðDqmÞ2:","rect":[135.21310424804688,587.0892944335938,303.81333863419436,557.227783203125]},{"page":379,"text":"(12.37a)","rect":[351.65325927734377,576.4302978515625,385.1292661270298,567.9539184570313]},{"page":380,"text":"12.2 Dielectric Instability of Cholesterics","rect":[53.812843322753909,44.274620056152347,195.13217623709955,36.68026351928711]},{"page":380,"text":"369","rect":[372.49737548828127,42.62367248535156,385.1889700332157,36.68026351928711]},{"page":380,"text":"When the electric field applied parallel to the helical axis is used to create the","rect":[65.76496887207031,68.2883529663086,385.17273748590318,59.35380554199219]},{"page":380,"text":"distortion, the electric free energy density is given by:","rect":[53.812950134277347,80.24788665771485,272.1276994473655,71.31333923339844]},{"page":380,"text":"gE ¼ ðea=42pÞE2 Wm2:","rect":[182.22723388671876,119.4399642944336,256.7411187855615,97.42593383789063]},{"page":380,"text":"(12.37b)","rect":[351.08612060546877,114.3116226196289,385.15142188874855,105.77549743652344]},{"page":380,"text":"It includes term (e|| \u0004 e⊥)/2 because in the problem discussed, the torque is con-","rect":[65.76561737060547,145.94119262695313,385.1685727676548,137.00657653808595]},{"page":380,"text":"trolled by anisotropy e||h \u0004 e⊥h defined with respect to the helical axis as e||h ¼ e⊥ and","rect":[53.81382369995117,157.90078735351563,385.1796502213813,148.96617126464845]},{"page":380,"text":"e⊥h ¼ (e|| þ e⊥)/2 (in addition, small anisotropy ea \u0005 <e> is assumed).","rect":[53.813961029052737,169.8604736328125,341.28313870932348,160.92591857910157]},{"page":380,"text":"On account of Eqs. (12.36) and (12.37) the total free energy density reads","rect":[65.76653289794922,181.76324462890626,363.8125039492722,172.8087615966797]},{"page":380,"text":"g ¼ 81\"K33\u00042wp\u00052 þ 2K22\bq20dw","rect":[127.62400817871094,228.8669891357422,254.11224624331738,199.00547790527345]},{"page":380,"text":"2 \u0004 e4aEp2#Wm2","rect":[260.05792236328127,228.8669891357422,310.83021187049016,199.00547790527345]},{"page":380,"text":"(12.38)","rect":[356.0715026855469,218.20750427246095,385.1596921524204,209.73114013671876]},{"page":380,"text":"Now we can find the period of the distortion using minimisation (12.38) with","rect":[65.76595306396485,255.3919677734375,385.11892024091255,246.45741271972657]},{"page":380,"text":"respect to w (∂g/∂w ¼ 0):","rect":[53.81393051147461,267.391357421875,161.02770860263895,257.4109802246094]},{"page":380,"text":"w2 ¼ \u00042KK2323\u00051=2ðP0dÞ","rect":[173.72927856445313,310.9722900390625,265.2495461856003,284.6438293457031]},{"page":380,"text":"(12.39)","rect":[356.0707702636719,303.2853698730469,385.1589597305454,294.80902099609377]},{"page":380,"text":"This period is determined solely by the elastic forces. In fact, the instability","rect":[65.76519012451172,337.46575927734377,385.1630486588813,328.53118896484377]},{"page":380,"text":"with the same period can be caused by other external factors, for example, by a","rect":[53.813167572021487,349.4253234863281,385.15708196832505,340.49078369140627]},{"page":380,"text":"magnetic field or by an electrohydrodynamic process caused by conductivity of the","rect":[53.813167572021487,361.32806396484377,385.16992986871568,352.39349365234377]},{"page":380,"text":"material [19].","rect":[53.813167572021487,372.6800231933594,108.61407132651095,364.35308837890627]},{"page":380,"text":"Now the threshold field Eth for the instability can be found from Eqs. (12.38)","rect":[65.76517486572266,385.2475280761719,385.15950904695168,376.3126220703125]},{"page":380,"text":"and (12.39):","rect":[53.81367874145508,396.80865478515627,102.91979081699441,388.27252197265627]},{"page":380,"text":"Eth ¼ 2pðK22K33Þ1=4\u0004ea2Pp0d\u00051=2","rect":[153.5090789794922,440.8282470703125,285.01008675178846,414.4995422363281]},{"page":380,"text":"(12.40)","rect":[356.0715026855469,433.14117431640627,385.1596921524204,424.6648254394531]},{"page":380,"text":"We can see that, for our one-dimensional distortion in the xy plane the threshold","rect":[65.76595306396485,467.3215637207031,385.14080134442818,458.38702392578127]},{"page":380,"text":"coherence length xth / E\u0004th1 / pffiPffiffi0ffiffidffiffi is determined by the geometrical average of","rect":[53.81393051147461,480.04217529296877,385.1505063614048,468.7986145019531]},{"page":380,"text":"the two characteristic lengths. The numerical coefficients in Eq. (12.40) should not","rect":[53.81367111206055,491.2406311035156,385.1346269375999,482.30609130859377]},{"page":380,"text":"be taken too seriously due to the qualitative nature of our consideration. Neverthe-","rect":[53.81270980834961,503.2001647949219,385.06795631257668,494.265625]},{"page":380,"text":"less, in experiment the distortion emerges at the fields higher than the Fredericks","rect":[53.81270980834961,515.15966796875,385.1247598086472,506.22515869140627]},{"page":380,"text":"transition threshold but lower than the helix unwinding one. As a rule, due to","rect":[53.81270980834961,527.1192016601563,385.1376580338813,518.1846923828125]},{"page":380,"text":"rotational symmetry of the planar cholesteric texture having helical axis along z","rect":[53.81270980834961,539.02197265625,385.1665078555222,530.0874633789063]},{"page":380,"text":"we observe not a one-dimensional stripe pattern but a two-dimensional grid in the","rect":[53.81272506713867,550.9815673828125,385.17346990777818,542.0470581054688]},{"page":380,"text":"xy plane, see Ref. [19].","rect":[53.81272506713867,562.9411010742188,146.90548368002659,554.006591796875]},{"page":381,"text":"370","rect":[53.81287384033203,42.55740737915039,66.5044798293583,36.73252868652344]},{"page":381,"text":"12 Electro-Optical Effects in Cholesteric Phase","rect":[223.29486083984376,44.276084899902347,385.1407103278589,36.664794921875]},{"page":381,"text":"12.3 Bistability and Memory","rect":[53.812843322753909,70.66844177246094,209.0457281103093,59.086421966552737]},{"page":381,"text":"12.3.1 Naive Idea","rect":[53.812843322753909,97.73860168457031,149.50449886226245,89.39572143554688]},{"page":381,"text":"Let an electro-optical cell based on a liquid crystal, possesses two optically different","rect":[53.812843322753909,127.5765609741211,385.154676986428,118.6021728515625]},{"page":381,"text":"stable configurations having the same equilibrium elastic energy. Then imagine that","rect":[53.812843322753909,139.53610229492188,385.1367631680686,130.60154724121095]},{"page":381,"text":"we can switch the cell between the two states by relatively short low-voltage pulses","rect":[53.812843322753909,151.43887329101563,385.12680448638158,142.5043182373047]},{"page":381,"text":"and keep the structure in either state for an infinitely long time. Such a cell would","rect":[53.812843322753909,163.3984375,385.13481989911568,154.46388244628907]},{"page":381,"text":"represent a bistable optical memory device. For instance, it may be a display","rect":[53.812843322753909,175.35800170898438,385.09987727216255,166.42344665527345]},{"page":381,"text":"consuming very little energy from the voltage source because the source is used","rect":[53.812843322753909,187.31753540039063,385.12383357099068,178.3829803466797]},{"page":381,"text":"only during switching. Such displays can also be useful for many applications from","rect":[53.812843322753909,199.27703857421876,385.13480328202805,190.3424835205078]},{"page":381,"text":"small smart cards and electronic books to gigantic advertising tableaux.","rect":[53.812843322753909,211.23660278320313,342.6960110237766,202.3020477294922]},{"page":381,"text":"The simplest but not the best idea of a bistable structure is shown in Fig. 12.16a.","rect":[65.76486206054688,223.19613647460938,385.1815456917453,214.26158142089845]},{"page":381,"text":"A nematic liquid crystal layer of thickness d is placed between two plates and the","rect":[53.81288146972656,235.15570068359376,385.17261541559068,226.2012176513672]},{"page":381,"text":"directors at the plates are aligned perpendicular to each other with j0 ¼ 0 and","rect":[53.81288146972656,247.0584716796875,385.1465691666938,238.12391662597657]},{"page":381,"text":"jd ¼ p/2. Assume the infinitely strong anchoring. Then the nematic is twisted left","rect":[53.814720153808597,259.0193176269531,385.11811692783427,250.0847625732422]},{"page":381,"text":"or right by þ p/2 or \u0004 p/2 and both the twisted structures have the same total","rect":[53.81305694580078,270.9788818359375,385.1827836758811,262.0443115234375]},{"page":381,"text":"energy including the elastic and surface terms. The elastic distortion energy Fd ofa","rect":[53.81405258178711,282.93841552734377,385.16245306207505,274.00384521484377]},{"page":381,"text":"structure twisted through angle j has been calculated in Section 8.3.2:","rect":[53.81462860107422,294.8981628417969,338.477186752053,285.963623046875]},{"page":381,"text":"K22j2","rect":[218.08389282226563,319.53143310546877,242.2999884851869,309.1287841796875]},{"page":381,"text":"Fd¼","rect":[194.52137756347657,325.8138427734375,215.2692797491303,317.64093017578127]},{"page":381,"text":"2d","rect":[225.27783203125,331.2523193359375,235.2967461686469,324.26007080078127]},{"page":381,"text":"(12.41)","rect":[356.0711669921875,325.9573059082031,385.15935645906105,317.48095703125]},{"page":381,"text":"This energy is shown by the dashed parabola in Fig. 12.16b. If we release the","rect":[65.76561737060547,354.6962890625,385.17331731988755,345.76171875]},{"page":381,"text":"anchoring condition at the top interface, the nematic would relax to the uniform","rect":[53.813594818115237,366.65582275390627,385.1644053328093,357.72125244140627]},{"page":381,"text":"structure with zero elastic energy. However, due to strong anchoring energy","rect":[53.813594818115237,378.6153869628906,385.1653985123969,369.68084716796877]},{"page":381,"text":"Ws \f Fd, the total free energy of the twisted structure F ¼ Fd þ Ws shows two","rect":[53.813594818115237,390.5755920410156,385.1541375260688,381.64105224609377]},{"page":381,"text":"minima almost exactly at the \u0007p/2 twist angles at the horizontal level of Fd(p/2).","rect":[53.814292907714847,402.5351257324219,385.16949124838598,393.6005859375]},{"page":381,"text":"If we realign the director at the top interface by an external force through an","rect":[53.813716888427737,414.49468994140627,385.1506280045844,405.56011962890627]},{"page":381,"text":"angle \u0007djs the surface energy will dramatically increase. Therefore we have two","rect":[53.813716888427737,426.4545593261719,385.14931574872505,417.22088623046877]},{"page":381,"text":"minima on the angular dependence of the total free energy with a barrier between","rect":[53.81341552734375,438.3572998046875,385.1741875748969,429.4227294921875]},{"page":381,"text":"these stable states. We say that the cell is bistable.","rect":[53.81341552734375,450.3168640136719,256.96790738608129,441.38232421875]},{"page":381,"text":"a","rect":[90.35135650634766,481.31317138671877,95.9066189035815,475.72442626953127]},{"page":381,"text":"n","rect":[108.5472640991211,504.9383850097656,113.12590514209745,500.8442687988281]},{"page":381,"text":"s","rect":[113.12594604492188,507.009765625,116.18137091591129,503.9359436035156]},{"page":381,"text":"Left","rect":[171.04884338378907,491.12017822265627,185.70049876551557,485.2014465332031]},{"page":381,"text":"n","rect":[189.54356384277345,502.83380126953127,194.1222048857498,498.73968505859377]},{"page":381,"text":"s","rect":[194.1223907470703,504.90484619140627,197.17781561805973,501.8310241699219]},{"page":381,"text":"F","rect":[289.7862548828125,500.4768981933594,294.6727027901631,494.7311096191406]},{"page":381,"text":"n","rect":[122.4233169555664,536.0140380859375,127.00195799854277,531.919921875]},{"page":381,"text":"n","rect":[111.69066619873047,556.0936279296875,116.26930724170683,551.99951171875]},{"page":381,"text":"s","rect":[116.2693099975586,558.1646728515625,119.324734868548,555.0908203125]},{"page":381,"text":"– /2","rect":[247.4659423828125,562.482421875,264.0646906484353,556.4405517578125]},{"page":381,"text":"0","rect":[282.9996032714844,562.0880737304688,287.44619089257597,556.2622680664063]},{"page":381,"text":"Fd","rect":[335.40374755859377,543.9076538085938,343.9523416473169,536.0847778320313]},{"page":381,"text":"/2 Angle","rect":[313.3066101074219,564.0794067382813,343.6187030951518,556.4405517578125]},{"page":381,"text":"Fig. 12.16 Bistable twist cell. Right- and left-handed twist-structures of a nematic liquid crystal","rect":[53.812843322753909,584.0003662109375,385.15429405417509,576.2705688476563]},{"page":381,"text":"with the same elastic energy (a) and the angular dependence of total free energy (b)","rect":[53.812843322753909,593.9085693359375,341.72528165442636,586.314208984375]},{"page":382,"text":"12.3 Bistability and Memory","rect":[53.812843322753909,44.274620056152347,152.5865225234501,36.68026351928711]},{"page":382,"text":"371","rect":[372.4981689453125,42.55594253540039,385.18979400782509,36.73106384277344]},{"page":382,"text":"The two stable twist states have different optical properties. For example, the","rect":[65.76496887207031,68.2883529663086,385.1707233257469,59.35380554199219]},{"page":382,"text":"electric vector of a linearly polarised light is rotated by the two structures in","rect":[53.812950134277347,80.24788665771485,385.13982478192818,71.31333923339844]},{"page":382,"text":"opposite directions. If the Mauguin regime (dDn > l) is not strictly fulfilled the","rect":[53.812950134277347,92.20748138427735,385.1746295757469,82.9442367553711]},{"page":382,"text":"two structures can be distinguished using crossed polarizers.","rect":[53.813873291015628,104.11019134521485,297.3038296272922,95.17564392089844]},{"page":382,"text":"The problem, however, arises with the mechanism of selection of the right and","rect":[65.7658920288086,116.0697250366211,385.1437920670844,107.13517761230469]},{"page":382,"text":"left structures. For example, if a nematic has positive dielectric anisotropy, the","rect":[53.813873291015628,128.02932739257813,385.1736530132469,119.09477233886719]},{"page":382,"text":"applied electric field will align the director into the third, ON- state along the cell","rect":[53.813873291015628,139.98886108398438,385.1218705899436,131.0343780517578]},{"page":382,"text":"normal, from which afterwards, in the absence of the field, it will relax to both","rect":[53.813873291015628,150.0,385.12786189130318,143.0138397216797]},{"page":382,"text":"twisted stable states with equal probability. Actually a multi-domain structure","rect":[53.813873291015628,163.907958984375,385.0871051616844,154.97340393066407]},{"page":382,"text":"forms with many defects (walls) between domains. Of course, it is possible to","rect":[53.813873291015628,175.86749267578126,385.14080134442818,166.9130096435547]},{"page":382,"text":"apply an in-plane electric field directed along the angle p/4. Then, in the middle of","rect":[53.813873291015628,187.8270263671875,385.1517270645298,178.89247131347657]},{"page":382,"text":"the cell, it will be parallel to the director in right-handed domains but perpendicular","rect":[53.813873291015628,199.72976684570313,385.1347898086704,190.7952117919922]},{"page":382,"text":"to the director in the left-handed domains (or vice versa for the field angle of \u0004 p/4).","rect":[53.813873291015628,211.6893310546875,385.2691616585422,202.75477600097657]},{"page":382,"text":"Due to this, the right domain will grow at the cost of the left one and finally the","rect":[53.81483840942383,223.64886474609376,385.17368353082505,214.7143096923828]},{"page":382,"text":"overall right-handed twist structure will be restored. However, this process requires","rect":[53.81483840942383,235.60842895507813,385.16766752349096,226.6738739013672]},{"page":382,"text":"a motion of the domain walls and, therefore, is very slow.","rect":[53.81483840942383,247.56796264648438,287.4183010628391,238.63340759277345]},{"page":382,"text":"12.3.2 Berreman–Heffner Model","rect":[53.812843322753909,291.7017822265625,225.22288724731025,281.1476745605469]},{"page":382,"text":"12.3.2.1 A Cell and Free Energy","rect":[53.812843322753909,319.7262268066406,198.32906428388129,310.07452392578127]},{"page":382,"text":"We would like to consider this particular model in more detail because it demon-","rect":[53.812843322753909,343.19012451171877,385.12191139070168,334.25555419921877]},{"page":382,"text":"strates interesting physical aspects of the bistability problem. Generally, chiral","rect":[53.812843322753909,355.1496887207031,385.0850053555686,346.21514892578127]},{"page":382,"text":"nematics better suited to bistable devices as they have an additional degree of","rect":[53.812843322753909,367.1092224121094,385.1487058242954,358.1746826171875]},{"page":382,"text":"freedom. By doping nematics with chiral compounds a variety of materials with","rect":[53.812843322753909,379.0687561035156,385.11586848310005,370.13421630859377]},{"page":382,"text":"variable pitch can be prepared. The principal idea of Berreman and Heffner [20]","rect":[53.812843322753909,391.0282897949219,385.15764747468605,382.09375]},{"page":382,"text":"was to design a cell having two stable textures (states) with low enough energy","rect":[53.81181716918945,402.9878234863281,385.1656121354438,394.05328369140627]},{"page":382,"text":"barrier between them. Then one can switch them by reasonable voltage. It has been","rect":[53.81181716918945,414.9473876953125,385.1158379655219,406.0128173828125]},{"page":382,"text":"found that, using fine tuning the helical pitch to the cell thickness, the barrier","rect":[53.81181716918945,426.90692138671877,385.1496823867954,417.97235107421877]},{"page":382,"text":"becomes especially low when, instead of cholesteric textures with directors parallel","rect":[53.81181716918945,438.8096923828125,385.16267259189677,429.8751220703125]},{"page":382,"text":"or perpendicular to the cell normal z, the other textures were used with the director","rect":[53.81181716918945,450.76922607421877,385.1079038223423,441.83465576171877]},{"page":382,"text":"strongly tilted with respect to z. For this purpose, the director at the transparent","rect":[53.81181716918945,462.7287902832031,385.144667220803,453.79425048828127]},{"page":382,"text":"electrodes was tilted using evaporation of silicon monoxide from a grazing direc-","rect":[53.81181716918945,474.6883239746094,385.15862403718605,465.7537841796875]},{"page":382,"text":"tion. The zenithal angles of the director about 55\u0003 with respect to z were found to be","rect":[53.81181716918945,486.6495056152344,385.1524127788719,477.6535339355469]},{"page":382,"text":"optimal.","rect":[53.81357955932617,498.6090393066406,86.94318051840549,489.67449951171877]},{"page":382,"text":"In the test cells to be discussed below, the values of the helical pitch and the","rect":[65.7656021118164,510.568603515625,385.1733783550438,501.634033203125]},{"page":382,"text":"tunable cell thickness are close to each other (about 28 mm). Therefore, as shown in","rect":[53.81357955932617,522.4384765625,385.1992119889594,513.5935668945313]},{"page":382,"text":"Fig. 12.17 the full pitch structure (n ¼ 2) is the most stable (n means a number of","rect":[53.81357955932617,534.430908203125,385.1504148086704,525.4963989257813]},{"page":382,"text":"half-pitches). The elastic energy of the two states (n ¼ 0 and n ¼ 2) is calculated","rect":[53.813594818115237,546.390380859375,385.1494073014594,537.4558715820313]},{"page":382,"text":"with allowance for the twist, bend and splay distortions. Solid lines in Fig. 12.18","rect":[53.81357955932617,558.3499755859375,385.15239802411568,549.4154663085938]},{"page":382,"text":"demonstrate dependencies of the elastic energy of the two states on thickness-to-","rect":[53.81357955932617,570.3095092773438,385.09563575593605,561.375]},{"page":382,"text":"pitch ratio in the absence of an external field. In the figure, the free energy is","rect":[53.81357955932617,582.26904296875,385.18426908599096,573.3345336914063]},{"page":382,"text":"normalized to the unit cell area and factor d/K22. It is seen that the free energy for","rect":[53.81357955932617,594.2296142578125,385.17949806062355,585.2741088867188]},{"page":383,"text":"372","rect":[53.813350677490237,42.55789566040039,66.5049566665165,36.73301696777344]},{"page":383,"text":"Fig. 12.17 Berreman–","rect":[53.812843322753909,67.58130645751953,132.7926992812626,59.85148620605469]},{"page":383,"text":"Heffner bistable cell. Director","rect":[53.812843322753909,75.73698425292969,155.3152322159498,69.89517211914063]},{"page":383,"text":"configuration of the cell with","rect":[53.812843322753909,87.4087142944336,153.3877920547001,79.81436157226563]},{"page":383,"text":"two stable states (unwound","rect":[53.812843322753909,97.04601287841797,146.60371917872355,89.79031372070313]},{"page":383,"text":"with n ¼ 0 and twisted with","rect":[53.812843322753909,105.69280242919922,150.78262848048136,99.76632690429688]},{"page":383,"text":"n ¼ 2 half-turns) in the","rect":[53.812843322753909,116.99797821044922,134.3292326667261,109.74227905273438]},{"page":383,"text":"absence of field and the","rect":[53.81199645996094,125.52867889404297,134.95197436594487,119.66146850585938]},{"page":383,"text":"barrier state B in a weak","rect":[53.81199645996094,135.47923278808595,137.38960785059855,129.63742065429688]},{"page":383,"text":"electric field","rect":[53.81199264526367,145.48057556152345,96.34579223780557,139.61337280273438]},{"page":383,"text":"12 Electro-Optical Effects in Cholesteric Phase","rect":[223.29534912109376,44.276573181152347,385.1411986091089,36.665283203125]},{"page":383,"text":"Fig. 12.18 Zero field free","rect":[53.812843322753909,265.45416259765627,144.162698242898,257.50421142578127]},{"page":383,"text":"energy of the states with","rect":[53.812843322753909,275.3624267578125,137.75427765040323,267.76806640625]},{"page":383,"text":"different number of half turns","rect":[53.812843322753909,283.6112365722656,155.3541611003808,277.7440185546875]},{"page":383,"text":"n as a function of cell","rect":[53.812843322753909,293.5897216796875,128.38448814597192,287.6632385253906]},{"page":383,"text":"thickness d normalized to","rect":[53.812843322753909,303.565673828125,141.59390777735636,297.62225341796877]},{"page":383,"text":"pitch P0. Solid lines show the","rect":[53.812843322753909,315.2095031738281,154.8359923102808,307.5975341796875]},{"page":383,"text":"energy of the two stable states","rect":[53.813350677490237,325.184814453125,155.2937362956933,317.5904541015625]},{"page":383,"text":"to be switched. Low energy","rect":[53.813350677490237,335.10400390625,148.7693838515751,327.5096435546875]},{"page":383,"text":"(n ¼ 1) state is excluded from","rect":[53.813350677490237,344.7413330078125,155.3106560551642,337.4856262207031]},{"page":383,"text":"consideration for topological","rect":[53.813350677490237,355.0559387207031,152.4245729848391,347.4615783691406]},{"page":383,"text":"reasons. B marks the high","rect":[53.813350677490237,364.9751892089844,142.1816152968876,357.3808288574219]},{"page":383,"text":"energy barrier state playing","rect":[53.813350677490237,374.95111083984377,147.7566122451298,367.35675048828127]},{"page":383,"text":"the dominant role in the field-","rect":[53.813350677490237,383.1999206542969,155.35633939368419,377.33270263671877]},{"page":383,"text":"on state","rect":[53.813350677490237,393.1504821777344,80.20935199045658,388.1722412109375]},{"page":383,"text":"W","rect":[237.28660583496095,337.3006591796875,244.83224645021944,331.5578918457031]},{"page":383,"text":"30","rect":[250.8040008544922,264.4026184082031,259.6925149282203,258.579833984375]},{"page":383,"text":"20","rect":[250.8040008544922,313.5897216796875,259.6925149282203,307.7669372558594]},{"page":383,"text":"10","rect":[250.8040008544922,362.7768249511719,259.6925149282203,356.95404052734377]},{"page":383,"text":"0","rect":[259.9507141113281,416.38299560546877,264.39496854150158,410.5602111816406]},{"page":383,"text":"U=0","rect":[277.6366271972656,278.53961181640627,294.9180703481422,272.6528625488281]},{"page":383,"text":"n=1","rect":[274.0835876464844,382.04071044921877,290.4777627309547,376.3619079589844]},{"page":383,"text":"B","rect":[310.88055419921877,349.5500793457031,316.6516903477533,343.80731201171877]},{"page":383,"text":"0.5","rect":[305.7177429199219,416.38299560546877,316.8283791860328,410.5602111816406]},{"page":383,"text":"d/P0","rect":[314.591064453125,429.97369384765627,330.3595885277082,421.9721984863281]},{"page":383,"text":"d","rect":[339.99444580078127,296.8802795410156,344.87832971595386,291.0335388183594]},{"page":383,"text":"exp","rect":[344.87933349609377,300.01153564453127,355.2086282684107,295.50048828125]},{"page":383,"text":"n","rect":[351.5558776855469,339.71533203125,356.43976160071949,335.3482666015625]},{"page":383,"text":"= 0","rect":[359.46124267578127,339.85931396484377,371.59504178368908,334.0365295410156]},{"page":383,"text":"n=","rect":[361.2533264160156,396.7039794921875,373.0274102867767,392.3369140625]},{"page":383,"text":"1","rect":[358.1527404785156,416.239013671875,362.59699490868908,410.5602111816406]},{"page":383,"text":"2","rect":[376.0488586425781,396.7039794921875,380.49311307275158,391.0251770019531]},{"page":383,"text":"the states (n ¼ 0) and (n ¼ 2) is equal at the ratio of d/Po \u0001 0.6. However, the","rect":[53.812843322753909,486.6495056152344,385.1749652691063,477.6950378417969]},{"page":383,"text":"optimum cell thickness for the bistable operation found from experiment is larger,","rect":[53.81421661376953,498.6090393066406,385.1809963753391,489.67449951171877]},{"page":383,"text":"dexp/Po ¼ 0.89. It is that thickness, at which the energy of both states would reach","rect":[53.81421661376953,511.4255065917969,385.1285332780219,501.6141357421875]},{"page":383,"text":"the barrier state (B) at relatively low voltage. The voltage dependence of the energy","rect":[53.81357192993164,522.4715576171875,385.16838923505318,513.5370483398438]},{"page":383,"text":"for the cell of that particular thickness is shown in Fig. 12.19. Here voltage U is","rect":[53.81456756591797,534.4310913085938,385.1892129336472,525.49658203125]},{"page":383,"text":"normalized to the Frederiks transition threshold U0. With increasing U/U0, the two","rect":[53.81459426879883,546.3910522460938,385.15422907880318,537.4561157226563]},{"page":383,"text":"stable states, indeed, merge at U/U0 \u0001 1.8 (cross point). The energy of the barrier","rect":[53.814414978027347,558.3507080078125,385.1236814102329,549.4160766601563]},{"page":383,"text":"state B is also changing and that curve also merges with the other two in point R at","rect":[53.81374740600586,570.3102416992188,385.18147142002177,561.375732421875]},{"page":383,"text":"U/U0 \u0001 2. Point R may be called the turn point, from which the system can relax to","rect":[53.81374740600586,582.2699584960938,385.14251032880318,573.33544921875]},{"page":383,"text":"one of the two states in the absence of voltage.","rect":[53.81361770629883,594.2294921875,243.08395047690159,585.2949829101563]},{"page":384,"text":"12.3 Bistability and Memory","rect":[53.81285095214844,44.274986267089847,152.5865225234501,36.68062973022461]},{"page":384,"text":"Fig. 12.19 Voltage","rect":[53.812843322753909,67.58130645751953,121.96253344797612,59.546695709228519]},{"page":384,"text":"dependence of the free energy","rect":[53.812843322753909,77.4895248413086,155.34061187891886,69.89517211914063]},{"page":384,"text":"for the uniform (n ¼ 0) and","rect":[53.812843322753909,87.07006072998047,149.38062042384073,79.81436157226563]},{"page":384,"text":"twisted (n ¼ 2) states. R is","rect":[53.81285095214844,97.04601287841797,146.34225924491205,89.79031372070313]},{"page":384,"text":"the turn point from the barrier","rect":[53.81285095214844,107.36067962646485,155.34570401770763,99.76632690429688]},{"page":384,"text":"state to one of the two stable","rect":[53.81285095214844,115.58409118652344,152.4706358649683,109.74227905273438]},{"page":384,"text":"initial states","rect":[53.81285095214844,125.50328063964844,94.96580203055658,119.66146850585938]},{"page":384,"text":"30","rect":[249.0263214111328,66.52861022949219,257.91483548486095,60.70585250854492]},{"page":384,"text":"373","rect":[372.4981689453125,42.55630874633789,385.1898245254032,36.73143005371094]},{"page":384,"text":"12.3.2.2 Backflow and Director Relaxation","rect":[53.812843322753909,299.6168212890625,242.00113267252994,292.10662841796877]},{"page":384,"text":"But how to force the system relax to a particular state selected by an experimental-","rect":[53.812843322753909,325.2790222167969,385.1407712539829,316.344482421875]},{"page":384,"text":"ist? Berreman and Heffner [20] suggested to exploit the backflow effect discussed in","rect":[53.812843322753909,337.23858642578127,385.1407403092719,328.2243347167969]},{"page":384,"text":"Section. 11.2.6. We know that, upon relaxation of the director from the field-ON","rect":[53.81185531616211,349.1981201171875,385.1079382891926,340.24365234375]},{"page":384,"text":"quasi-homeotropic state (barrier state B) to a field-OFF state, a flow appears within","rect":[53.81185531616211,361.15765380859377,385.1287774186469,352.20318603515627]},{"page":384,"text":"the cell. The direction of the flow depends on the curvature of the director field,","rect":[53.81184387207031,373.1171875,385.1108669808078,364.1826171875]},{"page":384,"text":"which is more pronounced near the electrodes. Moreover it has the opposite sign at","rect":[53.81184387207031,385.01995849609377,385.17854173252177,376.08538818359377]},{"page":384,"text":"the","rect":[53.81184387207031,394.91766357421877,66.03561437799299,388.044921875]},{"page":384,"text":"top","rect":[71.76824188232422,396.9794921875,84.51958552411566,389.0609130859375]},{"page":384,"text":"and","rect":[90.2910385131836,394.9475402832031,104.69477931073675,388.044921875]},{"page":384,"text":"bottom","rect":[110.3995361328125,394.91766357421877,138.79193829179367,388.044921875]},{"page":384,"text":"electrodes,","rect":[144.49868774414063,395.0,187.60140653158909,388.044921875]},{"page":384,"text":"see","rect":[193.32608032226563,394.91766357421877,206.05750311090316,390.2760925292969]},{"page":384,"text":"the","rect":[211.79212951660157,394.91766357421877,224.01589239312973,388.044921875]},{"page":384,"text":"molecules","rect":[229.74851989746095,394.91766357421877,270.3378030215378,388.044921875]},{"page":384,"text":"distribution","rect":[276.0843811035156,394.9475402832031,322.21318903974068,388.044921875]},{"page":384,"text":"in","rect":[327.91400146484377,394.8379821777344,335.6882256608344,388.044921875]},{"page":384,"text":"state","rect":[341.45172119140627,394.91766357421877,359.7774127788719,389.0609130859375]},{"page":384,"text":"B","rect":[365.5289001464844,394.8379821777344,371.6407851544734,388.2640686035156]},{"page":384,"text":"in","rect":[377.3674621582031,394.8379821777344,385.1416863541938,388.044921875]},{"page":384,"text":"Fig. 12.17. Due to this, the close-to-electrode flows create a strong torque exerted","rect":[53.81181716918945,408.9390563964844,385.1706170182563,400.0045166015625]},{"page":384,"text":"on the director mostly in the middle of the cell that holds the director to be more or","rect":[53.81281661987305,420.8985900878906,385.14669166413918,411.96405029296877]},{"page":384,"text":"less parallel to the boundaries in favour of the (n ¼ 2) initial state in Fig. 12.17.","rect":[53.81281661987305,432.8581237792969,377.5923733284641,423.923583984375]},{"page":384,"text":"Therefore, if we switch the field off abruptly, the backflow will bring the system","rect":[65.76580047607422,444.8176574707031,385.1745676863249,435.88311767578127]},{"page":384,"text":"into the twisted (n ¼ 2) state. However, if we smoothly reduce the field to zero, the","rect":[53.81378173828125,456.7771911621094,385.1725543804344,447.8426513671875]},{"page":384,"text":"backflow will be negligible and, according to Fig. 12.19, the system will follow","rect":[53.81377410888672,468.73675537109377,385.14071416809886,459.80218505859377]},{"page":384,"text":"curve B (state B) downward and smoothly transform into state (n ¼ 0). This","rect":[53.814796447753909,480.6395263671875,385.1217385684128,471.7049560546875]},{"page":384,"text":"selection of the final state has been confirmed experimentally using different","rect":[53.81675338745117,492.5990905761719,385.081923080178,483.66455078125]},{"page":384,"text":"forms of the voltage pulse either with the abrupt rear edge or the rear edge consisted","rect":[53.81675338745117,504.5586242675781,385.1575860123969,495.62408447265627]},{"page":384,"text":"of several steps down.","rect":[53.81675338745117,516.5181274414063,143.4245571663547,507.5836181640625]},{"page":384,"text":"12.3.2.3 Topological Problem and Trap States","rect":[53.81675338745117,558.4278564453125,257.0070687686081,549.0949096679688]},{"page":384,"text":"Now let us go back to Fig. 12.18 and have a look at the dashed curve (n ¼ 1) with","rect":[53.81675338745117,582.2672119140625,385.2033318620063,573.3327026367188]},{"page":384,"text":"the lowest free energy in the field-OFF regime. An interesting question arises why","rect":[53.81675338745117,594.226806640625,385.1247490983344,585.2723388671875]},{"page":385,"text":"374","rect":[53.81283950805664,42.55612564086914,66.50444549708291,36.73124694824219]},{"page":385,"text":"a","rect":[137.00454711914063,68.27505493164063,142.55980951637447,62.68629837036133]},{"page":385,"text":"n = 0","rect":[134.24905395507813,94.95986938476563,152.8973274825919,89.28907775878906]},{"page":385,"text":"b","rect":[199.36317443847657,68.23306274414063,205.46796818795478,60.92469024658203]},{"page":385,"text":"12 Electro-Optical Effects in Cholesteric Phase","rect":[223.29483032226563,44.274803161621097,385.1406798102808,36.66351318359375]},{"page":385,"text":"c","rect":[260.7247619628906,68.24972534179688,266.28002436012448,62.66096878051758]},{"page":385,"text":"n = 1","rect":[203.55543518066407,98.12001037597656,222.2037239669669,92.49720764160156]},{"page":385,"text":"n = 2","rect":[259.04705810546877,93.80574035644531,277.6953316329825,88.18293762207031]},{"page":385,"text":"Fig. 12.20 Three planar cholesteric structures with different number of helix half-turns. Struc-","rect":[53.812843322753909,156.40032958984376,385.1526193009107,148.67051696777345]},{"page":385,"text":"tures (a) and (c) can be transformed into each other by continuous distortion. The central structure","rect":[53.812843322753909,166.25186157226563,385.1780028083277,158.65750122070313]},{"page":385,"text":"(b) is topologically incompatible with the other two structures","rect":[53.81283950805664,176.227783203125,266.5918625163964,168.6334228515625]},{"page":385,"text":"this state is not used as one of the two stable states. Here we again meeta","rect":[53.812843322753909,211.74758911132813,385.15769231988755,202.8130340576172]},{"page":385,"text":"topological problem related to helix untwisting. Consider for clarity a non-tilted","rect":[53.812843322753909,223.7469940185547,385.15871516278755,214.75267028808595]},{"page":385,"text":"that is planar cholesteric texture with infinitely strong anchoring at two boundaries.","rect":[53.812843322753909,235.60992431640626,385.1556667854953,226.6753692626953]},{"page":385,"text":"In Fig. 12.20 we see three possible structures, uniform one (n ¼ 0) and two twisted,","rect":[53.812843322753909,247.5694580078125,385.1705593636203,238.63490295410157]},{"page":385,"text":"the p-state (n ¼ 1) and the 2p-state (n ¼ 2). Note that the direction of arrows at the","rect":[53.813812255859378,259.1305847167969,385.1765521831688,250.59446716308595]},{"page":385,"text":"opposite interfaces are the same in the (a) and (c) sketches but different in the","rect":[53.8157844543457,271.4885559082031,385.17456854059068,262.55401611328127]},{"page":385,"text":"central (b) sketch. Therefore, we can continuously transform structure (a) into (c)","rect":[53.8157844543457,283.44805908203127,385.1616147598423,274.51348876953127]},{"page":385,"text":"and vice versa. On the contrary, transformation of the central p-structure (n ¼ 1)","rect":[53.8157844543457,295.4076232910156,385.15459571687355,286.4531555175781]},{"page":385,"text":"into either left of right structure is impossible without break of anchoring, for","rect":[53.81679153442383,307.3671569824219,385.1268552383579,298.4326171875]},{"page":385,"text":"instance, at the bottom boundary. Such a transformation would take much higher","rect":[53.81679153442383,319.32672119140627,385.13079200593605,310.39215087890627]},{"page":385,"text":"energy than the continuous transition.","rect":[53.81679153442383,331.2294921875,205.81858487631565,322.294921875]},{"page":385,"text":"We meet the same problem in the Berreman–Heffner non-planar cell: the p-","rect":[65.76881408691406,343.1889953613281,385.1625913223423,334.25445556640627]},{"page":385,"text":"structure (n ¼ 1) is topologically different from the two stable states. However,","rect":[53.81675338745117,355.1485595703125,385.1556667854953,346.2139892578125]},{"page":385,"text":"despite a high barrier, both the uniform and the 2p states may little by little relax to","rect":[53.81675720214844,367.10809326171877,385.14565363935005,358.17352294921877]},{"page":385,"text":"the “forbidden” lowest energy p-state. This is possible via slow formation of","rect":[53.817779541015628,379.067626953125,385.15462623445168,370.133056640625]},{"page":385,"text":"intermediate defect states of the cholesteric structure. This will reduce the lifetimes","rect":[53.817779541015628,388.9952392578125,385.15765775786596,382.0926513671875]},{"page":385,"text":"of both stable states; they become quasi-stable. The topologically forbidden p-state","rect":[53.817779541015628,402.9867248535156,385.1366657085594,394.05218505859377]},{"page":385,"text":"behaves as a trap, and one needs strong voltage pulses to destroy the trap in order","rect":[53.818763732910159,414.9463195800781,385.1496518692173,406.01177978515627]},{"page":385,"text":"to continue the bistable switching. It is a disadvantage of the Berreman–Heffner","rect":[53.818763732910159,426.8490905761719,385.10088477937355,417.91455078125]},{"page":385,"text":"model.","rect":[53.818763732910159,436.77667236328127,81.28242917563205,429.87408447265627]},{"page":385,"text":"It would be better not to deal with such a trap state at all. To avoid it, there has","rect":[65.77078247070313,450.7681579589844,385.13776029692846,441.8336181640625]},{"page":385,"text":"been suggested another configuration with the same quasi-stable states, uniform","rect":[53.818763732910159,462.72772216796877,385.1676096785124,453.79315185546877]},{"page":385,"text":"0-state and twisted 2p-state, but now the ratio d/P0 \u0001 1 corresponds to the lowest","rect":[53.818763732910159,474.6898498535156,385.1302324063499,465.7328186035156]},{"page":385,"text":"energy 2p-structure [21]. The system may stay for a long time in either state without","rect":[53.81426239013672,486.6494140625,385.2009111172874,477.71484375]},{"page":385,"text":"trapping and be switched at a low voltage from the beginning. However, now","rect":[53.81426239013672,498.6089782714844,385.12328863098949,489.6744384765625]},{"page":385,"text":"another problem appears: the difference in energy of the 0- and 2p-states is larger","rect":[53.81426239013672,510.5685119628906,385.0953610977329,501.63397216796877]},{"page":385,"text":"than in the previous case. Thus, the reliable selection of a desired memory state","rect":[53.81525421142578,522.4712524414063,385.1829303569969,513.5367431640625]},{"page":385,"text":"using backflow becomes more difficult unless the liquid crystal has high ratio of","rect":[53.81525421142578,534.4307861328125,385.1501706680454,525.4962768554688]},{"page":385,"text":"elastic constants K33/K22 > 3. Such a material","rect":[53.81525421142578,546.0296630859375,247.6167588956077,537.455810546875]},{"page":385,"text":"voltage","rect":[53.81325912475586,558.3505859375,83.23790777398908,549.4160766601563]},{"page":385,"text":"bistability","rect":[88.59326934814453,558.3505859375,128.6758965592719,549.4160766601563]},{"page":385,"text":"demonstrated.","rect":[133.9645538330078,557.0,190.31434293295627,549.4160766601563]},{"page":385,"text":"In","rect":[195.5930633544922,557.0,203.8849267106391,549.6152954101563]},{"page":385,"text":"principle,","rect":[209.18753051757813,558.3505859375,247.2942318489719,549.4160766601563]},{"page":385,"text":"has been designed and the low-","rect":[251.72984313964845,546.3910522460938,385.12420020906105,537.45654296875]},{"page":385,"text":"in the bistable devices a dual-","rect":[252.5769500732422,557.0,385.13619361726418,549.4160766601563]},{"page":385,"text":"frequency addressing regime discussed in Section 7.2.4 should be very efficient.","rect":[53.81325912475586,570.3101196289063,385.1700100472141,561.3756103515625]},{"page":385,"text":"Indeed, using positive ea at low frequency, one can easily force the director to reach","rect":[53.81327438354492,582.2699584960938,385.1287774186469,573.3351440429688]},{"page":385,"text":"the uniform homeotropic state. Operating with high frequency and negative ea it is","rect":[53.81282424926758,594.2294921875,385.18878568755346,585.2750244140625]},{"page":386,"text":"12.3 Bistability and Memory","rect":[53.813682556152347,44.276573181152347,152.58736175684855,36.68221664428711]},{"page":386,"text":"375","rect":[372.4990234375,42.55789566040039,385.19064850001259,36.63142013549805]},{"page":386,"text":"easy to reach the planar 2p-state. However, at present, this technique is not used due","rect":[53.812843322753909,68.2883529663086,385.14173162652818,59.35380554199219]},{"page":386,"text":"to complexity of the corresponding addressing circuits.","rect":[53.812843322753909,80.24788665771485,275.80071683432348,71.31333923339844]},{"page":386,"text":"12.3.3 Bistability and Field-Induced Break of Anchoring","rect":[53.812843322753909,130.3584442138672,344.24680843257496,119.72067260742188]},{"page":386,"text":"Using field-induced break of anchoring discussed in Section 11.2.4 one can over-","rect":[53.812843322753909,157.90078735351563,385.1476987442173,148.9662322998047]},{"page":386,"text":"come topological problems [22]. The advantage is that we can design a cell with","rect":[53.81282424926758,169.8603515625,385.1138543229438,160.92579650878907]},{"page":386,"text":"two long living ground states having very high energy barrier between them. The","rect":[53.812835693359378,181.81988525390626,385.1437457866844,172.8853302001953]},{"page":386,"text":"long-life states are very important in the display technology, because the ratio of","rect":[53.812835693359378,193.77944946289063,385.1486753067173,184.8448944091797]},{"page":386,"text":"the switch-OFF and switch-ON times determines a number of addressed lines of the","rect":[53.812835693359378,203.7169952392578,385.17359197809068,196.7845001220703]},{"page":386,"text":"screen or the so-called multiplexing of the display. Note that in a standard display,","rect":[53.812835693359378,217.69851684570313,385.1666225960422,208.7639617919922]},{"page":386,"text":"switching of each pixel is controlled by a separate thin-film transistor that compli-","rect":[53.812835693359378,229.65805053710938,385.11483131257668,220.72349548339845]},{"page":386,"text":"cates technology and increases price of the display.","rect":[53.812835693359378,241.61761474609376,260.9589504769016,232.6830596923828]},{"page":386,"text":"Figure 12.21a and b shows schematically two ground states, the uniform one","rect":[65.76485443115235,253.52035522460938,385.1427387066063,244.58580017089845]},{"page":386,"text":"(n ¼ 0) and p-twisted (n ¼ 1). As mentioned before the transitions between them","rect":[53.81184387207031,265.08148193359377,385.1278147566374,256.54534912109377]},{"page":386,"text":"are topologically blocked. There is a small but principal difference between this","rect":[53.81184387207031,277.439453125,385.1427346621628,268.5048828125]},{"page":386,"text":"pair of states and the pair of the corresponding states in Fig. 12.20: at the top plate","rect":[53.81184387207031,289.3990173339844,385.1118549175438,280.4644775390625]},{"page":386,"text":"the director is slightly tilted and the anchoring energy is made weak to facilitate the","rect":[53.81181716918945,301.3585510253906,385.1705707378563,292.42401123046877]},{"page":386,"text":"break of anchoring. The optimum thickness-to-pitch ratio is d/Po \u0001 1/4. This","rect":[53.81181716918945,313.3180847167969,385.1887551699753,304.3636169433594]},{"page":386,"text":"means that the p/2-twist is the equilibrium state and costs no elastic energy and","rect":[53.814083099365237,325.27911376953127,385.1449822526313,316.34454345703127]},{"page":386,"text":"the elastic energies of the two non-equilibrium stable states (0 and p) are higher and","rect":[53.814083099365237,337.2386779785156,385.1459588151313,328.30413818359377]},{"page":386,"text":"nearly equal. With increasing voltage, at a certain critical value, the two non-","rect":[53.81411361694336,349.14141845703127,385.13411842195168,340.20684814453127]},{"page":386,"text":"equilibrium states merge into one. In the new state, the director is uniformly aligned","rect":[53.81411361694336,361.1009826660156,385.09621516278755,352.16644287109377]},{"page":386,"text":"along the field almost everywhere except at the bottom interface, Fig. 12.21c. When","rect":[53.81411361694336,373.0605163574219,385.1131219010688,364.1259765625]},{"page":386,"text":"the voltage isreduced the system reaches a bifurcation point, at which two scenarios","rect":[53.81411361694336,385.0200500488281,385.15195097075658,376.08551025390627]},{"page":386,"text":"are possible depending on the rate of the voltage decay: fast decay causesa","rect":[53.81411361694336,396.9795837402344,385.1599811382469,388.0450439453125]},{"page":386,"text":"backflow that drives the system into the p-twisted state (n ¼ 1); smooth decay","rect":[53.81411361694336,408.93914794921877,385.0952080827094,400.00457763671877]},{"page":386,"text":"results in the uniform state (n ¼ 0).","rect":[53.814144134521487,420.500244140625,197.88042874838596,411.964111328125]},{"page":386,"text":"In principle, topologically blocked states may exist for unlimited time. The main","rect":[65.76616668701172,432.8582458496094,385.12917414716255,423.9237060546875]},{"page":386,"text":"problem is to break anchoring without breakdown of the sample. Anchoring is","rect":[53.814144134521487,444.760986328125,385.18686308013158,435.826416015625]},{"page":386,"text":"broken when an electric field coherence length become comparable to a surface","rect":[53.814144134521487,456.7205505371094,385.08335149957505,447.7860107421875]},{"page":386,"text":"extrapolation length. Therefore, a critical voltage Ub necessary for the break of","rect":[53.814144134521487,468.6817321777344,385.15035377351418,459.74554443359377]},{"page":386,"text":"anchoring is proportional to the anchoring energy. The latter should be as low as","rect":[53.813533782958987,480.6412658691406,385.1404153262253,471.70672607421877]},{"page":386,"text":"a","rect":[194.59019470214845,505.9629211425781,200.14545709938228,500.3741760253906]},{"page":386,"text":"b","rect":[263.310791015625,505.1856994628906,269.4155847651032,497.8773193359375]},{"page":386,"text":"c","rect":[333.5985412597656,505.6059875488281,339.15380365699948,500.0172424316406]},{"page":386,"text":"Fig. 12.21 Two long-living","rect":[53.812843322753909,531.1171264648438,150.71832794337198,523.3873291015625]},{"page":386,"text":"topologically stable field-off","rect":[53.812843322753909,541.0253295898438,151.00430387122325,533.4309692382813]},{"page":386,"text":"states (a, b) and the field","rect":[53.812843322753909,550.6626586914063,138.88719696192667,543.4069213867188]},{"page":386,"text":"induced state (c) in a bistable","rect":[53.81283950805664,560.5819091796875,154.3768858649683,553.326171875]},{"page":386,"text":"device using the break of","rect":[53.813682556152347,570.896484375,140.16566556555919,563.3021240234375]},{"page":386,"text":"anchoring effect","rect":[53.813682556152347,580.8724975585938,109.48244957175318,573.2781372070313]},{"page":386,"text":"n = 0","rect":[190.2710418701172,535.1781616210938,208.91933065642002,529.5073852539063]},{"page":386,"text":"n = 1","rect":[262.5184631347656,537.0155029296875,280.7670784591544,531.3927001953125]},{"page":387,"text":"376","rect":[53.812191009521487,42.55728530883789,66.50379699854776,36.68160629272461]},{"page":387,"text":"P","rect":[141.53802490234376,66.38604736328125,146.86953156587917,60.643272399902347]},{"page":387,"text":"12 Electro-Optical Effects in Cholesteric Phase","rect":[223.294189453125,44.275962829589847,385.1400389411402,36.6646728515625]},{"page":387,"text":"ITO","rect":[136.62777709960938,90.78007507324219,149.95255406291299,84.74136352539063]},{"page":387,"text":"n = 0","rect":[138.0345916748047,117.98704528808594,156.47504666650156,112.16429138183594]},{"page":387,"text":"ITO","rect":[139.8298797607422,146.8257293701172,153.1546567240458,140.78701782226563]},{"page":387,"text":"elastic","rect":[178.88336181640626,110.93497467041016,203.76639798486094,105.0322265625]},{"page":387,"text":"back-","rect":[217.5531768798828,104.4363784790039,238.43157786707807,98.5896224975586]},{"page":387,"text":"flow","rect":[217.5531768798828,114.0342788696289,233.53969700724893,108.11553955078125]},{"page":387,"text":"E","rect":[217.87130737304688,131.0986328125,223.20281403658229,125.3558578491211]},{"page":387,"text":"n=1","rect":[263.3841247558594,116.09307861328125,279.7783303579078,110.41429138183594]},{"page":387,"text":"Break of","rect":[263.7638244628906,138.92977905273438,295.7448865828984,133.01104736328126]},{"page":387,"text":"anchoring","rect":[263.7638244628906,150.15139770507813,302.40326257728199,142.625]},{"page":387,"text":"A","rect":[141.47488403320313,170.24313354492188,147.2460201817377,164.50035095214845]},{"page":387,"text":"Fig. 12.22 Operation of BiNem® bistable device. Two stable states are the uniform (n ¼ 0) and","rect":[53.812843322753909,191.93917846679688,385.19330352930947,182.4327850341797]},{"page":387,"text":"p-twisted (n ¼ 1). Anchoring is strong at the top plate and weak at the bottom one. A strong field","rect":[53.81217575073242,201.79071044921876,385.1798147597782,194.19635009765626]},{"page":387,"text":"pulse E breaks anchoring and creates a transient quasi-homeotropic texture. If the rear edge of the","rect":[53.81217575073242,211.76663208007813,385.1553282477808,204.17227172851563]},{"page":387,"text":"pulse is short, the backflow develops and the pulse writes a signal in the form of the p-twisted","rect":[53.81217575073242,221.74258422851563,385.1781057754032,214.14822387695313]},{"page":387,"text":"texture. To erase the signal, a strong pulse with a step-like rear edge creates the same transient","rect":[53.812191009521487,231.71856689453126,385.13497642722197,224.12420654296876]},{"page":387,"text":"state, which relaxes to the uniform stable texture due solely to the elastic force","rect":[53.812191009521487,241.63778686523438,323.65441272043707,234.04342651367188]},{"page":387,"text":"possible. For instance, to have Ub \u0001 15 V the zenithal anchoring energy should be","rect":[53.812843322753909,266.16119384765627,385.1509174175438,257.1668701171875]},{"page":387,"text":"as low as 0.2 erg/cm2. The shape of the voltage pulses is also important because the","rect":[53.8140983581543,278.12078857421877,385.17261541559068,267.0698547363281]},{"page":387,"text":"system behaviour at the bifurcation point depends on the steepness of the rear edge.","rect":[53.81283950805664,290.080322265625,385.11492581869848,281.145751953125]},{"page":387,"text":"Figure 12.22 shows the operation principle of the recently described bistable","rect":[53.81283950805664,302.0398864746094,385.1049274273094,293.1053466796875]},{"page":387,"text":"display [23]. In this case the break of anchoring occurs at the bottom electrode","rect":[53.81283187866211,313.9994201660156,385.0819782085594,305.06488037109377]},{"page":387,"text":"where the zenithal anchoring of the director is weak. Beautiful colour images,","rect":[53.81283187866211,325.9021911621094,385.1775784065891,316.9676513671875]},{"page":387,"text":"stable in time and easily switchable have been demonstrated.","rect":[53.81283187866211,337.8617248535156,299.9078335335422,328.92718505859377]},{"page":387,"text":"12.4 Flexoelectricity in Cholesterics","rect":[53.812843322753909,376.6315612792969,244.57262305948897,365.3244323730469]},{"page":387,"text":"As has been mentioned in Section 11.3.1, the twist itself does not produce flexo-","rect":[53.812843322753909,403.61212158203127,385.1158383926548,394.67755126953127]},{"page":387,"text":"electric polarization. However, an interesting flexoelectric effect is observed when","rect":[53.812862396240237,415.5716857910156,385.15078059247505,406.63714599609377]},{"page":387,"text":"the twist distortion is combined with the splay-bend distortion [24,25]. In that case,","rect":[53.812862396240237,427.5312194824219,385.14165921713598,418.5368957519531]},{"page":387,"text":"the cholesteric axis h0 is homogeneously oriented in the plane of the cell along z,","rect":[53.81185531616211,439.4910583496094,385.18267484213598,430.55621337890627]},{"page":387,"text":"see Fig. 12.23a, and an electric field is applied to transparent electrodes ofa","rect":[53.81401443481445,451.4505920410156,385.1598590679344,442.51605224609377]},{"page":387,"text":"sandwich cell along the x-axis, E⊥h0. The dielectric anisotropy is negative,","rect":[53.81401443481445,463.4103698730469,385.16912503744848,454.4755859375]},{"page":387,"text":"ea 0. In the field-OFF state, the director components are parallel to the xy-","rect":[53.813350677490237,475.3133239746094,385.15865455476418,466.3588562011719]},{"page":387,"text":"plane, nx ¼ cosj, ny ¼ sinj and the conical distortions is absent, see Fig. 12.23","rect":[53.81280517578125,488.24615478515627,385.1525811295844,478.33831787109377]},{"page":387,"text":"(b) for E ¼ 0. If the cell is filled with a short pitch cholesteric P0 ¼ 2p/q0 it","rect":[53.8137321472168,499.2325134277344,385.17341477939677,490.2979736328125]},{"page":387,"text":"behaves like a uniaxial optical plate with the optical axis directed along h0. When","rect":[53.814659118652347,511.1921691894531,385.1131219010688,502.25762939453127]},{"page":387,"text":"the field is applied, a periodic splay-bend distortion appears due to the flexoelectric","rect":[53.814083099365237,523.1517944335938,385.1240924663719,514.21728515625]},{"page":387,"text":"torque Mf ¼ PfE in the surface regions. This distortion has been considered in","rect":[53.814083099365237,536.028076171875,385.14040461591255,526.1770629882813]},{"page":387,"text":"Section 11.3.2 for nematics. Interacting with the natural twist of the cholesteric, the","rect":[53.81344985961914,547.0711059570313,385.17221868707505,538.1365966796875]},{"page":387,"text":"director leaves the xy-plane as shown in the picture. For the conical distortion the","rect":[53.81344985961914,559.0306396484375,385.17221868707505,550.0961303710938]},{"page":387,"text":"new components of the director are given by","rect":[53.81344985961914,570.9334106445313,234.43657771161566,561.9989013671875]},{"page":387,"text":"nx ¼ cosj;ny ¼ sinjcosC;nz ¼ \u0004cosjsinC","rect":[122.35416412353516,595.7271118164063,316.6150389336774,585.8090209960938]},{"page":388,"text":"12.4 Flexoelectricity in Cholesterics","rect":[53.81536102294922,44.275291442871097,177.68881686209955,36.68093490600586]},{"page":388,"text":"a","rect":[106.7194595336914,68.23361206054688,112.27472193092525,62.64485549926758]},{"page":388,"text":"z","rect":[129.90591430664063,75.24951171875,133.90254598845127,70.99441528320313]},{"page":388,"text":"b","rect":[167.9212646484375,68.23361206054688,174.0260583979157,60.92523956298828]},{"page":388,"text":"h0","rect":[178.046142578125,79.28144073486328,186.26292409899728,71.4311752319336]},{"page":388,"text":"h0","rect":[237.7553253173828,84.06177520751953,245.97239675524728,76.2114486694336]},{"page":388,"text":"z","rect":[251.44598388671876,94.01220703125,255.4426155685294,89.8291015625]},{"page":388,"text":"x","rect":[256.428955078125,119.478759765625,260.42558675993566,115.295654296875]},{"page":388,"text":"y","rect":[272.1517333984375,107.99239349365235,276.14836508024816,102.10565185546875]},{"page":388,"text":"377","rect":[372.500732421875,42.55661392211914,385.19235748438759,36.73173522949219]},{"page":388,"text":"+/–U","rect":[113.38943481445313,182.12034606933595,132.29349790390567,176.08163452148438]},{"page":388,"text":"x","rect":[147.291259765625,155.39645385742188,151.73551419579844,151.141357421875]},{"page":388,"text":"E > 0","rect":[192.65834045410157,152.08384704589845,211.54642728173594,146.1970977783203]},{"page":388,"text":"E = 0","rect":[243.8092041015625,152.08384704589845,262.6972756704078,146.1970977783203]},{"page":388,"text":"E < 0","rect":[313.417724609375,159.0823211669922,332.3057961782203,153.19557189941407]},{"page":388,"text":"Fig. 12.23","rect":[53.812843322753909,214.611572265625,88.4846167006962,206.67855834960938]},{"page":388,"text":"Flexoelectric distortion in a cholesteric liquid crystal. (a) The d.c. field from the source","rect":[94.48435974121094,214.54385375976563,385.13903186106207,206.94949340820313]},{"page":388,"text":"U is applied to the cell along the x-axis. (b) The field induced director distortion for positive and","rect":[53.813697814941409,224.51980590820313,385.19574493555947,216.92544555664063]},{"page":388,"text":"negative field directed perpendicular to the plane of the figure along the x-axis; it is seen how the","rect":[53.813697814941409,234.49575805664063,385.1567930915308,226.90139770507813]},{"page":388,"text":"cholesteric quasi-layers are tilted though angle C from their field-OFF configuration within the x,","rect":[53.814552307128909,244.4716796875,385.194857331061,236.7841796875]},{"page":388,"text":"y-plane shown in the central sketch","rect":[53.81536102294922,254.39093017578126,174.60558075098917,246.79656982421876]},{"page":388,"text":"The turn of the director everywhere through angle C is equivalent to the turn of","rect":[65.76496887207031,273.3028869628906,385.1498044571079,264.2587585449219]},{"page":388,"text":"the optical axis about the x-axis through the same angle. The sign and the magnitude","rect":[53.81294250488281,285.262451171875,385.1029743023094,276.327880859375]},{"page":388,"text":"of the deviation angle C depend on polarity and strength of the applied field,","rect":[53.81393814086914,297.22198486328127,385.1120266487766,288.1778564453125]},{"page":388,"text":"respectively. It can be estimated as follows.","rect":[53.81393814086914,309.1815490722656,230.02138944174534,300.24700927734377]},{"page":388,"text":"For weak anchoring and ea ffi 0 by analogy with a nematic (see Eq. 10.77), the","rect":[65.76596069335938,321.0848693847656,385.1732868023094,312.14971923828127]},{"page":388,"text":"free energy of the distortion includes the elastic term due to the bend-distortion (we","rect":[53.81352615356445,333.04443359375,385.14145696832505,324.10986328125]},{"page":388,"text":"assume K ¼ K11 ¼ K33) and the flexoelectric term with an average coefficient e.","rect":[53.81352615356445,345.004150390625,385.18243070151098,336.069580078125]},{"page":388,"text":"The second elastic term is due to the cholesteric helical structure (modulus K22):","rect":[53.81376266479492,356.5653991699219,378.95027024814677,348.0291748046875]},{"page":388,"text":"g ¼ 21K\u0004qqjy\u00052 \u0007 12K22\u0004q0 \u0004 qqjz\u00052 \u0004 eEqqjy","rect":[127.56564331054688,394.0095520019531,309.73005935366896,368.1902160644531]},{"page":388,"text":"(12.42)","rect":[356.12762451171877,386.322021484375,385.15410743562355,377.8456726074219]},{"page":388,"text":"Minimisation with respect to ∂j/∂z results in ∂j/∂z ¼ q0. Minimization with","rect":[65.76529693603516,414.551025390625,385.11895075849068,404.6104736328125]},{"page":388,"text":"respect to ∂j/∂y results in ∂j/∂y ¼ eE/K, see analogy with Eq. (11.79) for","rect":[53.813961029052737,426.5110168457031,385.1796811660923,416.5704650878906]},{"page":388,"text":"nematics. These two derivatives can be imagined as two projections of wavevector","rect":[53.8129997253418,438.4705505371094,385.16973243562355,429.5360107421875]},{"page":388,"text":"k, which will show the direction of the field-induced helical axis. In zero field","rect":[53.8129997253418,449.0,385.11696711591255,441.49554443359377]},{"page":388,"text":"k ¼ q0||h0. In the field-ON state the components of vector k are |k|cosC ¼ q0 and","rect":[53.8129997253418,462.3901672363281,385.14626399091255,453.3460388183594]},{"page":388,"text":"|k|sinC ¼ eE /K , and these components define the position of the new optical axis.","rect":[53.814414978027347,474.2929382324219,385.1512722542453,465.2488098144531]},{"page":388,"text":"Therefore, the angle of the optical axis rotation is given by","rect":[53.815391540527347,486.2524719238281,291.6503838639594,477.31793212890627]},{"page":388,"text":"eE","rect":[231.8485870361328,504.32879638671877,242.41001122869216,497.54571533203127]},{"page":388,"text":"tanC ¼","rect":[192.48251342773438,510.9435119628906,225.01486995909125,503.9612121582031]},{"page":388,"text":"q0K;","rect":[227.82708740234376,519.7545776367188,246.54481446427247,511.14910888671877]},{"page":388,"text":"(12.43)","rect":[356.1277160644531,512.6073608398438,385.1541989883579,504.1309814453125]},{"page":388,"text":"which is linear in the electric field E for small distortions.","rect":[53.81339645385742,538.2376098632813,287.1380886604953,531.3350219726563]},{"page":388,"text":"The rise and decay of the flexoelectric distortion is controlled by periodicity of","rect":[65.7654037475586,552.2290649414063,385.1492856582798,543.2945556640625]},{"page":388,"text":"the helix,","rect":[53.81338119506836,562.070068359375,91.36762662436252,555.1973266601563]},{"page":388,"text":"g1","rect":[222.95529174804688,583.4727172851563,231.4240576990541,576.4773559570313]},{"page":388,"text":"t¼","rect":[198.31466674804688,587.9632568359375,213.1165697881928,583.4611206054688]},{"page":388,"text":"K22q02","rect":[215.9313201904297,597.7830200195313,238.50466988167129,586.75146484375]},{"page":388,"text":"(12.44)","rect":[356.1281433105469,589.636962890625,385.15462623445168,581.1605834960938]},{"page":389,"text":"378","rect":[53.81031799316406,42.56186294555664,66.50192398219034,36.73698425292969]},{"page":389,"text":"12 Electro-Optical Effects in Cholesteric Phase","rect":[223.2923126220703,44.280540466308597,385.13817736887457,36.66925048828125]},{"page":389,"text":"Therefore, for short pitch cholesterics with pitch about 0.3 mm, the characteristic","rect":[65.76496887207031,68.2883529663086,385.18161810113755,59.35380554199219]},{"page":389,"text":"time of the director switching is short, t < 100 ms (g1 \u0001 1 P or 0.1 Pa\u0006s in SI,","rect":[53.81391143798828,80.24788665771485,385.1572231819797,71.31333923339844]},{"page":389,"text":"q0 ¼ 2p/P0 \u0001 2 105 cm\u00041 or 2 107 m\u00041, K22 \u0001 3 10\u00047dyn or 3 10\u000412 N).","rect":[53.815406799316409,92.20772552490235,385.1657375862766,81.0770492553711]},{"page":389,"text":"Indeed,experiments show that the effect of the realignment of the helical axis is less","rect":[53.81393051147461,104.1104965209961,385.14081205474096,95.17594909667969]},{"page":389,"text":"than 100 ms, and the speed of the response is independent of the field strength.","rect":[53.81393051147461,116.0699691772461,370.3459133675266,107.13542175292969]},{"page":389,"text":"References","rect":[53.812843322753909,164.2294158935547,109.59614448282879,155.4442901611328]},{"page":389,"text":"1.","rect":[58.06126022338867,190.0,64.40706131055318,183.9311981201172]},{"page":389,"text":"2.","rect":[58.06126022338867,200.0,64.40706131055318,193.9071502685547]},{"page":389,"text":"3.","rect":[58.06126022338867,220.0,64.40706131055318,213.8023223876953]},{"page":389,"text":"4.","rect":[58.06126022338867,240.0,64.40706131055318,233.75425720214845]},{"page":389,"text":"5.","rect":[58.0612678527832,270.0,64.4070689399477,263.5237731933594]},{"page":389,"text":"6.","rect":[58.060420989990237,290.0,64.40622207715474,283.4697570800781]},{"page":389,"text":"7.","rect":[58.06123733520508,310.0,64.40703842236958,303.59100341796877]},{"page":389,"text":"8.","rect":[58.06123733520508,329.1925354003906,64.40703842236958,323.3676452636719]},{"page":389,"text":"9.","rect":[58.06123733520508,349.2121887207031,64.40703842236958,343.3195495605469]},{"page":389,"text":"10.","rect":[53.81290817260742,359.0636901855469,64.3892466743227,353.2388000488281]},{"page":389,"text":"11.","rect":[53.812904357910159,389.0,64.38924285962544,383.16668701171877]},{"page":389,"text":"12.","rect":[53.81290817260742,399.0,64.3892466743227,393.0859375]},{"page":389,"text":"13.","rect":[53.81205368041992,419.0,64.3883921821352,413.037841796875]},{"page":389,"text":"14.","rect":[53.811222076416019,439.0,64.3875605781313,432.9330139160156]},{"page":389,"text":"15.","rect":[53.81206130981445,459.0,64.38839981152974,452.78338623046877]},{"page":389,"text":"16.","rect":[53.811214447021487,479.0,64.38755294873677,472.7293701171875]},{"page":389,"text":"17.","rect":[53.81118392944336,499.0,64.38752243115865,492.7320251464844]},{"page":389,"text":"18.","rect":[53.810333251953128,519.0,64.38667175366841,512.6272583007813]},{"page":389,"text":"19.","rect":[53.810333251953128,529.0,64.38667175366841,522.6031494140625]},{"page":389,"text":"20.","rect":[53.810333251953128,549.0,64.38667175366841,542.4983520507813]},{"page":389,"text":"21.","rect":[53.8111572265625,568.1735229492188,64.38749572827779,562.4502563476563]},{"page":389,"text":"Oseen, C.W.: The theory of liquid crystals. 