Articles in JSON/article 11.json

{"lines":[{"page":1,"text":"research papers","rect":[44.794551849365237,50.97494125366211,154.82873011863493,36.115684509277347]},{"page":1,"text":"ISSN 2053-2733","rect":[44.794551849365237,123.79070281982422,94.21609491842269,119.01090240478516]},{"page":1,"text":"Crystallographic phase retrieval method for liquid","rect":[206.02037048339845,85.47140502929688,556.2493804017811,70.58029174804688]},{"page":1,"text":"crystal bicontinuous phases: indicator-based","rect":[206.02037048339845,103.37094116210938,515.9806426088123,88.51168060302735]},{"page":1,"text":"method","rect":[206.02037048339845,118.29231262207031,258.740591339281,106.4112777709961]},{"page":1,"text":"Toshihiko Oka*","rect":[206.020751953125,152.3378448486328,273.84436201892069,144.89231872558595]},{"page":1,"text":"Received 27 January 2022","rect":[44.794551849365237,187.40457153320313,123.936874337368,180.7574462890625]},{"page":1,"text":"Accepted 7 July 2022","rect":[44.794551849365237,197.37387084960938,109.83683771627425,190.72674560546876]},{"page":1,"text":"Department of Physics, Faculty of Science, Shizuoka University, Shizuoka, 422-8529, Japan, and Nanomaterials Research","rect":[206.020751953125,179.47427368164063,564.1769408656883,172.8201904296875]},{"page":1,"text":"Division, Research Institute of Electronics, Shizuoka University, Shizuoka, 422-8529, Japan. *Correspondence e-mail:","rect":[206.020751953125,189.38638305664063,559.8893177999067,182.7322998046875]},{"page":1,"text":"oka.toshihiko@shizuoka.ac.jp","rect":[206.020751953125,199.2442626953125,294.9963988735008,192.7086181640625]},{"page":1,"text":"Edited by I. Margiolaki, University of Patras,","rect":[44.794551849365237,231.18991088867188,177.75337724814893,224.53582763671876]},{"page":1,"text":"Greece","rect":[44.794551849365237,239.7169189453125,66.02368043331653,234.88137817382813]},{"page":1,"text":"Keywords: lyotropic liquid crystals; triply","rect":[44.794551849365237,263.0801086425781,168.71134446529894,256.3633117675781]},{"page":1,"text":"periodic minimal surfaces; crystallographic","rect":[44.794551849365237,273.0494689941406,173.21842454342394,266.3953857421875]},{"page":1,"text":"phase retrieval.","rect":[44.794551849365237,282.70379638671877,90.32274370322704,276.314453125]},{"page":1,"text":"Supporting information: this article has","rect":[44.794551849365237,304.65252685546877,163.85042811631173,298.1377868652344]},{"page":1,"text":"supporting information at journals.iucr.org/a","rect":[44.794559478759769,314.7402648925781,177.2016703930333,308.1976623535156]},{"page":1,"text":"An indicator-based crystallographic phase retrieval method has been developed","rect":[206.020751953125,226.2851104736328,528.4263331887822,217.51573181152345]},{"page":1,"text":"for diffraction data of bicontinuous cubic phases of lyotropic liquid crystals.","rect":[206.020751953125,238.2367706298828,528.4173601401681,229.46739196777345]},{"page":1,"text":"Such liquid crystals have large structural disorder; the number of independent","rect":[206.020751953125,250.18836975097657,528.4174210070194,241.4189910888672]},{"page":1,"text":"Bragg reflections that can be observed is limited. This paper proposes two","rect":[206.020751953125,262.0832824707031,528.3636500833135,253.3138885498047]},{"page":1,"text":"indicators to identify plausible combination(s) of crystallographic phases, i.e.","rect":[206.020751953125,274.0846862792969,528.3772600425118,264.947021484375]},{"page":1,"text":"electron-density distribution. The indicators are based on the characteristics of","rect":[206.020751953125,285.9866027832031,528.4143692492069,277.2171936035156]},{"page":1,"text":"the liquid crystals: amphiphilic molecules diffuse mainly in the direction parallel","rect":[206.020751953125,297.9382629394531,528.3726514576723,289.1688537597656]},{"page":1,"text":"to polar–nonpolar interfaces and the electron density in the direction parallel to","rect":[206.020751953125,309.8899230957031,528.4144923684697,301.1205139160156]},{"page":1,"text":"the interfaces is averaged. One indicator is the difference between the maximum","rect":[206.020751953125,321.841552734375,528.4234059357929,313.0721435546875]},{"page":1,"text":"and minimum electron density, and the other is calculated from the Hessian","rect":[206.020751953125,333.7931823730469,528.3338038919072,325.0237731933594]},{"page":1,"text":"matrix of the electron density. Using test data, the electron densities were","rect":[206.020751953125,345.7448425292969,528.3725914714664,336.9754333496094]},{"page":1,"text":"calculated for all possible phase combinations and indicators were obtained. The","rect":[206.020751953125,357.6397705078125,528.3816246745914,348.870361328125]},{"page":1,"text":"results indicated that the electron densities with the minimum indicators were","rect":[206.020751953125,367.5110168457031,528.3716149089664,360.8219909667969]},{"page":1,"text":"close to the true electron density. Therefore, this method is effective for phase","rect":[206.020751953125,381.5430603027344,528.3537926433414,372.7736511230469]},{"page":1,"text":"retrieval. The accuracy of the phase retrieval decreased when the volume","rect":[206.020751953125,393.4947204589844,528.4193444011539,384.7253112792969]},{"page":1,"text":"fraction of the region including the triply periodic minimal surface increased.","rect":[206.020751953125,405.4463806152344,522.9688738120431,396.6769714355469]},{"page":1,"text":"Published","rect":[44.794551849365237,747.3021850585938,73.86539454000472,742.2018432617188]},{"page":1,"text":"430","rect":[44.794551849365237,770.5427856445313,64.57316107852795,762.5875244140625]},{"page":1,"text":"under a CC BY 4.0 licence","rect":[76.22271728515625,747.3300170898438,157.332941755338,742.2018432617188]},{"page":1,"text":"1. Introduction","rect":[206.020751953125,432.5931396484375,275.43937403578328,424.8170166015625]},{"page":1,"text":"Restoring the phases of the structure factors is a crucial","rect":[206.020751953125,450.5911560058594,456.76522958267239,441.8217468261719]},{"page":1,"text":"problem","rect":[206.020751953125,462.486083984375,240.06940202954284,453.7166748046875]},{"page":1,"text":"in","rect":[247.75607299804688,461.0,255.51736712432905,453.7166748046875]},{"page":1,"text":"crystal","rect":[263.1025390625,462.486083984375,289.48837411392239,453.7166748046875]},{"page":1,"text":"structure","rect":[297.13525390625,461.0,333.42102286795088,454.6722717285156]},{"page":1,"text":"determination","rect":[341.07781982421877,461.0,398.63238787628219,453.7166748046875]},{"page":1,"text":"from","rect":[406.19964599609377,461.0,425.76166277173038,453.79632568359377]},{"page":1,"text":"X-ray","rect":[433.3817138671875,462.486083984375,456.6745934245915,453.94561767578127]},{"page":1,"text":"diffraction","rect":[206.020751953125,473.0,248.56811785675093,465.6683044433594]},{"page":1,"text":"(XRD)","rect":[254.43661499023438,474.4874572753906,283.93933289035069,465.34979248046877]},{"page":1,"text":"data.","rect":[289.8307189941406,473.0,309.7002275718086,465.6683044433594]},{"page":1,"text":"Direct","rect":[315.5965881347656,473.0,341.4709183214724,465.6683044433594]},{"page":1,"text":"methods","rect":[347.36529541015627,473.0,381.9314593551944,465.6683044433594]},{"page":1,"text":"solve","rect":[387.91241455078127,473.0,408.6079735515446,465.6683044433594]},{"page":1,"text":"the","rect":[414.5272216796875,473.0,427.4624352214665,465.6683044433594]},{"page":1,"text":"phase","rect":[433.38470458984377,474.4377136230469,456.6676659343571,465.6683044433594]},{"page":1,"text":"problem","rect":[206.020751953125,486.3893737792969,240.06940202954284,477.6199645996094]},{"page":1,"text":"by","rect":[248.60494995117188,486.3893737792969,258.436190104279,477.6199645996094]},{"page":1,"text":"combining","rect":[266.95281982421877,486.3893737792969,309.4026818034977,477.6199645996094]},{"page":1,"text":"the","rect":[317.8625793457031,485.0,330.797762369904,477.6199645996094]},{"page":1,"text":"intensity","rect":[339.32537841796877,486.3893737792969,374.098543619904,477.6199645996094]},{"page":1,"text":"information","rect":[382.58929443359377,485.0,430.74457049346969,477.6199645996094]},{"page":1,"text":"with","rect":[439.2174072265625,485.0,456.70950579620406,477.6199645996094]},{"page":1,"text":"constraints","rect":[206.020751953125,497.0,250.12057678683503,489.57159423828127]},{"page":1,"text":"based","rect":[256.19207763671877,497.0,279.5576198098759,489.57159423828127]},{"page":1,"text":"on","rect":[285.6390686035156,497.0,295.9877955911259,491.0]},{"page":1,"text":"the","rect":[302.1180419921875,497.0,315.05322501638838,489.57159423828127]},{"page":1,"text":"expected","rect":[321.1456298828125,498.34100341796877,357.4682338235478,489.57159423828127]},{"page":1,"text":"structural","rect":[363.5606689453125,497.0,402.4668287037661,489.57159423828127]},{"page":1,"text":"information","rect":[408.58013916015627,497.0,456.7274196145634,489.57159423828127]},{"page":1,"text":"(atomicity, positivity and zero density regions) (Sayre, 1952;","rect":[206.020751953125,510.3424072265625,456.7214372886055,501.2047424316406]},{"page":1,"text":"Giacovazzo,","rect":[206.020751953125,521.0,255.13224966653523,513.4749145507813]},{"page":1,"text":"2001).","rect":[261.5182189941406,522.2940673828125,286.16253836282427,513.1563720703125]},{"page":1,"text":"Closely","rect":[292.5505065917969,522.2442626953125,322.5339073894352,513.4749145507813]},{"page":1,"text":"related","rect":[329.0194091796875,521.0,357.4752223489384,513.4749145507813]},{"page":1,"text":"is","rect":[363.958740234375,521.0,370.16757385714757,513.4749145507813]},{"page":1,"text":"the","rect":[376.64312744140627,521.0,389.5783409831852,513.4749145507813]},{"page":1,"text":"charge-flipping","rect":[396.0101318359375,522.2442626953125,456.651766276154,513.4749145507813]},{"page":1,"text":"method, an efficient iterative phase retrieval method with","rect":[206.020751953125,534.1959228515625,456.7084987161259,525.4265747070313]},{"page":1,"text":"similar requirements that has become popular in recent years","rect":[206.020751953125,546.1475830078125,456.69259827121007,537.3782348632813]},{"page":1,"text":"(Oszla´nyi & Su¨t\u0002o\u0002, 2004, 2008). It may be possible to obtain the","rect":[206.020751953125,558.09228515625,456.741121744904,548.5265502929688]},{"page":1,"text":"structure using data from samples that satisfy the structural","rect":[206.02174377441407,569.994140625,456.7154859303286,561.2247924804688]},{"page":1,"text":"information; however, this method is limited to solid crystals","rect":[206.02174377441407,581.94580078125,456.76819031222569,573.1764526367188]},{"page":1,"text":"in which the atomic positions are almost fixed.","rect":[206.02174377441407,593.8974609375,397.054078890168,585.1281127929688]},{"page":1,"text":"Liquid","rect":[215.98834228515626,605.8490600585938,242.9026362893681,597.0797119140625]},{"page":1,"text":"crystals","rect":[249.96533203125,605.8490600585938,280.0890216110538,597.0797119140625]},{"page":1,"text":"have","rect":[287.1129150390625,604.0,306.2569298503727,597.0797119140625]},{"page":1,"text":"properties","rect":[313.33258056640627,605.8490600585938,354.8519610641788,597.0797119140625]},{"page":1,"text":"that","rect":[361.8629150390625,603.7687377929688,377.9020096300662,597.0797119140625]},{"page":1,"text":"are","rect":[384.96771240234377,604.0,397.9029259441227,599.0]},{"page":1,"text":"intermediate","rect":[404.9576416015625,604.0,456.732332682404,597.0797119140625]},{"page":1,"text":"between solids and liquids. In liquid crystals, the molecules","rect":[206.02174377441407,617.8007202148438,456.65075867160069,609.0313720703125]},{"page":1,"text":"move like a liquid in a certain direction, but are oriented like a","rect":[206.02174377441407,629.7523803710938,456.7851280925602,620.9830322265625]},{"page":1,"text":"solid crystal in other directions. For the latter reason, many","rect":[206.02174377441407,641.7040405273438,456.72143790701338,632.9346923828125]},{"page":1,"text":"liquid crystal phases have periodic structures and are subject","rect":[206.02174377441407,653.5989379882813,456.6846329210818,644.82958984375]},{"page":1,"text":"to","rect":[206.02174377441407,664.0,214.30053973175093,657.73681640625]},{"page":1,"text":"XRD","rect":[222.55743408203126,663.4005737304688,244.72777300122253,657.0101928710938]},{"page":1,"text":"measurements.","rect":[253.02548217773438,664.0,313.2917803061836,657.73681640625]},{"page":1,"text":"Bicontinuous","rect":[321.54571533203127,664.0,375.3812579391788,656.78125]},{"page":1,"text":"cubic","rect":[383.6670227050781,664.0,404.8800680158695,656.78125]},{"page":1,"text":"phases","rect":[413.2275390625,665.5505981445313,440.13192932589757,656.78125]},{"page":1,"text":"of","rect":[448.4505310058594,664.0,456.7293411730349,656.86083984375]},{"page":1,"text":"lyotropic","rect":[206.02174377441407,677.5022583007813,242.23785610180699,668.73291015625]},{"page":1,"text":"liquid","rect":[247.47244262695313,677.5022583007813,270.7543666360478,668.73291015625]},{"page":1,"text":"crystals","rect":[276.0128173828125,677.5022583007813,306.02011291964757,668.73291015625]},{"page":1,"text":"(LLCs)","rect":[311.23480224609377,677.5520629882813,341.23410217746007,668.4143676757813]},{"page":1,"text":"have","rect":[346.4019775390625,676.0,365.625643229279,668.73291015625]},{"page":1,"text":"structures","rect":[370.7527770996094,676.0,410.6600665329288,669.6884765625]},{"page":1,"text":"similar","rect":[415.88470458984377,676.0,443.3036212204288,668.73291015625]},{"page":1,"text":"to","rect":[448.4456787109375,676.0,456.7244594094853,669.6884765625]},{"page":1,"text":"triply periodic minimal surfaces (TPMSs) (Hyde et al., 1996),","rect":[206.02174377441407,689.5037231445313,456.79388601907427,680.3660278320313]},{"page":1,"text":"and diffraction data can be obtained as if they were 3D crystals","rect":[206.02174377441407,701.4055786132813,456.7692279098819,692.63623046875]},{"page":1,"text":"[Fig. 