Crystal Growth Rate Dispersion versus Size-Dependent Crystal

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Crystal Growth Rate Dispersion versus Size-Dependent Crystal Growth: Appropriate Modeling for Crystallization Processes Published as part of the Crystal Growth & Design virtual special issue of selected papers presented at the 11th International Workshop on the Crystal Growth of Organic Materials (CGOM11 Nara, Japan), a joint meeting with Asian Crystallization Technology Symposium (ACTS 2014). Sukanya Srisanga,†,# Adrian E. Flood,*,† Shaun C. Galbraith,† Supagorn Rugmai,‡ Siriwat Soontaranon,‡ and Joachim Ulrich§ †

School of Chemical Engineering, Suranaree University of Technology, 111 University Avenue, Muang District, Nakhon Ratchasima, 30000, Thailand ‡ Synchrotron Light Research Institute (Public Organazation) (SLRI), 111 University Avenue, Muang District, Nakhon Ratchasima 30000, Thailand § Martin-Luther-Universität Halle-Wittenberg, Zentrum für Ingenieurwissenschaften, Verfahrenstechnik/TVT, 06099 Halle (Saale), Germany ABSTRACT: Crystal growth rate dispersion (GRD) and size-dependent crystal growth (SDG) models are models to extend McCabe’s ΔL Law to more accurately account for variation in the crystal growth rates within a population of crystals. GRD is a phenomenon where the crystal growth rate either fluctuates randomly over time or varies over a population of crystals. SDG is where the growth rate of a crystal depends on its size, typically with growth rates assumed to increase monotonically with crystal size. Although it has been recognized for more than 30 years that, except for extremely small crystals, SDG is an artifact of GRD, it is still common in the literature for GRD in experimental results to be modeled using SDG models. This discussion will present some background and new experiments on the mechanism and extent of GRD to demonstrate that GRD is a real phenomenon, whereas SDG is largely an artifact, and some modeling work to demonstrate that SDG models cannot successfully replicate crystal size distribution data that originate due to GRD. More work needs to be done in accurate population balance modeling for processes where GRD is significant rather than assuming that SDG models are adequate.

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INTRODUCTION McCabe’s ΔL Law1 is probably the most significant historical insight into modeling the crystal growth of a population of particles. A statement of the law taken from the original paper is “...geometrically similar crystals of the same material suspended in the same solution grow at the same rate if the growth is measured as the increase in length of geometrically corresponding distances on all of the crystals.” This is significant mostly in that it shows that the most convenient independent variable representing the particle size in the population balance is a linear measure of crystal sizefor instance, a volume equivalent diameterrather than, for instance, the crystal volume. However, it also suggests that all crystals in a population, irrespective of their size, should have equal growth rates under equal conditions of growth, including equal conditions with respect to fluid dynamics. Even at this point, McCabe recognized the effect of size on the solubility and thus the growth rate and made the point that “it is very doubtful that the differences in solubility of the various faces of © XXXX American Chemical Society

a single crystal, or of different size particles of the same material, are large enough to influence crystal growth unless the crystals are less than 0.002 cm [20 μm] in diameter”. More recent work has further reduced the size range where the size dependency is likely to be significant, with Myerson and Ginde suggesting that size-dependent growth is only significant for crystals of size less than 1 μm.2 Later, researchers recognized the possibility of the growth rate depending on the size of the particle, over a wide range of crystal size (size-dependent growth or SDG) and growth being a probability distribution for a population of crystals (growth rate dispersion or GRD). Crystal GRD was first described in 1969.3,4 It describes the phenomenon that different crystals within a population (under uniform conditions) exhibit a range of growth rates. The phenomenon has been attributed to the crystals in a population having a range of internal lattice perfection5−7 and/or to surface Received: January 28, 2015

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DOI: 10.1021/acs.cgd.5b00126 Cryst. Growth Des. XXXX, XXX, XXX−XXX

