The primary crystallite size of titania powder relates to its properties in a number of applications. Transmission electron microscopy was used in this interlaboratory comparison (ILC) to measure primary crystallite size and shape distributions for a commercial aggregated titania powder. Data of four size descriptors and two shape descriptors were evaluated across nine laboratories. Data repeatability and reproducibility was evaluated by analysis of variance. Onethird of the laboratory pairs had similar size descriptor data, but 83% of the pairs had similar aspect ratio data. Scale descriptor distributions were generally unimodal and were welldescribed by lognormal reference models. Shape descriptor distributions were multimodal but data visualization plots demonstrated that the Weibull distribution was preferred to the normal distribution. For the equivalent circular diameter size descriptor, measurement uncertainties of the lognormal distribution scale and width parameters were 9.5% and 22%, respectively. For the aspect ratio shape descriptor, the measurement uncertainties of the Weibull distribution scale and width parameters were 7.0% and 26%, respectively. Both measurement uncertainty estimates and data visualizations should be used to analyze size and shape distributions of particles on the nanoscale.
This section reviews particle size and shape distributions by transmission electron microscopy (TEM), stakeholder needs for this information, morphology descriptions of powder aggregates, the relevance of primary crystallite size and shape distributions for titania applications, and the project objectives.
While many of the measurements methods for particle sizes in the nanoscale have focused on assessing an average particle size, the performance properties of nanoparticles often depend on size and shape distributions. Indeed, the nanoparticle size distribution is important to product performance in applications, in the environment, and for health, safety, and regulatory issues. Transmission electron microscopy (TEM) is a standard method for determining nanoparticle sizes.
This case study provides a scientific foundation for an International Organization for Standardization (ISO;
Although transmission electron microscopy (TEM) has been extensively applied to characterize nanomaterials, standard methods for imaging, analyzing and reporting size distributions are lacking. Exceptions to this circumstance are the average particle sizes and the associated measurement uncertainties for TEM analyses of reference materials [
Size and shape distribution measurements and analyses of titania powders are needed by multiple stakeholders, e.g., academia, industry, government and the public at large. Titania powder performance properties have been related to their physicochemical characteristics, including size, shape, surface structure and surface texture. In this work, the TEM measurements are not compared to traditional, onepoint estimates for particle size, such as xray diffraction (XRD) or specific surface area (BET) analysis. Neither method, XRD or BET, can provide information about particle shape. Our methods report the primary crystallite size and shape distribution, estimate parameters of references distributions fitted to the data, compute measurement uncertainties of these parameters, and visualize the correspondence between the data and the fitted reference distributions.
This protocol was developed based on an interlaboratory comparison (ILC) study that conformed to guidelines established by the Versailles Project on Advanced Materials and Standards (VAMAS) [
A recent study [
Aggregate particle size distributions are frequently measured via nonmicroscopy methods, such as those that measure hydrodynamic particle size (e.g., centrifugal liquid sedimentation); these have been the subject of multiple interlaboratory comparisons in the past. Here, the focus is on the measurement of size and shape distributions of primary crystallites in a titanium dioxide sample. This titania was a commercial powder sample consisting of primary crystallites aggregated to micronscale particles. The sizes and shapes of the primary crystallites are known to link with the performance of titania, as shown in
The primary crystallites of titania aggregates are tightly fused and it is not reasonable to use mechanical action to release primary crystallites for direct measurement [
Aggregated/agglomerated powder samples are representative of many commercial materials and mirror the majority of TEM sample preparation protocols. The measurement uncertainty of titania primary crystallite size and shape parameters via an interlaboratory comparison has not been reported to our knowledge. Current size measurement for similar powders may not be based on wellestablished protocols and the uncertainty in measurement is often not conveyed. The lack of confidence in reported values makes it difficult to draw reliable correlations between primary crystallite size and the behavior of various titania in their applications.
The protocol for this sample includes sample preparation, instrument factors, image capture, particle analysis, and data analysis, which has been subdivided into raw data triage, repeatability/reproducibility assessment, fitting distributions to data, estimating measurement uncertainty of distribution parameters, and visualization of results. This protocol is generally applicable to measuring primary crystallites of aggregated titania, but is not necessarily optimized for specific titania powder samples or specific titania application needs. The data analysis elements and statistics tools reported here can be used to improve elements of the protocol, leading to lower measurement uncertainty values, for example. Statistical protocols are used to distinguish between datasets, to select descriptors with similar means, to fit reference models to distribution data, and to compare twodimensional distributions. While measurement uncertainty is an important metric by which to judge data quality, visualization tools can provide additional information. Rather, measurement uncertainty estimates should be used along with visualization tools that can compare model predictions to the data. Our general approach can be used by stakeholders for evaluating descriptors of particle size and shape distributions of commercial materials.
