Comparing particle size distributions of an arbitrary shape

Powder Technology 294 (2016) 134–145

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Comparing particle size distributions of an arbitrary shape Otto Scheibelhofer a, Maximilian O. Besenhard b, Michael Piller b, Johannes G. Khinast a,b,⁎ a b

Institute for Process and Particle Engineering, Graz University of Technology, Graz, Austria Research Center Pharmaceutical Engineering GmbH, Graz, Austria

a r t i c l e

i n f o

Article history: Received 25 November 2015 Received in revised form 5 February 2016 Accepted 13 February 2016 Available online 16 February 2016 Keywords: Particle size distribution Statistics Process analytical technology Crystal growth

a b s t r a c t Since particle size distribution can provide crucial information regarding process and product quality, comparing one distribution with another and with modeled distributions is often necessary. Since only mean and median distribution values are often used for that purpose, important information may be missing. As such, methods are required that take into account the entire distribution and do not require modeling the distribution. The χ2-homogeneity test is a nonparametric homogeneity test based on the observed frequencies in the descriptive classes. In this work, we examined if it could be used for comparing particle size distributions and for large particle numbers. In conclusion, we demonstrate how this statistical test was successfully applied in several cases. © 2016 Elsevier B.V. All rights reserved.

1. Introduction Controlling particle size distribution (PSD) is essential for many industrial processes. Particle sizes can vary vastly, from rock debris in geological sciences to components in the millimeter range in the construction industry and crystals in the μm range (and below) in pharmaceutical applications [1–5]. Particle size often determines crucial quality attributes that relate to the processability of intermediates (e.g., handling and storing of cement, blending of active pharmaceutical ingredients (APIs) and excipients) and the quality of the final product (e.g., the strength of concrete or the bioavailability of dosage forms) [6–11]. Determining a particle size distribution typically involves sampling from a process stream. Sampling involves a mass reduction, and in that regard correct sampling of granular materials is challenging [12, 13]. Since positioning of the sampling tool defines the part of process stream that is sampled and analyzed, it has to be performed meticulously. Although sampling is generally performed in a non-destructive matter, sampling of the entire particle population is rarely achieved due to the huge number of particles and data processing involved [14]. The samples' particle sizes are determined via methods that are suitable for the size range in question [15]. There are several approaches to defining a single characteristic number or size of particles that represents their size, i.e., called equivalent diameters. Many different equivalent diameters exist, e.g., the diameter of sphere having the same volume, surface area, sedimentation velocity and many more. The individual size data for a number of particles are then grouped into ⁎ Corresponding author at: Institute for Process and Particle Engineering, Graz University of Technology, Graz, Austria. E-mail address: [email protected] (J.G. Khinast).

http://dx.doi.org/10.1016/j.powtec.2016.02.028 0032-5910/© 2016 Elsevier B.V. All rights reserved.

bins of a certain size range [16–18] called a histogram that determines a PSD [19,20]. Since the particle size is a continuous variable, the probability of finding a particle with a certain diameter is zero. However, there is a finite probability of finding a particle within a certain size range. Although PSDs are regarded as either continuous or discrete distributions (i.e., histograms), they are only truly continuous in simulated and modeled cases. Sampled PSDs always originate from a finite number of particles, leading to a discrete distribution. As a result, experimentally obtained particle sizes are typically split into size classes covering a certain range. The number of particles found within this range, compared to the total number of particles sampled, is the best guess with regard to the probability of finding a particle within this range [21,22]. A sample determines the number of particles in certain size classes, i.e., a PSD. A PSD can also be viewed as a multivariate description of the sample. However, since multivariate representations can be difficult to handle, simplifications have to be made [23]. Often, only a few relevant (integral) parameters (or moments), such as the mean, median diameter and the “broadness” of a PSD, are reported. However, this suffices only if PSDs have a known and common shape [19,24]. Various analytical distributions are used to approximate PSD shapes, e.g., normal distribution, log-normal distribution and others. PSDs however can take increasingly complex shapes, either due to combinations of several particle species, each having their own distribution, or due to sampling effects (e.g., a size cutoff in the measurement principle). Such complex PSDs are hard to describe by an analytical formula, and often only a visual representation of the PSD is informative [22,25]. Often, PSDs are compared with each other, e.g., for quality assurance purposes or if samples are drawn from the same population to ensure that sampling is representative and to understand the dependence of PSD on the sampling location. Theoretical and experimental

