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ERROR: type should be string, got "https://doi.org/10.1007/s11051-021-05220-6\nJ Nanopart Res (2021) 23: 120 https://doi.org/10.1007/s11051-021-05220-6\nJ Nanopart Res (2021) 23: 120 RESEARCH PAPER Dynamic light scattering distributions by any means Natalia Farkas\n& John A. Kramar Natalia Farkas\n& John A. Kramar Received: 20 February 2021 /Accepted: 22 April 2021\n# The Author(s) 2021\n/ Published online: 21 May 2021 Received: 20 February 2021 /Accepted: 22 April 2021\n# The Author(s) 2021\n/ Published online: 21 May 2021 Received: 20 February 2021 /Accepted: 22 April 2021\n# The Author(s) 2021\n/ Published online: 21 May 2021 identify and validate the harmonic mean as the central\nvalue of the intensity-weighted DLS size distribution that\nexpresses the inversion results consistently with the\ncumulant results. We also investigate the extent to which\nthe DLS polydispersity descriptors are representative of\nthe distributional quality and find them to be unreliable\nand misleading, both for monodisperse reference mate-\nrials and broad-distribution biomedical nanoparticles. These results overall are intended to bring essential im-\nprovements and to stimulate reexamination of the metro-\nlogical capabilities and role of DLS in nanoparticle\ncharacterization. Abstract Dynamic light scattering (DLS) is an essential\ntechnique for nanoparticle size analysis and has been\nemployed extensively for decades, but despite its long\nhistory and popularity, the choice of weighting and mean\nof the size distribution often appears to be picked ad hoc\nto bring the results into agreement with other methods\nand expectations by any means necessary. Here, we\ncritically discuss the application of DLS for nanoparticle\ncharacterization and provide much-needed clarification\nfor ambiguities in the mean-value practice of commercial\nDLS software and documentary standards. We address\nthe misleading way DLS size distributions are often\npresented, that is, as a logarithmically scaled histogram\nof measured relative quantities. Central values obtained\nincautiously from this representation often lead to signif-\nicant interpretation errors. Through the measurement of\nmonomodal nanoparticle samples having an extensive\nrange of sizes (5 to 250 nm) and polydispersity, we\nsimilarly demonstrate that the default outputs of a fre-\nquently used DLS inversion method are ill chosen, as\nthey are regularizer-dependent and significantly deviate\nfrom the cumulant z-average size. The resulting discrep-\nancies are typically larger than 15% depending on the\npolydispersity index of the samples. We explicitly Keywords Dynamic light scattering . Harmonic mean . NNLS inversion . Size distribution . Moment-ratio\ntransformation . Polydispersity index . Nanoparticles N. Farkas\nTheiss Research, 7411 Eads Ave, La Jolla, CA 92037, USA Introduction The significance of nanoparticles is apparent and indis-\nputable in a broad range of scientific and technical\nfields. They substantially enhance the performance of\nmany industrial and research-oriented products. Suc-\ncessful realization of such products is critically depen-\ndent on reliable dimensional measurements to correlate\nnanometer-scale physical-chemical properties with their\nintended functionality. Dynamic light scattering (DLS)\nis the method of choice in most cases. In recent years,\nfor instance, more than 50% of drug products containing\nnanomaterials submitted to the Center for Drug Evalu-\nation and Research within the US Food and Drug N. Farkas\nTheiss Research, 7411 Eads Ave, La Jolla, CA 92037, USA N. Farkas\nTheiss Research, 7411 Eads Ave, La Jolla, CA 92037, USA N. Farkas (*): J. A. Kramar\nNational Institute of Standards and Technology, Gaithersburg,\nMD 20899, USA\ne-mail: natalia.farkas@nist.gov J Nanopart Res (2021) 23: 120 120 Page 2 of 11 particle interactions, heterogeneity in particle size,\nshape, and composition, are adequately described in\nthe literature (ISO 22412:2017; Bhattacharjee 2016)\nand will not be considered here. First, we address the\nmisleading way DLS size distributions are often pre-\nsented, i.e., as a histogram of measured relative quanti-\nties based on a logarithmic scale. Obtaining central\nvalues incautiously from such graphical representation\nleads to significant interpretation errors. Based on ex-\ntensive measurements, we also demonstrate that the\ndefault output of the non-negatively constrained least-\nsquares (NNLS) inversion method given by common\nDLS analysis software is ill chosen, since it is\nregularizer-dependent and significantly deviates from\nthe cumulant z-average size. The resulting errors are\ntypically larger than 15% depending on the polydisper-\nsity index (PDI) of the samples. We explicitly identify\nand validate the harmonic mean as the physical-\nmeasurement-justified central value of the intensity-\nweighted DLS size distribution. Quantitative agreement\nbetween the cumulant and inversion results originating\nfrom the same DLS data is demonstrated. Importantly,\nwe investigate the extent to which the DLS polydisper-\nsity descriptors are representative of the distributional\nquality and find them to be completely unreliable and\nmisleading, both for reference materials and biomedical\nnanoparticles. In summary, the results critically advo-\ncate for essential improvements in existing DLS data\ninterpretation practices that behoove strong consider-\nation for adoption by the nano-bio community, regula-\ntory agencies, and instrument manufacturers. Administration relied on DLS for size characterization\n(D’Mello et al. 2017). Introduction Despite its overwhelming domi-\nnance in nanoparticle characterization, consistency and\nreproducibility of the DLS measurements remain a sig-\nnificant challenge, mainly because of questionable data\nanalysis and interpretation. In current practice, the choice of weighting and means\nof the size distribution by DLS is often unjustified or\nappears to be picked ad hoc (Bazylińska 2017; Myerson\net al. 2018; Takechi-Haraya et al. 2018; Chen et