This section provides an overview of the protocol's sample preparation, instrument factors and image acquisition, particle analysis, and data analysis methods. Additional details on the protocol steps, background data, and definitions are provided in the
The sample is a commercial titanium dioxide material (MT500BW, rutile, supplied by Tayca), which could be used for the applications cited in
Each collaborating laboratory used different TEM instruments. TEM operating conditions are shown in the
The primary crystallites of this commercial titania sample cannot be separated physically into individual nanoparticles. Manual particle tracing was used with the traced images being saved for each frame. Only those particles that have clear and distinguishable edges or boundaries were reported. This protocol assumes that all images were taken in digital format. Procedure steps were provided for the open source software, ImageJ (
There are a number of size and shape descriptors that could have been used for this study (see [
The descriptor definitions follow ISO 92766 [
Measured areas are based on the number of pixels associated with the primary crystallites. To reduce area measurement errors for a circle to less than 5%, the recommended pixel numbers per particle range from 100 to 200 (ISO 92766). At a specific magnification level, the number of pixels per nanometer can be estimated, which allows estimates of the number of pixels per particle. Thus, it is often possible to determine, during the image capture step, whether the error on smaller particles is sufficient, and adjust the magnification if needed.
Descriptor data can be analyzed directly, either by comparing grand mean values and standard deviations to individual datasets using oneway ANOVA or by comparing datasets pairwise to determine whether the sample population means were statistically similar. The conventional statistic is the pvalue. These methods provide qualitative dataset comparisons, addressing the repeatability (intralaboratory) or reproducibility (interlaboratory) of the data. Bivariate analysis can be used to compare descriptor pairs between two different datasets; the analysis returns a pvalue and an energy measure [
Cumulative distributions were constructed for each descriptor in a dataset by sorting them in numerical order and assigning a cumulative frequency fraction value to each point. Reference models (normal, lognormal, or Weibull) can be fitted to the cumulative distribution data; the resulting scale and breadth parameter with their standard errors can then be compared either across the grand dataset (ANOVA) or pairwise (pairwise ANOVA). Three copyrighted programs have been developed, programmed in R and implemented as Shiny applications, to provide consistent statistical analyses across the ILC. They are: ANOVA,
The industrial sample used in this ILC is not a certified reference material, so it is not possible to compute the bias or the relative bias of the datasets. We can determine the grand mean,
The relative coefficient of variation is defined as the ratio of the standard deviation, s, of the descriptor means for all datasets divided by the grand mean,
Single descriptor distributions were compared with two methods. Pairwise comparison of the experimental cumulative distributions was done using bivariate analysis [
where
The comparison of single descriptor distributions does not reveal potential variations in morphology, which might better be addressed by comparing sizesize or sizeshape distributions. These distributions can be compared using the bivariate analysis method. This has value as it reflects potential differences in morphology without relying on adherence to specific distribution models.
The major objective of this ILC is to report data quality effects for a standard protocol for measuring particle size and shape distributions of primary crystallites in a commercial titania powder. There are four major elements of the study: evaluating the quality of the raw data using ANOVA and/or bivariate analysis, generating reference model parameter values by fitting size and shape descriptor distribution data to reference models, using these values to estimate measurement uncertainties of these fitted parameters, and visualizing differences between data and model. Consumer, industrial, and regulatory stakeholders can follow this method to produce, report, and evaluate particle size and shape distributions by TEM.
Several factors might be responsible for interlaboratory variations. These include: the instrument used (TEM or STEM), the number of particles analyzed, the number of frames analyzed, the calibration method (gold nanoparticles, gratings, MAG*I*CAL^{®} [
This section addresses the effects of noninstrument factors on the reproducibility of descriptors. The oneway ANOVA compares the descriptor means of individual datasets (the datasets from each lab) to the grand mean of all the data for that descriptor. When the value of the statistic, p, is greater than 0.05, the null hypothesis cannot be rejected, i.e., the assumption that the mean of one dataset is the same as that of the grand mean cannot be rejected.
Seven factors, the laboratory doing the analysis, the number of particles reported, the instrument manufacturer (shown in the
Two factors, the calibration method and the software used, had some descriptors that conformed to the null hypothesis as shown in