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(i.e., discrete and continuous) PSDs have to be compared [26,27]. Comparing distributions only based on integral parameters may not reveal differences between them (see illustration in Fig. 1.). Although a visual comparison can be more informative, it does not have a comparative quantity. Another problem associated with comparing PSDs is that they may be obtained using different measurement techniques [28], in which case elaborate knowledge of the measurement process and particle shape is required. As such, a method of comparing particle size distributions that cannot be suitably represented via an analytical function, or whose complexity cannot be grasped by summary parameters, would be advantageous. Hence, a method for comparison that takes into account the entire distribution rather than the summary statistics only is beneficial. For these purposes, the χ2-homogeneity test can be used. This test is extensively applied in the psychological and medical sciences [29] or used for comparing cascade impactor profiles [30,31]. We applied the χ2-homogeneity test for comparing particle size distributions of an arbitrary shape. Thus, there is no necessity in calculating or selecting summary parameters, but the whole distribution is considered. Furthermore, this test allows to quantify the similarity (or difference) of PSDs, including consideration of the sample size. 2. Method The χ2-tests belong to a family of hypothesis tests, from which only the χ2-homogeneity test will be discussed below. The χ2-homogeneity test is a non-parametric test applied to 2 or more samples described by 2 or more categorical variables, to determine if they originate from the same population [32–35]. It is a hypothesis test whose null-hypothesis is: H0: All samples are drawn from populations that have the same proportions of observations between classes. In other words, the populations from which the samples originate are homogeneous, i.e., their size distributions are identical. Thus, the χ2-homogeneity test can determine if several samples of a limited size represent the same particle size distribution or if the difference between samples is too large to be explained by random sampling. 2.1. Prerequisites for the χ2-homogeneity test An agreement between the mathematical and experimental requirements for conducting the test will be established first. The population is defined as the total number of all particles, i.e., in the investigated process stream, or all particles inside a hopper or a

Fig. 1. Particle size distribution in three samples. Although their shapes are very different, they have the same summary statistics: x10 = 1.4, x50 = 5.0, and x90 = 8.6. The line connecting the actual data points is the interpolated spline setting of Excel (Microsoft Office 2013).

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heap. During the process, samples are obtained containing a reduced (compared to the population) number of particles. Although the sampled PSDs should be representative of the population, the real PSD of the population can only be established when measuring all particles of a population. As such, sampled PSDs are inherently only an approximation and contain an error. Correct sampling of process streams is an elaborate task, and numerous recommendations are available in the literature [36]. As a prerequisite for the χ2-homogeneity test the number of independent samples must be equal to or greater than 2. Repeated measurements of the same sample are not considered independent. If there are 2 samples, one is compared to the other. When there are more than 2 samples, all of them are compared and a single deviating one will dominate the outcome of the test (i.e., no distinction is made between reference and test samples). Suppose a number of M samples is measured, with sample m containing a number of Nm particles. The particle sizes are distributed in K different classes, where class k spans within a size region wk, represented by a mean size value of xk. This results in a particle count of nmk for class k and sample m, and a total test size of N ¼ ∑m Nm ¼ ∑m ∑k nmk . Observations (i.e., size measurements of single particles) should be unique and distinct. This is usually the case in particle size measurements. Every particle is only measured once, resulting in a single size value of this particle. In sieve analysis, a particle is found on top of a single non-passing mesh size. The size of the next particle measured does not relate to the previous one. Classes in PSDs are generally exclusive (i.e., not overlapping) and exhaustive (i.e., there is no particle that is not belonging to a class). The χ2-test can be applied on any scale, even non-ordinate (i.e., classes do not have to have an order). Since classes are categories for describing samples, the specific size and order of classes is of no importance to the test, and there is a significant amount of freedom in choosing the classes. For example, certain size classes may be neglected during an analysis, since (irrelevant) foreign particles (dust) or measurement uncertainties can obliterate those classes. Furthermore, this enables the pooling of classes, as discussed below. Since the χ2-homogeneity test is based on the frequency of particles per sample and size class, the absolute number of particles is important. If qmk is the relative distributional density of class k and sample m and if the total number of particles Nm in the sample m is known, the number of observations per class k can be calculated as nmk = qmk wk Nm. For every particle, the question if it belongs to class k can be answered modally. If a particle belonging to the population is measured (and hence is now part of sample m), the number of particles in class k increases by one with probability pk, which is the (unknown) fraction of particles of class k in the total population. Thus, sampling Nm particles, the number of particles in class k of sample m follows a binomial distribution and the expected value is bnmkN = pkNm with a variance of σnmk2 = Nmpk(1 − pk). For large samples, the binomial distribution can be approximated via a normal distribution according to the Moivre–Laplace theorem with N ðbnmk N; σ 2nmk Þ. In this notation, N ðμ; σ 2 Þ indicates the normal distribution with mean μ and variance σ2. The sum of squares of a number of f independent variables (e.g., sample and particle size class when comparing PSDs, see Section 2.2.), all of which follow a standard normal distribution (i.e., N ð0; 1Þ), is distributed as the χ2f (x)-distribution. The parameter f is generally referred to as the degrees of freedom. For the purposes of the χ2-homogeneity test, a test statistic, termed χ2, is calculated from the experimental data, indicating the difference between the observed and expected particle numbers. The exact calculation will be shown in Section 2.2. This χ2 test statistic is a sum of squared standardnormally distributed variables and hence follows the χ2-distribution. If the test statistic is much higher than the expected χ2crit, which is determined analytically based on the assumption of random sampling, this excess cannot be explained only by sampling effects and it is