1(a)]. However, the spatial resolution of the observed","rect":[206.02174377441407,713.4070434570313,456.75018572784469,704.2693481445313]},{"page":1,"text":"XRD data is low due to structural disorder, and the number of","rect":[206.02174377441407,723.2285766601563,456.7273575304568,716.53955078125]},{"page":1,"text":"independent reflections is small. The previously mentioned","rect":[206.02174377441407,737.2605590820313,456.64383196807906,728.4912109375]},{"page":1,"text":"constraints, except the positivity, are not satisfied. There is","rect":[206.02174377441407,749.1554565429688,456.75229065402257,740.3861083984375]},{"page":1,"text":"https://doi.org/10.1107/S2053273322006970","rect":[76.50749206542969,772.0558471679688,231.0165643637738,764.6498413085938]},{"page":1,"text":"Acta","rect":[446.87030029296877,771.0,462.36929611737187,765.000244140625]},{"page":1,"text":"Cryst.","rect":[464.99188232421877,772.2708740234375,484.46214543114658,764.9843139648438]},{"page":1,"text":"(2022).","rect":[487.07757568359377,771.3311767578125,511.51513663611089,765.0480346679688]},{"page":1,"text":"A78,","rect":[514.1472778320313,771.0,530.6979979642359,765.0321044921875]},{"page":1,"text":"430–436","rect":[533.344482421875,771.0,564.3504114096722,765.2391357421875]},{"page":2,"text":"thus a problem in restoring the electron density from XRD","rect":[44.794551849365237,80.42951965332031,295.5161183381366,71.6601333618164]},{"page":2,"text":"data, although the centrosymmetry of the LLC bicontinuous","rect":[44.794551849365237,92.38117980957031,295.49121277316319,83.6117935180664]},{"page":2,"text":"phase restricts the phase to 0 or \u0002. Luzzatti and colleagues first","rect":[44.794551849365237,104.33283996582031,295.4979874132693,95.5634536743164]},{"page":2,"text":"clarified the electron-density distribution of an LLC bicon-","rect":[44.794551849365237,116.28450012207031,295.4683548449099,107.5151138305664]},{"page":2,"text":"tinuous phase (Luzzati et al., 1988; Mariani et al., 1988). They","rect":[44.794551849365237,128.2859344482422,295.5657067058415,119.14825439453125]},{"page":2,"text":"proposed to use the average of the fourth moment of the","rect":[44.79457092285156,140.1878204345703,295.5161461589665,131.41844177246095]},{"page":2,"text":"electron density as an indicator for phase retrieval. They","rect":[44.79457092285156,152.1393585205078,295.5599083659977,143.36997985839845]},{"page":2,"text":"insisted that the combination of the phases at the smallest","rect":[44.79457092285156,164.0910186767578,295.5101639269412,155.32164001464845]},{"page":2,"text":"value was the most reliable. However, as they admitted, the","rect":[44.79457092285156,175.98597717285157,295.5151695964665,167.2165985107422]},{"page":2,"text":"electron density with the minimum indicator was not always","rect":[44.79457092285156,187.93763732910157,295.45346252902257,179.1682586669922]},{"page":2,"text":"the proper phase combination. Other methods have also been","rect":[44.79457092285156,199.8892364501953,295.5012233255009,191.11985778808595]},{"page":2,"text":"proposed for LLCs, such as looking for the position of the","rect":[44.79457092285156,211.84095764160157,295.5131554363102,203.0715789794922]},{"page":2,"text":"methyl group in the hydrocarbon chain of a lipid bilayer","rect":[44.79457092285156,223.7925567626953,295.4524249313663,215.02317810058595]},{"page":2,"text":"(Harper et al., 2000).","rect":[44.79457092285156,235.7939910888672,129.94603140481648,226.65631103515626]},{"page":2,"text":"We have performed X-ray crystallographic studies of LLC","rect":[54.76216125488281,247.6958770751953,295.5280196691302,238.92649841308595]},{"page":2,"text":"bicontinuous","rect":[44.795562744140628,258.0,97.1791629074405,250.8780975341797]},{"page":2,"text":"cubic","rect":[103.51837158203125,258.0,124.73134059887731,250.8780975341797]},{"page":2,"text":"phases","rect":[131.15316772460938,259.6474914550781,158.05749695285065,250.8780975341797]},{"page":2,"text":"using","rect":[164.450439453125,259.6474914550781,185.75596854666177,250.8780975341797]},{"page":2,"text":"single-crystal","rect":[192.08523559570313,259.6474914550781,245.0132917164614,250.8780975341797]},{"page":2,"text":"regions","rect":[251.31668090820313,259.6474914550781,280.8981036423038,250.8780975341797]},{"page":2,"text":"of","rect":[287.21942138671877,258.0,295.4982010363162,250.9577178955078]},{"page":2,"text":"samples (Oka, 2017; Oka et al., 2018, 2020). In these studies,","rect":[44.795562744140628,271.5921936035156,295.5627153647774,262.45452880859377]},{"page":2,"text":"the structures of the bicontinuous cubic phases were deter-","rect":[44.79457092285156,283.4940490722656,295.5349747179568,274.7246398925781]},{"page":2,"text":"mined using models based on available information. However,","rect":[44.79457092285156,295.44573974609377,295.6007402671211,286.67633056640627]},{"page":2,"text":"it would be better if the structure of the bicontinuous cubic","rect":[44.79457092285156,305.3170471191406,295.4664022443851,298.6280212402344]},{"page":2,"text":"phase could be revealed without a structural model. Here, a","rect":[44.79457092285156,319.3490905761719,295.560945963654,310.5796813964844]},{"page":2,"text":"simple indicator-based phase retrieval method is reported.","rect":[44.79457092285156,331.30072021484377,295.4872759116524,322.53131103515627]},{"page":2,"text":"Two valuable indicators have been identified based on the","rect":[44.79457092285156,341.1719665527344,295.5131554363102,334.4829406738281]},{"page":2,"text":"universal characteristics of the LLC bicontinuous structures.","rect":[44.79457092285156,353.1833801269531,295.55694754251177,346.43463134765627]},{"page":2,"text":"The phase combination in which these indicators are mini-","rect":[44.79457092285156,367.09893798828127,295.4793106554568,358.32952880859377]},{"page":2,"text":"mized is very close to the true phase combination, and makes","rect":[44.79457092285156,379.05059814453127,295.49325745089757,370.28118896484377]},{"page":2,"text":"it possible to retrieve the crystallographic phase of the TPMS-","rect":[44.79457092285156,391.00225830078127,295.50122227655057,382.23284912109377]},{"page":2,"text":"like structure. Although this method was designed to clarify","rect":[44.79457092285156,402.95391845703127,295.4693321941227,394.18450927734377]},{"page":2,"text":"Figure 1","rect":[44.794551849365237,663.7316284179688,77.03409446233377,655.6150512695313]},{"page":2,"text":"(a) A gyroid (G) surface (orange), one of the triply periodic minimal","rect":[44.794551849365237,673.4273681640625,295.4881434558877,665.6604614257813]},{"page":2,"text":"surfaces (TPMSs), and the networks (green and blue) through the centers","rect":[44.793704986572269,682.8864135742188,295.4805627034772,675.1195068359375]},{"page":2,"text":"of the two interwoven spaces where the surface divides. A single unit cell","rect":[44.793704986572269,692.30322265625,295.5211939929971,684.8493041992188]},{"page":2,"text":"is shown. (b) Two regions separated by a surface (an interface) of","rect":[44.793704986572269,701.8046264648438,295.5582819126903,694.0377197265625]},{"page":2,"text":"constant thickness from the TPMS: TPMS side (yellow) and network side","rect":[44.79370880126953,711.2637329101563,295.5313010428635,703.496826171875]},{"page":2,"text":"(blue). The volume fraction of the TPMS side is 0.4. (c) Structural model","rect":[44.79370880126953,720.7227783203125,295.495528709794,712.9558715820313]},{"page":2,"text":"with fluctuations. In the structure shown in (b), the TPMS side region is","rect":[44.7945671081543,730.181884765625,295.5606713460553,722.4149780273438]},{"page":2,"text":"set to an electron density of 1 and thenetwork side region is set to 0, and a","rect":[44.7945671081543,739.5986938476563,295.5363974784104,732.144775390625]},{"page":2,"text":"Gaussian function is convolved as a fluctuation.","rect":[44.7945671081543,747.3402099609375,210.68659797206224,741.6038208007813]},{"page":2,"text":"Acta Cryst. (2022). A78, 430–436","rect":[44.794551849365237,772.2708740234375,162.21811770850037,764.9843139648438]},{"page":2,"text":"research papers","rect":[452.9285888671875,50.97494125366211,562.962778580549,36.115684509277347]},{"page":2,"text":"the structure of the LLC bicontinuous cubic phases, it would","rect":[313.4498596191406,80.42756652832031,564.1912868020635,71.6581802368164]},{"page":2,"text":"be applicable to thermotropic liquid crystals, polymers, and","rect":[313.4498596191406,92.37922668457031,564.2002589700322,83.6098403930664]},{"page":2,"text":"other materials with TPMS-like structures.","rect":[313.4498596191406,102.31024932861328,487.8220232261055,95.5615005493164]},{"page":2,"text":"2. Indicators for bicontinuous structure (TPMS-like","rect":[313.4483947753906,143.3450164794922,547.6532369531976,134.36695861816407]},{"page":2,"text":"structure)","rect":[313.4483947753906,155.29661560058595,357.951396997353,146.9038543701172]},{"page":2,"text":"2.1. Bicontinuous structure (TPMS-like structure)","rect":[313.4483947753906,171.09088134765626,517.5158595412459,162.9871368408203]},{"page":2,"text":"An LLC consists of two or more components, amphiphilic","rect":[323.47186279296877,187.99440002441407,564.176088523682,179.2250213623047]},{"page":2,"text":"molecules and water (or oil). There are two types of LLC, oil-","rect":[313.44854736328127,199.99583435058595,564.1780411242069,190.858154296875]},{"page":2,"text":"in-water and water-in-oil types, commonly referred to as type I","rect":[313.44854736328127,211.89772033691407,564.2527179001163,203.1283416748047]},{"page":2,"text":"and","rect":[313.44854736328127,222.0,328.5332973001103,215.0800018310547]},{"page":2,"text":"type","rect":[334.3450622558594,223.84938049316407,352.0271935222477,216.0]},{"page":2,"text":"II,","rect":[357.846923828125,222.0,367.3686846030586,215.30892944335938]},{"page":2,"text":"respectively","rect":[373.1933898925781,223.84938049316407,421.4380821941227,215.0800018310547]},{"page":2,"text":"(Israelachvili,","rect":[427.3294677734375,223.89915466308595,482.4946306479805,214.761474609375]},{"page":2,"text":"2011).","rect":[488.4298095703125,223.89915466308595,512.9557512534493,214.761474609375]},{"page":2,"text":"In","rect":[518.8392333984375,222.0,527.635500669251,215.30892944335938]},{"page":2,"text":"type","rect":[533.5059814453125,223.84938049316407,551.0975670574039,216.0]},{"page":2,"text":"II","rect":[556.9501953125,221.6993408203125,564.1940020798038,215.30892944335938]},{"page":2,"text":"bicontinuous cubic phases of LLCs, three different TPMS-like","rect":[313.44854736328127,235.74427795410157,564.1860680339664,226.9748992919922]},{"page":2,"text":"structures, P surface, D surface and gyroid (G) surface, have","rect":[313.44854736328127,247.74571228027345,564.2039513347477,238.6080322265625]},{"page":2,"text":"been observed, while only the G surface has been observed in","rect":[313.44854736328127,259.6475524902344,564.2048976419072,250.87815856933595]},{"page":2,"text":"type I (Hyde et al., 1996). The space groups of the three","rect":[313.44854736328127,271.6489562988281,564.1808190105289,262.51129150390627]},{"page":2,"text":"TPMS-like structures of P, D and G are Im3m, Pn3m and Ia3d,","rect":[313.4475402832031,282.913818359375,564.2773454917306,274.7814636230469]},{"page":2,"text":"respectively.","rect":[313.44757080078127,295.50250244140627,363.23675711282427,286.73309326171877]},{"page":2,"text":"In","rect":[368.5459899902344,294.0,377.3422572610478,286.9620361328125]},{"page":2,"text":"type","rect":[382.646484375,295.50250244140627,400.2380699870915,287.6886901855469]},{"page":2,"text":"II,","rect":[405.6378173828125,294.0,415.15960867532427,286.9620361328125]},{"page":2,"text":"the","rect":[420.47479248046877,294.0,433.4099755046696,286.73309326171877]},{"page":2,"text":"nonpolar","rect":[438.7659606933594,295.50250244140627,475.62890808566319,286.73309326171877]},{"page":2,"text":"region","rect":[480.95404052734377,295.50250244140627,506.8234889505009,286.73309326171877]},{"page":2,"text":"contains","rect":[512.2123413085938,294.0,545.9605426071476,286.73309326171877]},{"page":2,"text":"the","rect":[551.28564453125,294.0,564.2208580730289,286.73309326171877]},{"page":2,"text":"TPMS as the center surface, and the polar region is located on","rect":[313.44757080078127,307.45416259765627,564.2417628762822,298.68475341796877]},{"page":2,"text":"the two networks separated by the TPMS [Fig. 1(a)]. In the","rect":[313.44757080078127,319.45556640625,564.2286095378727,310.3179016113281]},{"page":2,"text":"type I bicontinuous cubic phase, the positions of the polar and","rect":[313.44659423828127,331.30072021484377,564.1969630715947,322.53131103515627]},{"page":2,"text":"nonpolar regions are swapped with those of type II (Israe-","rect":[313.44659423828127,343.3021240234375,564.1472794054569,334.1644592285156]},{"page":2,"text":"lachvili, 2011). In both cases, the polar–nonpolar interface is","rect":[313.44659423828127,355.2537841796875,564.230928349335,346.1161193847656]},{"page":2,"text":"on the amphiphilic molecule, and the amphiphilic molecule","rect":[313.44659423828127,367.15570068359377,564.1373009441227,358.38629150390627]},{"page":2,"text":"can move like a liquid molecule in the direction parallel to the","rect":[313.44659423828127,379.10736083984377,564.2239098308414,370.33795166015627]},{"page":2,"text":"polar–nonpolar interface. Therefore, the electron density in","rect":[313.44659423828127,391.05902099609377,564.2000148294072,382.28961181640627]},{"page":2,"text":"the direction parallel to the interface is averaged, but not that","rect":[313.44659423828127,403.0106506347656,564.2110001085819,394.2412414550781]},{"page":2,"text":"in the perpendicular direction. This averaging leads to the","rect":[313.44659423828127,414.9623107910156,564.2199425456852,406.1929016113281]},{"page":2,"text":"separation of the polar and nonpolar regions. In the case of