Like Crystal Shape on Measured Particle Size

TRIBUTE TO FOUNDERS: ROGER SARGENT. PARTICLE TECHNOLOGY AND FLUIDIZATION

Effect of Needle-Like Crystal Shape on Measured Particle Size Distributions Ian de Albuquerque and Marco Mazzotti ETH Zurich, Institute of Process Engineering, Sonneggstrasse 3, CH-8092, Zurich, Switzerland

David R. Ochsenbein and Manfred Morari ETH Zurich, Automatic Control Laboratory, Physikstrasse 3, CH-8092, Zurich, Switzerland DOI 10.1002/aic.15270 Published online in Wiley Online Library (wileyonlinelibrary.com)

The effect crystal morphology has on measured particle size distributions (PSDs) is explored, with a focus on particles exhibiting a needle-like habit. An idealized in silico study was performed, targeted at modeling the measurement principles of various particle sizing devices, namely laser diffraction, Coulter counter, focused beam reflectance measurement, a single and a dual projection imaging devices. The evolution of a crystal population is measured, allowing for an evaluation of the introduced biases. Further, the consequences of these biases are highlighted by demonstrating how the real growth mechanism may be incorrectly interpreted depending on the chosen particle sizing technique. It is found that techniques which utilize a one-dimensional PSD are incapable of simultaneously describing the concentration profile and average length; in contrast, imaging techniques are able to reproduce both quantities. Finally, the dual projecC 2016 American tion imaging device is shown to be the only instrument to yield a nearly bias-free measurement. V Institute of Chemical Engineers AIChE J, 00: 000–000, 2016 Keywords: crystal growth, crystallization, particle technology

Introduction Process analytical technology (PAT) tools play a crucial role in various industries, enabling monitoring of critical quantities and thus facilitating the modeling, operation and control of entire processes. Consequently, PAT forms an important element in the Quality by Design approach, as showcased by the field of crystallization1–7. Among the various available instruments, those that allow insight into the (one-dimensional) particle size distribution (PSD) of crystals are of particular importance, due to the strong impact that the PSD has on downstream process steps and final product quality8,9. While several commercial devices exist, based on different physical principles, among the most widely used particle sizing techniques are Coulter counter (CC), laser diffraction (LD), focused beam reflectance measurement (FBRM) and imaging systems that are based on a single projection (SP) of crystals. Crystal shape is an important characteristic that, together with the crystal size, determines the product’s filterability, flowability, and bioavailability (note that the term “shape” is This article is dedicated to Professor Roger Sargent whose pioneering contributions have been the foundation of the work of generations of process systems engineers and in particular of the work of the senior author for the last four decades. Sargent’s ideas have influenced everything from research projects to text books to software to whole curricula. His impact on the field of process systems engineering remains unsurpassed. Correspondence concerning this article should be addressed to M. Mazzotti at [email protected]. C 2016 American Institute of Chemical Engineers V

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used interchangeably with “morphology” or “habit” in this work). It is also for this reason that crystal shape has been extensively studied, particularly for single crystals of nonequant morphology10–14. Accurate size and shape measurement of single crystals is possible using, e.g., confocal microscopy or tomography techniques14,15, which yield detailed information that can be used to investigate crystal habit over time. As shapes of crystals are typically distributed, it follows that populations of crystals are more effectively characterized by (multidimensional) particle size and shape distributions (PSSDs) than by simple 1D distributions. Still, the robust measurement of such PSSDs has eluded researchers for a long time, as commercially available techniques only allow for their reconstruction under rather restrictive assumptions16–19. However, the introduction of novel measurement techniques and the availability of greater computational power has given new momentum to the modeling and monitoring of PSSDs of particulate populations20,21,51, bringing promising opportunities for parameter estimation22,23, and ultimately also modelbased and model-free process control. The goal of this work is to investigate commonly used particle sizing techniques and to explore the consequences, possibly negative, in the interpretation of experiments. Given that PSD data is utilized with increasing frequency for the development of process models and process design, the study of the validity and applicability of different particle sizing techniques for quantitative purposes is of primary importance both for academia and industry. To achieve this goal, the relationship between the true size and shape distribution of an ensemble of growing needle-like

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Figure 1. Sources of bias among different particle sizing techniques. Biases in the top row are caused by effects inherent to the particle sizing technique; those in the middle row may be influenced to some degree by the choice of operating conditions or are dependent on compound properties. Effects described in the last row are influenced by choices made in the interpretation of raw data. The phenomena taken into account in this work are highlighted in boldface and italic.

crystals and the PSD measured by various particle sizing techniques, namely LD, CC, FBRM, single and double projection imaging devices, is studied. Choosing a system and designing a process in which only crystal growth occurs is ambitious due to the fact that guaranteeing the absence of nucleation, agglomeration, and breakage is challenging experimentally. Thus, to be able to decouple the various potential mechanisms affecting the measurements, our analysis is performed via the use of highly idealized simulations, in which we aim at capturing particle size and shape effects. As such, our work distinguishes itself from other comparative works, which have followed a purely experimental approach24–26. Figure 1 summarizes the major potential sources of measurement errors for various particle sizing techniques, highlighting the effects that are included in this work. As can be seen, our simulations are designed to elucidate the errors and biases that are intrinsic to the different particle measurement techniques. Whereas the inclusion of additional phenomena affecting measurements would be of great interest, we refrain from modeling them here for clarity’s sake. As an example, particle transparency and absorbance clearly play a critical role in practice, yet it is our aim to decouple those effects, which are compound-specific and likely cause further deterioration of the measurement quality, from those that we consider fundamentally associated with the different methods. Obviously, such a separation is not feasible in an experimental setting. In this study we focus on the highly relevant case of needles, treating three different morphologies, namely, that of vanillin, that of paracetamol, and a perfect cylinder for illustrative purposes. The conditions analyzed were chosen to highlight general trends that may be expected for elongated particles. The article is organized as follows: first, a motivational example on the effect of shape on ‘size’ measurements for a generic cylindrical particle is presented. Then the investigated particle sizing technologies and their idealized, virtual implementations are described, followed by the discussion of and the analysis of their differences in the case of vanillin crystals. Finally, a case study, using experimental growth rate data of paracetamol11, is presented, thus illustrating the impact of the instrument choice on parameter estimation. 2

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Motivation Describing a single cylinder as a sphere To demonstrate the consequences of reducing multiple characteristic sizes to a single one, we consider a cylindrical particle and the case where its two descriptors, the length (L1), and the width (L2) are mapped to the diameter of its volume equivalent sphere (D). While we will restrict our analysis to this simple case, extensions to more generalized situations, e.g., for arbitrary convex polytopes, are feasible. The shape of the cylinder will evolve due to the influence of the two growth Ð t rates, Gi 5dLi =dt with i5f1; 2g: in fact Li ðtÞ5Li ðt0 Þ1 t0 Gi ðÞd. If it is further assumed that the ratio R5G1 =G2 is constant and that particles grow indefinitely, the shape will evolve toward the steady state shape27–29: lim