The number of particles reported varied by a factor of 2 while the number of frames used varied by an order of magnitude. Each factor impacted reproducibility, but it was not possible to interpret in what ways this might be occurring using ANOVA. However, an individual lab could use oneway or pairwise ANOVA to assess this for their samples and procedures; this constitutes an intralaboratory repeatability evaluation. In general, when there are only a few frames, each frame may more closely resemble the total sample population. Conversely, when there are many frames, there can be more frametoframe variation in the frame means.
In some of the particle size and shape distribution case studies organized by JWG2, there were significant numbers of undersized particles, i.e., particles which had a pixel count less than that recommended for precision[
The grand mean average scale and width for the area distribution can be used to estimate what percentage of particles might be ‘missed’ for each laboratory at the ‘edges’ of the distribution. Given that the resolution between laboratories varies by an order of magnitude, it was possible that labs with high resolution values might miss significant cumulative fractions of small particles.
Two laboratories performed tests with both TEM and STEM (SEM in transmission mode; [
Industrial users may wish to compare distributions of unknown samples to a standard sample directly, without referring to reference distributions. Pairwise ANOVA, done using a commercial statistical package (Systat^{®} v13.1), can be used to identify pairs of datasets that are dissimilar. If the statistic, p, has a value less than 0.05, the null hypothesis is rejected. Bivariate analysis also compares descriptors of two datasets. The bivariate analysis tool generates a pseudo population by developing a ‘joint’ population for the two datasets being compared, and then determines whether the two initial datasets are dissimilar from the generated population. The bivariate method is independent of reference models and can be applied directly to single descriptor cumulative distributions (the aspect ratio cumulative distribution for example), or sizeshape distributions, as visualized by a twodimensional plot of a size descriptor against a shape descriptor.
When a number of laboratories are generating particle size distributions that are different from the grand mean as assessed by ANOVA, it can useful to find pairs of laboratories that have similar means. Pairwise ANOVA analysis can be used to determine which descriptor datasets have similar means. For size descriptors, onethird of the lab pairs with similar means (
Bivariate analysis, a pairwise approach, can be done on descriptor cumulative distributions; one variable is the descriptor value and the other variable is its position in the cumulative distribution. This was done for three descriptors, the Feret diameter, ECD, and the aspect ratio (
As shown by
The reference models were used empirically in this study; they should be considered when there is little prior experience with mathematical descriptions of a specific sample. The normal distribution is a classic choice. Lognormal distributions often fit data for aerosols or discrete particles synthesized in the liquid phase. The Weibull distribution has been used to model particle comminution from grinding, milling, and crushing [
We fitted reference models to cumulative distribution data rather than histogram data. Differential probability distributions lose information when the data are binned, often obscuring the details near the tails of the distributions. In general, parameter values from cumulative distribution fits have lower relative standard errors than those for binned differential distributions. Nonlinear regression and maximum likelihood methods can provide estimates for reference distribution parameters and their standard errors. These values can be converted to coefficients of variance (
Commercial statistical software can provide the coefficient of determination,
A Shiny App^{®},
We present examples of the fitted distributions of the equivalent circular diameter plus their data for three different cases: (A) a pair of laboratories, L3 and L1, with similar descriptor scales (assessed by ANOVA) and similar descriptor distributions (assessed by bivariate analysis), (B) a pair of laboratories, L3 and L4, with similar descriptor scales but dissimilar descriptor distributions, and (C) a pair of laboratories, L3 and L7, for which both the descriptor scales and descriptor distributions are dissimilar. In a quantile plot for a lognormal size distribution model,
For Case A, the L1 and L3 datasets are wellrepresented by their models over a five quantile range, −2.5
As shown in
Even better correspondence between model and data can be achieved by fitting a bimodal model to the data. The bimodal Weibull model has an average relative residual deviation of 1.8%, which may justify estimating five parameters rather than 2. The relative residual deviation plot is another tool that can be used to interpret distribution data: it helps define the descriptor range over which the model fits the data and complements the measurement uncertainties of the fitted parameters.
Since titania primary crystallites are asymmetric, pairs of size and shape descriptors can provide important information on their morphology. A specific manufacturing method is likely to generate specific crystallite morphologies. These can be compared for statistical similarity using bivariate analysis.
Nanoparticles in commercial products are expected to have variations in size and shape. These, in turn, may affect their performance properties in applications. Twodimensional plots of descriptors, such as Feret/minFeret (a sizesize distribution) or Feret/aspect ratio (a size shape distribution), can be used to explore particle morphology features. Four laboratories reported datasets with similar means as evaluated by pairwise ANOVA: L1, L2, L3, and L8. There are six unique descriptor pairs of the choice, equivalent circular diameter, Feret diameter, aspect ratio, and compactness.