an","rect":[313.44659423828127,426.85723876953127,564.2327907083135,418.08782958984377]},{"page":2,"text":"amphiphilic molecule and water, if the electron densities of","rect":[313.44659423828127,438.8088684082031,564.2079483507694,430.0394592285156]},{"page":2,"text":"the polar parts of the amphiphilic molecules and waters are","rect":[313.44659423828127,450.7605285644531,564.2239098308414,441.9911193847656]},{"page":2,"text":"almost the same, then the electron density can be described in","rect":[313.44659423828127,462.7121887207031,564.2019679544072,453.9427795410156]},{"page":2,"text":"two levels, polar and nonpolar, without considering fluctua-","rect":[313.44659423828127,474.6638488769531,564.1472794054569,465.8944396972656]},{"page":2,"text":"Figure 2","rect":[313.4483947753906,692.0529174804688,345.7443758344041,683.9363403320313]},{"page":2,"text":"Sliced contour plots of idealized electron densities of (a) sphere-like, (b)","rect":[313.4483947753906,701.8054809570313,564.2263512776959,694.03857421875]},{"page":2,"text":"cylinder-like and (c) minimal surface like structures. Only in (a) is there a","rect":[313.447509765625,711.2645874023438,564.2559104178636,703.4976806640625]},{"page":2,"text":"strictly convex upward density region where all eigenvalues of the","rect":[313.447509765625,720.6813354492188,564.2037253592698,713.2274169921875]},{"page":2,"text":"Hessian matrix are negative. The strictly convex upward density region","rect":[313.447509765625,730.1404418945313,564.2113332984942,722.6865234375]},{"page":2,"text":"forms a closed isoelectron density surface. On the other hand, there is no","rect":[313.447509765625,739.5995483398438,564.2130422828692,732.1456298828125]},{"page":2,"text":"closed isoelectron density surface in (b) and (c).","rect":[313.447509765625,749.1008911132813,481.67068305507,741.333984375]},{"page":2,"text":"Toshihiko Oka \u0002 Liquid crystal bicontinuous phases 431","rect":[352.1836853027344,772.2470092773438,564.2215909491334,762.4442138671875]},{"page":3,"text":"research papers","rect":[44.794551849365237,50.97494125366211,154.82873011863493,36.115684509277347]},{"page":3,"text":"tions. The region in the TPMS is a minimal surface like","rect":[44.794551849365237,80.42951965332031,295.49023673513838,71.6601333618164]},{"page":3,"text":"structure, and the regions on the networks are cylinder-like","rect":[44.794551849365237,92.38117980957031,295.5559715984196,83.6117935180664]},{"page":3,"text":"structures with branches (Fig. 2). If the electron densities of","rect":[44.794551849365237,104.38261413574219,295.4982315538943,95.24493408203125]},{"page":3,"text":"the polar parts of water and amphiphiles are different, then","rect":[44.794551849365237,116.28450012207031,295.4753444192509,107.5151138305664]},{"page":3,"text":"the electron density can be described in three continuous","rect":[44.794551849365237,128.2361602783203,295.4474505661319,119.4667739868164]},{"page":3,"text":"regions, located on or along the TPMS or the networks.","rect":[44.794551849365237,140.1878204345703,273.873170687043,131.41844177246095]},{"page":3,"text":"2.2. Indicator Iq","rect":[44.794551849365237,163.8312530517578,111.19913878113016,153.999755859375]},{"page":3,"text":"The bicontinuous structure consists of two (or three) flat","rect":[54.76140213012695,179.0380096435547,295.51037754998807,169.90032958984376]},{"page":3,"text":"electron-density regions if fluctuations are not considered [Fig.","rect":[44.794803619384769,190.93983459472657,295.6089189780586,181.9016876220703]},{"page":3,"text":"1(b)]. If electron density is restored using structure factors","rect":[44.794803619384769,202.94126892089845,295.5382708786319,193.8035888671875]},{"page":3,"text":"with incorrect phases, then the density in the flat region will","rect":[44.7938117980957,214.84315490722657,295.49151742446926,206.0737762451172]},{"page":3,"text":"undulate","rect":[44.7938117980957,224.71444702148438,80.49344108084147,218.0254364013672]},{"page":3,"text":"and","rect":[87.03865051269531,224.71444702148438,102.04378008819625,218.0254364013672]},{"page":3,"text":"the","rect":[108.55813598632813,224.71444702148438,121.49331901052897,218.0254364013672]},{"page":3,"text":"difference","rect":[127.92507934570313,224.71444702148438,168.91600272634927,218.0254364013672]},{"page":3,"text":"between","rect":[175.4353485107422,224.71444702148438,209.48399615264936,218.0254364013672]},{"page":3,"text":"the","rect":[216.03819274902345,224.71444702148438,228.97339103201333,218.0254364013672]},{"page":3,"text":"minimum","rect":[235.40518188476563,224.71444702148438,274.0077870393085,218.0254364013672]},{"page":3,"text":"and","rect":[280.48333740234377,224.71444702148438,295.4884974954228,218.0254364013672]},{"page":3,"text":"maximum","rect":[44.7938117980957,237.0,84.96173601391785,229.9203338623047]},{"page":3,"text":"electron","rect":[90.60731506347656,237.0,123.83294939971968,229.9203338623047]},{"page":3,"text":"densities","rect":[129.4536590576172,237.0,164.7144183395694,229.9203338623047]},{"page":3,"text":"will","rect":[170.45155334472657,237.0,184.8386853932192,229.9203338623047]},{"page":3,"text":"become","rect":[190.49819946289063,237.0,221.96045157888833,229.9203338623047]},{"page":3,"text":"more","rect":[227.70455932617188,237.0,248.81904838064615,231.0]},{"page":3,"text":"significant","rect":[254.54725646972657,238.68971252441407,295.4198624132693,229.9203338623047]},{"page":3,"text":"compared","rect":[44.7938117980957,250.64137268066407,85.1916224954228,241.8719940185547]},{"page":3,"text":"with","rect":[90.32369995117188,249.0,107.81574511505171,241.8719940185547]},{"page":3,"text":"the","rect":[113.03141784667969,249.0,125.96658561209147,241.8719940185547]},{"page":3,"text":"true","rect":[131.15240478515626,249.0,147.7090019939274,242.8275604248047]},{"page":3,"text":"density.","rect":[152.89779663085938,250.64137268066407,183.43351919778523,241.8719940185547]},{"page":3,"text":"Therefore,","rect":[188.57354736328126,249.0,231.28518850930866,241.8719940185547]},{"page":3,"text":"the","rect":[236.42523193359376,249.0,249.36043021658365,241.8719940185547]},{"page":3,"text":"difference","rect":[254.5462646484375,249.0,295.52734611013838,241.8719940185547]},{"page":3,"text":"between the minimum electron density in a unit cell, \u0003min, and","rect":[44.7938117980957,262.7655029296875,295.5740993020634,253.8236541748047]},{"page":3,"text":"the maximum electron density, \u0003max, would serve as an indi-","rect":[44.79402542114258,274.7631530761719,295.48938145623807,265.7754211425781]},{"page":3,"text":"cator:","rect":[44.79390335083008,284.41607666015627,67.76733572610552,278.6826477050781]},{"page":3,"text":"I\u0003 ¼ \u0003max \u0003 \u0003min:","rect":[135.68548583984376,306.1363525390625,204.63032550524808,296.2512512207031]},{"page":3,"text":"This indicator corresponds to the range in statistics. A smaller","rect":[44.793582916259769,325.0702819824219,295.51815979464757,316.3008728027344]},{"page":3,"text":"indicator is a better candidate for the proper phase solution.","rect":[44.793582916259769,337.0219421386719,295.5141924155586,328.2525329589844]},{"page":3,"text":"The actual electron density is close to the step-like electron","rect":[44.793582916259769,348.97357177734377,295.5131862161259,340.20416259765627]},{"page":3,"text":"density with the convolution of the fluctuation function [Fig.","rect":[44.793582916259769,360.9252014160156,295.6067217124336,351.8870544433594]},{"page":3,"text":"1(c)]. In this case, the electron density is almost constant at a","rect":[44.793582916259769,372.9266052246094,295.5559410808415,363.7889404296875]},{"page":3,"text":"maximum or minimum on the TPMS and the networks [Fig.","rect":[44.793582916259769,384.8285217285156,295.6136797202461,375.7903747558594]},{"page":3,"text":"1(a)]. Electron densities calculated with incorrect phases will","rect":[44.793582916259769,396.82989501953127,295.489259123688,387.6922302246094]},{"page":3,"text":"undulate on the TPMS or networks, which increases the","rect":[44.79359817504883,407.0,295.51318595388838,399.96240234375]},{"page":3,"text":"indicator I\u0003.","rect":[44.79359817504883,422.0278015136719,94.01995260842974,411.8572998046875]},{"page":3,"text":"2.3. Indicator IK","rect":[44.794551849365237,442.9466247558594,111.93651350502711,434.4955749511719]},{"page":3,"text":"A minimal surface like structure located on the TPMS and","rect":[54.76140213012695,457.4634094238281,295.5681178567509,450.71466064453127]},{"page":3,"text":"cylinder-like structures with junctions located on the networks","rect":[44.794803619384769,471.4356994628906,295.4337786911319,462.6662902832031]},{"page":3,"text":"are close to parts of the electron density of the LLC bicon-","rect":[44.794803619384769,483.3873596191406,295.47155919061307,474.6179504394531]},{"page":3,"text":"tinuous cubic phases [Figs. 2(b) and 2(c)]. The molecules","rect":[44.794803619384769,495.33203125,295.54403870089757,486.1943664550781]},{"page":3,"text":"diffuse along the structural motifs; therefore, the isoelectron","rect":[44.79378890991211,507.2339172363281,295.41388201690719,498.4645080566406]},{"page":3,"text":"density surfaces are not closed but infinitely connected.","rect":[44.79378890991211,519.185546875,295.4855058921211,510.4161682128906]},{"page":3,"text":"However, only the sphere-like structure in Fig. 2 has closed","rect":[44.79378890991211,531.13720703125,295.4725368020634,522.3678588867188]},{"page":3,"text":"isoelectron density surfaces, i.e. a region that is strictly convex","rect":[44.79378890991211,543.0888671875,295.46240470388838,534.3195190429688]},{"page":3,"text":"upward. An isolated density maximum/minimum is difficult to","rect":[44.79378890991211,555.0404663085938,295.5680873391728,546.2711181640625]},{"page":3,"text":"imagine","rect":[44.79378890991211,566.9921264648438,76.77248618582193,558.2227783203125]},{"page":3,"text":"in","rect":[82.33845520019531,565.0,90.09976458526656,558.2227783203125]},{"page":3,"text":"the","rect":[95.58909606933594,564.9118041992188,108.52427909353678,558.2227783203125]},{"page":3,"text":"LLC","rect":[114.05043029785156,564.9714965820313,133.11583400018484,558.3023681640625]},{"page":3,"text":"bicontinuous","rect":[138.62806701660157,565.0,191.00468628390534,558.2227783203125]},{"page":3,"text":"cubic","rect":[196.5587158203125,564.9118041992188,217.77173061352574,558.2227783203125]},{"page":3,"text":"phase","rect":[223.34368896484376,566.9921264648438,246.62666556814615,558.2227783203125]},{"page":3,"text":"because","rect":[252.16775512695313,564.9118041992188,284.2460350749821,558.2227783203125]},{"page":3,"text":"it","rect":[289.8259582519531,564.9118041992188,295.51733555780057,558.2227783203125]},{"page":3,"text":"seemingly","rect":[44.79378890991211,578.9437866210938,85.16572806814615,570.1744384765625]},{"page":3,"text":"contradicts","rect":[93.32510375976563,577.0,137.9423999069522,570.1744384765625]},{"page":3,"text":"molecular","rect":[146.15850830078126,577.0,186.4916400192569,570.1744384765625]},{"page":3,"text":"diffusion","rect":[194.74655151367188,577.0,230.44525408721968,570.1744384765625]},{"page":3,"text":"over","rect":[238.74594116210938,577.0,256.8550433395694,572.0]},{"page":3,"text":"a","rect":[265.0791015625,576.8634643554688,269.736483073029,572.3543090820313]},{"page":3,"text":"long","rect":[277.8779602050781,578.9437866210938,295.5610985515446,570.1744384765625]},{"page":3,"text":"distance.","rect":[44.79378890991211,589.0,79.9729479087227,582.0693359375]},{"page":3,"text":"Therefore,","rect":[86.01858520507813,589.0,128.67242605813679,582.0693359375]},{"page":3,"text":"few","rect":[134.7180633544922,589.0,149.10619044549734,582.14892578125]},{"page":3,"text":"density","rect":[155.16078186035157,590.8386840820313,184.13409659841958,582.0693359375]},{"page":3,"text":"regions","rect":[190.27029418945313,590.8386840820313,219.76111023410065,582.0693359375]},{"page":3,"text":"that","rect":[225.8893585205078,588.7583618164063,241.92849888787868,582.0693359375]},{"page":3,"text":"are","rect":[247.97415161132813,589.0,260.909334635529,584.0]},{"page":3,"text":"convex","rect":[267.00177001953127,589.0,295.458651041779,584.0]},{"page":3,"text":"upward or downward are expected. Convex regions can be","rect":[44.79378890991211,602.790283203125,295.5263085124821,594.0209350585938]},{"page":3,"text":"determined by the eigenvalues of the Hessian matrix of the","rect":[44.79378890991211,614.741943359375,295.5133690593571,605.9725952148438]},{"page":3,"text":"electron density, \u0003ðrÞ:","rect":[44.79378890991211,626.693603515625,132.47297842630085,617.4464721679688]},{"page":3,"text":"HðrÞ ¼ 0@\u0003\u0003\u0003zyxxxx \u0003\u0003\u0003xyzyyy \u0003\u0003\u0003yzxzzz A1;","rect":[115.35710144042969,670.758056640625,224.9692539901419,635.30126953125]},{"page":3,"text":"where subscripts indicate partial derivatives. If all eigenvalues","rect":[44.79429244995117,689.6814575195313,295.5159014938663,680.912109375]},{"page":3,"text":"of the Hessian matrix are positive, then the region is strictly","rect":[44.79429244995117,701.6331176757813,295.537752604279,692.86376953125]},{"page":3,"text":"convex downward, and if all are negative, then the region is","rect":[44.79429244995117,713.5280151367188,295.52585022433507,704.7586669921875]},{"page":3,"text":"strictly convex upward (Rockafellar & Wets, 2010). Let C be","rect":[44.79429244995117,725.5294799804688,295.5315880534977,716.3917846679688]},{"page":3,"text":"the","rect":[44.79428482055664,736.0,57.729464030060217,728.6619873046875]},{"page":3,"text":"electron-density","rect":[64.27467346191406,737.4313354492188,129.68200919607583,728.6619873046875]},{"page":3,"text":"regions","rect":[136.2481231689453,737.4313354492188,165.82946960910065,728.6619873046875]},{"page":3,"text":"that","rect":[172.37767028808595,736.0,188.50732579217556,728.6619873046875]},{"page":3,"text":"are","rect":[195.0296630859375,736.0,207.96484611013833,730.0]},{"page":3,"text":"convex","rect":[214.56678771972657,736.0,243.02363822439615,730.0]},{"page":3,"text":"upward","rect":[249.67733764648438,737.4313354492188,280.1046168801884,728.6619873046875]},{"page":3,"text":"or","rect":[286.7135314941406,736.0,295.5098284958194,730.0]},{"page":3,"text":"downward