L1 ðtÞ

t!1 L2 ðtÞ

5xss 5R

(1)

The diameter of the volume equivalent sphere is given by 13 3 D5 L1 L22 (2) 2 Further assuming that the original particle grows via a sizeindependent mechanism in both directions, the growth rate of the corresponding sphere, GD , is: 22=3 2 dD 1 3 2 5 L1 L2 (3) L2 G1 12L1 L2 G2 GD 5 dt 2 2 Equation 3 makes it immediately clear that the observed growth rate of such a crystal is a function not only of G1 and G2 but also of the crystal shape. Indeed, Eq. 3 can be simplified by substituting the aspect ratio x5L1 =L2 , thus yielding 1 3 22=3 x ðG1 12xG2 Þ (4) GD 5 2 2 that is, the growth rate of the volume equivalent sphere is a function of G1, G2 and of the time-varying aspect ratio x (cf. Figure 2a). From the perspective of a hypothetical observer measuring D, this is problematic as it implies that the measured growth rate GD is not constant even when the underlying facet growth rates, G1 and G2, are. By taking the derivative of

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where x5G1 =G2 . Note that a change in aspect ratio with changing operating conditions is often observed in practice, even for the same polymorph2,11,12,30–32.

Describing an ensemble of cylinders as an ensemble of spheres As seen above, whenever a particle grows and its shape evolves, a measurement of the growth rate of its volume equivalent sphere depends on the underlying particle’s shape. However, the consequences of this apparent size dependent growth on the measurement of entire ensembles of crystals are not obvious. As a simple illustration, let us consider a distribution of cylinders that grow in both directions, i.e., length and width at a constant rate. The cylinders 2D PSSD is represented in the space domain of coordinates (L2, L1), and will be shifted along a straight line with slope R, where R is the relative growth rate. The absolute speed with which the distribution moves along this line is determined by the exact values of G1 and G2, yet it is of no relevance for the analysis presented in this section; indeed, our focus lies on understanding how movements of the underlying 2D PSSD affect the measured 1D PSD. As the distribution evolves every cylinder is mapped into a sphere using Eq. 2, thus transforming the 2D PSSD into a 1D PSD. The properties of the evolving 1D PSD of spheres can then be investigated as the underlying cylinders move in the space domain of coordinates (L2, L1). Specifically, consider a Gaussian distribution with diagonal covariance matrix consisting of 00 5106 cylinders, where ij is the ij-cross moment of the distribution, defined as ð1 ð1 f ðL1 ; L2 Þ Li1 Lj2 dL1 dL2 (6) ij 5 0

Figure 2. (a) Equation 4 is plotted with respect to x/R, where the minimum can be seen to be at x=R51. (b) (left) Initial (dashed) and final (continuous) 1D PSDs of spheres corresponding to the rigidly shifted 2D PSSDs of cylinders. (right) The variation in standard deviation over average length is shown, and can be used as a measure of how much the distributions have broadened (red) or shrank (black). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Eq. 4 with respect to x and setting it to zero we can find the value of x for which GD does not change with the instantaneous morphology, namely: dGD G1 50 () x5 5R (5) dx G2 which implies that this situation occurs if and only if the aspect ratio of the particle equals its steady state shape xss 5R, which is where GD given by Eq. 4 attains its minimum value. Hence, whenever a particle’s shape evolves toward its steady state shape the growth rate of its volume equivalent sphere will appear to decrease over time and to be size dependent even in the absence of any measurement error. When the crystal attains its steady state shape, given by Eq. 5, the growth rate GD does not change any more and it is given by Eq. 4 AIChE Journal

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and f being the number density function with units ½m25 . Using the cross moment definition we can compute an average length and width, 10 =00 and 01 =00 , respectively. Furthermore, the standard deviation in the length, 11, and width direction, 22, can also be calculated, with the former being defined as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 20 2 10 (7) 11 5 00 00 and the latter being defined analogously. The considered initial distribution’s average length and width are 10 =00 530 lm and 01 =00 510 lm, respectively. Additionally, the standard deviation in the length direction is 11 57 lm, and that in the width direction is 22 51 lm. We then consider two cases, whereby the initial distributions are translated along two straight lines with slopes given by R 5 1 and R 5 6. Both translations are stopped before the cylinders reach their steady state shape. In the case when R 5 1 the cylinder’s final average sizes are 10 =00 541 lm and 01 =00 521 lm, while for R 5 6 the cylinders reach the final average sizes of 10 =00 5 72 lm and 01 =00 517 lm. The transformation of the bi-dimensional PSSDs into onedimensional distributions of volume equivalent spheres is illustrated in Figure 2b, along with a measure of how much the one-dimensional distribution’s width change as a function of the increase in average sphere diameter. As can be seen, the observed broadness of the distributions either increases or decreases, depending on the growth trajectory, i.e., the value of R, despite the fact that the initial distribution is the same in both cases.


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Figure 2a), yielding a broadening of the volume distribution. Conversely, when R 5 6, smaller particles appear to grow faster than larger ones, thus leading to a narrowing of the PSD. Interestingly, similar effects exist for the case of broader distributions where the simplified rationale above cannot be applied any longer. Generally speaking, it is difficult to predict whether the measured distribution of volumes will become narrower or broader over the course of an experiment. However, it is important to note that the width of the measured distribution will rarely remain constant, even if that of the underlying PSSD will.

Extended Virtual Test Bench Recently, we presented a simulation tool for the study of particle overlap effects in a stereoscopic imaging device, called the virtual test bench (VTB)20,22. The VTB combines particle models of convex, faceted crystals with a morphological population balance model and a simulated measurement environment which can be used to assess the features and limitations of the utilized imaging device. The extended virtual test bench (eVTB; licensed under GNU GPL v3.0) presented in this work is structurally identical, yet it contains an expanded set of sizing techniques. The different elements of the tool are visualized in Figure 3 and summarized in the following.