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unlikely that the samples originate from the same population (i.e. the null hypothesis is not supported by the data). Because the test statistic is only an approximation of the χ2-distribution, which is in better agreement for larger numbers, a minimal individual class count is necessary and nmk N 5 is typically established. Furthermore, a minimum sample size (often N N 20) is required, although no definitive numbers are recommended [37]. Whereas the latter is generally of no concern for PSDs, the former can occur regularly in the outer ranges of PSDs. As the χ2-homogeneity test is based on counting occurrences, it cannot be used on relative fractions (i.e., the particle size distribution density) directly, as often given for PSDs. Nonetheless, it is possible to draw some conclusions from the data, as shown below. If the number of particles in a sample is known, which is the case in many applications, the number of particles per size class can be calculated from the relative fractions. 2.2. Procedure of the χ2-test In the equations below, we only used number-based PSDs and skipped the common index ‘0’ for number-based particle size distributions. Following the measurement method, volume/mass-based distributions of particles were regularly obtained, which had to be converted into number-based distributions since the number of sampled particles was important. For a single sample m, the particle size distribution density is qmk ¼ qm ðxk Þ ¼

pmk wk

¼

nmk

xk

ship to the cumulative particle size distribution Q m ðxk Þ ¼ ∫ qm ðx0Þ dx0 in k0¼k

0 k0¼k

the continuous case and Q m ðxk Þ ¼ ∑k0¼1 qmk0 wk0 ¼ N1m ∑k0¼1 nmk0 in the discrete case. It is now assumed that M samples are available. Hence, we can obtain a more representative PSD by simply summing up all samples M

for every class, i.e., nk ¼ ∑m¼1 nmk . The said PSD has a number of N ¼ ∑k nk ¼ ∑m Nm particles and can be used to calculate the expected value of the number of particles in a size class for the drawn samples as Emk ¼

nk  N m : N

ð1Þ

On a side note, nk and Nm are the margins of a contingency table containing the nmk values. H0 can now be expressed as ∀(m, k) : bnmk N= Emk. In Eq. (1), Emk is chosen as the weighted mean of observed values. There are alternative possibilities to define Emk, e.g., minimizing variance or coefficient of variation. Furthermore, a theoretical or a simulated value can be used for Emk if the obtained PSDs should be compared to it [38]. Under the assumption of random sampling, nmk follows a normal p−E ffiffiffiffiffimk ffi distribution N ðbnmk N; σ 2n Þ, thus by normalization nmkσ−bnmk N ¼ nmk mk

nmk

Emk

follows N ð0; 1Þ. The variance is approximated according to a Poisson process via σn2mk ≈Emk [37]. Squaring this expression gives the Euclidian distance between the measured and expected particle numbers normalized by the expected particle number and can be used to quantify how similar the PSD sample m is to the estimated population distribution. 2

ξm ¼ N1m ∑k

χ2 ¼

X X ðnmk −Emk Þ2 X X 2 2 ¼ N m ξm ¼ n ξ : m k m k k k Emk

ð2Þ

This test statistic is also termed “Pearson cumulative test statistic”. Since χ2 as defined in Eq. (2) follows the χ2(m− 1)(k−1)(x)-distribution, an upper critical limit χ2crit can be defined based on the valid null hypothesis, i.e. all terms following a N ð0; 1Þ distribution. This limit is further chosen according to a previously to-be-specified confidence level of α. Then, if χ2 N χ2crit the H0 hypothesis is not confirmed by the data. Note that χ2crit denotes the (1-α)-quantile of the distribution χ2(M − 1)(K − 1)(x), with the number of degrees of freedom given by (M−1)(K−1), so that Z

χ 2crit 2 0 χ ðM−1ÞðK−1Þ ðxÞdx

  ¼ F ðM−1ÞðK−1Þ χ 2crit ¼ 1−α;

ð3Þ

nmk wk N m

¼ , where pmk is an estimated probability ∑k nmk of finding a particle in class k represented by particle size xk when drawing a single particle from the sample (i.e. pmk is an estimation for pk, based on the sample m). For completeness, we show the well-known relationwk

as measure describing the deviations of size class k from all samples. Both expressions can facilitate the interpretation of the PSD comparison. Hence ξ2m and ξ2k are standardized squared normal variables and their sum should follow a χ2-distribution. The number of summed variables is M ⋅ K, however those are not independent, due to the normalization step (where mean and standard deviation are estimated from the empirical values). Thus, only f = (M − 1)(K − 1) degrees of freedom remain. As such, the H0-hypothesis can be tested against. The test statistic χ2 of the experimental data set is given by

ðnmk −Emk Þ2 represents a distance from sample m to Emk 2

the overall population. ξk ¼ n1k ∑m

ðnmk −Emk Þ2 can be constructed Emk

where F f (χ2) denotes the cumulative χ2f (x)-distribution F f ðχ 2 Þ ¼ 2 ∫χ 0 χ f ðxÞ dx with f degrees of freedom. Therefore, the p-value (the 2

probability for obtaining the observations under the valid null hypothesis) can be calculated as p(χ2crit,(M,K))=1−F(M−1)(K−1)(χ2crit). 2.3. Significance and particle number While testing a hypothesis, Type I error (rejecting the correct null hypothesis) and Type II error (accepting a wrong null hypothesis) may occur. The probability of Type I error occurrence is exactly α and hence chosen by the experimenter as level of significance. The probability of Type II error occurrence is denoted as β. Often, the value 1 − β (the so-called power of a test) is mentioned, i.e., the probability of rejecting a wrong null hypothesis. It is evident that decreasing α leads to an increase in β, and vice versa. β can be determined when an alternative hypothesis is formed. The implicitly-stated alternative hypothesis H1 for the χ2-test is that nmk is unrestricted (i.e., ∃(m,k) : bnmk N≠Emk). Hence, under assumption of H1 the normalized variables would pN−E ffiffiffiffiffiffi mk . Summing up follow N ðemk ; 1Þ distributions, with emk ¼ bnmk Emk

those squared variables leads to the non-central χ2-distribution, with non-centrality parameter