and let K(r) be the determinant of H(r): K(r) =","rect":[44.79428482055664,749.4327392578125,295.54056186926439,740.2950439453125]},{"page":3,"text":"432 Toshihiko Oka \u0002 Liquid crystal bicontinuous phases","rect":[44.794551849365237,772.2470092773438,256.8675980616458,762.5875244140625]},{"page":3,"text":"det[H(r)]. Therefore, K(r) is the product of three eigenvalues.","rect":[313.4495849609375,80.47691345214844,564.2763689292306,71.3392333984375]},{"page":3,"text":"The following indicator was considered:","rect":[313.4495849609375,92.37879943847656,476.7445390952461,83.60941314697266]},{"page":3,"text":"IK ¼ R\u0002\u0002KðrÞ\u0002\u0002dr;","rect":[405.7005310058594,111.72029876708985,472.0458683700247,99.85128784179688]},{"page":3,"text":"C","rect":[428.01129150390627,117.37872314453125,432.3855929562514,113.0034408569336]},{"page":3,"text":"which is the integral of the absolute value of KðrÞ in the region","rect":[313.4471130371094,136.27955627441407,564.2291896340947,127.03238677978516]},{"page":3,"text":"C of the electron density in which the three eigenvalues are of","rect":[313.4471130371094,148.23121643066407,564.2064835070194,139.4618377685547]},{"page":3,"text":"the same sign in a unit cell, where r is the position in a unit cell.","rect":[313.4471130371094,160.18287658691407,564.1983659995431,151.4134979248047]},{"page":3,"text":"Thus, IK is an indicator that expresses the total (integrated)","rect":[313.4471435546875,172.1276092529297,564.216279911835,162.98992919921876]},{"page":3,"text":"convexity.","rect":[313.4479675292969,184.0294952392578,353.8935564292305,175.26011657714845]},{"page":3,"text":"With","rect":[360.33526611328127,181.94912719726563,380.1480433938603,175.26011657714845]},{"page":3,"text":"regard","rect":[386.61065673828127,184.0294952392578,413.08807635284469,175.26011657714845]},{"page":3,"text":"to","rect":[419.568603515625,183.0,427.8473842141728,176.21568298339845]},{"page":3,"text":"the","rect":[434.34881591796877,183.0,447.2839989421696,175.26011657714845]},{"page":3,"text":"geometrical","rect":[453.7724914550781,184.0294952392578,501.35946786392239,175.26011657714845]},{"page":3,"text":"meaning,","rect":[507.90966796875,184.0294952392578,544.8383806479806,175.26011657714845]},{"page":3,"text":"the","rect":[551.2880859375,183.0,564.2232994792789,175.26011657714845]},{"page":3,"text":"eigenvalues of the Hessian matrix are the principal curvatures,","rect":[313.4479675292969,195.98109436035157,564.2670915854806,187.2117156982422]},{"page":3,"text":"and K(r) is the Gaussian curvature if the 3D electron density is","rect":[313.4479675292969,207.98252868652345,564.2341632126163,198.8448486328125]},{"page":3,"text":"regarded as a hypersurface in a 4D hyperspace (Monga &","rect":[313.447998046875,219.93418884277345,564.2193043732929,210.7965087890625]},{"page":3,"text":"Benayoun, 1995; Goldman, 2005).","rect":[313.447998046875,231.88584899902345,452.195192171418,222.7481689453125]},{"page":3,"text":"3. Phase retrieval method based on the Iq and IK","rect":[313.4483947753906,264.73834228515627,536.5478296032263,253.84774780273438]},{"page":3,"text":"indicators","rect":[313.4483947753906,273.596435546875,358.6097264627512,265.8203125]},{"page":3,"text":"The structure factor FðhÞ can be divided into the amplitude","rect":[313.4483947753906,291.5377502441406,564.1737389324039,282.2905578613281]},{"page":3,"text":"jFðhÞj and the phase part exp½i\u0004ðhÞ\u0004, where \u0004ðhÞ is the phase.","rect":[313.4464111328125,303.4894104003906,564.2722795737618,294.2422180175781]},{"page":3,"text":"In a diffraction experiment, intensities are measured as the","rect":[313.4433898925781,315.4410705566406,564.2146935222477,306.6716613769531]},{"page":3,"text":"squares of the amplitudes and the phases are not observable.","rect":[313.4433898925781,327.3927001953125,564.2644670737618,318.623291015625]},{"page":3,"text":"The electron density \u0003ðrÞ can be obtained by Fourier trans-","rect":[313.4433898925781,339.3443298339844,564.2433487413944,330.0971374511719]},{"page":3,"text":"formation of the structure factors as follows:","rect":[313.4443664550781,349.2156066894531,496.536256624543,342.5265808105469]},{"page":3,"text":"1","rect":[375.4584655761719,365.4024658203125,380.11584708670088,358.82293701171877]},{"page":3,"text":"\u0003ðrÞ ¼ V X \u0002\u0002FðhÞ\u0002\u0002exp½i\u0004ðhÞ\u0004expð\u0003i2\u0002h \u0005 rÞ;","rect":[345.2707824707031,379.10601806640627,532.4153752059623,362.7398376464844]},{"page":3,"text":"h2Hobs","rect":[383.10345458984377,384.1294860839844,402.6604593817712,378.4043273925781]},{"page":3,"text":"where ris the position in the unit cell, h is the reciprocal-lattice","rect":[313.44781494140627,403.01092529296877,564.1960167644352,394.24151611328127]},{"page":3,"text":"vector and V is the volume of the unit cell. Hobs is the set of h","rect":[313.44781494140627,414.4432373046875,564.2288844583135,406.19317626953127]},{"page":3,"text":"for which the structure factor could be observed. F(000) was","rect":[313.44775390625,426.90740966796877,564.2188433883976,417.7697448730469]},{"page":3,"text":"not used in this paper; therefore, the integrated value of the","rect":[313.44873046875,438.8092956542969,564.2250084636539,430.0398864746094]},{"page":3,"text":"electron density \u0003ðrÞ in the unit cell is zero. High-electron-","rect":[313.44873046875,450.7609558105469,564.1680923937381,441.5137634277344]},{"page":3,"text":"density regions are thus positive, whereas low-electron-density","rect":[313.44775390625,462.71258544921877,564.1285118816227,453.94317626953127]},{"page":3,"text":"regions are expressed as negative in \u0003ðrÞ.","rect":[313.44775390625,474.66424560546877,482.2778032554024,465.41705322265627]},{"page":3,"text":"In the phase retrieval, the electron densities were calculated","rect":[323.4720458984375,486.61590576171877,564.1782863137822,477.84649658203127]},{"page":3,"text":"for all possible phase combinations by placing them into the","rect":[313.44873046875,498.56756591796877,564.2230553386539,489.79815673828127]},{"page":3,"text":"phase part exp½i\u0004ðhÞ\u0004 of the structure factor. The structures of","rect":[313.44873046875,510.5191955566406,564.2119156359256,501.2720031738281]},{"page":3,"text":"the bicontinuous cubic phasesare centrosymmetric,so that the","rect":[313.44671630859377,522.4140625,564.2210411784977,513.6447143554688]},{"page":3,"text":"phase values are 0 or \u0002 (if the unit-cell origin is placed on a","rect":[313.44671630859377,534.41552734375,564.2139000652164,525.27783203125]},{"page":3,"text":"symmetry center), i.e. the structure factors are real (positive","rect":[313.44671630859377,546.3671875,564.1969322917789,537.2294921875]},{"page":3,"text":"or negative, respectively). When n is the number of the","rect":[313.44671630859377,558.31884765625,564.2297692058414,549.18115234375]},{"page":3,"text":"independent","rect":[313.44671630859377,570.2206420898438,364.2410294054568,561.4512939453125]},{"page":3,"text":"structure-factor","rect":[369.9622497558594,569.0,433.2847003219913,561.5308837890625]},{"page":3,"text":"amplitude,","rect":[438.9909973144531,570.2206420898438,481.87274344094927,561.4512939453125]},{"page":3,"text":"the","rect":[487.4656066894531,569.0,500.400789713654,561.4512939453125]},{"page":3,"text":"corresponding","rect":[506.09613037109377,570.2206420898438,564.1612877605289,561.4512939453125]},{"page":3,"text":"number of phase combinations is 2n. The electron density was","rect":[313.44671630859377,582.172607421875,564.21457092746,573.022705078125]},{"page":3,"text":"calculated in 32 \u0006 32 \u0006 32 voxels as a unit cell. I","rect":[313.44744873046877,592.0438842773438,539.1174639938663,585.3548583984375]},{"page":3,"text":"\u0003","rect":[539.1199340820313,595.4683837890625,542.6732330095368,591.2770385742188]},{"page":3,"text":"obtained","rect":[313.44781494140627,603.9956665039063,349.23004412628219,597.306640625]},{"page":3,"text":"from","rect":[354.8427734375,603.9956665039063,374.47443132641788,597.38623046875]},{"page":3,"text":"the","rect":[380.0423889160156,603.9956665039063,392.9775719402165,597.306640625]},{"page":3,"text":"difference","rect":[398.6161804199219,603.9956665039063,439.488680338654,597.306640625]},{"page":3,"text":"between","rect":[445.16412353515627,603.9956665039063,479.30133318878219,597.306640625]},{"page":3,"text":"the","rect":[484.91705322265627,603.9956665039063,497.8522667644352,597.306640625]},{"page":3,"text":"maximum","rect":[503.4351806640625,603.9956665039063,543.5842946076679,597.306640625]},{"page":3,"text":"minimum","rect":[313.44781494140627,615.8905639648438,352.05035906079288,609.2015380859375]},{"page":3,"text":"electron","rect":[359.14886474609377,615.8905639648438,392.26104998565719,609.2015380859375]},{"page":3,"text":"densities.","rect":[399.41131591796877,615.8905639648438,436.4584062827461,609.2015380859375]},{"page":3,"text":"To","rect":[443.4673767089844,615.8905639648438,453.96539568878219,609.4304809570313]},{"page":3,"text":"obtain","rect":[461.0231018066406,615.8905639648438,486.89248919464156,609.2015380859375]},{"page":3,"text":"IK,","rect":[493.9860534667969,617.4635620117188,505.15533621438677,609.4304809570313]},{"page":3,"text":"the","rect":[512.164306640625,615.890625,525.0994591472477,609.2015991210938]},{"page":3,"text":"Hessian","rect":[532.1542358398438,616.0,564.219485044251,609.2015991210938]},{"page":3,"text":"matrices were obtained from the electron densities for all","rect":[313.4473571777344,627.84228515625,564.2316602467348,621.1532592773438]},{"page":3,"text":"voxel points in the unit cell, which yielded three eigenvalues at","rect":[313.4473571777344,641.8742065429688,564.2007462023319,633.1048583984375]},{"page":3,"text":"all points. The region where the signs of the three eigenvalues","rect":[313.4473571777344,653.8258666992188,564.2535113571476,645.0565185546875]},{"page":3,"text":"were the same was designated as C, and K(r) was obtained at","rect":[313.4473571777344,665.8273315429688,564.2085587023319,656.6896362304688]},{"page":3,"text":"each point within C. IK was then calculated by integration as","rect":[313.4483642578125,677.729248046875,564.2137164352726,668.9598388671875]},{"page":3,"text":"described in Section 2.3. The fourth moment of the electron","rect":[313.44757080078127,687.6602172851563,564.1273829934697,680.9114990234375]},{"page":3,"text":"density h½\u0003ðrÞ\u00044i was calculated at the same time (Luzzati et al.,","rect":[313.44757080078127,701.6823120117188,564.2775285971993,690.61767578125]},{"page":3,"text":"1988). F(000) was not used in this paper; therefore, h½\u0003ðrÞ\u00044i","rect":[313.4477233886719,713.5772094726563,564.261690068085,702.5693359375]},{"page":3,"text":"corresponds to the average of the fourth power of the electron","rect":[313.44775390625,725.479248046875,564.1255519387822,716.7098999023438]},{"page":3,"text":"density, and h½\u0003ðrÞ\u00044i was calculated as the average of the","rect":[313.44775390625,737.430908203125,564.2277550456852,726.416015625]},{"page":3,"text":"fourth power of the electron density \u0003ðrÞ in the unit cell.","rect":[313.4476623535156,749.3825073242188,546.5296648276681,740.1353759765625]},{"page":3,"text":"Acta Cryst. (2022). A78, 430–436","rect":[446.87060546875,772.2708740234375,564.3506555502972,764.9843139648438]},{"page":4,"text":"According","rect":[54.76140213012695,80.42951965332031,97.30578849295084,71.6601333618164]},{"page":4,"text":"to","rect":[102.89564514160156,79.0,111.1744334695439,72.61570739746094]},{"page":4,"text":"the","rect":[116.82598876953125,79.0,129.761156534943,71.6601333618164]},{"page":4,"text":"Babinet","rect":[135.3997802734375,79.0,167.3784805773318,71.6601333618164]},{"page":4,"text":"principle,","rect":[173.05789184570313,80.42951965332031,210.8414018882149,71.6601333618164]},{"page":4,"text":"when","rect":[216.43423461914063,79.0,238.0662105081181,71.6601333618164]},{"page":4,"text":"the","rect":[243.72972106933595,79.0,256.6649346111149,71.6601333618164]},{"page":4,"text":"electron","rect":[262.3035583496094,79.0,295.5163600442509,71.6601333618164]},{"page":4,"text":"density","rect":[44.794803619384769,92.38117980957031,73.76809928396647,83.6117935180664]},{"page":4,"text":"\u0003ðrÞ","rect":[79.9620132446289,92.33141326904297,95.6209739458932,83.1340103149414]},{"page":4,"text":"is","rect":[101.70741271972656,91.0,107.91626160128816,83.6117935180664]},{"page":4,"text":"inverted,","rect":[113.99573516845703,91.0,150.02476675637898,83.6117935180664]},{"page":4,"text":"the","rect":[156.07040405273438,91.0,169.00557181814615,83.6117935180664]},{"page":4,"text":"amplitudes","rect":[175.1547088623047,92.38117980957031,219.65060608371003,83.6117935180664]},{"page":4,"text":"of","rect":[225.78182983398438,91.0,234.06064000115993,83.6914291381836]},{"page":4,"text":"the","rect":[240.16598510742188,91.0,253.10115287283365,83.6117935180664]},{"page":4,"text":"structure","rect":[259.1935729980469,91.0,295.5132775066227,84.56736755371094]},{"page":4,"text":"factors are the same, but the phases are shifted by \u0002. There-","rect":[44.7938117980957,104.33283996582031,295.5560013292849,95.5634536743164]},{"page":4,"text":"fore,","rect":[44.79378890991211,114.20413970947266,63.3815172446602,107.5947494506836]},{"page":4,"text":"the","rect":[69.03108215332031,114.20413970947266,81.96626517752115,107.5151138305664]},{"page":4,"text":"structures","rect":[87.71832275390625,114.20413970947266,127.69516174777253,108.47068786621094]},{"page":4,"text":"with","rect":[133.3606414794922,114.20413970947266,150.85269427276655,107.5151138305664]},{"page":4,"text":"inverted","rect":[156.57789611816407,114.20413970947266,190.3499019143681,107.5151138305664]},{"page":4,"text":"electron","rect":[196.04823303222657,114.20413970947266,229.16047930694624,107.5151138305664]},{"page":4,"text":"densities","rect":[234.9513702392578,114.20413970947266,270.24109070285069,107.5151138305664]},{"page":4,"text":"were","rect":[275.9493713378906,114.20413970947266,295.5114159343571,109.0]},{"page":4,"text":"considered