Polytopic particle model The morphology of a large number of crystals can be described by that of convex polytopes P, whose half-space representation is given by PðLÞ5fx 2 R3 jAx MLg

Figure 3. Schematic outlining main eVTB steps from specification of the crystal system to the measured PSSD. Initially the simulations require specifying a crystal system, along with the associated crystallographic, thermodynamic, and kinetic data. For a given process the nD PBE can be solved resulting in what is here denoted as true PSSD. The response of an FBRM device is then directly calculated using the true PSSD. On the basis of the true PSSD a multitude of crystals are generated from which a random subset of 2000 crystals is obtained. For every particle within this subset a single particle contribution is computed for the CC, LD, SP, and DP. Assuming low suspension densities the overall signal can be determined, followed by estimating the measured PSSDs, for SP and DP, and measured PSDs, for the CC and LD. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

This somewhat contradicting behavior can be elucidated based on the properties of Eqs. 1 and 4 and the two values of R. First, we note that the Gaussian defined above has a narrow distribution of widths, approaching the monodisperse case. Given that the volume of a cylinder is L32 x=4, the distribution of volumes is thus largely determined by the distribution of lengths, hence smaller volumes are strongly correlated with smaller aspect ratios. As all cylinders will grow toward their steady state shape, xss 5R, we find that, in the case where R 5 1, smaller particles appear to grow more slowly than larger particles (cf. 4

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(8)

In Eq. 8, A 2 Rm33 is the matrix of unit normal vectors of all facets and M is a m 3 n Boolean matrix grouping together facets that grow with the same speed. Both matrices are constant and compound-specific and together describe the m potential facets, and their symmetrical relation; more details regarding their construction can be found elsewhere22,33. The elements of the characteristic size vector L 2 Rn represent the normal distances from the centroid to the facets of the polytope and is used to scale the n independently growing facet groups; generally it is a time-varying variable. McCrone34 studied vanillin crystals grown from different solvents and found that their morphology could be described by three sets of Miller index families. In the case of paracetamol, Boerrigter et al.11 reported four Miller index families as well as their growth rates. In this work, the habit of paracetamol is simplified, assuming that only two of the four facet groups are actually present. The resulting morphology still represents a satisfactory approximation of the original shape as grown from low supersaturations. A summary of the unit cell parameters and the modeled facets for both crystal types is reported in Table 1. Using this crystallographic information the matrices A and M can be calculated. For a given size vector L, the corresponding polytope can then be constructed on the basis of Eq. 8; visual examples for both crystal systems used in this work are provided in Figure 4. Numerically, P and its projections can be computed using, e.g., the freely available multi-parametric toolbox (MPT3) in Matlab35.

Morphological population balance equation The crystal population evolution, in terms of size, number, and shape can be modeled using a morphological population balance equation (PBE). For a well-mixed batch crystallizer


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Table 1. Miller Indices and Unit Cell Parameters for Paracetamol and Vanillin Crystals

Crystal

Unit cell ˚ ˚ ] c [A ˚] a [A] b [A

Facet group

Paracetamolf001gf110g 12.65 8.89 Vanillin f100gf120gf111g 7.88 14.04

7.24 13.53

Lattice system Monoclinic Orthorhombic

and with size-independent growth only, this PBE can be written as: n @f ðt; LÞ X @f ðt; LÞ 1 Gi 50 @t @Li i51

(9)

where Gi is the normal growth rate of the ith facet and f is the number density function. Equation 9 is associated to initial and boundary conditions, which are given by: f ð0; LÞ5f0 ðLÞ

(10a)

f ðt; 0Þ50

(10b)

where f0 ðLÞ denotes the seed distribution and no nucleation occurs. To account for concentration depletion in solution due to crystal growth, the PBE is coupled to a material balance ð dc d 1c VðLÞf ðt; LÞdL50 (11a) dt dt cð0Þ5c0

(11b)

here, V denotes the volume of a crystal with characteristic length vector L; c is the crystal density, c0 is the initial solution concentration, and is the size domain. The numerical solution of the PBE was obtained as reported elsewhere, i.e., using a customized finite volume high resolution scheme with a van Leer flux limiter22,36,37. As the disappearance of facets was avoided in all cases (distributions never crossed borders in the morphology domain map)38,39, no special attention needed to be paid to this case.

detectors. Incident light and particle interaction result in different levels of reflection, refraction, absorption, and diffraction, depending on the wavelength of light, the particle’s refractive index and its shape. Collecting the total scattering intensity due to all particles onto the photodiode detectors provides sufficient information to estimate the PSD assuming that particles are spherical, and that the scattering pattern is due to single scattering. The implementation of LD in this work assumes that the total scattering pattern is only given by single scattering and the scattering itself has been modeled by a Fraunhofer diffraction model, which has been shown by several authors to be a suitable approximation to light scattering by large crystals, i.e., greater than 20 lm40–42. More details regarding Fraunhofer diffraction and the exact mathematical and numerical methods employed can be found in the Appendix. Focused Beam Reflectance Measurement. The FBRM uses a rotating monochromatic laser beam that travels from the FBRM probe to the suspension and may be reflected whenever the laser encounters a particle. A chord length (s) measurement is obtained when the laser intercepts a particle, at a random position within the particle’s projected area, thus resulting in a wide range of chord lengths even for the same spherical particle43. Combining all measured chord lengths yields a chord length distribution (CLD), which may be used online to monitor a crystallization process. Although it is known that the CLD and the PSSD are connected (both s and D have units of length, e.g., lm), their exact relationship has only been established for systems consisting of spherical particles17. In this work, we follow the geometric approach to modeling the FBRM based on a dimensionless laser spot and ideal backscattering as described previously by Kempkes et al.16. Image Analysis. The state of the art methods to reconstruct PSSDs in crystallization processes are based on image

Particle sizing techniques As highlighted above, the VTB presented earlier was extended for this work to include simulations of additional, idealized measurement devices. In particular, four techniques were added: CC, LD, FBRM, and an imaging method based on SP. Coulter Counter. In a CC apparatus, particles are suspended in an electrolytic solution, where a sample tube containing an aperture is immersed. Two electrodes, placed in- and outside the sample tube, measure the instantaneous impedance of the solution. Due to an effect known as Coulter principle, whenever a particle passes through the small orifice into the sample tube, it results in a measurable increase in impedance which is proportional to the particle’s volume. As highlighted in Figure 1, the simulations performed in this work assume sufficiently dilute solutions, such that at any instance only one particle travels through the orifice, allowing precise particle volume measurements. PSDs are reconstructed by computing the volume equivalent lengths D, computed as 1 D5 6 V ðLÞ 3 for each of 2000 sampled particles, followed by simple binning of the space domain. Laser Diffraction. In an LD instrument, a filtered and expanded laser beam travels through a sample cell, in which the particles of interest are located. The resulting scattered beam is then focused by a lens onto an array of photodiode AIChE Journal

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Figure 4. Polytopic models of (a) vanillin, and (b) paracetamol. Miller indices for vanillin are f100g; f120g; f111g. For paracetamol, modeled facets are f110g; f001g. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]


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The quantification of the particle’s length and width are based on boundary curve calculations, as described in detail elsewhere for DP45. Particle sizing for SP is performed in an analogous fashion, that is, the measurement relies on the width calculation using a single boundary curve and an estimation of the maximum interparticle length, which is used to compute the length of the corresponding cylinder.