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Teor Fiz. 119, 638–648 (2001) [JETP 103, 469 (2006)]","rect":[68.59699249267578,281.08819580078127,279.03057950598886,273.1213073730469]},{"page":389,"text":"De Vries, Hl: Rotatory power and optical properties of certain liquid crystal. Acta Cryst. 4,","rect":[68.59614562988281,291.0641174316406,385.205111237311,283.4697570800781]},{"page":389,"text":"219–226 (1951)","rect":[68.59696197509766,300.701416015625,123.15902036292245,293.3948974609375]},{"page":389,"text":"Kats, E.I.: Optical properties of cholesteric liquid crystals. Zh. Eksp. Teor Fiz. 59, 1854–1856","rect":[68.59696197509766,311.01605224609377,385.1949514785282,303.0491638183594]},{"page":389,"text":"(1970) [JETP 32, 1004 (1971]","rect":[68.59696197509766,320.5965881347656,171.8727731095045,313.39166259765627]},{"page":389,"text":"Joannopoulos, J.D., Meade, R.D., Winn, J.N.: Photonic Crystals: Molding the Flow of Light.","rect":[68.59696197509766,330.9112243652344,385.1500880439516,323.3168640136719]},{"page":389,"text":"Princeton University Press, Princeton (1995)","rect":[68.59696197509766,340.8871765136719,221.438569007942,333.24200439453127]},{"page":389,"text":"Mauguin, C.: Sur les cristaux liquides de Lehman. Bull. Soc. Fr. Miner. 34, 71–117 (1911)","rect":[68.59696197509766,350.8631286621094,380.0447091446607,342.9978332519531]},{"page":389,"text":"Chilaya, G., Hauk, G., Koswig, H.D., Sikharulidze, D.: Electric-field controlled color effect in","rect":[68.59693908691406,360.7823791503906,385.16864532618447,353.1880187988281]},{"page":389,"text":"cholesteric liquid crystals and polymer-dispersed cholesteric liquid crystals. J. Appl. Phys. 80,","rect":[68.59693908691406,370.7583312988281,385.20508071973287,362.8760986328125]},{"page":389,"text":"1907–1909 (1996)","rect":[68.596923828125,380.3956604003906,131.65557950598888,373.13995361328127]},{"page":389,"text":"Fergason, J.L.: Liquid crystals in nondestructive testing. Appl. Optics 7, 1729–1737 (1968)","rect":[68.59693145751953,390.71026611328127,381.0075692520826,383.0989685058594]},{"page":389,"text":"De Gennes, P.-G.: Calcul de la distorsion d’une structure cholesteric par un champ magnetic.","rect":[68.59693908691406,400.6295166015625,385.1652552802797,393.03515625]},{"page":389,"text":"Sol. State Comms. 6, 163–165 (1968)","rect":[68.59693908691406,410.2668151855469,197.873627601692,402.95184326171877]},{"page":389,"text":"Meyer, R.B.: Effect of electric and magnetic field on the structure of cholesteric liquid","rect":[68.59608459472656,420.5814208984375,385.18463653712197,412.987060546875]},{"page":389,"text":"crystals. Appl. Phys. Lett. 12, 281–282 (1968)","rect":[68.59607696533203,430.5006408691406,227.04486173255138,422.8385314941406]},{"page":389,"text":"Meyer, R.B.: Distortion of a cholesteric structure by a magnetic field. Appl. Phys. Lett. 14,","rect":[68.59525299072266,440.4765930175781,385.20422622754537,432.8144836425781]},{"page":389,"text":"208–209 (1969)","rect":[68.59608459472656,450.11395263671877,123.15814298255136,442.8582458496094]},{"page":389,"text":"Blinov, L.M., Palto, S.P.: Cholesteric helix: topological problem, photonics and electro-","rect":[68.5960922241211,460.4285583496094,385.15341275794199,452.8341979980469]},{"page":389,"text":"optics. Liq. Cryst. 36, 1037–1045 (2009)","rect":[68.59608459472656,470.3477478027344,208.46519559485606,462.4824523925781]},{"page":389,"text":"Blinov, L.M., Belyayev, S.V., Kizel’, V.A.: High-order reflections from a cholesteric helix","rect":[68.59524536132813,480.32373046875,385.1905874648563,472.7293701171875]},{"page":389,"text":"induced by an electric field. Phys. Lett. 65A, 33–35 (1978)","rect":[68.5952377319336,490.2996520996094,270.3773507462232,482.6460266113281]},{"page":389,"text":"Helfrich, W.: Deformation of cholesteric liquid crystals with low threshold voltage. Appl.","rect":[68.59521484375,500.2756042480469,385.1804835517641,492.6812438964844]},{"page":389,"text":"Phys. Lett. 17, 531–532 (1970)","rect":[68.59520721435547,510.19482421875,175.21399015052013,502.53271484375]},{"page":389,"text":"Chandrasekhar, S.: Liquid Crystals. Cambridge University Press, Cambridge (1977)","rect":[68.5943603515625,520.1707763671875,355.9679269180982,512.576416015625]},{"page":389,"text":"Blinov, L.M.: Electro-Optical and Magneto-Optical Properties of Liquid Crystals. Wiley,","rect":[68.5943603515625,530.1466674804688,385.1389796455141,522.535400390625]},{"page":389,"text":"Chichester (1983). Chapter 6","rect":[68.59436798095703,540.06591796875,167.81903595118448,532.4715576171875]},{"page":389,"text":"Berreman, D.W., Heffner, W.R.: New bistable liquid-crystal twist cell. J. Appl. Phys. 52,","rect":[68.5943603515625,550.0418701171875,385.2033412177797,542.447509765625]},{"page":389,"text":"3032–3039 (1981)","rect":[68.59518432617188,559.67919921875,131.65384000403575,552.4743041992188]},{"page":389,"text":"Palto, S.P., Barnik, M.I.: Bistable switching of nematic liquid crystal layers with ground","rect":[68.59518432617188,569.9937744140625,385.19489044337197,562.3994140625]},{"page":389,"text":"2p-state. Zh. Eksp. Teor. Fiz. 127, 220–229 (2005)","rect":[68.59518432617188,579.9130249023438,243.35772794348888,572.2509765625]},{"page":390,"text":"References","rect":[53.812835693359378,42.52305221557617,91.48151858329095,36.68124008178711]},{"page":390,"text":"379","rect":[372.4981994628906,42.62464904785156,385.18979400782509,36.73204040527344]},{"page":390,"text":"22.","rect":[53.812843322753909,65.12664794921875,64.38918182446919,59.40336608886719]},{"page":390,"text":"23.","rect":[53.812843322753909,95.15609741210938,64.38918182446919,89.33122253417969]},{"page":390,"text":"24.","rect":[53.81285858154297,125.0,64.38919708325826,119.20237731933594]},{"page":390,"text":"25.","rect":[53.81201934814453,145.0,64.38835784985982,138.9959259033203]},{"page":390,"text":"Barberi, R., Durand, G.: Controlled textural bistability in nematic liquid crystals. In: Collings,","rect":[68.59687042236328,66.9469223022461,385.148226471686,59.35256576538086]},{"page":390,"text":"P.J., Patel, J.S. (eds.) Handbook of Liquid Crystal Research, pp. 567–589. Oxford University","rect":[68.59687042236328,76.9228744506836,385.1449331679813,69.27771759033203]},{"page":390,"text":"Press, New York (1997). Chapter XV","rect":[68.59687042236328,86.8988265991211,197.43613220793217,79.30447387695313]},{"page":390,"text":"Joubert, C., Angele, J., Boissier, A., Pecout, B., Forget, S.L., Dozov, I., Stoenescu, D.,","rect":[68.59687042236328,96.8747787475586,385.1271693427797,89.28042602539063]},{"page":390,"text":"Lallemand, S., Lagarde, P.M.: Reflective bistable nematic displays (BiNem®) fabricated by","rect":[68.59687042236328,106.79402923583985,385.2067007461063,99.19967651367188]},{"page":390,"text":"standard manufacturing equipment. J. SID 11, 17–24 (2003)","rect":[68.59689331054688,116.76998138427735,274.7407235489576,109.10789489746094]},{"page":390,"text":"Patel, J.S., Meyer, R.B.: Flexoelectric Electro-optics of a Cholesteric Liquid Crystal. Phys.","rect":[68.59688568115235,126.74593353271485,385.169466706061,119.15158081054688]},{"page":390,"text":"Rev. Lett. 58, 1538–1540 (1987)","rect":[68.59688568115235,136.3264617919922,180.9362038956373,128.7829132080078]},{"page":390,"text":"Patel, J.S., Lee, S.-D.: Fast linear effect based on cholesteric liquid crystals. J. Appl. Phys. 66,","rect":[68.5960464477539,146.64108276367188,385.2049891669985,138.98745727539063]},{"page":390,"text":"1879–1881 (1989)","rect":[68.59686279296875,156.2783966064453,131.65551847083263,149.07350158691407]},{"page":391,"text":"Chapter 13","rect":[53.812843322753909,72.10812377929688,121.10908599090695,59.25117874145508]},{"page":391,"text":"Ferroelectricity and Antiferroelectricity","rect":[53.812843322753909,91.18268585205078,327.19766074227706,76.0426254272461]},{"page":391,"text":"in Smectics","rect":[53.812843322753909,105.72347259521485,130.87025846393613,94.07360076904297]},{"page":391,"text":"13.1 Ferroelectrics","rect":[53.812843322753909,210.0841827392578,158.1371829960124,201.07196044921876]},{"page":391,"text":"13.1.1 Crystalline Pyro-, Piezo- and Ferroelectrics","rect":[53.812843322753909,241.9628448486328,311.4744480106608,231.3250732421875]},{"page":391,"text":"The discussion of ferroelectricity in liquid crystalline phase is based on the concepts","rect":[53.812843322753909,269.5052795410156,385.1776162539597,260.57073974609377]},{"page":391,"text":"developed for the solid crystals. Therefore, we have to start from a brief survey of","rect":[53.812843322753909,281.4648132324219,385.1487973770298,272.5302734375]},{"page":391,"text":"the elementary physics of ferroelectricity in crystals [1, 2].","rect":[53.812843322753909,293.42437744140627,290.7279934456516,284.48980712890627]},{"page":391,"text":"13.1.1.1 Polarization Catastrophe in Liquids and Solids","rect":[53.813812255859378,335.1846618652344,297.2878457461472,326.0010986328125]},{"page":391,"text":"In Section 7.2.1 we discussed polarization of molecular isotropic liquids. We","rect":[53.813812255859378,359.11669921875,385.1267780132469,350.18212890625]},{"page":391,"text":"introduced the equations for dielectric permittivity e and dielectric susceptibility","rect":[53.81281661987305,371.07623291015627,385.1227959733344,362.14166259765627]},{"page":391,"text":"wE and wrote the microscopic definition of the polarization vector P as a sum of","rect":[53.81282424926758,383.0369873046875,385.15035377351418,372.07904052734377]},{"page":391,"text":"dipole moments in the unit volume nv ¼ rNA/M (r is density, NA is Avogadro","rect":[53.8125,394.99664306640627,385.10549250653755,386.06195068359377]},{"page":391,"text":"number, M is molecular mass):","rect":[53.814414978027347,406.5577697753906,179.05529649326395,398.02166748046877]},{"page":391,"text":"e ¼ DE ¼ E þE4pP; wE ¼ EP ¼ e4\u0002p1; P ¼ Xpe ¼ ngEloc:","rect":[81.28602600097656,441.66998291015627,333.60862672013186,421.2141418457031]},{"page":391,"text":"n","rect":[272.3498840332031,445.29693603515627,275.83384773811658,442.0617980957031]},{"page":391,"text":"(13.1)","rect":[361.0563659667969,436.4853820800781,385.1057370742954,428.009033203125]},{"page":391,"text":"Here pe is the electric dipole induced by the electric field in a molecule having","rect":[65.76671600341797,469.8152770996094,385.16826716474068,460.8807373046875]},{"page":391,"text":"mean molecular polarizability g. Then we used the Lorentz approximation for the","rect":[53.81344223022461,481.77484130859377,385.17514837457505,472.84027099609377]},{"page":391,"text":"local field acting on a molecule and found corresponding field induced polarization.","rect":[53.81344985961914,493.7344055175781,385.0836147835422,484.79986572265627]},{"page":391,"text":"From that we have obtained the electric susceptibility of the dielectric (Eq. 7.18):","rect":[53.81344985961914,505.6939392089844,382.659803939553,496.7593994140625]},{"page":391,"text":"wE ¼ P=E ¼ nvg=½1 \u0002 ð4p=3Þnvg\u0003","rect":[151.07305908203126,531.1893920898438,287.8953088490381,519.9229125976563]},{"page":391,"text":"(13.2)","rect":[361.0557861328125,530.4054565429688,385.10515724031105,521.9290771484375]},{"page":391,"text":"This formula is very important for the further discussion because it predicts the","rect":[65.7661361694336,554.7229614257813,385.1729205913719,545.7884521484375]},{"page":391,"text":"“polarization catastrophe”. For small molecular polarizability g, susceptibility wE","rect":[53.81411361694336,566.6824951171875,385.1826684003752,555.6680908203125]},{"page":391,"text":"depends linearly on g. However, when g ! 3=4pnv, the denominator of (13.2)","rect":[53.812843322753909,578.9144897460938,385.1592038711704,568.98388671875]},{"page":391,"text":"tends to zero and wE diverges.","rect":[53.813350677490237,590.54541015625,174.70391507651096,579.58740234375]},{"page":391,"text":"L.M. Blinov, Structure and Properties of Liquid Crystals,","rect":[53.812843322753909,623.7623291015625,251.63688919141254,616.1510620117188]},{"page":391,"text":"DOI 10.1007/978-90-481-8829-1_13, # Springer ScienceþBusiness Media B.V. 2011","rect":[53.812843322753909,633.698486328125,351.58160919337197,625.4920043945313]},{"page":391,"text":"381","rect":[372.4981994628906,622.0606079101563,385.18979400782509,616.2357177734375]},{"page":392,"text":"382","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.73106384277344]},{"page":392,"text":"13 Ferroelectricity and Antiferroelectricity in Smectics","rect":[197.23812866210938,44.274620056152347,385.18039401053707,36.68026351928711]},{"page":392,"text":"Its physical sense is well seen from the equations (7.17) for P and Eloc. Fora","rect":[65.76496887207031,68.2883529663086,385.16077459527818,59.35380554199219]},{"page":392,"text":"fixed concentration of molecules and given local field, the polarization should be","rect":[53.814022064208987,80.24788665771485,385.14893377496568,71.31333923339844]},{"page":392,"text":"linear function of g. However, with increasing P, Eloc itself begins to grow and this","rect":[53.814022064208987,92.20772552490235,385.14618314849096,83.27293395996094]},{"page":392,"text":"results in the non-linear, avalanche-like increase of susceptibility. Finally, for","rect":[53.81427764892578,104.1104965209961,385.12127052156105,95.17594909667969]},{"page":392,"text":"small, densely packed molecules (nv is large) with high polarizability g, wE would","rect":[53.81427764892578,116.07039642333985,385.1387566666938,105.11235046386719]},{"page":392,"text":"tend to infinity. It means that an infinitesimally low field, even a small fluctuation of","rect":[53.813838958740237,128.02999877929688,385.1507505020298,119.09544372558594]},{"page":392,"text":"a local field, may create a finite polarization. In other words, polarization may","rect":[53.813838958740237,139.98953247070313,385.16460505536568,131.0549774169922]},{"page":392,"text":"appear spontaneously, without any applied field. The appearance of the spontane-","rect":[53.813838958740237,151.94906616210938,385.1556638321079,143.01451110839845]},{"page":392,"text":"ous polarization is a necessary (but not sufficient) condition for phenomenon called","rect":[53.813838958740237,163.90863037109376,385.1735772233344,154.9740753173828]},{"page":392,"text":"ferroelectricity.","rect":[53.813838958740237,175.86813354492188,115.94105191733127,166.93357849121095]},{"page":392,"text":"What’s about liquid ferroelectric? Let us examine the qualitative criterion","rect":[65.76586151123047,187.82766723632813,363.6376885514594,178.8931121826172]},{"page":392,"text":"k ¼ ð4=3Þpnvg ¼ ð4=3ÞprgNA=M","rect":[150.9032440185547,212.07640075683595,287.89265658278608,202.07814025878907]},{"page":392,"text":"(13.3)","rect":[361.0561218261719,211.2924041748047,385.1054929336704,202.8160400390625]},{"page":392,"text":"for the polarization catastrophe in liquids having non-polar molecules. In this case,","rect":[53.81446075439453,235.60992431640626,385.1413540413547,226.6753692626953]},{"page":392,"text":"the Lorentz formula for the local field is approximately valid. We can take, e.g.,","rect":[53.81446075439453,247.5694580078125,385.1404079964328,238.63490295410157]},{"page":392,"text":"liquid","rect":[53.81446075439453,259.5290222167969,77.13717738202581,250.59446716308595]},{"page":392,"text":"benzene","rect":[82.53335571289063,258.0,115.26279485895002,250.59446716308595]},{"page":392,"text":"(r \u0004 0.9g/cm3,","rect":[120.65499114990235,259.5290222167969,182.56348844076877,248.4784698486328]},{"page":392,"text":"M ¼ 78,","rect":[187.8919677734375,258.0,222.99349637533909,250.65475463867188]},{"page":392,"text":"electronic","rect":[228.3369140625,258.0,267.90589178277818,250.59498596191407]},{"page":392,"text":"polarizability","rect":[273.25628662109377,259.529541015625,326.74224177411568,250.59498596191407]},{"page":392,"text":"ge \u0004 1.25","rect":[332.0538330078125,259.43988037109377,372.13213435224068,250.53521728515626]},{"page":392,"text":"\u0005","rect":[377.4835205078125,257.2884216308594,385.1482622930756,252.4974365234375]},{"page":392,"text":"10\u000223 cm\u00023). Then k \u0004 0.09 <<1 and liquid benzene cannot be polarised sponta-","rect":[53.81438446044922,271.4891662597656,385.11052833406105,260.4381408691406]},{"page":392,"text":"neously. Even for hypothetical liquids consisted of smaller molecules with higher","rect":[53.81254577636719,283.4486999511719,385.12651954499855,274.51416015625]},{"page":392,"text":"electronic polarizability it would be difficult to reach criterion (13.3). More per-","rect":[53.81254577636719,295.3514709472656,385.15435157624855,286.41693115234377]},{"page":392,"text":"spective are liquids whose molecules carry permanent dipole moments pe which","rect":[53.81254577636719,307.31103515625,385.12102595380318,298.37646484375]},{"page":392,"text":"additionally contribute to g due to orientational polarizability gor. Let us take liquid","rect":[53.81405258178711,319.2710876464844,385.1146172623969,310.33642578125]},{"page":392,"text":"nitrobenzene (r \u0004 1.2g/cm3, M ¼ 123) with quite a large dipole moment, pe \u00044","rect":[53.81362533569336,331.2307434082031,385.1796807389594,320.1796875]},{"page":392,"text":"Debye ¼ 4\u0006\u0005 10\u000218 CGS). The application of the Lorenz formula for Eloc would","rect":[53.81493377685547,343.1903991699219,385.1387566666938,332.13934326171877]},{"page":392,"text":"result in equation (7.22) for orientational polarizability gor ¼ pe2=3kBT \u0004 1.3 \u0005","rect":[53.813838958740237,355.93206787109377,385.14835384581,344.6676025390625]},{"page":392,"text":"10\u000222cm\u00023","rect":[53.81450653076172,365.0876159667969,98.7830775293479,356.0587463378906]},{"page":392,"text":"at","rect":[103.88724517822266,366.0,111.13390977451394,359.19110107421877]},{"page":392,"text":"room","rect":[116.29218292236328,366.0,137.3353800642546,360.0]},{"page":392,"text":"temperature.","rect":[142.4617919921875,367.10968017578127,192.80721707846409,359.19110107421877]},{"page":392,"text":"Then,","rect":[197.97344970703126,366.0,220.97762723471409,358.17510986328127]},{"page":392,"text":"coefficient","rect":[226.12594604492188,366.0,268.36762864658427,358.17510986328127]},{"page":392,"text":"k \u0004 3.2","rect":[273.53887939453127,365.1275329589844,304.78810969403755,358.15521240234377]},{"page":392,"text":"would","rect":[309.9055480957031,365.0777282714844,334.7811211686469,358.17510986328127]},{"page":392,"text":"exceed","rect":[339.98419189453127,365.0777282714844,367.7365349870063,358.17510986328127]},{"page":392,"text":"the","rect":[372.8948059082031,365.0478515625,385.1185687847313,358.17510986328127]},{"page":392,"text":"criterion for the polarization catastrophe, however, this is incorrect result, because","rect":[53.81359100341797,379.0692138671875,385.1037067241844,370.1346435546875]},{"page":392,"text":"the Onsager reaction field discussed in Section 7.2.1 has not been taken into","rect":[53.81359100341797,390.97198486328127,385.15837946942818,382.01751708984377]},{"page":392,"text":"account. In reality, the dipole–dipole interaction in nitrobenzene and other known","rect":[53.8145751953125,402.9315490722656,385.1773308854438,393.99700927734377]},{"page":392,"text":"dipolar liquids is not sufficient to form a spontaneously polarised state.","rect":[53.8145751953125,414.8910827636719,340.25799222494848,405.95654296875]},{"page":392,"text":"In solid crystals, the situation is different because (i) their packing is denser; (ii)","rect":[65.76659393310547,426.8506164550781,385.1723874649204,417.91607666015627]},{"page":392,"text":"ionic crystals consist of small ions of high polarizability; (iii) different ions interact","rect":[53.8145751953125,438.8101501464844,385.1802812344749,429.8756103515625]},{"page":392,"text":"with each other forming large dipoles and (iv) there is a possibility to overcome the","rect":[53.8145751953125,450.7696838378906,385.17237127496568,441.83514404296877]},{"page":392,"text":"limitation posed by the Lorenz formula for the local field. Indeed due to crystal","rect":[53.8145751953125,462.729248046875,385.12153489658427,453.794677734375]},{"page":392,"text":"anisotropy, at least, for some directions the criterion for the polarisation catastrophe","rect":[53.8145751953125,474.6888122558594,385.15842474176255,465.7542724609375]},{"page":392,"text":"is weaker. On the other hand, in solids there are strong elastic forces counteracting","rect":[53.8145751953125,486.5915832519531,385.10967341474068,477.6371154785156]},{"page":392,"text":"the electric force and hindering displacement of ions. Nevertheless, a spontaneously","rect":[53.8145751953125,498.55108642578127,385.10662165692818,489.61651611328127]},{"page":392,"text":"polarised state is quite typical of many crystals, the molecular organic crystals","rect":[53.8145751953125,510.5106506347656,385.10165800200658,501.57611083984377]},{"page":392,"text":"included.","rect":[53.8145751953125,520.438232421875,90.79148526694064,513.53564453125]},{"page":392,"text":"13.1.1.2 Pyro-, Piezo- and Ferroelectrics","rect":[53.8145751953125,558.7471923828125,232.64892973052219,549.0955200195313]},{"page":392,"text":"Totally there are 32 crystallographic classes. Among them we can distinguish 11","rect":[53.8145751953125,582.2678833007813,385.16835871747505,573.3333740234375]},{"page":392,"text":"unpolar classes, 11 neutral-polar classes and ten polar classes. Unpolar classes have","rect":[53.8145751953125,594.2274169921875,385.11963689996568,585.2929077148438]},{"page":393,"text":"13.1 Ferroelectrics","rect":[53.813682556152347,42.55630874633789,117.63130648612298,36.68062973022461]},{"page":393,"text":"Fig. 13.1 Examples of non-","rect":[53.812843322753909,67.58130645751953,151.68374723059825,59.648292541503909]},{"page":393,"text":"polar, piezoelectric and","rect":[53.812843322753909,77.4895248413086,133.91718048243448,69.89517211914063]},{"page":393,"text":"pyroelectric crystals: calcite","rect":[53.812843322753909,87.4087142944336,149.7698302009058,79.81436157226563]},{"page":393,"text":"(a), ZnS (b) and tourmaline","rect":[53.812843322753909,97.04601287841797,148.74520251535894,89.79031372070313]},{"page":393,"text":"(c). An arrow shows the","rect":[53.812843322753909,107.02202606201172,137.2195372321558,99.76632690429688]},{"page":393,"text":"direction of the polar axis in","rect":[53.813682556152347,117.33663177490235,151.05338043360636,109.74227905273438]},{"page":393,"text":"tourmaline","rect":[53.813682556152347,125.50328063964844,90.62780139231205,119.66146850585938]},{"page":393,"text":"a","rect":[193.3865966796875,68.23336791992188,198.94185907692134,62.64461135864258]},{"page":393,"text":"C","rect":[226.73281860351563,76.64115905761719,232.5039547520502,70.60244750976563]},{"page":393,"text":"b","rect":[267.6892395019531,70.35690307617188,273.7940332514313,63.04853057861328]},{"page":393,"text":"c","rect":[350.14093017578127,68.65530395507813,355.6961925730151,63.06654739379883]},{"page":393,"text":"383","rect":[372.4990539550781,42.55630874633789,385.1906790175907,36.73143005371094]},{"page":393,"text":"no polar directions at all. They have a centre of symmetry and show no polar","rect":[53.812843322753909,209.70703125,385.1367429336704,200.77247619628907]},{"page":393,"text":"properties. The polarisation can be induced only by external electric field. An","rect":[53.812843322753909,221.66656494140626,385.12783137372505,212.7320098876953]},{"page":393,"text":"example is calcite having inversion centre and symmetry D3d shown in Fig. 13.1a.","rect":[53.812843322753909,233.6263427734375,385.18267484213598,224.69154357910157]},{"page":393,"text":"Piezoelectrics. In the neutral-polar classes there are polar directions (not axes),","rect":[65.7660140991211,245.58590698242188,385.1319851448703,236.57167053222657]},{"page":393,"text":"which can be described by several vectors with their vector sum equal to zero. Such","rect":[53.8139762878418,257.4886779785156,385.14189997724068,248.5541229248047]},{"page":393,"text":"crystals do not possess spontaneous polarization and do not manifest polar properties","rect":[53.8139762878418,269.4482116699219,385.1547891055222,260.513671875]},{"page":393,"text":"(such as pyroelectric, photogalvanic or linear electrooptical effects); however, the","rect":[53.8139762878418,281.40777587890627,385.1737750835594,272.47320556640627]},{"page":393,"text":"polarization can be induced not only by an electric field but also by a pure mechani-","rect":[53.8139762878418,293.3673095703125,385.10210548249855,284.4327392578125]},{"page":393,"text":"calstress.Thesecrystals arecalledpiezoelectrics.Examplesarecrystalsofquartzor","rect":[53.8139762878418,305.32684326171877,385.14885841218605,296.39227294921877]},{"page":393,"text":"ZnS having cubic symmetry with four polar direction but no polar axis, Fig. 13.1b.","rect":[53.8139762878418,317.286376953125,385.18365140463598,308.351806640625]},{"page":393,"text":"Such crystals are used in technics as microphones, mechanical micro-motors and","rect":[53.814979553222659,329.24591064453127,385.1438836198188,320.31134033203127]},{"page":393,"text":"sensors, etc.","rect":[53.814979553222659,339.1436462402344,102.27007718588595,333.2868957519531]},{"page":393,"text":"Pyroelectrics. In a crystal belonging to polar classes there is only one polar axis with","rect":[65.76702117919922,353.50665283203127,385.1317986588813,344.093994140625]},{"page":393,"text":"a symmetry of the polar vector. These crystals are also piezoelectric, but, in addition,","rect":[53.814979553222659,365.0677490234375,385.2078823616672,356.1331787109375]},{"page":393,"text":"manifest spontaneous polarization Ps and all other polar properties. Such crystals are","rect":[53.814979553222659,377.02886962890627,385.18103826715318,368.0927734375]},{"page":393,"text":"called pyroelectrics. An example is tourmaline having symmetry C3v and shown in","rect":[53.81330490112305,388.9884033203125,385.1774529557563,380.0538330078125]},{"page":393,"text":"Fig. 13.1c. Pyroelectric crystals are also used in techniques as piezoelectrics and also as","rect":[53.813777923583987,400.94793701171877,385.15854276763158,392.01336669921877]},{"page":393,"text":"detectors of infrared light or a heat flow. There are many organic pyroelectric crystals,","rect":[53.814781188964847,412.907470703125,385.1336636116672,403.972900390625]},{"page":393,"text":"e.g., p-nitroaniline, one of the best generators of the optical second harmonic.","rect":[53.814781188964847,424.86700439453127,353.0463528206516,415.93243408203127]},{"page":393,"text":"Ferroelectrics. Speaking in terms of the polarisation catastrophe we can say that,","rect":[65.76680755615235,436.8265686035156,385.1357388069797,427.81231689453127]},{"page":393,"text":"inthemostofpyroelectricsk > 1,andthecatastropheoccursuponcrystallisationof","rect":[53.813777923583987,448.7293395996094,385.1486753067173,439.7748718261719]},{"page":393,"text":"the substance. In this case the polarisation is forever fixed along the direction of the","rect":[53.813777923583987,460.6888732910156,385.17356146051255,451.75433349609377]},{"page":393,"text":"polar axis even upon variation of temperature or an external field. However, there are","rect":[53.813777923583987,472.6484069824219,385.16065252496568,463.7138671875]},{"page":393,"text":"some crystals belonging to the same point groups as pyroelectrics but having not so","rect":[53.813777923583987,484.6079406738281,385.15960017255318,475.67340087890627]},{"page":393,"text":"stable spontaneous polarisation. The direction of Ps, that is the direction of the vector","rect":[53.813777923583987,496.5674743652344,385.11358009187355,487.6329345703125]},{"page":393,"text":"of polar axis, can be inverted by an external electric field. In fact, this direction is","rect":[53.81356430053711,508.5282287597656,385.18426908599096,499.59368896484377]},{"page":393,"text":"degenerate and there are two equivalent energy states. One of the two minima of the","rect":[53.81356430053711,520.4877319335938,385.17136419488755,511.5332946777344]},{"page":393,"text":"freeenergymaybeselected byanexternalfield andthisisanothertypeofbistability","rect":[53.81356430053711,532.447265625,385.16433039716255,523.5127563476563]},{"page":393,"text":"in addition to discussed in Section 12.3.3. The switching between the two states is","rect":[53.81356430053711,544.3500366210938,385.18722929106908,535.41552734375]},{"page":393,"text":"characterised by a certain threshold and hysteresis. This possibility of the polarisa-","rect":[53.81456756591797,556.3095703125,385.1683591446079,547.3750610351563]},{"page":393,"text":"tion switching between two stable states is usually taken as a criterion to distinguish","rect":[53.81456756591797,568.2691650390625,385.1623467545844,559.3346557617188]},{"page":393,"text":"such (soft) ferroelectrics from the normal (rigid) pyroelectrics. The bistability is the","rect":[53.81456756591797,580.2286987304688,385.17237127496568,571.294189453125]},{"page":393,"text":"sufficient condition for ferroelectricity [3].","rect":[53.81456756591797,592.188232421875,224.17193265463596,583.2537231445313]},{"page":394,"text":"384","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.73106384277344]},{"page":394,"text":"13 Ferroelectricity and Antiferroelectricity in Smectics","rect":[197.23812866210938,44.274620056152347,385.18039401053707,36.68026351928711]},{"page":394,"text":"In ferroelectrics criterion k only slightly exceeds unity. For this reason, with","rect":[65.76496887207031,68.2883529663086,385.11696711591255,59.333885192871097]},{"page":394,"text":"increasing temperature, the ferroelectric state or phase can be destroyed by the","rect":[53.81295394897461,80.24788665771485,385.1716998882469,71.31333923339844]},{"page":394,"text":"phase transition into the non-ferroelectric (paraelectric) phase with zero spontane-","rect":[53.81295394897461,92.20748138427735,385.1528256973423,83.27293395996094]},{"page":394,"text":"ous polarization. Sometimes this ferroelectric transition at temperature Tc is taken","rect":[53.81295394897461,104.11019134521485,385.1419610123969,95.13580322265625]},{"page":394,"text":"as an additional criterion for ferroelectricity, although there are crystals, in which a","rect":[53.814083099365237,116.0702133178711,385.15695989801255,107.13566589355469]},{"page":394,"text":"ferroelectric phase survives up to the crystal melting. Experimentally, close to Tc","rect":[53.814083099365237,128.02981567382813,385.1820833351628,119.09526062011719]},{"page":394,"text":"the dielectric constant very often obeys the Curie law, e ¼ C/ (Tc \u0002 T), where C is","rect":[53.812843322753909,139.98959350585938,385.18915189849096,131.04507446289063]},{"page":394,"text":"Curie constant. A good example of a ferroelectric crystal is BaTiO3, in which two","rect":[53.814537048339847,151.94912719726563,385.15285578778755,142.99464416503907]},{"page":394,"text":"stable states are characterised by two different sites the Ti4þ ion occupies in the","rect":[53.814022064208987,163.908935546875,385.17432439996568,152.8579559326172]},{"page":394,"text":"crystallographic lattice. Among crystalline ferroelectrics there are some organic","rect":[53.813594818115237,175.86846923828126,385.17932928277818,166.9339141845703]},{"page":394,"text":"crystals and even polymers, e.g., poly-vinylidene-fluoride (PVDF), in which the","rect":[53.813594818115237,187.8280029296875,385.17234075738755,178.89344787597657]},{"page":394,"text":"spontaneous","rect":[53.813594818115237,199.73077392578126,103.12376798735812,191.81219482421876]},{"page":394,"text":"polarisation","rect":[108.47515106201172,199.73077392578126,155.7228173356391,190.7962188720703]},{"page":394,"text":"is","rect":[161.04135131835938,198.0,167.670856610405,190.7962188720703]},{"page":394,"text":"owed","rect":[172.93663024902345,198.0,194.56714716962348,190.7962188720703]},{"page":394,"text":"to","rect":[199.84286499023438,198.0,207.6171044450141,191.81219482421876]},{"page":394,"text":"collective","rect":[212.92767333984376,198.0,251.96611822320785,190.7962188720703]},{"page":394,"text":"alignment","rect":[257.2806396484375,199.73077392578126,297.37324388095927,190.7962188720703]},{"page":394,"text":"of","rect":[302.709716796875,198.0,311.00155006257668,190.7962188720703]},{"page":394,"text":"the","rect":[316.3041687011719,198.0,328.52796209527818,190.7962188720703]},{"page":394,"text":"C–F","rect":[333.8644104003906,198.0,350.96573228190496,190.85598754882813]},{"page":394,"text":"dipoles","rect":[356.29522705078127,199.73077392578126,385.1026650820847,190.7962188720703]},{"page":394,"text":"perpendicular to the backbone of the polymer.","rect":[53.813594818115237,211.69033813476563,240.39631314780002,202.7557830810547]},{"page":394,"text":"13.1.1.3 Simplest Description of a Proper Ferroelectric","rect":[53.813594818115237,253.45059204101563,295.0897907085594,244.26702880859376]},{"page":394,"text":"In the proper ferroelectrics, the spontaneous polarisation appears as a result of the","rect":[53.813594818115237,277.4394226074219,385.1733783550438,268.5048828125]},{"page":394,"text":"polarisation catastrophe or, in other words, due to electric dipole–dipole interac-","rect":[53.813594818115237,289.3989562988281,385.0887387832798,280.46441650390627]},{"page":394,"text":"tions. There are also improper ferroelectrics, in particular, liquid crystalline ones, in","rect":[53.813594818115237,301.3584899902344,385.1395196061469,292.4239501953125]},{"page":394,"text":"which a structural transition into a polar phase occurs due to other interactions and,","rect":[53.813594818115237,313.31805419921877,385.1385464241672,304.38348388671877]},{"page":394,"text":"consequently, Ps appears as a secondary phenomenon. We shall discuss this case","rect":[53.813594818115237,325.2792053222656,385.1449664898094,316.343017578125]},{"page":394,"text":"later. For simplicity, the square of spontaneous polarisation vector can be taken as a","rect":[53.81307601928711,337.23876953125,385.1579059429344,328.30419921875]},{"page":394,"text":"scalar order parameter for the transition from the higher symmetry paraelectric","rect":[53.81307601928711,349.1415100097656,385.1489642925438,340.20697021484377]},{"page":394,"text":"phase to the lower symmetry ferroelectric phase. Therefore, in the absence of an","rect":[53.81307601928711,361.10107421875,385.14898005536568,352.16650390625]},{"page":394,"text":"external field, we can expand the free energy density in a series over Ps2(T) and this","rect":[53.81307601928711,373.06060791015627,385.14606107817846,362.010498046875]},{"page":394,"text":"expansion for ferroelectrics is called Landau–Ginzburg expansion:","rect":[53.814144134521487,385.020751953125,322.2214799649436,376.086181640625]},{"page":394,"text":"g ¼ g0 þ 21 APs2 þ 41 BP4s þ 61 CPs6 þ \u0006\u0006\u0006","rect":[140.311279296875,411.64312744140627,297.0158227650537,399.26611328125]},{"page":394,"text":"(13.4)","rect":[361.0562438964844,409.90203857421877,385.1056150039829,401.4256896972656]},{"page":394,"text":"Here, g0 is free energy of the paraelectric phase, A ¼ a(T \u0002 Tc), B, C are","rect":[65.76659393310547,435.22259521484377,385.1658405132469,426.2388610839844]},{"page":394,"text":"Landau coefficients. As in Eq. (13.5) there is no any derivative, the conditions for","rect":[53.81405258178711,447.1429138183594,385.1807797989048,438.1485900878906]},{"page":394,"text":"the free energy minimum are given by the simplest Euler equation qg=qPs ¼ 0 and","rect":[53.81405258178711,459.43115234375,385.1469353776313,449.5005798339844]},{"page":394,"text":"stability condition q2g=qP2s>0:","rect":[53.814083099365237,471.78790283203127,179.01231248447489,459.84307861328127]},{"page":394,"text":"APs þ BPs3 þ CPs5 ¼ 0 and A þ 3BP2s þ 5CP4s >0","rect":[112.21357727050781,497.12451171875,326.77849665692818,485.3224182128906]},{"page":394,"text":"(13.5)","rect":[361.05682373046877,496.00018310546877,385.1061948379673,487.404296875]},{"page":394,"text":"Consider the case of small Ps. Then, the sixth order term is ignored and","rect":[65.76615142822266,520.714599609375,385.14421931317818,511.77960205078127]},{"page":394,"text":"g ¼ g0 þ 12 APs2 þ 14 BPs4 þ \u0006\u0006\u0006 The plot of this function is very similar to that in","rect":[53.813350677490237,534.7275390625,385.1417168717719,522.9855346679688]},{"page":394,"text":"Fig. 6.10b for the free energy close to the SmA-nematic transition discussed in","rect":[53.8138313293457,546.3348388671875,385.14177790692818,537.4003295898438]},{"page":394,"text":"Section 6.3. The phase transition occurs at A ¼ 0. For A > 0 there are two minima","rect":[53.8138313293457,558.2943725585938,385.1108783550438,549.35986328125]},{"page":394,"text":"corresponding to finite values of the spontaneous polarisation in the ferroelectric","rect":[53.813838958740237,570.1971435546875,385.08301580621568,561.2626342773438]},{"page":394,"text":"phase (curve 1 in the figure); for A < 0 only one minimum at zero Ps corresponds to","rect":[53.813838958740237,582.15673828125,385.1429070573188,573.22216796875]},{"page":394,"text":"the paraelectric phase (curve 3). Totally, we have only three solutions of Eq. (13.5):","rect":[53.814022064208987,594.1162719726563,385.18073899814677,585.1220092773438]},{"page":395,"text":"13.1 Ferroelectrics","rect":[53.812843322753909,42.55594253540039,117.63046725272455,36.68026351928711]},{"page":395,"text":"385","rect":[372.4981994628906,42.55594253540039,385.1898245254032,36.62946701049805]},{"page":395,"text":"Ps ¼ 0 (for the paraelectric phase) and Ps ¼ \u0007ð\u0002A=BÞ1=2 ¼ \u0007½ða=BÞðTc \u0002 TÞ\u00031=2","rect":[53.812843322753909,68.62700653076172,384.7055823572572,56.16347885131836]},{"page":395,"text":"(for the ferroelectric phase). Signs (\u0007) mean that Ps can look in two opposite","rect":[53.812843322753909,80.24800872802735,385.1816791362938,71.31346130371094]},{"page":395,"text":"directions and correspond to the two minima in the free energy. The square root","rect":[53.8139762878418,92.20760345458985,385.16279466220927,83.27305603027344]},{"page":395,"text":"dependence of Ps on temperature near the phase transition is a continuous function","rect":[53.8139762878418,104.11067962646485,385.16848078778755,95.17576599121094]},{"page":395,"text":"characteristic","rect":[53.81270217895508,114.00839233398438,107.25188482477033,107.13566589355469]},{"page":395,"text":"of","rect":[112.55249786376953,114.00839233398438,120.84435401765478,107.13566589355469]},{"page":395,"text":"the","rect":[126.03347778320313,114.00839233398438,138.25725591852035,107.13566589355469]},{"page":395,"text":"second","rect":[143.53697204589845,114.03827667236328,171.19971552899848,107.13566589355469]},{"page":395,"text":"order","rect":[176.44757080078126,114.03827667236328,197.48081336824073,107.13566589355469]},{"page":395,"text":"transition","rect":[202.7873992919922,114.00839233398438,240.50393000653754,107.13566589355469]},{"page":395,"text":"like","rect":[245.78065490722657,114.00839233398438,260.8015521831688,107.13566589355469]},{"page":395,"text":"SmA-nematic","rect":[266.05938720703127,114.04823303222656,321.5740436382469,107.13566589355469]},{"page":395,"text":"transition,","rect":[326.89556884765627,114.00839233398438,367.10064359213598,107.13566589355469]},{"page":395,"text":"see","rect":[372.38134765625,114.00839233398438,385.11277044488755,109.36681365966797]},{"page":395,"text":"Fig. 6.10a or SmC-SmA transition. Such dependence is in accordance with many","rect":[53.81270217895508,128.02981567382813,385.1754998307563,119.09526062011719]},{"page":395,"text":"experiments on ferroelectrics. When Ps is not small we should return back to the","rect":[53.81270217895508,139.98934936523438,385.1738971538719,131.05479431152345]},{"page":395,"text":"initial free energy expansion (13.5) and keep the sixth order term. A similar","rect":[53.81315231323242,151.94912719726563,385.15398536531105,142.95480346679688]},{"page":395,"text":"situation has been already discussed in Section 6.2 for the isotropic – nematic","rect":[53.813167572021487,163.90869140625,385.1012653179344,154.97413635253907]},{"page":395,"text":"phase transition. Now Eq. (13.5) has five solutions and can explain a jump-like","rect":[53.81318283081055,175.86822509765626,385.15796697809068,166.8739013671875]},{"page":395,"text":"growth of Ps with decreasing temperature, (as in Fig. 6.5), hysteresis of Ps(T) close","rect":[53.81318283081055,187.82827758789063,385.1624225444969,178.83395385742188]},{"page":395,"text":"to the transition temperature and specific features of other thermodynamic properties.","rect":[53.81462860107422,199.73104858398438,385.18133206869848,190.79649353027345]},{"page":395,"text":"To discuss the electric field switching we add the field term gE ¼ \u0002PE to","rect":[65.76665496826172,211.73045349121095,385.14394465497505,202.7560577392578]},{"page":395,"text":"(Eq. (13.4)) and make minimisation of free energy with respect to the total polari-","rect":[53.814083099365237,223.65048217773438,385.1330503067173,214.71592712402345]},{"page":395,"text":"zation P ¼ Ps","rect":[53.81505584716797,235.1655731201172,112.08608126986352,226.6754913330078]},{"page":395,"text":"P ¼ Ps þ Pin.","rect":[53.81306076049805,247.19161987304688,112.09697385336642,238.8543701171875]},{"page":395,"text":"þ Pin that includes the","rect":[115.38615417480469,235.23196411132813,211.8411334819969,226.6757049560547]},{"page":395,"text":"Then we obtain ∂gE/∂P","rect":[114.87619018554688,247.6096954345703,211.84650414861404,237.62928771972657]},{"page":395,"text":"spontaneous","rect":[216.10052490234376,235.61026000976563,265.4017373477097,227.69168090820313]},{"page":395,"text":"¼ \u0002E or","rect":[214.6844482421875,245.50802612304688,250.05277381745948,238.85443115234376]},{"page":395,"text":"and","rect":[269.7427673339844,234.0,284.14653864911568,226.6757049560547]},{"page":395,"text":"the","rect":[288.435791015625,234.0,300.65955389215318,226.6757049560547]},{"page":395,"text":"field","rect":[304.9757080078125,234.0,322.6942071061469,226.6757049560547]},{"page":395,"text":"induced","rect":[327.01031494140627,234.0,358.6149529557563,226.6757049560547]},{"page":395,"text":"terms","rect":[362.92315673828127,234.0,385.17877592192846,227.69168090820313]},{"page":395,"text":"E ¼ AP þ BP3 þ CP5","rect":[175.08958435058595,273.5652160644531,263.42857430061658,263.87054443359377]},{"page":395,"text":"(13.6)","rect":[361.0561828613281,274.49151611328127,385.10555396882668,265.95538330078127]},{"page":395,"text":"This equationimplicitly represents the dependence of polarization on the applied","rect":[65.76653289794922,298.8090515136719,385.17238703778755,289.87451171875]},{"page":395,"text":"electric field. Usually, the function P(E) can be found numerically with temperature","rect":[53.81450653076172,310.76861572265627,385.0727008648094,301.83404541015627]},{"page":395,"text":"dependent coefficient A and constant B and C.","rect":[53.815513610839847,322.7281494140625,240.40021939780002,313.78363037109377]},{"page":395,"text":"From Eq. (13.6) we can easily derive the Curie law for dielectric permittivity e","rect":[65.76750946044922,334.6876525878906,385.17622272854586,325.75311279296877]},{"page":395,"text":"or susceptibility w ¼ (e \u0002 1)/4p. For small fields, we can leave only the first","rect":[53.815513610839847,346.5904235839844,385.16328294345927,337.6558837890625]},{"page":395,"text":"term of the expansion E ¼ AP. In the paraelectric phase P ¼ Pin and","rect":[53.81548309326172,358.54998779296877,385.14626399091255,349.61541748046877]},{"page":395,"text":"Pin ¼ wparaE ¼ E=aðT \u0002 TcÞ. In the ferroelectric phase, for small fields, Ps>> Pin,","rect":[53.814414978027347,372.1153259277344,385.1832241585422,360.898681640625]},{"page":395,"text":"therefore P2 \u0004 Ps2 þ 2PsPin þ \u0006\u0006\u0006 and ðA þ BP2s þ 2BPsPinÞðPs þ PinÞ ¼ E. From","rect":[53.814537048339847,384.44281005859377,385.1669993269499,373.23480224609377]},{"page":395,"text":"here, using formula for (Ps)2¼\u0002A/B found above and leaving only linear terms in","rect":[53.815208435058597,395.6201171875,385.14104548505318,384.5691833496094]},{"page":395,"text":"Pin, we obtain E ¼ \u00022APin.","rect":[53.81315994262695,407.2015075683594,166.02312894369846,398.64520263671877]},{"page":395,"text":"Therefore, the Curie law is given by","rect":[65.76531219482422,419.5392761230469,212.23256770185004,410.604736328125]},{"page":395,"text":"wpara ¼ 1=aðT \u0002 TcÞðfor T > Tc in paraelectric phaseÞ","rect":[109.43841552734375,448.0109558105469,329.54208768950658,436.7942199707031]},{"page":395,"text":"(13.7)","rect":[361.0561218261719,446.0644226074219,385.1054929336704,437.58807373046877]},{"page":395,"text":"wpara ¼ 1=2aðT \u0002 TcÞðfor T < Tc in ferroelectric phaseÞ","rect":[105.8709487915039,475.2177429199219,333.11029447661596,464.0010070800781]},{"page":395,"text":"(13.8)","rect":[361.0557556152344,473.2144470214844,385.1051267227329,464.73809814453127]},{"page":395,"text":"The inverse susceptibility follows a linear dependence on temperature in both","rect":[65.76610565185547,497.7590637207031,385.12712946942818,488.82452392578127]},{"page":395,"text":"the paraelectric and ferroelectric phases; a sign of the slope of function 1/w(T) in the","rect":[53.814083099365237,509.7185974121094,385.17481268121568,500.7840576171875]},{"page":395,"text":"two phases is opposite and the magnitude of the slope is twice larger in the","rect":[53.81406784057617,521.6781005859375,385.11707342340318,512.7435913085938]},{"page":395,"text":"ferroelectric phase in agreement with experiment. Note that the Curie law is valid","rect":[53.81406784057617,533.6376342773438,385.1299981217719,524.703125]},{"page":395,"text":"for both second and first order transitions, but the critical temperatures Tc and Tc*","rect":[53.81406784057617,545.5404052734375,385.17937556317818,536.5465698242188]},{"page":395,"text":"the two cases may not coincide, like in Fig. 6.8b.","rect":[53.81462860107422,557.5003662109375,252.2657436897922,548.5658569335938]},{"page":395,"text":"Now, going back to Eq. (13.6) we may discuss the P(E) dependence even for","rect":[65.76665496826172,569.4598999023438,385.1823667129673,560.525390625]},{"page":395,"text":"strong fields. An example of such dependence is pictured qualitatively in Fig. 13.2.","rect":[53.815635681152347,581.41943359375,385.1843227913547,572.4849243164063]},{"page":395,"text":"The values of P at E ¼ 0 on the vertical axis corresponds to \u0007 Ps. In the field range","rect":[53.81566619873047,593.3795166015625,385.14954412652818,584.4445190429688]},{"page":396,"text":"386","rect":[53.812843322753909,42.55630874633789,66.50444931178018,36.68062973022461]},{"page":396,"text":"Fig. 13.2 Hysteresis type","rect":[53.812843322753909,67.58130645751953,143.1490568854761,59.648292541503909]},{"page":396,"text":"dependence of total","rect":[53.812843322753909,77.4895248413086,120.84483055319849,69.89517211914063]},{"page":396,"text":"polarization P on electric field","rect":[53.812843322753909,87.4087142944336,155.31436676173136,79.81436157226563]},{"page":396,"text":"E for a typical crystalline","rect":[53.812843322753909,97.3846664428711,141.14969775461675,89.79031372070313]},{"page":396,"text":"ferroelectric","rect":[53.812843322753909,105.60813903808594,95.35669848703862,99.76632690429688]},{"page":396,"text":"13","rect":[197.23812866210938,42.55630874633789,205.6991934218876,36.73143005371094]},{"page":396,"text":"Ferroelectricity","rect":[208.05731201171876,44.274986267089847,260.4398245254032,36.68062973022461]},{"page":396,"text":"and","rect":[262.8901672363281,43.0,275.1333364882938,36.68062973022461]},{"page":396,"text":"D","rect":[265.3824462890625,170.38592529296876,271.15358243759706,164.6431427001953]},{"page":396,"text":"Antiferroelectricity","rect":[277.5049743652344,44.274986267089847,343.0951284804813,36.68062973022461]},{"page":396,"text":"B","rect":[301.1794738769531,98.15411376953125,306.5109805404885,92.41133880615235]},{"page":396,"text":"G","rect":[298.810302734375,128.33250427246095,305.0290616312724,122.29379272460938]},{"page":396,"text":"O","rect":[317.8238830566406,128.1205291748047,324.042641953538,122.08181762695313]},{"page":396,"text":"in","rect":[345.5361633300781,43.0,352.14423126368447,36.68062973022461]},{"page":396,"text":"Smectics","rect":[354.54296875,43.0,385.18039401053707,36.68062973022461]},{"page":396,"text":"between points G and H the function P(E) has three values. In reality, a hysteresis is","rect":[53.812843322753909,224.33071899414063,385.18649686919408,215.3961639404297]},{"page":396,"text":"observed. With increasing field the curve follows the route DECFA, but, due toa","rect":[53.81282424926758,236.290283203125,385.1566547222313,227.35572814941407]},{"page":396,"text":"memory of state A, the back route AFBED does not coincide with the forward one.","rect":[53.81282424926758,248.24981689453126,385.1367458870578,239.3152618408203]},{"page":396,"text":"As a result, a ferroelectric loop forms with width G-H. A half of the width is called","rect":[53.81282424926758,260.2093811035156,385.1695489030219,251.2748260498047]},{"page":396,"text":"“coercive field”. We see that, upon application of the electric field, the spontaneous","rect":[53.81282424926758,272.1689147949219,385.1557656680222,263.234375]},{"page":396,"text":"polarization of the ferroelectric can be switched between two states \u0007 Ps","rect":[53.81282424926758,284.0716857910156,385.1724078689846,275.13714599609377]},{"page":396,"text":"corresponding to the two equivalent minima in the free energy. Therefore, the","rect":[53.812843322753909,296.031982421875,385.11292303277818,287.097412109375]},{"page":396,"text":"ferroelectric demonstrates a bistability again in agreement with experiment. It is","rect":[53.812843322753909,307.99151611328127,385.1845742617722,299.05694580078127]},{"page":396,"text":"very surprising how many experimental results on ferroelectricity can be explained","rect":[53.812843322753909,319.9510498046875,385.1377801041938,311.0164794921875]},{"page":396,"text":"using such a simple theoretical consideration!","rect":[53.812843322753909,331.9106140136719,238.39275489656104,322.97607421875]},{"page":396,"text":"13.1.2 Ferroelectric Cells with Non-ferroelectric Liquid Crystal","rect":[53.812843322753909,387.4059753417969,376.5247060949665,376.7681884765625]},{"page":396,"text":"13.1.2.1 Meyer’s Discovery","rect":[53.812843322753909,415.3467102050781,176.07842341474066,405.69500732421877]},{"page":396,"text":"Year 1975 has been marked off by an outstanding publication of R. Meyer and his","rect":[53.812843322753909,438.81060791015627,385.1616250430222,429.8162841796875]},{"page":396,"text":"French co-workers [4]. As has been discussed in Section 4.9, chirality of molecules","rect":[53.812843322753909,450.7701721191406,385.16559232817846,441.83563232421877]},{"page":396,"text":"removes the mirror symmetry of any phase. The idea of Meyer was to apply this","rect":[53.81280517578125,462.7297058105469,385.24228300200658,453.795166015625]},{"page":396,"text":"principle to the SmC phase by making it chiral. He believed that if chiral molecules","rect":[53.81280517578125,474.6892395019531,385.1626016055222,465.75469970703127]},{"page":396,"text":"formed a tilted smectic phase, its point group symmetry would reduce from C2h to C2","rect":[53.81280517578125,486.6487731933594,385.18130140630105,477.7142333984375]},{"page":396,"text":"and the new phase would belong to pyroelectric class with a specific polar axis [5].","rect":[53.812843322753909,498.6091613769531,385.26812406088598,489.6148376464844]},{"page":396,"text":"The chemists from Orsay have synthesised chiral compound p-decyloxybenzy-","rect":[65.7648696899414,510.5686950683594,385.15862403718605,501.6142272949219]},{"page":396,"text":"lidene-p0-amino-2methylbutylcinnamate (DOBAMBC), Fig. 13.3. Indeed, in the","rect":[53.81385040283203,522.5284423828125,385.17395818902818,512.8688354492188]},{"page":396,"text":"temperature range 95–117\bC, this substance showed a linear electro-optical effect","rect":[53.81322479248047,534.431396484375,385.15486009189677,525.4368896484375]},{"page":396,"text":"characteristic of a pyroelectric phase. The effect was observed in thick home-","rect":[53.81303787231445,546.3909301757813,385.1290219864048,537.4564208984375]},{"page":396,"text":"otropically oriented layers. Due to chiral structure of DOBAMBC molecules, the","rect":[53.81303787231445,558.3504638671875,385.1718219585594,549.39599609375]},{"page":396,"text":"SmC* phase had a spiral structure with the helical axis perpendicular to the limiting","rect":[53.81303787231445,570.31005859375,385.1200799088813,561.3755493164063]},{"page":396,"text":"glasses, Fig. 13.4a. Under a microscope the preparation showed a conoscopic cross","rect":[53.81303787231445,582.2695922851563,385.13995756255346,573.3350830078125]},{"page":396,"text":"typical of a uniaxial phase, and, upon application of the in-plane electric field Ex,","rect":[53.81203842163086,594.2291259765625,385.1832241585422,585.2946166992188]},{"page":397,"text":"13.1 Ferroelectrics","rect":[53.812843322753909,42.55594253540039,117.63046725272455,36.68026351928711]},{"page":397,"text":"C10H21O","rect":[104.17041015625,94.17250061035156,130.82183853429692,86.925048828125]},{"page":397,"text":"CH N","rect":[165.98190307617188,91.57061767578125,187.71998643953806,86.2867431640625]},{"page":397,"text":"p","rect":[278.8263244628906,66.96891784667969,283.09972288866666,61.70603561401367]},{"page":397,"text":"e","rect":[283.0997314453125,70.75443267822266,286.9884540717143,66.84226989746094]},{"page":397,"text":"O","rect":[262.3443908691406,77.20697021484375,267.7858049039258,71.923095703125]},{"page":397,"text":"CH CH C O CH CH C2H5","rect":[220.91441345214845,94.60890197753906,334.7948404153428,86.61016845703125]},{"page":397,"text":"*","rect":[303.38153076171877,98.13230895996094,308.0455610060683,94.68907165527344]},{"page":397,"text":"DOBAMBC","rect":[118.44107818603516,119.72282409667969,160.17390714462833,113.68411254882813]},{"page":397,"text":"Fig. 13.3 Chemical formula of DOBAMBC molecule which is chiral due to asymmetric","rect":[53.812843322753909,142.2867431640625,359.6129393317652,134.35372924804688]},{"page":397,"text":"C* and has a dipole moment pe. Below is a sequence of transition temperatures between","rect":[53.812843322753909,152.13821411132813,359.91632599024697,144.54379272460938]},{"page":397,"text":"(Cr), SmC*, SmA and isotropic (I) phases","rect":[53.81339645385742,162.1141357421875,197.85890658377924,154.519775390625]},{"page":397,"text":"387","rect":[372.4981994628906,42.55594253540039,385.1898245254032,36.73106384277344]},{"page":397,"text":"carbon","rect":[362.0776672363281,141.0,385.1510061660282,134.62466430664063]},{"page":397,"text":"crystal","rect":[362.52825927734377,152.13815307617188,385.15319542136259,144.54379272460938]},{"page":397,"text":"Fig. 13.4 Meyer’s experiment. Geometry of the cell with helical structure of DOBAMBC (a) and","rect":[53.812843322753909,341.7464904785156,385.1940054336063,333.8134765625]},{"page":397,"text":"two conoscopic images (b), one in the absence of the field (top) and the other at Ex ¼ 672 V/mm","rect":[53.81200408935547,351.5980224609375,385.16015861864079,344.0036315917969]},{"page":397,"text":"showing a shift of the conoscopic cross perpendicular to Ex","rect":[53.81283187866211,361.5739440917969,257.00330042283579,353.9795837402344]},{"page":397,"text":"the cross moved perpendicular to the field along the y-axis, Fig. 13.4b. The","rect":[53.812843322753909,402.3652038574219,385.14768255426255,393.4306640625]},{"page":397,"text":"direction of the cross shift changed with a change of the field polarity. The effect","rect":[53.81385040283203,414.3247375488281,385.1526933438499,405.39019775390627]},{"page":397,"text":"was clearly related to the pyroelectric nature of the SmC* phase and existence of","rect":[53.81385040283203,426.2842712402344,385.14974342195168,417.3497314453125]},{"page":397,"text":"the spontaneous polarisation interacting with the external field. Therefore, in 1975","rect":[53.81385040283203,438.2438049316406,385.15276423505318,429.2494812011719]},{"page":397,"text":"for the first time, a polar liquid phase (of course, anisotropic) with finite spontane-","rect":[53.81385040283203,450.203369140625,385.15667091218605,441.2489013671875]},{"page":397,"text":"ous polarisation was