As show in
The
An alternative model is plotted in
which assumes that the grand mean of the aspect ratio for the ILC should be a reasonable value for
Sizeshape distributions can be constructed, but do not yield power law descriptor correlations with high
Measurement uncertainty of ILC means is usually reported for the sizes of certified reference materials. A typical approach [
Two methods have been used to compute measurement uncertainties for this ILC. The first method (procedure A) follows the approach for certified reference materials outlined above and adds an assessment of the measurement uncertainty due to the width of the distribution. In Procedure A, the size and shape populations are modeled using the scale and width parameters fitted to a normal distribution. The second method (procedure B) is based on the scale and width parameters for the preferred reference distribution. In both procedures, the grand means and standard deviations of these parameters are then used to compute coefficients of variation for all descriptor distributions (
The coefficients of variation for the size descriptors of the full ILC are significantly larger than those reported by Rice for a gold nanoparticle certified reference material [
The aspect ratio data (
This project developed methods for measuring and analyzing primary crystallite size and shape distributions of an aggregate using transmission electron microscopy. Available sample preparation, instrument factors, image acquisition, and particle analysis techniques were modified to meet the measurement challenges of this commercial material. Statistical protocols were used to distinguish between datasets, to select descriptors with similar means, to fit reference models to distribution data, and to compare twodimensional distributions. There were significant interlaboratory variations with respect to descriptor mean values and fitted parameter estimates of their reference models. The study determined that: 1) calibration method and imaging software did not seem to affect data quality, and 2) some datasets had similar single descriptor means, fitted parameter estimates, and sizesize and/or sizeshape (two dimensional) distributions. Some evidence indicates that dataset differences in size descriptor means may be due to absolute differences in calibrations. While the equivalent circular diameter, D_{ecd}, is now a common choice for characterizing titania primary crystallite size, other size descriptors would also have reasonable reproducibility, as shown in the data tables in the
For all datasets, size descriptor distributions appear to conform well to lognormal distributions rather than normal distributions. Scale and width parameters of the aspect ratio shape descriptor have fairly similar measurement uncertainty values for either the normal or the Weibull distributions. However, comparing the residual deviations of the models from the data suggests that the Weibull is preferred. This result suggests that using measurement uncertainty as the only metric by which to judge data quality may not be sound. Rather, it should be used along with visualization tools that can compare model predictions to the data. This general approach can be used by stakeholders for evaluating descriptors of particle size and shape distributions of commercial materials.
The authors thank Toshiyuki Fujimoto and Naoyuki Taketoshi of AIST for their advice and counsel with respect to ISO/TC229 needs and requirements of this study. The authors also thank the following researchers at NIST for their useful critiques regarding this ILC: Angela Hight Walker, Jeff Fagan, Vince Hackley, and John Bonevich. The work by AIST was part of the research program of “strategic international standardization acceleration projects” supported by the Ministry of Economy, Trade, and Industry (METI) of Japan. The work by BAM on this project was supported by the SETNanoMetro Seventh Framework Programme project (project number 604577; call identifier FP7NMP2013_LARGE7). The work by KRISS was part of the project, “Nano Material technology Development Program (2014M3A7B6020163) of MSIP/NRF. The findings and conclusions in this report are those of the authors and do not necessarily represent the views of the National Institute for Occupational Safety and Health.
TEM images of titania sampled from a water dispersion. (a) Aggregate scale. (b) Primary crystallite scale.
Examples of distinguishable primary crystallites.
Example of primary crystallites that should not be traced.
Laboratory L1 Feret diameter. Left hand side = density distributions; right hand side = cumulative distributions. Black curve = empirical data; Red curve = maximum likelihood estimate fit; Blue curve = nonlinear regression fit. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Laboratory L1 aspect ratio data. Left hand side = density distributions; right hand side = cumulative distributions; Black curve = empirical data; Red curve = maximum likelihood estimate; Blue curve = nonlinear regression fit. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Quantile plot comparison of equivalent circular diameters, lognormal reference model. L3 with L1 = similar scale, similar width; L3 with L4 = similar scale, different width; L3 with L7 = different scale, different width.
Relative residual differences (%) between equivalent circular diameter data and lognormal models: laboratories L1, L3, L4, and L7.
Relative residual differences (%) between aspect ratio data and three models. Data from laboratory 1.
Feret/minFeret plot. Open blue diamonds = L3 data; black line diamonds = centroids of statistically similar datasets; black circles = centroids of statistically dissimilar datasets. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Titania applications dependent on primary crystallite size.
Application  Preferred primary crystallite size, nm  Reference 