equivalent, and the number of phase combinations","rect":[44.79378890991211,128.2361602783203,295.45858948214757,119.4667739868164]},{"page":4,"text":"\u0003","rect":[272.0520935058594,129.82376098632813,275.2565312118216,129.28623962402345]},{"page":4,"text":"to be calculated was reduced. In the space group Pn3m, if","rect":[44.79378890991211,140.1878204345703,295.52612462030057,131.41844177246095]},{"page":4,"text":"the origin of a structure is (0,0,0), then moving the origin to","rect":[44.793827056884769,152.1891326904297,295.4964625833134,143.05145263671876]},{"page":4,"text":"(1/2,1/2,1/2)","rect":[44.793827056884769,164.1407928466797,92.39167206271394,155.00311279296876]},{"page":4,"text":"will","rect":[103.06285095214844,163.0,117.44996774185202,155.32164001464845]},{"page":4,"text":"give","rect":[128.09228515625,164.0910186767578,144.6488976238102,155.32164001464845]},{"page":4,"text":"equivalent","rect":[155.3310089111328,164.0910186767578,197.86442723260525,155.32164001464845]},{"page":4,"text":"amplitudes","rect":[208.5047607421875,164.0910186767578,252.9997729538272,155.32164001464845]},{"page":4,"text":"of","rect":[263.6610107421875,163.0,271.93982090936307,155.40126037597657]},{"page":4,"text":"the","rect":[282.5751953125,163.0,295.51037833670088,155.32164001464845]},{"page":4,"text":"structure factors, although some phases are shifted by \u0002","rect":[44.793827056884769,175.98597717285157,295.5649755012218,167.2165985107422]},{"page":4,"text":"(Giacovazzo,","rect":[44.793827056884769,187.98741149902345,97.52771170266802,178.8497314453125]},{"page":4,"text":"2001).","rect":[102.78021240234375,187.98741149902345,127.42451651223833,178.8497314453125]},{"page":4,"text":"Therefore,","rect":[132.67999267578126,186.0,175.33384878762898,179.1682586669922]},{"page":4,"text":"it","rect":[180.64306640625,185.85726928710938,186.33542027459743,179.1682586669922]},{"page":4,"text":"was","rect":[191.57298278808595,186.0,206.47860901339753,180.0]},{"page":4,"text":"considered","rect":[211.78981018066407,186.0,255.89074969268843,179.1682586669922]},{"page":4,"text":"that","rect":[261.2258605957031,186.0,277.2649857042849,179.1682586669922]},{"page":4,"text":"the","rect":[282.5751953125,186.0,295.510408854279,179.1682586669922]},{"page":4,"text":"structure is equivalent, even when the translation operation of","rect":[44.793827056884769,199.8892364501953,295.4965530870974,191.11985778808595]},{"page":4,"text":"(1/2,1/2,1/2) is performed, and the number of phase combi-","rect":[44.793827056884769,211.89073181152345,295.4656082628787,202.7530517578125]},{"page":4,"text":"nations to be calculated is reduced. The electron densities and","rect":[44.793827056884769,221.71218872070313,295.4846217630009,215.02317810058595]},{"page":4,"text":"indicators","rect":[44.793827056884769,234.0,84.76770203097566,226.97483825683595]},{"page":4,"text":"were","rect":[91.73784637451172,234.0,111.29983756521647,228.0]},{"page":4,"text":"Phase retrieval","rect":[44.793827056884769,246.0,110.30764901626609,238.92649841308595]},{"page":4,"text":"Python3 script.","rect":[44.793827056884769,259.6474914550781,106.07122976907427,250.8780975341797]},{"page":4,"text":"4. Test data and quality of phase retrieval","rect":[44.794551849365237,303.362548828125,237.42017410218763,293.59014892578127]},{"page":4,"text":"4.1. Test data derived from constructed models","rect":[44.794551849365237,317.3128356933594,242.44001443740837,310.23980712890627]},{"page":4,"text":"Electron-density models were constructed to test the indi-","rect":[54.76140213012695,335.2660827636719,295.4815079210818,326.4966735839844]},{"page":4,"text":"cators for the phase retrieval. Two polar–nonpolar interface","rect":[44.794803619384769,347.21771240234377,295.50647208670088,338.44830322265627]},{"page":4,"text":"models for the LLC bicontinuous cubic phase were used: a","rect":[44.794803619384769,359.16937255859377,295.5611290691227,350.39996337890627]},{"page":4,"text":"parallel surface (PS) (Hyde et al., 1996) and a constant mean","rect":[44.794803619384769,371.1707458496094,295.50223040557906,362.0330810546875]},{"page":4,"text":"curvature","rect":[44.794803619384769,382.0,83.59836051443522,375.0]},{"page":4,"text":"surface","rect":[88.73839569091797,382.0,117.80720756033365,374.3829040527344]},{"page":4,"text":"(CMCS)","rect":[122.82882690429688,383.1224060058594,157.27951233371003,373.9847412109375]},{"page":4,"text":"(Anderson","rect":[162.3001251220703,383.1224060058594,206.37109637237593,373.9847412109375]},{"page":4,"text":"et","rect":[211.4584197998047,380.9922790527344,218.1847547291567,375.5972595214844]},{"page":4,"text":"al.,","rect":[223.23721313476563,381.0,235.08667166360554,374.3032531738281]},{"page":4,"text":"1990;","rect":[240.05653381347657,382.43560791015627,261.0794389975899,374.34307861328127]},{"page":4,"text":"Große-","rect":[266.04931640625,381.052001953125,295.5569778917849,374.3829040527344]},{"page":4,"text":"Brauckmann,","rect":[44.79481506347656,394.0,99.24338705667192,386.1981506347656]},{"page":4,"text":"1997).","rect":[104.36949157714844,395.0173034667969,128.89537222512898,385.879638671875]},{"page":4,"text":"In","rect":[134.04237365722657,394.0,142.83865618682905,386.4270935058594]},{"page":4,"text":"the","rect":[147.97271728515626,394.0,160.90790030935708,386.1981506347656]},{"page":4,"text":"PS","rect":[165.98025512695313,392.9468994140625,176.84649920440718,386.2778015136719]},{"page":4,"text":"model,","rect":[181.94970703125,394.0,209.5576341880196,386.1981506347656]},{"page":4,"text":"the","rect":[214.68174743652345,394.0,227.6169609783024,386.1981506347656]},{"page":4,"text":"polar–nonpolar","rect":[232.7460479736328,394.9675598144531,295.50341980441319,386.1981506347656]},{"page":4,"text":"interface is parallel to the TPMS, and in the CMCS model, the","rect":[44.79481506347656,406.9192199707031,295.5163292644352,398.1498107910156]},{"page":4,"text":"interface is a CMCS. The PS and CMCS models were gener-","rect":[44.79481506347656,418.8708801269531,295.54028477655057,410.1014709472656]},{"page":4,"text":"ated as model electron densities from the three TPMSs, i.e. the","rect":[44.79481506347656,429.0,295.521151041779,422.0531311035156]},{"page":4,"text":"G, P and D surfaces. The PS and CMCS models based on the G","rect":[44.79481506347656,441.0,295.51236467602726,434.0047607421875]},{"page":4,"text":"surface are referred to as the G-PS and G-CMCS models,","rect":[44.79481506347656,453.0,295.56109793313677,445.9563903808594]},{"page":4,"text":"respectively.","rect":[44.79481506347656,466.6774597167969,94.5272920859688,457.9080505371094]},{"page":4,"text":"The TPMSs and CMCSs were created using Surface Evolver","rect":[54.761409759521487,478.6291198730469,295.47140686496007,469.8597106933594]},{"page":4,"text":"(Brakke, 1992; Shearman et al., 2007). The PSs were generated","rect":[44.794795989990237,490.57379150390627,295.41763567901656,481.4361267089844]},{"page":4,"text":"as surfaces at fixed distances from the TPMSs [Fig. 1(b)].","rect":[44.79478454589844,502.525390625,295.5539873374336,493.3877258300781]},{"page":4,"text":"Stepwise electron densities were created for both models. The","rect":[44.794776916503909,514.42724609375,295.5223107097477,505.6578674316406]},{"page":4,"text":"electron density was calculated in 128 \u0006 128 \u0006 128 voxels as a","rect":[44.794776916503909,526.37890625,295.5648827312321,517.6095581054688]},{"page":4,"text":"unit cell. The density from the interface to the TPMS side was","rect":[44.79376983642578,538.33056640625,295.49939148410069,529.5612182617188]},{"page":4,"text":"set to 1.0, and the rest of the region was set to 0.0. The volume","rect":[44.79376983642578,550.2822875976563,295.5601525066227,541.512939453125]},{"page":4,"text":"fractions from the interface to the TPMS side (Vfrac) were 0.2,","rect":[44.79376983642578,562.3959350585938,295.4922807944649,553.1460571289063]},{"page":4,"text":"0.4, 0.6, 0.7 and 0.8. PS models with different Vfrac were","rect":[44.793766021728519,574.3016357421875,295.52112052420088,565.4165649414063]},{"page":4,"text":"generated by varying the constant distance from the TPMS to","rect":[44.7938117980957,586.080810546875,295.49844622589156,577.3114624023438]},{"page":4,"text":"the polar–nonpolar interface (Qiu & Caffrey, 1998), while the","rect":[44.7938117980957,598.082275390625,295.5104393718571,588.944580078125]},{"page":4,"text":"CMCS model requires only a volume fraction to generate the","rect":[44.7938117980957,609.984130859375,295.51343009451338,601.2147827148438]},{"page":4,"text":"CMCS. The CMCS with Vfrac \u0007 0:6 could not be generated for","rect":[44.7938117980957,622.0516357421875,295.5011309860538,613.1664428710938]},{"page":4,"text":"the P surface; therefore, only P-CMCS models with Vfrac ¼ 0.2","rect":[44.79367446899414,634.0033569335938,295.53540275076338,625.1182250976563]},{"page":4,"text":"and 0.4 were used. Using values from previous papers as a","rect":[44.79314041137695,645.8392333984375,295.559481119904,637.0698852539063]},{"page":4,"text":"reference (Oka, 2017; Oka et al., 2018, 2020), the lattice","rect":[44.79314041137695,657.8406982421875,295.56225821951338,648.7030029296875]},{"page":4,"text":"constant was set to 1 and the width of the Gaussian function","rect":[44.794132232666019,667.721923828125,295.50873064971969,660.9732055664063]},{"page":4,"text":"was set to 0.05 for the G surface based models and 0.07 for the","rect":[44.794132232666019,679.6168212890625,295.5137352702946,672.8182983398438]},{"page":4,"text":"other models.","rect":[44.794132232666019,691.5087890625,100.30453665383989,684.8197631835938]},{"page":4,"text":"The structure factors FðhÞ were obtained by Fourier trans-","rect":[54.7607307434082,705.5407104492188,295.4985672472537,696.2935791015625]},{"page":4,"text":"formation of the model densities \u0003ðrÞ:","rect":[44.79312515258789,717.442626953125,200.03022940286335,708.2452392578125]},{"page":4,"text":"FðhÞ ¼ \u0002\u0002FðhÞ\u0002\u0002exp½i\u0004ðhÞ\u0004 ¼ R \u0003ðrÞexpði2\u0002h \u0005 rÞdr: ð5Þ","rect":[69.99275207519531,736.1519775390625,295.52545392636196,724.2828979492188]},{"page":4,"text":"uc","rect":[180.93356323242188,741.7840576171875,187.0805782630176,738.80810546875]},{"page":4,"text":"Acta Cryst. (2022). A78, 430–436","rect":[44.794551849365237,772.2708740234375,162.21811770850037,764.9843139648438]},{"page":4,"text":"research papers","rect":[452.9285888671875,50.97494125366211,562.962778580549,36.115684509277347]},{"page":4,"text":"Integration is performed in a unit cell. The numbers of inde-","rect":[313.44842529296877,80.42677307128906,564.1669937609256,71.65738677978516]},{"page":4,"text":"pendent structure-factor amplitudes used were 22, 19 and 13","rect":[313.44842529296877,92.37843322753906,564.2266564128727,83.60904693603516]},{"page":4,"text":"for the G, D and P surface based models, respectively. The","rect":[313.44842529296877,104.33009338378906,564.2367272136539,95.56070709228516]},{"page":4,"text":"model density construction, except for the formation of the","rect":[313.44842529296877,116.28175354003906,564.2226891277164,107.51236724853516]},{"page":4,"text":"TPMS and CMCS, and other calculations were conducted","rect":[313.44842529296877,126.21271514892578,564.1958644387822,119.4639663696289]},{"page":4,"text":"using Mathematica 12.1 (Wolfram Research, Inc., USA).","rect":[313.44842529296877,140.2347869873047,543.9330461753243,131.09710693359376]},{"page":4,"text":"4.2. Rp value to evaluate the quality of phase retrieval","rect":[313.4483947753906,172.42835998535157,539.8410451226529,162.62831115722657]},{"page":4,"text":"The Rp value is defined as a quantity that evaluates how well","rect":[323.47186279296877,188.4662628173828,564.1795362232973,178.88511657714845]},{"page":4,"text":"the phase combination used in the calculation agrees with the","rect":[313.4480895996094,199.6061553955078,564.2224449870914,190.83677673339845]},{"page":4,"text":"true phase combination:","rect":[313.4480895996094,211.5578155517578,412.4204118979805,202.78843688964845]},{"page":4,"text":"Rp ¼ Ph\u0002\u0002FðP hÞ\u0002\u0002h\u0002\u0002\u0002\u0002\u0004FtrðuheðÞh\u0002\u0002ðÞ\u0002\u0003=2\u0004ÞcalcðhÞ\u0002\u0002:","rect":[367.4169921875,250.5520782470703,510.32073993395906,224.12576293945313]},{"page":4,"text":"Rp close to 0 indicates a good agreement with the true phase","rect":[313.4468994140625,274.3936767578125,564.1525597331852,264.8125305175781]},{"page":4,"text":"combination. From the Babinet principle, the inverted elec-","rect":[313.4479064941406,285.5335693359375,564.1993423937381,276.76416015625]},{"page":4,"text":"tron density was also treated in the same way, and the \u0002-","rect":[313.4479064941406,297.4852294921875,564.2539078234256,288.7158203125]},{"page":4,"text":"shifted phase combination from \u0004true was also treated as the","rect":[313.44696044921877,309.5985412597656,564.2269615886539,300.66748046875]},{"page":4,"text":"true phase. Of the Rp calculated from the original and \u0002-","rect":[313.4478759765625,322.2002868652344,564.2545792101444,312.619140625]},{"page":4,"text":"shifted phase combinations, that with the smaller value was","rect":[313.4476318359375,333.3401794433594,564.2099932907413,324.5707702636719]},{"page":4,"text":"adopted.","rect":[313.4476318359375,345.235107421875,348.9183672202461,336.4656982421875]},{"page":4,"text":"Figure 3","rect":[313.4483947753906,692.0529174804688,345.7443758344041,683.9363403320313]},{"page":4,"text":"Distributions of the I\u0003 and IK indicators of G-PS models at (a) Vfrac ¼ 