al. 2017;\nDeLoid et al. 2017). Ambiguities in mean-value terminol-\nogy are also prevalent in commercial DLS software and\ndocumentary standards (Zetasizer nano user manual 2013;\nASTM E2490-09:2015; ISO 22412:2017). A common\ndilemma is that the DLS software typically offers several\ntypes of central values based on intensity, volume, or\nnumber weighting as the output of the measurement\n(Bhattacharjee 2016). Unfortunately, however, instrument\nmanufacturers (Nobbmann and Morfesis 2009), best prac-\ntice guides (Nicolas et al. 2018), and documentary stan-\ndards (ASTM E2490-09:2015; ISO 22412:2017) fre-\nquently offer vague, inadequate, or even incorrect guid-\nance. For instance, the definition of the DLS size in the\ncurrent ISO 22412 (2017) as the “central value of the\nunderlying particle size distribution” is ambiguous, i.e.,\nwhether it refers to the mean, median, mode, or other\ncentral tendencies. Thus, the nano-bio community is left\nwith insufficient guidelines which lead to serious data\nanalysis and interpretation errors with implications beyond\njust irreproducibility, especially when the key conclusions\nof a study are critically dependent on such choices. Avail-\nable review articles merely identify some of the deficien-\ncies of DLS characterization and only briefly mention the\nrisk of conversion error between weighted means\n(Bhattacharjee 2016). In fact, they are frequently cited as\nan explanation for disagreement with other methods or\nwith expectations with no solution offered, even in cases\nwhere the inconsistent DLS results are simply due to\nquestionable data analysis or the lack of metrological rigor. The DLS measurement process A DLS measurement samples laser light scattered from\nthe simultaneous Brownian motion of a large ensemble\nof particles in solution. The fundamental quantity mea-\nsured is the time autocorrelation function (ACF) of the\nscattered light intensity. Decay rates for the ensemble of\nparticles are embedded within the ACF and are related\nto an apparent mean diffusion constant, which is then\nconverted to particle diameter through the Stokes-\nEinstein equation. Since the diameter is derived from\nscattered intensity and is inversely proportional to the\ndecay rate, the DLS-measured average hydrodynamic\nsize is naturally presented as an intensity-weighted Considering the topical urgency generated by the\naforesaid state of DLS characterization practice for\nnano-bio materials in particular, we herein present a\ncritical discussion of the data analysis and interpretation\nchallenges affecting DLS dimensional metrology. We\nprovide a complete and systematic articulation of how\nthese mathematical and statistical measurement issues\ncould lead to significant and often unrecognized errors. Sample and sample preparation related measurement\nlimitations, such as multiple-scattering effects, particle- J Nanopart Res (2021) 23: 120 Page 3 of 11 120 120 harmonic mean. There are two approaches commonly\nused to analyze the ACF: cumulant analysis and direct\ninversion—typically an NNLS inversion method\n(Zetasizer nano user manual 2013). In the cumulant\nmethod, the intensity-weighted, harmonic-averaged di-\nameter is referred to as the z-average and is calculated\nfrom the first cumulant of the ACF. A distributional\nwidth parameter, the PDI, is derived from a ratio of the\nfirst and second cumulants. The NNLS approach poses\na set of particle radii and the corresponding set of decay\nrates and performs a least-squares fitting to obtain the\nweightings for a logarithmically scaled histogram of\nparticle sizes. The theory and practice of the cumulant\nmethod are generally well documented and provide\ndeterministic and reliable size estimates for particle\nsamples with a coefficient of variation (CV) less than\n3% in an appropriate solution environment in terms of\ndilution and salt concentration. Inversion methods, on\nthe other hand, are mathematically ill posed and highly\ndependent on the inversion parameters. et al. 1999) well-approximated by such a form. Fig. 1a\ndemonstrates that reading off the mode from the loga-\nrithmically scaled histogram of the measured relative\nquantities leads to an overestimation of the actual value\nby a factor of exp(σ2), which specifically translates to a\n10% error in the case of this particular typical sample of\n100-nm mean-diameter gold nanoparticles. The DLS measurement process It is generally accepted that the NNLS distributions\nare broad due to the data weighting and smoothing\nnecessary for dealing with the inherent noise in the\nACF during the inversion process (Xu et al. 2018), but\nthe fact that the broadness of the distribution is\nregularizer-dependent is largely ignored. With simple\nmodification of the regularizer value (or other inversion\nparameters, such as the size range and the number of\nbins), very different size distributions may be produced\nfrom the same ACF data. This is demonstrated for the\n100-nm gold nanoparticle sample in Fig. 1b where a\nsingle set of ACF data is analyzed using a range of\nparameters in the inversion algorithm. With prior\nknowledge, it is possible to achieve good agreement\nbetween the NNLS distribution and atomic force micro-\nscope (AFM) size distribution (transformed to intensity-\nweighted for comparison). However, without prior\nknowledge, it is impossible to confidently obtain the\ncorrect width of the distribution from DLS measure-\nments (i.e., the shape-parameter in the case of a lognor-\nmal distribution). If left uncorrected, the impact of an\nincorrect—almost always too large—shape-parameter\nis amplified dramatically during conversions between\nweighted means (ISO 9276-2:2014), which has impor-\ntant consequences for data interpretation. Parametrization and graphical representation of DLS\ndata To understand the analysis of DLS data and avoid\nconfusion over “which mean do you mean?”, we must\nfirst address the easily misunderstood way NNLS size\ndistributions are characteristically presented, that is, as\nlogarithmically scaled relative quantities displayed as a\nhistogram. Simply