λ ¼ ∑m ∑k e2mk ¼ ∑m ∑k

ðbnmk N−Emk Þ2 Emk

ðpmk −pk Þ2 . Often the effect size w is used for quantifying pk the difference between the null and alternative hypotheses, as provided

¼ ∑m ∑k Nm

ðpmk −pk Þ2 . For (roughly) equal sample size it can be appk ffi qffiffiffiffi 2 proximated by w ¼ χN : This is sometimes referred to as Cohen's w, as by w2 ¼ ∑m ∑k

the effect size calculation is performed differently for different test statistics. The effect size can also be calculated when only relative possibilities (as in common particle frequency distributions) are provided. The test power 1 − β can then be calculated as   β ¼ 1−F λ;ðM−1ÞðK−1Þ χ 2crit ;

ð4Þ

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where Fλ,(M−1)(K−1)(χ2crit) is the non-central cumulative χ2-distribution with (M−1)(K−1) degrees of freedom and the non-centrality parameter λ=Nw2 in position χ2crit [39]. In practice, the reverse calculation is often relevant, i.e., determining the required number of observations N to achieve a certain level of α and β. Depending on the hypothesis to be tested, either α or β must be determined beforehand, whereas the complementary significance level is a result of the chosen significance level, the particle number and the effect size. A diagram visualizing this relationship is given in Fig. 2. Thus, it is important to examine not only the χ2-value, but also the sample size N and the effect size w, before proclaiming statistical inference.

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true that χ 2f ðxÞ ¼ χ 2f −1 ðxÞ þ N ð0; 1Þ2 (by definition of the χ2f (x)-distri-

bution). As such, a greater number of classes leads to a higher χ2crit, which is stricter in terms of refuting the null hypothesis. As the number of particle classes increases, the number of particles per class decreases. Based on that, to keep the number of particles per class roughly constant, it is recommended to divide the PSD into classes of unequal size. 2

It is suggested that NK N1 . Several further recommendations for selecting the number and size of classes exist, but have to be decided on case by case [40–43]. In contrast, manufacturers of particle size measurement devices often (and rightfully) suggest to have at least NK N1000, in order to have an accurate representation of the size classes. 2.5. Alternative and continuative tests

2.4. Size classes The effect of particle number seems to be contradictory. The following observations were made in that regard: First, for a single class and sample, the number of observations (i.e., particles) should be greater than 5; to achieve this it is possible to pool classes (i.e., add up to a single class). Nonetheless, oversimplification of the PSD should be avoided. The classes should ensure that the PSD is a distribution of sizes and should not withhold information by neglecting peculiarities. Second, intuitively it seems that nmk ~ N, i.e., a larger sample leads to more observations in all classes. It can easily be calculated that χ2 ~ N but χ2crit ≁ N, meaning that larger samples are more representative and the conclusion as to whether the sample is part of the population has a higher significance. Third, if the number of particles is very large, the sampling error becomes relatively small. For counting experiments, the standard deviapffiffiffi tion is usually assumed to follow  NN . Thus, if the sample is large enough, it will never be considered part of the population, because other kinds of errors start dominating over the sampling error. The resulting question is how the width of classes should be chosen. Apart from the sampling problems and the measurement set-up, the behavior of χ2 depending on class number can shed light on this issue. It is

The χ2-test statistic via Eq. (2) is only an approximation to the χ2distribution. The alternative is the G-statistic, which is more reliable when nmk N 2 Emk and can be performed by replacing the calculation of mk the Pearson-χ2 test statistic with G ¼ 2  ∑ ∑ nmk ln ðnEmk Þ. Empty cells m

k

are skipped, and G is once again compared to χ2crit [44]. If the prerequisite of large particle number is not fulfilled, neither the χ2-test statistic nor the G-statistic can be used. In this case Fisher's exact test, respectively the multivariate counterpart Freeman–Halton [45], the permutation tests or Monte-Carlo method can be applied. Since N is usually very large in terms of particle size measurements (at least several thousands up to millions), the levels of α and β are low. This means that the χ2-test can be considered significant although this is not justified, since the effect size is very small and might actually be below what is achievable using the measurement equipment. For the σ2

mk experimental relative error σ 2rel ¼ bnmk N, which is assumed to be the same

for every size class, the effect size w2exp. error = m ⋅ k ⋅ σ2rel is resulting. In such a case the χ2-test is not appropriate, but a multivariate T2-test should be used. For the T2-test, the variance of the samples is considered independent of the mean, whereas in contrast the χ2-test is based on mean and variance scaling with the number of observations, which is the usual case for PSD measurements. Neglecting a relationship between size classes (they have a relative order) leads to the unnecessarily discarding of some information. The Kruskal–Wallis Test is designed for observations on an ordinate scale and uses the order of the particle sizes and their origin to compare the PSDs instead of binning. However, it requires a greater computational effort and a table containing all measured particle sizes (not only the distribution) [34,46]. When the H0-hypothesis is rejected, at least one of the samples differs from all the others. An indication of the deviating samples is given by ξ2m. Deviating samples can be identified via a post-hoc analysis that excludes single measurements (or groups). Caution should be exercised when applying this method to a large number of samples since, due to the nature of random variables, multiple tests increase the chance of Type-I error. As such, it is recommended to make corrections to critical values, e.g., using the Bonferroni method [47]. Comparing χ2-test statistics of various tests can be cumbersome since the obtained χ2 changes depending on the number of particles, classes qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi χ2 , and samples. Cramer's V can be used, as defined by V ¼ Nð minðM;KÞ−1Þ which is between 0 and 1. Evidently, this is related to the effect 1 size via V ¼ w  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi [43]. minðM;KÞ−1