reported.","rect":[53.81385040283203,462.1629333496094,174.56438870932346,453.2283935546875]},{"page":397,"text":"The value of Ps in DOBAMBC was very small, about 18 CGSQ/cm2 (or","rect":[65.765869140625,474.1230773925781,385.12471900788918,463.0720520019531]},{"page":397,"text":"60 mC/m2). It is 2,500 times less than Ps in BaTiO3 (150 mC/m2). However,","rect":[53.813777923583987,485.9929504394531,385.1556057503391,474.9750061035156]},{"page":397,"text":"nowadays there are SmC* materials with Ps \u0004 5 mC/m2. The magnitude of Ps","rect":[53.813777923583987,497.9856872558594,385.1724078689846,486.9347229003906]},{"page":397,"text":"depends on the molecular structure. A molecule should have a large transverse","rect":[53.812843322753909,509.9453430175781,385.17063177301255,501.01080322265627]},{"page":397,"text":"dipole moment located close to the chiral moiety; otherwise the intra-molecular","rect":[53.812843322753909,521.9048461914063,385.1069272598423,512.9703369140625]},{"page":397,"text":"rotation of chiral moiety with respect to the dipolar part would prevent, at least,","rect":[53.812843322753909,533.8643798828125,385.1795925667453,524.9298706054688]},{"page":397,"text":"partial orientation of dipoles along the polar axis. For the same reason, a rotation of","rect":[53.812843322753909,545.8239135742188,385.1487973770298,536.889404296875]},{"page":397,"text":"molecules about their long molecular axes should be hindered. In DOBAMBC, the","rect":[53.812843322753909,557.783447265625,385.1706012554344,548.8289794921875]},{"page":397,"text":"smallness of Ps is explained by rather free rotation of the chiral tail about the","rect":[53.812843322753909,569.7435302734375,385.1731647319969,560.8085327148438]},{"page":397,"text":"–O–CH- bridge connecting the asymmetric carbon C* with the –C¼O dipolar","rect":[53.813411712646487,581.703125,385.09746681062355,572.7486572265625]},{"page":397,"text":"group shown by an arrow in Fig. 13.3.","rect":[53.81338882446289,593.6058349609375,208.79049344565159,584.6713256835938]},{"page":398,"text":"388","rect":[53.812591552734378,42.55716323852539,66.50419754176065,36.73228454589844]},{"page":398,"text":"13 Ferroelectricity and Antiferroelectricity in Smectics","rect":[197.23788452148438,44.275840759277347,385.18014986991207,36.68148422241211]},{"page":398,"text":"13.1.2.2 Goldstone Mode and Helicity of the Structure","rect":[53.812843322753909,68.68677520751953,292.6312946148094,59.035072326660159]},{"page":398,"text":"As we discussed in Section 4.9, on a large scale, the SmC* phase acquires a helical","rect":[53.812843322753909,92.20748138427735,385.2054277188499,83.27293395996094]},{"page":398,"text":"structure. This is not surprising, because, in the achiral SmC phase, the c-director of","rect":[53.812843322753909,104.11019134521485,385.1506589492954,95.17564392089844]},{"page":398,"text":"the infinite stack of layers may look over any direction. The rotation of the SmC*","rect":[53.8138313293457,116.0697250366211,385.1358269791938,107.13517761230469]},{"page":398,"text":"director about the smectic layer normal does not cost any energy. We meet the same","rect":[53.8138313293457,128.02932739257813,385.12683904840318,119.09477233886719]},{"page":398,"text":"situation in nematics: when there is no external field and limiting surfaces, the","rect":[53.8138313293457,139.98886108398438,385.17261541559068,131.05430603027345]},{"page":398,"text":"direction of the director n is not fixed and one can rotate it without spending any","rect":[53.8138313293457,151.94839477539063,385.14470759442818,143.0138397216797]},{"page":398,"text":"energy. It is an elastic mode, the so-called Goldstone mode, trying to restore the","rect":[53.81382369995117,163.907958984375,385.17456854059068,154.9435272216797]},{"page":398,"text":"continuous symmetry D1h ! Kh of the phase existing above the corresponding","rect":[53.813812255859378,175.86868286132813,385.1164483170844,166.9329376220703]},{"page":398,"text":"transition (isotropic phase in case of nematics) [6]. Due to this non-energy demand-","rect":[53.813472747802737,187.82821655273438,385.0975278457798,178.89366149902345]},{"page":398,"text":"ing rotation, any small amount of a chiral additive (considered as a perturbation)","rect":[53.813472747802737,199.73098754882813,385.1154721817173,190.7964324951172]},{"page":398,"text":"would easily twist the nematic structure into the cholesteric.","rect":[53.813472747802737,211.6905517578125,296.5459255745578,202.75599670410157]},{"page":398,"text":"The Goldstone mode in an achiral SmC tries to restore the symmetry of the","rect":[65.76549530029297,223.65008544921876,385.17322576715318,214.6956024169922]},{"page":398,"text":"smectic A phase C1h ! D1h by free rotation of the director along the conical","rect":[53.813472747802737,235.61026000976563,385.1554399258811,226.6750946044922]},{"page":398,"text":"surface with the smectic layer normal as a rotation axis. Thus, like chiral molecules","rect":[53.81362533569336,247.56979370117188,385.16241850005346,238.63523864746095]},{"page":398,"text":"convert a nematic into a cholesteric, they convert an achiral SmC into chiral SmC*","rect":[53.81362533569336,259.52935791015627,385.1355828385688,250.5948028564453]},{"page":398,"text":"without any phase transition. In addition, mixing left (L)- and right (R)-handed","rect":[53.81362533569336,271.4888916015625,385.17641535810005,262.5543212890625]},{"page":398,"text":"additives results in a partial or complete compensation of the helical pitch both in","rect":[53.81362533569336,283.44842529296877,385.14055720380318,274.51385498046877]},{"page":398,"text":"cholesterics and chiral smectic C*. For example, the L- and R- isomers of the same","rect":[53.81362533569336,295.3511962890625,385.1276019878563,286.4166259765625]},{"page":398,"text":"molecule taken in the equal amounts would give us a racemic mixture, that is","rect":[53.81362533569336,307.3107604980469,385.1863137637253,298.376220703125]},{"page":398,"text":"achiral SmC without helicity and polarity.","rect":[53.81362533569336,319.270263671875,223.73098416830784,310.335693359375]},{"page":398,"text":"What has been said above shows that the macroscopic helicity as such has no","rect":[65.76564025878906,331.2298278808594,385.1684502702094,322.2952880859375]},{"page":398,"text":"direct relation to the polarity of the SmC* phase. One can select chiral molecules","rect":[53.81362533569336,343.1893615722656,385.1605264102097,334.2348937988281]},{"page":398,"text":"without dipoles and construct a helical SmC* that will have a polar axis without","rect":[53.81362533569336,355.14892578125,385.12263352939677,346.21435546875]},{"page":398,"text":"polarisation. But, is it possible to have finite polarisation without helical structure in","rect":[53.81362533569336,367.10845947265627,385.1415642838813,358.17388916015627]},{"page":398,"text":"the bulk? Can we make a uniform polar phase with infinite helical pitch? The answer","rect":[53.81362533569336,379.0679931640625,385.1883481582798,370.1334228515625]},{"page":398,"text":"is “Yes”. To this effect we should prepare a mixture of left-handedand right-handed","rect":[53.81362533569336,390.9707336425781,385.1754082780219,382.03619384765627]},{"page":398,"text":"molecules of different chemical structure. An example is shown in Fig. 13.5 [7].","rect":[53.81362533569336,402.9302978515625,385.1563992073703,393.93597412109377]},{"page":398,"text":"In this case, R-DOBAMBC is mixed with L-HOBACPC (p-hexyloxybenzylidene-","rect":[53.814613342285159,414.8898620605469,385.1594174942173,405.9353942871094]},{"page":398,"text":"p0-amino-chloropropyl-cinnamate). The sample is rather thick, d ¼ 200 mm. With a","rect":[53.814613342285159,426.8512268066406,385.1604389019188,417.191650390625]},{"page":398,"text":"Fig. 13.5 Phase diagram for","rect":[53.812843322753909,459.2458801269531,153.12127774817638,451.3128662109375]},{"page":398,"text":"the mixture of two chemically","rect":[53.812843322753909,469.1541442871094,155.32283538966105,461.5597839355469]},{"page":398,"text":"different left- and right-","rect":[53.812843322753909,479.13006591796877,135.4841775284498,471.53570556640627]},{"page":398,"text":"handed compounds. Note that","rect":[53.812843322753909,489.0492858886719,155.34399131980005,481.4549255371094]},{"page":398,"text":"at a certain concentration of","rect":[53.812843322753909,497.2726745605469,149.73768704993419,491.4308776855469]},{"page":398,"text":"HOBACPC (c \u0004 15%) the","rect":[53.812843322753909,508.6625671386719,145.885354492898,501.3560485839844]},{"page":398,"text":"spontaneous polarization Ps","rect":[53.812843322753909,518.9771118164063,148.70093443037957,511.3827819824219]},{"page":398,"text":"vanishes in the helical","rect":[53.812843322753909,527.14306640625,129.82540611472192,521.3012084960938]},{"page":398,"text":"structure (wavevector","rect":[53.812843322753909,538.5328979492188,127.78205197913339,531.3280029296875]},{"page":398,"text":"q0 \u0004 0.8 mm\u00021); on the","rect":[53.812843322753909,548.7710571289063,134.499612305398,539.4141845703125]},{"page":398,"text":"contrary, at c ¼ 50–70%, the","rect":[53.81315612792969,558.7664184570313,154.1555266120386,551.1212768554688]},{"page":398,"text":"helicity is compensated","rect":[53.81315994262695,568.742431640625,134.0765585341923,561.1480712890625]},{"page":398,"text":"(q0 ¼ 0) but Ps remains finite","rect":[53.81315994262695,578.6253051757813,155.31241748117925,571.1235961914063]},{"page":398,"text":"(\u00045 nC/cm2)","rect":[53.81343078613281,588.3552856445313,98.37790006262948,579.2608642578125]},{"page":398,"text":"–1","rect":[207.14601135253907,463.60174560546877,213.8123930931379,459.3846435546875]},{"page":398,"text":"m","rect":[200.4874267578125,466.6006164550781,207.14581513970902,462.29754638671877]},{"page":398,"text":"1.0","rect":[203.1804656982422,488.2760009765625,214.29110196435313,482.50921630859377]},{"page":398,"text":"0.5","rect":[203.1804656982422,521.1593627929688,214.29110196435313,515.392578125]},{"page":398,"text":"no Ps","rect":[228.63760375976563,530.3702392578125,249.2920073509504,522.5499877929688]},{"page":398,"text":"0","rect":[216.71945190429688,562.9425048828125,221.1637063344703,557.1757202148438]},{"page":398,"text":"Right","rect":[187.08682250976563,570.4744873046875,205.7430927912586,562.9800415039063]},{"page":398,"text":"DOBAmBC","rect":[187.08682250976563,578.4647827148438,227.50076566513614,572.4260864257813]},{"page":398,"text":"no helicity","rect":[276.4695129394531,482.1322937011719,316.0033974965797,474.6298828125]},{"page":398,"text":"P","rect":[314.45947265625,496.9389343261719,319.7909793197854,491.1961669921875]},{"page":398,"text":"s","rect":[319.7899475097656,499.023681640625,322.7874212711236,495.7123718261719]},{"page":398,"text":"40","rect":[267.45819091796877,562.9425048828125,276.3466897329078,557.1757202148438]},{"page":399,"text":"13.1 Ferroelectrics","rect":[53.812843322753909,42.55594253540039,117.63046725272455,36.68026351928711]},{"page":399,"text":"389","rect":[372.4981994628906,42.62367248535156,385.1898245254032,36.73106384277344]},{"page":399,"text":"change in composition we observe two magic points: in one of them the helical pitch","rect":[53.812843322753909,68.2883529663086,385.12981501630318,59.35380554199219]},{"page":399,"text":"is compensated and the helical vector q ¼ 0 (no helicity), in the other the spontane-","rect":[53.812843322753909,80.24788665771485,385.1566403946079,71.31333923339844]},{"page":399,"text":"ous polarisation vanishes (no polarity). The two points do not degenerate into one","rect":[53.81385040283203,92.20748138427735,385.1417926616844,83.27293395996094]},{"page":399,"text":"because molecular interactions responsible for polarity and chirality are different.","rect":[53.81385040283203,104.11019134521485,385.1676296761203,95.17564392089844]},{"page":399,"text":"This result is very interesting from the practical point of view: it shows how to get","rect":[53.81385040283203,116.0697250366211,385.17060716220927,107.13517761230469]},{"page":399,"text":"rid of the undesirable helicalstructure inliquid crystals electroopticalcells. Itisalso","rect":[53.81385040283203,128.02932739257813,385.1527031998969,119.09477233886719]},{"page":399,"text":"significant from the theoretical point of view because it proves a possibility to have a","rect":[53.81385040283203,139.98886108398438,385.15769231988755,131.05430603027345]},{"page":399,"text":"uniform polar liquid crystalline structure without any limiting substrates.","rect":[53.81385040283203,151.94839477539063,346.0954250862766,143.0138397216797]},{"page":399,"text":"Such a polar phase would manifest the same Goldstone mode as an achiral SmC","rect":[65.765869140625,163.907958984375,385.14277908023146,154.97340393066407]},{"page":399,"text":"does. The origin of the Goldstone mode in SmC* is not a helical structure, as often","rect":[53.81385040283203,175.86749267578126,385.13674250653755,166.9329376220703]},{"page":399,"text":"stated in literature. On the contrary, the helicity originates from the Goldstone mode","rect":[53.81385040283203,187.8270263671875,385.1547321148094,178.87254333496095]},{"page":399,"text":"due to its gapless nature (the absence of any energy gap for the c-director rotation","rect":[53.81385040283203,199.72976684570313,385.11879817060005,190.7952117919922]},{"page":399,"text":"from one orientational state to the other). What is true that the same mode in the","rect":[53.8138313293457,211.29090881347657,385.17258489801255,202.75477600097657]},{"page":399,"text":"SmC* phase is much better seen in the low frequency dielectric spectrum due to the","rect":[53.8138313293457,223.64886474609376,385.17163885309068,214.7143096923828]},{"page":399,"text":"coupling of the director to the spontaneous electric polarisation of the chiral polar","rect":[53.8138313293457,235.60842895507813,385.13680396882668,226.6738739013672]},{"page":399,"text":"phase. Particularly, in the helical structure, the Goldstone mode has a characteristic","rect":[53.8138313293457,247.56796264648438,385.1815875835594,238.63340759277345]},{"page":399,"text":"(hydrodynamic) dependence of the relaxation frequency on the wavevector of the","rect":[53.8138313293457,259.52752685546877,385.1696246929344,250.5929718017578]},{"page":399,"text":"helix, o \u0004 Kq2, exactly like in cholesterics.","rect":[53.8138313293457,271.4891662597656,230.66766019369846,260.4381408691406]},{"page":399,"text":"13.1.2.3 Smectic C* Phase and Criteria for Ferroelectricity","rect":[53.813209533691409,307.7662048339844,313.43007746747505,298.114501953125]},{"page":399,"text":"Is the SmC* phase ferroelectric? To answer this question we should look more","rect":[53.813209533691409,331.2301330566406,385.13709295465318,322.29559326171877]},{"page":399,"text":"carefully at the criteria formulated for crystalline ferroelectrics:","rect":[53.813209533691409,343.1896667480469,309.9789262540061,334.255126953125]},{"page":399,"text":"1.","rect":[53.813209533691409,360.0,61.27886624838595,352.2825622558594]},{"page":399,"text":"2.","rect":[53.81318664550781,383.0,61.27884336020236,376.1448669433594]},{"page":399,"text":"3.","rect":[53.81399154663086,406.91790771484377,61.279648261325409,400.0650939941406]},{"page":399,"text":"4.","rect":[53.81399154663086,443.0,61.279648261325409,435.9437255859375]},{"page":399,"text":"All crystalline ferroelectrics without exceptions belong to one of the pyroelectric","rect":[66.27488708496094,361.1573486328125,385.17795599176255,352.2227783203125]},{"page":399,"text":"classes and possess spontaneous polarisation (polar class).","rect":[66.27488708496094,373.11688232421877,302.8098110726047,364.16241455078127]},{"page":399,"text":"Sometimes, a formation of domains with different direction of Ps is also taken as","rect":[66.27486419677735,384.6328430175781,385.1408730898972,376.065185546875]},{"page":399,"text":"a pre-requisite of the ferroelectricity.","rect":[66.2756576538086,396.9803161621094,215.18368192221409,388.0457763671875]},{"page":399,"text":"There is a distinct phase transition between the ferroelectric and the paraelectric","rect":[66.27566528320313,408.93988037109377,385.1498187847313,399.98541259765627]},{"page":399,"text":"phase (there would be no exception from this rule if we consider even melting to","rect":[66.2756576538086,420.8993835449219,385.14189997724068,411.96484375]},{"page":399,"text":"the liquid phase as such a transition).","rect":[66.2756576538086,432.85894775390627,216.0427212288547,423.92437744140627]},{"page":399,"text":"There are two equivalent stable states (bistability) differed by the spontaneous","rect":[66.27566528320313,444.8185119628906,385.15878690825658,435.8640441894531]},{"page":399,"text":"polarisation direction, between which we can switch the direction of Ps. It seems","rect":[66.27666473388672,456.7780456542969,385.11053861724096,447.843505859375]},{"page":399,"text":"there is no exception from this criterion among the crystalline ferroelectrics.","rect":[66.27620697021485,468.7383117675781,374.16900296713598,459.80377197265627]},{"page":399,"text":"Aswe haveseen, locally thesmectic C* layersare polar,belongingtopyroelectric","rect":[65.76653289794922,486.6492614746094,385.1593707866844,477.7147216796875]},{"page":399,"text":"class C2. Macroscopically SmC* either forms a helical structure or does not. So, we","rect":[53.81450653076172,498.6091613769531,385.2224811382469,489.67425537109377]},{"page":399,"text":"can discuss a structure without helicity. In a sense, the formation of a helix is","rect":[53.81296157836914,510.5686950683594,385.24142850981908,501.6341552734375]},{"page":399,"text":"equivalent to formation of ferroelectric domains which would reduce overall macro-","rect":[53.81296157836914,522.5281982421875,385.2623227676548,513.5936889648438]},{"page":399,"text":"scopic polarisation. Thus we can consider the (1) (very important) and (2) (additional)","rect":[53.81296157836914,534.4309692382813,385.15483985749855,525.4964599609375]},{"page":399,"text":"requirements fulfilled. As to the phase transition (3), we know that in the smectic A*","rect":[53.81296157836914,546.3905029296875,385.24346247724068,537.4559936523438]},{"page":399,"text":"phase, even chiral, there is no polar axis, therefore that phase can be considered as a","rect":[53.81296157836914,558.35009765625,385.15479314996568,549.4155883789063]},{"page":399,"text":"paraelectric phase. The two-component order parameter of the A*–C* transition is","rect":[53.81296157836914,570.3096313476563,385.2434426699753,561.3751220703125]},{"page":399,"text":"the same as the parameter of the A–C transition in an achiral substance, namely Wexp","rect":[53.81296157836914,582.2691650390625,385.17864314130318,573.0358276367188]},{"page":399,"text":"(ij), where we recognise the tilt W and azimuth j angles. The spontaneous","rect":[53.81296920776367,594.2286987304688,385.16571439849096,584.995361328125]},{"page":400,"text":"390","rect":[53.812843322753909,42.62367248535156,66.50444931178018,36.73106384277344]},{"page":400,"text":"13 Ferroelectricity and Antiferroelectricity in Smectics","rect":[197.23812866210938,44.274620056152347,385.18039401053707,36.68026351928711]},{"page":400,"text":"polarisation is not considered as an order parameter, but this is not a limitation for, at","rect":[53.812843322753909,68.2883529663086,385.2392411954124,59.35380554199219]},{"page":400,"text":"least, improper ferroelectricity. We just deal with a transition to improper pyroelectric","rect":[53.812843322753909,80.24788665771485,385.29212225152818,71.31333923339844]},{"page":400,"text":"phase. Thus, the third criterion for ferroelectricity is also fulfilled.","rect":[53.812843322753909,92.20748138427735,313.8636135628391,83.27293395996094]},{"page":400,"text":"The last criterion (4) isbistability. In the non-helical structure the direction of the","rect":[65.76486206054688,104.11019134521485,385.1726764507469,95.17564392089844]},{"page":400,"text":"polar axis is fixed in the sense that the three vectors, the polar axis Ps, the directorn","rect":[53.812843322753909,116.0697250366211,385.1868626529987,107.13517761230469]},{"page":400,"text":"and the smectic normal h form either right or left vector triple. This depends on","rect":[53.81417465209961,128.02999877929688,385.1699151139594,119.09544372558594]},{"page":400,"text":"molecular handedness and cannot be changed. In this sense there is no bistability.","rect":[53.81417465209961,139.98953247070313,385.1569790413547,131.0549774169922]},{"page":400,"text":"On the other hand, the Goldstone mode allows the thresholdless rotation of Ps","rect":[53.81417465209961,151.64414978027345,385.1724078689846,142.9945831298828]},{"page":400,"text":"together with n about h through any angle by an infinitesimally low electric field.","rect":[53.812843322753909,163.908935546875,385.1129116585422,154.97438049316407]},{"page":400,"text":"So, a number of possible states is infinite.","rect":[53.812843322753909,175.86846923828126,222.09969754721409,166.9339141845703]},{"page":400,"text":"The situation can be well modelled by a magnetic arrow, placed in a viscous","rect":[65.76486206054688,187.8280029296875,385.11688627349096,178.89344787597657]},{"page":400,"text":"liquid: by realignment of an external magnetic field the arrow will follow the field","rect":[53.812843322753909,199.73077392578126,385.11489192060005,190.7962188720703]},{"page":400,"text":"and eventually it takes the field direction (the time depends on liquid viscosity). Our","rect":[53.812843322753909,211.69033813476563,385.15667091218605,202.73585510253907]},{"page":400,"text":"case can also be modelled by a pyroelectric crystal installed on the needle like the","rect":[53.812843322753909,223.64987182617188,385.1715778179344,214.71531677246095]},{"page":400,"text":"magnetic arrow. Now the arrow is not magnetic but electric and follows an electric","rect":[53.812843322753909,235.60940551757813,385.1467975444969,226.6748504638672]},{"page":400,"text":"field E. If the electrodes are fixed in the smectic layer plane, we can switch the","rect":[53.812843322753909,247.56893920898438,385.17063177301255,238.63438415527345]},{"page":400,"text":"polarisation between two angular states, controlled by the positive and negative","rect":[53.81283950805664,259.52850341796877,385.17160833551255,250.5939483642578]},{"page":400,"text":"field. Since the polarisation direction is rigidly coupled to the director (and the","rect":[53.81283950805664,271.488037109375,385.17261541559068,262.553466796875]},{"page":400,"text":"optical axis) we would observe a linear electrooptical effect. The switching is faster","rect":[53.81283950805664,283.44757080078127,385.1347592910923,274.51300048828127]},{"page":400,"text":"if we have stronger E, higher Ps and lower viscosity and this is in agreement with","rect":[53.81283950805664,295.3517761230469,385.11589900067818,286.415771484375]},{"page":400,"text":"experiment. However, the two field controlled states are not intrinsically stable","rect":[53.813899993896487,307.31134033203127,385.1716693706688,298.37677001953127]},{"page":400,"text":"states and, in the absence of the field, they can easily be destroyed by thermal","rect":[53.813899993896487,319.2708740234375,385.1806779629905,310.3363037109375]},{"page":400,"text":"fluctuations or even by very weak chirality.","rect":[53.813899993896487,331.2304382324219,229.1612972786594,322.2958984375]},{"page":400,"text":"Therefore, in conclusion, we may say that the bulk smectic C* phase is, in","rect":[65.76591491699219,343.1899719238281,385.1437615495063,334.2355041503906]},{"page":400,"text":"principle, a liquid pyroelectric, which, due to its fluidity, allows a thresholdless","rect":[53.813899993896487,355.1495056152344,385.09601225005346,346.1950378417969]},{"page":400,"text":"realignment of its polarisation (and the director) by an external field. Strictly","rect":[53.81388473510742,367.1090393066406,385.09597102216255,358.17449951171877]},{"page":400,"text":"speaking, it is not a ferroelectric in both the uniform and helical states. It may be","rect":[53.81388473510742,379.0685729980469,385.14978826715318,370.134033203125]},{"page":400,"text":"called a helielectric [8] to distinguish it from the conventional pyroelectric, how-","rect":[53.81388473510742,390.9713439941406,385.1507505020298,382.0168762207031]},{"page":400,"text":"ever, this does not change anything. But why a large class of smectic C* materials is","rect":[53.81386947631836,402.930908203125,385.1855813418503,393.996337890625]},{"page":400,"text":"called ferroelectric and under this name is widely used in modern technology?","rect":[53.81386947631836,414.89044189453127,370.2373431987938,405.95587158203127]},{"page":400,"text":"13.1.2.4 Surface Stabilised Ferroelectric Cells","rect":[53.81386947631836,449.3347473144531,255.73108305083469,441.5157775878906]},{"page":400,"text":"We can answer the last question if consider a construction of the so-called “surface","rect":[53.81386947631836,474.6881408691406,385.1388324566063,465.75360107421877]},{"page":400,"text":"stabilised ferroelectric liquid crystal cell” or simply SSFLC cell [9]. Such SSFLC","rect":[53.81386947631836,486.647705078125,385.12681838687208,477.713134765625]},{"page":400,"text":"cell is only few micrometers thin and, due to anchoring of the director at the","rect":[53.81388473510742,498.6072692871094,385.1178668804344,489.6727294921875]},{"page":400,"text":"surfaces, the intrinsic helical structure of the SmC* is unwound by boundaries","rect":[53.81388473510742,510.5668029785156,385.15866483794408,501.63226318359377]},{"page":400,"text":"but a high value of the spontaneous polarisation is conserved. The cell is con-","rect":[53.81388473510742,522.5263061523438,385.11797462312355,513.591796875]},{"page":400,"text":"structed in a way to realise two stable states of the smectic C* liquid crystal using its","rect":[53.81388473510742,534.4290771484375,385.1507607852097,525.4945678710938]},{"page":400,"text":"interaction with the surfaces of electrodes, see Fig. 13.6a. First of all, in the SSFLC","rect":[53.81388473510742,546.388671875,385.12584182437208,537.4541625976563]},{"page":400,"text":"cell, the so-called bookshelf geometry is assumed: the smectic layers are vertical","rect":[53.81386947631836,558.3482055664063,385.180555892678,549.4136962890625]},{"page":400,"text":"(like books) with their normal hs parallel the z-axis. Then the director is free to","rect":[53.81386947631836,570.310302734375,385.14251032880318,561.3732299804688]},{"page":400,"text":"rotate along the conical surface about the hs axis as shown in Fig. 13.6b (Goldstone","rect":[53.81362533569336,582.2699584960938,385.16663397027818,573.3353881835938]},{"page":400,"text":"mode). It is important that, to have a bistability, the director should be properly","rect":[53.81388473510742,594.2294921875,385.1706475358344,585.2949829101563]},{"page":401,"text":"13.1 Ferroelectrics","rect":[53.812843322753909,42.55649185180664,117.63046725272455,36.68081283569336]},{"page":401,"text":"391","rect":[372.4981994628906,42.62422180175781,385.1898245254032,36.73161315917969]},{"page":401,"text":"Fig. 13.6 SSFLC cell. The structure of the cell with bookshelf alignment of smectic layers (a) and","rect":[53.812843322753909,171.08062744140626,385.19403595118447,163.14761352539063]},{"page":401,"text":"the cone of the director n motion with two stable states \u0007W in the electrode plane yz (b). Note that","rect":[53.81287384033203,180.93212890625,385.1533785268313,173.08377075195313]},{"page":401,"text":"in sketch (a) the director in the cell plane yz is turned to the reader through angle þW (shown by","rect":[53.813682556152347,190.9080810546875,385.2075552382938,183.05972290039063]},{"page":401,"text":"thicker right parts of the rod-like molecules) in agreement with sketch (b). The double-head arrow","rect":[53.812843322753909,200.9178924560547,385.179628250259,193.25579833984376]},{"page":401,"text":"shows the optimum angular position of polarizerP","rect":[53.812843322753909,210.8599853515625,227.28681310221703,203.265625]},{"page":401,"text":"anchored at the electrodes. In the ideal case, the zenithal anchoring energy Wz","rect":[53.812843322753909,235.61026000976563,385.1820833351628,226.6757049560547]},{"page":401,"text":"should be relatively strong, but the azimuthal one Wa should be zero. Thus, in the","rect":[53.812843322753909,247.56985473632813,385.17380560113755,238.6352996826172]},{"page":401,"text":"absence of the external field, to have minimum Wz the director n must be located","rect":[53.81406784057617,259.08837890625,385.1740655045844,250.59486389160157]},{"page":401,"text":"parallel to electrodes that is in plane yz intersecting the surface of the cone.","rect":[53.81338119506836,271.48907470703127,385.1143459847141,262.55450439453127]},{"page":401,"text":"Therefore, there are only two stable positions for n, either angle þW or \u0002W.","rect":[53.8133659362793,283.4486083984375,385.18304105307348,274.2152404785156]},{"page":401,"text":"Consequently, the spontaneous polarization Ps will be directed along the \u0007x-axis,","rect":[53.814369201660159,295.4084777832031,385.1243862679172,286.4736328125]},{"page":401,"text":"i.e. either up or down in sketch (a).","rect":[53.81344223022461,307.3680114746094,195.7933315804172,298.4334716796875]},{"page":401,"text":"In experiments, very often spatial domains appear in the initial field-off state of","rect":[65.76546478271485,319.32757568359377,385.1513608535923,310.39300537109377]},{"page":401,"text":"approximately equal total area with the director oriented at þW or \u0002W. Under a","rect":[53.81344223022461,331.2303466796875,385.16123235895005,321.9969787597656]},{"page":401,"text":"microscope, with a polariser P oriented, say, along the þW direction, as shown in","rect":[53.81344223022461,343.18988037109377,385.14434138349068,333.9565124511719]},{"page":401,"text":"Fig. 13.6b, and analyser A⊥P, the þW domains look black and \u0002W domains bright.","rect":[53.81344223022461,355.1494445800781,385.1194119026828,345.91607666015627]},{"page":401,"text":"When a sufficiently strong, square-wave electric field \u0007 Ex is applied to electrodes,","rect":[53.81444549560547,367.10968017578127,385.1520046761203,358.17437744140627]},{"page":401,"text":"Ps is switched along the x-axis and, since the director is rigidly fixed to both the Ps","rect":[53.81417465209961,379.0693359375,385.1724078689846,370.134765625]},{"page":401,"text":"and the conical surface, its projection on the yz-plane will oscillate between","rect":[53.812843322753909,391.0289611816406,385.1755303483344,382.09442138671877]},{"page":401,"text":"\u0007W positions through total angle 2W. Usually it sticks in one of the two stable","rect":[53.812862396240237,402.9884948730469,385.17359197809068,393.755126953125]},{"page":401,"text":"positions (memory states) as soon as the field is switched off. This process results in","rect":[53.812870025634769,414.94805908203127,385.19753352216255,406.01348876953127]},{"page":401,"text":"a true bistable switching of Ps like in solid state ferroelectrics and, due to director","rect":[53.812870025634769,426.890625,385.1071714004673,417.8963317871094]},{"page":401,"text":"switching, a fast electrooptical effect with a good contrast is observed.","rect":[53.8131217956543,438.810791015625,338.4627651741672,429.876220703125]},{"page":401,"text":"SSFLC cells are very convenient for measurements of the magnitude of Ps.","rect":[65.76514434814453,450.7703552246094,385.1832241585422,441.8358154296875]},{"page":401,"text":"Indeed, upon switching the polarization by external voltage, a change of the surface","rect":[53.814537048339847,462.7301025390625,385.08274114801255,453.7955322265625]},{"page":401,"text":"charge at the electrodes of area A creates an electric current i ¼ dQ/dt ¼ A dP/dt.","rect":[53.814537048339847,474.6896667480469,385.1832241585422,465.6754150390625]},{"page":401,"text":"Therefore, applying a step voltage of sufficient amplitude to switch the polarization","rect":[53.814537048339847,486.6492004394531,385.0817498307563,477.71466064453127]},{"page":401,"text":"from \u0002Ps (at t ¼ \u00021) to þ Ps (at t ¼ þ1) and measuring the time dependence of","rect":[53.814537048339847,498.6091613769531,385.1498960098423,489.6741943359375]},{"page":401,"text":"the current i(t) we find Ps by integrating the area under the i(t) function.","rect":[53.81309127807617,510.5688171386719,344.5595364144016,501.6341552734375]},{"page":401,"text":"1","rect":[227.37367248535157,523.00732421875,234.3415998951785,519.8628540039063]},{"page":401,"text":"2Ps ¼ A\u00021 ð iðtÞdt","rect":[179.45103454589845,547.043701171875,259.7966752774436,524.9215087890625]},{"page":401,"text":"\u00021","rect":[224.14474487304688,553.0482177734375,236.47798295181912,549.9037475585938]},{"page":401,"text":"Unfortunately, the ideal bookshelf structure is difficult to make. Usually the","rect":[65.76624298095703,571.6137084960938,385.1720355816063,562.67919921875]},{"page":401,"text":"electrodes are covered by polymer layers and rubbed unidirectionally. This pro-","rect":[53.81421661376953,583.5732421875,385.17302833406105,574.6387329101563]},{"page":401,"text":"vides a good alignment of the director along the electrodes and the “bookshelf”","rect":[53.81421661376953,595.5328369140625,385.1182330913719,586.5983276367188]},{"page":402,"text":"392","rect":[53.812843322753909,42.62367248535156,66.50444931178018,36.73106384277344]},{"page":402,"text":"13 Ferroelectricity and Antiferroelectricity in Smectics","rect":[197.23812866210938,44.274620056152347,385.18039401053707,36.68026351928711]},{"page":402,"text":"structure of the SmA* phase. Upon cooling the cell from the SmA* to SmC* phase","rect":[53.812843322753909,68.2883529663086,385.09693182184068,59.35380554199219]},{"page":402,"text":"the director keeps its in-plane direction. Although the rubbing is appropriate for","rect":[53.812843322753909,80.24788665771485,385.1776059707798,71.31333923339844]},{"page":402,"text":"strong zenithal anchoring it is incompatible with the requirement of zero azimuthal","rect":[53.812843322753909,92.20748138427735,385.1417680508811,83.27293395996094]},{"page":402,"text":"anchoring of the director necessary for the genuine bistability. Moreover, upon the","rect":[53.812843322753909,104.11019134521485,385.16858709527818,95.17564392089844]},{"page":402,"text":"passage of the SmA* \u0002 SmC* transition, the smectic layers become tilted or","rect":[53.812843322753909,116.0697250366211,385.1486753067173,107.13517761230469]},{"page":402,"text":"acquire a chevron structure. The chevrons and accompanying zigzag defects have","rect":[53.81285095214844,128.02932739257813,385.11792791559068,119.09477233886719]},{"page":402,"text":"been discussed in Section 8.5.4, see Figs. 8.33 and 8.34. These factors reduce the","rect":[53.81285095214844,139.98886108398438,385.17456854059068,130.99453735351563]},{"page":402,"text":"performance of the SSFLC cells. Nevertheless, the principle of the bistable switch-","rect":[53.81285095214844,151.94839477539063,385.09893165437355,143.0138397216797]},{"page":402,"text":"ing of the pyroelectric SmC* phase is realised in the SSFLC cells and such cells can","rect":[53.81285095214844,163.907958984375,385.12383357099068,154.97340393066407]},{"page":402,"text":"be considered as genuine ferroelectric cells.","rect":[53.81285095214844,175.90733337402345,231.44773526694065,166.89309692382813]},{"page":402,"text":"13.1.3 Phase Transition SmA*–SmC*","rect":[53.812843322753909,226.482666015625,249.74366512202807,217.94854736328126]},{"page":402,"text":"13.1.3.1 Simplification","rect":[53.812843322753909,256.05889892578127,155.43040818522526,246.87533569335938]},{"page":402,"text":"Due to low symmetry (C2) of the chiral smectic C* phase, its theoretical description","rect":[53.812843322753909,280.04791259765627,385.1556328873969,271.1131591796875]},{"page":402,"text":"is very complicated. Even description of the achiral smectic C phase is not at all","rect":[53.81376266479492,292.0074768066406,385.14362962314677,283.07293701171877]},{"page":402,"text":"simple. In the chiral SmC* phase two new aspects are very important, the spatially","rect":[53.81376266479492,303.91021728515627,385.15956965497505,294.97564697265627]},{"page":402,"text":"modulated (helical) structure and the presence of spontaneous polarisation. The","rect":[53.81376266479492,315.8697814941406,385.1466449566063,306.93524169921877]},{"page":402,"text":"strict theory of the SmA*–SmC* transition developed by Pikin [10] is based on","rect":[53.81376266479492,327.8293151855469,385.1705254655219,318.894775390625]},{"page":402,"text":"consideration of the two-component order parameter, represented by the c-director","rect":[53.81276321411133,339.78887939453127,385.1724790176548,330.85430908203127]},{"page":402,"text":"whose projections (x1, x2) ¼ (nznx, nzny) are combinations of the director compo-","rect":[53.81276321411133,352.6656494140625,385.1224301895298,342.4950866699219]},{"page":402,"text":"nents nx ¼ sinWcosj, ny ¼ sinWsinj;and nz ¼ cosW:","rect":[53.81345748901367,364.5260925292969,275.3173661954124,354.4754638671875]},{"page":402,"text":"x1 ¼ nznx ¼ 21sin2Wcosj; x2 ¼ nzny ¼ 21sin2Wsinj","rect":[95.33529663085938,389.68072509765627,315.0627925079658,377.9393310546875]},{"page":402,"text":"(13.9a)","rect":[356.6385803222656,388.0799865722656,385.1374753555454,379.6036376953125]},{"page":402,"text":"or","rect":[53.81456756591797,411.1295166015625,62.10642371980322,406.4879455566406]},{"page":402,"text":"c ¼ 12sin2Wðicosj þ jsinjÞ:","rect":[159.45762634277345,439.1630554199219,279.5685266224756,427.4216613769531]},{"page":402,"text":"(13.9b)","rect":[356.07061767578127,437.5055847167969,385.1588071426548,428.969482421875]},{"page":402,"text":"Here, W is the tilt angle of the director with respect to the smectic layer normal","rect":[65.76506042480469,462.7304992675781,385.181776595803,453.49713134765627]},{"page":402,"text":"(and the helical axis z) and j is the azymuthal angle counted from the x-axis. The","rect":[53.813045501708987,474.6900329589844,385.1468890972313,465.7554931640625]},{"page":402,"text":"free energy of the SmC* includes both the helicityand polarization. Then, assuming","rect":[53.81403732299805,486.6495666503906,385.16787043622505,477.71502685546877]},{"page":402,"text":"constant orientational order parameter Q, a linear relationship between the tilt and","rect":[53.81403732299805,498.609130859375,385.1448906998969,489.5948791503906]},{"page":402,"text":"polarisation and leaving only the lowest order terms in x1, x2 and gradients ∂x1/∂z,","rect":[53.81406784057617,510.5688171386719,385.18291898276098,500.6282653808594]},{"page":402,"text":"∂x2/∂z, one has fifteen terms in the equation for the free energy [11].","rect":[53.81426239013672,522.5284423828125,331.51116605307348,512.5877685546875]},{"page":402,"text":"However, many interesting effects in ferroelectric cells may be described with-","rect":[65.76458740234375,534.43115234375,385.1464780410923,525.4966430664063]},{"page":402,"text":"out account of the helicity, in the approximation of a uniform SmC* structure (e.g.,","rect":[53.81256866455078,546.3907470703125,385.11162992026098,537.4562377929688]},{"page":402,"text":"unwound by limiting surfaces or formed by mixtures with compensated helicity).","rect":[53.81256866455078,558.3502807617188,385.1643948128391,549.415771484375]},{"page":402,"text":"So, in this paragraph, we ignore all the space dependent terms i.e. consider a SmC*","rect":[53.81256866455078,570.309814453125,385.1375359635688,561.3753051757813]},{"page":402,"text":"structure with azimuthal angle j ! 0. Going back to Fig. 13.5 this approximation","rect":[53.81256866455078,582.2693481445313,385.1016472916938,573.2750854492188]},{"page":402,"text":"may correspond to a ferroelectric mixture with q0 \u0004 0. Then the free energy is:","rect":[53.812564849853519,594.2296142578125,374.67439134189677,585.2943725585938]},{"page":403,"text":"13.1 Ferroelectrics","rect":[53.812843322753909,42.55594253540039,117.63046725272455,36.68026351928711]},{"page":403,"text":"393","rect":[372.4981994628906,42.62367248535156,385.1898245254032,36.73106384277344]},{"page":403,"text":"gSmC ¼ g0 þ 1AW2 þ 1BW4 þ P2 \u0002 mPW \u0002 PE","rect":[122.18362426757813,76.71692657470703,316.80390160955155,59.44980239868164]},{"page":403,"text":"2","rect":[178.6021728515625,81.39361572265625,183.57927790692816,74.66033172607422]},{"page":403,"text":"(13.10)","rect":[356.0702819824219,76.2786636352539,385.1584714492954,67.80229949951172]},{"page":403,"text":"where W [in rad] is the tilt of the director, A ¼ a(T \u0002 T0) [in erg/cm3K] describes","rect":[53.81269454956055,109.2685775756836,385.15643705474096,98.21757507324219]},{"page":403,"text":"the elasticity for W-changes, P [in CGSQ/cm2 or statV/cm] is polarization, m [in","rect":[53.813594818115237,121.2282943725586,385.1735772233344,110.12071228027344]},{"page":403,"text":"statV/cm] is the polarization-tilt coupling constant (or piezocoefficient), E is an","rect":[53.813838958740237,133.1709747314453,385.1546563248969,124.1766586303711]},{"page":403,"text":"external electric field [statV/cm] applied perpendicular to the tilt plane, g0 and","rect":[53.81482696533203,145.1304473876953,385.14626399091255,136.1560516357422]},{"page":403,"text":"w? ¼ ðe? \u0002 1Þ=4p are the background energy of the SmA* phase and dielectric","rect":[53.814414978027347,157.38922119140626,385.18042791559068,147.43870544433595]},{"page":403,"text":"*","rect":[258.8683166503906,160.72137451171876,262.6010706934104,157.87962341308595]},{"page":403,"text":"susceptibility of the SmA* phase well above the A –C* transition.","rect":[53.814674377441409,169.0101318359375,322.2784695198703,160.01589965820313]},{"page":403,"text":"Equation (13.10) is principally different from the equation (13.4) for the free","rect":[65.76543426513672,180.96975708007813,385.13730657770005,172.0352020263672]},{"page":403,"text":"energy of a solid ferroelectric. Here, the leading term of the expansion is related to","rect":[53.81342697143555,192.9293212890625,385.19606867841255,183.99476623535157]},{"page":403,"text":"the tilt angle, but the appearance of the spontaneous polarisation (the secondary","rect":[53.81342697143555,204.88885498046876,385.08562556317818,195.9542999267578]},{"page":403,"text":"effect) is taken into account by coupling term mPW. Term P2/2w⊥ describes the","rect":[53.81342697143555,216.84841918945313,385.1759418316063,205.74122619628907]},{"page":403,"text":"energy of the polarised dielectric. For a racemic phase, with spontaneous polariza-","rect":[53.81417465209961,228.75164794921876,385.1629880508579,219.8170928955078]},{"page":403,"text":"tion Ps ¼ 0 and without coupling of the tilt to total polarisation P we would put","rect":[53.81417465209961,240.71142578125,385.1357255704124,231.77662658691407]},{"page":403,"text":"total polarization P ¼ 0 in Eq. (13.10) because there is no additional contribution to","rect":[53.81283950805664,252.67095947265626,385.1407097916938,243.7364044189453]},{"page":403,"text":"the field energy in the SmC phase above the background (SmA) term g0. Therefore,","rect":[53.81283950805664,264.67034912109377,385.1521267464328,255.6561279296875]},{"page":403,"text":"for the achiral SmC phase, minimisation of Eq. (13.10) with respect to the tilt angle","rect":[53.814292907714847,276.59027099609377,385.1421588726219,267.65570068359377]},{"page":403,"text":"would provide the result obtained in Section 6.4.","rect":[53.814292907714847,288.5498352050781,250.53736539389377,279.61529541015627]},{"page":403,"text":"qqWg ¼ AW þ BW3and Ws ¼ ða=BÞ1=2ðT0 \u0002 TÞ1=2","rect":[128.6982879638672,325.5408935546875,311.4634253748353,304.7663269042969]},{"page":403,"text":"13.1.3.2 Soft Mode for Smectic A*–Smectic C* Transition","rect":[53.812843322753909,379.19671630859377,308.1469456608112,371.6865234375]},{"page":403,"text":"Low field limit. Minimising Eq. (13.10) with respect to polarization P we find the","rect":[53.812843322753909,404.8021545410156,385.1735309429344,395.6484680175781]},{"page":403,"text":"relation between the tilt and polarization:","rect":[53.81181716918945,416.7616882324219,220.28303392002176,407.8271484375]},{"page":403,"text":"P ¼ w?mW þ w?E","rect":[184.32147216796876,442.84759521484377,254.67078393377029,433.43182373046877]},{"page":403,"text":"(13.11)","rect":[356.0713195800781,442.2667541503906,385.15950904695168,433.7904052734375]},{"page":403,"text":"For discussion of the soft mode close to the phase transition we can assume small","rect":[65.76575469970703,468.5683898925781,385.1376786954124,459.63385009765627]},{"page":403,"text":"W angles, and a weak field E << mW. Then, substituting P \u0004 w?mW in Eq. (13.10)","rect":[53.8137321472168,480.71038818359377,385.1590818008579,471.2945556640625]},{"page":403,"text":"and ignoring term BW4 we exclude P from the free energy difference between the","rect":[53.81325912475586,492.4876403808594,385.17270696832505,481.4366149902344]},{"page":403,"text":"SmC* and SmA* phases:","rect":[53.8129997253418,504.4471740722656,156.39909226963114,495.51263427734377]},{"page":403,"text":"Dg ¼ 12\u0002A \u0002 w?m2\u0003W2 \u0002 w?mEW","rect":[155.1508331298828,540.9617919921875,283.79275864909246,520.62548828125]},{"page":403,"text":"(13.12)","rect":[356.0712585449219,535.8468627929688,385.1594480117954,527.3704833984375]},{"page":403,"text":"Note that the fourth and fifth terms in expansion (13.10) merge with the second","rect":[65.76570892333985,566.5131225585938,385.1794060807563,557.57861328125]},{"page":403,"text":"term and this results in renormalisation of the transition temperature.","rect":[53.81370162963867,578.47265625,331.6556667854953,569.5381469726563]},{"page":404,"text":"394","rect":[53.812843322753909,42.62367248535156,66.50444931178018,36.73106384277344]},{"page":404,"text":"13 Ferroelectricity and Antiferroelectricity in Smectics","rect":[197.23812866210938,44.274620056152347,385.18039401053707,36.68026351928711]},{"page":404,"text":"Now we minimise the free energy with respect to the tilt angle","rect":[65.76496887207031,68.2883529663086,318.8289264507469,59.35380554199219]},{"page":404,"text":"qqDWg ¼ \u0002A \u0002 w?m2\u0003W \u0002 w?mE ¼0","rect":[152.48878479003907,105.27920532226563,288.1465691666938,84.4747543334961]},{"page":404,"text":"and arrive at the expression for the tilt angle linearly dependent on the field:","rect":[53.814491271972659,130.75094604492188,361.46235520908427,121.81639099121094]},{"page":404,"text":"W ¼ A w\u0002?wm?Em2 ¼ aðwT?\u0002mETcÞ ¼ ecE","rect":[150.56443786621095,167.70513916015626,288.4313124982234,145.047607421875]},{"page":404,"text":"(13.13)","rect":[356.0711669921875,160.1096649169922,385.15935645906105,151.63330078125]},{"page":404,"text":"where","rect":[53.813594818115237,189.1673583984375,78.1516345806297,182.2946319580078]},{"page":404,"text":"Tc ¼ T0 þ w?m2\u0004a","rect":[182.00091552734376,217.7394561767578,256.9909142594672,205.4595947265625]},{"page":404,"text":"We see, that the phase transition temperature increases, because the dipole–","rect":[65.76554107666016,241.278076171875,385.11562434247505,232.34352111816407]},{"page":404,"text":"dipole interactions (P2-term) stabilise the smectic C* phase. Note that the field","rect":[53.81351852416992,253.23794555664063,385.11711970380318,242.1869354248047]},{"page":404,"text":"induced tilt angle (or electroclinic coefficient ec) diverges at a temperature Tc. This","rect":[53.81308364868164,265.1976318359375,385.1889993106003,256.2231140136719]},{"page":404,"text":"means, that at Tc even infinitesimally low field would create a finite tilt. This is the","rect":[53.814353942871097,277.15716552734377,385.17307317926255,268.22259521484377]},{"page":404,"text":"soft mode of the director motion: any small force (not necessary electric) would","rect":[53.81327438354492,289.1167297363281,385.13625422528755,280.142333984375]},{"page":404,"text":"cause the tilt of the director, because, at the transition, the medium becomes soft","rect":[53.81327438354492,301.05633544921877,385.1451249844749,292.10186767578127]},{"page":404,"text":"with respect to the tilt. The corresponding dielectric susceptibility shows the","rect":[53.81326675415039,312.9790344238281,385.1153034038719,304.0245666503906]},{"page":404,"text":"Curie–Weiss law:","rect":[53.81326675415039,322.9166259765625,125.43879563877175,316.0040283203125]},{"page":404,"text":"P","rect":[202.22315979003907,347.5105285644531,208.3350447980281,340.9366149902344]},{"page":404,"text":"wsm ¼ E ¼ aðT \u0002 TcÞ","rect":[175.82598876953126,363.5940856933594,261.51095975981908,349.7431640625]},{"page":404,"text":"(13.14)","rect":[356.07098388671877,355.9981994628906,385.1591733535923,347.5218505859375]},{"page":404,"text":"In reality, a growth of the induced tilt at the phase transition is limited by two","rect":[65.76541900634766,387.1745300292969,385.15129939130318,378.239990234375]},{"page":404,"text":"factors.","rect":[53.81338119506836,398.0,83.52869077231174,390.19952392578127]},{"page":404,"text":"In","rect":[88.65013122558594,398.0,96.94198695233831,390.3987121582031]},{"page":404,"text":"a","rect":[102.13111877441406,398.0,106.58065069391096,392.0]},{"page":404,"text":"strictly","rect":[111.76081848144531,399.1340637207031,139.5231103043891,390.19952392578127]},{"page":404,"text":"compensated","rect":[144.72816467285157,399.1340637207031,196.9141015153266,390.19952392578127]},{"page":404,"text":"non-helical","rect":[202.1659393310547,398.0,247.12913377353739,390.19952392578127]},{"page":404,"text":"ferroelectrics","rect":[252.40985107421876,398.0,305.2298928652878,390.19952392578127]},{"page":404,"text":"only","rect":[310.4130859375,399.1340637207031,328.14153376630318,390.19952392578127]},{"page":404,"text":"BW4","rect":[333.35650634765627,397.0722351074219,348.70180921880105,388.08331298828127]},{"page":404,"text":"term","rect":[353.86224365234377,397.0725402832031,372.1978831160124,391.2157897949219]},{"page":404,"text":"in","rect":[377.3701171875,396.99285888671877,385.14434138349068,390.1998291015625]},{"page":404,"text":"expansion (13.10) is limiting. In the most practical cases, the helix cannot be","rect":[53.814476013183597,411.09393310546877,385.15235174371568,402.15936279296877]},{"page":404,"text":"precisely compensated over the whole range of the smectic C* phase and a finite","rect":[53.814476013183597,422.9966735839844,385.1393512554344,414.0621337890625]},{"page":404,"text":"wavevector qo ¼ 2p/P0 remains. Thus, in a more advanced theory, the space","rect":[53.814476013183597,434.9566345214844,385.0814594097313,426.0220947265625]},{"page":404,"text":"dependent, chiral terms must be added to expansion (13.10). They renormalize","rect":[53.81327438354492,446.9161682128906,385.1102985210594,437.98162841796877]},{"page":404,"text":"the transition temperature for the second time, and put a limit for the divergence of","rect":[53.813289642333987,458.875732421875,385.14620338288918,449.941162109375]},{"page":404,"text":"the induced tilt:","rect":[53.813289642333987,468.81329345703127,117.30622727939675,461.90069580078127]},{"page":404,"text":"W ¼ aðT \u0002 wT?chmÞEþ Kjq20 with Tch ¼ T0 þ w?am2 þ Kjaq02","rect":[95.16407775878906,510.1540222167969,312.59651253303846,485.0660705566406]},{"page":404,"text":"(13.15)","rect":[356.0714416503906,501.9512634277344,385.15963111726418,493.3553771972656]},{"page":404,"text":"Here, Kj is an effective elastic modulus for the azimuthal motion of the director","rect":[65.76587677001953,534.5711669921875,385.10637794343605,524.760009765625]},{"page":404,"text":"in the SmC* phase that includes factor sin2W [11]. Due to this factor, in the one-","rect":[53.81332015991211,545.6541137695313,385.1157163223423,534.6033325195313]},{"page":404,"text":"constant approximation, which will be used below, Kj \u0004 10\u00027 dyn is roughly one","rect":[53.812740325927737,558.490478515625,385.14450872613755,546.6558227539063]},{"page":404,"text":"order of magnitude smaller than <Kii> for nematics. The third term in the equation","rect":[53.81362533569336,569.573486328125,385.1772088151313,560.6388549804688]},{"page":404,"text":"for Tch determins the difference in the transition temperatures for a helical and","rect":[53.81350326538086,581.5331420898438,385.1432732682563,572.5985107421875]},{"page":404,"text":"unwound ferroelectric.","rect":[53.81333923339844,591.4039916992188,145.1671108772922,584.5014038085938]},{"page":405,"text":"13.1 Ferroelectrics","rect":[53.81254196166992,42.55728530883789,117.63016970633783,36.68160629272461]},{"page":405,"text":"395","rect":[372.4978942871094,42.62501525878906,385.18951934962197,36.63080978393555]},{"page":405,"text":"With the chiral term in the expansion, the Curie type temperature dependence of","rect":[65.76496887207031,68.2883529663086,385.1489194473423,59.35380554199219]},{"page":405,"text":"the low field soft-mode susceptibility in the smectic A phase becomes somewhat","rect":[53.812950134277347,80.24788665771485,385.172743392678,71.31333923339844]},{"page":405,"text":"smoothed:","rect":[53.812950134277347,90.18550109863281,95.46037156650613,83.27293395996094]},{"page":405,"text":"22","rect":[225.3343963623047,109.11090087890625,240.2040412195619,104.39761352539063]},{"page":405,"text":"w?m","rect":[220.3496856689453,115.51568603515625,236.72023362467838,108.14680480957031]},{"page":405,"text":"wsm ¼ aðT \u0002 TchÞ þ Kjq20 :","rect":[165.97003173828126,129.42898559570313,272.99830567520999,115.02794647216797]},{"page":405,"text":"(13.16)","rect":[356.0711669921875,121.2833023071289,385.15935645906105,112.74717712402344]},{"page":405,"text":"At a low field with ignored term BW4 in Eq. (13.10), the dynamics of the director","rect":[65.76558685302735,151.04232788085938,385.10796485749855,139.9347381591797]},{"page":405,"text":"soft mode can be investigated using the Landau–Khalatnikov equation, see Sec-","rect":[53.812923431396487,162.94509887695313,385.10799537507668,154.0105438232422]},{"page":405,"text":"tion 6.5.1. The corresponding equation describing the balance of the viscous and","rect":[53.812923431396487,174.9046630859375,385.1398858170844,165.91033935546876]},{"page":405,"text":"elastic torques reads:","rect":[53.812923431396487,186.86419677734376,138.32914598056864,177.9296417236328]},{"page":405,"text":"qW","rect":[184.83311462402345,206.284912109375,195.30495042155338,199.11337280273438]},{"page":405,"text":"gW qt ¼ \u0002aðT \u0002 TchÞW","rect":[173.8437957763672,219.987548828125,265.15616196940496,205.5365447998047]},{"page":405,"text":"(13.17)","rect":[356.0709533691406,214.74998474121095,385.15914283601418,206.27362060546876]},{"page":405,"text":"Here, gW is the rotational viscosity for the W-angle change. From this equation is","rect":[65.7654037475586,241.50497436523438,385.18759550200658,232.2716064453125]},{"page":405,"text":"clear, that the inverse relaxation time of the soft mode (tsm)\u00021 diverges at Tch (on","rect":[53.81389617919922,253.464599609375,385.14443293622505,242.41371154785157]},{"page":405,"text":"the SmA* side of the transition):","rect":[53.814537048339847,265.02569580078127,186.04527146884989,256.48956298828127]},{"page":405,"text":"tsmðSmA","rect":[168.57763671875,291.60943603515627,203.23920434392654,281.6589050292969]},{"page":405,"text":"Þ¼","rect":[208.2839813232422,291.60943603515627,222.51950863096625,281.6589050292969]},{"page":405,"text":"gW","rect":[242.38461303710938,284.7638854980469,251.21572647221667,277.71282958984377]},{"page":405,"text":"aðT \u0002 TchÞ","rect":[225.3343963623047,298.46783447265627,268.7613564883347,288.5173034667969]},{"page":405,"text":"(13.18)","rect":[356.07073974609377,290.8719482421875,385.1589292129673,282.3955993652344]},{"page":405,"text":"Indeed, the Curie-type behaviour is in agreement with the experimental data [12]","rect":[65.76517486572266,320.06414794921877,385.1579831680454,311.12957763671877]},{"page":405,"text":"obtained by the pyroelectric technique on a SmC* mixture with Ps \u0004 600 statV/cm","rect":[53.812129974365237,331.9668884277344,385.0916819441374,323.0323486328125]},{"page":405,"text":"or 2 mC/m2, see Fig. 13.7. However, the maximum time at Tch is limited by the","rect":[53.81456756591797,343.92724609375,385.1756061382469,332.87628173828127]},{"page":405,"text":"value of tsm \u0004 13ms. To account for this saturation the fourth order term BW4 in the","rect":[53.81386947631836,355.7972106933594,385.1754535503563,344.8358154296875]},{"page":405,"text":"free energy has to be taken into account.","rect":[53.813716888427737,367.846435546875,217.5485043099094,358.911865234375]},{"page":405,"text":"3","rect":[203.5019073486328,397.1465148925781,207.39062997503457,392.1005859375]},{"page":405,"text":"15","rect":[297.5343322753906,392.3370666503906,306.29496243798595,386.5702819824219]},{"page":405,"text":"2","rect":[203.5019073486328,422.2669677734375,207.39062997503457,417.3470153808594]},{"page":405,"text":"1","rect":[203.5019073486328,447.41259765625,207.39062997503457,442.4926452636719]},{"page":405,"text":"10","rect":[297.5343322753906,431.3095397949219,306.29496243798595,425.5427551269531]},{"page":405,"text":"–5","rect":[138.38755798339845,479.6620178222656,146.04610177679238,474.7210693359375]},{"page":405,"text":"20","rect":[119.5212631225586,518.55322265625,128.28187802636485,512.7864379882813]},{"page":405,"text":"0","rect":[160.09725952148438,479.6620178222656,163.98598214788613,474.6160888671875]},{"page":405,"text":"5","rect":[179.8982696533203,479.6341857910156,183.8318234369075,474.63629150390627]},{"page":405,"text":"T, °C","rect":[158.9453582763672,489.0,174.7947087295771,483.324951171875]},{"page":405,"text":"30","rect":[162.29481506347657,518.55322265625,171.0554299672828,512.7864379882813]},{"page":405,"text":"0","rect":[203.5019073486328,472.5575256347656,207.39062997503457,467.5115966796875]},{"page":405,"text":"10","rect":[197.85963439941407,479.6620178222656,205.518178192808,474.6160888671875]},{"page":405,"text":"40","rect":[205.06756591796876,518.55322265625,213.828180821775,512.7864379882813]},{"page":405,"text":"T (°C)","rect":[198.82362365722657,531.88037109375,220.06892222742963,524.3300170898438]},{"page":405,"text":"Tch","rect":[245.164794921875,496.4400329589844,256.2713316917707,488.61370849609377]},{"page":405,"text":"50","rect":[247.84112548828126,518.55322265625,256.60172513329845,512.7864379882813]},{"page":405,"text":"5","rect":[297.5343322753906,470.2828369140625,301.97858670556408,464.6360168457031]},{"page":405,"text":"0","rect":[297.5343322753906,509.25531005859377,301.97858670556408,503.488525390625]},{"page":405,"text":"60","rect":[290.6138610839844,518.55322265625,299.37446072900158,512.7864379882813]},{"page":405,"text":"Fig. 13.7 Experimental temperature dependence of the soft-mode relaxation time (main plot), and","rect":[53.812843322753909,551.8623657226563,385.1956839004032,543.9293823242188]},{"page":405,"text":"demonstration of the Curie type behaviour of the inverse relaxation time on both sides of the phase","rect":[53.812843322753909,561.7705688476563,385.1550535895777,554.1762084960938]},{"page":405,"text":"transition (inset) in accordance with Eqs. (13.18) and (13.19) depicted by solid lines. Experimental","rect":[53.812843322753909,571.7465209960938,385.15002159323759,564.13525390625]},{"page":405,"text":"parameters: chiral mixture with Ps \u0004 2 mC/m2, a ¼ 5\u0006104 J m\u00023 K\u00021, Tch ¼ 49\bC, cell thickness","rect":[53.81113052368164,581.6657104492188,385.1620224285058,572.2890625]},{"page":405,"text":"10 mm, the rotational viscosity found is gW ¼ 0.36 Pa\u0006s or 3.6 P","rect":[53.81301498413086,591.641357421875,271.2984097819045,584.0468139648438]},{"page":406,"text":"396","rect":[53.812843322753909,42.62367248535156,66.50444931178018,36.68026351928711]},{"page":406,"text":"13 Ferroelectricity and Antiferroelectricity in Smectics","rect":[197.23812866210938,44.274620056152347,385.18039401053707,36.68026351928711]},{"page":406,"text":"Strong field limit. For a very strong field, the helical structure is always","rect":[65.76496887207031,68.36803436279297,385.08115018950658,59.134674072265628]},{"page":406,"text":"unwound, the BW4 term in expansion (13.10) dominates over AW2 term, and the","rect":[53.81296157836914,80.24800872802735,385.17420232965318,69.19712829589844]},{"page":406,"text":"temperature dependencies of the induced tilt and susceptibility disappear:","rect":[53.81444549560547,92.20760345458985,350.3344865567405,83.27305603027344]},{"page":406,"text":"W ¼ \u0005w?BmE\u00061=3and","rect":[117.2566146850586,133.44786071777345,194.93190852216254,107.17605590820313]},{"page":406,"text":"w ¼ qqEP ¼ 31\u0005w4?Bm4\u00061=3E\u00022=3","rect":[207.66134643554688,133.4481658935547,321.2065589197572,106.49588012695313]},{"page":406,"text":"In fact, a strong field smears the soft mode behaviour and the phase transition","rect":[65.76496887207031,156.93719482421876,385.1538018327094,148.0026397705078]},{"page":406,"text":"vanishes. All these results are in agreement with experiments on dielectric and","rect":[53.812950134277347,168.89675903320313,385.1429070573188,159.9622039794922]},{"page":406,"text":"electrooptical