changing the logarithmic scale to\nlinear is incorrect, as is obtaining central values directly\nfrom such graphical representations (Hess 2004). These\npractices are, however, frequent in current DLS studies\n(DeLoid et al. 2017; Yeap et al. 2018; Maguire et al. 2018) and often lead to questionable study results. To\nprevent interpretation errors, the logarithmically spaced\ndiscrete data—the typical output of NNLS software—\nneeds to be transformed to a linearly scaled density\ndistribution as described in ISO 9276-1 (1998).We\nshow an example of a correct transformation for the\noutput from a NNLS inversion method analysis in Fig. 1a. Here, we utilize as a mathematical convenience the\nmodel assumption of a lognormal distribution, in terms\nof a scale-parameter (exp(μ)) and a shape-parameter (σ)\nrelated to the distribution width; however, the described\ngeneral trends in mean transformations apply to all\nmonomodal distributions. The lognormal is appropriate\nhere since both the NNLS-output distributions and typ-\nical actual particle size distributions are in general (Kiss Width-error and its consequence on transformation The AFM distribution was transformed from number to\nintensity-weighted and fitted with a lognormal function. All of the\nplotted DLS distributions (solid lines in different colors) are gen-\nerated from the same ACF measurement data with only a readjust-\nment of the parameters of the NNLS inversion algorithm. The red\nline (flattest curve) represents the density distribution obtained\nwith the default “general purpose” algorithm from a commonly\nused instrument (0.4 nm to 10 μm size range, 70 size bins, 0.01\nregularizer value). Good agreement is achievable between the\nNNLS and AFM size distributions, but only by the contortions\nof limiting the NNLS-inversion parameters to a size range of 10 to\n500 nm and using 100 size bins and a regularizer value of 0.001\n(black line) Fig. 1 a Graphical representation of the intensity-weighted size Fig. 1 a Graphical representation of the intensity-weighted size\ndata obtained for 100-nm gold nanoparticles by the NNLS inver-\nsion method. Histograms are of the measured relative quantities\n(ΔQ6) (open circles) and the corresponding density distribution\n(q6) (solid circles) calculated by dividing the relative quantities by\nthe corresponding logarithmically scaled size intervals. The solid\nline is a lognormal fit, in terms of a scale-parameter (exp(μ)) and a\nshape-parameter (σ), to the density distribution. The dotted line is\nonly to guide the eye between the discrete NNLS data points. Note\nthat the most frequently occurring size class in the logarithmically\nscaled histogram of the relative quantities differs from the actual\nmode (M) of the size distribution by a factor of exp(σ2). b Com-\nparison of intensity-weighted density distributions obtained for\n100-nm gold nanoparticles by AFM (dotted line) and DLS (solid lines). The AFM distribution was transformed from number to\nintensity-weighted and fitted with a lognormal function. All of the\nplotted DLS distributions (solid lines in different colors) are gen-\nerated from the same ACF measurement data with only a readjust-\nment of the parameters of the NNLS inversion algorithm. The red\nline (flattest curve) represents the density distribution obtained\nwith the default “general purpose” algorithm from a commonly\nused instrument (0.4 nm to 10 μm size range, 70 size bins, 0.01\nregularizer value). Width-error and its consequence on transformation Good agreement is achievable between the\nNNLS and AFM size distributions, but only by the contortions\nof limiting the NNLS-inversion parameters to a size range of 10 to\n500 nm and using 100 size bins and a regularizer value of 0.001\n(black line) by microscopy is only 0.2%, while it is 25% when calcu-\nlated from the NNLS-derived distribution using a typical\ndefault “general purpose” algorithm (Fig. 2). The larger the\nwidth-error, the broader the size range that is obtainable by\nchoosing different weightings, even for ideal standard\nparticles. Ad hoc or unjustified selection of these size\nvalues to align with other methods is seemingly a common\npractice especially in the case of industrial and pharmaceu-\ntical materials (Bazylińska 2017; Myerson et al. 2018;\nTakechi-Haraya et al. 2018; Chen et al. 2017; DeLoid\net al. 2017). DLS-measured diameter as an input, the study proposes to\nestimate the number of lipid and cytokine molecules per\nliposome. However, given the findings in Fig. 2, the choice\nof the number-weighted average size in this particular\nwork would introduce a bias in the calculated results. The latest revisions of ASTM (ASTM E3247-20:2020)\nand ISO (ISO 22412:2017) standards specifically depre-\ncate the use of number distributions, especially for\nmethods involving smoothing in the process of extracting\nthe intensity-weighted size distribution. Although ac-\nknowledged, the warning is often ignored by the measure-\nment community (Coleman et al. 2011; Di Francesco and\nBorchard 2018; Liu et al. 2019; Ulasov et al. 2011; Chen\net al. 2019). One of the arguments is that the effect of the\nshape-parameter on the transformation from intensity to\nvolume or number is minimal for particles with narrow\nsize distributions (ASTM E2490-09:2015). This is certain-\nly the case when the shape-parameter reflects the actual\nnarrow distribution as it does for the microscopy data, but\ndefinitely not for the vast majority of DLS-derived esti-\nmates where substantial width-error is unequivocally pres-\nent when using default parameters. For instance, the dif-\nference between number- and intensity-weighted arithmet-\nic means calculated from the size distribution of NIST\nStandard Reference Material® SRM 1963 (National\nInstitute of Standards and Technology 2003) we obtain Historically, the effect of the width-error was first\nencountered in the work of Thomas over 30 years ago\n(Thomas 1987), before the widespread use of inversion\nmethods. Width-error and its consequence on transformation We show in Fig. 2 that the typical positive error in