2.6. Summary Fig. 2. Interdependence between the probabilities for Type I error (α), Type II error (β), effect size (w) and number of observed particles N. The value for α or β should be set before conducting the experiments, dependent on the context. Thus for a given number of particles and effect size (which are both experimentally determined), a certain (α,β) pair will result.

Hence, when the experimental data (and the number of particles in the samples) is present in the form of particles per size class, the χ2homogeneity test is performed according to the following steps: 1. Definition of the α or β-level.

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2. Calculation of the expected frequencies for every size class and sample (Eq. (1)). 3. Calculation of the χ2 test statistic (Eq. (2)), and the effect size. 4. Calculation of the critical χ2crit limit (Eq. (3)). 5. Calculation of the complementary error level (Eqs. (3) and (4)). 6. Decision about the rejection or acceptance of hypothesis. 7. Interpretation and follow-up-tests. The origin and shape of the particle size distribution is of no concern to the test, and the test is carried out in the same fashion regardless of the number or size of samples available. Larger samples (i.e., more particles in the sample) are weighted stronger in the test intrinsically. 3. Applications To illustrate the advantages of the χ2-homogeneity test, some practical applications are described below. 3.1. Comparing PSDs before and after agitation Often, it is necessary to establish if the PSD of a certain material has changed during an operation step. Agitation of powder, as present for powder transport and blending, leads to shear forces inside the powder bed, which can lead to attrition and breakage of particles, ultimately changing the PSD [48,49]. Furthermore, the particle sizes present influence how homogeneity in a sample (to be understood as variation of concentration of different particle species inside a volume) is estimated [50]. In the example below, samples of two materials were drawn before and after being agitated in a blender (to mimic blending without a second component) to determine the effect of agitation in a blender on the particle size distribution. In this example, not the efficiency of blending is estimated, but the attrition of particles due to movement inside the blender. 3.1.1. Materials and experimental set-up The materials used were Acetylic Salicylic Acid (ASA; Rhodine 3020 crist., Rhodia Group) and Microcrystalline Cellulose (MCC; H102 AVO 1002, Meggle GmbH). MCC forms small flakey particles and ASA forms elongated needle-shaped crystals [51]. For every species separately, the following steps were performed: First, 120 g was taken from the package. Three samples were collected from this 120 g right before agitating. Next, the powders were agitated in a tumble-blender (Turbula Blender T2F, WAB AG Maschinenfabrik) [52] for 15 min at 50 rpm. Subsequently, three samples were drawn again. Finally, the samples were analyzed using a particle dispersion and imaging system (RODOS/QicPic, Sympatec GmbH) [53] that performed high-speed imaging of the dispersed particles. The particle diameter was set to be an “equivalent circle projection diameter”. Hence, the total number of (detected) particles and their assigned characteristic diameters are known. Additionally, sphericity of the particles was investigated. Due to the measurement technique, sphericity (or more precisely circularity) is determined here by the ratio of the circumference of the equivalent circle projection to the actual circumference, and thus ranges from 1 (perfect circles) to 0 [19]. All in all this resulted in twelve samples (before and after agitation × 2 substances × 3 samples). 3.1.2. Comparison of PSDs at various process steps The questions to be answered were: 1) Do the triplicate samples belong to the same population (i.e., do they have the same PSD according to the χ2-test)? If not, then the drawn samples are different from each other, which can either be caused by improper sampling, or by an already present heterogeneity of the material. 2) Are the samples obtained before and after blending from the same population? If not,

the blending step has altered the particle size distribution, leading, for example, to particle breakup due to the application of shear forces, etc. To answer these questions, Type II error should be minimized to reduce the probability of the wrong assertion that the samples originate from the same distribution. For that purpose, a limit of β = 0.01 was set (which in the end results in a shift of χ2crit). This rather unusual value is achievable due to a large number of particles N = 7,185,493 for all six measurements of MCC (roughly 1 million particles per sample). The effect size calculated based on the experimental data is w = 0.005321. In order to achieve β = 0.01, we chose α listed in Table 1. The interdependence between α, β, and N for a given w is shown in Fig. 2. The same procedure was repeated for the ASA particle measurements. The number of classes was initially set to eighty to preserve the overall shape of the distribution and yet have a reasonable amount of particles per class. To avoid classes with low (b 50) number of particles or with no particles, classes of large particle sizes were grouped into a single tail class. It was reported that the number of observations in the first class above the detection limit peaks due to observation effects [54]. Since the number of particles in this class can be very high and fluctuating and can significantly affect the expectation values, it was omitted.