properties owed to the soft mode in the chiral SmA* phase.","rect":[53.812950134277347,180.85629272460938,351.5892300179172,171.92173767089845]},{"page":406,"text":"It is worth to mention that, in the higher symmetry SmA* phase, the soft mode is","rect":[65.76496887207031,192.81585693359376,385.1847268496628,183.8813018798828]},{"page":406,"text":"the only one elastic mode for the director. It is related to the short-range elastic","rect":[53.812950134277347,204.775390625,385.09507024957505,195.84083557128907]},{"page":406,"text":"excitations of the director tilt i.e. to the amplitude of the two-component order","rect":[53.812950134277347,216.73489379882813,385.1458676895298,207.8003387451172]},{"page":406,"text":"parameter of the less symmetric phase SmC*. Therefore, the soft mode may be","rect":[53.812950134277347,228.6944580078125,385.1488422222313,219.75990295410157]},{"page":406,"text":"called the amplitude mode and the corresponding excitation amplitudons [6, 13].","rect":[53.812950134277347,240.59722900390626,385.1547512581516,231.6626739501953]},{"page":406,"text":"On the transition to the SmC* phase the continuous symmetry group D1 is broken","rect":[53.81294250488281,252.5567626953125,385.17949763349068,243.60227966308595]},{"page":406,"text":"and the reduced symmetry C2 of the SmC* phase allows the Goldstone mode. The","rect":[53.81380844116211,264.5174560546875,385.14646185113755,255.5826873779297]},{"page":406,"text":"latter is related to the long-range excitations of the director azimuth i.e. the phase of","rect":[53.813602447509769,276.47698974609377,385.1474851211704,267.54241943359377]},{"page":406,"text":"the SmC* order parameter. Such excitations may be called phasons. Note that in the","rect":[53.813602447509769,288.4365539550781,385.17237127496568,279.50201416015627]},{"page":406,"text":"SmC* phase the soft and Goldstone modes coexist and have very different relaxa-","rect":[53.813602447509769,300.3960876464844,385.1555112442173,291.4615478515625]},{"page":406,"text":"tion times.","rect":[53.813602447509769,310.2937927246094,96.38676114584689,303.42108154296877]},{"page":406,"text":"13.1.3.3 Goldstone and Soft Modes in Sm C* Phase","rect":[53.813602447509769,358.3142395019531,279.4845660991844,350.88372802734377]},{"page":406,"text":"In the SmC* phase the tilt is W ¼ Ws þ dW where Ws and dW are spontaneous and the","rect":[53.813602447509769,384.0572814941406,385.17444647027818,374.82275390625]},{"page":406,"text":"field induced tilt. In the absence of the field, Ws is constant and minimisation of","rect":[53.81368637084961,395.57232666015627,385.15096412507668,386.783447265625]},{"page":406,"text":"Eq. (13.10) with respect to P relates the spontaneous polarization to the tilt angle:","rect":[53.81411361694336,407.9764404296875,384.2013383633811,399.0418701171875]},{"page":406,"text":"Ps ¼ w?mWs","rect":[195.7104034423828,432.0215759277344,242.80148750200213,422.6058349609375]},{"page":406,"text":"Then we exclude P ¼ w?mW from Eq. (13.10) and minimize that equation with","rect":[65.76496887207031,455.94085693359377,385.1173028092719,446.5251159667969]},{"page":406,"text":"respect to W:","rect":[53.814292907714847,467.7180480957031,103.85310990879128,458.48468017578127]},{"page":406,"text":"\u0002A \u0002 w?m2 þ BW2\u0003W ¼ w?mE","rect":[161.26876831054688,494.4551696777344,277.72553246892655,481.94854736328127]},{"page":406,"text":"In the low field limit, expanding (Ws + dW)2 we shall find the soft mode suscepti-","rect":[65.76554107666016,517.9942626953125,385.0969174942173,506.9430847167969]},{"page":406,"text":"bility of the SmC* phase using exactly the same procedure as for crystalline","rect":[53.813838958740237,529.9534912109375,385.0999225444969,521.0189819335938]},{"page":406,"text":"ferroelectrics, see Eq. (13.8):","rect":[53.813838958740237,541.9130859375,170.83653123447489,532.9785766601563]},{"page":406,"text":"22","rect":[226.69395446777345,560.8570556640625,241.56350777229629,556.143798828125]},{"page":406,"text":"w?m","rect":[221.70913696289063,567.2617797851563,238.07970017741276,559.892822265625]},{"page":406,"text":"wsm ¼ 2aðTch \u0002 TÞ","rect":[181.26455688476563,580.5681762695313,256.0727578555222,566.7176513671875]},{"page":407,"text":"13.1 Ferroelectrics","rect":[53.812843322753909,42.55789566040039,117.63046725272455,36.68221664428711]},{"page":407,"text":"397","rect":[372.4981994628906,42.62562561035156,385.1898245254032,36.73301696777344]},{"page":407,"text":"By analogy, instead of Eq. (13.18) for the soft-mode relaxation times on the","rect":[65.76496887207031,68.2883529663086,385.1726459331688,59.35380554199219]},{"page":407,"text":"SmC* side of the A*–C* transition we have","rect":[53.812931060791019,78.22590637207031,232.00725591852035,71.25357818603516]},{"page":407,"text":"tsmðSmC","rect":[165.8565216064453,108.47362518310547,200.98115798648144,98.52310943603516]},{"page":407,"text":"Þ¼","rect":[206.01898193359376,108.47362518310547,220.31124141905219,98.52310943603516]},{"page":407,"text":"gW","rect":[242.61123657226563,101.57152557373047,251.44231948979479,94.52037048339844]},{"page":407,"text":"2aðT \u0002 TchÞ","rect":[223.12525939941407,115.27526092529297,271.4802590762253,105.32474517822266]},{"page":407,"text":"(13.19)","rect":[356.0711669921875,107.7361831665039,385.15935645906105,99.25981903076172]},{"page":407,"text":"If we take into account the chiral terms, the low","rect":[65.76558685302735,137.0,275.4301123614582,129.9211883544922]},{"page":407,"text":"consists of two parts (the soft and Goldstone modes):","rect":[53.81356430053711,150.81527709960938,268.5557695157249,141.88072204589845]},{"page":407,"text":"field","rect":[279.994140625,137.0,297.7126397233344,129.9211883544922]},{"page":407,"text":"susceptibility","rect":[302.3124694824219,138.85574340820313,355.67205134442818,129.9211883544922]},{"page":407,"text":"would","rect":[360.2599182128906,137.0,385.1354912858344,129.9211883544922]},{"page":407,"text":"w ¼ wsm þ wG ¼ 2aðTðch1=\u00022ÞTwÞ?2þm2Kjq02 þ ð1=K2jÞwq?202 m2","rect":[116.85955047607422,190.02125549316407,319.9604651697572,165.04611206054688]},{"page":407,"text":"(13.20)","rect":[356.0715026855469,181.8750457763672,385.1596921524204,173.398681640625]},{"page":407,"text":"The Goldstone mode does not show the explicit temperature dependence (in","rect":[65.76595306396485,213.61810302734376,385.17177668622505,204.6835479736328]},{"page":407,"text":"reality, parameters Kj, q0, m depend on temperature but not critically) and the total","rect":[53.81393051147461,226.45428466796876,385.15815599033427,216.6431121826172]},{"page":407,"text":"susceptibility manifests a quasi Curie type behaviour at temperature Tch with a cusp","rect":[53.8143310546875,237.53729248046876,385.1260918717719,228.60264587402345]},{"page":407,"text":"of amplitude","rect":[53.81411361694336,249.49685668945313,104.99466741754377,240.5623016357422]},{"page":407,"text":"wmsm ¼ w?2 m2\u0004Kjq02","rect":[182.85154724121095,275.95062255859377,255.61157295784316,263.6707763671875]},{"page":407,"text":"(13.21)","rect":[356.0715026855469,274.2648620605469,385.1596921524204,265.78851318359377]},{"page":407,"text":"qualitatively depicted in Fig. 13.8.","rect":[53.81393051147461,299.4897766113281,192.64728971030002,290.55523681640627]},{"page":407,"text":"13.1.3.4 Measurements of Landau Expansion Coefficients","rect":[53.81393051147461,337.1692810058594,306.0058633242722,327.9857177734375]},{"page":407,"text":"We can use Eqs. (13.13) and (13.14) and find parameters a, w⊥ and m in the SmA*","rect":[53.81393051147461,361.1580810546875,385.17815486005318,352.1638488769531]},{"page":407,"text":"phase. For this we need slow, automatically made temperature scans through the","rect":[53.81342697143555,373.1177062988281,385.15723455621568,364.18316650390627]},{"page":407,"text":"A* ! C* phase transition with simultaneous measurements of SSFLC cell capaci-","rect":[53.81342697143555,385.0204772949219,385.15731178132668,376.0261535644531]},{"page":407,"text":"tance, i.e. wsm(T) and the electrooptical response i.e., field induced angle W(T) at","rect":[53.81342697143555,396.9804992675781,385.23994309970927,387.74713134765627]},{"page":407,"text":"frequency 0.1–1 kHz. Then the asymptotic behaviour of capacitance at temper-","rect":[53.81447982788086,408.9400634765625,385.1583188614048,400.0054931640625]},{"page":407,"text":"ature T > Tc provides us the value of dielectric constant and susceptibility","rect":[53.81447982788086,420.89971923828127,385.25493708661568,411.96514892578127]},{"page":407,"text":"w? ¼ ðe? \u0002 1Þ=4p, and the ratio wsm/ec ¼ mw⊥ gives us the coupling constant m in","rect":[53.81353759765625,433.1981201171875,385.25734797528755,423.2475891113281]},{"page":407,"text":"the vicinity of the transition. After this we can substitute the m value into any of the","rect":[53.81400680541992,444.8190002441406,385.1618121929344,435.88446044921877]},{"page":407,"text":"two Eqs. (13.13) and (13.14) and find the Landau coefficient a, e.g., from the slope","rect":[53.81405258178711,456.778564453125,385.1608356304344,447.843994140625]},{"page":407,"text":"de\u00021/dT ¼ a/w⊥. Finally,with known coefficient a wecan find coefficient B fromthe","rect":[53.81406784057617,468.7384338378906,385.1586383648094,457.6307067871094]},{"page":407,"text":"temperature dependence of the spontaneous tilt angle measured by the electrooptical","rect":[53.8118782043457,480.6412048339844,385.23139817783427,471.7066650390625]},{"page":407,"text":"switching technique Ws ¼ ða=BÞ1=2ðT0 \u0002 TÞ1=2 in the SmC* phase.","rect":[53.8118782043457,492.939697265625,317.2392849495578,480.47607421875]},{"page":407,"text":"Fig. 13.8 Qualitative","rect":[53.812843322753909,539.8460083007813,128.68570849680425,531.8960571289063]},{"page":407,"text":"temperature dependence of","rect":[53.812843322753909,549.6974487304688,146.33886808020763,542.1030883789063]},{"page":407,"text":"dielectric susceptibility in the","rect":[53.812843322753909,559.6734008789063,154.9488615729761,552.0790405273438]},{"page":407,"text":"SmC* phase with the","rect":[53.812843322753909,569.6493530273438,126.85218188547612,562.0549926757813]},{"page":407,"text":"Goldstone mode plateau and","rect":[53.812843322753909,579.568603515625,151.4205831923954,571.9742431640625]},{"page":407,"text":"the soft mode cusp","rect":[53.812843322753909,589.5445556640625,118.36826843409463,581.9501953125]},{"page":407,"text":"(1/2) max","rect":[275.1379089355469,538.9946899414063,307.14298178449237,531.524169921875]},{"page":407,"text":"Tch","rect":[320.0860900878906,568.2902221679688,331.0528021304986,560.7178344726563]},{"page":407,"text":"max","rect":[352.3365478515625,527.1998901367188,363.2653267551955,524.278564453125]},{"page":407,"text":"T","rect":[370.92279052734377,585.9744873046875,376.25429719087915,580.4796752929688]},{"page":408,"text":"398","rect":[53.812843322753909,42.62525939941406,66.50444931178018,36.73265075683594]},{"page":408,"text":"13 Ferroelectricity and Antiferroelectricity in Smectics","rect":[197.23812866210938,44.276206970214847,385.18039401053707,36.68185043334961]},{"page":408,"text":"13.1.4 Electro-Optic Effects in Ferroelectric Cells","rect":[53.812843322753909,69.85308837890625,309.4211032840983,59.298980712890628]},{"page":408,"text":"In principle, the Pikin’s free energy [11] may be used for interpretation of almost all","rect":[53.812843322753909,97.47896575927735,385.14570481845927,88.54441833496094]},{"page":408,"text":"electro-optic effects at any temperature of the SmC* phase including the phase","rect":[53.812835693359378,109.43856048583985,385.09690130426255,100.50401306152344]},{"page":408,"text":"transition domain. However, for simplicity, it is more convenient to use different","rect":[53.812835693359378,121.3980941772461,385.1716142422874,112.46354675292969]},{"page":408,"text":"variables for different effects. For discussion of the pre-transitional electroclinic","rect":[53.812835693359378,133.35763549804688,385.09299505426255,124.42308044433594]},{"page":408,"text":"effect in the SmA* phase, one takes as a variable the tilt angle W(T,E) assuming","rect":[53.812835693359378,145.31716918945313,385.1705254655219,136.08380126953126]},{"page":408,"text":"helical vector q0 ¼ 0 i.e. constant azimuthal angle j. On the contrary, when","rect":[53.81379318237305,157.27734375,385.1790703873969,148.32286071777345]},{"page":408,"text":"discussing the Deformed Helix Ferroelectric (DHF) and Clark–Lagerwall effects","rect":[53.81434631347656,169.23687744140626,385.11337675200658,160.3023223876953]},{"page":408,"text":"observed in the SmC* phase well below the A*–C* transition, one assumes a tilt","rect":[53.81434631347656,181.1396484375,385.15717942783427,172.20509338378907]},{"page":408,"text":"angle W to be constant and operates with the director projection on the plane","rect":[53.81434631347656,193.09921264648438,385.1602252788719,183.8658447265625]},{"page":408,"text":"perpendicular to the helical axis, the c-director. The latter may be represented by","rect":[53.81434631347656,205.05874633789063,385.17107478192818,196.1241912841797]},{"page":408,"text":"a single variable, the azimuthal angle j (T, E, r).","rect":[53.81433868408203,217.01828002929688,253.00306363608127,208.08372497558595]},{"page":408,"text":"13.1.4.1 Electroclinic Effect in SmA","rect":[53.81433868408203,257.10516357421877,213.41317510559885,249.594970703125]},{"page":408,"text":"This electro-optical effect is related to the soft elastic mode just discussed [14]. For","rect":[53.81433868408203,282.7673645019531,385.1511472305454,273.83282470703127]},{"page":408,"text":"observation of the electroclinic effect one should use a proper chiral material in a","rect":[53.813350677490237,294.7269287109375,385.1592181987938,285.7923583984375]},{"page":408,"text":"standard planar cell with its normal along the x-axis and homogeneous alignment of","rect":[53.813350677490237,306.68646240234377,385.15117774812355,297.75189208984377]},{"page":408,"text":"*","rect":[147.05087280273438,310.3019714355469,150.78362684575417,307.4602355957031]},{"page":408,"text":"the director in the SmA phase (e.g., by rubbing polymer layers on both transparent","rect":[53.814353942871097,318.5908203125,385.14350755283427,309.6546630859375]},{"page":408,"text":"electrodes in one direction, e.g., along the z-axis). In such a bookshelf structure the","rect":[53.81363296508789,330.55035400390627,385.1743854351219,321.61578369140627]},{"page":408,"text":"smectic layer normal is also parallel to the z-axis and the layers themselves are","rect":[53.81464385986328,342.5099182128906,385.16443670465318,333.57537841796877]},{"page":408,"text":"located in the x,y-plane, see Fig.13.9a. The electrooptical effect is observed in","rect":[53.81464385986328,354.4694519042969,385.14348689130318,345.534912109375]},{"page":408,"text":"polarised light of a laser or using a polarising microscope with crossed polarisers.","rect":[53.81463623046875,366.42901611328127,385.1465725472141,357.49444580078127]},{"page":408,"text":"As discussed in the previous paragraph, the electric voltage applied to the electro-","rect":[53.81463623046875,378.3885192871094,385.18038307038918,369.4539794921875]},{"page":408,"text":"des (along the x-axis) induces a tilt Wy of the director from the smectic normal","rect":[53.81463623046875,391.26531982421877,385.18183763095927,381.1147155761719]},{"page":408,"text":"according","rect":[53.81315231323242,402.30841064453127,93.27861109784613,393.37384033203127]},{"page":408,"text":"to","rect":[99.4681396484375,401.0,107.24237910321722,394.38983154296877]},{"page":408,"text":"Eq.","rect":[113.40303802490235,402.30841064453127,127.05025907065158,393.57305908203127]},{"page":408,"text":"(13.13).","rect":[133.2288360595703,401.90997314453127,164.74985929037815,393.4336242675781]},{"page":408,"text":"The","rect":[170.95530700683595,401.0,186.49382818414535,393.37384033203127]},{"page":408,"text":"electroclinic","rect":[192.70724487304688,401.0,242.2991412944969,393.37384033203127]},{"page":408,"text":"coefficient","rect":[248.558349609375,401.0,290.8288713223655,393.37384033203127]},{"page":408,"text":"is","rect":[296.98956298828127,401.0,303.6190530215378,393.37384033203127]},{"page":408,"text":"field","rect":[309.79168701171877,401.0,327.5917901139594,393.37384033203127]},{"page":408,"text":"independent,","rect":[333.75244140625,402.30841064453127,385.1390652229953,393.37384033203127]},{"page":408,"text":"a","rect":[73.71866607666016,436.5453186035156,79.273928473894,430.9565734863281]},{"page":408,"text":"I","rect":[103.26487731933594,442.0932312011719,105.48700453442265,436.3504638671875]},{"page":408,"text":"I0","rect":[99.09878540039063,545.8456420898438,104.65439596179023,537.9959716796875]},{"page":408,"text":"P","rect":[125.89656066894531,530.2628173828125,131.22806733248073,524.52001953125]},{"page":408,"text":"A","rect":[138.79209899902345,448.75860595703127,144.12360566255885,443.0158386230469]},{"page":408,"text":"x","rect":[152.1280517578125,459.9090576171875,156.12468343962315,455.7259521484375]},{"page":408,"text":"y","rect":[145.0756072998047,485.7331237792969,149.07223898161534,479.84637451171877]},{"page":408,"text":"LC","rect":[145.07080078125,502.99346923828127,155.28619657333926,496.9547424316406]},{"page":408,"text":"n(E)","rect":[162.10525512695313,495.7286071777344,177.20454234461713,488.21820068359377]},{"page":408,"text":"b","rect":[191.8179473876953,436.5453186035156,197.92274113717353,429.2369384765625]},{"page":408,"text":"3","rect":[195.60353088378907,460.13616943359377,200.0477853139625,454.369384765625]},{"page":408,"text":"2","rect":[195.60353088378907,472.42315673828127,200.0477853139625,466.80035400390627]},{"page":408,"text":"1","rect":[195.60353088378907,484.8541259765625,200.0477853139625,479.2313232421875]},{"page":408,"text":"0","rect":[195.60354614257813,497.42901611328127,200.04780057275156,491.6622314453125]},{"page":408,"text":"–1","rect":[191.60690307617188,509.71600341796877,200.4954171499,504.09320068359377]},{"page":408,"text":"–2","rect":[191.60690307617188,522.14697265625,200.4954171499,516.524169921875]},{"page":408,"text":"–60","rect":[195.9991912841797,541.1549072265625,210.1312814077125,535.3881225585938]},{"page":408,"text":"–40","rect":[220.75991821289063,541.1549072265625,234.89200833642344,535.3881225585938]},{"page":408,"text":"–20","rect":[245.92111206054688,541.1549072265625,259.2538705190406,535.3881225585938]},{"page":408,"text":"0","rect":[275.0933532714844,541.1549072265625,279.5376077016578,535.3881225585938]},{"page":408,"text":"20","rect":[297.6327209472656,541.1549072265625,306.5212197622047,535.3881225585938]},{"page":408,"text":"E (V/µm)","rect":[261.66455078125,554.0577392578125,293.19795054774218,546.515380859375]},{"page":408,"text":"40","rect":[322.3934326171875,541.1549072265625,331.2819619497047,535.3881225585938]},{"page":408,"text":"1.0","rect":[354.362548828125,472.33038330078127,365.47318509423595,466.5635986328125]},{"page":408,"text":"0.6","rect":[354.362548828125,496.8259582519531,365.47318509423595,491.0591735839844]},{"page":408,"text":"0.2","rect":[354.362548828125,521.5142822265625,365.47318509423595,515.7474975585938]},{"page":408,"text":"60","rect":[347.1541748046875,541.1549072265625,356.0427041372047,535.3881225585938]},{"page":408,"text":"Fig. 13.9 Electroclinic effect in SmA* phase. The geometry of a bookshelf cell placed between","rect":[53.812843322753909,574.0244750976563,385.1238150039188,565.9898681640625]},{"page":408,"text":"polarizer (P) and analyser (A); W is field controlled tilt angle of the director (a). Typical linear field","rect":[53.812843322753909,583.9326782226563,385.18134063868447,576.0843505859375]},{"page":408,"text":"dependence of angle W(E) and characteristic soft-mode relaxation time independent of the field (b)","rect":[53.813682556152347,593.9086303710938,385.1720284805982,586.060302734375]},{"page":409,"text":"13.1 Ferroelectrics","rect":[53.812843322753909,42.55594253540039,117.63046725272455,36.68026351928711]},{"page":409,"text":"proportional to the tilt-polarisation","rect":[53.812843322753909,68.2883529663086,197.41130915692816,59.35380554199219]},{"page":409,"text":"dependent on temperature","rect":[53.81380844116211,80.24788665771485,158.37680853082504,71.31333923339844]},{"page":409,"text":"coupling","rect":[201.59805297851563,68.2883529663086,236.52736750653754,59.35380554199219]},{"page":409,"text":"constant","rect":[240.73898315429688,67.0,273.9960160977561,60.369773864746097]},{"page":409,"text":"m","rect":[278.1837158203125,68.1987075805664,283.6784398014362,61.714439392089847]},{"page":409,"text":"¼","rect":[287.0200500488281,64.82210540771485,294.68479183409127,62.491355895996097]},{"page":409,"text":"Ps/Ww⊥","rect":[298.009521484375,68.1987075805664,326.74341269683858,59.05499267578125]},{"page":409,"text":"and","rect":[330.8643798828125,67.0,345.26812068036568,59.35380554199219]},{"page":409,"text":"399","rect":[372.4981994628906,42.62367248535156,385.1898245254032,36.73106384277344]},{"page":409,"text":"critically","rect":[349.44390869140627,68.2883529663086,385.16558161786568,59.35380554199219]},{"page":409,"text":"w?m","rect":[192.82008361816407,108.2607421875,209.19064683268619,101.34498596191406]},{"page":409,"text":"ec ¼ aðT \u0002 TchÞ ¼ aðT \u0002 TchÞ","rect":[157.98362731933595,122.02037811279297,279.3540078555222,108.23883819580078]},{"page":409,"text":"(13.22)","rect":[356.07110595703127,114.4245376586914,385.1592954239048,105.94817352294922]},{"page":409,"text":"The optical appearance of the electroclinic effect corresponds to a rotation ofa","rect":[65.76555633544922,151.55215454101563,385.1583942241844,142.6175994873047]},{"page":409,"text":"birefringent plate in the zy-plane through angle \u0007W counted from the z-axis.","rect":[53.813533782958987,163.51171875,385.1244778206516,154.27835083007813]},{"page":409,"text":"Therefore, the optical transmission depends on orientation of the polariser with","rect":[53.813533782958987,175.47128295898438,385.11553278974068,166.53672790527345]},{"page":409,"text":"respect to the director (induced optical axis) and on the birefringence of a cell. As","rect":[53.813533782958987,187.43081665039063,385.1753274356003,178.4962615966797]},{"page":409,"text":"seen from Fig.13.9a, the maximal switched transmission (and optical contrast)","rect":[53.813533782958987,199.390380859375,385.1772702774204,190.45582580566407]},{"page":409,"text":"is observed when the polariser is oriented along one of the \u0007W positions (-W in","rect":[53.813533782958987,211.29312133789063,385.14238825849068,202.05975341796876]},{"page":409,"text":"the figure). In this case, the switching angle is 2W and the transmission (see","rect":[53.8125114440918,223.25265502929688,385.14441717340318,214.019287109375]},{"page":409,"text":"Section 11.1.1) is given by:","rect":[53.8125114440918,235.21218872070313,164.8995042569358,226.2776336669922]},{"page":409,"text":"T ¼ I=I0 ¼ sin24W \u0006 sin2d=2","rect":[165.06375122070313,266.5701904296875,273.9283074479438,253.47018432617188]},{"page":409,"text":"For proper selected birefringence Dn at a given wavelength l and cell thickness","rect":[65.76587677001953,294.1049499511719,385.1477090273972,284.8417053222656]},{"page":409,"text":"d, d/2 ¼ pDnd/l ¼ p/2 and the transmission is T ¼ sin24W. In the ideal case, when","rect":[53.814857482910159,304.9787902832031,385.15642634442818,295.013427734375]},{"page":409,"text":"there are neither light scattering nor reflections and the induced angle reaches","rect":[53.814598083496097,318.0239562988281,385.0858193789597,309.08941650390627]},{"page":409,"text":"W ¼ 22.5\b, the transmission is complete, T ¼ 1. Such wide induced angles W(E)","rect":[53.814598083496097,329.98370361328127,385.15902076570168,320.7501220703125]},{"page":409,"text":"can, in principle, be reached very close to the phase transition (from the SmA*","rect":[53.813167572021487,341.886474609375,385.1440362077094,332.951904296875]},{"page":409,"text":"side), but, in this case, its time characteristics are not very attractive. The reason is","rect":[53.813167572021487,353.8460388183594,385.18588651763158,344.9114990234375]},{"page":409,"text":"in the properties of the soft mode.","rect":[53.813167572021487,365.8055725097656,190.80500455405002,356.87103271484377]},{"page":409,"text":"The dynamics of the electroclinic effect is, in fact, the dynamics of the elastic","rect":[65.76519012451172,377.76513671875,385.09433782770005,368.83056640625]},{"page":409,"text":"soft mode. From Eqs. (13.18) and (13.19) follows that the switching time of the","rect":[53.813167572021487,389.72467041015627,385.17493475152818,380.79010009765627]},{"page":409,"text":"effect is defined only by viscosity gW and the term a(T \u0002 Tc) and is independent of","rect":[53.81417465209961,401.68499755859377,385.1513608535923,392.7496337890625]},{"page":409,"text":"any characteristic size q\u00021 of the cell or material. It means that the relaxation of the","rect":[53.81350326538086,413.6445617675781,385.1710895366844,402.5936279296875]},{"page":409,"text":"order parameter amplitude is not of the hydrodynamic type controlled by term Kq2","rect":[53.81333541870117,425.6042175292969,385.18130140630105,414.5532531738281]},{"page":409,"text":"(K is elastic coefficient). For the same reason tW is independent of the electric field","rect":[53.812843322753909,437.50726318359377,385.2000664811469,428.57269287109377]},{"page":409,"text":"in agreement with the experimental data, shown in Fig.13.9b. At present, the","rect":[53.8134880065918,449.4668273925781,385.1164935894188,440.53228759765627]},{"page":409,"text":"electroclinic effect is the fastest one among the other electro-optical effects in","rect":[53.81345748901367,461.4263610839844,385.1393975358344,452.4918212890625]},{"page":409,"text":"liquid crystals.","rect":[53.81345748901367,473.3858947753906,112.92155118490939,464.45135498046877]},{"page":409,"text":"The coefficient gW is rotational viscosity of the director similar to coefficient g1","rect":[65.76549530029297,485.34588623046877,385.18130140630105,476.410888671875]},{"page":409,"text":"for nematics. In fact, it does not include a factor of sin2j and, in the same","rect":[53.812843322753909,497.2257385253906,385.1299518413719,486.2544860839844]},{"page":409,"text":"temperature range, can be considerably larger than the viscosity gj for the Gold-","rect":[53.8139762878418,510.1418151855469,385.11858497468605,500.3304443359375]},{"page":409,"text":"stone mode. This may be illustrated by Fig. 13.10: the temperature dependence of","rect":[53.81362533569336,521.2247314453125,385.15142188874855,512.2902221679688]},{"page":409,"text":"viscosities gW and gj have been measured for a chiral mixture that shows the","rect":[53.813594818115237,534.0610961914063,385.1159137554344,524.1929321289063]},{"page":409,"text":"nematic, smectic A* and smectic C* phases [15]. The pyroelectric and electro-","rect":[53.81289291381836,545.0873413085938,385.1785825332798,536.0930786132813]},{"page":409,"text":"optic techniques were the most appropriate, respectively, for the measurements of","rect":[53.81290817260742,557.046875,385.15081153718605,548.1123657226563]},{"page":409,"text":"gW and gj describing the viscous relaxation of the amplitude and phase of the SmC","rect":[53.81290817260742,569.8833618164063,385.1433589142158,560.0722045898438]},{"page":409,"text":"order parameter. The result of measurements clearly shows that gW is much larger","rect":[53.81346130371094,580.9663696289063,385.09539161531105,572.03173828125]},{"page":409,"text":"than gp and, in fact, corresponds to nematic viscosity g1.","rect":[53.81432342529297,592.926025390625,282.0889858772922,583.9913940429688]},{"page":410,"text":"400","rect":[53.81314468383789,42.55630874633789,66.50475067286416,36.73143005371094]},{"page":410,"text":"Fig. 13.10 Comparison of","rect":[53.812843322753909,67.58130645751953,145.9987496720045,59.648292541503909]},{"page":410,"text":"the temperature dependencies","rect":[53.812843322753909,77.4895248413086,155.30591280936518,69.89517211914063]},{"page":410,"text":"of viscosity coefficients g1","rect":[53.812843322753909,87.4087142944336,144.67245996310454,79.81436157226563]},{"page":410,"text":"(nematic), gW (soft mode) and","rect":[53.812843322753909,97.3082275390625,155.2159628310673,89.79006958007813]},{"page":410,"text":"gj (Goldstone mode) of the","rect":[53.81258010864258,108.12533569335938,148.54732653879644,99.76602172851563]},{"page":410,"text":"same chiral mixture within","rect":[53.812957763671878,115.58378601074219,145.89224762110636,109.74197387695313]},{"page":410,"text":"the ranges of the N* and","rect":[53.812957763671878,127.2555160522461,138.22144073634073,119.66116333007813]},{"page":410,"text":"SmC* phases [15]. Note that","rect":[53.812957763671878,137.23147583007813,152.51133445944849,129.58631896972657]},{"page":410,"text":"g1 and gW curves may be","rect":[53.81296157836914,147.20687866210938,137.84342334055425,139.61251831054688]},{"page":410,"text":"bridged through the SmA*","rect":[53.81314468383789,157.182861328125,145.18423980860636,149.5885009765625]},{"page":410,"text":"phase (black points) where","rect":[53.81314468383789,167.10208129882813,145.7028898932886,159.49078369140626]},{"page":410,"text":"measurement have not been","rect":[53.81314468383789,175.3254852294922,149.14146942286417,169.48367309570313]},{"page":410,"text":"made","rect":[53.81314468383789,185.3268585205078,72.21597430490971,179.45965576171876]},{"page":410,"text":"13","rect":[197.23843383789063,42.55630874633789,205.69949859766886,36.73143005371094]},{"page":410,"text":"Ferroelectricity and","rect":[208.05760192871095,44.274986267089847,275.13364166407509,36.68062973022461]},{"page":410,"text":"10","rect":[261.424072265625,80.44779968261719,270.31257108056408,74.6810302734375]},{"page":410,"text":"1","rect":[265.86834716796877,110.7965087890625,270.3126015981422,105.1737060546875]},{"page":410,"text":"0.1","rect":[259.20196533203127,141.4308319091797,270.3126015981422,135.6640625]},{"page":410,"text":"0.01","rect":[254.75770568847657,171.92112731933595,270.3126015981422,166.15435791015626]},{"page":410,"text":"Antiferroelectricity","rect":[277.5052795410156,44.274986267089847,343.09543365626259,36.68062973022461]},{"page":410,"text":"80 °C","rect":[296.9797058105469,66.52964782714844,317.0587948399408,60.490936279296878]},{"page":410,"text":"Iso N* A*","rect":[280.604736328125,164.75704956054688,331.45624224141747,157.52659606933595]},{"page":410,"text":"in","rect":[345.5364685058594,43.0,352.1445364394657,36.68062973022461]},{"page":410,"text":"Smectics","rect":[354.54327392578127,43.0,385.1806991863183,36.68062973022461]},{"page":410,"text":"a","rect":[89.21609497070313,242.18698120117188,94.77135736793697,236.5982208251953]},{"page":410,"text":"I0","rect":[132.55149841308595,250.83489990234376,138.10691823962228,242.9845733642578]},{"page":410,"text":"P","rect":[173.0840301513672,258.7486267089844,178.4155368149026,253.00584411621095]},{"page":410,"text":"b","rect":[209.28570556640626,242.18698120117188,215.39049931588446,234.87860107421876]},{"page":410,"text":"cy","rect":[249.51121520996095,254.55250549316407,256.50522522620175,246.9721221923828]},{"page":410,"text":"E","rect":[99.21598815917969,286.8092346191406,104.54749482271508,281.06646728515627]},{"page":410,"text":"x","rect":[104.54727935791016,288.80938720703127,107.54475311926814,285.67205810546877]},{"page":410,"text":"ITO","rect":[180.1780548095703,276.6720275878906,193.50283177287393,270.63330078125]},{"page":410,"text":"h","rect":[175.00001525878907,305.5322265625,179.4442696889625,299.7894592285156]},{"page":410,"text":"ITO","rect":[179.79437255859376,319.6204528808594,193.11914952189736,313.58172607421877]},{"page":410,"text":"+Ex","rect":[222.68392944335938,290.8543701171875,235.6798773258111,283.1120300292969]},{"page":410,"text":"–E","rect":[311.0809020996094,315.6976013183594,320.8566531479104,309.954833984375]},{"page":410,"text":"x","rect":[320.8566589355469,317.6971740722656,323.85413269690488,314.5598449707031]},{"page":410,"text":"x","rect":[90.72064208984375,336.7327880859375,94.71727377165439,332.5496826171875]},{"page":410,"text":"y","rect":[97.50931549072266,359.52874755859377,101.5059471725333,353.6419982910156]},{"page":410,"text":"I","rect":[132.17381286621095,348.3569030761719,134.39594008129766,342.6141357421875]},{"page":410,"text":"A","rect":[169.2512664794922,339.0867004394531,174.5827731430276,333.34393310546877]},{"page":410,"text":"E>Eu E>0","rect":[208.75376892089845,359.7084045410156,252.38778165185313,351.882080078125]},{"page":410,"text":"E=0","rect":[269.11370849609377,358.53369140625,283.55753568017345,352.6469421386719]},{"page":410,"text":"E<0 E<-Eu","rect":[303.349609375,359.96337890625,349.5932311058332,351.8814697265625]},{"page":410,"text":"Fig. 13.11 Deformed Helix Ferroelectric effect. Scheme of observation of the effect (a) and the","rect":[53.812843322753909,381.5364074707031,385.1542296149683,373.6033935546875]},{"page":410,"text":"picture of distortion of the helical structure (b) in the zero, positive and negative field. P and A are","rect":[53.812843322753909,391.4446716308594,385.1475157477808,383.8503112792969]},{"page":410,"text":"polarizers and analyser, ITO means indium–tin oxide electrodes, I0 and I are intensities of","rect":[53.812843322753909,401.3638610839844,385.1555184708326,393.7525634765625]},{"page":410,"text":"incoming and outgoing beams. Note that at E ¼ 0 the helix is harmonic, for 0 < |E| < |Eu|","rect":[53.8132438659668,411.3397521972656,385.1545016784386,403.7452087402344]},{"page":410,"text":"anharmonic and for |E| > |Eu| unwound","rect":[53.81393051147461,420.957275390625,189.44136566309855,413.7210693359375]},{"page":410,"text":"13.1.4.2 Helix Distortion and Deformed Helix Ferroelectric effect","rect":[53.812843322753909,459.9668273925781,339.65303932038918,452.4566345214844]},{"page":410,"text":"To consider this effect we should leave our discussion of the phase transition and","rect":[53.812843322753909,485.6289978027344,385.14373103192818,476.6944580078125]},{"page":410,"text":"consider the field interaction with the helical structure deeply in the SmC* phase.","rect":[53.812843322753909,497.5885314941406,385.1775784065891,488.65399169921877]},{"page":410,"text":"Now the amplitude of the two-dimensional order parameter W is considered con-","rect":[53.812843322753909,509.548095703125,385.1168149551548,500.3147277832031]},{"page":410,"text":"stant, but the variation of the azimuthal angle j is essential. The helical structure","rect":[53.81185531616211,521.5076293945313,385.08799017145005,512.5731201171875]},{"page":410,"text":"under discussion has the axis of the helix h||z, the electric field Ex is applied","rect":[53.81185531616211,533.4678955078125,385.0992364030219,524.5326538085938]},{"page":410,"text":"perpendicular to the helical axis and the boundary conditions are not taken into","rect":[53.81417465209961,545.3706665039063,385.1550225358344,536.4361572265625]},{"page":410,"text":"account. This corresponds to a thick cell with the geometry shown in Fig. 13.11a.","rect":[53.81417465209961,557.3302001953125,383.3134731819797,548.3956909179688]},{"page":410,"text":"In the absence of field, the azimuth of the c-director is changed along the z-axis","rect":[65.76622772216797,569.2897338867188,385.1291543398972,560.355224609375]},{"page":410,"text":"as j(z) ¼ q0z and the polarization vector has projections onto the x,y-plane, Px ¼","rect":[53.814205169677737,581.2498168945313,385.14807918760689,572.3153076171875]},{"page":410,"text":"sinq0z and Py ¼ cosq0z. Therefore, for E ¼ 0 the projections of the c-director and","rect":[53.814231872558597,594.1258544921875,385.14492121747505,584.2748413085938]},{"page":411,"text":"13.1 Ferroelectrics","rect":[53.812843322753909,42.55594253540039,117.63046725272455,36.68026351928711]},{"page":411,"text":"401","rect":[372.4981994628906,42.55594253540039,385.1898245254032,36.73106384277344]},{"page":411,"text":"Ps-vector follow the harmonic law. This is shown for the cy(z)-component in the","rect":[53.812843322753909,69.20487976074219,385.1737445659813,59.35380554199219]},{"page":411,"text":"middle of Fig. 13.11b. When the electric field is applied along the x-axis, it interacts","rect":[53.81399154663086,80.24788665771485,385.16376127349096,71.31333923339844]},{"page":411,"text":"with the spontaneous polarisation tending to install the Ps-vector parallel to the","rect":[53.81399154663086,92.20772552490235,385.1752399273094,83.27293395996094]},{"page":411,"text":"field. The distortion begins at an infinitesimally small field, then develops as shown","rect":[53.813472747802737,104.1104965209961,385.1503838639594,95.17594909667969]},{"page":411,"text":"in the figure. Now the form of the helix has no longer a sine-form and, with","rect":[53.813472747802737,116.0699691772461,385.11647883466255,107.13542175292969]},{"page":411,"text":"increasing field, the probability for the c-director to be aligned along þy becomes","rect":[53.813472747802737,128.02957153320313,385.0896035586472,119.09501647949219]},{"page":411,"text":"larger than along –y; it is shown by wider and narrower dashed areas, respectively.","rect":[53.813472747802737,139.98910522460938,385.0985989144016,131.05455017089845]},{"page":411,"text":"Then, the narrow areas transform into peaks called p-solitons [13]. In fact, the latter","rect":[53.813472747802737,151.94863891601563,385.1154721817173,143.0140838623047]},{"page":411,"text":"are defects (walls) that, finally, at a certain critical field disappear, the helix","rect":[53.813472747802737,163.908203125,385.12944880536568,154.97364807128907]},{"page":411,"text":"unwinds and the structure becomes uniform. Like in cholesterics, due to similar","rect":[53.813472747802737,174.0,385.15227638093605,166.9331817626953]},{"page":411,"text":"topological problems, the helix unwinding has to be assisted by other structural","rect":[53.813472747802737,187.8272705078125,385.14533860752177,178.89271545410157]},{"page":411,"text":"defects or thermal fluctuations.","rect":[53.813472747802737,197.69808959960938,178.58054776694065,190.7954864501953]},{"page":411,"text":"In the stationary regime, for the balance of the elastic and electric torques we","rect":[65.76549530029297,211.68960571289063,385.16928899957505,202.7550506591797]},{"page":411,"text":"have a sine-Gordon equation [16]:","rect":[53.813472747802737,223.64910888671876,192.07743699619364,214.7145538330078]},{"page":411,"text":"Kj q2qjzð2zÞ þ PsEsinjðzÞ ¼0","rect":[159.73919677734376,261.6516418457031,279.25432673505318,238.90127563476563]},{"page":411,"text":"(13.23)","rect":[356.0729675292969,256.4670715332031,385.1611569961704,247.99072265625]},{"page":411,"text":"We guess, that in the low field regime the solution should have a form j(z) ¼","rect":[65.76737213134766,286.15228271484377,385.1492083379975,277.1380615234375]},{"page":411,"text":"q0z þ dj(z) with dj << q0z. Substituting this in Eq. (13.23) we find","rect":[53.81536102294922,298.072509765625,334.8151482682563,288.8391418457031]},{"page":411,"text":"x2 q2dqjz2ðzÞ þ sinðq0z þ djðzÞÞ ¼0","rect":[149.26168823242188,336.0736999511719,289.7327813248969,313.3232116699219]},{"page":411,"text":"(13.24)","rect":[356.0716247558594,330.8890075683594,385.1598142227329,322.41265869140627]},{"page":411,"text":"with field coherence length x given by","rect":[53.81400680541992,360.5343933105469,208.89263239911566,351.2810974121094]},{"page":411,"text":"x2 ¼ Kj","rect":[199.84524536132813,390.8042907714844,235.12979065592129,375.79522705078127]},{"page":411,"text":"PsE","rect":[222.04904174804688,397.6493835449219,237.45070641912185,389.4553527832031]},{"page":411,"text":"(13.25)","rect":[356.071044921875,390.9135437011719,385.15923438874855,382.3176574707031]},{"page":411,"text":"Neglecting dj in the second term of Eq. (13.24) and substituting solution of the","rect":[65.7654800415039,422.2024230957031,385.17319524957505,412.96905517578127]},{"page":411,"text":"form dj ¼ Asinqoz therein we get","rect":[53.811458587646487,434.1630859375,194.84776170322489,424.9285888671875]},{"page":411,"text":"\u0002 Aq02x2 sinq0z þ sinq0z ¼0","rect":[161.49517822265626,461.296142578125,279.70659724286568,449.3570861816406]},{"page":411,"text":"and obtain the amplitude A of the j-angle modulation by electric field: A ¼ q\u000202x\u00022:","rect":[53.81468963623047,485.5552978515625,385.15577637833499,473.6165466308594]},{"page":411,"text":"Finally the field dependence of the twist angle is given by:","rect":[65.7660140991211,496.7952575683594,302.6035600430686,487.8607177734375]},{"page":411,"text":"j ¼ q0z þ KPjsqE20 sinq0z","rect":[171.97543334960938,534.3567504882813,267.0057717715378,511.09228515625]},{"page":411,"text":"(13.26)","rect":[356.071044921875,526.1543579101563,385.15923438874855,517.6182250976563]},{"page":411,"text":"According to this result, for E ! 0, there is a small modulation of the helical","rect":[65.7654800415039,557.8973999023438,385.109510970803,548.962890625]},{"page":411,"text":"structure that is a deflection from the harmonic low without a change in the","rect":[53.813472747802737,569.85693359375,385.1165241069969,560.9224243164063]},{"page":411,"text":"structure period. With increasing field, the helix becomes distorted stronger and","rect":[53.813472747802737,581.8563232421875,385.14434138349068,572.8421020507813]},{"page":411,"text":"the soliton structure appears. Now a solution of Eq. (13.23) may be found in the","rect":[53.813472747802737,593.7760009765625,385.17517889215318,584.8414916992188]},{"page":412,"text":"402","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.73106384277344]},{"page":412,"text":"13 Ferroelectricity and Antiferroelectricity in Smectics","rect":[197.23812866210938,44.274620056152347,385.18039401053707,36.68026351928711]},{"page":412,"text":"form of the elliptic functions or numerically. However, the critical field for the","rect":[53.812843322753909,68.2883529663086,385.1726459331688,59.35380554199219]},{"page":412,"text":"complete helix unwinding can be estimated just from comparison of the electric","rect":[53.812843322753909,80.24788665771485,385.1477435894188,71.31333923339844]},{"page":412,"text":"energy –PsE gained and the elastic energy Kjq20 lost due to unwinding,","rect":[53.812843322753909,92.98324584960938,340.6740383675266,81.72531127929688]},{"page":412,"text":"Eu \u0004 Kjq02\u0004Ps:","rect":[188.57168579101563,117.69774627685547,250.3968499867334,105.41787719726563]},{"page":412,"text":"(13.27a)","rect":[351.6530456542969,116.0120620727539,385.1290525039829,107.53569793701172]},{"page":412,"text":"The exact solution of the same equation would give us a slightly lower critical","rect":[65.7661361694336,140.21603393554688,385.1092363125999,131.28147888183595]},{"page":412,"text":"field:","rect":[53.81411361694336,150.15357971191407,74.32974107334207,143.2410125732422]},{"page":412,"text":"Eu ¼ p2Kjq02","rect":[191.8581085205078,182.12191772460938,244.96225044807754,165.38619995117188]},{"page":412,"text":"16Ps","rect":[220.68951416015626,189.06405639648438,239.45940071000994,180.5941925048828]},{"page":412,"text":"(13.27b)","rect":[351.0867004394531,182.3285675048828,385.1520017227329,173.7924346923828]},{"page":412,"text":"For typical parameters Kj ¼ 10\u00027 dyn (10\u000212 N), pitch \u0004 0.3 mm, i.e. q0 ¼ 2 \u0005","rect":[65.76619720458985,212.51089477539063,385.14829281065377,200.5265350341797]},{"page":412,"text":"104 cm\u00021 ¼ 2 \u0005 106 m\u00021, Ps ¼ 300 statC/cm2 (\u00041 mC/m2) the threshold field for","rect":[53.814414978027347,223.14915466308595,385.1799253067173,212.44631958007813]},{"page":412,"text":"helix unwinding is about Eu ¼ 9 statV/cm (\u0004 0.28 V/mm).","rect":[53.814205169677737,235.49673461914063,290.63055081869848,226.5621795654297]},{"page":412,"text":"The Deformed Helix Ferroelectric (DHF) electrooptical effect [17] is observed","rect":[65.76790618896485,247.45639038085938,385.11687556317818,238.52183532714845]},{"page":412,"text":"in short pitch materials. It is a particular case of a more general phenomenon of the","rect":[53.81488800048828,259.4159240722656,385.17466009332505,250.4813690185547]},{"page":412,"text":"field induced helix distortion discussed above. The geometry of the cell showing","rect":[53.81488800048828,271.37548828125,385.1109246354438,262.44091796875]},{"page":412,"text":"DHF-effect is the same as presented in Fig. 13.11a; the helical axis h||z is in the cell","rect":[53.81488800048828,283.33502197265627,385.1228166348655,274.40045166015627]},{"page":412,"text":"plane and smectic layers are perpendicular to the substrates. To study the new","rect":[53.81488800048828,295.2945556640625,385.16268682434886,286.3599853515625]},{"page":412,"text":"regime, the equilibrium pitch of the helix should be shorter than the visible light","rect":[53.81488800048828,307.1973571777344,385.1238847500999,298.2628173828125]},{"page":412,"text":"wavelength P0 < l and the layer thickness d is much larger than P0. A light beam","rect":[53.81488800048828,319.15771484375,385.14237164140305,309.9044189453125]},{"page":412,"text":"with aperture a>> P0 and wavelength l passes through the cell along x. Due to the","rect":[53.81350326538086,331.1173400878906,385.17417181207505,321.8640441894531]},{"page":412,"text":"shortness of the pitch, the helical structure is not seen under a microscope and the","rect":[53.81344985961914,343.076904296875,385.17221868707505,334.142333984375]},{"page":412,"text":"cell behaves as a uniaxial plate with its optical axis directed along z in the absence","rect":[53.81344985961914,355.03643798828127,385.10645330621568,346.10186767578127]},{"page":412,"text":"of field.","rect":[53.8134651184082,364.96405029296877,85.12244077231174,358.06146240234377]},{"page":412,"text":"In an electric field \u0007E the helical structure becomes strongly deformed, and","rect":[65.7654800415039,378.9555358886719,385.14434138349068,370.02099609375]},{"page":412,"text":"cosj(z) function oscillates between the two situations pictured in the sketches for","rect":[53.8134651184082,390.9150695800781,385.17818580476418,381.98052978515627]},{"page":412,"text":"E > 0 and E < 0 in Fig. 13.11b. These oscillations cause variation of the local","rect":[53.8134651184082,402.8178405761719,385.11347825595927,393.88330078125]},{"page":412,"text":"refractive index which, being averaged over the entire cell, results in either clock-","rect":[53.813472747802737,414.7773742675781,385.11251197663918,405.84283447265627]},{"page":412,"text":"or anticlockwise deviation of the optical axis from the z-axis in the plane of the cell","rect":[53.813472747802737,426.7369079589844,385.20808274814677,417.76251220703127]},{"page":412,"text":"zy. The axis rotation angle a is proportional to PsE/Kjq02. As usual, the cell is","rect":[53.814476013183597,439.57415771484377,385.18759550200658,427.6468200683594]},{"page":412,"text":"placed between two crossed polarisers and the first of them (P) is installed at the","rect":[53.81394577026367,450.6570739746094,385.17270696832505,441.7225341796875]},{"page":412,"text":"same angle a to the z-axis. As the optical transmission is proportional to sin2a and","rect":[53.81394577026367,462.61663818359377,385.1458367448188,451.5658874511719]},{"page":412,"text":"the helix distortion has no threshold, the DFH effect provides a smooth variation of","rect":[53.81399154663086,474.5763854980469,385.1489499649204,465.641845703125]},{"page":412,"text":"the angle a and transmission Tthat is the so-called grey scale. The effect takes place","rect":[53.81399154663086,486.57574462890627,385.12293279840318,477.5814514160156]},{"page":412,"text":"up to the fields of helix unwinding Eu. The characteristic response times of the","rect":[53.81399154663086,498.4391784667969,385.17310369684068,489.504150390625]},{"page":412,"text":"effect in low fields E/Eu << 1 are independent of spontaneous polarization and","rect":[53.8133659362793,510.3988342285156,385.1436394791938,501.46417236328127]},{"page":412,"text":"field strength and determined only by the rotational viscosity gj and helix pitch P0:","rect":[53.81271743774414,523.235107421875,385.151991439553,513.423828125]},{"page":412,"text":"gj","rect":[204.43240356445313,546.713134765625,214.05773865396817,538.2744750976563]},{"page":412,"text":"tc ¼ Kjq02 ¼ 4p2Kj","rect":[177.92323303222657,560.486572265625,258.9207147281869,546.1649169921875]},{"page":412,"text":"(13.28)","rect":[356.0715026855469,552.3408813476563,385.1596921524204,543.864501953125]},{"page":412,"text":"Therefore at relatively low field a fast and reversible switching could be obtained","rect":[65.76595306396485,582.0997924804688,385.1638115983344,573.165283203125]},{"page":412,"text":"inthe DHFmode.Notethat theoptics ofthe DHF effect isalmost the sameasthatof","rect":[53.81393051147461,594.059326171875,385.14885841218605,585.1248168945313]},{"page":413,"text":"13.1 Ferroelectrics","rect":[53.81399917602539,42.55777359008789,117.6316269206933,36.68209457397461]},{"page":413,"text":"403","rect":[372.4993591308594,42.55777359008789,385.19098419337197,36.73289489746094]},{"page":413,"text":"the linear electro-optical effect in cholesterics described in Section 12.4. Moreover,","rect":[53.812843322753909,68.2883529663086,385.1815151741672,59.35380554199219]},{"page":413,"text":"formula (13.28) is identical to Eq. (12.44). However, as the rotational viscosity of","rect":[53.81282424926758,80.24788665771485,385.1487058242954,71.31333923339844]},{"page":413,"text":"SmC* phase sin2W times less than the rotational nematic viscosity [8, 15], the DHF","rect":[53.81382369995117,92.20772552490235,385.14947861979558,81.15672302246094]},{"page":413,"text":"effect is faster than flexoelectric switching of cholesterics.","rect":[53.81361770629883,104.1104965209961,286.65557523276098,95.17594909667969]},{"page":413,"text":"13.1.4.3 Frederiks Transition and Clark–Lagerwall Bistability","rect":[53.81361770629883,146.39569091796876,326.9204033952094,136.74398803710938]},{"page":413,"text":"The switching of the director in the surface stabilised ferroelectric liquid crystal","rect":[53.81361770629883,169.85958862304688,385.12263352939677,160.92503356933595]},{"page":413,"text":"cells (SSFLC) [8] has briefly been discussed in Section 13.1.2. Due to its impor-","rect":[53.81361770629883,181.81912231445313,385.1733640274204,172.8845672607422]},{"page":413,"text":"tance for ferroelectric liquid crystal displays we shall discuss this effect in more","rect":[53.81462097167969,193.7786865234375,385.13867986871568,184.84413146972657]},{"page":413,"text":"detail. The geometry of a planar cell of thickness d is shown in Fig.13.1.2. Now, the","rect":[53.81462097167969,205.73822021484376,385.1753619976219,196.7837371826172]},{"page":413,"text":"helical structure is considered to be unwound. We are interested in the field and","rect":[53.81462097167969,215.67576599121095,385.14556208661568,208.76319885253907]},{"page":413,"text":"time behaviour of the director or c-director given by angle j(x), and this process is","rect":[53.81462097167969,229.65728759765626,385.19028104888158,220.7227325439453]},{"page":413,"text":"considered to be independent of z and y- coordinates. The smectic C* equilibrium","rect":[53.814605712890628,241.61685180664063,385.09171246171555,232.6822967529297]},{"page":413,"text":"tilt angle W is assumed constant.","rect":[53.815589904785159,253.51962280273438,183.07497830893284,244.2862548828125]},{"page":413,"text":"(i) Case of infinitely strong zenithal anchoring (Frederiks transition). First we","rect":[65.76761627197266,265.8775939941406,385.2290424175438,256.32550048828127]},{"page":413,"text":"shall explain why, for the infinite zenithal anchoring of SmC* liquid crystal at the","rect":[53.815574645996097,277.438720703125,385.16040838434068,268.504150390625]},{"page":413,"text":"boundaries of a SSFLC cell, the bistability is absent. Let define an easy axis parallel to","rect":[53.815574645996097,289.3982849121094,385.25701228192818,280.4637451171875]},{"page":413,"text":"z fixed, for instance, by a rubbing procedure. Consider rather an artificial but simple","rect":[53.815574645996097,301.3578186035156,385.23706854059068,292.42327880859377]},{"page":413,"text":"caseoftheinfinitelystrongzenithalanchoringstrengthWz ! 1 andextremelyweak","rect":[53.815574645996097,313.319580078125,385.2676629166938,304.3828125]},{"page":413,"text":"(hardly possible!)azimuthal strengthWa ! 0.Inthezerofield,due to strongzenithal","rect":[53.81438446044922,325.2792053222656,385.23774583408427,316.34454345703127]},{"page":413,"text":"anchoring, the director n ought to be in one of the two stable states, left or right in the","rect":[53.81332015991211,337.23876953125,385.1621478862938,328.30419921875]},{"page":413,"text":"zy-plane. Hence, angle j is either 0 or p for the c-director coincides with either þy","rect":[53.81332015991211,349.1415100097656,385.15906561090318,340.20697021484377]},{"page":413,"text":"or –y. Then the n director forms either þW or \u0002W angle counted from the z-axis and the","rect":[53.81130599975586,361.10107421875,385.15808904840318,351.8677062988281]},{"page":413,"text":"polarisation vector Ps⊥n is looking either up or down.","rect":[53.81130599975586,373.0611877441406,266.90740628744848,364.12603759765627]},{"page":413,"text":"Assume that j ¼ 0, W ¼ þW and Ps is looking up. Then, with the electric field","rect":[65.7656478881836,385.0207824707031,385.11626521161568,375.7873840332031]},{"page":413,"text":"directed down, the situation becomes unstable and, at a certain threshold field, due","rect":[53.813289642333987,395.0,385.1402057476219,388.0457763671875]},{"page":413,"text":"to some j-fluctuation in the bulk and PsE ¼ PsEcosj interaction energy, a torque","rect":[53.813289642333987,408.9400634765625,385.1305621929344,400.00531005859377]},{"page":413,"text":"appears, which drives the director along the conical surface with apex angle 2W.","rect":[53.81362533569336,420.89959716796877,385.1823696663547,411.6662292480469]},{"page":413,"text":"Fig. 13.12 Clark–Lagerwall","rect":[53.812843322753909,461.68316650390627,152.60007957663599,453.7501525878906]},{"page":413,"text":"effect in thin SSFLC cell.","rect":[53.812843322753909,469.87274169921877,141.82995864942036,463.9970703125]},{"page":413,"text":"Application of the electric","rect":[53.812843322753909,481.5673522949219,143.57970568918706,473.9729919433594]},{"page":413,"text":"field E between the ITO","rect":[53.812843322753909,489.82464599609377,136.32773377043217,483.9320373535156]},{"page":413,"text":"electrodes causes up-down","rect":[53.812843322753909,501.4625549316406,145.20508331446573,493.8681945800781]},{"page":413,"text":"switching of spontaneous","rect":[53.812843322753909,511.4385070800781,140.4025772380761,503.8441467285156]},{"page":413,"text":"polarization Ps accompanied","rect":[53.812843322753909,521.4144287109375,151.6510519423954,513.8193359375]},{"page":413,"text":"by conical motion of the","rect":[53.81315994262695,531.3896484375,137.84174487375737,523.7952880859375]},{"page":413,"text":"director n. The projection of","rect":[53.81315994262695,541.308837890625,151.26692288977794,533.7144775390625]},{"page":413,"text":"the n-vector on plane xy is","rect":[53.81315994262695,551.2847900390625,144.9261825847558,543.6904296875]},{"page":413,"text":"c-director forming an angle j","rect":[53.81400680541992,561.2608032226563,155.34938227857928,553.6664428710938]},{"page":413,"text":"with respect to y. W is the tilt","rect":[53.81400680541992,571.1799926757813,152.2568483754641,563.3316650390625]},{"page":413,"text":"angle between n and the","rect":[53.814002990722659,581.1559448242188,137.33321521067144,573.5615844726563]},{"page":413,"text":"smectic layer normalz","rect":[53.81399917602539,591.1318969726563,131.27678378104486,583.5375366210938]},{"page":413,"text":"glass","rect":[203.99928283691407,453.2623596191406,222.73204230867254,445.7372131347656]},{"page":413,"text":"ITO","rect":[205.3909912109375,467.24847412109377,218.77039376979446,461.18499755859377]},{"page":413,"text":"Sm","rect":[193.3865203857422,482.7689208984375,205.4255708285336,476.7054443359375]},{"page":413,"text":"layers","rect":[193.3865203857422,493.9722900390625,214.79196357332098,486.4953308105469]},{"page":413,"text":"E","rect":[254.4180145263672,517.5086059570313,259.7713800720934,511.7422790527344]},{"page":413,"text":"x","rect":[319.2777099609375,501.2461853027344,323.2907276114069,497.0459289550781]},{"page":413,"text":"d","rect":[379.9913635253906,473.92388916015627,384.4538391527126,468.0451354980469]},{"page":414,"text":"404","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.73106384277344]},{"page":414,"text":"13 Ferroelectricity and Antiferroelectricity in Smectics","rect":[197.23812866210938,44.274620056152347,385.18039401053707,36.68026351928711]},{"page":414,"text":"The beginning of this motion is shown in Fig. 13.12 by curved arrows. In the strong","rect":[53.812843322753909,68.2883529663086,385.1357354264594,59.35380554199219]},{"page":414,"text":"enough field, everywhere except surfaces, the final situation will correspond to","rect":[53.812843322753909,80.24788665771485,385.13774958661568,71.31333923339844]},{"page":414,"text":"j ¼ p, W ¼ \u0002W, Ps directed down. Then, with a change in the field polarity, the","rect":[53.812843322753909,92.20772552490235,385.1736224956688,82.97412109375]},{"page":414,"text":"process reverses.","rect":[53.812862396240237,104.1104965209961,121.69062467123752,97.40709686279297]},{"page":414,"text":"The balance of the volume torques is expressed by the same sine-Gordon","rect":[65.76488494873047,116.0699691772461,385.16664973310005,107.13542175292969]},{"page":414,"text":"equation (13.23) rewritten here for convenience:","rect":[53.812862396240237,128.02957153320313,249.2080217129905,119.09501647949219]},{"page":414,"text":"q2jðxÞ","rect":[172.59783935546876,152.8546142578125,200.33525480376438,141.29690551757813]},{"page":414,"text":"Kj qx2 þ PsEsinjðxÞ ¼0","rect":[159.11546325683595,164.06712341308595,279.87746516278755,149.64927673339845]},{"page":414,"text":"(13.29)","rect":[356.0729675292969,158.8627166748047,385.1611569961704,150.3863525390625]},{"page":414,"text":"However the physics of the two phenomena is different. Now, there is no helix","rect":[65.76738739013672,186.52496337890626,385.13237849286568,177.5904083251953]},{"page":414,"text":"and the elastic term Kjq2j=qx2 corresponds to the uniform rotation of the","rect":[53.81536102294922,199.25970458984376,385.11548650934068,187.26504516601563]},{"page":414,"text":"c-director in the bulk with fixed js angles at the boundaries. Thus, we deal with a","rect":[53.8134880065918,210.44384765625,385.15894354059068,201.5092315673828]},{"page":414,"text":"kind of the Frederiks transition, like in nematics, however, with another electrical","rect":[53.81318283081055,220.31466674804688,385.1490922696311,213.4120635986328]},{"page":414,"text":"torque and the confinement for the director motion along the cone surface. The first","rect":[53.81318283081055,234.30618286132813,385.15998704502177,225.3716278076172]},{"page":414,"text":"integral of Eq. (13.29) is given by:","rect":[53.81318283081055,246.26571655273438,193.8460374600608,237.33116149902345]},{"page":414,"text":"x22 \u0005qqjx\u00062 ¼ cosj þC","rect":[173.1076202392578,285.3525390625,267.4808528107002,259.5330810546875]},{"page":414,"text":"(13.30)","rect":[356.07135009765627,277.6656188964844,385.1595395645298,269.18927001953127]},{"page":414,"text":"where coherence length x is defined by Eq.(13.25). The range of the field-induced","rect":[53.81376266479492,307.82098388671877,385.1267022233344,298.56768798828127]},{"page":414,"text":"j(x)-variation is 0-p. Due to the symmetry of our cell with respect to the middle","rect":[53.81475067138672,319.7805480957031,385.1675494976219,310.84600830078127]},{"page":414,"text":"plane yz at d/2, the maximal c-director deviation from the z-axis is j(d/2) ¼ jm.","rect":[53.81475067138672,331.74005126953127,385.1832241585422,322.78558349609377]},{"page":414,"text":"Therefore,","rect":[53.814537048339847,341.6385498046875,95.75361295248752,334.76580810546877]},{"page":414,"text":"and","rect":[53.812843322753909,398.0,68.21658412030706,390.31317138671877]},{"page":414,"text":"Eq.