Comparison of Particle Sizing Technologies for Vanillin

Figure 5. Example of a vanillin crystal imaged in a DP setup where its projections onto the xz and yz plane are available. An imaging system based on SP would have obtained images with boundaries based on a single projection. In this work widths and lengths were estimated based on boundary curve calculations which rely on the projection’s area center, shown by black points20.

analysis20,23. Imaging systems can be divided into in and ex situ devices, which offer different advantages and disadvantages. While in situ probes enable direct insight into events occurring in the reactor the resulting image quality tends to be limited due to potentially high suspension densities, crystals that lie outside the device’s focal range, and insufficient illumination. All of these effects can cause particles to appear blurred and potentially be incorrectly classified and characterized by the image analysis software44. In contrast, ex situ systems can alleviate a number of the above issues by ensuring that only a fraction of particles is measured at optimal conditions, the drawbacks being mainly related to the necessary sampling, which might be undesired or cumbersome in many circumstances. In this work, our goal is to understand the difference in measurement accuracy when using either one or two orthogonal projections of crystals, assuming almost perfect conditions for either system. Hence, the single and dual projection (SP and DP) techniques that we have implemented are idealized versions of ex situ imaging devices with a flow cell whose geometry is identical to the one used previously20, which has a field of vision of 2 3 2 3 2.6 mm. It is assumed that crystal optical properties are favorable, allowing crystal boundaries to be accurately detected, and that crystals are always within the depth of field and well-illuminated, thus resulting in no blurriness. The two methods differ only in the number of projections used for the shape approximation; an example for the case of a vanillin crystal is shown in Figure 5. Our implementation of the two imaging devices simulates 10 particles per image, such that overlapping particle projections, an unavoidable occurrence for most systems, can indeed occur. To reconstruct the crystal’s length and width, we assume that its underlying shape is that of a cylinder, a choice motivated by the fact that a majority of elongated crystals can be well approximated this way22,45. Generic shapes have the advantage of enabling fast and robust particle reconstruction also when dealing with imperfect crystals20,46, while allowing for an intuitively understandable representation even of entire populations. 6

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Similar to the case discussed in the motivation section, we compare the results obtained with the various devices for a PSSD that is rigidly shifted along straight lines, the equivalent of particle growth with constant relative growth rates. In particular, we have considered the PSSD of flat vanillin needles for which we give initial conditions and relative growth rates in Table 2. It should be noted that through our choice of conditions one aspect ratio is kept constant, namely that between facet families {120} and {100}, hence obtaining a problem with only two independent characteristic lengths.

1D sizing techniques The one-dimensional distributions of CC, LD, and FBRM are compared in Figure 6 in terms of how the measured standard deviations and skewness (Þ evolve during crystal growth. Figure 6 demonstrates that there are discrepancies between the measurements obtained using the different techniques. With respect to the average size, the values are comparable in absolute terms, yet the observed changes of the mean size differ quite significantly: the FBRM measurements suggest a much larger increase in average size than the other two techniques. As also has been found in other studies26, the estimated mean size is not representative of the largest dimension, i.e., the length of the needles, but rather of their width. The standard deviations diverge drastically, ranging from a few micrometers (CC) up to over a hundred micrometers (FBRM), thus providing completely different impressions of the population’s width. Furthermore, it can be seen that all measured 1D distributions broaden with increasing size of the underlying distribution, an effect that is particularly noticeable for LD and FBRM. Still, even in our implementation of a CC, where the volume of each sampled crystal is measured precisely, the PSD broadened by 27%. The reason for the better performance of CC stems from the fact that the device—as modeled here—is only affected by the particle model mismatch. In contrast, the other two devices are affected by additional, random events, namely the orientation of the particle with respect to the probe at the time of measurement (LD and FBRM) and the exact location where the laser hits the particle (FBRM). Each additional effect decreases the measurement precision, which ultimately manifests itself in a distribution that is broader than it should be. Table 2. Mean Sizes (cf. Eq. 8) and Constant Relative Growth Rates of Different Vanillin Facets

Initial Final

Lf111g ½lm

Lf120g ½lm

Lf100g ½lm

Gf111g = Gf120g

Gf120g = Gf100g

60 163

20 30

10 15

10

2

Initial seed sizes are Gaussian with diagonal covariance matrix. The standard deviations are rf111g 57 lm, rf120g 53 lm, and rf100g 51:5 lm.


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cant tailing toward smaller sizes and thus in measurements that underestimate the average particle length in comparison with DP, a consequence of the fact that DP utilizes two projections to estimate the cylinder’s length, thereby mitigating the dependence on particle orientation. A comparison of the standard deviation along the length, 11, for the two devices, reveals that there is again an increase, which is particularly pronounced for the SP case; in fact, the DP device performs better than both LD and FBRM. For both imaging techniques, greater uncertainties are present when measuring a crystal’s width. The two devices estimate similar average values and standard deviations for distributions of smaller particles, the reason being that the DP technique simply uses an averaged value of the width of two projections, a difference that vanishes for large sample sets. However, as particles increase in size, the SP technique suffers from overestimation of particle width due to more frequent particle overlap, whereas such cases are typically discarded in the stereoscopic case20. While improved segmentation could mitigate this issue for the case of needles, the extent to which this effect can be reduced is limited, particularly in a more realistic setting with more complex backgrounds than what is assumed here.

Estimation of Growth Kinetics of Paracetamol In the previous sections, the results of the various sizing techniques were presented, assessed and compared based on dimensionless relative growth rates. Nevertheless, it is not clear at this point whether even the best-performing devices represent useful tools for the determination of, e.g., growth kinetics. In this section, the quantitative consequences of the instrument choice on parameter estimation shall be studied using specific, absolute values for the growth rates. Particularly, we compare estimated and real values of growth rates for a compound whose real habit and kinetics are known. Figure 6. Particle size simulations of vanillin crystals as measured by the CC, LD, and FBRM. The (a) standard deviation and (b) skewness of the distributions are shown as functions of the mean size. Note that the skewness of a normalized distribution is defined as c5ðl3 23l1 l2 12l31 Þ=ðl2 2l21 Þð3=2Þ 53.

Similarly, an analysis of the skewness, i.e., a measure of the distribution’s asymmetry, shown in Figure 6b reveals that the measurements obtained using CC did not introduce any asymmetry in the obtained distribution. In contrast, the other two techniques show significant tailing toward larger sizes as indicated by the positive skew. Noticeably, this bias increases for both LD and FBRM with changing particle size, yet the effect is more pronounced for the LD instrument. In summary, we find that particle sizing measurements can differ substantially, even when the measurement techniques assume the same particle shape.