0:4","rect":[313.4483947753906,702.96240234375,564.2581076834886,694.03857421875]},{"page":4,"text":"and (b) Vfrac ¼ 0:8. The indicators were calculated from the electron","rect":[313.4488830566406,711.4166259765625,564.1999807594317,703.4976806640625]},{"page":4,"text":"densities using all phase combinations. The colors of the points indicate","rect":[313.44818115234377,720.6815795898438,564.1671653006761,713.2276611328125]},{"page":4,"text":"Rp that compare the phase combinations with the true phase combina-","rect":[313.44818115234377,730.8426513671875,564.1657648228465,722.6873779296875]},{"page":4,"text":"tion. The black circles show the true phase combination indicators of the","rect":[313.448486328125,739.6004028320313,564.2029929373948,732.146484375]},{"page":4,"text":"model.","rect":[313.448486328125,747.2911376953125,336.84982123866379,741.6055297851563]},{"page":4,"text":"Toshihiko Oka \u0002 Liquid crystal bicontinuous phases 433","rect":[352.1836853027344,772.2470092773438,564.2215909491334,762.6353149414063]},{"page":5,"text":"research papers","rect":[44.794551849365237,50.97494125366211,154.82873011863493,36.115684509277347]},{"page":5,"text":"5. Results","rect":[44.794551849365237,249.57989501953126,88.62776235142306,241.78286743164063]},{"page":5,"text":"5.1. Phase retrieval for constructed models","rect":[44.794551849365237,265.4844970703125,224.09200723525994,258.4114685058594]},{"page":5,"text":"Phase retrieval was performed for model densities with","rect":[54.76140213012695,283.4377746582031,295.4825465676884,274.6683654785156]},{"page":5,"text":"known true phase combinations. The electron density was","rect":[44.794803619384769,295.3894348144531,295.5004290817569,286.6200256347656]},{"page":5,"text":"calculated for all phase combinations, and the corresponding","rect":[44.794803619384769,307.341064453125,295.5581993816227,298.5716552734375]},{"page":5,"text":"IK and I\u0003 indicators were calculated.","rect":[44.794803619384769,320.63690185546877,194.45636160501179,310.46673583984377]},{"page":5,"text":"Fig. 3(a) shows the distribution of IK and I\u0003 for all phase","rect":[54.76179122924805,332.5886535644531,295.5656151531071,322.0998840332031]},{"page":5,"text":"combinations at Vfrac ¼ 0:4 of the G-PS model. The points are","rect":[44.79347610473633,343.31170654296877,295.5199913738102,334.3699951171875]},{"page":5,"text":"distributed from the lower left to upper right. Therefore, there","rect":[44.79462814331055,355.0910339355469,295.558992838654,346.3216247558594]},{"page":5,"text":"is an approximate positive correlation between IK and I\u0003. The","rect":[44.79462814331055,368.38690185546877,295.53076407888838,358.2732849121094]},{"page":5,"text":"Rp values of the points tend to be smaller in the lower left","rect":[44.793460845947269,379.80615234375,295.5222488878787,370.22503662109377]},{"page":5,"text":"region; points with Rp of approximately 0.1 or less are","rect":[44.79472732543945,391.7579040527344,295.5154442546696,382.1766357421875]},{"page":5,"text":"concentrated in the lower left region. IK and I\u0003 for the true","rect":[44.794063568115237,404.2419128417969,295.5167870281071,394.1282958984375]},{"page":5,"text":"phase combination are in the lower left corner,which indicates","rect":[44.79341506958008,414.7926940917969,295.4901446579288,406.0232849121094]},{"page":5,"text":"that both indicators of the true phase are almost the smallest,","rect":[44.79341506958008,426.7443542480469,295.54486258157427,417.9749450683594]},{"page":5,"text":"as expected; the same is true when Vfrac is 0.2 and 0.6 in the","rect":[44.79341506958008,438.914306640625,295.5172753093571,429.9266052246094]},{"page":5,"text":"other","rect":[44.79391860961914,449.0,66.52438538058503,441.8782958984375]},{"page":5,"text":"models","rect":[74.18418884277344,449.0,103.15847198947175,441.8782958984375]},{"page":5,"text":"(Fig.","rect":[110.76652526855469,450.69744873046877,129.29254324563679,441.5597839355469]},{"page":5,"text":"S1","rect":[136.8159942626953,448.6270446777344,146.6472649333805,441.9181213378906]},{"page":5,"text":"in","rect":[154.25830078125,449.0,162.0206019876103,441.8782958984375]},{"page":5,"text":"the","rect":[169.66249084472657,449.0,182.5976738689274,441.8782958984375]},{"page":5,"text":"supporting","rect":[190.21865844726563,450.647705078125,233.78914115408365,441.8782958984375]},{"page":5,"text":"information).","rect":[241.4668731689453,450.69744873046877,295.5215166343086,441.5597839355469]},{"page":5,"text":"However, when Vfrac is 0.8, IK for the true phase combination","rect":[44.79391860961914,462.76104736328127,295.5709254739384,453.8299255371094]},{"page":5,"text":"is roughly minimized, whereas I\u0003 is clearly not minimized [Fig.","rect":[44.793827056884769,475.8951721191406,295.6124590171211,465.5129089355469]},{"page":5,"text":"3(b)]. The tendency that the indicator for the true phase","rect":[44.7935676574707,486.5524597167969,295.55786368826338,477.414794921875]},{"page":5,"text":"combination is not minimized when Vfrac is large is also true","rect":[44.79257583618164,498.61602783203127,295.5121483562321,489.6849670410156]},{"page":5,"text":"for the G-CMCS model and the other surface-based models","rect":[44.793766021728519,508.3287048339844,295.5073870895694,501.5799560546875]},{"page":5,"text":"(Fig. S1). For example, in the G-CMCS model, both IK and I\u0003","rect":[44.793766021728519,523.7018432617188,295.0291290056306,513.213134765625]},{"page":5,"text":"are not minimized when Vfrac is 0.7 or 0.8.","rect":[44.794551849365237,534.470947265625,217.23249991067585,525.4833374023438]},{"page":5,"text":"The Rp values of the phase combinations with the minimum","rect":[54.76054000854492,547.0160522460938,295.527684500246,537.4349365234375]},{"page":5,"text":"indicators in the G-PS and G-CMCS models are shown in Fig.","rect":[44.794273376464847,558.1559448242188,295.61334402688677,549.3865966796875]},{"page":5,"text":"4. When the volume fraction is smaller than 0.6, R of the","rect":[44.794273376464847,568.02734375,295.5198693034977,561.3381958007813]},{"page":5,"text":"p","rect":[262.254638671875,570.9193725585938,265.9806767081857,566.7028198242188]},{"page":5,"text":"phase combinations at the minimum indicators is approxi-","rect":[44.79350662231445,582.059326171875,295.4335037706912,573.2899780273438]},{"page":5,"text":"mately 0.1 or less. Therefore, in this range of volume fractions,","rect":[44.79350662231445,594.0109252929688,295.55792410501177,585.2415771484375]},{"page":5,"text":"both IK and I\u0003 indicators are useful for phase retrieval. When","rect":[44.79350662231445,607.3068237304688,295.59308123565719,597.1365356445313]},{"page":5,"text":"Vfrac is 0.7 or 0.8, some of the phase combinations at the","rect":[44.79401397705078,617.857666015625,295.51343009451338,609.0883178710938]},{"page":5,"text":"minimum indicator are Rp > 0.2, which indicates that the phase","rect":[44.794918060302737,630.6210327148438,295.4984764812321,621.0399169921875]},{"page":5,"text":"retrieval by the indicator is not realized. In particular, the","rect":[44.79404830932617,641.7609252929688,295.50866935232588,632.9915771484375]},{"page":5,"text":"phase combination with the minimum I\u0003 differs significantly","rect":[44.79404830932617,655.0568237304688,295.4618859050602,644.9431762695313]},{"page":5,"text":"from the true combination for Vfrac ¼ 0:8 with all the models.","rect":[44.79428482055664,665.825927734375,295.565309358918,656.8948364257813]},{"page":5,"text":"This indicates that when Vfrac is large, the constraint that the","rect":[44.7941780090332,677.6159057617188,295.5149559734196,668.8465576171875]},{"page":5,"text":"difference","rect":[44.7945671081543,687.4872436523438,85.77554557302897,680.7982177734375]},{"page":5,"text":"between","rect":[91.73858642578125,687.4872436523438,125.78721880889937,680.7982177734375]},{"page":5,"text":"the","rect":[131.77513122558595,687.4872436523438,144.71031424978677,680.7982177734375]},{"page":5,"text":"minimum","rect":[150.68927001953126,687.4872436523438,189.29179888012878,680.7982177734375]},{"page":5,"text":"and","rect":[195.25682067871095,687.4872436523438,210.26195025421186,680.7982177734375]},{"page":5,"text":"maximum","rect":[216.21006774902345,687.4872436523438,256.35916643383976,680.7982177734375]},{"page":5,"text":"electron","rect":[262.3072814941406,687.4872436523438,295.52005267120406,680.7982177734375]},{"page":5,"text":"density is minimal does not hold. On the other hand, the Rp","rect":[44.7945671081543,702.3310546875,295.5415897941232,692.693115234375]},{"page":5,"text":"value for minimum IK is relatively small, even when the","rect":[44.794551849365237,713.5866088867188,295.5151085613102,704.6448974609375]},{"page":5,"text":"volume fraction is 0.7 or 0.8. Including other models, Rp > 0.2","rect":[44.793704986572269,726.1776733398438,295.5365624187321,716.5965576171875]},{"page":5,"text":"with minimum IK is limited to three conditions: the G-CMCS","rect":[44.79329299926758,737.489990234375,295.44952654815719,728.5482177734375]},{"page":5,"text":"model with Vfrac ¼ 0:8 and the P-PS model with Vfrac = 0.7 and","rect":[44.79385757446289,749.430908203125,295.57367205596969,740.4999389648438]},{"page":5,"text":"434 Toshihiko Oka \u0002 Liquid crystal bicontinuous phases","rect":[44.794551849365237,772.2470092773438,256.8675980616458,762.6353149414063]},{"page":5,"text":"Rp of","rect":[305.3502502441406,104.52960205078125,322.62005967372439,96.95068359375]},{"page":5,"text":"minimum","rect":[305.35040283203127,111.1877670288086,336.15731526391667,105.83631896972656]},{"page":5,"text":"IK","rect":[305.35040283203127,121.89453125,312.0688550574289,114.96881866455078]},{"page":5,"text":"0","rect":[305.3497009277344,137.2439727783203,309.07576516976629,132.00401306152345]},{"page":5,"text":"0","rect":[305.3497009277344,155.14344787597657,309.07576516976629,149.9034881591797]},{"page":5,"text":"0","rect":[305.3504943847656,164.0927276611328,309.0765586267975,158.85276794433595]},{"page":5,"text":"0.015","rect":[305.3497619628906,173.0420684814453,322.0740867029694,167.6508026123047]},{"page":5,"text":"0.017","rect":[305.3497009277344,190.94154357910157,322.07402566781317,185.6219482421875]},{"page":5,"text":"0.005","rect":[305.35076904296877,199.9478302001953,322.0750937830475,194.5565643310547]},{"page":5,"text":"0.8.","rect":[402.47149658203127,78.3465347290039,416.34112723001177,71.79686737060547]},{"page":5,"text":"Therefore,","rect":[421.3846435546875,78.3465347290039,464.0961931479805,71.65750885009766]},{"page":5,"text":"IK","rect":[469.12579345703127,80.60186767578125,477.53732378541118,71.8864517211914]},{"page":5,"text":"is","rect":[483.2257995605469,79.0,489.4346331833194,71.6601333618164]},{"page":5,"text":"useful","rect":[494.49505615234377,79.0,518.8119825123598,71.6601333618164]},{"page":5,"text":"for","rect":[523.9420776367188,79.0,535.842256474335,71.7397689819336]},{"page":5,"text":"phase","rect":[540.8738403320313,80.42951965332031,564.1567711589664,71.6601333618164]},{"page":5,"text":"retrieval to some extent, even if","rect":[402.4709777832031,90.30081939697266,564.2313858507694,83.6117935180664]},{"page":5,"text":"by fluctuations leads to a failure in the","rect":[402.4710998535156,223.7928009033203,564.2265343425602,215.02342224121095]},{"page":5,"text":"phase retrieval of IK because the elec-","rect":[402.4710998535156,235.86004638671876,564.2080704210819,226.9750213623047]},{"page":5,"text":"tron density is out of the constraint","rect":[402.4705810546875,247.69606018066407,564.2468887804569,238.9266815185547]},{"page":5,"text":"based on the eigenvalues.","rect":[313.4479675292969,259.6476745605469,418.1683672202461,250.87828063964845]},{"page":5,"text":"The previously used indicator, the fourth moment of the","rect":[323.4712829589844,271.5425720214844,564.2262291667789,262.7731628417969]},{"page":5,"text":"electron density h½\u0003ðrÞ\u00044i (Luzzati et al., 1988), was compared","rect":[313.4479675292969,283.5441589355469,564.1947047708135,272.4796142578125]},{"page":5,"text":"with I\u0003. In all models, when Vfrac = 0.2, the Rp value of the","rect":[313.448486328125,296.84686279296877,564.2292809245914,286.6766357421875]},{"page":5,"text":"minimum h½\u0003ðrÞ\u00044i phase combination is larger than that of I\u0003","rect":[313.447265625,308.7419128417969,563.7395171892243,296.3828430175781]},{"page":5,"text":"(Figs. 3 and S2). When Vfrac = 0.7 for D-PS and D-CMCS and","rect":[313.4483947753906,319.399169921875,564.2042872903447,310.2615051269531]},{"page":5,"text":"Vfrac = 0.6 for P-PS, the Rp value for the minimum h½\u0003ðrÞ\u00044i is","rect":[313.4480895996094,332.1127624511719,564.244111943085,320.2862548828125]},{"page":5,"text":"larger than that based on I\u0003. Although there are conditions in","rect":[313.4471435546875,344.59686279296877,564.205324888001,334.4832763671875]},{"page":5,"text":"which the Rp values are similar for both indicators, I\u0003 gives","rect":[313.4481201171875,356.5485534667969,564.2605914352726,346.4349365234375]},{"page":5,"text":"better results than h½\u0003ðrÞ\u00044i under several conditions. Both I\u0003","rect":[313.44769287109377,368.50018310546877,563.7395171892243,356.1412658691406]},{"page":5,"text":"and h½\u0003ðrÞ\u00044i are indicators based on electron density: h½\u0003ðrÞ\u00044i","rect":[313.4483947753906,379.0510559082031,564.261690068085,368.0362548828125]},{"page":5,"text":"is","rect":[313.44775390625,390.0,319.65658752902257,382.2333068847656]},{"page":5,"text":"calculated","rect":[324.9996643066406,390.0,365.9786403176884,382.2333068847656]},{"page":5,"text":"from","rect":[371.3216857910156,390.0,390.8836720490741,382.3129577636719]},{"page":5,"text":"the","rect":[396.2376708984375,390.0,409.17285392263838,382.2333068847656]},{"page":5,"text":"entire","rect":[414.5288391113281,390.0,438.32824943045088,382.2333068847656]},{"page":5,"text":"electron","rect":[443.69219970703127,390.0,476.80438494659469,382.2333068847656]},{"page":5,"text":"calculated","rect":[313.4471130371094,400.87396240234377,354.4281032083134,394.1849365234375]},{"page":5,"text":"from","rect":[362.8262939453125,400.87396240234377,382.388280203371,394.26458740234377]},{"page":5,"text":"the","rect":[390.8004150390625,400.87396240234377,403.7356285808415,394.1849365234375]},{"page":5,"text":"maximum","rect":[412.093017578125,400.87396240234377,452.2370656037616,394.1849365234375]},{"page":5,"text":"and","rect":[460.62530517578127,400.87396240234377,475.7100856301884,394.1849365234375]},{"page":5,"text":"density, while