Crystal Growth & Design

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effects induced by the crystal’s growth history.8−10 A good review of what is known about GRD has already been published11unfortunately there has not been a significant jump in understanding the basic mechanisms of the phenomenon since that time. Three major models have been proposed to model GRD and help integrate it into the population balance equation, namely, the growth diffusivity model,12 the constant (or inherent) crystal growth model,13 and the common history (CH) growth model.14 The growth diffusivity model assumed that all crystals in a population grow at the same time-averaged growth rate, but at any point in time the growth rate of a crystal randomly fluctuates around this mean growth rate. The constant crystal growth model assumes that each crystal in the population has its own inherent growth rate, which is a constant value under a given set of conditions, but which is different than the growth rate of other crystals within the population. The common history model can be used where the seed crystal population follows a particular behavior; if the nuclei from which the population was produced nucleated at the same instant and have experienced the same conditions since that time (thus the name “common history”) then the size distribution of the crystals at any time will be proportional to the distribution of inherent crystal growth rates. Where this occurs the modeling of the GRD within the population balance (where nucleation is negligible) is significantly simplified. The growth diffusivity model is not currently in use. Modeling of processes with GRD has been attempted by a number of studies. For example, Zumstein and Rousseau used the growth diffusivity model in combination with the CCG model to model both batch and MSMPR crystallizers.15 The CCG model has been used for MSMPR crystallizers16−18 and also in batch crystallizers.19−22 The common history model also has been used in population balance models for batch crystallizers.23,24 SDG had been proposed far before GRD, and there is a very wide range of empirical models proposed for the relation between the crystal growth rate and the crystal sizein fact any function relating growth rate to size is possible; however, simple monotonically increasing functions are generally used. A large number of papers have applied SDG models to the population balance equation for batch25−32 and continuous32−34 crystallizers. It has been recognized for a significant period of time that SDG is not a physically correct model of crystal growth in the vast majority of cases, and that dispersion of growth rates can appear to be size-dependent if care is not taken to investigate individual crystals rather than a population. For instance, Girolami and Rousseau35 have made the comment “...quantitative and qualitative results show that what has often been referred to as size dependent growth is in fact a manifestation of growth rate dispersion.” One of the authors of the current work has also commented, “it was found that there is no proof for size dependent growth. Size dependent growth was expected, as it could be easily explained by the proven phenomena of GRD.”36 It is important to note that the discussion presented here is not intended for very small crystalsthose less than ca. 2 μmwhich grow at a reduced rate compared to their larger brethren because they have a larger solubility due to the GibbsThompon effect. However, most papers presenting sizedependent growth models consider size dependency over a far wider range of crystal size than this. It is also still evident

that SDG models are presented in articles as being suitable to model systems where GRD is the underlying phenomenon. The objectives of the current paper are to present evidence from carefully performed experiments to demonstrate that growth rate dispersion is a real phenomenon that may sometimes be mistaken for a size-dependent crystal growth rate, to present modeling work showing that size-dependent growth models are not equivalent to growth rate dispersion modelsand thus should not be used in their placeand to call for more work on modeling of batch and continuous crystallizers using models that can accurately model the phenomenon of growth rate dispersion.

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RESULTS AND DISCUSSION Two mechanisms have been proposed to be responsible for crystal growth rate dispersion. The first proposed mechanism is that differences in internal crystalline perfection create differences in the stability of the crystals within the population and differences in the rate that entities can be incorporated into the surface of the crystals. An initial concept was that crystals with higher mosaic spread, indicating larger concentrations of dislocations in the crystal, could grow at enhanced rates due to the relationship between crystal growth rates and dislocation density predicted by the Burton−Cabrera−Frank model; however, later the relationship in real crystals was shown to be inverse to thishigher mosaic spread correlating with lower crystal growth rates.5 Other studies have questioned this result, asking whether radiation damage might be responsible for the variation in mosaic spread.37 Certainly, however, crystals that have suffered mechanical damage that changes their shape grow significantly faster while the shape of the crystal is corrected, and perhaps for some time after this.38,39 It is evident that there is a range of lattice perfection in a population of crystals, with Figure 1 showing some of our recent results on the mosaic

Figure 1. Range of mosaic spread (estimated as the full diffraction peak width at half the maximum intensity) of potassium dihydrogen phosphate crystals grown from aqueous solution at 3.0% relative supersaturation at 25 °C.

spread of potassium dihydrogen phosphate (KDP) grown at a relative supersaturation of 3.0% at 25 °C. The diffraction measurements were carried out at BL1.3W of the Synchrotron Light Research Institute. It is also predictable that such variation in the lattice perfection will result in differences in the thermodynamic stability in the population leading to some degree of GRD. Individual crystals were measured multiple B



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Figure 2. AFM micrographs after growth of KDP in aqueous solution at 25 °C and 1.48% relative supersaturation. (Left) Crystal 1; (right) crystal 2.