2D sizing techniques Initial and final 2D PSSDs, as measured by SP and DP, are given on the left-hand side of Figure 7, whereas the right-hand side shows how the standard deviation evolves as a function of average length (L1 ) and width (L2 ). It is evident that a technique based on SP results in broad distributions with signifiAIChE Journal

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Figure 7. Simulation of vanillin crystals measured using SP and DP. Left: initial and final number based particle size distribution measured using DP (red) and SP (blue). Top right: standard deviation along L1 plotted as a function of average length. Bottom right: standard deviation along L2 plotted as a function of average width. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]


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an independent device, white noise was added. The amount of white noise added is in accordance with the signal to noise ratio typically encountered in experiments and is not expected to significantly alter the estimated parameters. The output of the eVTB consists of a set of PSDs for every particle sizing technique at each time step, together with the solute concentration.

Parameter estimation

Figure 8. Facet-specific paracetamol growth rate as a function of supersaturation47. The birth and spread growth model reported in Eq. 12 was used to fit the experimental data. Note that data obtained at supersaturations above 1.15 were not used, as those conditions lead to prismatic rather than elongated morphologies. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary. com.]

Setup for in silico experiments The true behavior of the system is modeled using the thermodynamic and kinetic data of paracetamol in ethanol reported by Boerrigter et al.11 Given that these data points are described well by a birth and spread model (cf. Figure 8), all growth rates in this work are assumed to follow the corresponding kinetic law47,48: kj GF 5ki ðS21Þ2=3 ðln SÞ1=6 exp 2 2 (12) T ln S Here, GF may represent any growth rate considered here, i.e., G1, G2, GD ; Gf001g , and Gf110g . As an example, the kinetics of paracetamol, where two facets are considered, are modeled as hlmi 0:74 104 ½K2 Gf001g 50:12 ðS21Þ2=3 ðln SÞ1=6 exp 2 s T 2 ln S (13a) hlmi 4 7:58 10 ½K2 ðS21Þ2=3 ðln SÞ1=6 exp 2 Gf110g 54:56 s T 2 ln S (13b) where ki and kj are substituted with the parameters that were used to describe the experimental data of Boerrigter et al.11. In an attempt to exclude or to minimize nucleation and agglomeration, growth rate kinetics are often estimated using seeded desupersaturation experiments at low supersaturations1,22. To mimic such conditions, our in silico experiments were performed isothermally at low supersaturations, i.e., S 1:13. The initial conditions encompass various seed sizes and supersaturations as reported in Table 3, i.e., assuming that these properties may be chosen freely. For the measurement of the solute concentration, which is assumed to be obtained by 8

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Growth kinetics parameter estimation was based on the concentration profile and average length as modeled on the basis of Eq. 12, i.e., employing the correct functional form of the growth rate. The fitting is performed by finding the maximum likelihood estimate (MLE); given that the independent variables are deterministic and assuming that errors are normally distributed with zero mean and given variance, and that errors at different times are not correlated, the objective function for finding the MLE reduces to49: " # Nv Nt X Nt X 2 (14) ln ðykl 2^ y kl ðpÞÞ UðpÞ5 2 k51 l51 where Nt is the number of observations, Nv is the number of measured outputs, ykl is the experimental output k at time l, and y^kl ðpÞ is the corresponding model estimate given a parameter set p. Here, the output vector yl at time l consist of a con that is: centration and average characteristic size L, " # cðtl Þ yl 5 (15) lÞ Lðt For CC and LD measurements the average size equals the D, average size of the measured equivalent sphere, that is, L5 and only one growth rate is estimated, that is, GD . In contrast, for SP and DP the average particle (cylinder) length was used L1 , and two growth rates are estias measured output, i.e., L5 mated, G1 and G2, with a total of four parameters. Note that the system is structurally observable, allowing for the estimation of both growth rates. The estimated parameter values for all cases are reported in Table 4. A consequence of the unclear relationship between the CLD and the PSD is that performing the parameter estimation on the basis of the CLD is generally not recommended. For this reason and for the sake of clarity, we do not show the results obtained with the FBRM; however, it is worth mentioning that Table 3. Operating Conditions for Seeded Desupersaturation of Paracetamol in Ethanol Run

Lf001g ½lm

Lf110g ½lm

S0 ½2

F1 F2 F3 F4 F5 F6 F7 F8 F9 F10

30 30 45 20 20 25 50 50 60 60

15 15 15 15 15 15 30 30 30 30

1:08 1:12 1:08 1:08 1:12 1:08 1:08 1:13 1:08 1:13

Seed crystal sizes are given as distances from the crystal center to the crystallographic facets. Seed sizes are normally distributed with diagonal covariance matrix and with mean sizes as indicated below. The standard deviations are rf001g 54 lm, and rf110g 53 lm. Temperature, seed, and solvent mass were fixed at 258C, 2 g of seeds and 2 kg of ethanol, respectively.


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Table 4. Modeled and Estimated Parameters Using Various Measurement Sizing Techniques. Parameters Obtained by DP and SP Cannot be Directly Compared to the Real Parameters Since Both Paracetamol Facets Contribute to the Particle’s Length k1 ½lm=s DP SP CC LD

0:38 0:23 0:10 0:15

k2 ½K 2 1.11104 0.89104 1.65104 1.73104

k3 ½lm=s 2.19 1.59 2 2

k4 ½K 2 2.21104 2.00104 2 2

Parameters obtained by DP and SP cannot be directly compared to the real parameters since both paracetamol facets contribute to the particle’s length.

the obtained results are qualitatively similar to those obtained using the CC and LD device.

Results and discussion

Figure

In the following, particle sizing techniques that use one or two characteristic descriptors are presented separately. Figures 9 and 10 represent concatenated measurements of in silico concentration profiles (c), and of average sizes for CC and LD, and for SP and DP, respectively. In addition, the standard deviations of the distributions, which are not fitted, are shown for illustration purposes. Coulter Counter and Laser Diffraction. As can be seen in Figure 9, the maximum likelihood parameters obtained using a size-independent growth model can fit the concentration profiles well; however, they cannot simultaneously describe the average size accurately. Vice versa, a good fit of the average size alone would be feasible—provided that the outputs are weighted accordingly—yet this would come at the price of a badly-fitted concentration profile. This analysis reveals that estimating parameters that are able to describe both the solute concentration and the average size using a size-independent growth mechanism is not possible when data is obtained by a 1D sizing technique. Considering a laboratory experimental setting, to fit both

Figure 9. Paracetamol seeded desupersaturation experiments and fits using PSDs obtained with CC, LD, and concentration measurements. Note that the standard deviation r was not used for the fitting. Modeled standard deviations increase by small amounts due to numerical diffusion. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

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10. Paracetamol seeded desupersaturation experiments and fits of concentration, DP and SP measurement techniques. Note that the standard deviation r11 was not used for the fitting. Modeled standard deviations increase by small amounts due to numerical diffusion. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

outputs with satisfactory accuracy, other phenomena would need to be included in the model, such as agglomeration, breakage, growth rate dispersion, or size-dependent growth, all of which are known to be absent here. Furthermore, the introduction of these mechanisms might be erroneously justified by the observed increase in the standard deviation, , which is particularly evident in the case of the LD measurement and cannot be replicated by a size-independent growth model. A crystallization model which includes mechanisms that are absent would be useless for process design, if not harmful. Whenever considering parameter estimation based on 1D PSDs, obtained either by CC or LD, it is thus crucial to assess