Lithium ion electrodes  <1500  [ 
Powder cosmetics  25–200  [ 
Nanocomposite fibers  35  [ 
Dye cell photoanodes  50  [ 
Photocatalysts  20  [ 
IRreflective nanocomposites  41  [ 
Slurry polishing compound  10–70  [ 
Light emitting diodes  30  [ 
Conductive ceramics  <100  [ 
Statistical analysis methods for this ILC.
Data element analyzed  Objective  Specific metrics  Statistical method 

Descriptor datasets  Repeatability or reproducibility  Grand mean values and standard deviations Pairwise pvalues  ANOVA of all datasets Pairwise ANOVA 
Descriptor distributions  Repeatability or reproducibility Coefficient of variance; measurement uncertainty  Bivariate pairwise pvalues\Fitted scale and breadth parameters plus their standard errors  Bivariate analysis\Curvefitting methods: nonlinear regression, maximum likelihood estimation 
Sizesize and sizeshape distributions  Repeatability or reproducibility  Bivariate pairwise pvalues  Bivariate analysis 
Factors that might affect size descriptor measurements.
Lab Code  Instrument type  Number of particles reported  Number of frames reported  Calibration method  Resolution, nm per pixel  Image analysis software 

L1  TEM  1033  83  Magical  0.151  ImageJ 
L1a  STEM  722  4  Other  0.435  ImageJ 
L2  TEM  573  13  Magical  0.2  Other 
L2a  STEM  515  8  Other  0.6  ImageJ 
L3  TEM  519  41  Grating  0.405  ImageJ 
L4  TEM  524  50  Gold np  0.225  ImageJ 
L5  TEM  500  10  Gold np  0.532  ImageJ 
L6  TEM  532  83  Magical  0.151  ImageJ 
L7  TEM  566  10  Grating  0.5  Other 
L8  TEM  517  13  Other  1.43  ImageJ 
L9  TEM  872  25  Grating  0.578  Other 
Descriptor pvalues for calibration method and software categories.
Descriptor  pvalues  