Figure 3. Examples of the effect of size-dependent growth models on the relationship between particle size and time for crystals in a batch cell (left) exponential increase to a maximum growth rate; (right) the growth rate is a linear function of particle size.

times, and no evidence of change in the mosaic spread due to radiation damage was noticed. Varying degrees of irregularity in the surface of crystals due to variations in crystal growth rate history8−10 have also been proposed as a cause of a spread in growth rates. Surface analysis of populations of KDP crystals after growth, Figure 2, has also revealed a range of the number concentration of surface features at the growing surface of the crystals, with an expectation that the larger the concentration of these features the smaller the growth rate of the crystal will be, as described previously.8,9 A large number of experiments, including early work in growth cells by the group of Berglund,40−46 have shown that individual crystals within a population grow at a constant rate over time if the solution conditions remain constant; these results preclude the mechanism of size-dependent growth since the size of the crystals does change significantly over the time of the experiment, whereas the growth rate is constant. However, the individuals within the population (under the same conditions) do not all grow at the same rate; there is a distribution of growth rates within the population. In an article now 30 years old Berglund and Larson make the very important conclusion “...this demonstrates how the phenomenon of growth dispersion can be misinterpreted as size dependent growth... Crystals don’t grow faster because they are larger, rather they become larger because they grow faster.”16

Our groups have also performed studies on crystal growth of a number of different solutes in small cells, with results always indicating that the crystals grow via a constant crystal growth mechanism but with significant amounts of growth rate dispersion. It is instructive to first consider what SDG and GRD would appear as in a small cell crystallizer operated at sufficiently low crystal population such that the growth occurs at essentially constant supersaturation, which is the typical operation for this type of research crystallizer. In a small cell crystallizer the relationship between size and time is measured for individual crystals; in the case of SDG, if the different crystals in the same cell have different initial sizes they must then grow at rates (dL/ dt) that are representative of the modeled growth rate for that value of size. In addition, as the experiments progress the growth rate of the individual crystals must accelerate as the size of the crystals increase, as the models predict an increasing growth rate at increasing sizes, so that the slopes of the plots of size vs time cannot be linear. Examples of predicted size vs time plots for two different but common SDG models (for typical values of constants in the models) are shown in Figure 3. If a set of crystals is simultaneously nucleated at essentially zero size within a small cell, via a secondary nucleation experiment within the cell, for instance, and they follow the inherent crystal growth model over an extended period of time, then results from a small cell experiment will appear as in C



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or during their growth, or if they had different nonzero sizes at the initiation of the batch, then the plots of size vs time do not extrapolate back to the origin. Examples from the current work are shown in Figure 5. Many other species, including sucrose,49 potassium aluminum sulfate,50,51 potassium nitrate,52 fructose, 16 citric acid,16 glucose, 41 ammonium dihydrogen phosphate,53 ammonium aluminum sulfate,54 potassium sodium tartrate,55 and sodium chlorate5 among others have been tested using similar methods under many conditions, and GRD is almost always found and SDG never found. Constant history crystal growth is usually investigated through seeded, stirred batch experiments. If the crystal size distribution is plotted on a log size scale, then the growth of common history seed crystals will not change the shape of the size distribution, and thus if the distributions are first normalized by dividing the size variable by the geometric mean size (as in Figure 6) then the distributions for all times

Figure 4 (which is a prediction rather than an experimental result). Since the crystals are all at essentially zero size at the

Figure 4. Expected sizes of individual particles in a batch growth cell vs time for a growth rate described by common history seed. The growth rate distribution is shown in the inset. Note that the particle size distribution at any time (see the distribution to the right of the plot) is a linear expansion of the growth rate distribution.