the\nshape-parameter, which we refer to as width-error, will\nalmost always lead to significant underestimation when\ntransforming from the intensity-weighted to the volume-\nand number-weighted central values of the distribution. The calculated central values of the transformed NNLS\nnumber-weighted distribution wildly undershoot the cor-\nresponding size estimates obtained by microscopy even for\ncertified reference particles with narrow distributions. For\nexample, this discrepancy has been reported in a recent\nstudy involving unilamellar liposomes (Barberio et al. 2020). In this case, the number-weighted arithmetic mean\nis shown to underestimate the corresponding z-average\nsize of the liposomes by as much as 35%. Using the J Nanopart Res (2021) 23: 120 120 Page 4 of 11 Fig. 1 a Graphical representation of the intensity-weighted size\ndata obtained for 100-nm gold nanoparticles by the NNLS inver-\nsion method. Histograms are of the measured relative quantities\n(ΔQ6) (open circles) and the corresponding density distribution\n(q6) (solid circles) calculated by dividing the relative quantities by\nthe corresponding logarithmically scaled size intervals. The solid\nline is a lognormal fit, in terms of a scale-parameter (exp(μ)) and a\nshape-parameter (σ), to the density distribution. The dotted line is\nonly to guide the eye between the discrete NNLS data points. Note\nthat the most frequently occurring size class in the logarithmically\nscaled histogram of the relative quantities differs from the actual\nmode (M) of the size distribution by a factor of exp(σ2). b Com-\nparison of intensity-weighted density distributions obtained for\n100-nm gold nanoparticles by AFM (dotted line) and DLS (solid\nlines). The AFM distribution was transformed from number to\nintensity-weighted and fitted with a lognormal function. All of the\nplotted DLS distributions (solid lines in different colors) are gen-\nerated from the same ACF measurement data with only a readjust-\nment of the parameters of the NNLS inversion algorithm. The red\nline (flattest curve) represents the density distribution obtained\nwith the default “general purpose” algorithm from a commonly\nused instrument (0.4 nm to 10 μm size range, 70 size bins, 0.01\nregularizer value). Good agreement is achievable between the\nNNLS and AFM size distributions, but only by the contortions\nof limiting the NNLS-inversion parameters to a size range of 10 to\n500 nm and using 100 size bins and a regularizer value of 0.001\n(black line)\n120 Page 4 of 11\nJ Nanopart Res (2021) 23: 120 lines). Width-error and its consequence on transformation Thomas proposed to calculate the number-\nweighted arithmetic mean from the direct output of the\ncumulant analysis (i.e., the intensity-weighted z-average\nand PDI) with the key assumption that the relative\nvariance of the distribution is equal to the cumulant\nPDI. The calculations largely underestimated the exper-\nimental mean size obtained by microscopy. Thomas\nspeculated that the PDI might overestimate the relative\nvariance of the distribution due to noise in the ACF,\nwhich led him to propose an empirical reduction of the\nPDI. A decade later, Hanus and Ploehn (1999)\nreinvestigated Thomas’ approach with expanded data Page 5 of 11 120 Page 5 of 11 120 120 J Nanopart Res (2021) 23: 120 Fig. 2 Relationship between the differently weighted Pythagore-\nan means calculated from the measured number-weighted SEM\nand intensity-weighted NNLS distributions. Red and black points\nindicate instrument output and calculated data, respectively. (In\nthis case, the default DLS instrument output is the arithmetic\nmean.) This example is for SRM 1963 100-nm polystyrene nano-\nparticles. The density distributions are fitted by lognormal func-\ntions with shape-parameter, σ, which equals 0.017, 0.2, and 0.115\nfor the SEM, NNLS (reg = 0.01), and NNLS (reg = 0.001) distributions, respectively. The moment-ratio transformations be-\ntween means possess an underlying scale-factor of\nexp(0.5σ2) from H to G and G to A; transformations between\nweightings possess a further scale factor of exp(3σ2) from number\nto volume and volume to intensity. The larger the width-error, the\nmore severe the underestimation of the volume- and number-\nweighted mean values from the default DLS-output NNLS-de-\nrived size distribution. The dotted line indicates values that are in\nagreement between methods distributions, respectively. The moment-ratio transformations be-\ntween means possess an underlying scale-factor of\nexp(0.5σ2) from H to G and G to A; transformations between\nweightings possess a further scale factor of exp(3σ2) from number\nto volume and volume to intensity. The larger the width-error, the\nmore severe the underestimation of the volume- and number-\nweighted mean values from the default DLS-output NNLS-de-\nrived size distribution. The dotted line indicates values that are in\nagreement between methods of 2% (Fig. 2) reveal that while the arithmetic mean of the\nNNLS-derived intensity-weighted size distribution de-\npends on the regularizer and significantly deviates from\nthe cumulant z-average and scanning electron microsco-\npy (SEM) results, the harmonic mean in contrast does\ncoincide irrespective of the regularizer value used. Width-error and its consequence on transformation Our measurement and analysis of SRM 1963 with a CV 120 Page 6 of 11 J Nanopart Res (2021) 23: 120 in all cases, and the degree to which it does so is related\nto the PDI (Fig. 3b). The harmonic mean of the\nintensity-weighted size distribution, on the other hand,\nagrees with the z-average within 2% on average. This is\nsignificant because we can now state a definition of the\nDLS size that encompasses both the cumulant and dis-\ntributional result in a consistent manner. proposes the combination of analytical ultracentrifuga-\ntion and DLS to determine porosity of polyplex nanopar-\nticles (Niebel et al. 2014). The DLS-measured size is one\nof the inputs meant to be an independent variable. The\nauthors’ choice for DLS size is the arithmetic mean of the\nintensity-weighted