3.1.3. Results Before presenting the results of the χ2-homogeneity test, a “classical” comparison shall be presented. The obtained x10, x50, and x90 of the PSDs were used to calculate mean and standard deviation, of powders obtained under the same circumstances (e.g., three samples of one powder before agitation). These were then compared to the PSDs of the same substance after the agitation steps, and the obtained results are shown in Fig. 3. These indicate that MCC should be considered to have the same PSD before and after agitation, whereas the PSD for ASA changed. For the χ2-homogeneity test, first the groups of three samples obtained under the same circumstances were compared. The numbers indicate that the sample triplets can all be considered homogenous. Although the samples were homogeneous, the effect of size on ASA was a magnitude larger than on MCC, suggesting that a larger (although not significantly) heterogeneity existed in the sample triplets. This is a result that could not be obtained from the summary parameters, as not enough samples were present to reliably determine the uncertainties on the summary parameters. Here the χ2-homogeneity test is superior as it can exploit the knowledge of all size classes, and the particle numbers. The large difference in the particle number between ASA and MCC was a result of using roughly the same bulk volume of material. Next, the samples before and after agitation were compared for both substances. The resulting numbers are summarized in Table 1. The obtained particle size distributions for the latter comparison, as well as ξ2m and ξ2k ,are shown in Fig. 4 for MCC and Fig. 5 for ASA.A comparison before and after agitation showed that MCC did not undergo a significant change in particle size during the blending process. However, for ASA such a change was detected. This is in agreement with the classical analysis. However, the χ2homogeneity test allows to access more information, as it can exhibit which size classes, and which samples are the origin of the deviating behavior. The change in PSD can be attributed to the needle-like shape and overall larger size of ASA particles, leading to more likely breaking events during blending. This assumption was further supported by looking at the sphericity of the particles, as shown in Fig. 6. Attrition across all particle sizes would lead to spheronization of the particles, which is not reflected in the data. In contrast, ablation of particles and breakage along crystal interfaces would lead to sharply bounded particles of prismatic form. This can explain the observed decrease in sphericity for particles of larger size [55,56].

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Table 1 Summary of the comparison of sampled particle size distributions of two materials, before and after agitation, showing the particle number N, number of samples m, number of classes k, effect size w, Type I error probability α, Type II error probability β, the test statistics χ2 and χ2crit. The H0 hypothesis is accepted if χ2 b χ2crit. The last set of three is the comparison of particle sphericity distributions. In the cases marked with *, the effect size was that large, that α and β could not be balanced, but are both much smaller than required. Samples

N

m

k

w

α

β

χ2

Particle Size Distribution MCC before agitation MCC after agitation MCC before vs. after ASA before agitation ASA after agitation ASA before vs. after*

3,504,983 3,680,510 7,185,493 134,608 36,998 171,606

3 3 6 3 3 6

39 39 39 54 54 54

0.004993 0.004377 0.005321 0.029470 0.059575 0.085178

2.40E −03 2.50E−02 2.00E −08 1.56E −04 1.44E −05 1.00E −10

1.02E−02 9.99E−03 1.01E−02 1.04E−02 1.00E−02 4.40E−82

148,323 40,129 188,452

3 3 6

42 42 42

0.024723 0.056314 0.070226

2.00E −03 3.40E −06 1.00E −10

1.06E−02 0.99E−02 8.84E−56

Sphericity ASA before agitation ASA after agitation ASA before vs. after*

The same χ2-homogeneity test procedure as applied for the particle sizes, was applied for particle shapes (i.e., particle sphericity). χ2 numbers of particle shape test are shown in Table 1 as well. The result is in agreement with the PSD test, i.e., the shape of the particles is consistent within the samples drawn at the same stage, but changed during the agitation of the material. As the χ2-homogeneity test is suitable for any PSD or particle size and shape distribution (PSSD), evidently the same procedure can also be used to assess the homogeneity of a blend of different materials, e.g., by comparing the particle size and/or shape and data on (blend) composition data.

χ2crit

H0 accepted

87 71 203 117 131 1245

115 102 317 167 178 439

Yes Yes Yes Yes Yes No

91 127 929

123 152 361

Yes Yes No

3.2. Pressure titration for dispersion in particle measurement Small particles are likely to agglomerate due to their high surface-tovolume ratio and the resulting comparatively large adhesion forces. In the presented case, fine particles adhered to larger carrier particles. Thus, the material investigated actually consists of a blend of small and large particles. To achieve a bimodal distribution in the measured PSD, that shows the size and number of carrier and fine particles in the blend individually, the particles have to be separated, for example, via mechanical interaction (particle–particle collision, particle–wall collision, centrifugal forces, etc.), which may be accomplished by

Fig. 3. Comparison of summary parameters of the obtained PSDs. Top row: ASA, bottom row: MCC; from left to right: x10, x50, x90. Every subfigure shows the comparison of the mean size of the respective quantile, with the error bars indicating 2 times the standard deviation (as determined by the three samples). Values before agitation (blue) and after agitation (red) are compared. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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Fig. 4. Left: particle size distribution of ASA samples. Experimentally obtained PSD values are indicated by squares (before agitation) and diamonds (after agitation), respectively. The lines indicate the expected values Emk, when H0 is true. The expected distributions just differ from each other by a multiplication factor (i.e., the relative number of particles in the sample). Right: ξ2k indicates in which size classes the samples are different. Differences appear mainly in larger size classes. ξ2m indicates the disparity between samples. ASA after agitation is less homogeneous than before.

controlling the air flow transporting the particles. In our set-up, the air pressure was set to determine the air flow. Determining the optimal pressure range is crucial: too low pressure may fail to separate particles, whereas too high pressure can result in attrition of the carrier particles.