initiation of the batch, then the size of a crystal in the population at any time, t, is equal to its growth rate multiplied by the crystallization time. Thus, the crystal size distribution has the same shape as the distribution of crystal growth rates in the population but is stretched by the amount of time the crystals have been growing. This is termed common history seed experiments23 and relies on all of the crystals in the population nucleating at the same time and all crystals in the population growing under identical conditions during the entire period of their crystallization. It is important to note that the extent of GRD is not an inherent property of the solute but also depends on the method of preparation of the nuclei or seed and also their subsequent growth history. This means that not all seeded experiments, nor experiments initiated by nucleation, will follow this result, although some results have already been demonstrated in the literature.14,23,24,47 When GRD via the inherent crystal growth model is the mechanism causing the spreading of the size distributions the small cell data (of crystal size vs time) will also appear as a series of straight lines of different slopes. However, if different crystals within the cell have different conditions of nucleation

Figure 6. Batch crystallization of glucose monohydrate at 25 °C; example of measured particle size distributions normalized against the mean particle size for various times: ● 0 h, replicate 1; ○ 0 h, rep. 2; ■ 1 h; □ 2 h; ⧫ 3 h, rep. 1; ◊ 3 h, rep. 2; ▼ 4 h, rep. 1; ▽ 4 h, rep. 2; ▲ 6 h, rep. 1; △ 6 h, rep. 2. This figure shows a system characterized by common history crystals.

during the experiment will align. This is perhaps the mechanism most commonly mistaken for size-dependent growth since in this case at a particular point in time the growth rate of any crystal in a distribution will be proportional to its size. However, this is still not true SDG, since for the SDG

Figure 5. Batch small cell results demonstrating crystals growing via the constant crystal growth model without any evidence of size-dependent crystal growth. (Left) Crystal growth of potassium dihydrogen phosphate at 25 °C, 4.0% relative supersaturation; (right) sucrose at 25 °C, 2.4% relative supersaturation. D



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mechanism the growth rate of all crystals in the population will increase as the crystals in the population grow larger. This does not occur with common history crystals; the growth rate of each crystal remains at the same constant value over time assuming that the conditions of the solution surrounding the crystal remain constant. An interesting set of experiments using bimodal crystal seed distributions to decipher whether a spread in sizes during batch crystallization is due to SDG or GRD was performed by AddaiMensah et al. using the solute D-fructose.48 If the SDG mechanism is predominant in the system, then the fine crystals (the small mode in the bimodal distribution) will grow by a smaller amount than the large crystals (the large mode in the bimodal distribution). Thus, for SDG the two modes in the bimodal distribution will move further apart. If GRD is the predominant mechanism, then the widths of the two peaks in the distribution will increase; however, the two modes will stay approximately the same distance apart. We have repeated these types of experiments both in seeded batch crystallizers and in small cell crystallizers. Both types of crystallizers showed the same result, that there was no discernible increase in the distance between the two peaks over the time of the experimentindicating that SDG growth was not significanthowever the results in small cell crystallizers were easier to analyze due to the difficulties in accurately sampling from and measuring the number-based size distributions in seeded batch crystallizers. Figure 7 shows results from a bimodal seed

Figure 8. Crystal growth of a bimodal seed of hexamethylenetetramine in a small cell. The constant ΔL between the two peaks is evidence that there is no size-dependent crystal growth in this system; however, significant increases in peak width are evidence of GRD. The conditions are 25 °C and a relative supersaturation of 0.22%.