NNLS distribution seemingly un-\naware that it overestimates the z-average by at least\n20% depending on the PDI of their samples. Such dis-\ncrepancy may introduce significant unrecognized error/\nuncertainty into their porosity calculation. Another work\nproposes the combination of DLS and TEM to determine\nthe shell volume of nanoparticles as the difference be-\ntween the hydrodynamic volume and the core volume\n(Chen et al. 2017). Here as well, the intensity-weighted\narithmetic mean was unjustifiably used, in this case for\nthe hydrodynamic diameter. Polydispersity descriptors: are they good indicators\nof the broadness of the true distribution? Having demonstrated the effect of the width-error on the\nmean value when transforming between weightings and\nPythagorean means, our final goal in this work is to\naddress the unjustified dependence on DLS to provide a\nserviceable indication of the distributional property of\nnanoparticle samples. In a recent study on quantifying\nthe influence of nanoparticle polydispersity on cellular\ndelivered dose (Johnston et al. 2018), for instance, poly-\ndispersity is defined as the shape-parameter of a lognor-\nmal function fitted to the intensity-weighted NNLS size\ndistribution, which as we have shown is unreliable and\nalways too broad. Such improper valuation of the poly-\ndispersity renders the quantification that is supposed to\nbe the independent variable for the dose-delivery exper-\niment questionable at best. Even recent editorials on best\npractices for nanoparticulate systems encourage the use\nof the intensity-weighted distribution “as it is considered\nalways correct” (Nicolas et al. 2018). This is concerning\nbecause such a statement is misleading. Width-error and its consequence on transformation This is\nnot unexpected since based on the underlying 1/D depen-\ndence of the Stokes-Einstein equation used for determin-\ning size from the ACF, the mathematically natural output\nof the DLS method would be the harmonic mean of the\nintensity-weighted distribution. Unfortunately, one of the\nmost popular software programs (Zetasizer nano user\nmanual 2013) for the NNLS inversion method chooses\nas its default output the arithmetic mean. Since there is no\nrequirement in international documentary standards for\nconsistent representation of the cumulant and inversion\nmethods, this unguided choice is consistent with standard\npractices, but not without negative implications for cer-\ntain applications. A recently published work, for instance, of 2% (Fig. 2) reveal that while the arithmetic mean of the\nNNLS-derived intensity-weighted size distribution de-\npends on the regularizer and significantly deviates from\nthe cumulant z-average and scanning electron microsco-\npy (SEM) results, the harmonic mean in contrast does\ncoincide irrespective of the regularizer value used. This is\nnot unexpected since based on the underlying 1/D depen-\ndence of the Stokes-Einstein equation used for determin-\ning size from the ACF, the mathematically natural output\nof the DLS method would be the harmonic mean of the\nintensity-weighted distribution. Unfortunately, one of the\nmost popular software programs (Zetasizer nano user\nmanual 2013) for the NNLS inversion method chooses\nas its default output the arithmetic mean. Since there is no\nrequirement in international documentary standards for\nconsistent representation of the cumulant and inversion\nmethods, this unguided choice is consistent with standard\npractices, but not without negative implications for cer-\ntain applications. A recently published work, for instance, sets and additional analytical effort, in essence reaching\nthe same conclusion that a reduction of the PDI was\nnecessary to properly fit their data. Unfortunately,\namount of reduction in the PDI needed is not consistent\nbetween different cases. We note that the inconsistency\nbetween DLS-measured polydispersity and the relative\nvariance of the actual distribution—which prompted the\nwidth-error correction in these historic DLS studies—\npersists in present-day implementations of the inversion\nmethod through the indeterminacy of the regularizer\nvalue and other inversion parameters. Comparison of the outputs of the cumulant and NNLS\ninversion methods While the width-error causes inaccuracies in transforming\nbetween weightings of the distributions, it also causes\nerrors when comparing the different Pythagorean means. Width-error and its consequence on transformation Samples with low PDI may measure\ninconsistently by the two methods, while samples with\nhigh PDI may measure consistently. It becomes appar-\nent in Fig. 4c that sensitivity of DLS to the tail of large\nparticles significantly biases the size measurements by\nboth the cumulant and NNLS inversion methods. shown, the intensity-weighted distribution is indeed the\ncorrect one from which to calculate the harmonic mean,\nbut in context, this statement appears to endorse reliance\non the distribution itself, a practice all too often taken. As we have shown above, the width of the NNLS-\ndetermined intensity-weighted size distribution is nei-\nther quantitative nor qualitatively reliable, as it is\nregularizer-dependent. If alternatively using the\ncumulant analysis, the PDI is also known to significant-\nly overestimate the relative variance of the actual size\ndistribution (often by an order of magnitude) due to the\ninescapable need for ACF noise reduction by truncation\n(Koppel 1972), hence it is not a quantitative measure of\npolydispersity. But the PDI is at least deterministic, so\ndoes it provide at least a qualitative indication of the\ndistributional property of nanoparticle samples? To in-\nvestigate the question methodically, gold reference and\nfunctional polyplex nanoparticles in a broad range of\nsize and polydispersity were measured by DLS and\nAFM. The ratio of the z-average and AFM size is\ncorrelated in Fig. 4a and b with the measures of poly-\ndispersity, i.e., the relative variance of the AFM\nnumber-weighted distribution (Q) and the PDI from\nthe cumulant analysis. The deviation