Figure 11. Comparison between the real paracetamol length growth rate and that estimated using DP and SP instruments. The bounding box is shown in gray to illustrate at which direction the length grows. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]


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9

to which extent crystal shape has evolved, and only perform parameter estimation in cases where crystal growth does not lead to a large change in crystal shape. Single and Dual Projection measurements. The results for the two imaging devices shown in Figure 10 immediately make clear that incorporating a more adequate particle model allows for precise description of both fitted outputs (c and L1 ). Indeed, model fits for SP and DP, shown by solid red and blue lines, respectively, are in excellent agreement with the simulated measurements points, indicated by the corresponding markers. An important advantage gained by using DP is illustrated in the third row of Figure 10 where the evolution of 11, which was not used for fitting, is shown. Consistent with the previous assessment, the width of the distribution measured by SP increases considerably whereas that of the DP distribution does not. In an experimental setting where the true kinetics are unknown, the broadening of the former would undoubtedly call the completeness of the pure size-independent growth model into question. Comparison with True Growth Rate in Length Direction. The single most important feature of imaging techniques is that they allow for the direct and robust measurement of multiple characteristic sizes. Therefore, given that the SP and DP measurements of lengths and widths are markedly different, it is of interest to compare estimated and true growth rates of the crystal length. For the cylindrical generic particle the growth of a single facet contributes to the cylinder’s length, while for the modeled paracetamol crystal the growth of either one of its two facet families will influence the particle’s length. As a consequence, care must be taken when comparing modeled and estimated parameters and some post-processing of the data is required22,50. Within certain limits and making use of Eq. 8, it is possible to compare the growth rate of the length of a crystal using the real parameters with that given by the estimated parameters, even when particle models differ. Namely, one can compare the growth rates of the bounding boxes of the two objects, defined as the smallest-volume box containing the whole crystal. Clearly, the growth rate of the length of the bounding box of a cylinder is identical to G1. To obtain the corresponding information for the paracetamol crystals, consider the distance from the centroid to the furthest real vertex, whose position is determined by the intersection of two {110} facets and one {001} facet (shown in the inset of Figure 11). Through geometric considerations, it can be shown that the rate of change of this distance, Gv , is related to the ‘real’ G1 by the relationship29: G1 52 zT Gv

(16a)

52:22 Gf001g 11:53 Gf110g

(16b)

where z is the unit normal vector pointing in the length direction of the bounding box as indicated in Figure 11. Note that, while Eq. 16a has general validity, Eq. 16b is specific to the case of paracetamol. Also in Figure 11, real and estimated length growth rates are plotted as functions of supersaturation. Both SP and DP lead to imperfect results, however, parameter estimation based on DP measurements results in an improved estimation over the supersaturation spectrum considered. For process characterization of elongated particles, an unavoidable step in the development of model-based process control, stereoscopic imaging provides advantages when compared with other approaches. Although there are still many improvements to be made, from more sophisticated image analysis techniques to better equipment design, the potential of this method is unparalleled. 10

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Conclusion Within the crystallization community, treating crystals of all shapes as if they were spheres seems to be widely accepted even though there is a general consensus that this simplification introduces errors, whose exact extent is largely unknown. The goal of this work was the development of an in silico tool that allows for the quantification of these errors in an idealized setting and for the analysis of their consequences. Moreover, we have attempted to demonstrate the repercussions that the choice of the measurement technique can have on the mechanistic interpretation of crystallization processes. Inferring an incorrect mechanism from experimental data might lead to unnecessarily complex, overparameterized process models, whose predictions outside of the verified region might be entirely incorrect. Note that, while the presented case study has focused on batch experiments, which are used frequently for kinetic parameter estimation, there is no restriction and the methodology can be applied also to other cases. All sizing techniques relying on a single size descriptor, namely CC, LD, and FBRM, were shown to be inherently limited for parameter estimation of particles whose shape evolve. Estimations based on 2D PSD information resulted in a major improvement, yet it was shown that SP devices are prone to yield biased distributions, both in terms of average size and distribution width. Whereas the presented conclusions strictly apply only to morphologies of elongated particles, we expect that similar issues arise for other nonequant shapes, such as platelets, agglomerates, and dendrites. It is important to highlight that even in the ideal framework implemented here the effects observed were significant. Although data processing of the different devices might be improved (e.g., better segmentation of particles in SP), the performance of all methods is expected to deteriorate further under laboratory conditions. Ultimately, different applications require different particle sizing techniques and there is no one solution that fulfills all demands. Particle volumes can be accurately assessed using CC, while broad distributions, ranging from nano- up to micrometers, can be conveniently measured by LD. Even the FBRM, whose outputs remain difficult to interpret, can be used in situ for reproducibility testing or for model-free process control5,52. Still, when it comes to evolving particle shape, imaging methods—particularly those making use of multiple crystal projections—remain the only tools capable of dealing with the emerging issues in a satisfactory manner.

Acknowledgments The authors are thankful to Dr. Thomas Vetter at the University of Manchester for helpful discussions. Furthermore, the authors gratefully acknowledge the financial support of the Swiss National Science Foundation (project number 200021-135218).

Literature Cited 1. Vetter T, Mazzotti M, Brozio J. Slowing the growth rate of Ibuprofen crystals using the polymeric additive pluronic F127. Cryst Growth Des. 2011;11(9):3813–3821. 2. Kempkes M, Vetter T, Mazzotti M. Monitoring the particle size and shape in the crystallization of paracetamol from water. Chem Eng Res Des. 2010;88(4):447–454. 3. Nagy ZK, Fujiwara M, Woo XY, Braatz RD. Determination of the kinetic parameters for the crystallization of paracetamol from water using metastable zone width experiments. Ind Eng Chem Res. 2008;47:1245–1252.