I\u0003 is","rect":[482.19921875,392.4034729003906,564.244111943085,382.2332763671875]},{"page":5,"text":"minimum","rect":[484.0694885253906,400.87396240234377,522.5765431428241,394.1849365234375]},{"page":5,"text":"electron","rect":[531.0155639648438,400.87396240234377,564.1278712747197,394.1849365234375]},{"page":5,"text":"densities.","rect":[313.4471130371094,413.0,350.550905062043,406.1365966796875]},{"page":5,"text":"Therefore,","rect":[357.38970947265627,413.0,400.10028250344927,406.1365966796875]},{"page":5,"text":"h½\u0003ðrÞ\u00044i","rect":[406.94403076171877,414.8562316894531,438.88241760714757,403.8912048339844]},{"page":5,"text":"uses","rect":[445.7361145019531,412.8257141113281,462.89279358371007,408.31658935546877]},{"page":5,"text":"information","rect":[469.74749755859377,412.8257141113281,517.8956325051885,406.1366882324219]},{"page":5,"text":"from","rect":[524.7911376953125,412.8257141113281,544.3531544709491,406.2163391113281]},{"page":5,"text":"the","rect":[551.29345703125,412.8257141113281,564.2286095378727,406.1366882324219]},{"page":5,"text":"intermediate-electron-density region, which is not used in I\u0003.","rect":[313.44757080078127,428.20184326171877,564.2772844565743,418.0883483886719]},{"page":5,"text":"Figure 4","rect":[313.4483947753906,720.4310913085938,345.7443758344041,712.3145141601563]},{"page":5,"text":"Rp of the phase combinations with the minimum indicators in models","rect":[313.4483947753906,730.8426513671875,564.2203698323834,722.6873779296875]},{"page":5,"text":"b(oarsaendgoensq(uaa)rGes-)P,Sanadndh½\u0003(bðr)Þ\u0004G4i-C(sMkyC-bSluseurtfraicaensg:leI\u0003s)(.black circles) and IK","rect":[313.4481201171875,749.102294921875,563.5763162802579,731.8757934570313]},{"page":5,"text":"Acta Cryst. (2022). A78, 430–436","rect":[446.87060546875,772.2708740234375,564.3506555502972,764.9843139648438]},{"page":6,"text":"This may cause the phase retrieval results of I\u0003 to be better","rect":[44.794551849365237,81.83036041259766,295.46252624972569,71.6601333618164]},{"page":6,"text":"under several conditions.","rect":[44.79390335083008,90.30081939697266,147.02241690286335,83.6117935180664]},{"page":6,"text":"5.2. Phase retrieval for experimental data","rect":[44.794551849365237,124.00282287597656,217.99712889376543,115.1615219116211]},{"page":6,"text":"The experimental data of LLC bicontinuous cubic phases","rect":[54.76140213012695,140.1879425048828,295.5442218063663,131.41856384277345]},{"page":6,"text":"acquired in previous X-ray single-crystal structure analyses","rect":[44.794803619384769,152.13954162597657,295.5412921188663,143.3701629638672]},{"page":6,"text":"(Oka, 2017; Oka et al., 2018, 2020) were used as test data. In","rect":[44.794803619384769,164.14097595214845,295.5131862161259,155.0032958984375]},{"page":6,"text":"the","rect":[44.794803619384769,173.96249389648438,57.72998282888834,167.2734832763672]},{"page":6,"text":"previously derived from the model analysis were taken as the","rect":[44.794376373291019,187.9944610595703,295.51294181326338,179.22508239746095]},{"page":6,"text":"true combinations.","rect":[44.794376373291019,197.86575317382813,120.57632620462114,191.17674255371095]},{"page":6,"text":"Monoolein, a type of lipid, and water form three type II","rect":[54.7609748840332,211.8977813720703,295.54077332003819,203.12840270996095]},{"page":6,"text":"bicontinuous cubic phases with TPMSs of G, D and P (Oka,","rect":[44.794376373291019,223.8992156982422,295.49603445657427,214.76153564453126]},{"page":6,"text":"2017). The number of independent structure factors is from 8","rect":[44.794376373291019,235.7941131591797,295.5587792156071,226.65643310546876]},{"page":6,"text":"to 14. Vfrac can be estimated to be around 0.43 to 0.54 (Table","rect":[44.794376373291019,247.91439819335938,295.52051017263838,238.60809326171876]},{"page":6,"text":"S2). The scatter plot shows that IK and I\u0003 are roughly posi-","rect":[44.7940788269043,261.0486755371094,295.5054337023318,250.5596923828125]},{"page":6,"text":"tively correlated, and that the Rp value is smaller toward the","rect":[44.79404067993164,272.4111328125,295.5156883952946,262.8299865722656]},{"page":6,"text":"lower left [Fig. S3(a)]. This is the same as when Vfrac \b 0:6 in","rect":[44.79433059692383,283.71282958984377,295.55621600128219,274.463134765625]},{"page":6,"text":"the model densities. The phase combinations obtained from","rect":[44.793094635009769,295.5027770996094,295.5136464143085,286.7333679199219]},{"page":6,"text":"the structural models in the previous paper are also located at","rect":[44.793094635009769,307.4544372558594,295.5475479601443,298.6850280761719]},{"page":6,"text":"the lower left of the scatter plot. In the phase combinations","rect":[44.793094635009769,319.4060974121094,295.4569720505069,310.6366882324219]},{"page":6,"text":"with the minimum indicators, the calculated R values were","rect":[44.793094635009769,329.2206726074219,295.5200829265446,322.53155517578127]},{"page":6,"text":"p","rect":[239.14950561523438,332.1694641113281,242.87554365154507,327.952880859375]},{"page":6,"text":"almost zero (Table 1). This shows a good agreement between","rect":[44.79375076293945,343.3024597167969,295.56805682159469,334.164794921875]},{"page":6,"text":"the structure shown by the indicator-based phase retrieval","rect":[44.79375076293945,355.2043762207031,295.54025399673489,346.4349670410156]},{"page":6,"text":"method and the structural model in the previous paper (Oka,","rect":[44.79375076293945,367.20574951171877,295.4943865073555,358.0680847167969]},{"page":6,"text":"2017). For the two type II bicontinuous cubic phases of","rect":[44.79375076293945,379.15740966796877,295.4973770617068,370.0197448730469]},{"page":6,"text":"phytantriol with TPMSs of G and D, the number of inde-","rect":[44.79375076293945,391.059326171875,295.5541092394412,382.2899169921875]},{"page":6,"text":"pendent reflections is 21, and the Vfrac values are 0.57 and 0.66","rect":[44.79375076293945,403.1267395019531,295.4893517253727,394.1918029785156]},{"page":6,"text":"(Oka et al., 2018). These data are also similar to those of","rect":[44.79384231567383,415.01239013671877,295.49939122186307,405.8747253417969]},{"page":6,"text":"monoolein, and the structure shown by the phase retrieval","rect":[44.79384231567383,426.8575744628906,295.5412915943911,418.0881652832031]},{"page":6,"text":"method and the structure model in the previous paper (Oka et","rect":[44.79384231567383,438.85894775390627,295.54107797134426,429.7212829589844]},{"page":6,"text":"al., 2018) are in good agreement (Table 1).","rect":[44.79384231567383,450.81060791015627,220.4816606772774,441.6729431152344]},{"page":6,"text":"Hexaethylene","rect":[54.760440826416019,462.7124938964844,111.15854117849772,453.9430847167969]},{"page":6,"text":"glycol","rect":[117.44898986816406,462.7124938964844,141.2483991383364,453.9430847167969]},{"page":6,"text":"monododecyl","rect":[147.51795959472657,462.7124938964844,202.40535104263328,453.9430847167969]},{"page":6,"text":"ether","rect":[208.50772094726563,461.0,229.72073547824128,453.9430847167969]},{"page":6,"text":"(C12EO6)","rect":[235.97238159179688,462.7624206542969,274.3148822067569,453.62457275390627]},{"page":6,"text":"and","rect":[280.4898986816406,461.0,295.49502825714156,453.9432678222656]},{"page":6,"text":"water form a type I bicontinuous cubic phase of G TPMS, with","rect":[44.793521881103519,474.6643371582031,295.56976580596969,465.8949279785156]},{"page":6,"text":"Vfrac 0:72 (Oka et al., 2020). In the scatter plot of IK and I\u0003,","rect":[44.793521881103519,487.9601135253906,295.56695730813677,477.528076171875]},{"page":6,"text":"the distribution of points is split in two on the lower left side","rect":[44.79385757446289,498.5676574707031,295.50049064138838,489.7982482910156]},{"page":6,"text":"[Fig. S3(c)]. IK is a minimum near the point of the phase","rect":[44.79385757446289,510.6349792480469,295.5627770183415,501.4313659667969]},{"page":6,"text":"combination that was considered true in the previous paper","rect":[44.794559478759769,522.4141845703125,295.51913635714757,513.6448364257813]},{"page":6,"text":"(Oka et al., 2020) (I\u0003 20), whereas I\u0003 is not a minimum near","rect":[44.794559478759769,535.7667236328125,295.48166077121007,525.2779541015625]},{"page":6,"text":"the point. On the other hand, close to the point where I\u0003 is","rect":[44.79416275024414,547.718505859375,295.53366272433507,537.5482177734375]},{"page":6,"text":"corresponding electron densities are shown in Fig. 5. The","rect":[44.794132232666019,570.220947265625,295.5226464031071,561.4017944335938]},{"page":6,"text":"electron density of the minimum IK is in very good agreement","rect":[44.794132232666019,582.172607421875,295.5405899523318,573.4032592773438]},{"page":6,"text":"with the electron density in the previous paper (Oka et al.,","rect":[44.794315338134769,594.1740112304688,295.56744558938677,585.0363159179688]},{"page":6,"text":"2020). A comparison of the electron density of the minimum","rect":[44.79434585571289,606.1256713867188,295.52786760571476,596.9879760742188]},{"page":6,"text":"IK [Fig. 5(b)] with Figs. 1(b) and 1(c) indicates that the width","rect":[44.79434585571289,618.020751953125,295.50821185089156,608.883056640625]},{"page":6,"text":"Figure 5","rect":[44.794551849365237,729.8905029296875,77.03409446233377,721.77392578125]},{"page":6,"text":"Electron densities of C12EO6 calculated with phase combinations of (a)","rect":[44.794551849365237,739.6429443359375,295.57293941246157,731.8760375976563]},{"page":6,"text":"minimum I\u0003 and (b) minimum IK.","rect":[44.79411697387695,750.2025146484375,162.76690497889818,741.3353881835938]},{"page":6,"text":"Acta Cryst. (2022). A78, 430–436","rect":[44.794551849365237,772.2708740234375,162.21811770850037,764.9843139648438]},{"page":6,"text":"research papers","rect":[452.9285888671875,50.97494125366211,562.962778580549,36.115684509277347]},{"page":6,"text":"of the high-electron-density region on the TPMS side is wider,","rect":[313.4481201171875,80.42744445800781,564.2840593589181,71.6580581665039]},{"page":6,"text":"which is consistent with the large volume fraction on the","rect":[313.4481201171875,92.37910461425781,564.2243981120914,83.6097183227539]},{"page":6,"text":"TPMS side of C12EO6. With the minimum I\u0003, the high-","rect":[313.4481201171875,105.73368072509766,564.1757217882694,95.5613784790039]},{"page":6,"text":"electron-density region on the TPMS side is narrower and the","rect":[313.4473876953125,116.28437805175781,564.2217125652164,107.5149917602539]},{"page":6,"text":"electron-density","rect":[313.4473876953125,128.2360382080078,378.85473868826338,119.4666519165039]},{"page":6,"text":"regions","rect":[384.91033935546877,128.2360382080078,414.5244769333194,119.4666519165039]},{"page":6,"text":"on","rect":[420.5293273925781,127.0,430.87808489776656,121.0]},{"page":6,"text":"the","rect":[436.95257568359377,127.0,449.8877587077946,119.4666519165039]},{"page":6,"text":"network","rect":[455.9234619140625,127.0,489.53424334503219,119.4666519165039]},{"page":6,"text":"sides","rect":[495.6197204589844,127.0,515.2803057907413,119.4666519165039]},{"page":6,"text":"are","rect":[521.3856201171875,127.0,534.3208336589664,121.0]},{"page":6,"text":"larger","rect":[540.3565673828125,128.2360382080078,564.2515582321476,119.4666519165039]},{"page":6,"text":"compared with the minimum IK. If the volume fraction is","rect":[313.4473876953125,140.1876983642578,564.2396563766788,131.41831970214845]},{"page":6,"text":"known in advance, then the electron density indicates that the","rect":[313.4476318359375,152.13954162597657,564.2229943034977,143.3701629638672]},{"page":6,"text":"phase combination with the minimum I\u0003 is not close to the","rect":[313.4476318359375,165.43540954589845,564.2290367839664,155.3218231201172]},{"page":6,"text":"true phase combination.","rect":[313.447998046875,175.9861602783203,412.4203203452461,167.21678161621095]},{"page":6,"text":"The fourth moment of the electron density h½\u0003ðrÞ\u00044i was also","rect":[323.4713134765625,187.93775939941407,564.2191798684697,176.9229736328125]},{"page":6,"text":"calculated","rect":[313.44708251953127,197.80911254882813,354.4280726907353,191.12010192871095]},{"page":6,"text":"in","rect":[360.5045471191406,198.0,368.2658412454228,191.12010192871095]},{"page":6,"text":"the","rect":[374.3214416503906,197.80911254882813,387.2566246745915,191.12010192871095]},{"page":6,"text":"experimental","rect":[393.3490295410156,199.8894805908203,446.6541761158755,191.12010192871095]},{"page":6,"text":"data,","rect":[452.7515563964844,198.0,472.6210649741524,191.12010192871095]},{"page":6,"text":"and","rect":[478.6876220703125,198.0,493.6927516458134,191.12010192871095]},{"page":6,"text":"the","rect":[499.75433349609377,197.80911254882813,512.6895470378727,191.12010192871095]},{"page":6,"text":"results","rect":[518.781982421875,198.0,545.1678790329288,191.12010192871095]},{"page":6,"text":"are","rect":[551.2861938476563,198.0,564.2214073894352,193.0]},{"page":6,"text":"summarized in Table 1. The Rp values for the phase combi-","rect":[313.44708251953127,212.65284729003907,564.1817032335819,203.07176208496095]},{"page":6,"text":"nation with the minimum indicator h½\u0003ðrÞ\u00044i are comparable","rect":[313.4473876953125,223.7928009033203,564.1882042644352,212.77783203125]},{"page":6,"text":"with