between DLS\nand AFM measurements would be expected to increase\nwith increase of the actual polydispersity of the sample\nbecause the DLS-derived mean, originating from an\nintensity-weighted measurement which scales as the\nsixth power of the particle radius, is extremely sensitive\nto and skewed by the positive tail of the distribution. The gold nanoparticle suspensions are free from large-\nsize outliers and aggregates and the nanoparticles ex-\nhibit a narrow size distribution with a relatively small Q\nthat is roughly constant for the entire series. These observations present a major challenge for\nstandardization in nanoparticle metrology. Width-error and its consequence on transformation As previously In our view, the effect of the inevitable width-error in\nDLS behooves the selection of the harmonic mean of the\nintensity-weighted size distribution for obtaining size\nvalues that are metrologically comparable between\nmethods. To demonstrate that this is routinely possible,\nwe measure a variety of monomodal nanoparticles with\nan extensive range of size and polydispersity including\nsome of the most significant reference material types—\ni.e., polystyrene, silica, and gold—as well as functional\npolyplex nanoparticles (Farkas et al. 2019). The results\nof the cumulant and NNLS inversion methods for the 44\nsamples in total are summarized in Fig. 3. We find that\nthe arithmetic mean of the NNLS intensity-weighted\nsize distribution overestimates the cumulant z-average Fig. 3 a Relative differences between the intensity-weighted\nNNLS (reg = 0.01) means (arithmetic or harmonic) and the\ncumulant z-average as a function of the z-average diameter for a\nlarge set of different nanoparticle types and sizes (polystyrene,\nsilica, gold, and polyplex nanoparticles in the 5 to 250 nm size\nrange). b Same relative differences as a function of the PDI\nobtained from the cumulant method silica, gold, and polyplex nanoparticles in the 5 to 250 nm size\nrange). b Same relative differences as a function of the PDI\nobtained from the cumulant method Fig. 3 a Relative differences between the intensity-weighted\nNNLS (reg = 0.01) means (arithmetic or harmonic) and the\ncumulant z-average as a function of the z-average diameter for a\nlarge set of different nanoparticle types and sizes (polystyrene, Fig. 3 a Relative differences between the intensity-weighted\nNNLS (reg = 0.01) means (arithmetic or harmonic) and the\ncumulant z-average as a function of the z-average diameter for a\nlarge set of different nanoparticle types and sizes (polystyrene, silica, gold, and polyplex nanoparticles in the 5 to 250 nm size\nrange). b Same relative differences as a function of the PDI\nobtained from the cumulant method 11 120 Page 7 of 11 120 Page 7 of 11 120 J Nanopart Res (2021) 23: 120 120 Consequently, their AFM and DLS sizes agree on aver-\nage. The polyplex nanoparticles in contrast have widely\nvarying Q, which is often the case for functional, multi-\ncomponent nanoparticles. We find that the increase in Q\nparallels the increase in the z-average/AFM size ratio,\nalso as expected. In contrast, no discernable trend is\nobserved when the same ratios are plotted as a function\nof PDI (Fig. 4b). 1 Disclaimer: certain commercial equipment, instruments, or materials\nare identified in this paper in order to adequately specify the experi-\nmental procedure. Such identification does not imply recommendation\nor endorsement by NIST, nor does it imply that the materials or\nequipment identified are necessary the best available for the purpose. Width-error and its consequence on transformation For example,\nthe mean size of ERM-FD304, a certified colloidal silica\nnanoparticle suspension (Institute for Reference Mate-\nrials and Measurements of the European Commission’s\nJoint Research Centre 2012), can be measured consis-\ntently by DLS using both the cumulant and NNLS\ninversion methods when expressed as the intensity-\nweighted harmonic mean, but the positive skewing in\nthe size distribution renders its use very limited for\ntraceability transfer to microscopy or to other individual\nsizing techniques. Furthermore, the findings in Fig. 4\ndemonstrate that PDI values are generally unreliable\npredictors of nanoparticle polydispersity. We see that\nboth the mean and distributional DLS measurement\nresults may be unknowably significantly biased, which\nwould lead to ambiguous interpretation of structure-\nfunction relationships of nanoparticle formulations or\nto inconclusive interlaboratory comparisons in terms of\nmetrological compatibility. Cross-examination by a\nsizing method for individual particles is therefore a must Fig. 4 a Correlation between the relative variance of the AFM\nnumber-weighted distribution (Q) and the ratio of DLS z-average\nand AFM sizes. b The ratio of DLS z-average and AFM sizes as a\nfunction of the PDI obtained from the cumulant method. c\nDistributional details of the AFM and DLS size measurements\nfor the data point marked by the arrow in Fig. 4b. A PDI value of\n0.1 or below is widely assumed to indicate a nearly monodisperse\nsample Distributional details of the AFM and DLS size measurements\nfor the data point marked by the arrow in Fig. 4b. A PDI value of\n0.1 or below is widely assumed to indicate a nearly monodisperse\nsample Distributional details of the AFM and DLS size measurements\nfor the data point marked by the arrow in Fig. 4b. A PDI value of\n0.1 or below is widely assumed to indicate a nearly monodisperse\nsample Distributional details of the AFM and DLS size measurements\nfor the data point marked by the arrow in Fig. 4b. A PDI value of\n0.1 or below is widely assumed to indicate a nearly monodisperse\nsample Fig. 4 a Correlation between the relative variance of the AFM\nnumber-weighted distribution (Q) and the ratio of DLS z-average\nand AFM sizes. b The ratio of DLS z-average and AFM sizes as a\nfunction of the PDI obtained from the cumulant method. Materials and methods From the NNLS intensity-weighted distribution, the\nintensity-weighted arithmetic and harmonic diameters\nmay be calculated