(13.30)","rect":[76.47061920166016,399.24774169921877,119.210008009354,390.3729553222656]},{"page":414,"text":"takes","rect":[127.45110321044922,397.1859130859375,147.95677579985813,390.31317138671877]},{"page":414,"text":"corresponding integral","rect":[53.814205169677737,411.1505126953125,144.38456590244364,402.2159423828125]},{"page":414,"text":"qqjx jd=2 ¼ 0; C ¼ \u0002cosjm","rect":[165.6304168701172,377.7539367675781,272.8779108450995,356.96942138671877]},{"page":414,"text":"the form: qj=qx ¼ \u0002p2ffiffi\u0004x\u0003ðcosj \u0002 cosjmÞ1=2.","rect":[156.28347778320313,400.53564453125,361.33562894369848,387.12255859375]},{"page":414,"text":"The","rect":[369.60955810546877,397.1859130859375,385.14807928277818,390.31317138671877]},{"page":414,"text":"j","rect":[161.835205078125,429.1570739746094,166.47584473306973,424.4298400878906]},{"page":414,"text":"ð0 ðcosj \u0002dcjosjmÞ1=2 ¼ dpx2","rect":[161.2698974609375,458.80194091796877,279.3666619401313,428.3258056640625]},{"page":414,"text":"(13.31)","rect":[356.07281494140627,445.5534362792969,385.1610044082798,437.07708740234377]},{"page":414,"text":"may be reduced to the Legendre form of the 1st kind elliptic integral. Its solution","rect":[53.81523895263672,481.321533203125,385.15667048505318,470.90802001953127]},{"page":414,"text":"may be found in the form of elliptic functions, which would give us the angle j as a","rect":[53.8138542175293,493.2810974121094,385.16065252496568,484.3465576171875]},{"page":414,"text":"function of dp2/x and jm.","rect":[53.813838958740237,505.6131286621094,161.83137174155002,495.66259765625]},{"page":414,"text":"From Eq. (13.29) is seen that, at j ¼ 0, there is no electric torque exerted on the","rect":[65.7652816772461,517.2003784179688,385.1739887066063,508.265869140625]},{"page":414,"text":"director. Thus, there should be a threshold for the distortion as in the case of the","rect":[53.81325912475586,527.1279907226563,385.1730121441063,520.2254028320313]},{"page":414,"text":"Frederiks transition in nematics. We can find the threshold field Ec, considering a","rect":[53.81325912475586,541.1197509765625,385.16147649957505,532.1849365234375]},{"page":414,"text":"small distortion j ! 0. The equation","rect":[53.813655853271487,553.0225219726563,205.84034052899848,544.0880126953125]},{"page":414,"text":"x2 qq2xj2 þ j ¼0","rect":[187.8928985595703,590.0236206054688,251.10083857587348,567.2532958984375]},{"page":414,"text":"(13.32)","rect":[356.07196044921877,584.8757934570313,385.1601499160923,576.3994140625]},{"page":415,"text":"13.1 Ferroelectrics","rect":[53.81367874145508,42.55813980102539,117.63130648612298,36.68246078491211]},{"page":415,"text":"405","rect":[372.4990539550781,42.55813980102539,385.1906790175907,36.63166427612305]},{"page":415,"text":"has solutions jðxÞ ¼ jm sinqx þ C where constant C ¼ 0 is found from the","rect":[53.812843322753909,68.70973205566406,385.11817205621568,58.67649459838867]},{"page":415,"text":"boundary","rect":[53.814205169677737,80.24788665771485,91.53170100263128,71.31333923339844]},{"page":415,"text":"condition:","rect":[96.58047485351563,79.0,137.1876817471702,71.31333923339844]},{"page":415,"text":"j¼0","rect":[142.293212890625,80.16820526123047,168.2856377702094,71.37310028076172]},{"page":415,"text":"for","rect":[173.33441162109376,78.18606567382813,184.94101844636573,71.31333923339844]},{"page":415,"text":"d","rect":[190.04454040527345,78.28567504882813,195.0216454606391,71.29341888427735]},{"page":415,"text":"¼","rect":[198.37123107910157,76.7816390991211,206.03597286436469,74.45088958740235]},{"page":415,"text":"0.","rect":[209.36068725585938,79.0,216.82635159994846,71.37310028076172]},{"page":415,"text":"Substituting","rect":[221.9358367919922,80.24788665771485,270.20378962567818,71.31333923339844]},{"page":415,"text":"j(x)","rect":[275.2953796386719,80.16820526123047,293.0556882461704,71.37310028076172]},{"page":415,"text":"into","rect":[298.1811218261719,78.18606567382813,313.7295769791938,71.31333923339844]},{"page":415,"text":"(13.32)","rect":[318.79925537109377,79.8494644165039,347.88848243562355,71.37310028076172]},{"page":415,"text":"we","rect":[352.9800720214844,78.18606567382813,364.5767291851219,73.54448699951172]},{"page":415,"text":"find","rect":[369.6683044433594,78.21595001220703,385.18093195966255,71.31333923339844]},{"page":415,"text":"\u0002x2q2 þ 1 ¼ 0. As minimum value of q is fixed by rigid boundary conditions,","rect":[56.0240478515625,93.73799896240235,385.1329006722141,82.51895141601563]},{"page":415,"text":"qmin ¼ p/d we arrived at the threshold condition 1/xc ¼ qmin. Hence, the threshold","rect":[53.812984466552737,105.6081314086914,385.1400994401313,96.44449615478516]},{"page":415,"text":"field for the “quasi-Frederiks” transition is:","rect":[53.814205169677737,117.65731048583985,227.7371507413108,108.72276306152344]},{"page":415,"text":"E ¼ p2Kj","rect":[195.93643188476563,147.02978515625,240.85095154459317,131.83114624023438]},{"page":415,"text":"F","rect":[202.10997009277345,149.807861328125,206.38827752240719,145.2061309814453]},{"page":415,"text":"Psd2","rect":[221.42588806152345,155.50894165039063,239.52433846077285,146.22811889648438]},{"page":415,"text":"(13.33)","rect":[356.0715026855469,148.77354431152345,385.1596921524204,140.29718017578126]},{"page":415,"text":"Forinstance,forKj ¼ 10\u00027dyn,d ¼ 2\u0005 10\u00024cm,Ps ¼ 300statC/cm2(1mC/m2),","rect":[65.76595306396485,179.9193115234375,385.1832546761203,167.9917755126953]},{"page":415,"text":"the threshold field is EF* \u0004 0.1 statV/cm, i.e. 3 kV/m. Due to a high value of Ps, the","rect":[53.814537048339847,191.00241088867188,385.1752094097313,182.00808715820313]},{"page":415,"text":"Frederiks type distortion in SmC* can be observed at extremely low voltage across","rect":[53.814476013183597,202.96197509765626,385.11655058013158,194.0274200439453]},{"page":415,"text":"the cell (Uc ¼ dEc \u0004 30 mV for 10 mm thick cell). However, independently of the","rect":[53.814476013183597,214.92172241210938,385.17493475152818,205.9672393798828]},{"page":415,"text":"field magnitude, after switching the field off, the distortion relaxes to the initial","rect":[53.81320571899414,226.88128662109376,385.16499192783427,217.9467315673828]},{"page":415,"text":"uniformstructure,j(x) ¼ 0.Therelaxationtimeofthedistortedstructureisowedto","rect":[53.81320571899414,238.70433044433595,385.1420830827094,229.84947204589845]},{"page":415,"text":"pure elastic, nematic-like torque and for small distortion only fundamental Fourier","rect":[53.81418991088867,250.74359130859376,385.1111997207798,241.8090362548828]},{"page":415,"text":"harmonic is important,","rect":[53.81418991088867,262.703125,144.88328214194065,253.76856994628907]},{"page":415,"text":"tF ¼ gj\u0004Kjq2min ¼ gjd2\u0004Kjp2:","rect":[154.52891540527345,289.5522155761719,284.49732911270999,276.9340515136719]},{"page":415,"text":"(13.34)","rect":[356.0710754394531,287.5281066894531,385.15926490632668,279.0517578125]},{"page":415,"text":"For larger distortions, the relaxation rate will be determined by the sum of the","rect":[65.7655258178711,313.0926818847656,385.1732868023094,304.15814208984377]},{"page":415,"text":"rates m=tF of each harmonic with number m. Evidently, there is no bistability in this","rect":[53.81350326538086,325.68817138671877,385.14609159575658,315.4503479003906]},{"page":415,"text":"case.","rect":[53.81327819824219,334.95025634765627,73.48279996420627,330.3086853027344]},{"page":415,"text":"(ii) Case of finite anchoring energy (Clark-Lagerwall bistability). In reality,","rect":[65.76529693603516,349.37005615234377,385.1043362190891,339.8179626464844]},{"page":415,"text":"both the zenithal and azimuthal anchoring strengths are always finite, therefore the","rect":[53.81229019165039,360.8744201660156,385.1691058941063,351.93988037109377]},{"page":415,"text":"surface terms should be added to the balance of torques. But what is the anchoring","rect":[53.81229019165039,372.8339538574219,385.17507258466255,363.8994140625]},{"page":415,"text":"energy for the smectic C* phase? In Section 10.2.3 we introduced the zenithal and","rect":[53.81229019165039,384.7934875488281,385.1431511979438,375.85894775390627]},{"page":415,"text":"azimuthal anchoring energies Wz and Wa for the director n in both nematics and","rect":[53.81230926513672,396.7530212402344,385.1448601823188,387.8184814453125]},{"page":415,"text":"cholesterics. But in this paragraph we operate with another variable, the c-director","rect":[53.81403732299805,408.7132873535156,385.17376075593605,399.77874755859377]},{"page":415,"text":"or angle j and should reconsider the problem. In Fig. 13.13 the easy axis z is","rect":[53.81403732299805,420.6728515625,385.18866361724096,411.73828125]},{"page":415,"text":"situated in the substrate plane xy and the end of the director n ¼ 1 is confined to","rect":[53.81405258178711,432.6323547363281,385.1439141373969,423.69781494140627]},{"page":415,"text":"move along the semicircle (dot line). The conical angle W between n and the easy","rect":[53.81405258178711,444.5919189453125,385.10906306317818,435.3585510253906]},{"page":415,"text":"axis z is assumed to be fixed. The projection of the director on the substrate plane","rect":[53.81403732299805,456.4947204589844,385.14002264215318,447.5601806640625]},{"page":415,"text":"Fig. 13.13 Geometry for","rect":[53.812843322753909,493.70782470703127,141.28254788977794,485.7748107910156]},{"page":415,"text":"discussion of the anchoring","rect":[53.812843322753909,503.6160583496094,147.54202789454386,496.0216979980469]},{"page":415,"text":"energies for the c-director.","rect":[53.812843322753909,513.5919799804688,144.6382014472719,505.9976501464844]},{"page":415,"text":"a and b are the angles the","rect":[53.812843322753909,523.51123046875,142.4865507819605,515.6544189453125]},{"page":415,"text":"director n forms with the easy","rect":[53.81284713745117,533.4871826171875,155.30591339259073,525.892822265625]},{"page":415,"text":"axis z coinciding with the","rect":[53.812843322753909,543.463134765625,141.58036181711675,535.8687744140625]},{"page":415,"text":"normal to smectic layers; W is","rect":[53.812843322753909,553.3823852539063,155.29152377127924,545.5340576171875]},{"page":415,"text":"the director tilt angle and AC","rect":[53.812835693359378,563.3583374023438,154.39886653150897,555.7639770507813]},{"page":415,"text":"is the c-director forming","rect":[53.812835693359378,573.3342895507813,137.3176779189579,565.7399291992188]},{"page":415,"text":"angle j with y-axis","rect":[53.812835693359378,583.3102416992188,120.2965439128808,575.7158813476563]},{"page":415,"text":"0","rect":[228.42037963867188,563.4398193359375,232.41701132048252,557.76904296875]},{"page":415,"text":" y","rect":[240.7561798095703,571.3368530273438,246.29551467075255,566.08203125]},{"page":415,"text":"/2x","rect":[284.4948425292969,482.4070129394531,303.4201637186041,475.6636657714844]},{"page":415,"text":"z","rect":[373.82965087890627,523.8909912109375,376.93903032735497,520.4437255859375]},{"page":415,"text":"Easy","rect":[350.9089660644531,545.5311279296875,366.45744711149816,538.5086669921875]},{"page":415,"text":"axis","rect":[350.9089660644531,553.5054931640625,363.80768267110497,547.8506469726563]},{"page":416,"text":"406","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.68026351928711]},{"page":416,"text":"13 Ferroelectricity and Antiferroelectricity in Smectics","rect":[197.23812866210938,44.274620056152347,385.18039401053707,36.68026351928711]},{"page":416,"text":"OB forms angles a and b with respect to n and easy axis z, respectively. Each","rect":[53.812843322753909,68.2883529663086,385.1198357682563,59.04502868652344]},{"page":416,"text":"arbitrary point C on the semicircle is characterised by the only variable j used in","rect":[53.81181716918945,80.24788665771485,385.14165583661568,71.31333923339844]},{"page":416,"text":"Eq. (13.29). Now we would like to make find the strength of anchoring for the c-","rect":[53.81181716918945,92.20748138427735,385.15862403718605,83.27293395996094]},{"page":416,"text":"director in the SmC phase, no matter chiral or achiral.","rect":[53.81280517578125,104.11019134521485,271.8219875862766,95.17564392089844]},{"page":416,"text":"In the Rapini approximation, the zenithal and azimuthal anchoring energies are","rect":[65.76482391357422,116.0697250366211,385.16065252496568,107.13517761230469]},{"page":416,"text":"defined for the director n in terms of angles a and b:","rect":[53.81280517578125,128.02932739257813,266.59199388095927,118.78599548339844]},{"page":416,"text":"Wz ¼ 12Wz0sin2a","rect":[138.94915771484376,154.45538330078126,202.73318833659244,142.26046752929688]},{"page":416,"text":"and","rect":[212.70233154296876,151.16436767578126,227.1060723405219,144.2617645263672]},{"page":416,"text":"Wa ¼ 12Wa0sin2b","rect":[237.1160125732422,154.45538330078126,300.0495335514362,142.26046752929688]},{"page":416,"text":"(13.35)","rect":[356.0708312988281,152.62754821777345,385.15902076570168,144.03164672851563]},{"page":416,"text":"Let express the same energies using angles W and j, which describe the motion","rect":[65.76526641845703,177.96499633789063,385.1759881120063,168.73162841796876]},{"page":416,"text":"of the director in the smectic C* (or C) phase. Using the elementary geometry and","rect":[53.8122673034668,189.924560546875,385.1431511979438,180.99000549316407]},{"page":416,"text":"the fact that all angles OAC, OAB, OBC and CBA are equal to p/2, we find","rect":[53.8122673034668,201.88409423828126,360.6497735123969,192.9296112060547]},{"page":416,"text":"Wz ¼ 12Wz0sin2W \u0006 sin2j and Wa ¼ 12Wa0 1\u0002sinsi2nW2\u0006Wco\u0006ssi2nj2j","rect":[114.02629089355469,232.5313720703125,324.91242920572599,216.15853881835938]},{"page":416,"text":"(13.36)","rect":[356.0715026855469,228.58009338378907,385.1596921524204,220.04396057128907]},{"page":416,"text":"Typically W \u0004 0.5 or less, and the j-angles for the anchored director at the","rect":[65.76595306396485,256.072021484375,385.1746295757469,246.83865356445313]},{"page":416,"text":"surface are even smaller (the case of the break of anchoring is discussed below).","rect":[53.813899993896487,268.03155517578127,385.1626858284641,259.09698486328127]},{"page":416,"text":"Then the denominator of Wa is close to unity and we can approximately write","rect":[53.813899993896487,279.99127197265627,367.79468572809068,271.05657958984377]},{"page":416,"text":"Wz \u0004 21Wz0sin2W \u0006 sin2j ¼ 12Wzjsin2j","rect":[145.57789611816407,310.44091796875,293.3672358673408,298.1894226074219]},{"page":416,"text":"(13.37a)","rect":[351.6531066894531,308.78350830078127,385.12911353913918,300.3071594238281]},{"page":416,"text":"and","rect":[53.81418991088867,335.9437255859375,68.21793452313909,329.0411376953125]},{"page":416,"text":"Wa \u0004 21Wa0sin2W \u0006 cos2j ¼ 12Wajcos2j:","rect":[142.1807098388672,368.36932373046877,296.7893823353662,356.1738586425781]},{"page":416,"text":"(13.37b)","rect":[351.08660888671877,366.71124267578127,385.15191016999855,358.17510986328127]},{"page":416,"text":"Note that the amplitudes of the anchoring energy Wzjand Waj defined for the","rect":[65.76610565185547,396.6021423339844,385.17429388238755,386.90716552734377]},{"page":416,"text":"azimuthal angle j include factor sin2W \u0004 0.1, which, unfortunately, is often for-","rect":[53.813594818115237,407.86322021484377,385.15206275788918,396.8121643066406]},{"page":416,"text":"gotten in the literature. This is very important for our discussion of bistability. For","rect":[53.8132438659668,419.8227844238281,385.14815650788918,410.88824462890627]},{"page":416,"text":"simplicity and just to begin with, let assume that the azimuthal anchoring energy is","rect":[53.8132438659668,431.7823181152344,385.1849404727097,422.8477783203125]},{"page":416,"text":"negligible.","rect":[53.8132438659668,443.7418518066406,96.39436765219455,434.80731201171877]},{"page":416,"text":"If we are only interested in the Frederiks-type threshold we should add the","rect":[65.76525115966797,455.7013854980469,385.11325872613755,446.766845703125]},{"page":416,"text":"surface term \u0007 Wzjj to Eq. (13.32). Then, a finite anchoring only increases the","rect":[53.8132438659668,468.29705810546877,385.1714557476219,458.60833740234377]},{"page":416,"text":"apparent cell thickness by two extrapolation lengths bj: dapp ¼ d + 2bj where","rect":[53.81368637084961,480.5508728027344,385.11109197809068,469.87457275390627]},{"page":416,"text":"bj ¼ Kj=Wzj. For realistic values of Kj ¼ 10\u00027 dyn and Wz (measured for","rect":[53.81405258178711,492.4573974609375,385.12432227937355,480.5660705566406]},{"page":416,"text":"nematics) \u0004 0.1 erg/cm2 (Wzj \u0004 0.01 erg/cm2), the value of bj \u0004 2 \u0005 10\u00025 cm","rect":[53.814353942871097,504.57305908203127,385.1755442488249,492.8065185546875]},{"page":416,"text":"(0.2 mm) is considerably less than a typical cell thickness (few micrometers) and the","rect":[53.813838958740237,515.896728515625,385.17261541559068,506.96221923828127]},{"page":416,"text":"second term in dapp is of minor importance.","rect":[53.813838958740237,528.7862548828125,229.4363979866672,518.9017944335938]},{"page":416,"text":"To discuss a bistability, we should leave the small \u0002 j approximation and go","rect":[65.76528930664063,539.8160400390625,385.1710137467719,530.8815307617188]},{"page":416,"text":"back to the equation (13.29) with the Rapini surface energy added.","rect":[53.81428527832031,551.7755737304688,323.3673367073703,542.841064453125]},{"page":416,"text":"Kj q2qjxð2xÞ þ PsEsinjðxÞ \u0007 21Wzjsin2j ¼0","rect":[132.38087463378907,588.7765502929688,306.6126641373969,566.0062255859375]},{"page":416,"text":"(13.38)","rect":[356.0711669921875,583.572021484375,385.15935645906105,575.0956420898438]},{"page":417,"text":"13.1 Ferroelectrics","rect":[53.812843322753909,42.55594253540039,117.63046725272455,36.68026351928711]},{"page":417,"text":"407","rect":[372.4981994628906,42.55594253540039,385.1898245254032,36.73106384277344]},{"page":417,"text":"Now we are interested in the strong field case when distortion has a specific form.","rect":[65.76496887207031,68.2883529663086,385.1777615120578,59.35380554199219]},{"page":417,"text":"Like in nematics, well above the Frederiks transition, the director alignment is uniform","rect":[53.812950134277347,80.24788665771485,385.14285992265305,71.31333923339844]},{"page":417,"text":"throughout the SSFLC cell thickness except two layers of thickness bj adjacent to","rect":[53.812950134277347,92.20772552490235,385.25734797528755,82.51783752441406]},{"page":417,"text":"electrodes. To have a bistability we should break the initial zenithal anchoring, that is","rect":[53.814022064208987,104.1104965209961,385.23846830474096,95.17594909667969]},{"page":417,"text":"reach the second critical field Ec, whose coherence length becomes comparable with","rect":[53.814022064208987,116.07039642333985,385.1773614030219,107.13542175292969]},{"page":417,"text":"the surface extrapolation length, xc ¼ bj, see Section 11.2.4. Using the coherence","rect":[53.813594818115237,128.02999877929688,385.15955389215318,118.34022521972656]},{"page":417,"text":"length from Eq. (13.25) the condition for the break of anchoring is given by","rect":[53.81376266479492,139.98953247070313,350.6144951920844,130.99520874023438]},{"page":417,"text":"\u0005PKsEjc\u00061=2 ¼ WKjzj or Ec ¼ \u0002PWsKzj\u0003j2","rect":[149.54327392578126,177.72097778320313,287.27620003303846,150.25244140625]},{"page":417,"text":"(13.39)","rect":[356.0715026855469,169.6887664794922,385.1596921524204,161.21240234375]},{"page":417,"text":"Hence, for typical values of Kj ¼ 10\u00027 dyn, Ps ¼ 300 statC/cm2, Wzj ¼ 0.01","rect":[65.76595306396485,198.227294921875,385.1605156998969,186.2429656982422]},{"page":417,"text":"erg/cm2, the break of anchoring occurs at Ec ¼ 3 statV/cm (or 0.1 V/mm).","rect":[53.814720153808597,209.253662109375,352.8257107308078,198.20265197753907]},{"page":417,"text":"Thus, we see, that for a typical value of Wzj \u0004 0.01 erg/cm2 (corresponding to","rect":[65.76619720458985,221.84893798828126,385.1429375748969,210.16224670410157]},{"page":417,"text":"nematic anchoring as high as Wz \u0004 0.1 erg/cm2) we only need a voltage as low as","rect":[53.81405258178711,233.17294311523438,385.14194120513158,222.12196350097657]},{"page":417,"text":"U ¼ 0.2 V to break the zenithal anchoring in 2 mm thick cells. As soon as the initial","rect":[53.814083099365237,245.13247680664063,385.1489396817405,236.1979217529297]},{"page":417,"text":"anchoring is broken, the director is driven by the same electric field into the new","rect":[53.816078186035159,257.092041015625,385.16293096497386,248.15748596191407]},{"page":417,"text":"stable position at j ¼ p. When the field is switched off the director is still held in","rect":[53.816078186035159,269.05157470703127,385.20073786786568,260.11700439453127]},{"page":417,"text":"the new position by the zenithal anchoring until the field of the opposite polarity","rect":[53.816078186035159,281.0111083984375,385.1190117936469,272.0765380859375]},{"page":417,"text":"switches it back to j ¼ 0. Thus, in SSFLC cells, we have real bistable switching at","rect":[53.816078186035159,292.91387939453127,385.18379075595927,283.97930908203127]},{"page":417,"text":"rather low fields [18]. Such cells are unequivocally ferroelectric!","rect":[53.816078186035159,304.8734436035156,314.7965329727329,295.93890380859377]},{"page":417,"text":"13.1.5 Criteria for Bistability and Hysteresis-Free Switching","rect":[53.812843322753909,352.037109375,361.06544246577809,341.3993225097656]},{"page":417,"text":"13.1.5.1 Cells with No Insulating Layers","rect":[53.812843322753909,379.97784423828127,231.88971342192844,370.3261413574219]},{"page":417,"text":"In real SmC* cells both Wzjand Waj are finite. A finite azimuthal strength would","rect":[53.812843322753909,404.1972351074219,385.13762751630318,394.4457702636719]},{"page":417,"text":"create an additional surface torque trying to move the director from its angular","rect":[53.8137321472168,415.458251953125,385.10479102937355,406.523681640625]},{"page":417,"text":"positions at j ¼ 0 or j ¼ p to the easy axis z, Fig.13.13. On the cone surface, the","rect":[53.8137321472168,427.4178161621094,385.17444647027818,418.4633483886719]},{"page":417,"text":"minimum anchoring energy Waj corresponds to director position at j ¼ p/2 and the","rect":[53.814735412597659,440.076171875,385.17444647027818,430.3246765136719]},{"page":417,"text":"minima of Wzj are at j ¼ 0 and j ¼ p. But what will happen if Wzj¼ Waj? It","rect":[53.813716888427737,451.97930908203127,385.1811967618186,442.2843322753906]},{"page":417,"text":"follows from Eqs. (13.37) that the sum Wj of the two anchoring energies with equal","rect":[53.814476013183597,463.2403259277344,385.14045579502177,453.55047607421877]},{"page":417,"text":"amplitudes W","rect":[53.813594818115237,475.1998596191406,108.83748843093059,466.26531982421877]},{"page":417,"text":"Wj ¼ Wsin2W\u0002sin2j þ cos2j\u0003 ¼ Wsin2W","rect":[135.72183227539063,505.67852783203127,303.27835435221746,493.4548034667969]},{"page":417,"text":"becomes independent of j and the director is free to take any position on the cone","rect":[53.813899993896487,533.2410888671875,385.1168903179344,524.3065795898438]},{"page":417,"text":"as though there were no anchoring at the substrates. Thus, due to a competition","rect":[53.813899993896487,545.1438598632813,385.1089715104438,536.2093505859375]},{"page":417,"text":"between equal Wzjand Waj, there is no stable states at j ¼ 0 and j ¼ p, and, of","rect":[53.813899993896487,557.8023681640625,385.1507505020298,548.107421875]},{"page":417,"text":"course, no bistable switching. In fact, what does control the bistability is the","rect":[53.812923431396487,569.0632934570313,385.1139606304344,560.1287841796875]},{"page":417,"text":"difference Wzja between Wzj and Waj! Hence, the criterion for bistability is very","rect":[53.812923431396487,581.7214965820313,385.11797419599068,571.9700317382813]},{"page":417,"text":"simple:","rect":[53.81301498413086,592.9826049804688,83.23766191074441,584.048095703125]},{"page":418,"text":"408","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.73106384277344]},{"page":418,"text":"13 Ferroelectricity and Antiferroelectricity in Smectics","rect":[197.23812866210938,44.274620056152347,385.18039401053707,36.68026351928711]},{"page":418,"text":"1 > Wj ¼ Wj \u0002 Wj>0","rect":[166.8766632080078,67.37557220458985,272.1154717545844,59.575477600097659]},{"page":418,"text":"za","rect":[199.5039825439453,69.83734893798828,205.6705939539369,66.66494750976563]},{"page":418,"text":"(13.40)","rect":[356.0712890625,68.74015045166016,385.15947852937355,60.26378631591797]},{"page":418,"text":"As to dynamics of the response of a SSFLC cell to the alternating field, it is","rect":[65.76567840576172,89.3165512084961,385.1854287539597,80.38200378417969]},{"page":418,"text":"controlled by Eq. (13.38) with the viscous gjdj/dt torque added. When the helical","rect":[53.81364059448242,102.15332794189453,385.1816850430686,92.3221664428711]},{"page":418,"text":"structure of the SmC* is unwound in a thin cell (a typical case) one can neglect the","rect":[53.81301498413086,113.23616790771485,385.2305377788719,104.30162048339844]},{"page":418,"text":"elastictorque ∂2j/∂x2. If,in addition, theanchoring energies Wzjand Waj are reason-","rect":[53.81301498413086,125.89447784423828,385.15926490632668,114.14500427246094]},{"page":418,"text":"ably weak, the electric field torque would solely be balanced by the viscous torque:","rect":[53.814476013183597,137.15554809570313,384.549208236428,128.2209930419922]},{"page":418,"text":"dj","rect":[194.29283142089845,156.2340545654297,206.19009658511426,147.35926818847657]},{"page":418,"text":"gj dt ¼ PsEsinj","rect":[182.511474609375,167.95455932617188,256.4913355987861,154.18092346191407]},{"page":418,"text":"(13.41)","rect":[356.0712585449219,162.71705627441407,385.1594480117954,154.24069213867188]},{"page":418,"text":"Here, the viscosity coefficient gj corresponds to the azimuthal motion of the","rect":[65.76570892333985,188.36477661132813,385.1728900737938,178.55320739746095]},{"page":418,"text":"director. From this equation, for small distortions, we immediately find the time of","rect":[53.81313705444336,199.44772338867188,385.1480649551548,190.51316833496095]},{"page":418,"text":"the response to an external field.","rect":[53.81313705444336,211.35049438476563,184.7040218880344,202.4159393310547]},{"page":418,"text":"gj","rect":[224.9945831298828,238.08609008789063,234.62002503092129,229.64747619628907]},{"page":418,"text":"tj ¼ PSE","rect":[199.1635284423828,250.48410034179688,238.13043969548904,237.5375518798828]},{"page":418,"text":"(13.42)","rect":[356.0708923339844,243.7134246826172,385.1590818008579,235.237060546875]},{"page":418,"text":"For larger j the response is not exponential, however, controlled by the same","rect":[65.76534271240235,278.00634765625,385.12632024957505,269.07177734375]},{"page":418,"text":"physical parameters with a numerical factor of 1–2. The viscosity gj can be found","rect":[53.81332015991211,290.8436279296875,385.12883845380318,281.0313720703125]},{"page":418,"text":"from the measurements of kinetics of the optical transmission.","rect":[53.81393051147461,301.8699035644531,305.35686917807348,292.93536376953127]},{"page":418,"text":"13.1.5.2 Role of Aligning Layers in Bistability","rect":[53.81393051147461,344.09832763671877,255.94414607099066,334.4466247558594]},{"page":418,"text":"If one were capable to align a SmC* without insulating layers, e.g., by rubbing","rect":[53.81393051147461,367.6189880371094,385.1398552995063,358.6844482421875]},{"page":418,"text":"uncovered electrodes or by smectic layer shearing, then, in a cell with Wzja not","rect":[53.81393051147461,380.27789306640627,385.136610580178,370.52642822265627]},{"page":418,"text":"exceeding 0.01 erg/cm2, the bistable switching would be observed at voltages less","rect":[53.81368637084961,391.5391540527344,385.1412393008347,380.4881591796875]},{"page":418,"text":"than 1V for Ps as low as 20 nC/cm2. However, as a rule, there are insulating","rect":[53.81332015991211,403.49871826171877,385.1696709733344,392.4477844238281]},{"page":418,"text":"alignment layers covering the electrodes, which have their own capacitance. For","rect":[53.81394577026367,415.458251953125,385.14983497468605,406.523681640625]},{"page":418,"text":"a given a.c. voltage U across the electrodes, such layers may dramatically change","rect":[53.81394577026367,427.4178161621094,385.0851520366844,418.4832763671875]},{"page":418,"text":"both the amplitude and phase of field ELC on the liquid crystal layer. Field ELC may","rect":[53.81394577026367,439.3210754394531,385.16787043622505,430.38604736328127]},{"page":418,"text":"even have the opposite sign with respect to voltage applied! Thus, the criterion for","rect":[53.81411361694336,451.2806396484375,385.17986427156105,442.3460693359375]},{"page":418,"text":"the bistability (13.20) should also change.","rect":[53.81411361694336,463.2402038574219,222.6822933724094,454.3056640625]},{"page":418,"text":"Consider a two layer SSFLC cell, Fig.13.14, consisting of a liquid crystal layer","rect":[65.7661361694336,475.1997375488281,385.1220639785923,466.26519775390627]},{"page":418,"text":"(white) and a single alignment insulating layer (grey); the latter mimics two","rect":[53.81411361694336,487.1592712402344,385.1509942155219,478.2247314453125]},{"page":418,"text":"alignment layers of a typical experimental cell. For simplicity, both the liquid","rect":[53.81411361694336,499.1188049316406,385.1131524186469,490.18426513671877]},{"page":418,"text":"crystal and aligning layers are assumed to be nonconductive and having constant","rect":[53.81411361694336,511.0783386230469,385.1718889004905,502.143798828125]},{"page":418,"text":"dielectric permittivities. The x-component of the dielectric displacement Dx and the","rect":[53.81411361694336,523.037841796875,385.17517889215318,514.1033325195313]},{"page":418,"text":"total potential difference along the close contour (the Maxwell and Kirchhof","rect":[53.81444549560547,534.9412841796875,385.13741432038918,526.0067749023438]},{"page":418,"text":"equations, respectively) are given by:","rect":[53.81444549560547,546.9008178710938,204.776991439553,537.96630859375]},{"page":418,"text":"eILEIL ¼ eLCELC þ 4pPx","rect":[158.55067443847657,575.3080444335938,255.97866196527859,566.93359375]},{"page":418,"text":"dILEIL þ dLCELC ¼ U;","rect":[158.54978942871095,590.5530395507813,247.62205445450685,581.8176879882813]},{"page":418,"text":"Dx","rect":[269.6880187988281,575.2034301757813,279.93955368891138,567.0929565429688]},{"page":418,"text":"(13.43)","rect":[356.07220458984377,590.373779296875,385.1603940567173,581.8973999023438]},{"page":419,"text":"13.1 Ferroelectrics","rect":[53.812660217285159,42.55679702758789,117.6302841472558,36.68111801147461]},{"page":419,"text":"409","rect":[372.4980163574219,42.62452697753906,385.18964141993447,36.73191833496094]},{"page":419,"text":"Fig. 13.14 A model showing","rect":[53.812843322753909,67.58130645751953,155.34737152247355,59.648292541503909]},{"page":419,"text":"distribution of the electric","rect":[53.812843322753909,75.76238250732422,142.61684558176519,69.89517211914063]},{"page":419,"text":"field strength over an aligning","rect":[53.812843322753909,87.4087142944336,155.34485382227823,79.81436157226563]},{"page":419,"text":"insulating layer (EIL) anda","rect":[53.812843322753909,97.3846664428711,146.30146167063237,89.79006958007813]},{"page":419,"text":"ferroelectric liquid crystal","rect":[53.81265640258789,107.36043548583985,142.5997438832766,99.76608276367188]},{"page":419,"text":"layer (ELC) in a SSFLC cell.","rect":[53.81265640258789,117.33638763427735,150.9491145332094,109.74191284179688]},{"page":419,"text":"The electric voltage U is","rect":[53.81264114379883,127.25545501708985,138.41064913749018,119.66110229492188]},{"page":419,"text":"switched ON with polarity","rect":[53.812660217285159,137.23141479492188,144.75562042384073,129.6201171875]},{"page":419,"text":"shown by (+) and (\u0002) signs","rect":[53.812660217285159,147.20736694335938,148.3168838787011,139.61300659179688]},{"page":419,"text":"x","rect":[206.69448852539063,66.24563598632813,210.24349745883849,62.630401611328128]},{"page":419,"text":"EIL dLC","rect":[250.65284729003907,88.2877197265625,278.56821007577659,80.60535430908203]},{"page":419,"text":"ELC dIL","rect":[244.89060974121095,109.08648681640625,273.3483987762232,101.40412139892578]},{"page":419,"text":"+","rect":[352.8476867675781,72.3355712890625,357.35588730466056,68.12046813964844]},{"page":419,"text":"–","rect":[358.7906799316406,129.0,362.78731161345129,128.0]},{"page":419,"text":"U","rect":[371.4070739746094,97.17179107666016,377.1782101231439,91.75694274902344]},{"page":419,"text":"Here, suffices (IL) and (LC) mark insulating and liquid crystal layers, d is their","rect":[65.76496887207031,167.309814453125,385.1109555801548,158.35533142089845]},{"page":419,"text":"thickness, Px is x-component of spontaneous polarization parallel to the cell normal.","rect":[53.811954498291019,179.2694091796875,385.1771511604953,170.3347930908203]},{"page":419,"text":"From this set we immediately find the electric field in each layer,","rect":[53.81338882446289,191.22897338867188,317.2934231331516,182.29441833496095]},{"page":419,"text":"eILU \u0002 4pPxdIL","rect":[120.54090881347656,210.80242919921876,185.90173111088854,202.45301818847657]},{"page":419,"text":"ELC¼","rect":[88.53665924072266,217.7086639404297,114.10102871153264,209.43670654296876]},{"page":419,"text":"ðeLCdIL þ eILdLCÞ","rect":[116.91556549072266,225.34954833984376,190.08099760161594,215.39903259277345]},{"page":419,"text":"and","rect":[196.72842407226563,216.12014770507813,211.13216486981879,209.21754455566407]},{"page":419,"text":"eLCU þ 4pPxdLC","rect":[245.1602783203125,210.9070281982422,315.16377197916349,202.45301818847657]},{"page":419,"text":"EIL ¼","rect":[217.80050659179688,217.54132080078126,241.0427630254975,209.43667602539063]},{"page":419,"text":"ðeLCdIL þ eILdLCÞ","rect":[243.80072021484376,225.34954833984376,317.02289213286596,215.39903259277345]},{"page":419,"text":"(13.44)","rect":[356.0712585449219,217.75367736816407,385.1594480117954,209.27731323242188]},{"page":419,"text":"For Px ¼6 0, even without an applied voltage (U ¼ 0, short circuited electrodes),","rect":[65.76569366455078,246.88970947265626,385.1248440315891,237.62646484375]},{"page":419,"text":"there are “built-in” electric fields in both liquid crystal and aligning layers and these","rect":[53.81388473510742,258.8492431640625,385.1268695659813,249.91468811035157]},{"page":419,"text":"fields are opposite to each other.","rect":[53.81388473510742,270.8088073730469,184.66795011069065,261.874267578125]},{"page":419,"text":"Now, consider a switching process. Let, in the beginning, the director is close to","rect":[65.76590728759766,282.7683410644531,385.13979426435005,273.83380126953127]},{"page":419,"text":"j ¼ 0 and polarization P is almost parallel to the electrodes i.e. Px component is","rect":[53.81388473510742,294.72833251953127,385.18829740630346,285.7933349609375]},{"page":419,"text":"small). When the voltage is switched ON along positive x, at first, the field ELC","rect":[53.81362533569336,306.6878662109375,385.16990436186486,297.7333984375]},{"page":419,"text":"drives the polarization vector into the þ x-direction along the field, as shown in","rect":[53.812843322753909,318.6475524902344,385.1417168717719,309.7130126953125]},{"page":419,"text":"Fig.13.14. Then the counteracting polarization term in Eqs. (13.44) reduces the ELC","rect":[53.812862396240237,330.6070861816406,385.16990436186486,321.67254638671877]},{"page":419,"text":"field and increases field EIL and a charge on the insulating capacitor. This effect is","rect":[53.812843322753909,342.5102233886719,385.18597807036596,333.57568359375]},{"page":419,"text":"detrimental for devices because it increases the total voltage necessary to switch the","rect":[53.81328201293945,354.4697570800781,385.1710590191063,345.53521728515627]},{"page":419,"text":"director completely from j ¼ 0 to j ¼ p. With sufficient voltage the state j ¼ p","rect":[53.81328201293945,366.4293212890625,385.1869847233112,357.4947509765625]},{"page":419,"text":"is achieved and the director is temporary anchored at the new position. Let us see","rect":[53.81427764892578,378.38885498046877,385.1153034038719,369.45428466796877]},{"page":419,"text":"now whether this state is stable or not.","rect":[53.81427764892578,388.2865905761719,209.07806058432346,381.41387939453127]},{"page":419,"text":"When the voltage is switched OFF then, according to Eq. (13.44), the field ELC","rect":[65.76629638671875,402.30792236328127,385.16990436186486,393.35345458984377]},{"page":419,"text":"changes its sign and drives the Ps vector in the opposite direction. Now, if Ps is large","rect":[53.812843322753909,414.26812744140627,385.12314642145005,405.33355712890627]},{"page":419,"text":"and the torque Ps \u0005 ELC exerted on the polarization (and, consequently on the","rect":[53.81417465209961,426.227783203125,385.1737750835594,417.29315185546877]},{"page":419,"text":"director) in the OFF state is high enough, it would break anchoring at j ¼ p and,","rect":[53.81303024291992,438.13055419921877,385.1408962776828,429.17608642578127]},{"page":419,"text":"under the influence of ELC, the director leaves the j ¼ p position and moves back","rect":[53.8129997253418,450.0904235839844,385.1189812760688,441.15557861328127]},{"page":419,"text":"to j ¼ 0. The cycle is over and we are in the initial situation. It means that there is","rect":[53.81302261352539,462.0499572753906,385.1847268496628,453.11541748046877]},{"page":419,"text":"no bistability for high enough Ps.","rect":[53.81302261352539,474.0094909667969,188.11469693686252,465.074951171875]},{"page":419,"text":"Therefore, if we would like to work in the bistable regime, we should avoid the","rect":[65.7655029296875,485.9692077636719,385.17127264215318,477.03466796875]},{"page":419,"text":"action of the reversed field. According to the first of the last two equations, it means","rect":[53.8134880065918,497.92877197265627,385.1185342227097,488.99420166015627]},{"page":419,"text":"the condition of eILU>>4pPxdIL to be fulfilled, that is the alignment layers must be","rect":[53.8134880065918,509.8886413574219,385.1513141460594,500.9341735839844]},{"page":419,"text":"asthinaspossible,theirdielectricconstant (eIL)largeandPsoftheSmC*small.The","rect":[53.813472747802737,521.8482666015625,385.14673650934068,512.85400390625]},{"page":419,"text":"latter wouldbeincontradiction with a lowthresholdfield forthe bistabilitygivenby","rect":[53.81386947631836,533.7510375976563,385.16765681317818,524.8165283203125]},{"page":419,"text":"Eq. (13.39). Indeed, in experiments, the genuine bistable switching is always","rect":[53.81386947631836,545.7105712890625,385.08407987700658,536.7760620117188]},{"page":419,"text":"observed for liquid crystals with intermediate values of Ps \u0004 20–40 nC/cm2.","rect":[53.81487274169922,557.6701049804688,361.7322048714328,546.61962890625]},{"page":419,"text":"There is, however,another way to fight with the inverse field. Both the alignment","rect":[65.76665496826172,569.6300659179688,385.1734452969749,560.695556640625]},{"page":419,"text":"and liquid crystal materials can be made conductive. The conductivity s would","rect":[53.81462860107422,581.589599609375,385.1395196061469,572.6550903320313]},{"page":419,"text":"screen the built-in field EIL and accelerate its relaxation. Then, for a voltage pulse of","rect":[53.81462860107422,593.5494995117188,385.1487973770298,584.6146850585938]},{"page":420,"text":"410","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.73106384277344]},{"page":420,"text":"13 Ferroelectricity and Antiferroelectricity in Smectics","rect":[197.23812866210938,44.274620056152347,385.18039401053707,36.68026351928711]},{"page":420,"text":"duration exceeding tj given by Eq. (13.42), the built-in field would not prevent","rect":[53.812843322753909,69.16504669189453,385.10420091220927,59.35380554199219]},{"page":420,"text":"bistable switching if","rect":[53.81315994262695,80.24788665771485,135.5451901992954,71.31333923339844]},{"page":420,"text":"eLC=sLC > eIL=sIL","rect":[180.6407470703125,104.49595642089844,257.8410011304198,94.56536102294922]},{"page":420,"text":"(13.45)","rect":[356.0715026855469,103.7122573852539,385.1596921524204,95.11637115478516]},{"page":420,"text":"This","rect":[65.76595306396485,127.0,83.48444761382297,119.09526062011719]},{"page":420,"text":"is","rect":[88.65068054199219,127.0,95.28018583403781,119.09526062011719]},{"page":420,"text":"an","rect":[100.48922729492188,127.0,109.91585627606878,121.0]},{"page":420,"text":"additional","rect":[115.04725646972656,127.0,155.1189103848655,119.09526062011719]},{"page":420,"text":"criterion","rect":[160.30606079101563,127.0,194.27181330731879,119.09526062011719]},{"page":420,"text":"for","rect":[199.447021484375,127.0,211.0536130508579,119.09526062011719]},{"page":420,"text":"bistability","rect":[216.21389770507813,128.02981567382813,256.31841364911568,119.09526062011719]},{"page":420,"text":"valid","rect":[261.47271728515627,127.0,281.47072688153755,119.09526062011719]},{"page":420,"text":"for","rect":[286.6797790527344,127.0,298.2863706192173,119.09526062011719]},{"page":420,"text":"conductive","rect":[303.4466552734375,127.0,347.35569036676255,119.09526062011719]},{"page":420,"text":"(s","rect":[352.5567626953125,127.63138580322266,362.5308100860908,119.15502166748047]},{"page":420,"text":"> 0)","rect":[365.8704833984375,127.63138580322266,385.1517575821079,119.15502166748047]},{"page":420,"text":"materials [19]. In experiments with very conductive alignment layers the bistability","rect":[53.813961029052737,139.98934936523438,385.16179743817818,131.05479431152345]},{"page":420,"text":"is much easier to observe.","rect":[53.81295394897461,149.88705444335938,158.48048062826877,143.0143280029297]},{"page":420,"text":"13.1.5.3 V-Shape Effect","rect":[53.81295394897461,195.69329833984376,161.01383338777198,186.50973510742188]},{"page":420,"text":"In some applications the hysteresis and the threshold character of the director","rect":[53.81295394897461,219.68212890625,385.10503516999855,210.74757385253907]},{"page":420,"text":"switching are undesirable because they do not allow a grey scale to be realised.","rect":[53.81295394897461,231.64163208007813,385.0910610726047,222.7070770263672]},{"page":420,"text":"The hysteresis-free switching means the zero coercive field for the director switch-","rect":[53.81295394897461,243.6011962890625,385.09710059968605,234.66664123535157]},{"page":420,"text":"ing and the absence of bistability. Then one can observe the hysteresis-free V-shape","rect":[53.81295394897461,255.56072998046876,385.07920110895005,246.6261749267578]},{"page":420,"text":"switching. In this case, the curve of the optical transmission as a function of the total","rect":[53.81295394897461,267.520263671875,385.1567521817405,258.585693359375]},{"page":420,"text":"voltage on the cell T(U) acquires a shape of the letter “V” (no hysteresis) instead of","rect":[53.81295394897461,279.4798278808594,385.15081153718605,270.5452880859375]},{"page":420,"text":"letter “W” characteristic of hysteresis. For the first time the hysteresis-like switch-","rect":[53.81295394897461,291.3825988769531,385.10006080476418,282.44805908203127]},{"page":420,"text":"ing was observed in a chiral material having both SmC* and antiferroelectric","rect":[53.81295394897461,303.3421325683594,385.05725897027818,294.4075927734375]},{"page":420,"text":"SmC*A phase at temperatures close to the phase transition between them [20] and","rect":[53.81295394897461,315.3034362792969,385.14458552411568,306.307373046875]},{"page":420,"text":"explained by a kind of frustration between the two phases having very low energy","rect":[53.81272506713867,327.2629699707031,385.16753474286568,318.32843017578127]},{"page":420,"text":"barrier between them. However, the absence of hysteresis is also a characteristic","rect":[53.81272506713867,339.2225341796875,385.17746771051255,330.2879638671875]},{"page":420,"text":"feature of the SmC* phase well below the phase transition temperature when a","rect":[53.81272506713867,351.18206787109377,385.15854681207505,342.24749755859377]},{"page":420,"text":"special condition of Wzj¼Waj discussed above is fulfilled, although this case is","rect":[53.81272506713867,363.783935546875,385.18661893950658,354.08892822265627]},{"page":420,"text":"rather incident. Numerous experiments and modelling have unequivocally shown","rect":[53.812923431396487,375.1015930175781,385.17470637372505,366.16705322265627]},{"page":420,"text":"that the hysteresis-free switching with a clear V-shape of the T(U) transmission","rect":[53.812923431396487,387.0043640136719,385.17168513349068,378.06982421875]},{"page":420,"text":"curve in the SmC* phase may always be achieved when one uses relatively","rect":[53.81290817260742,398.9638977050781,385.1288689713813,390.02935791015627]},{"page":420,"text":"thick alignment layers and selects proper parameters for a liquid crystal and the","rect":[53.81290817260742,410.9234619140625,385.1706928081688,401.9888916015625]},{"page":420,"text":"layers [21].","rect":[53.81290817260742,422.88299560546877,99.60725064780002,413.94842529296877]},{"page":420,"text":"13.2 Introduction to Antiferroelectrics","rect":[53.812843322753909,473.02642822265627,258.8218601200358,464.01422119140627]},{"page":420,"text":"13.2.1 Background: Crystalline Antiferroelectrics","rect":[53.812843322753909,504.9051208496094,307.6328952762858,494.267333984375]},{"page":420,"text":"and Ferrielectrics","rect":[95.61687469482422,516.5062255859375,184.1642063114421,508.2111511230469]},{"page":420,"text":"Liquid crystal ferri and antiferroelectrics have many features discovered for years","rect":[53.812843322753909,546.3910522460938,385.1348000918503,537.45654296875]},{"page":420,"text":"of comprehensive studies of corresponding crystalline substances. Thus, it would be","rect":[53.812843322753909,558.3505859375,385.1487201519188,549.4160766601563]},{"page":420,"text":"convenient and instructive to begin with a short introduction in the structure and","rect":[53.812843322753909,570.3101196289063,385.14275446942818,561.3756103515625]},{"page":420,"text":"properties of antiferroelectric crystals. A difference between ferro-, ferri and anti-","rect":[53.812843322753909,582.2696533203125,385.1089109024204,573.3351440429688]},{"page":420,"text":"ferroelectrics is schematically shown in Fig. 13.15, where the three very simplified","rect":[53.812843322753909,594.229248046875,385.17656794599068,585.2349853515625]},{"page":421,"text":"13.2 Introduction to Antiferroelectrics","rect":[53.814552307128909,42.55618667602539,184.14294894217768,36.68050765991211]},{"page":421,"text":"411","rect":[372.4999084472656,42.45458984375,385.1915335097782,36.73130798339844]},{"page":421,"text":"Fig. 13.15","rect":[53.812843322753909,194.37643432617188,88.54130310206338,186.44342041015626]},{"page":421,"text":"Schematic structure of a ferroelectric (a), an antiferroelectric (b) and a ferrielectric (c).","rect":[94.48435974121094,193.9700469970703,385.1915003974672,186.71435546875]},{"page":421,"text":"Note that period of each structure is different: l, 2l and 3l, respectively","rect":[53.813716888427737,204.28466796875,295.93231720118447,196.67337036132813]},{"page":421,"text":"structures are depicted. In ferroelectrics, the dipoles are oriented parallel to each","rect":[53.812843322753909,247.56985473632813,385.0999688248969,238.6352996826172]},{"page":421,"text":"other everywhere, the period of the lattice is l, and, if each layer has spontaneous","rect":[53.812843322753909,259.5294189453125,385.15765775786596,250.57493591308595]},{"page":421,"text":"polarization P0 then the resulting spontaneous polarization of such a ferroelectric","rect":[53.81282424926758,271.4891662597656,385.08432806207505,262.55438232421877]},{"page":421,"text":"structure, Ps ¼ P0. An antiferroelectric may be represented as a combination of two","rect":[53.813167572021487,283.44879150390627,385.1515740495063,274.51422119140627]},{"page":421,"text":"dipolar sublattices built in each other, as shown in Fig. 13.15b. The periods of each","rect":[53.8127326965332,295.4083557128906,385.1017693620063,286.4140319824219]},{"page":421,"text":"sublattice and the entire structure are equal to 2l. An antiferroelectric has a higher","rect":[53.812747955322269,307.3678894042969,385.1296323379673,298.4134216308594]},{"page":421,"text":"translational symmetry than a ferroelectric. In sketch (b), we can recognise the","rect":[53.81270980834961,319.32745361328127,385.17050970270005,310.39288330078127]},{"page":421,"text":"additional planes of reflection situated exactly in the middle between any pair of","rect":[53.81270980834961,331.230224609375,385.14861427156105,322.295654296875]},{"page":421,"text":"dipolar layers. Therefore, despite each layer is polar and have finite local polariza-","rect":[53.81270980834961,343.18975830078127,385.1605466446079,334.25518798828127]},{"page":421,"text":"tion P0 ¼6 0, the macroscopic spontaneous polarization is absent Ps ¼ 0.","rect":[53.81270980834961,355.1500549316406,346.48724027182348,345.8868103027344]},{"page":421,"text":"In some crystals the location of dipole moments can even be more complicated. For","rect":[65.76656341552735,367.1095886230469,385.26494727937355,358.175048828125]},{"page":421,"text":"example, in Fig. 13.15c, one layer with the dipoles looking down alternates with two","rect":[53.814537048339847,379.0691223144531,385.18325129560005,370.0747985839844]},{"page":421,"text":"layers where the dipoles are looking up. Therefore we have three-layer periodicity 3l","rect":[53.814537048339847,391.0286865234375,385.2090898282249,382.07421875]},{"page":421,"text":"with two antiparallel layers and one extra polar layer. Such a structure may be","rect":[53.814537048339847,402.98822021484377,385.2649310894188,394.05364990234377]},{"page":421,"text":"considered as a mixture of the ferroelectric and antiferroelectric structures and is","rect":[53.814537048339847,412.91583251953127,385.23996366606908,406.01324462890627]},{"page":421,"text":"calledferrielectric.Incase(c),theferroelectricfractionisonepartperperiod,qF ¼ 1/3","rect":[53.814537048339847,426.85052490234377,385.18023005536568,417.91595458984377]},{"page":421,"text":"and the spontaneous polarization is finite, Ps ¼ (1/3)P0. For pure antiferroelectric","rect":[56.47730255126953,438.8108825683594,385.1510700054344,429.876220703125]},{"page":421,"text":"phase qF ¼ 0/2 and for pure ferroelectric one qF ¼ 1/1 ¼ 1. More generally, for","rect":[53.814205169677737,450.7705383300781,385.1800778946079,441.83587646484377]},{"page":421,"text":"different ferrielectric structures qF ¼ n/m, where m is the number of layers in the","rect":[53.81433868408203,462.7301940917969,385.1748737163719,453.7955322265625]},{"page":421,"text":"unit cell (period) and n < m is the ferroelectric layer fracture per unit cell, both","rect":[53.814144134521487,474.6897277832031,385.12709895185005,465.75518798828127]},{"page":421,"text":"being integers. Then, for both n and m ! 1, n/m ! 1, the difference betweenn","rect":[53.814144134521487,486.6492919921875,385.17986384442818,477.7147216796875]},{"page":421,"text":"and m become smaller and smaller and the so-called Devil’s staircase forms.","rect":[53.81511688232422,496.6764831542969,362.00542874838598,489.65435791015627]},{"page":421,"text":"With increasing temperature the order of dipoles in each sublattice decreases","rect":[65.76717376708985,510.5683898925781,385.12421049224096,501.63385009765627]},{"page":421,"text":"and, at a certain temperature, a phase transition into the paraelectric phase occurs. It","rect":[53.815147399902347,522.4711303710938,385.17793138095927,513.53662109375]},{"page":421,"text":"may be either second or first order transition. In the paraelectric phase local","rect":[53.815147399902347,534.4306640625,385.11323411533427,525.4961547851563]},{"page":421,"text":"polarization P0 vanishes. The nature of the spontaneous polarization is similar in","rect":[53.815147399902347,546.3910522460938,385.1418084245063,537.4556884765625]},{"page":421,"text":"solid ferro- and antiferroelectrics. In both cases, the dipole-dipole interactions are","rect":[53.81290054321289,558.3505859375,385.16074407770005,549.4160766601563]},{"page":421,"text":"dominant. For example, if dipoles are situated in the points of the body-centred","rect":[53.81290054321289,570.3101196289063,385.1109551530219,561.3756103515625]},{"page":421,"text":"cubic lattice, they preferably orient parallel to each other and such a structure is","rect":[53.81290054321289,582.2696533203125,385.1846047793503,573.3351440429688]},{"page":421,"text":"ferroelectric. However, the same dipoles placed into the points of a simple cubic","rect":[53.81290054321289,594.229248046875,385.13974798395005,585.2947387695313]},{"page":422,"text":"412","rect":[53.81200408935547,42.45556640625,66.50361007838174,36.73228454589844]},{"page":422,"text":"Fig. 13.16 Typical","rect":[53.812843322753909,67.58130645751953,121.08258537497584,59.648292541503909]},{"page":422,"text":"hysteresis-type dependence of","rect":[53.812843322753909,77.4895248413086,155.3456735001295,69.89517211914063]},{"page":422,"text":"the total (spontaneous and","rect":[53.812843322753909,87.4087142944336,143.43335479884073,79.81436157226563]},{"page":422,"text":"induced) polarization P asa","rect":[53.812843322753909,97.3846664428711,149.642922852273,89.79031372070313]},{"page":422,"text":"function of the external field","rect":[53.812843322753909,105.63353729248047,151.51874298243448,99.76632690429688]},{"page":422,"text":"for a ferroelectric (dashed","rect":[53.812843322753909,116.99797821044922,142.97305816798136,109.725341796875]},{"page":422,"text":"curve) and antiferroelectric","rect":[53.812843322753909,126.91716766357422,146.81356189035894,119.66146850585938]},{"page":422,"text":"(solid curve). Arrows show","rect":[53.812843322753909,136.8931121826172,146.66039063078373,129.6204833984375]},{"page":422,"text":"the direction of the field","rect":[53.81200408935547,145.48057556152345,136.44957489161417,139.61337280273438]},{"page":422,"text":"cycling","rect":[53.81200408935547,157.1837158203125,78.82292694239541,149.58935546875]},{"page":422,"text":"13","rect":[197.2373046875,42.55716323852539,205.69836944727823,36.73228454589844]},{"page":422,"text":"Ferroelectricity","rect":[208.05645751953126,44.275840759277347,260.4389700332157,36.68148422241211]},{"page":422,"text":"and Antiferroelectricity","rect":[262.8893127441406,44.275840759277347,343.09430450587197,36.68148422241211]},{"page":422,"text":"P","rect":[281.01776123046877,66.38568115234375,286.34926789400415,60.642906188964847]},{"page":422,"text":"F","rect":[274.50665283203127,105.55206298828125,279.39053674720386,99.80928802490235]},{"page":422,"text":"AF","rect":[276.89263916015627,152.55157470703126,287.1080355265007,146.8087921142578]},{"page":422,"text":"in Smectics","rect":[345.53533935546877,43.0,385.17957003592769,36.68148422241211]},{"page":422,"text":"EAF","rect":[345.0870361328125,125.78497314453125,358.0800882293482,118.0426254272461]},{"page":422,"text":"lattice prefer to align anti-parallel to each other and form an antiferroelectric","rect":[53.812843322753909,235.61026000976563,385.05616033746568,226.6757049560547]},{"page":422,"text":"structure. Very often the difference in electrostatic energy between the parallel","rect":[53.812843322753909,247.56979370117188,385.1686235196311,238.63523864746095]},{"page":422,"text":"and anti-parallel dipolar structures is small and the phase transitions occur between","rect":[53.812843322753909,259.52935791015627,385.1756219010688,250.5948028564453]},{"page":422,"text":"ferro- and antiferroelectric phases.","rect":[53.812843322753909,271.4888916015625,192.48293729330784,262.5543212890625]},{"page":422,"text":"Such a transition of an antiferroelectric into a ferroelectric state can also be","rect":[65.76486206054688,281.42645263671877,385.1487506694969,274.51385498046877]},{"page":422,"text":"observed in an electric field E exceeding a certain threshold EAF, Fig. 13.16,","rect":[53.812843322753909,295.4084777832031,385.1835598519016,286.473388671875]},{"page":422,"text":"because in the presence of the field the ferroelectric structure becomes more","rect":[53.814903259277347,307.3680114746094,385.1388629741844,298.4334716796875]},{"page":422,"text":"favourable. When the polarity of the field changes, all dipoles are realigned","rect":[53.814903259277347,319.32757568359377,385.13689509442818,310.39300537109377]},{"page":422,"text":"following the field. At large fields \u0007E the two opposite ferroelectric states are","rect":[53.814903259277347,331.2303466796875,385.16367376520005,322.2957763671875]},{"page":422,"text":"energetically equivalent. If the switching between \u0007E is fast enough the polariza-","rect":[53.814903259277347,343.18988037109377,385.1666501602329,334.25531005859377]},{"page":422,"text":"tion follows the dashed curve with the hysteresis characteristic of typical ferro-","rect":[53.814903259277347,355.1494445800781,385.17766700593605,346.21490478515627]},{"page":422,"text":"electrics (like in Fig. 13.2). For slow field cycling, the antiferroelectric state has","rect":[53.814903259277347,367.10894775390627,385.13290800200658,358.17437744140627]},{"page":422,"text":"enough time to recover at E < EFA and one observes a double hysteresis loop","rect":[53.814903259277347,379.0693359375,385.1271905045844,370.1339111328125]},{"page":422,"text":"indicative of the antiferroelectric nature of the ground state. The solid line in","rect":[53.8132438659668,391.0289001464844,385.1391533952094,382.0943603515625]},{"page":422,"text":"Fig. 13.16 shows this type of tri-stable switching. Note that in low fields between","rect":[53.8132438659668,402.9884338378906,385.17702570966255,394.05389404296877]},{"page":422,"text":"\u0007EFA the antiferroelectric behaves as a conventional linear unpolar dielectric.","rect":[53.814247131347659,414.94830322265627,368.57854886557348,406.01373291015627]},{"page":422,"text":"A difference between ferro- and antiferroelectrics may also be discussed","rect":[65.7654800415039,426.8510437011719,372.92559138349068,417.91650390625]},{"page":422,"text":"in","rect":[377.3671569824219,425.0,385.14141169599068,417.91650390625]},{"page":422,"text":"terms of the soft elastic mode [3]. In the infinite ferroelectric crystal, there is no","rect":[53.81345748901367,438.81060791015627,385.17022028974068,429.87603759765627]},{"page":422,"text":"spatial modulation of the spontaneous polarization (only dipole density is peri-","rect":[53.81346130371094,450.7701721191406,385.1195005020298,441.83563232421877]},{"page":422,"text":"odic). Therefore, at the transition from a paraelectric to the ferroelectric phase,","rect":[53.81346130371094,462.7297058105469,385.1811794808078,453.795166015625]},{"page":422,"text":"both the wavevector q for oscillations of ions responsible for polarization and","rect":[53.81346130371094,474.6892395019531,385.14336482099068,465.75469970703127]},{"page":422,"text":"the correspondent oscillation frequency o ¼ Kq2 tend to zero. We may say that the","rect":[53.81344985961914,486.6495056152344,385.1582416362938,475.59857177734377]},{"page":422,"text":"soft elastic mode in ferroelectrics condenses at q ! 