3.2.1. Materials and experimental set-up The RODOS device (Sympatec GmbH) was used for particle dispersion. The applied pressure was varied as needed. To measure particle sizes, the Helos Analyzer (Sympatec GmbH) was used, which is based

Fig. 5. Left: particle size distribution of MCC samples. Experimentally obtained PSD values are indicated by diamonds (before agitation) and squares (after agitation), respectively. The lines indicate the expected values Emk, when H0 is true. The expected distributions just differ from each other by a multiplication factor (i.e., the relative number of particles in the sample). Right: ξ2k indicates in which size classes the samples are different. Differences appear mainly in larger size classes. ξ2m indicates the disparity between samples. As a result, all samples originate from the same population.

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Fig. 6. Upper left: sphericity of particles of ASA samples; lower left: mean particle sphericity for different particle size classes. The agitated material exhibits a lower sphericity at larger particle sizes. Right: ξ2k indicates the sphericity values which are considered different between samples. ξ2m indicates the disparity between samples. As a result, samples cannot be considered as originating from the same distribution.

Fig. 7. PSDs obtained at various pressure settings. The pressure is indicated in bar. The bimodal distribution can clearly be seen. However, detecting PSDs with a greater similarity based on visual observations is an overwhelming task.

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on laser diffraction, calculating the particle size according to Mie-theory (i.e., using the detected diffraction pattern when particles cross the laser beam) [53]. Consequently, particle measurements are based on cross section and have to be converted into number distributions. In our set-up, since the number of measured particles is irrelevant, only the relative size distributions are provided. A pressure titration was performed, i.e., in order to determine the optimal pressure setting several dispersion pressure settings were tested and the PSDs obtained. The generated PSDs with a bimodal distribution are shown in Fig. 7.

3.2.2. Determining optimal dispersion settings Ideally, a range of parameter settings should exist that is sufficient for dispersion and does not damage the particles. Over a certain pressure region, a stable plateau is anticipated, where the obtained PSDs are the same or similar. Pair-wise χ2-tests were performed to locate this region. In this case, a hypothesis test is not required and, instead of calculating χ2, the effect sizes w that do not depend on the particle number were compared (similarly to a squared deviation procedure).

3.2.3. Results The squared effect size w2 for pairs of pressure settings is shown in Fig. 8. The lowest value is found in the region of 0.7–1.0 bar. PSDs obtained in these settings confirm that at that level the carrier particles and fines are separated and PSDs are similar. Although the results of repeated experiments slightly varied in terms of the magnitude of effect size, the minimum was in the same region.

3.3. Crystals in suspensions in various process settings Process settings, i.e., seed loadings, in a continuously operated tubular crystallizer were varied [57,58], leading to a variation in the measured product crystals' PSD. For each seed loading, duplicate PSD measurements were performed. Since PSDs were utilized for a control strategy, a method for determining the reliability of the obtained measurements was required.

3.3.1. Materials and set-up Acetylic salicylic acid (ASA) crystals are grown in ethanol suspension. To measure their size in suspension, the Particle Counter TCC (Klotz GmbH) was used, which measured the product crystals in suspension by applying the extinction of laser light principle to particles that pass through the measurement chamber. The measurement chamber had a volume of 1 mm3 and a bypass was created to dilute the particle suspension and pump it through the measurement chamber [57, 59]. Dilution is necessary to prevent particle occlusion, i.e. overlapping of particles during the measurement process and misinterpreting a particle agglomerate to be a single larger particle. 3.3.2. Quantifying the reliability of PSD measurement setup Measurements obtained in the same process settings were combined into lots. Size classes above 300 μm were grouped into a tail class. This is the origin of the spike at the end of the PSDs in Fig. 9. All measurements inside a lot (generally pairs) were compared via the χ2-homogeneity test to determine if they came from, or were representative of, the same distribution. Unlike in Section 3.1., the level of β (i.e., the probability of wrongly accepting the H0-hypothesis) and the level of α (the probability of wrongly rejecting the H0-hypothesis) were deemed equally important and α and β were chosen to be equal. 3.3.3. Results The results for the individual lots are provided in Table 2. Although in most process settings the measurements seemed to be adequate, in some cases the measurement pairs diverged. Lot D was not included in further studies, since the divergence was assigned to unstable state of the crystallization or PSD measurement process. The divergence in lot F was assigned to the occurrence of agglomeration, which was discussed to yield fluctuations in product crystal size distribution [57]. The particle size distributions of two lots (one considered homogeneous, the other considered heterogeneous) and the χ2-test statistics are shown in Fig. 9. Both look similar at first glance. However, for lot F a large contribution to the χ2-value (i.e., a large difference between the distributions) can be found in the upper particle size range. Both distributions had a much larger, respectively lower number of particles in those size classes than expected if both originated from the same

Fig. 8. Pairwise effect calculation between PSDs obtained at various pressure settings. The pressure is indicated in bar. A minimum is found between 0.8 and 0.9 bar, indicated by the darker column. Since a plateau is between 0.7 and 1.0 bar, this pressure range is considered optimal for dispersion.