 
Calibration method  Software  
Area  0.0375  0.0218 
Feret  0.0394  0.0565 
minFeret  0.0538  0.225 
ECD  0.353  0.144 
Aspect ratio  <0.001  0.061 
Compactness  <0.001  <0.001 
% similar  33%  67% 
Estimates of undersized primary crystallites not reported.
Laboratory  Primary crystallite area, nm^{2}  Undersized primary crystallites not reported, %  

 
Area @ 5% error  Minimum reported area  
L1  4.6  20  ∼0 
L2  8.0  80  ∼0 
L3  33  168  ∼0 
L4  10  43  ∼0 
L5  57  142  ∼0 
L6  4.6  262  0.0016 
L7  50  129  ∼0 
L8  409  414  0.30 
L9  67  271  0.0024 
Pairwise comparison of descriptors: ANOVA and bivariate methods (36 unique pairs).
Descriptor  Pairwise methods  

 
ANOVA, p > 0.05  Bivariate, p > 0. 05  

 
Number  %  Number  %  
Area  13  36  –  – 
Feret  12  33  6  17 
Minferet  12  33  –  – 
D_{ecd}  11  31  6  17 
Aspect ratio  30  83  21  58 
Compactness  17  47  –  – 
Reference model parameters and relative standard errors: size and shape descriptors.
Scale  RSE, scale  Width  RSE, width  

Size descriptor  
ln(Area), ln(nm^{2})  7.053  0.00142%  0.6310  0.01580% 
ln(Feret), ln(nm)  3.894  0.00278%  0.3571  0.05600% 
ln(minFeret), ln(nm)  3.426  0.00292%  0.3231  0.06190% 
ln(D_{ecd}), ln(nm)  3.647  0.00274%  0.3156  0.03170% 
Shape descriptor  
Aspect ratio  0.7104  0.0141%  4.781  0.00274% 
Compactness  0.8216  0.0122%  9.963  0.00853% 
Bivariate pvalues for sizeshape and sizesize descriptor pairs.
Sizeshape distribution  Sizesize distribution  

 
Shape  Aspect ratio  Compactness  

 
Size  ECD  Feret  ECD  Feret  FeretminFeret 
L1/L2  0.189  0.181  0.060  0.051  
L1/L3  0.774  0.852  0.781  0.845  0.593 
L2/L3  0.125  0.119  0.193  0.132  0.115 
L2/L8  0.596  0.365  0.539  0.384  0.371 
L3/L8  0.320  0.231  0.323  0.221  0.325 
L1/L8  0.200  0.200  0.054 
Note: entries in italics show pairs with p < 0.05.
Measurement uncertainties (U_{ILC}) for selected descriptors and distributions: all ILC datasets.
Descriptor  ECD, nm and ln(nm)  Aspect ratio  


 
Distribution  Normal distribution  Lognormal distribution  Normal distribution  Weibull distribution  



 
Parameter  Scale  Width  Scale  Width  Scale  Width  Scale  Width 
39.8  12.9  3.58  0.318  0.632  0.148  0.693  
s  4.12  1.88  0.161  0.033  0.0188  0.00559  0.023  0.609 
c_{v}, %  10.3%  14.6%  4.49%  10.5%  2.97%  3.78%  3.32%  12.3% 
U_{ILC}  22%  31%  9.5%  22%  6.3%  8.0%  7.0%  26% 