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4. Li H, Kawajiri Y, Grover MA, Rousseau RW. Application of an empirical FBRM model to estimate crystal size distributions in batch crystallization. Cryst Growth Des. 2014;14(2):607–616. 5. Nagy ZK, Fevotte G, Kramer H, Simon LL. Recent advances in the monitoring, modelling and control of crystallization systems. Chem Eng Res Des. 2013;91(10):1903–1922. 6. Woo XY, Tan RBH, Braatz RD. Equation—micromixing simulation of impinging jet crystallizers. Cryst Growth Des. 2009;9(1):156–164. 7. de Albuquerque I, Mazzotti M. Crystallization process design using thermodynamics to avoid oiling out in a mixture of vanillin and water. Cryst Growth Des. 2014;14:5617–5625. 8. Wibowo C, Chang WC, Ng KM. Design of integrated crystallization systems. AIChE J. 2001;47(11):2474–2492. 9. Yu ZQ, Chow PS, Tan RBH. Interpretation of focused beam reflectance measurement (FBRM) data via simulated crystallization. Org Proc Res Dev. 2008;12(4):646–654. 10. Berglund KA, Murphy VG. Modeling growth rate dispersion in a batch sucrose crystallizer. Ind Eng Chem Fund. 1986;25:174–176. 11. Boerrigter SXM, Cuppen HM, Ristic RI, Sherwood JN. Explanation for the supersaturation-dependent morphology of monoclinic paracetamol. Cryst Growth Des. 2002;2(5):357–361. 12. Lu JJ, Ulrich J. The influence of supersaturation on crystal morphology—experimental and theoretical study. Cryst Res Technol. 2005; 40(9):839–846. 13. Lovette MA, Browning AR, Griffin DW, Sizemore JP, Snyder RC, Doherty MF. Crystal shape engineering. Cryst Growth Des. 2008; 47(1):9812–9833. 14. Singh MR, Chakraborty J, Nere N, Tung Hh, Bordawekar S, Ramkrishna D. Image-analysis-based method for 3D crystal morphology measurement and polymorph identification using confocal microscopy. Cryst Growth Des. 2012;12:3735–3748. 15. Kovacevic T, Reinhold A, Briesen H. Identifying faceted crystal shape from three-dimensional tomography data. Cryst Growth Des. 2014;14:1666–1675. 16. Kempkes M, Eggers J, Mazzotti M. Measurement of particle size and shape by FBRM and in situ microscopy. Chem Eng Sci. 2008; 63(19):4656–4675. 17. Worlitschek J, Hocker T, Mazzotti M. Restoration of PSD from chord length distribution data using the method of projections onto convex sets. Part Part Sys Char. 2005;22(2):81–98. ˇık J, Morbidelli M. Modeling focused beam reflec18. Vaccaro A, Sefc tance measurement and its application to sizing of particles of variable shape. Part Part Sys Char. 2006;23(5):360–373. 19. Rawlings JB, Miller SM, Witkowskit WR. Model identification and control of solution crystallization processes: a review. Ind Eng Chem Res. 1993;32:1275–1296. 20. Schorsch S, Ochsenbein DR, Vetter T, Morari M, Mazzotti M. High accuracy online measurement of multidimensional particle size distributions during crystallization. Chem Eng Sci. 2014;105:155–168. 21. Ma CY, Wang XZ. Model identification of crystal facet growth kinetics in morphological population balance modeling of l–glutamic acid crystallization and experimental validation. Chem Eng Sci. 2012;70:22–30. 22. Ochsenbein DR, Schorsch S, Vetter T, Mazzotti M, Morari M. Growth rate estimation of L-glutamic acid from online measurements of multidimensional particle size distributions and concentration. Ind Eng Chem Res. 2014;53:9136–9148. 23. Borchert C, Temmel E, Eisenschmidt H, Lorenz H, Seidelmorgenstern A, Sundmacher K. Image-based in situ identification of face specific crystal growth rates from crystal populations. Cryst Growth Des. 2014;14:952–971. 24. Bosquillon C, Lombry C, Preat V, Vanbever R. Comparison of particle sizing techniques in the case of inhalation dry powders. J Pharm Sci. 2001;90(12):2032–2041. 25. Silva AFT, Burggraeve A, Denon Q, Van Der Meeren P, Sandler N, Van Den Kerkhof T, Hellings M, Vervaet C, Remon JP, Lopes JA, De Beer T. Particle sizing measurements in pharmaceutical applications: comparison of in-process methods versus off-line methods. Eur J Pharm Biopharm. 2013;85(3 PART B):1006–1018. http://dx. doi.org/10.1016/j.ejpb.2013.03.032. 26. Hamilton P, Littlejohn D, Nordon A, Sefcik J, Slavin P. Validity of particle size analysis techniques for measurement of the attrition that occurs during vacuum agitated powder drying of needle-shaped particles. Analyst 2012;137(1):118. 27. Doremus RH, Roberts BW, Turnbull D. Growth and Perfection of Crystals. New York: Wiley, 1958. 28. Chernov A. The kinetics of the growth forms of crystals. Sov Phys Crystallogr. 1963;7:728–730.