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Fig. 9. Particle size distribution and statistics for the crystals in solution. Top: particle size distributions measured (circles) and expected (lines) for slot E (top left) and slot F (top right). Lower left: χ2-statistics of the pairwise tests of samples from a single lot for all lots. The critical χ2crit is indicated by the red line. It can be seen that lot E is considered homogeneous and lot F not. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

population, likely due to aggregation at lower seed loadings (and higher supersaturation levels) [57]. The above indicates that the χ2-homogeneity tests can quickly and reliably determine the reliability of the obtained particle size distributions. 3.4. Remark on size classes Finally, we assessed the influence of size classification on the χ2homogeneity test by calculating the resulting χ2 and χ2crit-values for

Table 2 Summary of the comparison of sampled particle size distributions of crystals in suspension, showing the particle number N, number of samples m, number of classes k, and effect size w, Type I error probability α, Type II error probability β, the test statistics χ2 and χ2crit. The H0 hypothesis is accepted if χ2 b χ2crit. Samples N

m k

w

α

β

χ2

H0 χ2crit accepted

Lot A Lot B Lot C Lot D Lot E Lot F

2 2 2 2 2 2

0.02808 0.03378 0.03014 0.04407 0.04045 0.05528

7.73E-02 3.34E-02 8.32E-02 8.26E-02 3.47E-02 2.56E-03

7.73E-02 3.37E-02 8.33E-02 8.26E-02 3.48E-02 2.47E-03

28.9 39.8 27.9 56.9 39.3 70.6

39.2 43.2 38.9 49.0 43.0 53.5

36,602 34,886 30,666 29,277 24,034 23,093

29 29 29 29 29 29

Yes Yes Yes No Yes No

various classes. Two experimental test sets were used: 1) ASA particles from the package, as described in Section 3.1.1.; and 2) crystal sizes in suspension, considered heterogeneous, as described in Section 3.3.1. To change the number of size classes, the class with the lowest number of particles is added to the neighboring class of smaller particles. This leads to a change in χ2 and χ2crit. Both values are shown in Fig. 10 for both datasets. Since χ2 and χ2crit clearly scale with the number of classes, a change in the number of classes does not change the result with regard to the H0-hypothesis, except when a border case right from the beginning. 4. Conclusion The large number of data points contained in particle size distribution measurements, and the need for robust online PSD comparison, dictates the use of more sophisticated methods, than simply comparing summary parameters. Our investigation indicates that the χ2homogeneity test is suitable for comparing particle size distributions, but also for similarly obtained and classified parameters (such as the particle sphericity, and composition). The χ2-homogeneity test is useful as a hypothesis test for determining the homogeneity of populations (i.e., if samples indeed originate from the same population) if the effect of large particle numbers on

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Fig. 10. Obtained χ2-values (diamonds) and χ2crit-values (red line) for PSDs classified into different numbers of classes. Left: for ASA before blending, H0 is always accepted, although a number of 80 classes is possible. When too many nmk equal zero, the resulting χ2 is undefined. Right: for ASA crystals in suspension lot F, H0 is never accepted. Remarkably, beyond a certain number of classes, χ2-values and χ2crit-values follow the same trend. Hence, pooling classes has not influenced the outcome of the hypothesis test. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Type I and Type II error levels is considered. At a mathematical level, performing the test is rather straightforward. The great advantage of the χ2-homogeneity test is that it is nonparametric, i.e., the shape of particle size distribution is irrelevant. Furthermore, depending on the experimental needs, in most cases the size classification can be adjusted without changing the test results. In our study, the test was successfully used for: • Testing the homogeneity of samples before and after agitation (to mimic a blending environment). We showed that samples obtained at one time step were homogeneous. Furthermore, for one of the investigated materials agitation altered the PSD. • PSDs obtained in various operational settings of the measurement device were compared. We determined the optimal settings for the dispersion of particles in the studied setup and for the studied material. • Particle size measurements of crystals in solutions were compared. The test was used to detect inconsistency in the measurements, delivering a quality criterion for the experimentally obtained PSDs.

The χ2-homogeneity test can be used to compare particle size distributions of arbitrary shapes. It can eliminate the task of creating a model for complex shapes of PSDs and further quantify the extent of similarity of PSDs more rigorously than via visual superimposing. It relieves the experimenter from choosing summary parameters to describe PSDs, but uses the whole information present. Furthermore, it can give a definite answer, in the form of a hypothesis test, if samples taken from a larger bulk, can be considered equal, or different. This can be used in various applications, e.g., determining the quality of sampling, quantifying heterogeneity within a larger bulk, and more. Taking to use the intermediate results of the test, it is possible to explain where differences in PSD originate from. A Matlab (Matlab 2014b, The MathWorks, Inc.) file incorporating the χ2-homogeneity test as described here is added as separate download. For a minimal application, the file expects a matrix of observation frequencies, where rows represent samples, and columns represent categories. Thus, performing a χ2-homogeneity test on own data can be done within a Matlab environment. Instructions and explanations are contained inside the Matlab file as comments to the code. Acknowledgment Research Center Pharmaceutical Engineering is funded by the Austrian COMET Program (grant number 834224) under the auspices of the Austrian Federal Ministry of Transport, Innovation and

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