geometric mean and geometric standard deviation of the crystal growth rate distribution. We have chosen for simplicity an isothermal non-nucleating batch system where the crystal suspension density is sufficiently small that the supersaturation for the growth is constant. The growth rate distribution is lognormal with a geometric mean of 1 μm/min and a geometric standard deviation of 0.3. The results of the simulation over a period of 120 min are shown in Figure 9a, including an inset showing the crystal growth rate distribution used in the model. It can be seen that the crystal size distribution widens over time due to the GRD, and the population density at the mode decreases over time, which is necessary to keep the total number of crystals (which is the integral of the distribution) equal to a constant. It is possible to match two crystal size distributions in a batch using a SDG model even when the underlying mechanism is GRD. Consider the data for 60 min in Figure 9a. This result can be exactly replicated if a SDG model of G = 0.013L (with L in μm and G in μm·min−1) is used, as is shown in Figure 9b. However, if the batch is continued beyond 60 min (or if in fact a smaller time is used), then the results for the two simulations are different! A comparison of the results at 120 min shows that the width of the crystal size distribution using the SDG model is much larger than the width for the original GRD simulation. This is because each crystal in the GRD simulation is growing at a constant rate throughout the simulation, whereas the growth rate of the SDG crystalsand particularly the large onescontinues to accelerate throughout the batch as their size becomes larger. It is important to note here that only one SDG model could predict the given result at 60 minthere is no other SDG model that could exactly predict the 60 min result and give a better prediction for the 120 min data. Figure 9c shows that two different SDG models can replicate the two predicted size distributions, although obviously this is a nonphysical solution in an isothermal system at constant concentrationwhere the growth rate for the GRD model is at a constant level. It is important to note that the example here was for a batch crystallizer; however, inaccuracies in modeling continuous crystallizers may also result in inaccurate simulation results. Since continuous crystallizers are typically modeled at steady state, then an SDG model can completely fit data from a

Figure 7. Crystal growth of a bimodal seed of sucrose, demonstrating evidence of crystal growth rate dispersion but no size-dependent growth. The conditions are 25 °C and a relative supersaturation of 2.4%.

experiment for sucrose grown from aqueous solutions, and Figure 8 shows results from a bimodal seed experiment for hexamethylenetetramine. Both of these experiments provide evidence for the appearance of GRD in the system and show no evidence of SDG. These results concur with the earlier results for the crystal growth of fructose in aqueous ethanol solutions.48 Finally, we would like to demonstrate that SDG models in the population balance cannot correctly represent data where the underlying mechanism is GRD. We can demonstrate this by performing simulations using the two mechanisms and showing that the predicted crystal size distributions cannot be identical at all times. The simplest form of GRD to model is the common history mechanism. In the common case where the crystal growth rate distribution is log-normal the crystal size distribution at any time can be calculated directly using the E



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in the work of Fabian56,57 and later confirmed in the case of sucrose by Pantaraks.58

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CONCLUSIONS There are two types of models which have up to the present been used to extend McCabe’s ΔL Law: size-dependent crystal growth and crystal growth rate dispersion. Growth rate dispersion is a real and measurable phenomenon that affects essentially all populations of crystals, although is unfortunately not a specific property of the solute but is dependent on conditions that the population of crystals have experienced in their past. Size-dependent crystal growth, on the other hand, is only a real mechanism for crystals that are small enough that their surface energy has a significant effect on their thermodynamic stabilitytypically crystals that are smaller than approximately 1 μm in size. Even in this case the relationship is not fundamentally one of size-dependent crystal growth but rather size-dependent solubility leading to differences in the driving force for crystal growth. Here we have presented a wide range of experimental evidence that GRD is the predominant mechanism for the size spread in populations of crystals, not SDG, and also simulations that show SDG models cannot correctly simulate systems where GRD occurs. Thus, we believe that experimental and modeling research must begin to focus on the GRD mechanism rather than the SDG model.

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AUTHOR INFORMATION

Corresponding Author

*Fax: +66 44 224601. Phone: +66 44 224497. E-mail: adrianfl@sut.ac.th. Present Address #

Almendra (Thailand) Ltd. 7/313 Moo 6 Mapyangporn, PluakDaeng, Rayong, 21140, Thailand. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Suranaree University of Technology is acknowledged for funding this research through the Concept Paper grant. Figure 9. Modeling using a growth rate dispersion (common history seed crystals) model and size-dependent crystal growth model. (a) Common history seed crystals having a growth rate distribution with a mode of 1 μm/min and a geometric standard deviation of 0.3. (b) Size-dependent growth model that allows the data at 60 min to agree with the CH model. (c) Size-dependent growth model requiring two distinct models such that data 60 min and at 120 min both agree with the CH model.

ABBREVIATIONS CH, common history crystal growth; GRD, growth rate dispersion; MSMPR, mixed suspension−mixed product removal crystallizer; SDG, size-dependent growth

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REFERENCES

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Crystal Growth Rate Dispersion versus Size-Dependent Crystal

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