0. In antiferroelectrics,","rect":[53.81345748901367,498.4994812011719,385.1284451058078,489.67449951171877]},{"page":422,"text":"the sign of the local polarization P0 alternates in space with wavevector","rect":[53.8134880065918,510.5688171386719,385.08855567781105,501.634033203125]},{"page":422,"text":"q0 ¼ 2p/2l ¼ p/l and the corresponding ion oscillation frequency is finite, o ¼","rect":[53.813472747802737,522.4718017578125,385.1476519415131,513.517333984375]},{"page":422,"text":"Kq02 ¼ Kp2/l2. It means that in antiferroelectrics the soft mode condenses at a","rect":[53.813777923583987,534.3218383789063,385.1570514507469,523.3803100585938]},{"page":422,"text":"finite wavevector p/l and rather high frequency. As a result, in the temperature","rect":[53.81322479248047,546.3909301757813,385.12021673395005,537.4364624023438]},{"page":422,"text":"dependence of the dielectric permittivity at low frequencies, the Curie law at the","rect":[53.8142204284668,558.3504638671875,385.17200506402818,549.4159545898438]},{"page":422,"text":"phase transition between a paraelectric and antiferroelectric phases is not well","rect":[53.8142204284668,570.31005859375,385.1550431973655,561.3755493164063]},{"page":422,"text":"pronounced.","rect":[53.8142204284668,582.2695922851563,103.42202420737033,573.3350830078125]},{"page":423,"text":"13.2 Introduction to Antiferroelectrics","rect":[53.81199645996094,42.55685806274414,184.14040072440424,36.68117904663086]},{"page":423,"text":"13.2.2 Chiral Liquid Crystalline Antiferroelectrics","rect":[53.812843322753909,69.93675231933594,310.46388892862958,59.298980712890628]},{"page":423,"text":"413","rect":[372.49737548828127,42.55685806274414,385.1889700332157,36.73197937011719]},{"page":423,"text":"13.2.2.1 Discovery and Polymorphism","rect":[53.812843322753909,97.87738800048828,222.58052279372357,88.2256851196289]},{"page":423,"text":"The very first observation of antiferroelectric properties in a chiral liquid crystalline","rect":[53.812843322753909,121.3980941772461,385.1686481304344,112.46354675292969]},{"page":423,"text":"mixture was reported in 1982 [22]. The pyroelectric technique clearly showed the","rect":[53.812843322753909,133.35763549804688,385.1706012554344,124.42308044433594]},{"page":423,"text":"absence of spontaneous polarization in the zero field and a growth of pyroelectric","rect":[53.81184387207031,145.31716918945313,385.1785358257469,136.3826141357422]},{"page":423,"text":"coefficient g with characteristic saturation at the field strength of about 0.5 V/mm. In","rect":[53.81184387207031,157.27670288085938,385.1496514420844,148.28237915039063]},{"page":423,"text":"addition, the value of g was two orders of magnitude higher than the value of g for","rect":[53.81183624267578,169.23623657226563,385.17757545320168,160.3016815185547]},{"page":423,"text":"the electroclinic effect in the SmA* phase. The original picture presented in that","rect":[53.81090545654297,181.13900756835938,385.13386399814677,172.20445251464845]},{"page":423,"text":"paper is reproduced here without any changes, Fig. 13.17. We can see that, in the","rect":[53.81090545654297,193.09857177734376,385.17359197809068,184.1640167236328]},{"page":423,"text":"field absence (ground state), the tilt of molecules alternates from layer to layer,","rect":[53.8109245300293,205.05810546875,385.1130032112766,196.12355041503907]},{"page":423,"text":"however, in the strong field, the tilt within the smectic layers is uniform. Nowadays","rect":[53.8109245300293,217.01763916015626,385.15378202544408,208.0830841064453]},{"page":423,"text":"such pictures are called anticlinic and synclinic, respectively. The local polarization","rect":[53.8109245300293,228.97720336914063,385.07708064130318,220.02272033691407]},{"page":423,"text":"is always perpendicular to the tilt plane and also alternates in the zero field as shown","rect":[53.80991744995117,240.93673706054688,385.1508416276313,232.00218200683595]},{"page":423,"text":"by symbols l and \u000B.","rect":[53.80991744995117,252.89627075195313,136.3404964974094,243.9617156982422]},{"page":423,"text":"More impressive results, particularly, the tristable switching, were demons-","rect":[65.76513671875,264.85736083984377,385.1042417129673,255.9228057861328]},{"page":423,"text":"trated by Fukuda group [23]. In Fig. 13.18 we can see the chemical formula and","rect":[53.81311798095703,276.7601318359375,385.1429375748969,267.8255615234375]},{"page":423,"text":"the phase diagram of MHPOBC. Different antiferroelectric and ferrielectric phases","rect":[53.81311798095703,288.7196960449219,385.13412870513158,279.7652282714844]},{"page":423,"text":"in single compound 4-(L-methylheptyloxy-carbonyl)-40-octylbiphenyl-4-carboxyl-","rect":[53.81311798095703,300.67974853515627,385.15654884187355,291.0201416015625]},{"page":423,"text":"ate (MHPOBC) were unequivocally shown to exist by optical and electrooptical","rect":[53.81374740600586,312.6392822265625,385.1785112149436,303.684814453125]},{"page":423,"text":"techniques, dielectric spectroscopy, X-ray analysis, etc. In this compound, addi-","rect":[53.81374740600586,324.5988464355469,385.13665138093605,315.664306640625]},{"page":423,"text":"tionally to the known SmA* and SmC* (SmCb* in the figure) phases, new phases","rect":[53.81374740600586,337.4287109375,385.16717924224096,327.62384033203127]},{"page":423,"text":"have been revealed: antiferroelectric SmCA* and SmCa* and ferrielectric SmCg*","rect":[53.813411712646487,349.3883361816406,385.17937556317818,339.5240173339844]},{"page":423,"text":"phase. This work stimulated fast development of investigations in this area, see","rect":[53.81462860107422,360.4778747558594,385.11463201715318,351.5433349609375]},{"page":423,"text":"review articles [24, 25]. As we see, MHPOBC reveals rich polymorphism and","rect":[53.81462860107422,372.3806457519531,385.14251032880318,363.3863220214844]},{"page":423,"text":"becomes a model compound for further studies. Other liquid crystals made up of","rect":[53.81362533569336,384.3402099609375,385.1485532364048,375.3857421875]},{"page":423,"text":"chiral molecules that include three-benzene-ring cores and long tails with asym-","rect":[53.81362533569336,396.29974365234377,385.1186765274204,387.36517333984377]},{"page":423,"text":"metric carbon atoms and dipole moments also show a variety of similar phases","rect":[53.81362533569336,408.2593078613281,385.1424905215378,399.32476806640627]},{"page":423,"text":"(often called subphases). It has taken about 20 years to understand the structure of","rect":[53.81362533569336,420.2188415527344,385.14852271882668,411.2843017578125]},{"page":423,"text":"these subphases although many subtle details are still under question.","rect":[53.81362533569336,432.1783752441406,333.62652249838598,423.24383544921877]},{"page":423,"text":"Fig. 13.17 Field induced","rect":[53.812843322753909,511.618896484375,141.7969717422001,503.6858825683594]},{"page":423,"text":"switching between the","rect":[53.812843322753909,521.5271606445313,130.19515368723394,513.9328002929688]},{"page":423,"text":"antiferroelectric structure (left","rect":[53.812843322753909,531.429443359375,155.30337242331567,523.8181762695313]},{"page":423,"text":"sketch) and two ferroelectric","rect":[53.812843322753909,541.0836791992188,151.49166247629644,533.81103515625]},{"page":423,"text":"structures with opposite tilt","rect":[53.812843322753909,551.3982543945313,147.21545128073755,543.8038940429688]},{"page":423,"text":"and spontaneous polarization","rect":[53.812843322753909,561.3742065429688,153.2642568984501,553.7798461914063]},{"page":423,"text":"Ps. The directions of \u0007Ps","rect":[53.812843322753909,570.9347534179688,141.45032407881707,563.698486328125]},{"page":423,"text":"coincides with the field \u0007Ex","rect":[53.812843322753909,580.9161376953125,150.963566536117,573.6742553710938]},{"page":423,"text":"directions [22]","rect":[53.812843322753909,590.7280883789063,103.75853818518807,583.650146484375]},{"page":424,"text":"414","rect":[53.81307601928711,42.45538330078125,66.50468200831338,36.73210144042969]},{"page":424,"text":"13 Ferroelectricity and Antiferroelectricity in Smectics","rect":[197.23837280273438,44.275657653808597,385.18063815116207,36.68130111694336]},{"page":424,"text":"Fig. 13.18 Chemical formula and the phase sequence of the compound MHPOBC demonstrating","rect":[53.812843322753909,157.70394897460938,385.1754202285282,149.75401306152345]},{"page":424,"text":"a variety of transitions between ferro-, ferri- and antiferroelectric phases","rect":[53.812843322753909,167.6121826171875,301.29074557303707,160.017822265625]},{"page":424,"text":"Fig. 13.19","rect":[53.812843322753909,287.8431396484375,88.4846167006962,279.80853271484377]},{"page":424,"text":"Intermolecularinteractions responsible forformation ofdifferent liquid crystal phases:","rect":[94.42766571044922,287.7754211425781,385.1821871205813,280.1810607910156]},{"page":424,"text":"attractive anisotropic van der Waals and repulsive steric interactions for nematics (a), van der","rect":[53.812843322753909,297.7513732910156,385.1973885880201,290.1570129394531]},{"page":424,"text":"Waals (bifilic) and steric for SmA* (b), steric quadrupolar interaction for SmC*(c) and SmC*A (d)","rect":[53.81200408935547,307.72735595703127,385.1722421036451,300.13299560546877]},{"page":424,"text":"owed to molecular biaxiality. The density is increasing in a sequence: orthogonal (b), synclinic (c)","rect":[53.81307601928711,317.7030029296875,385.1714181290357,310.108642578125]},{"page":424,"text":"and anticlinic (d) phases. An interlayer steric correlations in SmC* (e) are shown by displacements","rect":[53.812191009521487,327.6222229003906,385.1146896648339,320.0278625488281]},{"page":424,"text":"of “grey molecules”. Note that the displacement of “gray molecules” may influence the next to","rect":[53.81307601928711,337.5981750488281,385.1680044570438,330.0038146972656]},{"page":424,"text":"nearest layer via a kind of relay race mechanism","rect":[53.81307601928711,347.5741271972656,220.0358757817267,339.9797668457031]},{"page":424,"text":"On account of the new experimental data and theoretical works, the same phase","rect":[65.76496887207031,384.3973083496094,385.0990680523094,375.4428405761719]},{"page":424,"text":"sequence for (L)-MHPOBC well purified from the right-handed (D)-enantiomer may","rect":[53.812950134277347,396.3569641113281,385.1663750748969,387.4024963378906]},{"page":424,"text":"be re-written as follows (for decreasing temperature) [26]:","rect":[53.813594818115237,408.3164978027344,288.8842912442405,399.3819580078125]},{"page":424,"text":"Iso ! SmA ! SmC a ! SmC ! SmC FI2 ! SmC FI1 ! SmC A (13.46)","rect":[61.40367889404297,431.9083251953125,385.1596921524204,423.24468994140627]},{"page":424,"text":"Here, ferrielectric SmC*FI1 and SmC*FI2 phases replace SmC*g and SmC*b","rect":[65.76595306396485,456.9685363769531,385.17287086388446,447.1042175292969]},{"page":424,"text":"phases. Further on we shall repeatedly refer to this phase sequence.","rect":[53.812843322753909,468.05816650390627,325.4742092659641,459.12359619140627]},{"page":424,"text":"13.2.2.2 Molecular Interactions","rect":[53.812843322753909,508.1450500488281,194.3961526309128,500.6348571777344]},{"page":424,"text":"A variety of different phases emerging in narrow temperature interval and also","rect":[53.812843322753909,533.8072509765625,385.1516961198188,524.8727416992188]},{"page":424,"text":"easily converted into each other by electric field (see below) testifies that different","rect":[53.812843322753909,545.7667846679688,385.1716142422874,536.832275390625]},{"page":424,"text":"inter-layer interactions have comparable energy. Moreover, molecules in one layer","rect":[53.812843322753909,557.6695556640625,385.11788307038918,548.7350463867188]},{"page":424,"text":"may interact even with next-to-nearest layers. Figure 13.19 may help to understand","rect":[53.812843322753909,569.6290893554688,385.1466912370063,560.694580078125]},{"page":424,"text":"some hierarchy of interactions beginning from the nematics and going down with","rect":[53.812843322753909,581.588623046875,385.11489192060005,572.6541137695313]},{"page":424,"text":"temperature. It is instructive to assume that our molecules are elongated, biaxial,","rect":[53.812843322753909,593.5481567382813,385.11883206869848,584.6136474609375]},{"page":425,"text":"13.2 Introduction to Antiferroelectrics","rect":[53.812843322753909,42.55594253540039,184.14123995780268,36.68026351928711]},{"page":425,"text":"415","rect":[372.4981994628906,42.55594253540039,385.1898245254032,36.62946701049805]},{"page":425,"text":"chiral with relatively long tails and having transverse dipole moments. At relatively","rect":[53.812843322753909,68.2883529663086,385.0929802995063,59.35380554199219]},{"page":425,"text":"high temperature of the nematic phase they rotate free and may be represented by","rect":[53.812843322753909,80.24788665771485,385.1676873307563,71.31333923339844]},{"page":425,"text":"ellipsoids with tails showing translational invariance, sketch (a). Between such","rect":[53.812843322753909,92.20748138427735,385.12087336591255,83.27293395996094]},{"page":425,"text":"ellipsoids the most important are the attractive anisotropic Van der Waals interac-","rect":[53.812843322753909,104.11019134521485,385.0890134414829,95.17564392089844]},{"page":425,"text":"tions (of Maier-Saupe type, decaying with distance as r\u00026) and repulsive steric","rect":[53.812843322753909,116.07039642333985,385.1312640972313,104.97955322265625]},{"page":425,"text":"ones. In the SmA* phase (b), due to decreasing thermal motion, the molecule form","rect":[53.814308166503909,128.02999877929688,385.13724468827805,119.09544372558594]},{"page":425,"text":"layers because the specific core-to-core and tail-to-tail Wan-der-Waals interactions","rect":[53.814308166503909,139.98953247070313,385.17108549224096,131.0549774169922]},{"page":425,"text":"may be more preferable, as very often observed in lyotropic systems (biphilic","rect":[53.814308166503909,151.94906616210938,385.15915716363755,143.01451110839845]},{"page":425,"text":"effect). However, molecular rotation about the layer normal is still free and, for a","rect":[53.814308166503909,163.90863037109376,385.1571735210594,154.9740753173828]},{"page":425,"text":"moment, the chirality may be ignored.","rect":[53.814308166503909,175.86813354492188,208.44500394369846,166.93357849121095]},{"page":425,"text":"Remember now that our molecules are biaxial i.e. they have either a lath-like","rect":[65.7663345336914,187.82766723632813,385.1731342144188,178.8931121826172]},{"page":425,"text":"form or a special form of the tails anti-symmetrically bent in the figure plane. Then,","rect":[53.814308166503909,199.73043823242188,385.1343044808078,190.79588317871095]},{"page":425,"text":"with decreasing temperature, mostly due to steric reasons, they may acquire a","rect":[53.814308166503909,211.69000244140626,385.15915716363755,202.7554473876953]},{"page":425,"text":"collective tilt and form a SmC* phase (c). Now, due to chiral symmetry C2 and","rect":[53.814308166503909,223.6495361328125,385.14626399091255,214.71498107910157]},{"page":425,"text":"transverse molecular dipoles each smectic layer acquires spontaneous polarization","rect":[53.814414978027347,235.61013793945313,385.0806511979438,226.6755828857422]},{"page":425,"text":"Ps and the helical arrangement of the layers on a micrometer scale. At even lower","rect":[53.814414978027347,247.56985473632813,385.0993589004673,238.6352996826172]},{"page":425,"text":"temperature, specific packing of molecular tails can stabilise the antiferroelectric","rect":[53.813228607177737,259.5294189453125,385.1481403179344,250.59486389160157]},{"page":425,"text":"phase. Indeed, the anticlinic arrangement of cores emerges (d) that increases","rect":[53.813228607177737,271.48895263671877,385.12518705474096,262.55438232421877]},{"page":425,"text":"density and reduces entropy. The steric forces can also provide the molecular","rect":[53.813228607177737,283.448486328125,385.1520932754673,274.513916015625]},{"page":425,"text":"interaction not only within the smectic layers but also between near neighbour","rect":[53.813228607177737,295.35125732421877,385.1063474258579,286.41668701171877]},{"page":425,"text":"(NN) layers as qualitatively pictured in Fig. 13.19e. Moreover, it is also seen how","rect":[53.813228607177737,307.3108215332031,385.1231970782551,298.37628173828127]},{"page":425,"text":"the distorted part is advanced up beyond the boundary of a neighbour layer. It","rect":[53.813201904296878,319.27032470703127,385.1779008633811,310.33575439453127]},{"page":425,"text":"means that the steric correlations may also be installed between next-to-nearest","rect":[53.813201904296878,331.2298889160156,385.0953202969749,322.29534912109377]},{"page":425,"text":"neighbours (NNN).","rect":[53.813201904296878,343.1894226074219,130.02661557699924,334.2548828125]},{"page":425,"text":"Similarly, the electrostatic correlations may be installed between NN and NNN","rect":[65.76622009277344,355.14898681640627,385.1371028107002,346.21441650390627]},{"page":425,"text":"layers. Note that P0 is a large collective dipole moment lying, due to chirality, in the","rect":[53.8151741027832,367.10968017578127,385.17102850152818,358.1739501953125]},{"page":425,"text":"layer plane, perpendicularly to the tilt plane. The energy between permanent","rect":[53.81325912475586,379.0692138671875,385.1412187344749,370.1346435546875]},{"page":425,"text":"molecular dipoles decay with distance as r\u00023 (see Section 3.2). It seems that the","rect":[53.81325912475586,390.97198486328127,385.1737750835594,379.9212646484375]},{"page":425,"text":"negative repulsive forces between parallel permanent dipoles may provide the long-","rect":[53.813045501708987,402.9318542480469,385.15291725007668,393.997314453125]},{"page":425,"text":"range interaction necessary for the antiparallel dipole packing i.e. antiferroelectric","rect":[53.813045501708987,414.8913879394531,385.1787799663719,405.95684814453127]},{"page":425,"text":"order of the layers. However, it is known from electrodynamics, that the same","rect":[53.813045501708987,426.8509216308594,385.1270221538719,417.9163818359375]},{"page":425,"text":"molecular dipoles oriented even in the same direction within a thin, infinitely wide","rect":[53.813045501708987,438.8104553222656,385.12403143121568,429.87591552734377]},{"page":425,"text":"(smectic) layer do not create an electric field outside the layer (due to complete","rect":[53.813045501708987,450.77001953125,385.1777728862938,441.83544921875]},{"page":425,"text":"compensation of the fields of individual dipoles). Therefore dipolar smectic layers","rect":[53.813045501708987,462.72955322265627,385.1002236758347,453.79498291015627]},{"page":425,"text":"cannot interact directly with each other. Nevertheless, there is a long-range interac-","rect":[53.813045501708987,474.6891174316406,385.08818946687355,465.75457763671877]},{"page":425,"text":"tion owed to fluctuations of P0 coupled to the director fluctuations. The latter are","rect":[53.813045501708987,486.5928039550781,385.16440618707505,477.6573486328125]},{"page":425,"text":"long-wave Goldstone excitations requiring very low energy (Kq2) in the limit of","rect":[53.81362533569336,498.5523376464844,385.15303932038918,487.50152587890627]},{"page":425,"text":"wavevectors q ! 0 as any hydrodynamic mode. It is that long-range coherent","rect":[53.814205169677737,510.5119934082031,385.0813432461936,501.57745361328127]},{"page":425,"text":"polarization wave that, according to [27], installs necessary correlations responsible","rect":[53.814205169677737,522.4714965820313,385.17295110895005,513.5369873046875]},{"page":425,"text":"for antiferroelectricity. There are other electrostatic interactions such as quadrupole–","rect":[53.814205169677737,534.4310302734375,385.26650324872505,525.4965209960938]},{"page":425,"text":"quadrupole or flexoelectric ones. The latter emerges due to spatial modulation of","rect":[53.814205169677737,546.390625,385.1511167129673,537.4561157226563]},{"page":425,"text":"the tilt of molecules in the layered structure. Consider briefly few interesting","rect":[53.814205169677737,558.3501586914063,385.13515559247505,549.4156494140625]},{"page":425,"text":"models that have been suggested for understanding polymorphism of tilted anti-","rect":[53.814205169677737,570.3096923828125,385.10823951570168,561.3751831054688]},{"page":425,"text":"ferroelectric smectics.","rect":[53.814205169677737,580.1506958007813,142.6057552864719,573.2779541015625]},{"page":426,"text":"416","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.68026351928711]},{"page":426,"text":"13 Ferroelectricity and Antiferroelectricity in Smectics","rect":[197.23812866210938,44.274620056152347,385.18039401053707,36.68026351928711]},{"page":426,"text":"13.2.2.3 Models","rect":[53.812843322753909,66.46558380126953,127.26098264800265,59.035072326660159]},{"page":426,"text":"Continuous models. With discovery of antiferroelectrics a question has arised","rect":[53.812843322753909,92.20748138427735,385.0999383073188,83.18328857421875]},{"page":426,"text":"about the possible structures and order parameters describing the new phases.","rect":[53.812843322753909,104.11019134521485,385.1596340706516,95.17564392089844]},{"page":426,"text":"Since all the structures are based on the single tilted SmC* layers of the same C2","rect":[53.812843322753909,116.0697250366211,385.18130140630105,107.13517761230469]},{"page":426,"text":"symmetry, it was suggested to use the same c-director to characterised each pair of","rect":[53.812843322753909,128.02999877929688,385.15068946687355,119.09544372558594]},{"page":426,"text":"neighbour layers (bilayer model [28]). Taking two neighbour layers i and i + 1, one","rect":[53.81385040283203,139.98953247070313,385.14670599176255,131.0549774169922]},{"page":426,"text":"writes two order parameters, ferroelectric and antiferroelectric, both in terms of the","rect":[53.81485366821289,151.94906616210938,385.17261541559068,143.01451110839845]},{"page":426,"text":"director components x(nxnz, nynz), see Eqs. (13.9a, b) where z is the normal to","rect":[53.81485366821289,164.83863830566407,385.14299861005318,154.65533447265626]},{"page":426,"text":"smectic layers:","rect":[53.813106536865237,175.86846923828126,113.94450242588113,166.9339141845703]},{"page":426,"text":"1","rect":[148.69346618652345,196.7962646484375,153.6705712418891,190.06298828125]},{"page":426,"text":"xF ¼ 2\u0002xi þ xiþ1\u0003 and xAF ¼ 2\u0002xi \u0002 xiþ1\u0003","rect":[125.75218200683594,210.3992919921875,313.2790912456688,195.09410095214845]},{"page":426,"text":"(13.47)","rect":[356.0712585449219,205.28468322753907,385.1594480117954,196.80831909179688]},{"page":426,"text":"Then, for paraelectric SmA phase both xF ¼ 0 and xAF ¼ 0, for ferroelectric","rect":[65.76569366455078,233.96685791015626,385.0864948101219,224.71316528320313]},{"page":426,"text":"SmC* phase xF 6¼ 0 but xAF ¼ 0 as discussed in Section 13.1, for antiferroelectric","rect":[53.81432342529297,245.926025390625,385.15174139215318,236.66290283203126]},{"page":426,"text":"SmC*A phase xF ¼ 0 but xAF ¼6 0, and for ferrielectric phases SmC*FI both xF ¼6 0","rect":[53.813899993896487,257.8291015625,385.1796807389594,248.56585693359376]},{"page":426,"text":"and xAF ¼6 0. Now the Landau expansion of the free energy in the vicinity of","rect":[53.81493377685547,269.78875732421877,385.14742408601418,260.5255126953125]},{"page":426,"text":"transitions between the paraelectric, ferroelectric and antiferroelectric phases will","rect":[53.81355667114258,281.748291015625,385.14546067783427,272.813720703125]},{"page":426,"text":"operate with two order parameters and both coefficients at the x2 terms in the free","rect":[53.81355667114258,293.7078552246094,385.1377643413719,282.6571044921875]},{"page":426,"text":"energy are considered to be dependent on temperature:","rect":[53.813838958740237,305.6675720214844,275.1437211758811,296.7330322265625]},{"page":426,"text":"21aFxF2 ¼ 21aFðT \u0002 TFÞxF2 and 12aAFx2AF ¼ 21aAFðT \u0002 TAFÞxA2F","rect":[89.38606262207031,340.19830322265627,349.10728692182127,319.86199951171877]},{"page":426,"text":"The two polarizations PF and PAF may be taken as secondary order parameters","rect":[65.76496887207031,363.7655029296875,385.1219827090378,354.8309326171875]},{"page":426,"text":"coupled with the genuine order parameters. As a result, depending of the model, the","rect":[53.81303024291992,375.7250671386719,385.1727985210594,366.79052734375]},{"page":426,"text":"theory predicts transitions from the smectic A phase into either the synclinic","rect":[53.81303024291992,387.6846008300781,385.1787799663719,378.75006103515627]},{"page":426,"text":"ferroelectric phase at temperature TF or into an anticlinic antiferroelectric phase","rect":[53.81303024291992,399.5877380371094,385.0993121929344,390.65283203125]},{"page":426,"text":"at TAF. One intermediate ferrielectric phase is also predicted that has a tilt plane in","rect":[53.8132438659668,411.54742431640627,385.13897028974068,402.59295654296877]},{"page":426,"text":"the i þ 1 layer turned through some angle j with respect to the tilt plane in the","rect":[53.813011169433597,423.5069580078125,385.1737445659813,414.5723876953125]},{"page":426,"text":"i layer. The models based on the two order parameters are of continuous nature","rect":[53.81399917602539,435.4665222167969,385.1388629741844,426.531982421875]},{"page":426,"text":"(j may take any values) and, although conceptually are very important, cannot","rect":[53.81301498413086,447.4260559082031,385.130995345803,438.49151611328127]},{"page":426,"text":"explain a variety of intermediate phases and their basic properties.","rect":[53.812015533447269,459.3855895996094,321.6509975960422,450.4510498046875]},{"page":426,"text":"Discrete models. Themostadvanced are discrete models that explicitlytake into","rect":[65.7640380859375,471.3451232910156,385.15187922528755,462.33087158203127]},{"page":426,"text":"account the interactions between nearest neighbour (NN) layers and even next to","rect":[53.812007904052737,483.3046569824219,385.13991633466255,474.3701171875]},{"page":426,"text":"nearest neighbour (NNN) layers. Among those approaches the most successful are","rect":[53.812007904052737,495.2074279785156,385.1598590679344,486.27288818359377]},{"page":426,"text":"Ising models [24] and the XY models, particularly, the so-called clock model [6, 26].","rect":[53.812007904052737,507.206787109375,385.1518215706516,498.2124938964844]},{"page":426,"text":"Consider one of the most successive Ising models. From the electrooptic study it","rect":[65.76197814941406,519.12646484375,385.1667619473655,510.19195556640627]},{"page":426,"text":"was clear that between ferroelectric and antiferroelectric phases there are interme-","rect":[53.80996322631836,531.0860595703125,385.06722389070168,522.1515502929688]},{"page":426,"text":"diate “subphases” of the mixed type. Thus, from the very beginning it was tempting","rect":[53.80996322631836,543.0455932617188,385.0990838151313,534.111083984375]},{"page":426,"text":"to classify the new phases using analogy with their crystalline counterparts. Such an","rect":[53.80996322631836,555.005126953125,385.1478203873969,546.0706176757813]},{"page":426,"text":"analogy is based on the assumption (even counter-intuitive) that there are SmC-like","rect":[53.80996322631836,566.9646606445313,385.16977728082505,558.0301513671875]},{"page":426,"text":"(achiral) correlations of the type shown in Fig.13.19c, d. For certain molecular","rect":[53.80996322631836,578.9241943359375,385.1497739395298,569.9896850585938]},{"page":426,"text":"parameters synclinic and anticlinic order may compete but the in-plane tilt configuration","rect":[53.80995559692383,590.8269653320313,385.2025078873969,581.8924560546875]},{"page":427,"text":"13.2 Introduction to Antiferroelectrics","rect":[53.81232833862305,42.55594253540039,184.1407364177636,36.68026351928711]},{"page":427,"text":"Fig. 13.20 Ising model.","rect":[53.812843322753909,67.58130645751953,137.76697036817036,59.648292541503909]},{"page":427,"text":"Sequence of ferroelectric","rect":[53.812843322753909,77.4895248413086,139.65294787668706,69.89517211914063]},{"page":427,"text":"(qF ¼ 1), ferrielectric","rect":[53.812843322753909,87.31558227539063,128.1699003913355,79.81430053710938]},{"page":427,"text":"(qF ¼ n/m ¼ 2/8 ¼ 1/4, 1/3","rect":[53.81232833862305,97.29147338867188,150.9775595107548,89.79006958007813]},{"page":427,"text":"and 3/4) and antiferroelectric","rect":[53.81147384643555,107.02178192138672,153.3830351569605,99.76608276367188]},{"page":427,"text":"(qF ¼ 0) phases","rect":[53.81147384643555,117.33626556396485,108.8668106365136,109.74191284179688]},{"page":427,"text":"417","rect":[372.4976806640625,42.55594253540039,385.18930572657509,36.73106384277344]},{"page":427,"text":"always remains. As mentioned above, such structures may be described in terms of","rect":[53.812843322753909,187.48818969726563,385.1487058242954,178.5536346435547]},{"page":427,"text":"ferroelectric fraction qF, in analogy with the microscopic Ising model describing the","rect":[53.812843322753909,199.44781494140626,385.1722797222313,190.51316833496095]},{"page":427,"text":"interaction energy of the up and down (or \u0007) spins in one-dimensional lattice. In","rect":[53.8134880065918,211.40737915039063,385.1483086686469,202.4728240966797]},{"page":427,"text":"Fig. 13.20, the “ferroelectric states” marked off by letter F have the same sign of the","rect":[53.81248092651367,223.31015014648438,385.1732562847313,214.37559509277345]},{"page":427,"text":"molecular tilt in the neighbour smectic layers whereas the “antiferroelectric states”","rect":[53.81248092651367,235.26968383789063,385.10950506402818,226.3351287841797]},{"page":427,"text":"with opposite tilt in neighbour layers marked by AF (here tilt angles \u0007W play the","rect":[53.81248092651367,247.22921752929688,385.17417181207505,237.995849609375]},{"page":427,"text":"role of up and down spins). Microscopically, the interaction between nearest and","rect":[53.812442779541019,259.18878173828127,385.1403740983344,250.2542266845703]},{"page":427,"text":"next to nearest layers is taken into account. The genuine ferroelectric F/F/F/F and","rect":[53.812442779541019,271.1483154296875,385.14238825849068,262.2137451171875]},{"page":427,"text":"antiferroelectric AF/AF/AF/AF. ...phases correspond to qF ¼ 1 and qF ¼ 0/1. The","rect":[53.812442779541019,283.10784912109377,385.1484760112938,274.17327880859377]},{"page":427,"text":"sequences AF/AF/F/AF/AF/F... , AF/F/AF/F/AF/F... and AF/F/F/F/AF/FFF mean","rect":[53.81462860107422,295.0682373046875,385.2689141373969,286.1336669921875]},{"page":427,"text":"ferrielectric phases with qF ¼ 1/3, 1/2, and 3/4, respectively. As in the case of","rect":[53.81560516357422,307.02801513671877,385.1487973770298,298.09326171875]},{"page":427,"text":"crystalline antiferroelectrics, for both n and m increasing the difference between","rect":[53.81392288208008,318.9307861328125,385.1776055436469,309.9962158203125]},{"page":427,"text":"qF values becomes step-by-step smaller down to zero and the Devil staircase forms.","rect":[53.81392288208008,330.890625,385.1094021370578,321.9560546875]},{"page":427,"text":"Note that in the limit of n/m ! 1 a ferrielectric becomes a ferroelectric.","rect":[53.81331253051758,340.887939453125,345.17998929526098,333.91558837890627]},{"page":427,"text":"In the Ising model, all the molecules are in the tilt plane but, despite such a","rect":[65.76533508300781,354.8097229003906,385.15723455621568,345.87518310546877]},{"page":427,"text":"severe simplification, the electrooptic measurements and resonant X-ray scattering","rect":[53.813316345214847,366.7692565917969,385.1621941666938,357.834716796875]},{"page":427,"text":"[29] have confirmed the sequence of ferro- ferri- and antiferroelectric phases.","rect":[53.813316345214847,378.72882080078127,385.1611599495578,369.79425048828127]},{"page":427,"text":"However, the same X-ray experiments clearly showed that the tilt planes in","rect":[53.812320709228519,390.6883239746094,385.1392449479438,381.7537841796875]},{"page":427,"text":"different layers are not at all parallel. Moreover, in frameworks of the Ising models","rect":[53.812320709228519,402.64788818359377,385.18597807036596,393.71331787109377]},{"page":427,"text":"the structure of the SmC*a phase could not be understood. Therefore, another","rect":[53.812320709228519,414.5514831542969,385.10503516999855,405.5563049316406]},{"page":427,"text":"approach has been developed.","rect":[53.812984466552737,426.5110168457031,174.68894620444065,417.57647705078127]},{"page":427,"text":"In the discrete clock model [26], one operates with the c-director lying in the XY","rect":[65.76499938964844,438.4705505371094,385.1488318438801,429.5360107421875]},{"page":427,"text":"plane and the tilt plane in layer i þ 1 is allowed to be at a discrete angle j with","rect":[53.81296157836914,450.4300842285156,385.11797419599068,441.49554443359377]},{"page":427,"text":"respect to the neighbour layer i. Therefore, for a ferroelectric structure j ¼ 0, for","rect":[53.81295394897461,462.3896484375,385.18062721101418,453.455078125]},{"page":427,"text":"antiferroelectricj ¼ p,forferrielectricstructureitcouldbe2p/3orp/2(byanalogy","rect":[53.81295394897461,474.3492126464844,385.12688532880318,465.4146728515625]},{"page":427,"text":"with a clock hand in the x, y plane). Correspondingly, SmC*, SmC*A, SmC*FI1 and","rect":[53.81393814086914,486.3087463378906,385.14626399091255,477.37420654296877]},{"page":427,"text":"SmC*FI2 phases have unit cells of one, two, three or four smectic layers and the order","rect":[53.814414978027347,498.2691345214844,385.1453794082798,489.2746887207031]},{"page":427,"text":"parameter is abruptly changed from layer to layer. Note that in any continuous model","rect":[53.813446044921878,510.1719055175781,385.1194291836936,501.2174377441406]},{"page":427,"text":"a number of order parameters corresponds to a great number of layers in a unit cell","rect":[53.813453674316409,522.1314086914063,385.11945970127177,513.1968994140625]},{"page":427,"text":"and even phenomenological theory becomes very complicated. Alternatively, in the","rect":[53.813453674316409,534.0909423828125,385.17124212457505,525.1564331054688]},{"page":427,"text":"discrete clock model the interlayer interactions can be separated from the molecular","rect":[53.813453674316409,546.050537109375,385.15230689851418,537.1160278320313]},{"page":427,"text":"interactions within the smectic layer. Due to complexity of both types of interactions","rect":[53.813453674316409,558.0100708007813,385.1682473574753,549.0755615234375]},{"page":427,"text":"they are modelled by phenomenological approximations based on the symmetry","rect":[53.813453674316409,569.9696044921875,385.17022028974068,561.0350952148438]},{"page":427,"text":"arguments. The intra-layer interactions were considered most important: they","rect":[53.813453674316409,581.9291381835938,385.1085137467719,572.9746704101563]},{"page":427,"text":"induce both smectic order and tilt. Inter-layer interactions between nearest layers","rect":[53.81246566772461,593.888671875,385.10052885161596,584.9342041015625]},{"page":428,"text":"418","rect":[53.814292907714847,42.55771255493164,66.50589889674112,36.73283386230469]},{"page":428,"text":"13 Ferroelectricity and Antiferroelectricity in Smectics","rect":[197.23959350585938,44.276390075683597,385.18185885428707,36.68203353881836]},{"page":428,"text":"responsible for ferro- or antiferroelectricity are usually weak because transitions","rect":[53.812843322753909,68.2883529663086,385.09894193755346,59.35380554199219]},{"page":428,"text":"between synclinic and anticlinic structures easily occur. Due to this, long range","rect":[53.812843322753909,80.24788665771485,385.14575994684068,71.31333923339844]},{"page":428,"text":"interactions between next to nearest layers especially electrostatic and chiral ones","rect":[53.812843322753909,92.20748138427735,385.12286771880346,83.27293395996094]},{"page":428,"text":"become very important.","rect":[53.812843322753909,104.11019134521485,149.08858914877659,95.17564392089844]},{"page":428,"text":"All these interactions are taken into account in the discrete phenomenological","rect":[65.76486206054688,116.0697250366211,385.178602767678,107.13517761230469]},{"page":428,"text":"model. The chiral interaction is also included although its contribution is consid-","rect":[53.812843322753909,128.02932739257813,385.1098874649204,119.09477233886719]},{"page":428,"text":"ered small. As the primary and secondary order parameters, the tilt vector xi and","rect":[53.812843322753909,139.98886108398438,385.14626399091255,130.73556518554688]},{"page":428,"text":"polarization vector Pi for a single smectic layer i are taken into account. Both","rect":[53.814414978027347,151.94937133789063,385.13698664716255,143.0145721435547]},{"page":428,"text":"parameters may vary from layer to layer, the tilt magnitude being constant and only","rect":[53.81307601928711,163.908935546875,385.12310114911568,154.97438049316407]},{"page":428,"text":"directions of xi and Pi change. Then Landau expansion for the free energy is written","rect":[53.81307601928711,175.86868286132813,385.15776911786568,166.61517333984376]},{"page":428,"text":"as a sum of different energy terms for N layers, where N is a number of layers in a","rect":[53.81293869018555,187.82821655273438,385.1607135601219,178.89366149902345]},{"page":428,"text":"unit cell. Further on, the polarization perpendicular to the tilt plane is excluded due","rect":[53.81393814086914,199.73098754882813,385.14185369684068,190.7964324951172]},{"page":428,"text":"to weakness of the chiral interaction within a smectic layer and possible stable","rect":[53.81393814086914,211.6905517578125,385.1726764507469,202.75599670410157]},{"page":428,"text":"structures have been found by minimisation of the free energy with respect to all tilt","rect":[53.81393814086914,223.65008544921876,385.1587358243186,214.7155303955078]},{"page":428,"text":"order parameters xi.","rect":[53.81393814086914,235.60964965820313,134.0185970833469,226.35635375976563]},{"page":428,"text":"As a result of numerical calculations [30], five phases shown in Fig. 13.21 have","rect":[65.76556396484375,247.56979370117188,385.1195453472313,238.63523864746095]},{"page":428,"text":"been found. In the first two rows we find the symbols and types of the phases","rect":[53.812557220458987,259.52935791015627,385.14938749419408,250.5948028564453]},{"page":428,"text":"whereas the third column represents the corresponding unit cells for the first four","rect":[53.812557220458987,271.4888916015625,385.1673520645298,262.5543212890625]},{"page":428,"text":"phases in terms of smectic layer numbers (m) per one period of the structure. The","rect":[53.812557220458987,283.44842529296877,385.1464008159813,274.51385498046877]},{"page":428,"text":"SmC*a phase is incommensurate in the sense that it has a short-pitch helical","rect":[53.81254959106445,295.3517761230469,385.1167131192405,286.35687255859377]},{"page":428,"text":"structure with a period not coinciding with integer number of the smectic layers.","rect":[53.81269454956055,307.31134033203127,385.0958218147922,298.37677001953127]},{"page":428,"text":"In the fourth column, a top view of the dielectric ellipsoid is presented for different","rect":[53.81269454956055,319.2708740234375,385.1714311368186,310.3363037109375]},{"page":428,"text":"layers within each unit cell. All these phases are in agreement with sequence","rect":[53.81269454956055,331.2304382324219,385.12464178277818,322.2958984375]},{"page":428,"text":"Fig. 13.21 Classification and structure of ferro-, ferri and antiferroelectric phases. The third","rect":[53.812843322753909,544.15380859375,385.18988556055947,536.2208251953125]},{"page":428,"text":"column represents the number (m) of the smectic layers l in a unit cell (for SmC*a abbreviation","rect":[53.812843322753909,554.06201171875,385.1523794570438,546.4507446289063]},{"page":428,"text":"IC means incommensurate). In the right column the orientation of the dielectric ellipsoid is","rect":[53.81350326538086,564.0377807617188,385.1590927410058,556.4434204101563]},{"page":428,"text":"presented for different layers within the unit cell viewed along the z-axis. The long-pitch helical","rect":[53.81350326538086,573.9569702148438,385.18282798972197,566.3626098632813]},{"page":428,"text":"structure due to the molecular chirality is ignored for clarity, although it slightly influences the","rect":[53.813472747802737,583.9329223632813,385.1548399665308,576.3385620117188]},{"page":428,"text":"value of angle j for the ellipsoids in the xy plane for each structure, see the next figure","rect":[53.813472747802737,593.9088745117188,352.3521361091089,586.3145141601563]},{"page":429,"text":"13.2 Introduction to Antiferroelectrics","rect":[53.813228607177737,42.55765151977539,184.1416366863183,36.68197250366211]},{"page":429,"text":"419","rect":[372.49859619140627,42.62538146972656,385.1902212539188,36.73277282714844]},{"page":429,"text":"(13.46). Note that in our simplified picture the molecular chirality is ignored and its","rect":[53.812843322753909,68.2883529663086,385.14972318755346,59.35380554199219]},{"page":429,"text":"role in the formation of long-pitch helix will be discussed below. The structure and","rect":[53.812843322753909,80.24788665771485,385.1427849870063,71.31333923339844]},{"page":429,"text":"properties of the phases pictured in Fig. 13.21 may be summarised in order of","rect":[53.812843322753909,92.20748138427735,385.14971290437355,83.27293395996094]},{"page":429,"text":"increasing temperature:","rect":[53.81385040283203,104.11019134521485,148.3669114834983,95.17564392089844]},{"page":429,"text":"1.","rect":[53.81385040283203,119.9364013671875,61.27950711752658,113.20311737060547]},{"page":429,"text":"2.","rect":[53.812496185302737,167.718505859375,61.27815289999728,160.9852294921875]},{"page":429,"text":"3.","rect":[53.81462860107422,191.75758361816407,61.280285315768768,184.90478515625]},{"page":429,"text":"4.","rect":[53.8140983581543,216.0,61.27975507284884,208.82415771484376]},{"page":429,"text":"5.","rect":[53.81456756591797,251.49891662597657,61.280224280612518,244.52658081054688]},{"page":429,"text":"SmC*A: optically uniaxial antiferroelectric phase with period of 2l and zero","rect":[66.27552795410156,122.0785140991211,385.26384821942818,113.0835952758789]},{"page":429,"text":"spontaneous polarization Ps. It manifests Bragg diffraction and optical rotatory","rect":[66.27518463134766,134.03817749023438,385.26381770185005,125.10356140136719]},{"page":429,"text":"power (ORP). NN interactions prevail. The helical structure is shown in Fig. 13.22:","rect":[66.27517700195313,145.99771118164063,385.2070756680686,137.04322814941407]},{"page":429,"text":"it is similar to that of SmC*, but the sign of helicity may be opposite.","rect":[66.274169921875,157.90045166015626,340.08965726401098,148.9658966064453]},{"page":429,"text":"SmC*FI1: biaxial ferrielectric phase with 3l periodicity and finite Ps. It manifests","rect":[66.274169921875,169.8604736328125,385.1374856387253,160.86569213867188]},{"page":429,"text":"ORP, which may change sign at a certain temperature.","rect":[66.27629852294922,181.82000732421876,286.4425015022922,172.8655242919922]},{"page":429,"text":"SmC*FI2: uniaxial antiferroelectric phase with 4l periodicity and zero Ps. However,","rect":[66.27630615234375,193.77975463867188,385.1817898323703,184.78524780273438]},{"page":429,"text":"on account of chirality the phase acquires small Ps and ferrielectric properties.","rect":[66.27574920654297,205.7393798828125,372.5499233772922,196.8047332763672]},{"page":429,"text":"SmC*: the helical, optically uniaxial phase, with period P0 >> l and finite Ps. It","rect":[66.27577209472656,217.69894409179688,385.1812577969749,208.7445831298828]},{"page":429,"text":"manifests Bragg diffraction and ORP and familiar to us from Sections 4.9 and","rect":[66.27620697021485,229.65859985351563,385.14638606122505,220.70411682128907]},{"page":429,"text":"13.1. The near-neighbour (NN) interactions prevail.","rect":[66.27620697021485,241.6181640625,273.8623013069797,232.68360900878907]},{"page":429,"text":"SmC*a: It is the most symmetric, antiferroelectric-type phase (Ps ¼ 0) that","rect":[66.2762451171875,253.52127075195313,385.1409745938499,244.52658081054688]},{"page":429,"text":"borders SmA phase. It is helical but the helicity originates not from the molecu-","rect":[66.2767562866211,265.4808349609375,385.1569760879673,256.5462646484375]},{"page":429,"text":"lar chirality but is due to specific NNN interactions. The pitch is short and","rect":[66.2767562866211,277.44036865234377,385.14592829755318,268.50579833984377]},{"page":429,"text":"incommensurate to the layer periodicity. In Fig. 13.21 the top view on the first","rect":[66.2767562866211,289.3999328613281,385.16389329502177,280.46539306640627]},{"page":429,"text":"five layers is shown and one may conclude that the helical pitch may be as short","rect":[66.27775573730469,301.3594665527344,385.1460099942405,292.4249267578125]},{"page":429,"text":"as 5l , but it vary with temperature. Due to short helical pitch the phase does not","rect":[66.27775573730469,313.3190002441406,385.1350541836936,304.3246765136719]},{"page":429,"text":"show ORP.","rect":[66.27775573730469,323.2565612792969,111.82822080160861,316.3240661621094]},{"page":429,"text":"Figure 13.21 presents the picture of the dielectric ellipsoid orientation within","rect":[65.76811981201172,343.1894836425781,385.13103571942818,334.25494384765627]},{"page":429,"text":"each unit cell that is at the nanometer scale. The weak molecular chirality results in","rect":[53.815086364746097,355.1490173339844,385.14202204755318,346.2144775390625]},{"page":429,"text":"additional weak twisting of all structures with characteristic pitch of about","rect":[53.815086364746097,367.1085510253906,385.1012407071311,358.17401123046877]},{"page":429,"text":"P0 \u0004 0.1–1 mm. An example of a such twisted structure is shown in Fig. 13.22; it","rect":[53.815086364746097,379.0693359375,385.1726518399436,370.134765625]},{"page":429,"text":"is an antiferroelectric double-layer cell describing two geared helices upon rotation","rect":[53.8139762878418,390.97210693359377,385.11794367841255,382.03753662109377]},{"page":429,"text":"about z-axis. The helices are shifted in phase by j ¼ p and have the same","rect":[53.8139762878418,402.9316711425781,385.12989080621568,393.99713134765627]},{"page":429,"text":"handedness. On the molecular scale, due to molecular chirality, the c-director","rect":[53.8139762878418,414.8912048339844,385.1736997207798,405.9367370605469]},{"page":429,"text":"turns from layer to layer by a small angle dj ¼ 2pl/P0, therefore, for l \u0004 1nm,","rect":[53.8139762878418,426.8507385253906,385.1258205940891,417.61737060546877]},{"page":429,"text":"Fig. 13.22 Chiral antiferroelectric SmC*A phase. Alternating tilt plane (a) and layer polarization","rect":[53.812843322753909,564.1054077148438,385.16818756251259,556.1724243164063]},{"page":429,"text":"(b) and the long-pitch helical structure (c). Note that the unit cell consisting of two layers rotates as","rect":[53.8132438659668,573.9569091796875,385.2037704753808,566.362548828125]},{"page":429,"text":"a whole forming two geared helices of the same handedness. This type of rotation is controlled by","rect":[53.813228607177737,583.932861328125,385.2087454238407,576.3385009765625]},{"page":429,"text":"molecular chirality inherent in all phases shown in Fig. 13.21","rect":[53.813228607177737,593.9088134765625,264.19772857813759,586.314453125]},{"page":430,"text":"420","rect":[53.812843322753909,42.55728530883789,66.50444931178018,36.73240661621094]},{"page":430,"text":"13 Ferroelectricity and Antiferroelectricity in Smectics","rect":[197.23812866210938,44.275962829589847,385.18039401053707,36.68160629272461]},{"page":430,"text":"dj \u0004 1\b. Consequently, to have a more correct structures in Fig. 13.21 one should","rect":[53.812843322753909,68.2883529663086,385.1784600358344,59.05499267578125]},{"page":430,"text":"present each ellipsoid configurations in the forth column as an overlapping stack of","rect":[53.81276321411133,80.24788665771485,385.1476987442173,71.31333923339844]},{"page":430,"text":"the same configurations. Such stuck will consist of the same sketches turned m-","rect":[53.81276321411133,92.20748138427735,385.1585630020298,83.27293395996094]},{"page":430,"text":"times through angle dj about the z-axis according to the number of layers (m) in a","rect":[53.81272506713867,104.11019134521485,385.16245306207505,94.8768310546875]},{"page":430,"text":"unit cell [30, 31]. More recent publications confirm the picture presented here, see","rect":[53.81371307373047,116.0697250366211,385.1117633648094,107.13517761230469]},{"page":430,"text":"[32] and references therein.","rect":[53.812713623046878,127.4217300415039,164.71058316733127,119.09477233886719]},{"page":430,"text":"13.2.2.4 Electric Field Switching","rect":[53.81172180175781,169.93899536132813,198.92818537763129,160.60601806640626]},{"page":430,"text":"Experimental data. Upon application of the external electric field, the transition","rect":[53.81172180175781,193.77841186523438,385.15456477216255,184.76417541503907]},{"page":430,"text":"temperaturesbetween different phaseschange [33].Itisseen inthe field–temperature","rect":[53.81172180175781,205.73794555664063,385.24411810113755,196.8033905029297]},{"page":430,"text":"phase diagram Fig. 13.23.First, we notice that all transition temperatures are shifted","rect":[53.811737060546878,217.697509765625,385.17253962567818,208.76295471191407]},{"page":430,"text":"considerably that confirms a subtle difference between the interactions responsible","rect":[53.81272888183594,229.65704345703126,385.17054022027818,220.7224884033203]},{"page":430,"text":"for antiferroelectricity. In addition, the temperature range of the polar SmC* phase","rect":[53.81272888183594,241.61660766601563,385.09388006402818,232.6820526123047]},{"page":430,"text":"becomes wider at the cost of the antiferroelectric SmCA* and unpolar SmCa*","rect":[53.81272888183594,253.52127075195313,385.17937556317818,244.52694702148438]},{"page":430,"text":"phases. The range of ferrielectric SmCFI2* phase remains unchanged. It can be","rect":[53.81462860107422,265.4809265136719,385.1521686382469,256.5462646484375]},{"page":430,"text":"understood as follows. As the smectic C* phase has high spontaneous polarization,","rect":[53.81338119506836,277.4404602050781,385.0836147835422,268.50592041015627]},{"page":430,"text":"due to the –PsE term, its free energy is reduced by the electric field. Therefore, the","rect":[53.81338119506836,289.4002685546875,385.17298162652818,280.4654541015625]},{"page":430,"text":"field stabilised the smectic C* phase and expands it temperature range. To some","rect":[53.81219482421875,301.35980224609377,385.1460651226219,292.42523193359377]},{"page":430,"text":"extent, the ferrielectric phases with lower spontaneous polarization are also stabi-","rect":[53.81219482421875,313.3193664550781,385.17000709382668,304.38482666015627]},{"page":430,"text":"lised by the field but not the antiferroelectric ones.","rect":[53.81219482421875,325.2789001464844,257.6694607308078,316.3443603515625]},{"page":430,"text":"In the field-off state the macroscopic polarization of the antiferroelectric phase is","rect":[65.76421356201172,337.23846435546877,385.1839944277878,328.30389404296877]},{"page":430,"text":"zero. With increasing field, the induced polarization, at first, increases linearly with","rect":[53.81220245361328,349.1412048339844,385.1142510514594,340.2066650390625]},{"page":430,"text":"field and then, at a certain threshold, the antiferroelectric (AF) structure with","rect":[53.81220245361328,360.70233154296877,385.1142510514594,352.16619873046877]},{"page":430,"text":"alternating molecular tilt transforms in the ferroelectric one (F) with a uniform","rect":[53.81220245361328,373.060302734375,385.16205547929368,364.125732421875]},{"page":430,"text":"tilt, see Fig. 13.24a. Correspondingly, the macroscopic polarization jumps from a","rect":[53.81220245361328,385.01983642578127,385.1570514507469,376.08526611328127]},{"page":430,"text":"low value to the level of the local polarization P0 [34]. The process is quite similar","rect":[53.81220245361328,396.9804992675781,385.1347592910923,388.0447998046875]},{"page":430,"text":"to that observed in crystalline antiferroelectrics. With a certain precaution we can","rect":[53.81281661987305,408.9400634765625,385.12282649091255,400.0054931640625]},{"page":430,"text":"speak about a field-induced AF-F non-equilibrium “phase transition”. The magni-","rect":[53.81281661987305,420.89959716796877,385.17058692781105,411.96502685546877]},{"page":430,"text":"tude of the switched polarization in some antiferroelectric materials can be quite","rect":[53.81281661987305,432.859130859375,385.12879217340318,423.924560546875]},{"page":430,"text":"0.4","rect":[217.16676330566407,478.477783203125,228.277399571775,472.71099853515627]},{"page":430,"text":"MHPOBC","rect":[332.10137939453127,467.7059020996094,371.5691273313637,460.912353515625]},{"page":430,"text":"Fig. 13.23 Electric","rect":[53.812843322753909,560.307861328125,121.5987105109644,552.3748779296875]},{"page":430,"text":"field–temperature phase","rect":[53.812843322753909,570.1593017578125,135.40548083567144,562.56494140625]},{"page":430,"text":"diagram of MHPOBC","rect":[53.812843322753909,580.13525390625,129.04275691236834,572.5239868164063]},{"page":430,"text":"(Adapted from [33])","rect":[53.812843322753909,590.1112060546875,123.51344388587167,582.516845703125]},{"page":430,"text":"T (°C)","rect":[293.9324645996094,588.1343994140625,315.3304273543828,580.592041015625]},{"page":430,"text":"A","rect":[364.005126953125,544.9506225585938,369.3366336166604,539.2078247070313]},{"page":430,"text":"C*","rect":[342.7384338378906,555.7662353515625,351.61896196797997,549.7275390625]},{"page":430,"text":"122","rect":[329.28985595703127,574.3781127929688,342.62262967431408,568.7553100585938]},{"page":430,"text":"124","rect":[371.8979797363281,574.8756103515625,385.23075345361095,569.2528076171875]},{"page":431,"text":"13.2 Introduction to Antiferroelectrics","rect":[53.81283950805664,42.55661392211914,184.14123995780268,36.68093490600586]},{"page":431,"text":"421","rect":[372.4981994628906,42.45501708984375,385.1898245254032,36.73173522949219]},{"page":431,"text":"Fig. 13.24 Tri-stable switching of an antiferroelectric liquid crystal. Typical hysteresis-type","rect":[53.812843322753909,197.72064208984376,385.12975451731207,189.78762817382813]},{"page":431,"text":"dependence [34] of the total polarization P as a function of the external field (a) and optical","rect":[53.812843322753909,207.62887573242188,385.1753511830813,200.03451538085938]},{"page":431,"text":"transmission-voltage curves [35] measured at three different temperatures (T1 > T2 > T3) in the","rect":[53.81196975708008,217.60482788085938,385.1547178962183,209.9596710205078]},{"page":431,"text":"same SmC*A phase (b)","rect":[53.813289642333987,227.58035278320313,133.72425931555919,219.9353485107422]},{"page":431,"text":"large, of about several hundred of nC/cm2. Upon a change in the field polarity, the","rect":[53.812843322753909,259.529541015625,385.1735309429344,248.4784698486328]},{"page":431,"text":"process reverses. Therefore, we have three distinct states, one stable antiferro-","rect":[53.81379318237305,271.48907470703127,385.15166602937355,262.55450439453127]},{"page":431,"text":"electric state and two (plus and minus) ferroelectric states with a certain memory.","rect":[53.81379318237305,283.4486083984375,385.1706509163547,274.5140380859375]},{"page":431,"text":"It should be noted that, in some materials, such a switching process between the","rect":[53.81379318237305,295.4081726074219,385.17356146051255,286.4736328125]},{"page":431,"text":"antiferroelectric and ferroelectric states could proceed via intermediate ferrielectric","rect":[53.81379318237305,307.3677062988281,385.14865911676255,298.43316650390627]},{"page":431,"text":"states.","rect":[53.81379318237305,317.2654113769531,78.46041532065158,311.4086608886719]},{"page":431,"text":"By a proper treatment of the electrodes, one can obtain a texture with a uniform","rect":[65.76580047607422,331.2300109863281,385.1566233503874,322.29547119140627]},{"page":431,"text":"orientation of the smectic normal in one direction within the cell plane. Between the","rect":[53.81379318237305,343.1895446777344,385.1725543804344,334.2550048828125]},{"page":431,"text":"crossed polarizers such a cell will be black if a polarizer is installed parallel to the","rect":[53.81379318237305,355.14910888671877,385.1715778179344,346.21453857421877]},{"page":431,"text":"smectic normal. Upon application of the electric field, the antiferroelectric structure","rect":[53.81379318237305,367.108642578125,385.08701360895005,358.174072265625]},{"page":431,"text":"becomes distorted. At low voltages of any polarity, the electrooptic response is","rect":[53.81379318237305,379.06817626953127,385.18649686919408,370.13360595703127]},{"page":431,"text":"proportional to E2: the bottom part of the curves has symmetric parabolic form [35]","rect":[53.81379318237305,391.0289611816406,385.1584714492954,379.97796630859377]},{"page":431,"text":"shown in Fig. 13.24b. Above the AF-F transition, the director acquires one of the","rect":[53.81264877319336,402.9884948730469,385.1714252300438,394.053955078125]},{"page":431,"text":"two symmetric angular positions (\u0007W on the conical surface) typical of the SmC*","rect":[53.81265640258789,414.94805908203127,385.13457575849068,405.7146911621094]},{"page":431,"text":"phase. At these two extreme positions the transmission is maximum. With increas-","rect":[53.81264114379883,426.8507995605469,385.10469947663918,417.916259765625]},{"page":431,"text":"ing temperature from T3 to T1 the AF–F threshold decreases due to a decrease of the","rect":[53.81264114379883,438.81036376953127,385.17212713434068,429.87579345703127]},{"page":431,"text":"potential barrier separating structures with alternating and uniform tilt. It is natural","rect":[53.81337356567383,450.7704162597656,385.14228684970927,441.83587646484377]},{"page":431,"text":"because within the SmC*A phase T1 is closer to the range of the SmC* phase than T2","rect":[53.81337356567383,462.7301940917969,385.18130140630105,453.73565673828127]},{"page":431,"text":"or T3.","rect":[53.812843322753909,474.27178955078127,76.63705869223361,465.97442626953127]},{"page":431,"text":"At high frequencies of the a.c. field, the total polarisation of the entire sample is","rect":[65.76506805419922,486.6494140625,385.18378080474096,477.71484375]},{"page":431,"text":"switched very fast and the ground, antiferroelectric state may be bypassed. Then the","rect":[53.81304931640625,498.6089782714844,385.1708453960594,489.6744384765625]},{"page":431,"text":"switching occurs between the two ferroelectric states as in an SSFLC cell. With","rect":[53.81304931640625,510.5685119628906,385.1270379166938,501.63397216796877]},{"page":431,"text":"increasing frequency (for example, from 100 Hz to 10 kHz) the double hysteresis","rect":[53.81304931640625,522.4712524414063,385.1658059512253,513.5367431640625]},{"page":431,"text":"loop is substituted by a single loop typical of ferroelectrics as shown in Fig. 13.16","rect":[53.81304931640625,534.4307861328125,385.15187922528755,525.4962768554688]},{"page":431,"text":"by the solid and the dashed curves.","rect":[53.81304931640625,546.3903198242188,195.13894315268284,537.455810546875]},{"page":431,"text":"Theoretical consideration. We shall consider a simple and instructive theory of","rect":[65.76506805419922,558.349853515625,385.1498960098423,549.3356323242188]},{"page":431,"text":"the switching of a helix free antiferroelectric phase [36]. The smectic layers normal","rect":[53.81304931640625,570.3093872070313,385.1807695157249,561.3748779296875]},{"page":431,"text":"h is aligned along the rubbing direction z in the plane of the cell (bookshelf","rect":[53.81304931640625,582.2689208984375,385.14690528718605,573.314453125]},{"page":431,"text":"geometry in Fig. 13.25). The tilt has the amplitude \u0007 W and its phase F changes","rect":[53.81306838989258,594.228515625,385.1658059512253,584.9951782226563]},{"page":432,"text":"422","rect":[53.81285095214844,42.45556640625,66.50445694117471,36.73228454589844]},{"page":432,"text":"Fig. 13.25 Geometry for","rect":[53.812843322753909,67.58130645751953,140.2951363907545,59.648292541503909]},{"page":432,"text":"discussion of the electric field-","rect":[53.812843322753909,75.76238250732422,155.40237516028575,69.89517211914063]},{"page":432,"text":"induced ferroelectric–","rect":[53.812843322753909,85.68157196044922,127.13563293604776,79.81436157226563]},{"page":432,"text":"antiferroelectric transition.","rect":[53.812843322753909,95.63212585449219,144.57560989453754,89.79031372070313]},{"page":432,"text":"Antiferroelectric structure","rect":[53.812843322753909,105.60813903808594,142.34017321848394,99.76632690429688]},{"page":432,"text":"(a) in the zero field and","rect":[53.812843322753909,116.99797821044922,133.63287872462198,109.74227905273438]},{"page":432,"text":"ferroelectric structure at the","rect":[53.81199645996094,125.50328063964844,148.26375720285894,119.66146850585938]},{"page":432,"text":"field exceeding the F–AF","rect":[53.81199645996094,137.23178100585938,140.0159085612014,129.63742065429688]},{"page":432,"text":"transition threshold (b)","rect":[53.81199645996094,146.8690643310547,132.1945046768873,139.61337280273438]},{"page":432,"text":"13 Ferroelectricity and Antiferroelectricity in Smectics","rect":[197.23814392089845,44.275840759277347,385.18042452811519,36.68148422241211]},{"page":432,"text":"by p in each subsequent layer. An applied electric field is within the smectic layer","rect":[53.812843322753909,191.8526611328125,385.1198667129673,182.91810607910157]},{"page":432,"text":"plane parallel to the x-axis. Due to chirality, each layer possesses polarization P0","rect":[53.81284713745117,203.81222534179688,385.18130140630105,194.87767028808595]},{"page":432,"text":"perpendicular to the tilt plane. The total polarization in the ground state is zero,","rect":[53.812843322753909,215.71527099609376,385.1497463753391,206.7807159423828]},{"page":432,"text":"sketch (a). With increasing external filed a transition is observed from the antiferro-","rect":[53.812843322753909,227.6748046875,385.15071998445168,218.74024963378907]},{"page":432,"text":"electric (AF) ground state to the ferroelectric (F) field-induced state, sketch (b).","rect":[53.812843322753909,239.63433837890626,385.1716579964328,230.6997833251953]},{"page":432,"text":"After transition, all vectors P0 are oriented along the field and the director azimuth","rect":[53.812843322753909,251.59420776367188,385.1169365983344,242.6593475341797]},{"page":432,"text":"is the same in all smectic layers, F ¼ 0.","rect":[53.812923431396487,263.5537414550781,216.65684171225315,254.5893096923828]},{"page":432,"text":"For simplicity we disregard a change in the W angle and focus our attention only","rect":[65.76593017578125,275.5133056640625,385.1288689713813,266.2799377441406]},{"page":432,"text":"on azimuthal motion of the director. The density of the bulk free energy can be","rect":[53.814910888671878,287.47283935546877,385.1517719097313,278.53826904296877]},{"page":432,"text":"taken in the form:","rect":[53.814910888671878,297.4104309082031,126.12329728672097,290.49786376953127]},{"page":432,"text":"F ¼ 12K\"\u0005qqFyi\u00062 þ \u0005qqFxi\u00062# þ W cosðFiþ1 \u0002 FiÞ \u0002 P0EcosFi","rect":[74.3205795288086,347.44287109375,335.0881145552864,317.5813903808594]},{"page":432,"text":"(13.48)","rect":[356.0715026855469,336.7838439941406,385.1596921524204,328.3074951171875]},{"page":432,"text":"Here, the first term describes the nematic-like elastic energy in one constant","rect":[65.76595306396485,376.9723815917969,385.172743392678,368.037841796875]},{"page":432,"text":"approximation (K \u0004 KNsin2W). This term allows a discussion of distortions below","rect":[53.81393051147461,388.93194580078127,385.1193213458332,377.8807678222656]},{"page":432,"text":"the AF-F threshold (a kind of the Frederiks transition as in nematics in a sample of a","rect":[53.81330490112305,400.8912658691406,385.1581500835594,391.95672607421877]},{"page":432,"text":"finite size). In fact, the most important specific properties of the antiferroelectric are","rect":[53.81330490112305,412.8507995605469,385.16419256402818,403.916259765625]},{"page":432,"text":"taken into account by the interaction potential W between molecules in neighbour","rect":[53.81330490112305,424.8103332519531,385.1063474258579,415.87579345703127]},{"page":432,"text":"layers: the second term in the equation corresponds to interaction of only the nearest","rect":[53.81230545043945,436.7698974609375,385.16010911533427,427.8353271484375]},{"page":432,"text":"layers (i) and (i þ 1). Let count layers from the top of our sketch (a); then for","rect":[53.81230545043945,448.7294616699219,385.17803321687355,439.794921875]},{"page":432,"text":"odd layers i, i þ 2, etc. the director azimuth is 0, and for even layers i þ 1, i þ 3,","rect":[53.81229782104492,460.6322021484375,385.1750454476047,451.6976318359375]},{"page":432,"text":"etc. the director azimuth is p. The third term describes interaction of the external","rect":[53.81332015991211,470.559814453125,385.1103654629905,463.6572265625]},{"page":432,"text":"field Ex with the layer polarization P0 of the layer i as in the case of ferroelectric","rect":[53.814308166503909,484.5523376464844,385.08692205621568,475.61676025390627]},{"page":432,"text":"cells. Although for substances with high P0 the dielectric anisotropy can be","rect":[53.813777923583987,496.5119934082031,385.15296209527818,487.57733154296877]},{"page":432,"text":"neglected, the quadratic-in-field effects are implicitly accounted for by the highest","rect":[53.814144134521487,508.4715270996094,385.1858354336936,499.5369873046875]},{"page":432,"text":"order terms proportional to P2.","rect":[53.814144134521487,520.4310302734375,177.86195035483127,509.3802795410156]},{"page":432,"text":"The solution of Eq. (13.48) depends on further simplifications. If we assume","rect":[65.76559448242188,532.3908081054688,385.0996784038719,523.456298828125]},{"page":432,"text":"that the director in the odd layers with Fi ¼ 0 is unaffected by an external field and","rect":[53.81357192993164,544.3505859375,385.14495173505318,535.3859252929688]},{"page":432,"text":"only the azimuth in the even layers Fiþ1 is changed from p to 0, then, for an","rect":[53.81411361694336,556.35693359375,385.15285578778755,547.2889404296875]},{"page":432,"text":"infinitely thick sample (x ! 