directly as the arithmetic and harmon-\nic averages of the logarithmically scaled discrete data as\nlong as the varying bin size is taken into consideration. Width-error and its consequence on transformation c J Nanopart Res (2021) 23: 120 120 Page 8 of 11 of 100 nm, 70 nm, 46 nm, and 31 nm, respectively, were\nobtained from Thermo Scientific1. European Reference\nMaterial, ERM®-FD100 and ERM-FD304, silica nano-\nparticle reference materials with certified sizes of 19.0\nand 42.1 nm, respectively, were provided by the Insti-\ntute for Reference Materials and Measurements\n(IRMM). Negatively charged, citrate-stabilized gold\nnanoparticles with nominal sizes of 5, 10, 30, 40, 50,\n60, 80, 100, 150, 200, and 250 nm were obtained from\nTed Pella Inc., Redding CA, USA. for credible reporting of nanoparticle size measurements\n(both mean size and distribution), and for reversing the\nalarming practice of the exclusive reliance on DLS-only\nin measurement-critical application fields. Conclusions In conclusion, we have investigated the nanoparticle\nsize and size-distribution output results from DLS using\nboth the cumulant and NNLS inversion data analysis\nmethods and compared them with individual particle\nsizing techniques. The NNLS distributions are shown\nto be seriously influenced by excessive smoothing,\nwhich is unavoidable because of the mathematically\nill-posed nature of the data inversion step. We resolve\nambiguities associated with distribution functions and\ntheir Pythagorean means and demonstrate that the\nintensity-weighted harmonic mean is a simple and trans-\nparent way of reporting the output data that for\nmonomodal samples generally renders a reliably consis-\ntent mean value. This definition addresses previous\ndiscrepancies between cumulant and NNLS inversion\nresults for DLS and most significantly (from the per-\nspective of measurement science) presents a single,\nconsistent DLS-derived size, which can be arrived at\nusing either analysis method. Polydispersity descriptors\nattainable from DLS measurements using either the\ninversion or cumulant methods were shown to be gen-\nerally unreliable in terms of their serviceability in de-\nscribing the distributional properties of both narrow-\ndistribution standard particles and the more-typical\nbroad-distribution biomedical nanoparticles. These re-\nsults advocate for crucial refinements of existing DLS\ndata analysis and reporting practices for correctly ad-\ndressing the limitations and improving the comparabil-\nity of DLS in nanoparticle size measurements. Dynamic light scattering DLS measurements were performed using a ZetaSizer\nNano-ZS (Malvern Instruments) (Zetasizer nano user\nmanual 2013). Each sample was loaded into a dispos-\nable micro cuvette and measured at 25 °C. The z-aver-\nage size was obtained using the cumulant analysis with a\nrepeatability of 1.6%. Intensity-weighted size distribu-\ntions were obtained by the regularized non-negatively\nconstrained least-squares (NNLS) method. In the NNLS\ninversion process, finding the original distribution is\nmathematically ill-posed since many different distribu-\ntions produce indistinguishable autocorrelation func-\ntions (ACF) given even a small amount of noise. The\nmathematical problem is addressed by the use of a\nregularizer parameter, which estimates the level of noise\nin the ACF and controls the degree of smoothness in the\ndistribution, but this does not solve the distribution\nindeterminacy. Regularizer values of 0.01 and 0.001\nare used for the default “general purpose” and the “mul-\ntiple narrow modes” algorithms, respectively. Polyplex nanoparticles The polyplex nanoparticles were prepared as reported in\nRef. (Farkas et al. 2019). Oligo DNA was complexed\nwith synthetic polycationic branched polyethylenimine\nmacromolecules through electrostatic interactions. Dynamic light scattering Reference nanoparticles Polystyrene latex (PSL), silica, and gold nanoparticles\nwere used in this study. SRMs 1963, 1963a, and 1964,\nPSL nanoparticles with NIST-certified sizes of 100.7,\n101.8, and 60.39 nm, respectively, were obtained from\nNIST. Additional PSL nanosphere size standards\n3100A, 3070A, 3050A, and 3030A, with reference sizes Page 9 of 11 120 120 J Nanopart Res (2021) 23: 120 Alternatively, the discrete NNLS data may be trans-\nformed to a density distribution first and fitted with a The expanded uncertainty calculated at the 95% confi-\ndence interval for the SEM size measurements is 3.2%. formed to a density distribution first and fitted with a\nlognormal function of f x\nð Þ ¼\nexp −1\n2\nlnx−μ\nσ\nð\nÞ\n2\n\u0001\n\u0003\nxσ ffiffiffiffi\n2π\np\nwhere\nexp(μ) and σ are the scale-parameter and shape-param-\neter, respectively. The arithmetic and harmonic means\nare then calculated as exp(μ + 0.5 σ2) and exp(μ −\n0.5 σ2), respectively (ISO 9276-2:2014). The mathemat-\nical convenience of the lognormal distribution function,\nto which the NNLS size distribution conforms well, is\nthat the transformations between means possess a scale-\nfactor of exp (0.5σ2). For example, the intensity-\nweighted harmonic mean is obtained from the\nintensity-weighted arithmetic mean by dividing the lat-\nter by exp(σ2). Transformation from number- to\nvolume- or volume- to intensity-weighting of respective\nmeans requires a multiplication by exp(3σ2). Atomic force microscopy Samples were prepared on poly-L-lysine-coated mica\nin the same manner as prepared for SEM. AFM images\nof the samples were acquired under ambient condi-\ntions. Size information of particles is typically obtained\nfrom AFM height data. The details of the AFM size\nmeasurement of polyplex and polystyrene nanoparti-\ncles are described in Ref. (Farkas et al. 2019) and Ref. (Dagata et al. 2016). AFM imaging was performed in\nTappingMode using Veeco OTESP cantilevers with a\nVeeco MultiMode AFM and Nanoscope IV controller. Calibration of the height was performed using a set of\nNANO2 step height reference standards over the range\nof 7 to 700 nm. Particle size analysis using AFM image\ndata was performed using resources available