those of I\u0003 for four of the six experimental data sets. On","rect":[313.4479675292969,237.0885467529297,564.1823146340947,226.9750213623047]},{"page":6,"text":"the other hand, the Rp value of the minimum indicator h½\u0003ðrÞ\u00044i","rect":[313.447998046875,248.5078887939453,564.261690068085,236.68121337890626]},{"page":6,"text":"is approximately 0.1 for monoolein P and 0.2 for D, which are","rect":[313.44775390625,259.6476135253906,564.2220787761539,250.8782196044922]},{"page":6,"text":"larger than that of I\u0003. For the difference of Rp","rect":[313.44775390625,272.9436340332031,516.2863407706857,262.7731018066406]},{"page":6,"text":"0.1, the","rect":[532.8906860351563,269.4623718261719,564.2294640300602,262.7733459472656]},{"page":6,"text":"electron densities show similar structures, but there are non-","rect":[313.44744873046877,281.4140319824219,564.1918961046756,274.7250061035156]},{"page":6,"text":"negligible differences in the details (Fig. S6). Considering the","rect":[313.44744873046877,295.4958190917969,564.2187828777164,286.358154296875]},{"page":6,"text":"results with the model structures, IK is the best indicator with","rect":[313.44744873046877,306.833740234375,564.1978785989385,298.6283264160156]},{"page":6,"text":"the widest range of applicable conditions, followed by I\u0003 and","rect":[313.44769287109377,320.693603515625,564.205080747376,310.58001708984377]},{"page":6,"text":"then","rect":[313.4478759765625,329.2206726074219,331.6246058938603,322.5316467285156]},{"page":6,"text":"h½\u0003ðrÞ\u00044i.","rect":[336.779541015625,331.25128173828127,370.9984453452461,320.2862548828125]},{"page":6,"text":"Regardless","rect":[376.0817565917969,331.3010559082031,420.6692218063663,322.5316467285156]},{"page":6,"text":"of","rect":[425.8580322265625,330.0,434.1368118761599,322.6112976074219]},{"page":6,"text":"which","rect":[439.2788391113281,330.0,463.0712916848759,322.5316467285156]},{"page":6,"text":"indicator","rect":[468.21630859375,330.0,504.5697345016788,322.5316467285156]},{"page":6,"text":"is","rect":[509.66796875,330.0,515.8778399704288,322.5316467285156]},{"page":6,"text":"used,","rect":[520.9949951171875,330.0,541.9820574057931,322.5316467285156]},{"page":6,"text":"care","rect":[547.1011962890625,330.0,564.1753258464664,324.0]},{"page":6,"text":"should be taken because there are volume fractions for which","rect":[313.44793701171877,341.1723327636719,564.1715724465947,334.4833068847656]},{"page":6,"text":"phase retrieval is difficult.","rect":[313.44793701171877,355.2043762207031,419.6481645835274,346.4349670410156]},{"page":6,"text":"Unobservable structure factors cause truncation errors in","rect":[323.4712829589844,365.0786437988281,564.2003810403447,358.32989501953127]},{"page":6,"text":"the Fourier synthesis of electron density, \u0003ðrÞ. The truncation","rect":[313.44793701171877,379.0509338378906,564.217043638001,369.8037414550781]},{"page":6,"text":"errors may make the indicatorsinaccurate. However,given the","rect":[313.44793701171877,391.0025634765625,564.2223229167789,382.233154296875]},{"page":6,"text":"observed amplitudes, the amplitudes of unobservable struc-","rect":[313.44793701171877,402.9542236328125,564.2103287218631,394.184814453125]},{"page":6,"text":"ture factors are probably less than 1% of the maximum","rect":[313.44793701171877,414.9058837890625,564.2033131623554,406.136474609375]},{"page":6,"text":"amplitude. The amplitude decreases rapidly at high scattering","rect":[313.44793701171877,426.8575134277344,564.2083458659977,418.0881042480469]},{"page":6,"text":"angles due to the structural disorder of the liquid crystal (this","rect":[313.44793701171877,438.85888671875,564.1834429977726,429.7212219238281]},{"page":6,"text":"is why the number of observable amplitudes is small). For this","rect":[313.44793701171877,450.810546875,564.188447880585,441.6728820800781]},{"page":6,"text":"reason, the effect of the truncation error on the resultant","rect":[313.44793701171877,461.0,564.2123428820194,453.8863220214844]},{"page":6,"text":"structure model, i.e. indicators, seems insignificant. On the","rect":[313.44793701171877,474.60736083984377,564.2240319011539,465.83795166015627]},{"page":6,"text":"other hand, it may affect the phase retrieval of structure","rect":[313.44793701171877,486.5589904785156,564.1188072917789,477.7895812988281]},{"page":6,"text":"factors with weak amplitudes","rect":[313.44793701171877,498.5106506347656,440.64718811496007,489.7412414550781]},{"page":6,"text":"deviate slightly from 0.","rect":[313.44732666015627,510.46270751953127,407.7015703452461,501.69329833984377]},{"page":6,"text":"6. Conclusion","rect":[313.4483947753906,540.3280029296875,376.494977063127,532.531005859375]},{"page":6,"text":"The indicators I\u0003 and IK were determined to be useful for the","rect":[313.4483947753906,559.613525390625,564.2264122722477,549.4999389648438]},{"page":6,"text":"crystallographic phase retrieval in LLC bicontinuous phases.","rect":[313.4483337402344,570.220947265625,564.2674577964181,561.4515991210938]},{"page":6,"text":"The phase combination with the minimum indicator was very","rect":[313.4483337402344,582.172607421875,564.1808800456852,573.4032592773438]},{"page":6,"text":"close to the true phase combination when the volume fraction","rect":[313.4483337402344,594.124267578125,564.1609523294072,585.3549194335938]},{"page":6,"text":"was less than 0.6. Even for volume fractions of approximately","rect":[313.4483337402344,606.075927734375,564.2514977214664,597.3065795898438]},{"page":6,"text":"0.7 to 0.8, IK was effective to determine the proper phase","rect":[313.4483337402344,618.1432495117188,564.1521324870914,609.2015991210938]},{"page":6,"text":"combination as an indicator, although there were exceptions.","rect":[313.4476318359375,629.922607421875,564.2677019370431,621.1532592773438]},{"page":6,"text":"In the scatter plots of IK and I\u0003, when Vfrac is small, the points","rect":[313.4476318359375,643.2750244140625,564.1780108688663,633.1048583984375]},{"page":6,"text":"extending to the lower left do not spread much [Fig. 3(a)]. On","rect":[313.44769287109377,653.8756713867188,564.1879298684697,644.7379760742188]},{"page":6,"text":"the other hand, when Vfrac is large, the points in the lower left","rect":[313.44769287109377,665.939208984375,564.1776749132694,657.0081787109375]},{"page":6,"text":"corner branch and spread out [Fig. 3(b)], which seems to be","rect":[313.4483642578125,677.779052734375,564.2383751628727,668.641357421875]},{"page":6,"text":"related to the certainty of the indicator.","rect":[313.4473876953125,689.680908203125,476.88958914407427,680.9115600585938]},{"page":6,"text":"The model used for the phase retrieval was a simple","rect":[323.470703125,701.632568359375,564.1559777019352,692.8632202148438]},{"page":6,"text":"structure with different electron densities on the TPMS side","rect":[313.4473876953125,711.5068359375,564.2107262370914,704.7581176757813]},{"page":6,"text":"and the network side. On the other hand, the sample of","rect":[313.4473876953125,725.4791259765625,564.2117325304569,716.7097778320313]},{"page":6,"text":"experimental data used for the phase retrieval has a more","rect":[313.4473876953125,737.4307250976563,564.1918053386539,728.661376953125]},{"page":6,"text":"complicated structure. In the case of a type I bicontinuous","rect":[313.4473876953125,749.3823852539063,564.1998004196476,740.613037109375]},{"page":6,"text":"Toshihiko Oka \u0002 Liquid crystal bicontinuous phases 435","rect":[352.1836853027344,772.2470092773438,564.2215909491334,762.6353149414063]},{"page":7,"text":"research papers","rect":[44.794551849365237,50.97494125366211,154.82873011863493,36.115684509277347]},{"page":7,"text":"cubic phase such as C12EO6, the polar region is located on the","rect":[44.794551849365237,80.42951965332031,295.514559244904,71.6601333618164]},{"page":7,"text":"TPMS side and the nonpolar region is located on the network","rect":[44.7941780090332,92.38117980957031,295.52862811065719,83.6117935180664]},{"page":7,"text":"side. The hydrophobic chain of the amphiphilic molecule is","rect":[44.7941780090332,104.33283996582031,295.5256976364444,95.5634536743164]},{"page":7,"text":"located in the nonpolarregion, water is located in the TPMS in","rect":[44.7941780090332,116.28450012207031,295.54959368682906,107.5151138305664]},{"page":7,"text":"the polar region, and the hydrophilic part of the amphiphilic","rect":[44.7941780090332,128.2361602783203,295.44412441235388,119.4667739868164]},{"page":7,"text":"molecule is located in the polar–nonpolar interface side of the","rect":[44.7941780090332,140.1878204345703,295.5087609050602,131.41844177246095]},{"page":7,"text":"polar region. Therefore, the electron density differs among the","rect":[44.7941780090332,152.1393585205078,295.5097985027165,143.36997985839845]},{"page":7,"text":"three regions, with the hydrophilic part of the amphiphilic","rect":[44.7941780090332,164.0910186767578,295.44607753735388,155.32164001464845]},{"page":7,"text":"molecule having the highest electron density [Fig. 5(b)]. Phase","rect":[44.7941780090332,176.03575134277345,295.566317057404,166.8980712890625]},{"page":7,"text":"retrieval was possible even in such a system.","rect":[44.794193267822269,187.93763732910157,227.16655142434773,179.1682586669922]},{"page":7,"text":"It takes a long time to calculate the indicators of electron","rect":[54.76079177856445,199.8892364501953,295.51776385284469,191.11985778808595]},{"page":7,"text":"densities for all phase combinations; especially for IK, it was","rect":[44.794193267822269,211.95684814453126,295.50900452121007,203.0715789794922]},{"page":7,"text":"necessary to calculate the eigenvalues of the Hessian matrix at","rect":[44.79361343383789,223.7928009033203,295.5461136339724,215.02342224121095]},{"page":7,"text":"each voxel, which takes a long time. On the other hand, the","rect":[44.79361343383789,235.7444610595703,295.5151695964665,226.97508239746095]},{"page":7,"text":"structures of the LLC bicontinuous cubic phase have highly","rect":[44.79361343383789,247.6961212158203,295.5579247234196,238.92674255371095]},{"page":7,"text":"symmetric space groups of Ia3d, Im3m and Pn3m. Therefore,","rect":[44.79361343383789,259.6477355957031,295.5627458823555,250.8783416748047]},{"page":7,"text":"the computational time can be reduced by calculating only in","rect":[44.79263687133789,271.5426330566406,295.54803729034469,262.7732238769531]},{"page":7,"text":"the asymmetric units. In addition, the Babinet principle does","rect":[44.79263687133789,283.4942932128906,295.48239319308507,274.7248840332031]},{"page":7,"text":"not allow positive and negative electron densities to be","rect":[44.79263687133789,295.4459533691406,295.5181603191227,286.6765441894531]},{"page":7,"text":"distinguished in this paper; therefore, even when they are","rect":[44.79263687133789,307.3976135253906,295.5101952312321,298.6282043457031]},{"page":7,"text":"reversed, the number of combinations can be reduced. For an","rect":[44.79263687133789,317.2688903808594,295.5241420266728,310.5798645019531]},{"page":7,"text":"LLC bicontinuous cubic phase, the largest number of inde-","rect":[44.79263687133789,331.3008728027344,295.5480667589724,322.5314636230469]},{"page":7,"text":"pendent structure factors so far is 21 for phytantriol and","rect":[44.79263687133789,343.2525329589844,295.4853541848759,334.4831237792969]},{"page":7,"text":"C12EO6–8 (Oka et al., 2018, 2020). When the number of voxels","rect":[44.79263687133789,355.25421142578127,295.4777240036319,346.1165466308594]},{"page":7,"text":"was 323, the computation times of the Python script were 16 to","rect":[44.79416275024414,367.0994567871094,295.55682635284469,356.55462646484377]},{"page":7,"text":"20 min using an AMD 3950X CPU with 16 cores and 32","rect":[44.79452133178711,379.05108642578127,295.52099845388838,370.2319030761719]},{"page":7,"text":"threads. Calculation of 230 phase combinations under the same","rect":[44.79452133178711,391.002685546875,295.56396720388838,380.4578552246094]},{"page":7,"text":"conditions would thus take 10 to 14 days. It may be possible to","rect":[44.793880462646487,402.954345703125,295.56717181182906,394.1849365234375]},{"page":7,"text":"increase the calculation speed by revising the algorithm. If the","rect":[44.793880462646487,414.906005859375,295.5094628093571,406.1365966796875]},{"page":7,"text":"IK calculation is eliminated, then the calculation time will be","rect":[44.793880462646487,426.2936706542969,295.5219444988102,418.08831787109377]},{"page":7,"text":"much shorter.","rect":[44.79444885253906,436.7289733886719,101.24627859719927,430.0399475097656]},{"page":7,"text":"Phase retrieval was performed using models and experi-","rect":[54.761043548583987,450.7610168457031,295.4622513292849,441.9916076660156]},{"page":7,"text":"mental data of LLC bicontinuous cubic phases. TPMS-like","rect":[44.79444885253906,462.65594482421877,295.5607628581852,453.88653564453127]},{"page":7,"text":"structures have been observed in thermotropic liquid crystals,","rect":[44.79444885253906,474.60760498046877,295.5588396323555,465.83819580078127]},{"page":7,"text":"polymers and other systems; it should also be possible to apply","rect":[44.79444885253906,486.5592041015625,295.4792198894352,477.789794921875]},{"page":7,"text":"this phase retrieval method to these systems.","rect":[44.79444885253906,498.5108642578125,228.34109671243366,489.741455078125]},{"page":7,"text":"436 Toshihiko Oka \u0002 Liquid crystal bicontinuous phases","rect":[44.794551849365237,772.2470092773438,256.8675980616458,762.5875244140625]},{"page":7,"text":"Funding information","rect":[313.4483947753906,80.45883178710938,407.125348156877,70.92682647705078]},{"page":7,"text":"The following funding is acknowledged:","rect":[313.4483947753906,96.68605041503906,485.89496023782427,87.91666412353516]},{"page":7,"text":"(grant Nos. 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