1), the free energy is independent of both x and Fi.","rect":[53.81405258178711,568.2130737304688,385.1832241585422,559.2486572265625]},{"page":432,"text":"The corresponding torque balance equation reduces to the form with index (i þ 1)","rect":[53.814537048339847,580.1727294921875,385.15337501374855,571.2382202148438]},{"page":432,"text":"omitted:","rect":[53.815574645996097,590.1102905273438,87.18208940097878,583.19775390625]},{"page":433,"text":"13.2 Introduction to Antiferroelectrics","rect":[53.812843322753909,42.55594253540039,184.14123995780268,36.68026351928711]},{"page":433,"text":"423","rect":[372.4981994628906,42.55594253540039,385.1898245254032,36.73106384277344]},{"page":433,"text":"q2F","rect":[164.66741943359376,68.5971908569336,180.76449346497385,59.44980239868164]},{"page":433,"text":"K qy2 þ ð2W \u0002 P0EÞsinF ¼0","rect":[155.944091796875,84.25201416015625,283.0485772233344,67.80216217041016]},{"page":433,"text":"(13.49)","rect":[356.0716552734375,77.0156021118164,385.15984474031105,68.53923797607422]},{"page":433,"text":"If we disregard the elastic, nematic-like term we would see that the distortion has","rect":[65.76610565185547,107.7376937866211,385.13006986724096,98.80314636230469]},{"page":433,"text":"a threshold character with the threshold field Eth ¼ 2W=P0. It is easy to understand:","rect":[53.814083099365237,120.02665710449219,385.2067704922874,110.09606170654297]},{"page":433,"text":"the AF–F threshold is achieved when the field energy is sufficient for the director in","rect":[53.81418991088867,131.65750122070313,385.14211360028755,122.72294616699219]},{"page":433,"text":"even layers to overcome potential barrier W and change its azimuth from p to 0.","rect":[53.81418991088867,143.61703491210938,385.1749233772922,134.68247985839845]},{"page":433,"text":"Above the threshold, Ex > Eth the uniform ferroelectric structure is installed.","rect":[53.81418991088867,155.13888549804688,364.8373684456516,146.6420135498047]},{"page":433,"text":"At fields below EA\u0002F, the macroscopic polarization is absent due to alternating","rect":[65.76507568359375,167.5364990234375,385.0884942155219,158.6017608642578]},{"page":433,"text":"\u0007 P0, the first order term in polarization is absent and the distortion is controlled by","rect":[53.813350677490237,179.49618530273438,385.16692439130318,170.56163024902345]},{"page":433,"text":"a higher order term proportional to P02E2. This explains the parabolic form of the","rect":[53.81308364868164,191.45584106445313,385.17346990777818,180.3479766845703]},{"page":433,"text":"electrooptical response at the fields below AF–F threshold.","rect":[53.8137321472168,203.35861206054688,291.2037319710422,194.42405700683595]},{"page":433,"text":"To describe the dynamics of F at constant W, the viscous torque with viscosity","rect":[65.76573944091797,215.31814575195313,385.1057976823188,206.08477783203126]},{"page":433,"text":"coefficient gj should be added to the balance equation","rect":[53.81374740600586,228.15472412109376,273.48711482099068,218.34315490722657]},{"page":433,"text":"qF","rect":[156.9070587158203,250.6559600830078,169.03129339173166,243.494384765625]},{"page":433,"text":"q2F","rect":[190.95086669921876,250.6559600830078,207.1045203204426,241.50863647460938]},{"page":433,"text":"gj qt ¼ K qy2 þ ð2W \u0002 P0EÞsinF:","rect":[145.12460327148438,266.31085205078127,293.9008325306787,249.9176788330078]},{"page":433,"text":"(13.50)","rect":[356.07183837890627,259.131103515625,385.1600278457798,250.53521728515626]},{"page":433,"text":"Assuming small field-induced F angles, sinF ! F, the inverse switching time","rect":[65.7662582397461,289.85321044921877,385.16101873590318,280.8887634277344]},{"page":433,"text":"can be found in the vicinity of the AF–F transition. t\u0002AF1 ¼ g\u0002j1ðP0E \u0002 2WÞ. It shows","rect":[53.814231872558597,303.8358154296875,385.1427041445847,291.3309631347656]},{"page":433,"text":"a divergence of the switching time at the threshold field, as observed in experi-","rect":[53.81481170654297,313.7162780761719,385.1795896133579,304.78173828125]},{"page":433,"text":"ments. When the field is switched off (Ex ¼ 0), the inverse time for the back","rect":[53.81481170654297,325.277587890625,385.1192559342719,316.74127197265627]},{"page":433,"text":"relaxation from the ferroelectric to the antiferroelectric state is controlled solely","rect":[53.81427764892578,337.6355895996094,385.1093682389594,328.7010498046875]},{"page":433,"text":"by the interlayer potential t\u0002FA1 ¼ 2W=gj. Surprisingly, this simple theory [36]","rect":[53.81427764892578,351.2209777832031,385.15923438874855,339.1128845214844]},{"page":433,"text":"explains the most important experimental facts and can be applied to both chiral","rect":[53.813411712646487,361.55487060546877,385.1801896817405,352.62030029296877]},{"page":433,"text":"and achiral (banana-type) antiferroelectrics.","rect":[53.813411712646487,373.514404296875,229.9541439583469,364.579833984375]},{"page":433,"text":"13.2.3 Polar Achiral Systems","rect":[53.812843322753909,423.2282409667969,205.97582130167647,412.5904541015625]},{"page":433,"text":"13.2.3.1 The Problem","rect":[53.812843322753909,449.3362121582031,152.2818777741923,441.5172424316406]},{"page":433,"text":"As we know, chiral tilted mesophases, manifest ferroelectric (C*, F*, I* and other","rect":[53.812843322753909,474.6896667480469,385.1377194961704,465.755126953125]},{"page":433,"text":"less","rect":[53.812843322753909,485.0,68.72424711333469,477.71466064453127]},{"page":433,"text":"symmetric","rect":[75.62052917480469,486.6492004394531,117.86719549371564,477.71466064453127]},{"page":433,"text":"phases),","rect":[124.73062896728516,486.6492004394531,157.05193753744846,477.71466064453127]},{"page":433,"text":"antiferroelectric","rect":[163.92831420898438,485.0,228.01551092340316,477.71466064453127]},{"page":433,"text":"(SmCA*,","rect":[234.90481567382813,486.2507629394531,270.9790005257297,477.7744140625]},{"page":433,"text":"SmCa*)","rect":[277.84442138671877,486.2510681152344,310.21062599031105,477.77471923828127]},{"page":433,"text":"and","rect":[317.0431823730469,485.0,331.44692317060005,477.7149658203125]},{"page":433,"text":"ferrielectric","rect":[338.3412170410156,485.0,385.15488470270005,477.7149658203125]},{"page":433,"text":"(SmCFI*) properties. These properties owe to a tilt of elongated chiral molecules,","rect":[53.81405258178711,498.6091613769531,385.1562771370578,489.67462158203127]},{"page":433,"text":"and polar ordering of the molecular short axes (and transverse dipole moments)","rect":[53.813419342041019,510.5686950683594,385.08757911531105,501.6341552734375]},{"page":433,"text":"perpendicular to the tilt plane. The head-to-tail symmetry n ¼ \u0002n is conserved. The","rect":[53.813419342041019,522.5281982421875,385.1462787456688,513.5936889648438]},{"page":433,"text":"PS vector lies in the plane of a smectic layer perpendicularly to the tilt plane. Such","rect":[53.81343460083008,534.431396484375,385.14336482099068,525.4968872070313]},{"page":433,"text":"materials belong to improper ferro-, ferri and antiferroelectrics.","rect":[53.81245803833008,546.3909301757813,309.1409878304172,537.4564208984375]},{"page":433,"text":"Since discovery of chiral ferroelectrics in 1975, a search for the achiral analo-","rect":[65.76448059082031,558.3504638671875,385.11348853913918,549.356201171875]},{"page":433,"text":"gues of liquid crystal ferro- and antiferroelectrics was a challenge to researchers,","rect":[53.81244659423828,570.31005859375,385.1632351448703,561.3755493164063]},{"page":433,"text":"both theoreticians and experimentalists and recently there was a great progress in","rect":[53.81244659423828,582.2695922851563,385.1413506608344,573.3350830078125]},{"page":433,"text":"this area. The idea was to find a way to break non-polar symmetry D1h or C2h of","rect":[53.81244659423828,594.2291259765625,385.1521237930454,585.2946166992188]},{"page":434,"text":"424","rect":[53.812828063964847,42.4560546875,66.50443405299112,36.73277282714844]},{"page":434,"text":"13 Ferroelectricity and Antiferroelectricity in Smectics","rect":[197.23812866210938,44.276329040527347,385.18039401053707,36.68197250366211]},{"page":434,"text":"achiral compounds and conserve a liquid crystalline state. For example, in materials","rect":[53.812843322753909,68.2883529663086,385.18158353911596,59.35380554199219]},{"page":434,"text":"belonging to point group C1v the axis C1 is polar axis and polarization vector is","rect":[53.812843322753909,80.24800872802735,385.1876565371628,71.30349731445313]},{"page":434,"text":"parallel to C1. In biaxial smectics of C2v symmetry the polar axis and polarization","rect":[53.814022064208987,92.20772552490235,385.0815056901313,83.27305603027344]},{"page":434,"text":"vector is parallel to rotation axis C2. There were many theoretical suggestions to use","rect":[53.8133659362793,104.11067962646485,385.1306842632469,95.17594909667969]},{"page":434,"text":"such a reduced symmetry reviewed in [37], however, only in 1992 the first polar","rect":[53.8137321472168,116.0702133178711,385.1406186660923,107.13566589355469]},{"page":434,"text":"smectic liquid crystal showing all polar properties was synthesised [38]).","rect":[53.81374740600586,128.02981567382813,347.9686550667453,119.09526062011719]},{"page":434,"text":"13.2.3.2 Achiral Ferroelectrics","rect":[53.81376266479492,162.44920349121095,190.730915204155,154.93899536132813]},{"page":434,"text":"It was an achiral lamellar mesophase formed by polyphilic compounds. The basic","rect":[53.81376266479492,188.05459594726563,385.12769354059068,179.1200408935547]},{"page":434,"text":"chemical idea to form “building elements” of a polar phase was quite remarkable:","rect":[53.81376266479492,200.01412963867188,385.107801986428,191.07957458496095]},{"page":434,"text":"the so-called polyphilic effect has been realised. The word“polyphilic” is a general-","rect":[53.81376266479492,211.97369384765626,385.1525815567173,203.0391387939453]},{"page":434,"text":"isation of the well-known term “biphilic” or “amphiphilic”. According to this","rect":[53.81376266479492,223.9332275390625,385.14068998442846,214.99867248535157]},{"page":434,"text":"concept, chemically different moieties of a molecule tend to segregate to form","rect":[53.81376266479492,235.89279174804688,385.13773296952805,226.95823669433595]},{"page":434,"text":"polar aggregates, lamellas or smectic layers. The latter can form a polar phase. As","rect":[53.81376266479492,247.85232543945313,385.17355741606908,238.9177703857422]},{"page":434,"text":"shown in Fig.13.26a, a compound studied was made up of three distinct parts:","rect":[53.81376266479492,259.8118591308594,385.156630111428,250.87730407714845]},{"page":434,"text":"perfluoroalkyl and alkyl chains and a biphenyl rigid core. A fluorinated chains does","rect":[53.81475067138672,271.714599609375,385.12372221099096,262.780029296875]},{"page":434,"text":"not like hydrocarbon chains and prefers to have another fluorinated chain as a","rect":[53.81475067138672,283.6741638183594,385.1615375347313,274.7396240234375]},{"page":434,"text":"neighbour. The same is also true for the hydrocarbon chains preferring to be close to","rect":[53.81475067138672,295.6336975097656,385.1407097916938,286.69915771484377]},{"page":434,"text":"each other. To some extent such a tendency is also characteristic of biphenyl","rect":[53.81475067138672,307.59326171875,385.10786302158427,298.65869140625]},{"page":434,"text":"moieties. In this way the head-to-tail symmetry is broken at the molecular scale","rect":[53.81475067138672,319.55279541015627,385.1088031597313,310.61822509765627]},{"page":434,"text":"and a polar smectic layer forms according to sketches (b) and (c) in the same figure.","rect":[53.81475067138672,331.51239013671877,385.1137966683078,322.57781982421877]},{"page":434,"text":"In principle, such polar layers may be stacked either in a ferro- or antiferroelectric","rect":[53.81475067138672,343.471923828125,385.15061224176255,334.537353515625]},{"page":434,"text":"structure.","rect":[53.81475067138672,353.3696594238281,91.31325955893283,347.5129089355469]},{"page":434,"text":"Indeed, upon cooling from the isotropic through the smectic A phase, a metasta-","rect":[65.76677703857422,367.334228515625,385.15569434968605,358.399658203125]},{"page":434,"text":"ble polar phase forms at temperature 82\bC, which existed down to the room","rect":[53.81475067138672,379.2937927246094,385.1002268660124,370.3592529296875]},{"page":434,"text":"temperature before the next heating cycle. The phase manifests all polar properties,","rect":[53.81315231323242,391.2556457519531,385.15398831869848,382.32110595703127]},{"page":434,"text":"namely, pyroelectric and piezoelectric effects, repolarization current and optical","rect":[53.81315231323242,403.2152099609375,385.12312181064677,394.2806396484375]},{"page":434,"text":"Fig. 13.26 Polyphilic effect. (a) An appropriate molecule having well-defined moieties hardly","rect":[53.812843322753909,564.1054077148438,385.16193145899697,556.1724243164063]},{"page":434,"text":"compatible with each other: hydrophobic perfluoro- and alkyl- chains and a hydrophilic biphenyl","rect":[53.812835693359378,573.9569091796875,385.19649986472197,566.362548828125]},{"page":434,"text":"core. Below are shown a steric model and its schematic version used in sketches (b) and (c). The","rect":[53.812835693359378,583.59423828125,385.1957030036402,576.3385009765625]},{"page":434,"text":"latter illustrate unfavourable (b) and favourable (c) packing of molecules in aggregates or layers","rect":[53.81281280517578,593.9088134765625,383.68201144217769,586.314453125]},{"page":435,"text":"13.2 Introduction to Antiferroelectrics","rect":[53.813697814941409,42.55752944946289,184.14209444999018,36.68185043334961]},{"page":435,"text":"425","rect":[372.4990539550781,42.55752944946289,385.1906790175907,36.63105392456055]},{"page":435,"text":"second harmonic generations. Ferroelectricity was demonstrated by the measure-","rect":[53.812843322753909,68.2883529663086,385.09395728913918,59.35380554199219]},{"page":435,"text":"ments of the hysteresis in the acoustically induced piezo-electric response. The","rect":[53.812843322753909,80.24788665771485,385.1427692241844,71.31333923339844]},{"page":435,"text":"subsequent X-ray investigations an infra-red spectroscopy have shown that the","rect":[53.812843322753909,92.20748138427735,385.16965521051255,83.27293395996094]},{"page":435,"text":"phase consists of polar liquid-like layers having some blurred disordered structure.","rect":[53.812843322753909,104.11019134521485,385.1785854866672,95.17564392089844]},{"page":435,"text":"The phase consists of mesoscopic domains with high spontaneous polarization","rect":[53.812843322753909,116.0697250366211,385.0979851823188,107.13517761230469]},{"page":435,"text":"within each of those domains but almost averaged over a macroscopic sample.","rect":[53.812843322753909,128.02932739257813,385.1656460335422,119.09477233886719]},{"page":435,"text":"Due to smallness of the overall Ps value and difficulties in the chemical synthesis","rect":[53.812843322753909,139.98959350585938,385.13412870513158,131.05430603027345]},{"page":435,"text":"the polyphilic polar materials have not found practical applications. However, their","rect":[53.813228607177737,151.94912719726563,385.10930763093605,143.0145721435547]},{"page":435,"text":"investigations stimulated activity in the search of new polar achiral liquid crystals,","rect":[53.813228607177737,163.90869140625,385.1003078987766,154.97413635253907]},{"page":435,"text":"especially based on bent-core molecules. In particular, the ferroelectric phase was","rect":[53.813228607177737,175.86822509765626,385.1461221133347,166.9336700439453]},{"page":435,"text":"reported in an achiral compound [39]. Later, it has been understood that, in sucha","rect":[53.813228607177737,187.8277587890625,385.1590045757469,178.89320373535157]},{"page":435,"text":"compound, the conglomerates of left and right chiral domains emerged as a result of","rect":[53.81223678588867,199.73052978515626,385.1461118301548,190.7959747314453]},{"page":435,"text":"spontaneous break of the mirror symmetry discussed in Section 4.11.","rect":[53.81223678588867,211.69009399414063,332.6147427132297,202.7555389404297]},{"page":435,"text":"13.2.3.3 Achiral Antiferroelectrics","rect":[53.81223678588867,251.77699279785157,206.763630047905,244.26678466796876]},{"page":435,"text":"Achiral smectic materials with anticlinic molecular packing are very rare [40] and","rect":[53.81223678588867,277.43914794921877,385.14409724286568,268.50457763671877]},{"page":435,"text":"their antiferroelectric properties have unequivocally been demonstrated only in","rect":[53.81220626831055,289.3987121582031,385.1391228776313,280.46417236328127]},{"page":435,"text":"1996 [41]. The antiferroelectric properties have been observed in mixtures of two","rect":[53.81220626831055,301.3582458496094,385.15007868817818,292.4237060546875]},{"page":435,"text":"achiral components, although no one of the two manifested this behaviour. In","rect":[53.811214447021487,313.31781005859377,385.14617243817818,304.38323974609377]},{"page":435,"text":"different mixtures of a rod like mesogenic compound (monomer) with the polymer","rect":[53.811214447021487,325.27734375,385.08431373445168,316.3427734375]},{"page":435,"text":"comprised by chemically same rod-like mesogenic molecules a characteristic anti-","rect":[53.811214447021487,337.2369079589844,385.10625587312355,328.3023681640625]},{"page":435,"text":"ferroelectric hysteresis of the pyroelectric coefficient proportional to the spontane-","rect":[53.811214447021487,349.13970947265627,385.1540769180454,340.20513916015627]},{"page":435,"text":"ous polarization value has been observed; for an example see Fig.13.27a. Upon","rect":[53.811214447021487,361.0992431640625,385.1142205338813,352.1646728515625]},{"page":435,"text":"application of a low voltage the response is linear, at a higher field a field-induced","rect":[53.810237884521487,373.05877685546877,385.1212090592719,364.12420654296877]},{"page":435,"text":"AF–F transition occurs.","rect":[53.810237884521487,382.9565124511719,148.85602994467502,376.08380126953127]},{"page":435,"text":"The absolute value of the Ps has been measured by the pyroelectric technique as","rect":[65.76224517822266,396.9804992675781,385.13910307036596,388.0433349609375]},{"page":435,"text":"explained in Section 11.3.1 but with an applied d.c. electric field, exceeding the","rect":[53.81321334838867,408.9400634765625,385.1729205913719,400.0054931640625]},{"page":435,"text":"AF–F transition threshold. The value of the observed polarization dramatically","rect":[53.81321334838867,420.89959716796877,385.13616267255318,411.96502685546877]},{"page":435,"text":"a","rect":[57.072601318359378,450.8563537597656,62.62786371559322,445.2676086425781]},{"page":435,"text":"1","rect":[81.64617156982422,465.66094970703127,86.09037035951471,460.0382080078125]},{"page":435,"text":"0.5","rect":[74.97907257080078,478.59564208984377,86.08956927308893,472.82891845703127]},{"page":435,"text":"0","rect":[80.92597961425781,490.9424743652344,85.3701784039483,485.1757507324219]},{"page":435,"text":"–0.5","rect":[69.73555755615235,503.82757568359377,86.08956927308893,498.06085205078127]},{"page":435,"text":"–1","rect":[77.201171875,516.0615844726563,86.08956927308893,510.4388427734375]},{"page":435,"text":"–200 –100 0","rect":[93.10115051269531,541.2894897460938,146.23091143617487,535.5227661132813]},{"page":435,"text":"100","rect":[158.8769073486328,541.2894897460938,172.20950335512019,535.5227661132813]},{"page":435,"text":"Voltage (V)","rect":[124.01792907714844,554.142578125,163.9997360711523,546.5283203125]},{"page":435,"text":"b","rect":[213.98377990722657,450.7154235839844,220.08857365670478,443.40704345703127]},{"page":435,"text":"400","rect":[228.88180541992188,464.12213134765627,242.21440142640925,458.35540771484377]},{"page":435,"text":"300","rect":[228.88180541992188,479.9041442871094,242.21440142640925,474.1374206542969]},{"page":435,"text":"200","rect":[228.88180541992188,495.54302978515627,242.21440142640925,489.77630615234377]},{"page":435,"text":"100","rect":[228.88180541992188,511.5394287109375,242.21440142640925,505.772705078125]},{"page":435,"text":"0","rect":[237.32180786132813,527.8221435546875,241.7660066510186,522.055419921875]},{"page":435,"text":"0","rect":[242.19204711914063,537.0736694335938,246.6362459088311,531.3069458007813]},{"page":435,"text":"20","rect":[266.5368347167969,537.1871948242188,275.4252168560967,531.4204711914063]},{"page":435,"text":"40","rect":[293.10211181640627,537.1871948242188,301.9905244732842,531.4204711914063]},{"page":435,"text":"60","rect":[319.66900634765627,537.1871948242188,328.5573884869561,531.4204711914063]},{"page":435,"text":"80","rect":[346.2350769042969,537.1871948242188,355.1234590435967,531.4204711914063]},{"page":435,"text":"monomer content (%)","rect":[272.2287902832031,549.9923706054688,349.0750534539648,542.4820556640625]},{"page":435,"text":"100","rect":[370.579833984375,537.1871948242188,383.91084307679986,531.4204711914063]},{"page":435,"text":"Fig. 13.27 Achiral antiferroelectric. Voltage dependence of pyroelectric coefficient describing","rect":[53.812843322753909,574.0244750976563,385.15768951563759,566.0914916992188]},{"page":435,"text":"the double hysteresis loop (a) and dependence of the field-induced polarization on the content ofa","rect":[53.812843322753909,583.9326782226563,385.1754698493433,576.3383178710938]},{"page":435,"text":"monomer in the polymer–monomer mixtures (b)","rect":[53.812843322753909,593.9086303710938,220.27847379309825,586.3142700195313]},{"page":436,"text":"426","rect":[53.81537628173828,42.55752944946289,66.50698227076455,36.68185043334961]},{"page":436,"text":"13 Ferroelectricity and Antiferroelectricity in Smectics","rect":[197.24066162109376,44.276206970214847,385.1829269695214,36.68185043334961]},{"page":436,"text":"depends on the mixture composition: there is a very sharp maximum for the","rect":[53.812843322753909,68.2883529663086,385.11487615777818,59.35380554199219]},{"page":436,"text":"polymer/monomer ratio 70:30 as shown in Fig.13.27b. The macroscopic polariza-","rect":[53.812843322753909,80.24788665771485,385.1626218399204,71.31333923339844]},{"page":436,"text":"tion measured in the mesophase reaches the value of about 400 nC/cm2, that","rect":[53.81185531616211,92.20748138427735,385.14033372470927,81.15672302246094]},{"page":436,"text":"requires a dipole moment projection onto the smectic plane of about 1 D per","rect":[53.814476013183597,104.1104965209961,385.12349830476418,95.17594909667969]},{"page":436,"text":"mesogenic unit. Indeed, polar intermolecular hydrogen bonds provide such dipole","rect":[53.814476013183597,116.0699691772461,385.12448919488755,107.13542175292969]},{"page":436,"text":"moments located at a small angle to the long axes of the tilted mesogenic groups.","rect":[53.814476013183597,128.02957153320313,385.1811794808078,119.09501647949219]},{"page":436,"text":"The arrows in Fig.13.28 picture schematically the dipolar parts of the mesogenic","rect":[53.814476013183597,139.98910522460938,385.13440740777818,131.05455017089845]},{"page":436,"text":"groups.","rect":[53.814476013183597,151.94863891601563,83.43522305990939,145.24522399902345]},{"page":436,"text":"As shown by the X-ray diffraction, polymer-monomer mixture consists of SmC","rect":[65.7665023803711,163.908203125,385.14439651187208,154.97364807128907]},{"page":436,"text":"bilayers. A bilayer is the principal unit cell having either non-polar C2h (a) or polar","rect":[53.814476013183597,175.86868286132813,385.1397946914829,166.9331817626953]},{"page":436,"text":"C2v (b) symmetry. The former is incompatible with both ferroelectricity or anti-","rect":[53.81393051147461,187.82827758789063,385.10942970124855,178.8937225341797]},{"page":436,"text":"ferroelectricity, because such a structure has an inversion centre. On the contrary, in","rect":[53.813358306884769,199.73104858398438,385.14031306317818,190.7765655517578]},{"page":436,"text":"sketch (b) each bilayer is polar with P0 vector located in the tilt plane along the","rect":[53.813358306884769,211.69094848632813,385.17469061090318,202.7560577392578]},{"page":436,"text":"y-axis. In a stack of such layers the direction of P0 alternates and the structure (b) is","rect":[53.8139762878418,223.65048217773438,385.1892434512253,214.71592712402345]},{"page":436,"text":"antiferroelectric in its ground state. Only strong electric field Ey causes the transi-","rect":[53.81356430053711,236.5266876220703,385.1235593399204,226.65565490722657]},{"page":436,"text":"tion to the ferroelectric structure shown in sketch (c) as observed in experiment.","rect":[53.813594818115237,247.56979370117188,385.1693996956516,238.63523864746095]},{"page":436,"text":"Note that both the P0 and Ps ¼ S P0 vectors are always lying in the tilt plane.","rect":[53.813594818115237,259.529541015625,369.16399808432348,250.55514526367188]},{"page":436,"text":"The suggested bilayer antiferroelectric structure is compatible with the X-ray","rect":[65.76563262939453,271.48907470703127,385.1046990495063,262.55450439453127]},{"page":436,"text":"diffraction data and the optical observation of the influence of a rather weak electric","rect":[53.8136100769043,283.4486083984375,385.1485065288719,274.5140380859375]},{"page":436,"text":"field (below the AF–F transition) on freely suspended films of the same mixture","rect":[53.8136100769043,295.35137939453127,385.1175922222313,286.41680908203127]},{"page":436,"text":"[42]. Only the structures with odd number of bilayers appeared to be field-sensitive","rect":[53.8136100769043,307.3109436035156,385.1385883159813,298.3564758300781]},{"page":436,"text":"due to a finite polarization P0 of single bilayers. Therefore, the antiferroelectricity","rect":[53.812618255615237,319.2710876464844,385.1587761979438,310.3359375]},{"page":436,"text":"of the polymer-monomer mixtures is confirmed by all possible experiments. The","rect":[53.81295394897461,331.23065185546877,385.14292181207505,322.29608154296877]},{"page":436,"text":"role of the monomer admixture is explained as follows. As X-ray analysis shows,","rect":[53.81295394897461,343.190185546875,385.13494534994848,334.255615234375]},{"page":436,"text":"pure polymer has only the bilayer smectic C phase shown in Fig.13.28a, which is","rect":[53.81295394897461,355.1497497558594,385.18759550200658,346.2152099609375]},{"page":436,"text":"too symmetric to manifest polar properties. From the polarization and electrooptical","rect":[53.81295394897461,367.1092834472656,385.1777177579124,358.17474365234377]},{"page":436,"text":"measurements it is evident that the monomer additive changes the packing of the","rect":[53.81295394897461,379.06878662109377,385.1697162456688,370.13421630859377]},{"page":436,"text":"mesogenic groups and provokes the alternating tilt structure (b) in side-chain","rect":[53.81295394897461,390.9715881347656,385.08013239911568,382.03704833984377]},{"page":436,"text":"polymer bilayers. This results in antiferroelectricity, although the molecular mech-","rect":[53.81295394897461,402.93115234375,385.11101661531105,393.99658203125]},{"page":436,"text":"anism of such polymer–monomer interaction is not clear.","rect":[53.81295394897461,414.8906555175781,284.7795071175266,405.95611572265627]},{"page":436,"text":"At","rect":[65.76496887207031,424.7883605957031,75.70922715732644,417.97540283203127]},{"page":436,"text":"the","rect":[81.56926727294922,425.0,93.79303777887189,417.9156494140625]},{"page":436,"text":"optimum","rect":[99.63914489746094,426.8501892089844,135.71320294023117,417.9156494140625]},{"page":436,"text":"concentration","rect":[141.55633544921876,425.0,196.0377129655219,417.9156494140625]},{"page":436,"text":"of","rect":[201.93955993652345,425.0,210.23140846101416,417.9156494140625]},{"page":436,"text":"the","rect":[216.04368591308595,425.0,228.2674487896141,417.9156494140625]},{"page":436,"text":"mixture","rect":[234.11355590820313,425.0,265.27222479059068,417.9156494140625]},{"page":436,"text":"the","rect":[271.1024169921875,425.0,283.32617986871568,417.9156494140625]},{"page":436,"text":"pyroelectric","rect":[289.17230224609377,426.8501892089844,336.99829900934068,417.9156494140625]},{"page":436,"text":"coefficient","rect":[342.871337890625,425.0,385.1130510098655,417.9156494140625]},{"page":436,"text":"reaches the value of 4 nC/cm2K exceeding that observed in the famous ferroelectric","rect":[53.81295394897461,438.8108825683594,385.08557928277818,427.75994873046877]},{"page":436,"text":"crystalline copolymers PVDF-TrFE. On cooling down to the glassy state and","rect":[53.81342697143555,450.7704162597656,385.1423272233344,441.8159484863281]},{"page":436,"text":"a","rect":[107.73909759521485,491.4689025878906,113.29435999244869,485.8801574707031]},{"page":436,"text":"x","rect":[170.80523681640626,518.362060546875,174.3542457498541,514.7388916015625]},{"page":436,"text":"b","rect":[204.13888549804688,491.4689025878906,210.24367924752509,484.1605224609375]},{"page":436,"text":" y","rect":[198.64976501464845,514.9635620117188,204.1970954812994,509.7087097167969]},{"page":436,"text":"P0","rect":[257.75982666015627,517.2543334960938,265.5852423160455,509.6938781738281]},{"page":436,"text":"–P0","rect":[252.18057250976563,547.3322143554688,264.002600714483,539.7718505859375]},{"page":436,"text":"c","rect":[271.4414978027344,491.4689025878906,276.9967601999682,485.8801574707031]},{"page":436,"text":"E","rect":[323.61004638671877,498.48040771484377,328.94155305025415,492.9855651855469]},{"page":436,"text":"P0","rect":[323.5390319824219,520.3380737304688,331.364417120733,512.7776489257813]},{"page":436,"text":"P0","rect":[322.2901611328125,545.0827026367188,330.11545471838925,537.5222778320313]},{"page":436,"text":"Fig. 13.28 Non-polar dipolar structure of a bilayer and a lamellar phase (a) and antiferroelectric","rect":[53.812843322753909,574.0244750976563,385.1187071540308,566.0745239257813]},{"page":436,"text":"phase (b) formed by polar bilayers. A strong electric field E applied along the y-axis converts the","rect":[53.812843322753909,583.9326782226563,385.1559080817652,576.3383178710938]},{"page":436,"text":"antiferroelectric phase (b) into polar ferroelectric phase (c)","rect":[53.814517974853519,593.9086303710938,255.40037625891856,586.3142700195313]},{"page":437,"text":"13.2 Introduction to Antiferroelectrics","rect":[53.8138542175293,42.55710220336914,184.14226229666986,36.68142318725586]},{"page":437,"text":"427","rect":[372.49920654296877,42.55710220336914,385.1908316054813,36.73222351074219]},{"page":437,"text":"subsequent removing the field, the pyroelectric coefficient may be kept stable for","rect":[53.812843322753909,68.2883529663086,385.1766599258579,59.35380554199219]},{"page":437,"text":"years, and this material may be very useful as an easily formed pyroelectric glass.","rect":[53.812843322753909,80.24788665771485,384.6191372444797,71.31333923339844]},{"page":437,"text":"13.2.3.4 Ferro- and Antiferroelectric Compounds Based on the Bent-Shape","rect":[53.812843322753909,122.0081787109375,381.07544744684068,112.82462310791016]},{"page":437,"text":"Molecules","rect":[96.18294525146485,132.29441833496095,138.93230070220188,125.1129150390625]},{"page":437,"text":"As discussed in Section 4.11, achiral molecules of the bent or banana shape may","rect":[53.812843322753909,157.8997802734375,385.16460505536568,148.96522521972657]},{"page":437,"text":"form locally chiral phases in the form of the left- and right-handed domains. This is","rect":[53.812843322753909,169.85934448242188,385.18552030669408,160.92478942871095]},{"page":437,"text":"a result of spontaneous break of the mirror symmetry [43].","rect":[53.812843322753909,181.81887817382813,290.7270168831516,172.8843231201172]},{"page":437,"text":"Assuming the head-to-tail symmetry of a bent-shape molecule, the highest","rect":[65.76485443115235,193.77841186523438,385.10200364658427,184.84385681152345]},{"page":437,"text":"symmetry of a uniaxial non-tilted smectic A layer is D1h. Then, according to","rect":[53.81283187866211,205.7393798828125,385.14296809247505,196.8033905029297]},{"page":437,"text":"molecular packing presented in Fig.13.29a, the highest symmetry of the biaxial","rect":[53.814083099365237,217.69894409179688,385.12702806064677,208.76438903808595]},{"page":437,"text":"polar layer is C2v: there is a rotation axis C2 parallel to x, and two symmetry planes xz","rect":[53.81406784057617,229.65866088867188,385.1415139590378,220.7239227294922]},{"page":437,"text":"and xy. The layer polarization is possible along the C2 axis. Had the layer consisted","rect":[53.813655853271487,241.61834716796876,385.1551446061469,232.6836700439453]},{"page":437,"text":"of the rod-like molecules tilted within the xz plane the symmetry would be C2h as in","rect":[53.8133659362793,253.52108764648438,385.14373103192818,244.58653259277345]},{"page":437,"text":"SmC. However, when the bent-core molecules are tilted in the y-direction (forward","rect":[53.81386947631836,265.4609069824219,385.10393611005318,256.5462646484375]},{"page":437,"text":"Fig. 13.29","rect":[53.812843322753909,491.6106872558594,88.5979971328251,483.5760803222656]},{"page":437,"text":"Bent-shape molecules form polar smectic layers in the polar plane xz with polarization","rect":[94.59689331054688,491.54296875,385.16779083399697,483.9486083984375]},{"page":437,"text":"Px (a). Upon cooling, the molecules can spontaneously acquire a tilt forward or back within the tilt","rect":[53.812843322753909,501.51873779296877,385.1820650502688,493.92437744140627]},{"page":437,"text":"plane yz. The stack of the layers may be either synclinic SmCS or anticlinic SmCA (b). Addition-","rect":[53.812801361083987,511.4379577636719,385.1666268692701,503.84356689453127]},{"page":437,"text":"ally, depending on the direction of polarization Px, both the synclinic and anticlinic structure may","rect":[53.81338119506836,521.4138793945313,385.1940969863407,513.8193359375]},{"page":437,"text":"have uniform (ferroelectric PF) or alternating (antiferroelectric PA) distribution of polarization","rect":[53.81290817260742,531.3895263671875,385.16864532618447,523.795166015625]},{"page":437,"text":"within the stack. In the field absence there are four structures marked by symbols below. Note that","rect":[53.81370162963867,541.365478515625,385.1542330190188,533.7711181640625]},{"page":437,"text":"the leftmost structure is chiral SmC* and rightmost structure is also chiral because, for any pair of","rect":[53.81370162963867,551.2847290039063,385.21347135169199,543.6903686523438]},{"page":437,"text":"neighbours, the directions of the tilt and polarization change together leaving the same handedness","rect":[53.81370162963867,561.2606811523438,385.17456515311519,553.6663208007813]},{"page":437,"text":"of the vector triple. In the electric field, the phase transitions from chiral SmCAPA* to chiral","rect":[53.81370162963867,571.2366333007813,385.1819124623782,563.5908813476563]},{"page":437,"text":"SmCSPF* and from racemic SmCSPA to racemic SmCAPF structures are possible (shown by ark","rect":[53.81344223022461,581.2119140625,385.1848692634058,573.5667724609375]},{"page":437,"text":"arrows)","rect":[53.8138542175293,590.7925415039063,80.53560727698495,583.587646484375]},{"page":438,"text":"428","rect":[53.812843322753909,42.55594253540039,66.50444931178018,36.73106384277344]},{"page":438,"text":"13 Ferroelectricity and Antiferroelectricity in Smectics","rect":[197.23812866210938,44.274620056152347,385.18039401053707,36.68026351928711]},{"page":438,"text":"or backward with respect to the drawing plane), both the reflection planes are lost.","rect":[53.812843322753909,68.2883529663086,385.1407742073703,59.35380554199219]},{"page":438,"text":"Now each smectic monolayer becomes chiral, either left- or right-handed with","rect":[53.812843322753909,80.24788665771485,385.1148308854438,71.31333923339844]},{"page":438,"text":"symmetry C2. In the right Cartesian frame forward (or backward) deviation corre-","rect":[53.812843322753909,92.20772552490235,385.0994809707798,83.27317810058594]},{"page":438,"text":"sponds to the right (or left) sign of chirality because we have three non-coplanar","rect":[53.81330871582031,104.1104965209961,385.1064389785923,95.17594909667969]},{"page":438,"text":"vectors, the smectic layer normal h, the director n (along the molecular long axis)","rect":[53.81330871582031,116.0699691772461,385.1262143692173,107.13542175292969]},{"page":438,"text":"and polarization P0 (along x) that form a left or right triples.","rect":[53.81330871582031,128.02999877929688,293.8243374397922,119.09501647949219]},{"page":438,"text":"In a stack of subsequent layers the tilt may be constant (synclinic structure) or","rect":[65.76539611816406,139.98953247070313,385.14824806062355,131.0549774169922]},{"page":438,"text":"alternating (anticlinic structure). Both synclinic and anticlinic multilayer stacks can","rect":[53.81337356567383,151.94906616210938,385.12542048505318,143.01451110839845]},{"page":438,"text":"further be subdivided into ferroelectric and antiferroelectric structures. The molec-","rect":[53.81337356567383,161.87667846679688,385.1601804336704,154.9740753173828]},{"page":438,"text":"ular projections onto the tilt plane zy are shown in Fig. 13.29b. In ferroelectric","rect":[53.81337356567383,175.8681640625,385.08753240777818,166.93360900878907]},{"page":438,"text":"(symbol F) phases spontaneous polarization has the same direction in each layer","rect":[53.81437683105469,187.82766723632813,385.1203549942173,178.8931121826172]},{"page":438,"text":"(synclinic chiral SmCsPF* and anticlinic achiral SmCAPF phases). In the antiferro-","rect":[53.81437683105469,199.73126220703126,385.1536801895298,190.7369384765625]},{"page":438,"text":"electric (symbol A) phases the direction of polarization alternates (achiral synclinic","rect":[53.8138542175293,211.69082641601563,385.17957342340318,202.7562713623047]},{"page":438,"text":"SmCsPA and chiral anticlinic SmCAPA* phases). In fact we have a conglomerate of","rect":[53.8138542175293,223.65057373046876,385.15044532624855,214.7160186767578]},{"page":438,"text":"chiral and achiral phases both in either synclinic or anticlinic form.","rect":[53.8136100769043,235.61013793945313,325.03204007651098,226.6755828857422]},{"page":438,"text":"As was said, each smectic layer is chiral, left or right, but the pair of layers might","rect":[65.76561737060547,247.56967163085938,385.1136308438499,238.63511657714845]},{"page":438,"text":"be homogeneously chiral or racemic. The leftmost structure is typical chiral SmC*","rect":[53.8136100769043,259.52923583984377,385.1375359635688,250.5946807861328]},{"page":438,"text":"structure and the rightmost structure is also chiral because in any pair of neighbours","rect":[53.8136100769043,271.48876953125,385.1624795352097,262.55419921875]},{"page":438,"text":"the direction of the tilt and polarization change together leaving the same handed-","rect":[53.8136100769043,283.4482727050781,385.15645728913918,274.51373291015627]},{"page":438,"text":"ness of the vector triple. The two middle stacks are racemic because left and right","rect":[53.8136100769043,295.3510437011719,385.1266313321311,286.41650390625]},{"page":438,"text":"vector triples alternate from layer to layer. As usual, the asterisks are added to the","rect":[53.8136100769043,307.31060791015627,385.1734088726219,298.37603759765627]},{"page":438,"text":"symbols of each homogeneously chiral subphase. The electric field exceeding some","rect":[53.8136100769043,319.2701416015625,385.14453924371568,310.3355712890625]},{"page":438,"text":"threshold (E > Etr) causes transitions between different structures: it transforms","rect":[53.8136100769043,330.8323059082031,385.14303983794408,322.295166015625]},{"page":438,"text":"SmCAPA* into SmCsPF* (both are homogeneously chiral) and the direction of the","rect":[53.81315231323242,343.1903991699219,385.17261541559068,334.1960754394531]},{"page":438,"text":"tilt is controlled by the sign of E. The racemic SmCsPA phase may be transformed","rect":[53.8128776550293,355.1500549316406,385.13271418622505,346.21539306640627]},{"page":438,"text":"into SmCAPF. As the field interacts with polarization, the final state is always","rect":[53.81376266479492,367.10968017578127,385.08075346099096,358.175048828125]},{"page":438,"text":"ferroelectric (PF) be it synclinic or anticlinic.","rect":[53.81356430053711,379.0693359375,235.25240750815159,370.1346435546875]},{"page":438,"text":"At present, eight different phases are known in banana compounds dependent on","rect":[65.7650375366211,390.97210693359377,385.1688164811469,382.03753662109377]},{"page":438,"text":"particular in-plane packing symmetry and they usually labelled as B1, B2, ...B8,","rect":[53.81301498413086,402.9316711425781,385.1832241585422,393.99713134765627]},{"page":438,"text":"etc., counted from the isotropic phase [44]. Among them the B2 phase is especially","rect":[53.814537048339847,414.8915710449219,385.1621941666938,405.9569091796875]},{"page":438,"text":"interesting, because it has low viscosity and can easily be switched by an electric","rect":[53.814414978027347,426.85113525390627,385.1503375835594,417.91656494140627]},{"page":438,"text":"field with rather short switching times [45]. In fact, the B2 phase is basically a","rect":[53.814414978027347,438.8108825683594,385.16040838434068,429.8163757324219]},{"page":438,"text":"conglomerate","rect":[53.813594818115237,450.7704162597656,108.29398382379377,441.83587646484377]},{"page":438,"text":"of","rect":[114.026611328125,449.0,122.31847511140478,441.83587646484377]},{"page":438,"text":"chiral","rect":[128.01824951171876,449.0,150.8034349209983,441.83587646484377]},{"page":438,"text":"and","rect":[156.51019287109376,449.0,170.9139336686469,441.83587646484377]},{"page":438,"text":"achiral","rect":[176.61868286132813,449.0,203.85339219638895,441.83587646484377]},{"page":438,"text":"antiferroelectric","rect":[209.58602905273438,449.0,273.70410955621568,441.83587646484377]},{"page":438,"text":"structures","rect":[279.3720703125,449.0,318.2104531680222,442.8518371582031]},{"page":438,"text":"SmCAPA*","rect":[323.894287109375,450.27276611328127,365.0703057389594,441.7762145996094]},{"page":438,"text":"and","rect":[370.7422180175781,448.73858642578127,385.1459588151313,441.83599853515627]},{"page":438,"text":"SmCAPA mixed with some percentage of the two ferroelectric structures.","rect":[53.81411361694336,462.7301940917969,348.2805447151828,453.795654296875]},{"page":438,"text":"Since the discovery of spontaneous break of mirror symmetry [39, 43], many","rect":[65.76509094238281,474.6897277832031,385.17678156903755,465.75518798828127]},{"page":438,"text":"new, so-called banana-form compounds have been synthesised and hundreds of","rect":[53.81206512451172,486.5924987792969,385.1469663223423,477.657958984375]},{"page":438,"text":"papers published on that subject [44]. It became a hot topic in modern physics and","rect":[53.81206512451172,498.5520324707031,385.1429375748969,489.61749267578127]},{"page":438,"text":"chemistry of liquid crystals. In the present book there is no space for discussion of","rect":[53.81207275390625,510.5115966796875,385.1469968399204,501.5770263671875]},{"page":438,"text":"different aspects of this fascinating phenomenon and I have decided to finish my","rect":[53.81207275390625,522.4710693359375,385.13201228192818,513.5365600585938]},{"page":438,"text":"narration here. I believe very soon the books shall appear devoted solely to this","rect":[53.81207275390625,534.4306640625,385.14093412505346,525.4961547851563]},{"page":438,"text":"important subject related not only to liquid crystals, but to the general problems of","rect":[53.81207275390625,546.3901977539063,385.1479428848423,537.4556884765625]},{"page":438,"text":"chirality of the matter.","rect":[53.81207275390625,558.3497314453125,144.32967801596409,549.4152221679688]},{"page":438,"text":"In conclusion of this chapter it should be stated that bistable and tristable","rect":[65.76409149169922,570.3092651367188,385.14597356988755,561.374755859375]},{"page":438,"text":"switching of ferro- and antiferroelectric liquid crystals is very fast and provides","rect":[53.81207275390625,582.2120361328125,385.10214628325658,573.2775268554688]},{"page":438,"text":"long memory states. The latter allows one to design displays without semiconductor","rect":[53.81207275390625,594.1715698242188,385.11312232820168,585.237060546875]},{"page":439,"text":"References","rect":[53.80617141723633,42.52720260620117,91.47485812186517,36.68539047241211]},{"page":439,"text":"429","rect":[372.49151611328127,42.62879943847656,385.1831106582157,36.73619079589844]},{"page":439,"text":"thin-film transistors used in each small pixels of a matrix with thousands rows and","rect":[53.812843322753909,68.2883529663086,385.14077082685005,59.35380554199219]},{"page":439,"text":"columns. Such displays have been constructed and their feasibility demonstrated.","rect":[53.812843322753909,80.24788665771485,385.1009487679172,71.31333923339844]},{"page":439,"text":"The beautiful pictures may be seen in references [8] (black and white) and [24] (in","rect":[53.812843322753909,92.20748138427735,385.1725701432563,83.27293395996094]},{"page":439,"text":"colour). However, some disadvantages, such as not enough tolerance of smectic","rect":[53.81282424926758,104.11019134521485,385.1785358257469,95.17564392089844]},{"page":439,"text":"structures to mechanical shots and temperature variations are still to be overcome.","rect":[53.81282424926758,116.0697250366211,385.1596951058078,107.13517761230469]},{"page":439,"text":"Today smectic materials are indispensable for temperature stabilised optical space","rect":[53.81282424926758,128.02932739257813,385.1367877788719,119.09477233886719]},{"page":439,"text":"modulators, image processors or image projectors and, in nearest future, will be","rect":[53.81282424926758,139.98886108398438,385.15171087457505,131.05430603027345]},{"page":439,"text":"very useful as electronic paper.","rect":[53.81282424926758,151.94839477539063,179.69276090170627,143.0138397216797]},{"page":439,"text":"References","rect":[53.812843322753909,200.1083221435547,109.59614448282879,191.3231964111328]},{"page":439,"text":"1.","rect":[58.06126022338867,226.0,64.40706131055318,219.8101348876953]},{"page":439,"text":"2.","rect":[58.06126022338867,246.0,64.40706131055318,239.70530700683595]},{"page":439,"text":"3.","rect":[58.06126022338867,256.0,64.40706131055318,249.68125915527345]},{"page":439,"text":"4.","rect":[58.06126022338867,276.0,64.40706131055318,269.5764465332031]},{"page":439,"text":"5.","rect":[58.0604133605957,296.0,64.4062144477602,289.4267578125]},{"page":439,"text":"6.","rect":[58.05955123901367,305.2724914550781,64.40535232617818,299.3968200683594]},{"page":439,"text":"7.","rect":[58.05869674682617,325.224365234375,64.40449783399068,319.51800537109377]},{"page":439,"text":"8.","rect":[58.05869674682617,345.1195373535156,64.40449783399068,339.2946472167969]},{"page":439,"text":"9.","rect":[58.05869674682617,365.13922119140627,64.40449783399068,359.24658203125]},{"page":439,"text":"10.","rect":[53.80956268310547,385.0,64.38590118482076,379.1417541503906]},{"page":439,"text":"11.","rect":[53.8095588684082,405.0,64.38589737012349,399.09368896484377]},{"page":439,"text":"12.","rect":[53.8095588684082,425.0,64.38589737012349,418.9888916015625]},{"page":439,"text":"13.","rect":[53.8095588684082,454.6849365234375,64.38589737012349,448.86004638671877]},{"page":439,"text":"14.","rect":[53.808712005615237,485.0,64.38505050733052,478.7879333496094]},{"page":439,"text":"15.","rect":[53.807857513427737,505.0,64.38419601514302,498.5815124511719]},{"page":439,"text":"16.","rect":[53.807003021240237,535.0,64.38334152295552,528.50341796875]},{"page":439,"text":"17.","rect":[53.807003021240237,564.250244140625,64.38334152295552,558.4253540039063]},{"page":439,"text":"Smolensky,","rect":[68.59698486328125,227.35369873046876,108.33017227246724,219.75933837890626]},{"page":439,"text":"G.A.,","rect":[113.00660705566406,226.0,131.5025050361391,219.8101348876953]},{"page":439,"text":"Krainik,","rect":[136.17471313476563,226.0,164.1808497627016,219.75933837890626]},{"page":439,"text":"N.N.:","rect":[168.85897827148438,226.0,187.61718467917505,219.9286651611328]},{"page":439,"text":"Ferroelectrics","rect":[192.2538604736328,226.0,238.90822299002924,219.75933837890626]},{"page":439,"text":"and","rect":[243.63121032714845,226.0,255.87437957911417,219.75933837890626]},{"page":439,"text":"Antiferroelectrics.","rect":[260.5118713378906,226.0,322.48926040723287,219.75933837890626]},{"page":439,"text":"Nauka,","rect":[327.2401428222656,226.0,351.45569107129537,219.75933837890626]},{"page":439,"text":"Moscow","rect":[356.0729064941406,226.0,385.1366326962134,219.9286651611328]},{"page":439,"text":"(1968) (in Russian)","rect":[68.59698486328125,236.9342803955078,134.947846351692,229.6785888671875]},{"page":439,"text":"Kittel, Ch: Introduction to Solid State Physics, 4th edn. 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Cryst. 40, 33–48 (1977)","rect":[68.59613800048828,297.0719299316406,353.47613614661386,289.4098205566406]},{"page":439,"text":"ˇ","rect":[151.24044799804688,298.894775390625,154.05798429114513,297.4808654785156]},{"page":439,"text":"Musˇevicˇ, I., Blinc, R., Zeksˇ, B.: The Physics of Ferroelectric and Antiferroelectric Liquid","rect":[68.59527587890625,306.9911804199219,385.16791290430947,299.1241760253906]},{"page":439,"text":"Crystals. World Scientific, Singapore (2000)","rect":[68.59442138671875,316.96710205078127,220.81245511634044,309.37274169921877]},{"page":439,"text":"Blinov, L.M., Beresnev, L.A.: Ferroelectric liquid crystals, Usp. Fiz. Nauk 143, 391–428","rect":[68.59442138671875,326.94305419921877,385.17218536524697,319.34869384765627]},{"page":439,"text":"(1984) (Sov. Phys.Uspekhi, 27, 7 (1984))","rect":[68.59442138671875,336.9190368652344,210.14813321692638,329.3246765136719]},{"page":439,"text":"Lagerwall, S.T.: Ferroelectric and Antiferroelectric Liquid Crystals. Wiley-VCH, New York","rect":[68.59442138671875,346.8382263183594,385.14841217188759,339.2438659667969]},{"page":439,"text":"(1999)","rect":[68.59442138671875,356.4755554199219,91.1516351090162,349.2706298828125]},{"page":439,"text":"Clark,","rect":[68.59442138671875,365.07147216796877,89.55249282910786,359.19580078125]},{"page":439,"text":"N.A.,","rect":[93.9150161743164,365.07147216796877,112.41091415479146,359.24658203125]},{"page":439,"text":"Lagerwall,","rect":[116.68629455566406,366.7901611328125,153.22119400098286,359.19580078125]},{"page":439,"text":"S.T.:","rect":[157.58456420898438,365.07147216796877,174.05827049460474,359.24658203125]},{"page":439,"text":"Submicrosecond","rect":[178.3167266845703,365.07147216796877,234.77745575098917,359.19580078125]},{"page":439,"text":"bistable","rect":[239.15353393554688,365.03759765625,265.5689940192652,359.19580078125]},{"page":439,"text":"electro-optic","rect":[269.8553771972656,366.7901611328125,312.8207030036402,359.19580078125]},{"page":439,"text":"switching","rect":[317.1544494628906,366.7901611328125,350.09335846095009,359.19580078125]},{"page":439,"text":"in","rect":[354.4270935058594,365.0,361.0351919570438,359.19580078125]},{"page":439,"text":"liquid","rect":[365.3029479980469,366.7901611328125,385.12717193751259,359.19580078125]},{"page":439,"text":"crystals. Appl. Phys. Lett. 36, 899–901 (1980)","rect":[68.59442138671875,376.7093811035156,227.0431832657545,368.8440856933594]},{"page":439,"text":"Pikin, S.A., Indenbom, V.L.: Thermodynamic states and symmetry of liquid crystals. Usp.","rect":[68.59358978271485,386.6853332519531,385.172884674811,379.0909729003906]},{"page":439,"text":"Fiz. Nauk 125, 251–277 (1984)","rect":[68.59358978271485,396.3226318359375,176.40200894934825,388.9991760253906]},{"page":439,"text":"Pikin, S.A.: Structural Transformations in Liquid Crystals. Gordon & Breach, New York","rect":[68.59358978271485,406.63726806640627,385.14667266993447,399.04290771484377]},{"page":439,"text":"(1991)","rect":[68.59358978271485,416.2178649902344,91.15080350501229,409.012939453125]},{"page":439,"text":"Blinov, L.M.: Pyroelectric studies of polar and ferroelectric mesophases. In: Vij, J. (ed.)","rect":[68.59358978271485,426.532470703125,385.1433419571607,418.9381103515625]},{"page":439,"text":"Advances in Chemical Physics. Advances in Liquid Crystals, vol. 113, pp. 77–158. Wiley,","rect":[68.59358978271485,436.5084533691406,385.1391322334047,428.86328125]},{"page":439,"text":"New York (2000)","rect":[68.59358978271485,446.145751953125,129.21630948645763,438.8900451660156]},{"page":439,"text":"Musˇevicˇ, I., Zˇeksˇ, B., Blinc, R., Rasing, Th: Ferroelectric liquid crystals: From the plane wave","rect":[68.59358978271485,456.40362548828127,385.1535277106714,446.0]},{"page":439,"text":"to the multisoliton limit. In: Kumar, S. (ed.) Liquid Crystals in the Nineties and Beyond, pp.","rect":[68.59274291992188,466.37957763671877,385.166292877936,458.78521728515627]},{"page":439,"text":"183–224. World Scientific, Singapore (1995)","rect":[68.59274291992188,476.3555603027344,222.68066495520763,468.71038818359377]},{"page":439,"text":"Garoff, S., Meyer, R.B.: Electroclinic effect at the A-C phase change in a chiral liquid crystal.","rect":[68.59274291992188,486.3315124511719,385.1703517158266,478.7371520996094]},{"page":439,"text":"Phys. Rev. Lett. 38, 848–851 (1977)","rect":[68.59274291992188,496.2507629394531,193.394379554817,488.3685302734375]},{"page":439,"text":"Pozhidayev, E.P., Blinov, L.M., Beresnev, L.A., Belyayev, V.V.: The dielectric anomaly near","rect":[68.59188842773438,506.2266845703125,385.1788339005201,498.63232421875]},{"page":439,"text":"the transition from the smectic A* to smectic C* phase and visco-elastic properties of","rect":[68.59188842773438,516.2025756835938,385.14840787512949,508.6082458496094]},{"page":439,"text":"ferroelectric liquid crystals. Mol. Cryst. Liq. 