in\nNanoscope Version 5. A first-order image flattening\nroutine is applied to obtain a global background suit-\nable for obtaining particle height measurements. Parti-\ncles in each image are identified by visual inspection\nand excluded before the background is calculated. No\nfiltering of the data was performed. In addition to\nrepeatability of replicate measurements of the modal\nheight values, other major uncertainty components\narise from particle-substrate and particle-tip deforma-\ntions, background flatness, and calibration. The com-\nbined standard uncertainty is estimated as the square\nroot of the sum of the squares of all the individual\nuncertainty components. The expanded uncertainty cal-\nculated at the 95% confidence interval for the AFM\nheight measurements is 2.8%. Conflict of interest\nThe authors declare no competing interests. Dagata JA et al (2016) Method for measuring the diameter of\npolystyrene latex reference spheres by atomic force micros-\ncopy. Special Publication (NIST SP) 260–185 https://doi. org/10.6028/NIST.SP.260-185 Open Access This article is licensed under a Creative Commons\nAttribution 4.0 International License, which permits use, sharing,\nadaptation, distribution and reproduction in any medium or format,\nas long as you give appropriate credit to the original author(s) and\nthe source, provide a link to the Creative Commons licence, and\nindicate if changes were made. The images or other third party\nmaterial in this article are included in the article's Creative Com-\nmons licence, unless indicated otherwise in a credit line to the\nmaterial. If material is not included in the article's Creative Com-\nmons licence and your intended use is not permitted by statutory\nregulation or exceeds the permitted use, you will need to obtain\npermission directly from the copyright holder. To view a copy of\nthis licence, visit http://creativecommons.org/licenses/by/4.0/. DeLoid GM, Cohen GM, Pyrgiotakis P, Demokritou P (2017) An\nintegrated dispersion preparation, characterization and\nin vitro dosimetry methodology for engineered\nnanomaterials. Nat Protoc 12:355–371. https://doi. org/10.1038/nprot.2016.172 Di Francesco T, Borchard G (2018) A robust and easily reproduc-\nible protocol for the determination of size and size distribu-\ntion of iron sucrose using dynamic light scattering. J\nPharmaceut Biomed 152:89–93. https://doi.org/10.1016/j. jpba.2018.01.029 Farkas N, Scaria PV, Woodle MC, Dagata JA (2019) Physical-\nchemical measurement method development for self-assem-\nbled, core-shell nanoparticles. Sci Rep 9:1655. https://doi. org/10.1038/s41598-018-38194-y Hanus LH, Ploehn HJ (1999) Conversion of intensity-averaged\nphoton correlation spectroscopy measurements to number-\naveraged particle size distributions. 1. Theoretical develop-\nment. Langmuir 15:3091–3100. https://doi.org/10.1021\n/la980958w Code availability\nNot applicable. Code availability\nNot applicable. D’Mello SR et al (2017) The evolving landscape of drug products\ncontaining nanomaterials in the United States. Nat\nNanotechnol 12:523–529. https://doi.org/10.1038\n/nnano.2017.67 Declarations 120 Page 10 of 11 Data availability\nThe datasets generated during the current\nstudy are available from the corresponding author on reasonable\nrequest. Coleman VA, Jämting ÅK, Catchpoole HJ, Roy M, Herrmann J\n(2011) Nanoparticles and metrology: a comparison of\nmethods for the determination of particle size distributions. Instrumentation, Metrology, and Standards for\nNanomanufacturing, Optics, and Semiconductors V. Proc\nSPIE 8105:810504. https://doi.org/10.1117/12.894297 Scanning electron microscopy Well-dispersed individual nanoparticles are attached to\na substrate by optimizing the surface zeta potential of\nthe substrate and the concentration of the nanoparticle\nsuspensions. In particular, a 20-μL droplet of the refer-\nence nanoparticle suspension was deposited onto poly-\nL-lysine-coated silicon and incubated for between 1 and\n30 min. Samples were immersed in water to remove\nunattached particles, dried with air, and imaged in a\nvacuum. Particle size by SEM is reported as the area-\nequivalent diameter. Imaging was performed on uncoat-\ned samples using an FEI Helios Dual-Beam SEM. A\nworking distance of 4 mm, beam current of 0.34 nA, and\naccelerating voltage of 5 or 15 keV were used to obtain\nall images. The magnification of the instrument was\ncalibrated using a VLSI Standards NanoLattice as a\npitch standard, which had been previously calibrated\nwith NIST’s Calibrated Atomic Force Microscope. In\naddition to repeatability of replicate measurements of\nthe area-equivalent diameter values, other uncertainty\ncomponents arise from the choice of the thresholding\nlevel for determining the particle boundary, non-\nuniformity of the background (which affects the\nthreshold-determination operation), digitization of the\nparticle projection area and signal intensity, and beam\nalignment and instrument stage instabilities. The com-\nbined standard uncertainty is estimated as the square\nroot of the sum of the squares of all the individual\nuncertainty components (Montoro Bustos et al. 2018). Acknowledgements\nThe authors wish to thank George W. Mulholland (NIST) for providing the SRM 1963 reference mate-\nrial, and András E. Vladár (NIST) and Prem Kavuri (NIST) for\ntheir assistance with SEM imaging. The authors also acknowledge\ninsightful discussions with John A. Dagata (NIST). Author contribution\nN.F. conceived the study and performed\nthe measurements. N.F. and J.A.K. analyzed the results and pre-\npared the manuscript. Author contribution\nN.F. conceived the study and performed\nthe measurements. N.F. and J.A.K. analyzed the results and pre-\npared the manuscript. Funding\nThe research leading to these results received funding\nfrom the NIST Measurement Science and Engineering Research\nGrant Programs under Grant Agreement No. 70NANB17H294. J Nanopart Res (2021) 23: 120 120 Page 10 of 11 Declarations Conflict of interest\nThe authors declare no competing interests. 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