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29. Zhang Y, Sizemore JP, Doherty MF. Shape evolution of 3dimensional faceted crystals. AIChE J. 2006;52(5):1906–1915. 30. Van der Heijden AEDM, Geertman RM, Bennema P. Solventdependent growth morphology of caprolactam. J Phys D Appl Phys. 1991;24:123–126. 31. Zhang X, Zhang P, Wei K, Wang Y, Ma R. The study of continuous membrane crystallization on lysozyme. Desalination 2008;219: 101–117. 32. Di G, Curcio E, Drioli E. Trypsin crystallization by membrane-based techniques. J Struct Biol. 2005;150:41–49. 33. Zhang Y, Doherty MF. Simultaneous prediction of crystal shape and size for solution crystallization. AIChE J. 2004;50(9):2101–2112. 34. McCrone WC. Vanillin I (3-methoxy-4-hydroxybenzaldehyde). Anal Chem. 1950;22(3):500. 35. Herceg M, Kvasnica M, Jones C, Morari M. Multi-parametric Toolbox 3.0. 2013. http://control.ee.ethz.ch/~mpt. 36. Ochsenbein DR, Schorsch S, Salvatori F, Vetter T, Morari M, Mazzotti M. Modeling the facet growth rate dispersion of lglutamic acid—combining single crystal experiments with nD particle size distribution data. Chem Eng Sci. 2015;133:30–43. 37. van Leer B. Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a second-order scheme. J Comp Phys. 1974;14(4):361–370. 38. Borchert C, Sundmacher K. Morphology evolution of crystal populations: modeling and observation analysis. Chem Eng Sci. 2012;70: 87–98. http://dx.doi.org/10.1016/j.ces.2011.05.057. 39. Ramkrishna D, Singh MR. Population balance modeling: current status and future prospects. Annu Rev Chem Biomol Eng. 2014;5:123–146. 40. Lee Black D, McQuay MQ, Bonin MP. Laser-based techniques for particle-size measurement: a review of sizing methods and their industrial applications. Prog Energ Combust. 1996;22(3):267–306. 41. de Boer GBJ, de Weerd C, Thoenes D, Goossens HWJ. Laser diffraction spectrometry: Fraunhofer diffraction versus Mie scattering. Part Part Sys Char. 1987;4(1–4):14–19. 42. Jones A. Error contour charts relevant to particle sizing by forwardscattered lobe methods. J Phys D Appl Phys. 1977;10:L163–L165. 43. Ruf A, Worlitschek J, Mazzotti M. Modeling and experimental analysis of PSD measurements through FBRM. Part Part Sys Char. 2000;17(4):167–179. 44. Patience DB. Crystal engineering through particle size and shape monitoring, modeling, and control. Ph.D. thesis, University of WisconsinMadison, 2002. 45. Schorsch S, Vetter T, Mazzotti M. Measuring multidimensional particle size distributions during crystallization. Chem Eng Sci. 2012;77:130–142. 46. Schorsch S, Hours JH, Vetter T, Mazzotti M, Jones CN. An optimization-based approach to extract faceted crystal shapes from stereoscopic images. Comp Chem Eng. 2015;75:171–183. http://dx. doi.org/10.1016/j.compchemeng.2015.01.016. 47. Ristic RI, Finnie S, Sheen DB, Sherwood JN. Macro- and micromorphology of monoclinic paracetamol grown from pure aqueous solution. J Phys Chem B. 2001;105(38):9057–9066. 48. Mersmann A. Crystallization Technology Handbook, 2nd ed. New York: CRC Press, 2001. 49. Bard Y. Nonlinear Parameter Estimation. New York: Academic Press, 1974. 50. Kitamura M, Ishizu T. Growth kinetics and morphological change of polymorphs of L-glutamic acid. J Cryst Growth. 2000;209(1): 138–145. 51. Borchert C, Sundmacher K. Efficient formulation of crystal shape evolution equations. Chem Eng Sci. 2012;84:85–99. 52. Saleemi AN, Rielly CD, Nagy ZK. Comparative investigation of supersaturation and automated direct nucleation control of crystal size distributions using ATR-UV/vis spectroscopy and FBRM. Cryst Growth Des. 2012;12(4):1792–1807. 53. Randolph A, Larson M. Theory of Particulate Processes: Analysis and Techniques of Continuous Crystallization. New York: Academic Press, 1971. 54. Heffels C, Heitzmann D, Hirleman ED, Scarlett B. Forwards light scattering for arbitrary sharp-edged convex crystals in Fraunhofer and anomalous diffraction approximations. Appl Optics. 1995;34(28):6553–6560. 55. Phillips DL. A Technique for the numerical solution of certain integral equations of the first kind. J ACM. 1962;9:84–97. 56. Twomey S. On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature. J ACM. 1963;10:97–101. 57. Hansen C. Analysis of discrete ill–posed problems by means of the L-curve. SIAM Rev. 1992;34(4):561–580.


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Figure A1. Intensity Iðh; /Þ of the diffraction pattern due to a single particle in the Fraunhofer diffraction approximation shown by the contour plot on the right. From the left hand side an expanded laser beam of wavelength k and intensity y0 travels through the medium and interacts with an elongated particle. Within the framework of Fraunhofer diffraction the diffraction pattern arises exclusively due to the particle’s projected area Sj . The diffraction pattern is calculated by integrating over the entire particle’s projected area, shown enclosed in a rectangle at the center of the Cartesian coordinates defined by green arrows, over the azimuthal angle /. For clarity objects are not shown in scale. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary. com.]

Appendix Laser Diffraction. As we aim at simulating an ideal measurement, dilute solutions leading to single scattering were assumed, while noise was not added to the diffraction patterns. Predicting the intensity of the resulting interaction between particle and light is possible by solving the Maxwell equations; however, several simplifications apply for particles of relevance to crystallization processes which are considered in this work. Most crystals are large in comparison with the wavelength of light since LD lasers typically emit light with submicron wavelengths. Several authors have shown that for particles larger than 20 m, light scattering may be well approximated by Fraunhofer diffraction in the near-forward direction40–42. Simulating light scattering with Fraunhofer diffraction is a significant simplification to solving the Maxwell equations, or even Mie theory which is only valid for spherical particles. Fraunhofer diffraction describes the complex amplitude of the diffraction electric field K in the far-field approximation, and it depends only on a particle’s projected area S19. For a given particle j with projected area given by Sj , the field Kj can be written in Cartesian coordinates as:

Kj ð ; Þ5

K0 F

ðð

eðikAð ; ÞxÞ eðikBð ; ÞzÞ dxdz

(A1a)

Sj

Að ; Þ5sin cos

(A1b)

Bð ; Þ5sin sin

(A1c)

where i is an imaginary unit, F is the lens’s focal point, k52= is the angular wavenumber, and angles and are defined in Figure A1. Equation A1c can be transformed into a contour integral using Green’s theorem54. In the absence of multiple scattering, the intensity I of a diffraction pattern due to an ensemble of N crystals is obtained using the superposition principle.

Ið ; Þ5

N 1X jKj ð ; Þj2 N j51

(A2)

where a jth particle’s diffracted intensity is the square of the absolute value of its field amplitude Kj. Note that the projected area Sj is obtained at a random orientation for every particle.

12

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From the collected intensity distribution over angles the underlying equivalent spherical diameter PSD can be estimated. The Fraunhofer diffraction intensity pattern due to an ensemble of spheres is given by

Ið Þ5

I0 2 k F2

ð1 0

2

J1 ð Þ 2 f ð Þd

(A3)

where 5kD=2 is the dimensionless size parameter, J1 is the Bessel function of first kind, and f ð Þ represents the spheres PSD as a function of . By integrating I with respect to , the dependence of I on the azimuthal angle was removed in Eq. A3. Given a known intensity distribution I, measured by a LD device or simulated using Fraunhofer diffraction as done here, it is possible to recover the equivalent sphere PSD. The PSD is obtained by solving the inverse problem posed by Eq. A3 for f ð Þ. Commercial LD devices contain an array of photo detectors arranged along , hence I is measured at discrete angles . Numerically the integrand in Eq. A3 is evaluated over angles covering all d detectors and p discretization sizes used in f ð Þ, yielding a linear system of equations, I5Af, where I 2 Rd31 ; A 2 Rd3p , and f 2 Rp31 . Matrix A consist of the scattering coefficients ajk at detector j due to particle’s of kth size, and is typically ill-conditioned. The regularization method developed independently by Phillips and Twomey was used to recover the PSD from LD measurements55,56. A further condition ensuring non-negativity of f ð Þ was imposed19, resulting in the following quadratic programming problem:

minimizef subject to

1 c> f1 f > ðA> A1L> LÞf 2 fk ð k Þ 0; k51; . . . ; p

(A4)

where and L are the regularization parameter and matrix, respectively, and c522A> y. Matrix L consists of the coefficients for the second derivative finite difference approximation, and the regularization parameter was found using the L-curve approach57. Equation A4 was solved in Matlab using quadprog and the activeset algorithm. Once f ð Þ is obtained, the distribution f(D) can easily be recovered by changing the size domain from to D. Manuscript received Dec. 7, 2015, and revision received Mar. 17, 2016.


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