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Appl.","rect":[68.59516906738281,236.31124877929688,385.1813380439516,228.71688842773438]},{"page":441,"text":"Phys. 45, 597–625 (2006)","rect":[68.59516143798828,246.23046875,156.97442716223888,238.58531188964845]},{"page":441,"text":"Blinov, L.M., Barnik, M., Soto Bustamante, E., Pelzl, G., Weissflog, W.: Dynamics of electro-","rect":[68.59516906738281,256.2064208984375,385.1533212052076,248.612060546875]},{"page":441,"text":"optical switching in the antiferroelectric B2 phase of an achiral bent-core shape compound.","rect":[68.59516143798828,266.1824035644531,385.180453034186,258.5856628417969]},{"page":441,"text":"Phys. Rev. E 67, 1217061–8 (2003)","rect":[68.59767150878906,276.1559753417969,191.19012540442638,268.5023498535156]},{"page":442,"text":"Index","rect":[53.812843322753909,70.80704498291016,92.61071616219891,59.714962005615237]},{"page":442,"text":"A","rect":[53.812843322753909,221.50906372070313,59.88789154631106,215.7434539794922]},{"page":442,"text":"Abrikosov vortices, 65","rect":[53.812843322753909,231.58660888671876,131.72068542628214,225.66014099121095]},{"page":442,"text":"Adsorption, 259–260, 279, 280","rect":[53.812843322753909,243.28128051757813,160.36563629298136,235.63612365722657]},{"page":442,"text":"AFM. See Atomic force microscope","rect":[53.812843322753909,253.2572021484375,177.48323962473394,245.662841796875]},{"page":442,"text":"Anchoring","rect":[53.81283950805664,263.17645263671877,90.01775879053995,255.58209228515626]},{"page":442,"text":"azimuthal, 361, 362, 364, 405, 406","rect":[65.76494598388672,271.4337158203125,185.23184723048136,265.5072326660156]},{"page":442,"text":"break, 314–315, 374–376, 406, 407, 409","rect":[65.76494598388672,281.4774169921875,204.37927765040323,275.4831848144531]},{"page":442,"text":"energy, 272, 274, 275, 280, 308, 313, 327,","rect":[65.76494598388672,293.10430908203127,211.1946437080141,285.4591369628906]},{"page":442,"text":"329–331, 362, 370, 375, 376, 391,","rect":[77.71704864501953,301.3725891113281,195.33438369824848,295.37835693359377]},{"page":442,"text":"405–407","rect":[77.71704864501953,311.28082275390627,107.33079284815713,305.3543395996094]},{"page":442,"text":"transition, 265, 280","rect":[65.76494598388672,321.25677490234377,132.2131094375126,315.3302917480469]},{"page":442,"text":"zenithal, 274, 362, 364, 376, 391, 392,","rect":[65.76494598388672,331.3004455566406,197.93954727246723,325.3570251464844]},{"page":442,"text":"403–407","rect":[77.71704864501953,341.1519470214844,107.33079284815713,335.3270568847656]},{"page":442,"text":"Anharmonicity of helix, 364–366","rect":[53.812835693359378,352.8465881347656,168.11090606837198,345.2522277832031]},{"page":442,"text":"Anisotropy","rect":[53.812835693359378,362.82257080078127,91.8707250991337,355.22821044921877]},{"page":442,"text":"of conductivity, 176, 183, 334–337","rect":[65.76494598388672,372.74176025390627,186.12366241602823,365.14739990234377]},{"page":442,"text":"optical, 27, 32, 37, 57, 264, 278, 291, 292,","rect":[65.76494598388672,382.7177429199219,211.53479263379536,375.07257080078127]},{"page":442,"text":"300, 302, 318, 335, 345, 354, 355, 366","rect":[77.71704864501953,391.0,210.66498321680948,385.04852294921877]},{"page":442,"text":"tensor, 36, 59–60","rect":[65.76494598388672,401.0187072753906,125.03981537012979,395.02447509765627]},{"page":442,"text":"Anticlinic, 413–416, 418, 425, 427, 428","rect":[53.812835693359378,410.8702087402344,190.67064422755167,404.9437255859375]},{"page":442,"text":"Antiferroelectric phase, 51, 70, 412, 414,","rect":[53.812835693359378,422.5648498535156,194.54158279492817,414.919677734375]},{"page":442,"text":"416–421, 426","rect":[77.71704864501953,430.8221130371094,124.50930542139932,424.9464416503906]},{"page":442,"text":"Arrhenius, 171, 174, 175","rect":[53.812835693359378,440.7980651855469,139.1799520888798,434.87158203125]},{"page":442,"text":"Atomic force microscope (AFM), 62, 278","rect":[53.812835693359378,452.4360046386719,197.0155996718876,444.8416442871094]},{"page":442,"text":"Avogadro number, 118, 160, 381","rect":[53.812835693359378,462.4119567871094,167.5025381484501,454.8175964355469]},{"page":442,"text":"B","rect":[53.812835693359378,480.48687744140627,59.44790827955585,474.72125244140627]},{"page":442,"text":"Backflow effect, 241, 315–318, 373","rect":[53.812835693359378,490.5644226074219,176.45267242579386,484.637939453125]},{"page":442,"text":"Benard convection, 335","rect":[53.812835693359378,500.5403747558594,135.1584219375126,494.6138916015625]},{"page":442,"text":"Benzene, 7, 19, 20, 154, 296, 382, 413","rect":[53.812835693359378,510.58404541015627,186.4214834609501,504.5898132324219]},{"page":442,"text":"Berreman, 280–281, 371–375","rect":[53.812835693359378,520.435546875,155.53604644923136,514.509033203125]},{"page":442,"text":"Biaxiality, 38–39, 59, 121, 165, 414","rect":[53.812835693359378,532.1301879882813,177.5289053359501,524.4850463867188]},{"page":442,"text":"local biaxiality, 32, 201","rect":[65.76494598388672,542.1060791015625,146.99713653712198,534.51171875]},{"page":442,"text":"Biaxial liquid crystals","rect":[53.812835693359378,552.08203125,128.86336977958,544.4876708984375]},{"page":442,"text":"cholesteric, 60","rect":[65.76494598388672,560.2826538085938,115.57695526270791,554.4069213867188]},{"page":442,"text":"ferroelectric phases, 51, 419, 424, 427","rect":[65.76494598388672,571.9772338867188,196.78799957423136,564.3320922851563]},{"page":442,"text":"nematic, 32, 37–40, 54, 71, 267","rect":[65.76494598388672,580.2344970703125,174.4135488906376,574.3079833984375]},{"page":442,"text":"smectic, 48, 49, 108, 166","rect":[65.76494598388672,590.2781982421875,152.03824371485636,584.3347778320313]},{"page":442,"text":"tilted smectic, 386, 427","rect":[237.39942932128907,221.61575317382813,317.66876739649697,215.74008178710938]},{"page":442,"text":"Biphenyl, 17, 19, 20, 51, 128, 296, 424","rect":[225.4473419189453,233.31039428710938,359.9250540175907,225.6652374267578]},{"page":442,"text":"Birefringence, 34, 47, 72, 120, 263, 264,","rect":[225.4473419189453,243.286376953125,364.70300552442037,235.6920166015625]},{"page":442,"text":"285–294, 297, 303, 318, 399","rect":[249.3515625,251.6113739013672,347.6903433242313,245.61717224121095]},{"page":442,"text":"Bloch‐De Broglie waves, 349","rect":[225.44735717773438,263.1815490722656,326.9014944472782,255.58718872070313]},{"page":442,"text":"Blue phase, 3, 57, 63–65, 72, 209, 219","rect":[225.44650268554688,273.1575012207031,357.77255767970009,265.5123291015625]},{"page":442,"text":"Bohr magneton, 76, 155","rect":[225.44650268554688,283.1334533691406,308.3776907363407,275.48828125]},{"page":442,"text":"Boodjoom, 216–217","rect":[225.44650268554688,293.1094055175781,295.33493560938759,285.5150451660156]},{"page":442,"text":"Bragg","rect":[225.44650268554688,303.02862548828127,246.14227813868448,295.60357666015627]},{"page":442,"text":"diffraction, 3, 43, 62, 68, 80–81, 96,","rect":[237.3986053466797,311.3536682128906,361.547884674811,305.4102478027344]},{"page":442,"text":"349, 350, 354, 419","rect":[249.35069274902345,321.3296203613281,313.58884185938759,315.33538818359377]},{"page":442,"text":"law, 43","rect":[237.3986053466797,331.23779296875,262.6269888320438,325.36212158203127]},{"page":442,"text":"peaks, 101","rect":[237.3986053466797,342.875732421875,273.88104766993447,335.2813720703125]},{"page":442,"text":"reflections, 64, 67, 80, 108, 226, 343–347,","rect":[237.3986053466797,351.13299560546877,382.1903407294985,345.25732421875]},{"page":442,"text":"350, 356, 357, 365","rect":[249.35069274902345,361.10894775390627,313.58884185938759,355.1824645996094]},{"page":442,"text":"scattering, 80","rect":[237.3986053466797,372.74688720703127,283.41583770899697,365.15252685546877]},{"page":442,"text":"Break of anchoring, 314–315, 374–376, 406,","rect":[225.44650268554688,382.7228698730469,378.9776942451235,375.07769775390627]},{"page":442,"text":"407","rect":[249.35072326660157,390.9801330566406,262.04233307032509,385.1552429199219]},{"page":442,"text":"Brillouin zone, 349","rect":[225.44651794433595,401.0238342285156,291.7246145644657,395.0804138183594]},{"page":442,"text":"Buger law, 295","rect":[225.44651794433595,412.593994140625,277.96014923243447,404.9488220214844]},{"page":442,"text":"C","rect":[225.44651794433595,430.85260009765627,231.5215661678931,424.8753356933594]},{"page":442,"text":"c‐director, 48, 229, 388, 389, 392, 398, 400,","rect":[225.44651794433595,440.8709411621094,376.5967738349672,434.9275207519531]},{"page":442,"text":"401, 403–405, 416, 417, 419","rect":[249.349853515625,450.7901306152344,347.6886343398563,444.7958984375]},{"page":442,"text":"Characteristic lengths","rect":[225.44564819335938,462.41705322265627,299.5536239910058,454.82269287109377]},{"page":442,"text":"diffusion, 185, 267","rect":[237.3977508544922,470.67431640625,302.3728384414188,464.7478332519531]},{"page":442,"text":"magnetic and electric field coherence, 309,","rect":[237.3977508544922,482.312255859375,384.3013636787172,474.7178955078125]},{"page":442,"text":"314, 375","rect":[249.34983825683595,490.56951904296877,279.2047171035282,484.6430358886719]},{"page":442,"text":"nematic, 100","rect":[237.3977508544922,500.5454406738281,281.5839895644657,494.6697692871094]},{"page":442,"text":"short‐range positional in isotropic liquids, 98","rect":[237.3977508544922,512.2401123046875,385.2617544570438,504.645751953125]},{"page":442,"text":"smectic, 223, 253, 255","rect":[237.3986053466797,520.440673828125,314.94855255274697,514.51416015625]},{"page":442,"text":"surface correlation, 262","rect":[237.3986053466797,530.4166259765625,318.2906851211063,524.5408935546875]},{"page":442,"text":"surface extrapolation, 313, 332, 407","rect":[237.3986053466797,542.1112060546875,360.71787017970009,534.516845703125]},{"page":442,"text":"for tilt in de Vries smectics, 47","rect":[237.3986053466797,550.3685302734375,343.85663360743447,544.4927978515625]},{"page":442,"text":"Chatelain (rubbing) method, 278","rect":[225.44650268554688,562.00634765625,337.4947561660282,554.4119873046875]},{"page":442,"text":"Chirality, 14, 21–22, 55–58, 65, 71, 72, 133,","rect":[225.44650268554688,571.9822998046875,378.0715052802797,564.337158203125]},{"page":442,"text":"223, 253, 333, 351, 386, 389, 390, 415,","rect":[249.35072326660157,580.3073120117188,384.4139125068422,574.3131103515625]},{"page":442,"text":"418, 419, 422, 428","rect":[249.35072326660157,590.2832641601563,313.5888723769657,584.3906860351563]},{"page":442,"text":"433","rect":[372.4973449707031,624.0540771484375,385.18893951563759,618.2291870117188]},{"page":443,"text":"434","rect":[53.81319808959961,42.55905532836914,66.50480407862588,36.73417663574219]},{"page":443,"text":"Chiral nematic phase, 56, 71","rect":[53.812843322753909,67.6270980834961,151.94263214259073,59.9819450378418]},{"page":443,"text":"Cholesteric phase, 2, 3, 55, 57, 63–65, 71, 123,","rect":[53.812843322753909,77.54634857177735,213.51721450879536,69.90119171142578]},{"page":443,"text":"197, 220, 229, 342–378","rect":[77.7170639038086,85.87135314941406,158.8773397841923,79.97874450683594]},{"page":443,"text":"Clausius‐Mosotti, 157–161","rect":[53.812843322753909,95.77957153320313,146.18994659571573,89.85309600830078]},{"page":443,"text":"Clock model (for antiferroelectrics),","rect":[53.81283950805664,107.07878875732422,177.6609065253969,99.82308959960938]},{"page":443,"text":"416, 417","rect":[77.71705627441406,115.67477416992188,107.57194275050088,109.79910278320313]},{"page":443,"text":"Coercive field, 386, 410","rect":[53.81283950805664,125.65072631835938,136.2913565078251,119.77505493164063]},{"page":443,"text":"Cole‐Cole diagram, 168–170","rect":[53.81283950805664,137.34536743164063,152.92920440821573,129.75100708007813]},{"page":443,"text":"Complex conductivity, 181","rect":[53.81199264526367,147.26455688476563,146.9395956435673,139.67019653320313]},{"page":443,"text":"Compressibility modulus, 95, 189, 194","rect":[53.81199264526367,157.24053955078126,186.87418121485636,149.5953826904297]},{"page":443,"text":"Convolution operation, 91, 92","rect":[53.81199264526367,167.21649169921876,156.58775848536417,159.62213134765626]},{"page":443,"text":"Correlations","rect":[53.81199264526367,175.4737548828125,95.72813113211908,169.59808349609376]},{"page":443,"text":"bond, 106","rect":[65.76409912109375,185.39297485351563,99.8672842422001,179.51730346679688]},{"page":443,"text":"interlayer positional, 107","rect":[65.76409912109375,197.08761596679688,151.41551727442667,189.49325561523438]},{"page":443,"text":"length, 47, 59, 98, 100, 262, 264","rect":[65.76409912109375,207.0635986328125,177.3571524062626,199.41844177246095]},{"page":443,"text":"liquid‐like, 101, 106, 107","rect":[65.76409912109375,216.9827880859375,152.66014618067667,209.388427734375]},{"page":443,"text":"molecular, 50, 72, 99","rect":[65.76409912109375,225.3078155517578,138.6308340468876,219.31361389160157]},{"page":443,"text":"nematic, 59","rect":[65.76409912109375,235.2837677001953,105.66312164454385,229.28956604003907]},{"page":443,"text":"positional, 71, 72, 88, 98, 106, 107, 264","rect":[65.76409912109375,246.91067504882813,202.56436676173136,239.31631469726563]},{"page":443,"text":"quasi‐long range, 101, 106, 107","rect":[65.76409912109375,256.8299560546875,174.2426504531376,249.235595703125]},{"page":443,"text":"short‐range molecular, 99","rect":[65.76409912109375,266.805908203125,153.6416220107548,259.2115478515625]},{"page":443,"text":"Cotton‐Mouton effect, 120, 124","rect":[53.8111457824707,275.0632019042969,162.57227081446573,269.1875305175781]},{"page":443,"text":"Criteria","rect":[53.8111457824707,285.0391540527344,79.77816912912846,279.1634826660156]},{"page":443,"text":"for bistability in ferroelectrics, 407–410","rect":[65.76325225830078,296.67706298828127,202.3240713515751,289.08270263671877]},{"page":443,"text":"for ferroelectricity onset, 389–390","rect":[65.76325225830078,306.65301513671877,182.95161956934855,299.05865478515627]},{"page":443,"text":"Critical field","rect":[53.8111457824707,314.9102783203125,96.79761261134073,309.03460693359377]},{"page":443,"text":"for break of anchoring in SmC*, 407","rect":[65.76325225830078,326.6049499511719,192.2562460341923,319.0105895996094]},{"page":443,"text":"for cholesteric helix unwinding, 359, 361","rect":[65.76325225830078,336.524169921875,207.20950836329386,328.8789978027344]},{"page":443,"text":"for Frederiks transition, 306, 307","rect":[65.76325225830078,344.78143310546877,178.8310751357548,338.90576171875]},{"page":443,"text":"for SmC* helix unwinding, 401, 402","rect":[65.76325225830078,356.47607421875,191.5184530654423,348.8817138671875]},{"page":443,"text":"Cross section, 55, 58, 79, 86, 208, 228, 245,","rect":[53.8111457824707,364.7443542480469,204.9055354316469,358.7501220703125]},{"page":443,"text":"280, 287, 301, 302, 346","rect":[77.71536254882813,374.6525573730469,159.11678070216105,368.7768859863281]},{"page":443,"text":"Cross‐section for light scattering, 208, 302","rect":[53.8111457824707,386.34716796875,199.84668487696573,378.7528076171875]},{"page":443,"text":"Curie law, 120, 131, 155, 160, 384, 385, 412","rect":[53.8111457824707,394.6044616699219,207.54877227930948,388.677978515625]},{"page":443,"text":"Curie‐Weiss law, 394","rect":[53.8111457824707,404.5914001464844,127.90643829493448,398.6479797363281]},{"page":443,"text":"Cyanobiphenyl, 23, 127, 128, 132, 173, 200","rect":[53.81114959716797,416.2182922363281,204.4900869765751,408.6239318847656]},{"page":443,"text":"Cyclohexane, 19, 20, 27, 147, 154","rect":[53.81114959716797,426.1942443847656,171.35233062891886,418.549072265625]},{"page":443,"text":"D","rect":[53.81114959716797,444.2691650390625,59.88619782072512,438.5035400390625]},{"page":443,"text":"De Broglie waves, 75, 349","rect":[53.81114959716797,456.0653991699219,145.1839346328251,448.42022705078127]},{"page":443,"text":"Debye","rect":[53.81114959716797,466.0412902832031,75.91146228098393,458.4469299316406]},{"page":443,"text":"diagram, 168–170","rect":[65.76325225830078,475.9605407714844,127.72113556055948,468.3661804199219]},{"page":443,"text":"dispersion law, 167","rect":[65.76325225830078,485.9364929199219,132.38147491602823,478.3421325683594]},{"page":443,"text":"formula, 183","rect":[65.76325225830078,494.1937561035156,109.83611816553995,488.3180847167969]},{"page":443,"text":"screening length, 185","rect":[65.76325225830078,505.8883972167969,139.06488556055948,498.24322509765627]},{"page":443,"text":"Debye‐Waller factor (DWF), 95, 96","rect":[53.81114959716797,515.8075561523438,176.75304168848917,508.16241455078127]},{"page":443,"text":"Defects","rect":[53.8111457824707,524.031005859375,79.6850937175683,518.1891479492188]},{"page":443,"text":"disclinations, 41, 61","rect":[65.76325225830078,534.0408325195313,134.55176300196573,528.1651000976563]},{"page":443,"text":"edge dislocation, 226, 227","rect":[65.76325225830078,545.7354125976563,155.83218902735636,538.1410522460938]},{"page":443,"text":"fan‐shape texture, 41, 46, 228","rect":[65.76325225830078,555.6546630859375,167.7267813125126,548.060302734375]},{"page":443,"text":"focal‐conic pair, 227–228","rect":[65.76325225830078,565.630615234375,153.60777801661417,558.0362548828125]},{"page":443,"text":"lattice of defects, 65","rect":[65.76240539550781,573.887939453125,135.45709747462198,567.96142578125]},{"page":443,"text":"t\u0001 and l\u0001lines, 218–219","rect":[65.76240539550781,583.8748779296875,154.28804535059855,577.6605834960938]},{"page":443,"text":"Index","rect":[365.8709716796875,42.55059051513672,385.16220611720009,36.68337631225586]},{"page":443,"text":"steps, 226–227","rect":[237.3977508544922,67.6346664428711,288.42307037501259,60.04030990600586]},{"page":443,"text":"walls, 68, 71, 209, 217–218, 359, 363,","rect":[237.3977508544922,75.90303039550781,368.950472565436,69.90882110595703]},{"page":443,"text":"371, 401","rect":[249.34983825683595,85.81124877929688,279.2047171035282,79.98637390136719]},{"page":443,"text":"zigzag, 230, 392","rect":[237.3977508544922,97.50582122802735,293.9896292129032,89.91146850585938]},{"page":443,"text":"de Gennes, P.G., 3, 207, 208, 302, 323,","rect":[225.44564819335938,105.70639038085938,360.2829310615297,99.83071899414063]},{"page":443,"text":"358–361","rect":[249.34983825683595,115.68240356445313,278.96356720118447,109.75592803955078]},{"page":443,"text":"Degree of ionisation, 177, 178","rect":[225.44564819335938,127.37703704833985,329.61835235743447,119.78268432617188]},{"page":443,"text":"Density","rect":[225.44564819335938,137.35293579101563,251.77649444727823,129.75857543945313]},{"page":443,"text":"autocorrelation function, 93","rect":[237.3977508544922,145.62123107910157,332.5264028945438,139.67782592773438]},{"page":443,"text":"charge density wave, 47","rect":[237.3977508544922,157.24813842773438,320.1774954238407,149.65377807617188]},{"page":443,"text":"correlation function, 77, 88, 93, 94, 97–99,","rect":[237.3977508544922,165.57313537597657,384.4917628486391,159.62973022460938]},{"page":443,"text":"101, 102, 106, 107, 140","rect":[249.34983825683595,175.48135375976563,330.7512564101688,169.60568237304688]},{"page":443,"text":"of elastic energy, 190, 199, 223","rect":[237.3977508544922,187.1192626953125,345.59283966212197,179.52490234375]},{"page":443,"text":"of photonic states, 353","rect":[237.3977508544922,197.09521484375,314.94855255274697,189.45005798339845]},{"page":443,"text":"of the quadrupole moment, 324","rect":[237.3977508544922,207.07119750976563,345.19601959376259,199.47683715820313]},{"page":443,"text":"wave, 46–49, 84, 96, 101, 122, 124, 125,","rect":[237.3977508544922,215.33946228027345,377.67297622754537,209.3452606201172]},{"page":443,"text":"164, 234, 253","rect":[249.34983825683595,225.24771118164063,296.4246878066532,219.3212432861328]},{"page":443,"text":"Devil’s staircase, 411, 417","rect":[225.44564819335938,235.22366333007813,316.4207815566532,229.34799194335938]},{"page":443,"text":"p‐decyloxybenzylidene‐p’‐","rect":[225.44564819335938,246.91830444335938,317.1602250217034,239.32394409179688]},{"page":443,"text":"amino–2methylbutyl cinnamate","rect":[249.35067749023438,256.8375244140625,357.4890684821558,249.2431640625]},{"page":443,"text":"(DOBAMBC), 21, 22, 66, 386, 387","rect":[249.35067749023438,266.4748229980469,370.8575491347782,259.2021789550781]},{"page":443,"text":"de Vries phase, 46","rect":[225.4464874267578,276.7894592285156,288.79620880274697,269.1950988769531]},{"page":443,"text":"Dichroism, 34, 294–299, 303","rect":[225.4464874267578,285.1144714355469,325.4851126113407,279.1710510253906]},{"page":443,"text":"Dielectric","rect":[225.4464874267578,294.9320983886719,258.97770068430426,289.0903015136719]},{"page":443,"text":"ellipsoid, 45, 58, 59, 285–286, 303, 319,","rect":[237.39859008789063,306.6606140136719,376.0882289130922,299.01544189453127]},{"page":443,"text":"330, 418, 419","rect":[249.35067749023438,315.0,296.4255422988407,309.0929870605469]},{"page":443,"text":"losses, 167, 182–184","rect":[237.39859008789063,325.0,308.87262481837197,319.0181884765625]},{"page":443,"text":"permittivity, 59, 158, 160–165, 167–169,","rect":[237.39859008789063,336.53173828125,378.0562464912172,328.8865661621094]},{"page":443,"text":"172, 182, 183, 186, 187, 285, 294, 297,","rect":[249.35067749023438,344.8567810058594,384.4138819892641,338.862548828125]},{"page":443,"text":"301, 318, 343, 345–348, 351, 364, 381,","rect":[249.35067749023438,354.76495361328127,384.18799087598287,348.8384704589844]},{"page":443,"text":"385, 412","rect":[249.35067749023438,364.6842041015625,279.2055715957157,358.7577209472656]},{"page":443,"text":"susceptibility, 36, 157, 161, 162, 394, 397","rect":[237.39859008789063,376.37884521484377,381.5625357070438,368.7336730957031]},{"page":443,"text":"susceptibility tensor, 59","rect":[237.39859008789063,386.3547668457031,318.76193756251259,378.7095947265625]},{"page":443,"text":"Differential cross section, 79, 86, 301, 302","rect":[225.4464874267578,394.6797790527344,371.5369924941532,388.7363586425781]},{"page":443,"text":"Dimer, 20, 24, 27, 39, 128","rect":[225.4464874267578,404.5989990234375,316.13815826563759,398.65557861328127]},{"page":443,"text":"Dipole‐dipole correlations, 164","rect":[225.4477081298828,416.214111328125,332.22641510157509,408.6197509765625]},{"page":443,"text":"Dipole moment, 23, 66, 70, 71, 78, 158, 160,","rect":[225.4477081298828,426.1900939941406,379.997591706061,418.544921875]},{"page":443,"text":"162–164, 169, 171, 266, 267, 296, 321,","rect":[249.35191345214845,434.5151062011719,384.18921157910787,428.5716857910156]},{"page":443,"text":"324, 327, 381, 382, 387, 411, 413, 415,","rect":[249.35191345214845,444.3665466308594,384.4151332099672,438.4400634765625]},{"page":443,"text":"423, 426","rect":[249.35191345214845,454.342529296875,279.2067922988407,448.46685791015627]},{"page":443,"text":"Disclination","rect":[225.4477081298828,464.28460693359377,266.9238638320438,458.44281005859377]},{"page":443,"text":"core, 214","rect":[237.39979553222657,474.1361083984375,269.2370962539188,468.4128112792969]},{"page":443,"text":"energy, 214–215","rect":[237.39979553222657,485.9323425292969,294.6000112929813,478.28717041015627]},{"page":443,"text":"strength, 211–212, 214, 217","rect":[237.39979553222657,495.9082946777344,333.1900991836063,488.3139343261719]},{"page":443,"text":"Distortion","rect":[225.4477081298828,504.1316833496094,259.79965728907509,498.2898864746094]},{"page":443,"text":"bend, 195, 196, 199, 207, 221, 280,","rect":[237.39979553222657,514.1525268554688,359.0953700263735,508.1583251953125]},{"page":443,"text":"317–320, 323, 325, 327, 331, 368, 371,","rect":[249.35191345214845,524.060791015625,384.18921157910787,518.13427734375]},{"page":443,"text":"376, 377","rect":[249.35191345214845,534.0367431640625,279.2067922988407,528.1610107421875]},{"page":443,"text":"splay, 195, 196, 198, 199, 207, 221–223,","rect":[237.39979553222657,545.7313232421875,377.433871002936,538.086181640625]},{"page":443,"text":"227, 317–320, 323–325, 327, 368, 371,","rect":[249.35191345214845,554.0,383.90661880567037,548.0054321289063]},{"page":443,"text":"376","rect":[249.35191345214845,563.9078979492188,262.04352325587197,558.0321655273438]},{"page":443,"text":"twist, 196, 198, 199, 205, 207, 234, 249,","rect":[237.39979553222657,573.9515380859375,376.2586390693422,567.9573364257813]},{"page":443,"text":"307, 315, 317, 321, 324, 368, 376","rect":[249.35191345214845,583.8030395507813,365.13657135157509,577.8765258789063]},{"page":444,"text":"Index","rect":[53.812828063964847,42.56230926513672,73.10407013087198,36.69509506225586]},{"page":444,"text":"Distribution function, 28–34, 36, 100, 101,","rect":[53.812843322753909,66.0,200.54641983106098,60.03274154663086]},{"page":444,"text":"112, 133, 134, 140, 165–167, 171, 173","rect":[77.7170639038086,75.82766723632813,210.3824209609501,69.90119171142578]},{"page":444,"text":"Dupin cyclides, 227, 228","rect":[53.812843322753909,87.52230072021485,139.57593292384073,79.92794799804688]},{"page":444,"text":"DWF. See Debye‐Waller factor","rect":[53.812843322753909,97.49825286865235,161.47741788489513,89.90390014648438]},{"page":444,"text":"E","rect":[53.812843322753909,115.57318115234375,59.44791590895038,109.80756378173828]},{"page":444,"text":"Einstein, 36, 173, 176, 185, 193","rect":[53.812843322753909,125.71846008300781,163.31009430079386,119.72425079345703]},{"page":444,"text":"Elastomers, 53, 260","rect":[53.812843322753909,135.62667846679688,121.67654937891885,129.70021057128907]},{"page":444,"text":"Electric conductivity, 164, 176–187, 335–337","rect":[53.812843322753909,147.26455688476563,210.59561676173136,139.61940002441407]},{"page":444,"text":"Electroclinic coefficient, 394, 398","rect":[53.812843322753909,155.58958435058595,169.93848175196573,149.64617919921876]},{"page":444,"text":"Electrohydrodynamics, 233, 234, 239,","rect":[53.812843322753909,167.21649169921876,184.3460108955141,159.62213134765626]},{"page":444,"text":"334–339","rect":[77.7170639038086,175.54148864746095,107.3308081069462,169.6488800048828]},{"page":444,"text":"Electrostatic interaction, 25, 26, 415","rect":[53.812843322753909,185.39297485351563,178.20918792872355,179.4665069580078]},{"page":444,"text":"Elementary cell, 15, 84–86, 108, 135","rect":[53.812843322753909,197.08761596679688,180.7009787002079,189.4424591064453]},{"page":444,"text":"Elliptic polarization, 288, 293","rect":[53.812843322753909,207.0635986328125,156.28738922266886,199.46923828125]},{"page":444,"text":"Enthalpy, 42, 121","rect":[53.812843322753909,216.9827880859375,114.59631866602823,209.388427734375]},{"page":444,"text":"Entropy, 27, 112, 119, 133–135, 146, 258,","rect":[53.812843322753909,226.95877075195313,199.01663467481098,219.31361389160157]},{"page":444,"text":"259, 415","rect":[77.7170639038086,235.2837677001953,107.57195037989541,229.28956604003907]},{"page":444,"text":"Equation of state, 25, 133, 137–140, 142","rect":[53.812843322753909,246.91067504882813,193.27664703516886,239.26551818847657]},{"page":444,"text":"Euler","rect":[53.812843322753909,255.07740783691407,72.21566861731698,249.235595703125]},{"page":444,"text":"angles, 16, 28–30, 140","rect":[65.76494598388672,266.805908203125,143.2023672500126,259.2115478515625]},{"page":444,"text":"equation, 201–205, 214, 223, 272, 273,","rect":[65.76494598388672,276.7818908691406,199.97952529981098,269.13671875]},{"page":444,"text":"276, 308, 328, 360, 384","rect":[77.71705627441406,285.0391540527344,159.11847442774698,279.1634826660156]},{"page":444,"text":"form for equation of motion, 236","rect":[65.76494598388672,296.67706298828127,179.17206329493448,289.08270263671877]},{"page":444,"text":"Euler‐Lagrange variation procedure, 200","rect":[53.812843322753909,306.65301513671877,193.16243499903605,299.05865478515627]},{"page":444,"text":"Ewald sphere, 76, 85, 86","rect":[53.81199645996094,316.62896728515627,138.8575949111454,308.9837951660156]},{"page":444,"text":"Excluded volume, 25, 138, 139, 141, 142","rect":[53.81199645996094,324.9540100097656,195.4274649062626,318.95977783203127]},{"page":444,"text":"F","rect":[53.81199645996094,344.6798400878906,59.00709340879672,338.9142150878906]},{"page":444,"text":"Ferrielectrics, 410–412","rect":[53.81199645996094,354.75738525390627,132.76392120509073,348.8817138671875]},{"page":444,"text":"Ferroelectrics and ferroelectricity","rect":[53.81199645996094,366.3952941894531,168.06773895411417,358.8009338378906]},{"page":444,"text":"phase, 51, 384, 385, 412, 416, 425, 426","rect":[65.76409912109375,376.3712463378906,201.0912832656376,368.72607421875]},{"page":444,"text":"properties, 72, 423","rect":[65.76409912109375,386.34716796875,129.83301300196573,378.7528076171875]},{"page":444,"text":"transition, 384","rect":[65.76409912109375,394.6044616699219,114.99229950098916,388.7287902832031]},{"page":444,"text":"Ferrofluid, 156","rect":[53.81199645996094,404.5236511230469,104.85255951075479,398.59716796875]},{"page":444,"text":"Ferromagnetism, 155–156","rect":[53.81199645996094,416.2182922363281,143.52643341212198,408.5731201171875]},{"page":444,"text":"Fick law, 174","rect":[53.81199645996094,424.4755554199219,100.66093963770791,418.5998840332031]},{"page":444,"text":"Field induced cholesteric‐nematic transition,","rect":[53.81199645996094,434.44305419921877,205.9784114082094,428.5758361816406]},{"page":444,"text":"358–361","rect":[77.71705627441406,444.3707580566406,107.33080047755166,438.44427490234377]},{"page":444,"text":"Fingerprint texture, 219–220","rect":[53.81283950805664,456.0653991699219,152.08055633692667,448.4710388183594]},{"page":444,"text":"First order transition, 124–127","rect":[53.81283950805664,464.3226013183594,158.1420645156376,458.4469299316406]},{"page":444,"text":"Flexible chain, 39, 51","rect":[53.81283950805664,474.3096008300781,128.3785604873173,468.31536865234377]},{"page":444,"text":"Flexoelectric","rect":[53.81283950805664,484.1839294433594,98.1826948859644,478.3421325683594]},{"page":444,"text":"coefficient, 323","rect":[65.76494598388672,494.1937561035156,118.7879385390751,488.3180847167969]},{"page":444,"text":"converse effect, 327–332","rect":[65.76494598388672,504.1697082519531,151.74042266993448,498.2940368652344]},{"page":444,"text":"domains, 332–334","rect":[65.76494598388672,514.0889282226563,128.57315582423136,508.2132263183594]},{"page":444,"text":"effect, 331–332","rect":[65.76494598388672,524.0648803710938,119.22621673731729,518.1891479492188]},{"page":444,"text":"instability, 333, 334","rect":[65.76494598388672,535.7594604492188,134.13886016993448,528.1651000976563]},{"page":444,"text":"polarization, 322–324","rect":[65.76494598388672,545.7354125976563,140.46857208399698,538.1410522460938]},{"page":444,"text":"Flexoelectricity in cholesterics, 376–378","rect":[53.81283950805664,555.6546630859375,192.8087667861454,548.060302734375]},{"page":444,"text":"Floquet‐Bloch theorem, 348, 352","rect":[53.81283950805664,565.630615234375,167.3316397109501,557.9854736328125]},{"page":444,"text":"Fluctuations","rect":[53.81199264526367,573.8540649414063,95.73659213065423,568.01220703125]},{"page":444,"text":"of director, 207, 208, 300, 337, 339, 415","rect":[65.76409912109375,583.8748779296875,204.71686309962198,577.8806762695313]},{"page":444,"text":"435","rect":[372.4981994628906,42.57077407836914,385.18979400782509,36.6442985534668]},{"page":444,"text":"of mass density, 300, 301","rect":[237.3986053466797,67.6346664428711,324.97406524805947,60.04030990600586]},{"page":444,"text":"thermal, 77, 95, 96, 102, 223, 307, 401","rect":[237.3986053466797,75.90303039550781,370.9701284804813,69.90882110595703]},{"page":444,"text":"Focal‐conic domains, 46, 219, 227–228","rect":[225.44650268554688,85.87898254394531,360.81683868555947,79.93557739257813]},{"page":444,"text":"Form‐factor, 87–91, 99, 100","rect":[225.44650268554688,95.85487365722656,322.1438345351688,89.91146850585938]},{"page":444,"text":"Fourier","rect":[225.44650268554688,105.67253112792969,250.37281888587169,99.83071899414063]},{"page":444,"text":"direct transform, 94","rect":[237.3986053466797,115.75013732910156,305.1108450332157,109.80673217773438]},{"page":444,"text":"expansion, 103","rect":[237.3986053466797,127.37703704833985,288.9485220351688,119.78268432617188]},{"page":444,"text":"harmonics, 96, 122, 125, 207, 300, 316,","rect":[237.3986053466797,135.7019805908203,373.7089564521547,129.70777893066407]},{"page":444,"text":"331, 349","rect":[249.35069274902345,145.62123107910157,279.2055715957157,139.72862243652345]},{"page":444,"text":"integral, 82, 93","rect":[237.3986053466797,157.24813842773438,289.3064321914188,149.65377807617188]},{"page":444,"text":"inverse transform, 93","rect":[237.3986053466797,165.57313537597657,310.2086538711063,159.62973022460938]},{"page":444,"text":"law, 172","rect":[237.3986053466797,175.48135375976563,266.85751861720009,169.60568237304688]},{"page":444,"text":"series, 132, 224","rect":[237.3986053466797,185.40057373046876,291.10101837305947,179.52490234375]},{"page":444,"text":"Frank","rect":[225.44650268554688,195.34266662597657,245.17772430567667,189.5008544921875]},{"page":444,"text":"energy, 197–200, 204, 206, 222, 272, 307,","rect":[237.3986053466797,207.07119750976563,382.6023890693422,199.47683715820313]},{"page":444,"text":"308, 367, 368","rect":[249.35069274902345,215.271728515625,296.4255422988407,209.39605712890626]},{"page":444,"text":"formula, 199–200","rect":[237.3986053466797,225.31544494628907,298.39362091212197,219.37203979492188]},{"page":444,"text":"Frederiks transition","rect":[225.44650268554688,235.21519470214845,291.88959259180947,229.34799194335938]},{"page":444,"text":"dynamics, 315–316","rect":[237.3986053466797,246.91830444335938,304.05829376368447,239.2731475830078]},{"page":444,"text":"electrooptical response, 318–322","rect":[237.3986053466797,256.8375244140625,349.8843130507938,249.2431640625]},{"page":444,"text":"in SmC*, 403","rect":[237.3986053466797,265.09478759765627,284.75775665430947,259.2191162109375]},{"page":444,"text":"theory, 307–312","rect":[237.3986053466797,276.7894592285156,293.18243927149697,269.1950988769531]},{"page":444,"text":"threshold field, 307, 309–312","rect":[237.3986053466797,285.1144714355469,337.7054495254032,279.1710510253906]},{"page":444,"text":"Friedel theorem, 89","rect":[225.44650268554688,295.0337219238281,292.7052359023563,289.0903015136719]},{"page":444,"text":"Frustrated phase, 47","rect":[225.44650268554688,306.6606140136719,294.46091217188759,299.0662536621094]},{"page":444,"text":"G","rect":[225.44650268554688,324.9277038574219,232.0545983159064,318.94195556640627]},{"page":444,"text":"Gauss theorem, 234","rect":[225.44650268554688,334.81304931640627,293.48029083399697,328.9373779296875]},{"page":444,"text":"Gay‐Berne, 26","rect":[225.44650268554688,346.5077209472656,275.54111236720009,338.9133605957031]},{"page":444,"text":"Gibbs free energy, 112","rect":[225.4464874267578,356.483642578125,303.9034475722782,348.8892822265625]},{"page":444,"text":"Goldstone","rect":[225.4464874267578,364.6842041015625,260.23839709543707,358.80853271484377]},{"page":444,"text":"mode, 28, 388–390, 396–397","rect":[237.39859008789063,374.7279052734375,337.42282623438759,368.78448486328127]},{"page":444,"text":"theorem, 233","rect":[237.39859008789063,384.6360778808594,282.4343313613407,378.7604064941406]},{"page":444,"text":"Grandjean zones, 61, 62","rect":[225.4464874267578,396.3307189941406,308.39545196680947,388.7363586425781]},{"page":444,"text":"Grey scale, 402, 410","rect":[225.4464874267578,406.24993896484377,296.1420950332157,398.65557861328127]},{"page":444,"text":"Group","rect":[225.4464874267578,416.2259216308594,247.03067535548136,408.6823425292969]},{"page":444,"text":"continuous symmetry, 396","rect":[237.39859008789063,426.20184326171877,328.09113067774697,418.60748291015627]},{"page":444,"text":"continuous translation, 17","rect":[237.39859008789063,434.4591064453125,325.78632110743447,428.58343505859377]},{"page":444,"text":"infinite, 11, 16","rect":[237.39859008789063,444.3783264160156,287.38068145899697,438.5026550292969]},{"page":444,"text":"point symmetry, 12, 15, 17","rect":[237.39859008789063,456.0729675292969,330.03461212305947,448.42779541015627]},{"page":444,"text":"space symmetry, 12","rect":[237.39859008789063,466.0489196777344,305.50766510157509,458.5053405761719]},{"page":444,"text":"Guest‐host effect, 156, 298, 321–322","rect":[225.4464874267578,474.3172302246094,352.2635549941532,468.322998046875]},{"page":444,"text":"Gyromagnetic ratio, 153","rect":[225.44732666015626,485.9440612792969,309.1721548476688,478.29888916015627]},{"page":444,"text":"H","rect":[225.44732666015626,504.0757141113281,232.05542229051577,498.3100891113281]},{"page":444,"text":"Hedgehogs, 215–216, 227, 230","rect":[225.44732666015626,515.8152465820313,332.0001272597782,508.17010498046877]},{"page":444,"text":"Helfrich, W., 251, 336–339, 367,","rect":[225.44732666015626,524.1401977539063,338.9872767646547,518.14599609375]},{"page":444,"text":"Helical twisting power, 56","rect":[225.44732666015626,535.76708984375,316.21429962305947,528.1219482421875]},{"page":444,"text":"Helix untwisting, 374","rect":[225.44732666015626,545.7430419921875,299.65598053126259,538.148681640625]},{"page":444,"text":"Helmholtz","rect":[225.44732666015626,553.9097900390625,261.2884154059839,548.0679321289063]},{"page":444,"text":"free energy, 112, 305","rect":[237.3994140625,565.6382446289063,310.4751333632938,557.9931030273438]},{"page":444,"text":"monolayer, 267","rect":[237.3994140625,575.6141967773438,290.9326223769657,568.0198364257813]},{"page":444,"text":"wave equation, 351","rect":[237.3994140625,585.5333862304688,304.13019317774697,577.8882446289063]},{"page":445,"text":"436","rect":[53.81208419799805,42.55752944946289,66.50369018702432,36.68185043334961]},{"page":445,"text":"Hexatic phase, 72, 105–107","rect":[53.812843322753909,67.6270980834961,148.9651846328251,59.9819450378418]},{"page":445,"text":"Hilbert transform, 297","rect":[53.812843322753909,75.89540100097656,130.4007543960087,69.95199584960938]},{"page":445,"text":"Hill equation, 347–349","rect":[53.812843322753909,87.52230072021485,132.76477569727823,79.92794799804688]},{"page":445,"text":"Hooke law, 103, 189–190, 193, 195, 221","rect":[53.812843322753909,95.84730529785156,193.84271759180948,89.85309600830078]},{"page":445,"text":"Huygens principle, 79, 81","rect":[53.812843322753909,107.41744232177735,142.36978668360636,99.82308959960938]},{"page":445,"text":"Hybrid cell, 217, 268, 269, 325–326","rect":[53.812843322753909,117.3934555053711,178.13727325587198,109.74829864501953]},{"page":445,"text":"Hydrocarbon chains (tails), 19, 20","rect":[53.812843322753909,127.3694076538086,170.6365255752079,119.77505493164063]},{"page":445,"text":"Hydrodynamic relaxation times, 233","rect":[53.812843322753909,137.34536743164063,179.05867523341105,129.75100708007813]},{"page":445,"text":"Hydrodynamic variables, 233–234","rect":[53.812843322753909,147.26455688476563,171.79315704493448,139.67019653320313]},{"page":445,"text":"Hydrogen bond, 26–27, 259, 426","rect":[53.812843322753909,157.24053955078126,166.9364981093876,149.5953826904297]},{"page":445,"text":"Hysteresis, 51, 117, 119, 361, 383, 385,","rect":[53.812843322753909,167.21649169921876,190.0098292548891,159.5713348388672]},{"page":445,"text":"407–410, 412, 421, 425","rect":[77.7170639038086,175.4737548828125,158.83588165430948,169.5472869873047]},{"page":445,"text":"I","rect":[53.812843322753909,195.267333984375,57.070355253212827,189.50172424316407]},{"page":445,"text":"Improper ferroelectric, 384, 423","rect":[53.812843322753909,207.0635986328125,163.70776886134073,199.46923828125]},{"page":445,"text":"Incommensurate structure, 47, 56, 67, 418","rect":[53.812843322753909,215.26409912109376,198.48782867579386,209.33763122558595]},{"page":445,"text":"Instability","rect":[53.812843322753909,226.95877075195313,88.24939483790323,219.36441040039063]},{"page":445,"text":"Benard, 335","rect":[65.76494598388672,235.21603393554688,107.45856231837198,229.28956604003907]},{"page":445,"text":"Carr‐Helfrich, 336, 339","rect":[65.76494598388672,245.2597198486328,146.20349640040323,239.31631469726563]},{"page":445,"text":"due to flow, 247","rect":[65.76409912109375,255.11126708984376,121.50647491602823,249.235595703125]},{"page":445,"text":"hydrodynamic, 234, 335","rect":[65.76409912109375,266.805908203125,149.14879364161417,259.1607360839844]},{"page":445,"text":"Landau‐Peierls, 102–105","rect":[65.76409912109375,275.0632019042969,151.17352813868448,269.13671875]},{"page":445,"text":"Interaction","rect":[53.8120002746582,285.0052795410156,90.6176504531376,279.1634826660156]},{"page":445,"text":"intermolecular, 24–37, 57, 111, 140, 146,","rect":[65.76410675048828,295.0,207.62662002637348,289.0318908691406]},{"page":445,"text":"414","rect":[77.7162094116211,304.8327331542969,90.4078115859501,299.10943603515627]},{"page":445,"text":"between layers, 106, 107, 414, 417","rect":[65.76410675048828,316.62896728515627,185.4010977187626,309.03460693359377]},{"page":445,"text":"of order parameters, 124, 126–127","rect":[65.76410675048828,326.6049499511719,184.0270285048954,319.0105895996094]},{"page":445,"text":"Inversion","rect":[53.8120002746582,334.7716064453125,85.78638214259073,328.9298095703125]},{"page":445,"text":"centre, 9, 15, 23, 31, 383, 426","rect":[65.76410675048828,344.84918212890627,168.4645742812626,338.8549499511719]},{"page":445,"text":"as group operation,","rect":[65.76410675048828,356.47607421875,131.48970291211567,348.8817138671875]},{"page":445,"text":"point for ea, 172","rect":[65.76410675048828,366.3952941894531,121.5637182631962,358.8009338378906]},{"page":445,"text":"symmetry, 9","rect":[65.76549530029297,376.36761474609377,108.40166992335245,368.82403564453127]},{"page":445,"text":"Ising model, 416, 417","rect":[53.81338882446289,386.34356689453127,128.70062774805948,378.74920654296877]},{"page":445,"text":"Isomer, 22, 27, 56, 67,","rect":[53.81338882446289,394.600830078125,131.2304866035219,388.6743469238281]},{"page":445,"text":"Isomerization, 19–20, 27","rect":[53.81338882446289,404.5877990722656,139.1415762343876,398.6443786621094]},{"page":445,"text":"K","rect":[53.81338882446289,424.370361328125,60.42148445482242,418.604736328125]},{"page":445,"text":"Kerr, 120, 124","rect":[53.81338882446289,434.4479064941406,103.66431945948526,428.6230163574219]},{"page":445,"text":"Kirchhof equation, 408","rect":[53.81338882446289,446.0858459472656,133.23322052149698,438.4914855957031]},{"page":445,"text":"Kirkwood factors, 164","rect":[53.81338882446289,454.3431091308594,130.74060577540323,448.4674377441406]},{"page":445,"text":"Kramers‐Kronig relations, 297–299","rect":[53.81338882446289,466.0377502441406,175.41925567774698,458.4433898925781]},{"page":445,"text":"L","rect":[53.813385009765628,484.1126708984375,59.4484575959621,478.3470458984375]},{"page":445,"text":"Lagrange form of equation of motion, 235","rect":[53.813385009765628,495.9089050292969,198.8276876845829,488.26373291015627]},{"page":445,"text":"Landau","rect":[53.813385009765628,504.15771484375,79.26428741602823,498.2904968261719]},{"page":445,"text":"coefficients, 118, 120, 124","rect":[65.76549530029297,514.0853881835938,156.4571737685673,508.2096862792969]},{"page":445,"text":"expansion, 113–115, 118, 119, 121, 122,","rect":[65.76549530029297,525.780029296875,205.07785293652973,518.1348876953125]},{"page":445,"text":"128–130, 261, 397, 416, 418","rect":[77.71759796142578,534.1050415039063,176.05638641505167,528.16162109375]},{"page":445,"text":"Landau‐de Gennes equation, 115–116","rect":[53.813385009765628,545.73193359375,183.51904815821573,538.0867919921875]},{"page":445,"text":"Landau‐Khalatnikov equations, 130–132, 315,","rect":[53.813385009765628,555.6511840820313,212.27231094434223,548.0060424804688]},{"page":445,"text":"395","rect":[77.71759796142578,563.9761962890625,90.40920013575479,557.9819946289063]},{"page":445,"text":"Landau‐Peierls","rect":[53.813385009765628,573.8759765625,105.08746798514642,568.0087280273438]},{"page":445,"text":"instability, 102–105","rect":[65.76549530029297,585.5223388671875,133.89825958399698,577.877197265625]},{"page":445,"text":"Index","rect":[365.869873046875,42.54906463623047,385.16110748438759,36.68185043334961]},{"page":445,"text":"theorem, 58, 105","rect":[237.3999786376953,66.0,295.4073843398563,59.9860954284668]},{"page":445,"text":"Lande factor, 155","rect":[225.4478759765625,75.83181762695313,286.11798614649697,69.90534210205078]},{"page":445,"text":"Langevin","rect":[225.4478759765625,87.52645111083985,257.5068716445438,79.93209838867188]},{"page":445,"text":"equation, 160","rect":[237.3999786376953,97.50240325927735,283.8520864882938,89.90805053710938]},{"page":445,"text":"formula, 155, 167","rect":[237.3999786376953,105.70297241210938,298.6361135879032,99.77649688720703]},{"page":445,"text":"time for chemical relaxation, 187","rect":[237.3999786376953,115.67892456054688,351.54404205470009,109.80325317382813]},{"page":445,"text":"Langmuir‐Blodgett films, 259","rect":[225.4478759765625,127.3735580444336,327.4114126601688,119.72840118408203]},{"page":445,"text":"Langmuir trough, 259","rect":[225.44705200195313,137.34951782226563,300.61859649805947,129.70436096191407]},{"page":445,"text":"Laplace‐Young formula, 258","rect":[225.44705200195313,147.26876831054688,324.6361441054813,139.6236114501953]},{"page":445,"text":"Larmor theorem, 153","rect":[225.44703674316407,155.52603149414063,298.35187286524697,149.5995635986328]},{"page":445,"text":"Latent heat, 43, 119","rect":[225.44703674316407,165.56971740722657,293.9343313613407,159.62631225585938]},{"page":445,"text":"Lattice of liquid threads, 50","rect":[225.44703674316407,177.19662475585938,320.63153595118447,169.5514678955078]},{"page":445,"text":"Layer","rect":[225.44703674316407,187.11587524414063,245.25439542395763,179.6908416748047]},{"page":445,"text":"compressibility, 105","rect":[237.3991241455078,197.09182739257813,306.9626516738407,189.44667053222657]},{"page":445,"text":"displacement, 220, 224, 234, 303","rect":[237.3991241455078,207.06781005859376,350.97543853907509,199.47344970703126]},{"page":445,"text":"Legendre polynomial, 29–31, 34, 57","rect":[225.44703674316407,216.98699951171876,350.14374298243447,209.3418426513672]},{"page":445,"text":"Lennard‐Jones, 25, 26, 146","rect":[225.44703674316407,225.24429321289063,318.3487600722782,219.3178253173828]},{"page":445,"text":"Lenz law, 153","rect":[225.4478759765625,235.22024536132813,274.27924102930947,229.2937774658203]},{"page":445,"text":"Leslie, 3, 194, 239","rect":[225.4478759765625,245.26393127441407,289.3467153945438,239.32052612304688]},{"page":445,"text":"coefficients, 240–250, 254, 318","rect":[237.3999786376953,255.11544799804688,345.02902740626259,249.18898010253907]},{"page":445,"text":"Light scattering by nematics, 206–207,","rect":[225.4478759765625,266.81005859375,358.96863052442037,259.2156982421875]},{"page":445,"text":"299–304","rect":[249.35208129882813,275.1351013183594,278.9658255019657,269.2424621582031]},{"page":445,"text":"Liquid crystals","rect":[225.4478759765625,286.7619934082031,276.36743624686519,279.1676330566406]},{"page":445,"text":"banana‐like, 17, 19, 56, 69","rect":[237.39996337890626,295.0302734375,328.7888540664188,289.0360412597656]},{"page":445,"text":"bent‐core, 17, 19, 56, 69, 425","rect":[237.39996337890626,305.0062255859375,338.40060943751259,299.0119934082031]},{"page":445,"text":"calamitic, 17, 54","rect":[237.3999786376953,315.0,294.57565826563759,308.9879455566406]},{"page":445,"text":"discotic, 50, 54, 102","rect":[237.3999786376953,324.8904113769531,306.90680450587197,318.96392822265627]},{"page":445,"text":"Local field, 157–161","rect":[225.4478759765625,334.8096008300781,296.41257232813759,328.88311767578127]},{"page":445,"text":"London dispersion forces, 146","rect":[225.4478759765625,346.5042724609375,329.2821096816532,338.909912109375]},{"page":445,"text":"Lorentzian, 98, 99, 101","rect":[225.4478759765625,354.82928466796877,305.3770803847782,348.8858642578125]},{"page":445,"text":"Lorentz‐Lorentz, 160","rect":[225.4478759765625,364.6807861328125,298.29681915430947,358.80511474609377]},{"page":445,"text":"Lyotropic","rect":[225.4478759765625,376.3753967285156,258.91986987375739,368.7810363769531]},{"page":445,"text":"nematic, 38, 54, 55","rect":[237.39996337890626,384.6326599121094,303.2423147597782,378.7061767578125]},{"page":445,"text":"phase, 38, 53–55","rect":[237.39996337890626,396.32733154296877,295.31512970118447,388.6821594238281]},{"page":445,"text":"M","rect":[225.4478759765625,414.4022216796875,233.46050921849654,408.6365966796875]},{"page":445,"text":"Magnetic moment, 23–24, 76, 151–153, 155,","rect":[225.4478759765625,426.1984558105469,379.7718836982485,418.55328369140627]},{"page":445,"text":"156","rect":[249.35208129882813,434.4557189941406,262.04367584376259,428.52923583984377]},{"page":445,"text":"Maier‐Saupe theory, 146–147","rect":[225.4478759765625,446.0936584472656,327.34027618555947,438.4992980957031]},{"page":445,"text":"Mandelshtam‐Brillouin doublet, 300","rect":[225.447021484375,454.3509216308594,350.0701040664188,448.4752502441406]},{"page":445,"text":"Mass action law, 177","rect":[225.447021484375,464.3268737792969,298.2392935195438,458.4512023925781]},{"page":445,"text":"Mathieu 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282, 383, 425","rect":[65.76495361328125,475.9605407714844,157.64624542384073,468.31536865234377]},{"page":447,"text":"Selective reflection, 63, 343, 344, 347, 355","rect":[53.81284713745117,484.2178039550781,201.14969391016886,478.29132080078127]},{"page":447,"text":"Shear","rect":[53.81284713745117,494.1937561035156,73.09562772376229,488.3180847167969]},{"page":447,"text":"force, 237","rect":[65.76495361328125,504.1697082519531,100.37834686427041,498.2940368652344]},{"page":447,"text":"modulus, 50, 189, 194","rect":[65.76495361328125,514.1566162109375,142.23863739161417,508.16241455078127]},{"page":447,"text":"quasi‐long range, 106","rect":[65.76495361328125,525.7835083007813,139.86023468165323,518.1891479492188]},{"page":447,"text":"rate tensor, 237, 241","rect":[65.76495361328125,534.0408325195313,135.9512228408329,528.2159423828125]},{"page":447,"text":"viscosity, 238","rect":[65.76495361328125,545.7354125976563,113.06656402735635,538.1410522460938]},{"page":447,"text":"Short‐range order","rect":[53.81284713745117,555.6546630859375,114.00489896399667,548.060302734375]},{"page":447,"text":"positional order, 98, 106, 261","rect":[65.76410675048828,565.630615234375,166.82227081446573,558.0362548828125]},{"page":447,"text":"smectic, 126, 339","rect":[65.76410675048828,574.0,126.09407562403604,568.01220703125]},{"page":447,"text":"Sine‐Gordon equation, 401, 404","rect":[53.8120002746582,585.5258178710938,163.4226278701298,577.9314575195313]},{"page":447,"text":"Index","rect":[365.87078857421877,42.54882049560547,385.1620230117313,36.68160629272461]},{"page":447,"text":"Six‐fold rotation axis, 49, 94","rect":[225.44566345214845,66.0,323.91559356837197,60.04030990600586]},{"page":447,"text":"Smectic monolayer, 105, 106, 428","rect":[225.4456787109375,77.5539779663086,343.1010794082157,69.90882110595703]},{"page":447,"text":"Smoluchowski, 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117","rect":[225.4456787109375,157.24813842773438,320.27228302149697,149.65377807617188]},{"page":447,"text":"STM. See Scanning tunnel microscopy","rect":[225.4456787109375,167.22409057617188,358.5704397597782,159.62973022460938]},{"page":447,"text":"Stokes","rect":[225.4456787109375,175.48135375976563,247.99443514823236,169.60568237304688]},{"page":447,"text":"formula, 181","rect":[237.3977813720703,185.40057373046876,281.47064727930947,179.52490234375]},{"page":447,"text":"law, 174, 239","rect":[237.3977813720703,195.4442596435547,284.01996368555947,189.5008544921875]},{"page":447,"text":"Strain, 190, 323","rect":[225.4456787109375,205.4202423095703,280.1109671035282,199.47683715820313]},{"page":447,"text":"tensor, 191–193, 195, 197","rect":[237.3977813720703,215.33946228027345,326.6172232070438,209.3452606201172]},{"page":447,"text":"Stripe domains, 200","rect":[225.4456787109375,226.96640014648438,294.04633087305947,219.37203979492188]},{"page":447,"text":"Structure factor, 87–88, 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393,","rect":[237.3977813720703,384.7038269042969,383.054232331061,378.7095947265625]},{"page":447,"text":"394, 397","rect":[249.34986877441407,394.6797790527344,279.2047476211063,388.7871398925781]},{"page":447,"text":"for smectic layer formation, 124","rect":[237.3977813720703,406.24993896484377,348.08551544337197,398.65557861328127]},{"page":447,"text":"soft mode, 128, 395, 396","rect":[237.3977813720703,414.5749816894531,322.8783010879032,408.58074951171877]},{"page":447,"text":"structural at N‐Iso transition, 162, 163","rect":[237.3977813720703,424.483154296875,368.36411041407509,418.60748291015627]},{"page":447,"text":"Switching, 130, 172, 320, 357, 366, 370,","rect":[225.4456787109375,436.17779541015627,365.0406214912172,428.5326232910156]},{"page":447,"text":"374–376, 383, 385, 390, 391, 397, 399,","rect":[249.34986877441407,444.4460754394531,384.1871669013735,438.45184326171877]},{"page":447,"text":"402, 403, 405, 407, 409, 410, 412, 413,","rect":[249.34986877441407,454.4220275878906,384.4130580146547,448.42779541015627]},{"page":447,"text":"420–423, 428","rect":[249.34986877441407,464.3302307128906,296.1421255507938,458.5053405761719]},{"page":447,"text":"Symmetry","rect":[225.4456787109375,475.9681701660156,260.7706350722782,468.4245910644531]},{"page":447,"text":"axes, 23, 28, 36, 102, 143, 166","rect":[237.3977813720703,484.2253723144531,342.87347931055947,478.3497009277344]},{"page":447,"text":"bilateral, 8","rect":[237.3977813720703,494.20135498046877,274.0858206191532,488.32568359375]},{"page":447,"text":"conical, 33, 44, 51, 121, 197, 201, 266, 323","rect":[237.3977813720703,504.2450256347656,385.18646759180947,498.25079345703127]},{"page":447,"text":"cubic, 65, 383","rect":[237.3977813720703,514.0965576171875,285.9457144179813,508.1700439453125]},{"page":447,"text":"cylindrical, 23, 33, 44, 58, 99, 141, 143,","rect":[237.3977813720703,525.7910766601563,375.0111415107485,518.1459350585938]},{"page":447,"text":"261, 264, 272, 335","rect":[249.34986877441407,534.0484619140625,313.5880178847782,528.1219482421875]},{"page":447,"text":"groups, 7–17, 44, 46–49, 55, 56, 65, 108,","rect":[237.3977813720703,545.7429809570313,378.5791957099672,538.0978393554688]},{"page":447,"text":"165, 254, 386, 396","rect":[249.34986877441407,554.0112915039063,313.5880178847782,548.01708984375]},{"page":447,"text":"head‐to‐tail symmetry, 28, 32, 49, 62, 63,","rect":[237.3977813720703,565.624755859375,379.619173737311,558.0303955078125]},{"page":447,"text":"65, 144, 195, 196, 212, 221, 345, 423,","rect":[249.3517303466797,573.9497680664063,380.16662857129537,567.95556640625]},{"page":447,"text":"424, 427","rect":[249.3517303466797,583.8013305664063,279.20660919337197,577.9764404296875]},{"page":448,"text":"Index","rect":[53.813690185546878,42.55895233154297,73.104932252454,36.69173812866211]},{"page":448,"text":"mirror, 21, 55, 58, 65, 69–72, 195, 197, 327,","rect":[65.76496887207031,66.0,213.51977798535786,59.9819450378418]},{"page":448,"text":"333, 386, 425, 427, 428","rect":[77.71707916259766,75.82766723632813,159.1185049453251,69.90119171142578]},{"page":448,"text":"operations, 7–13","rect":[65.76496887207031,87.52230072021485,122.5091299453251,79.92794799804688]},{"page":448,"text":"polar, 14, 44, 51, 266, 323","rect":[65.76496887207031,97.49825286865235,156.3999533095829,89.85309600830078]},{"page":448,"text":"rotational, 7, 8, 15, 17, 71, 279","rect":[65.76496887207031,105.76649475097656,171.80756134180948,99.77228546142578]},{"page":448,"text":"six‐fold, 106","rect":[65.76496887207031,115.67477416992188,109.21425384669229,109.79910278320313]},{"page":448,"text":"spherical, 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428","rect":[53.81370544433594,187.11166381835938,188.97166961817667,179.51730346679688]},{"page":448,"text":"T","rect":[53.81370544433594,205.24331665039063,59.44877803053241,199.4777069091797]},{"page":448,"text":"Tensor","rect":[53.81370544433594,215.23023986816407,77.32701963050057,209.55775451660157]},{"page":448,"text":"of curvature distortion, 196–197","rect":[65.76580810546875,225.3078155517578,176.26822418360636,219.36441040039063]},{"page":448,"text":"of density of the quadrupolar moment,","rect":[65.76580810546875,236.93472290039063,197.56474563672504,229.34036254882813]},{"page":448,"text":"of dielectric anisotropy, 59–60","rect":[65.76580810546875,246.91067504882813,170.69661468653605,239.26551818847657]},{"page":448,"text":"of elasticity, 189–193, 197–198","rect":[65.76580810546875,256.8299560546875,174.17325348048136,249.235595703125]},{"page":448,"text":"of momentum density flux, 236, 317","rect":[65.76580810546875,266.805908203125,190.6139883437626,259.2115478515625]},{"page":448,"text":"of polarizability, 22, 23, 146, 161","rect":[65.76580810546875,276.7818908691406,180.81436676173136,269.1875305175781]},{"page":448,"text":"strain, 191–193, 195, 197","rect":[65.76580810546875,285.1069030761719,153.05951446680948,279.1126708984375]},{"page":448,"text":"stress, 190–191, 194, 237, 238, 240, 243,","rect":[65.76580810546875,295.026123046875,206.6646144111391,289.13348388671877]},{"page":448,"text":"253, 318","rect":[77.7179183959961,305.0,107.57280487208291,299.0078430175781]},{"page":448,"text":"symmetric, 36, 191, 236","rect":[65.76580810546875,316.62896728515627,148.92374176173136,309.03460693359377]},{"page":448,"text":"of viscous stress, 237, 238, 240, 243, 253,","rect":[65.76580810546875,324.8862609863281,209.72329971387348,318.95977783203127]},{"page":448,"text":"318","rect":[77.7179183959961,334.80548095703127,90.4095205703251,328.9805908203125]},{"page":448,"text":"Theorem","rect":[53.81370544433594,344.74755859375,84.45970390672672,338.90576171875]},{"page":448,"text":"of average, 31","rect":[65.76580810546875,356.47607421875,114.55826324610635,348.8817138671875]},{"page":448,"text":"of convolution, 92, 97–99","rect":[65.76580810546875,364.7443542480469,154.3261465468876,358.8009338378906]},{"page":448,"text":"of multiplication, 93","rect":[65.76580810546875,376.3712463378906,135.91315216212198,368.7768859863281]},{"page":448,"text":"Thermal diffusion coefficient, 172, 174","rect":[53.81370544433594,384.62847900390627,188.1788839736454,378.7528076171875]},{"page":448,"text":"Thomson formula, 79","rect":[53.81370544433594,394.6722106933594,128.0401358047001,388.7287902832031]},{"page":448,"text":"Topology","rect":[53.81370544433594,406.2423400878906,86.76957458643838,398.6479797363281]},{"page":448,"text":"limitation, 361–364","rect":[65.76580810546875,414.4996032714844,132.99239105372355,408.6239318847656]},{"page":448,"text":"trap, 373–375","rect":[65.76580810546875,426.1942443847656,113.05303711085245,418.549072265625]},{"page":448,"text":"Transition","rect":[53.81370544433594,434.4176330566406,88.69023651270791,428.5758361816406]},{"page":448,"text":"anchoring, 265, 280","rect":[65.76580810546875,446.0894470214844,134.0263418593876,438.44427490234377]},{"page":448,"text":"cholesteric‐nematic, 358–361","rect":[65.76580810546875,454.3467102050781,166.41193145899698,448.42022705078127]},{"page":448,"text":"ferroelectric, 384","rect":[65.76496124267578,464.3226013183594,124.5660147109501,458.4469299316406]},{"page":448,"text":"second order, 42, 115–117, 121, 125, 126,","rect":[65.76496124267578,474.2418518066406,210.23266861035786,468.31536865234377]},{"page":448,"text":"129–131, 261, 385","rect":[77.71707153320313,484.2855529785156,141.67261261134073,478.29132080078127]},{"page":448,"text":"439","rect":[372.4990539550781,42.63514709472656,385.19064850001259,36.74253845214844]},{"page":448,"text":"virtual second order, 116, 117, 261","rect":[237.3994598388672,66.0,357.31989044337197,60.03866195678711]},{"page":448,"text":"weak first order transition, 124–127","rect":[237.3994598388672,75.83358764648438,359.6280874648563,69.95791625976563]},{"page":448,"text":"wetting transition, 260","rect":[237.3994598388672,87.5282211303711,314.4975637832157,79.93386840820313]},{"page":448,"text":"Tricritical point, 126, 127","rect":[225.44735717773438,97.5041732788086,313.7606253066532,89.90982055664063]},{"page":448,"text":"Triple point, 126, 129","rect":[225.44735717773438,107.4233627319336,300.44882721095009,99.82901000976563]},{"page":448,"text":"Twist structure, 293–294, 370, 371","rect":[225.44735717773438,115.74842834472656,345.31195587305947,109.80502319335938]},{"page":448,"text":"Two‐dimensional crystal, 1, 50, 104","rect":[225.44735717773438,127.37532806396485,349.10675567774697,119.73017120361328]},{"page":448,"text":"Two‐dimensional liquid, 1, 71, 101","rect":[225.44735717773438,137.35128784179688,346.2748159804813,129.75692749023438]},{"page":448,"text":"V","rect":[225.44735717773438,155.5531768798828,231.52240540129155,149.6605682373047]},{"page":448,"text":"Van‐der‐Waals, 25, 56, 57, 138, 140, 414, 415","rect":[225.44735717773438,165.50372314453126,384.67877716212197,159.57725524902345]},{"page":448,"text":"Vector","rect":[225.44735717773438,175.47967529296876,248.51223844153575,169.77333068847657]},{"page":448,"text":"of displacement, 157, 164, 285","rect":[237.39942932128907,187.11758422851563,343.2152456679813,179.47242736816407]},{"page":448,"text":"of reciprocal lattice, 84, 85, 103","rect":[237.39942932128907,197.09353637695313,347.18097442774697,189.44837951660157]},{"page":448,"text":"Vesicle, 54","rect":[225.44735717773438,205.350830078125,263.8741506972782,199.4243621826172]},{"page":448,"text":"Virtual polar cholesteric, 197","rect":[225.44735717773438,216.98870849609376,325.7694143691532,209.39434814453126]},{"page":448,"text":"Viscosity","rect":[225.44735717773438,226.96469116210938,257.41327423243447,219.37033081054688]},{"page":448,"text":"coefficients, 238, 240–242, 245, 248, 252,","rect":[237.39942932128907,235.22195434570313,381.4703089912172,229.2954864501953]},{"page":448,"text":"254, 302, 338, 408, 423","rect":[249.35154724121095,245.19790649414063,330.7529653945438,239.2714385986328]},{"page":448,"text":"rotational, 315, 365, 395, 399, 402, 403","rect":[237.39942932128907,255.1848907470703,373.1234182754032,249.19068908691407]},{"page":448,"text":"second viscosity, 238, 239","rect":[237.39942932128907,266.8117980957031,327.8076528945438,259.2174377441406]},{"page":448,"text":"for shear, 247","rect":[237.39942932128907,275.069091796875,284.7577261367313,269.19342041015627]},{"page":448,"text":"tensor, 238, 242, 243, 245","rect":[237.39942932128907,285.0450439453125,326.84479278712197,279.1185607910156]},{"page":448,"text":"Volterra process, 209–211, 217–219","rect":[225.44735717773438,296.6829833984375,349.7159475722782,289.088623046875]},{"page":448,"text":"W","rect":[225.44735717773438,314.9415588378906,233.90842712697833,309.0489501953125]},{"page":448,"text":"Waveguide regime, 293, 321, 356","rect":[225.44735717773438,326.61083984375,341.68552154688759,318.9656677246094]},{"page":448,"text":"Wave‐like distortion, 222–224","rect":[225.44735717773438,334.8113708496094,329.8332571425907,328.9356994628906]},{"page":448,"text":"Wedge, of Cano, 61","rect":[225.44818115234376,346.50604248046877,294.2925161757938,338.91168212890627]},{"page":448,"text":"Williams domains, 336","rect":[225.44818115234376,354.7632751464844,304.98138946680947,348.8876037597656]},{"page":448,"text":"X","rect":[225.44818115234376,374.556884765625,231.5232293759009,368.791259765625]},{"page":448,"text":"X‐ray","rect":[225.44818115234376,386.35308837890627,245.1150869765751,378.92803955078127]},{"page":448,"text":"diffraction, 64, 75–109, 123, 344,","rect":[237.40028381347657,394.6781005859375,352.58252212598287,388.6838684082031]},{"page":448,"text":"diffractogram, 43, 94, 108","rect":[237.40028381347657,406.2482604980469,327.18576568751259,398.6539001464844]},{"page":448,"text":"reflection, 64","rect":[237.40028381347657,414.5055236816406,282.4546255507938,408.6298522949219]},{"page":448,"text":"resonant scattering, 417","rect":[237.40028381347657,426.2001647949219,318.7467092910282,418.6058044433594]},{"page":448,"text":"scattering, 34, 77–83, 124, 417","rect":[237.40028381347657,436.1761169433594,343.4437307754032,428.5817565917969]},{"page":448,"text":"Y","rect":[225.44818115234376,454.2510070800781,231.5232293759009,448.4853820800781]},{"page":448,"text":"Young","rect":[225.44818115234376,466.0472106933594,248.4453634658329,458.6221618652344]},{"page":448,"text":"law, 260","rect":[237.40028381347657,474.2477722167969,266.85919708399697,468.3721008300781]},{"page":448,"text":"modulus, 190","rect":[237.40028381347657,484.2914733886719,283.79572052149697,478.3480529785156]}]}
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