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Two dimensional population balance modelling of crystal growth behaviour under the influence of impurities
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Two dimensional population balance modelling of crystal growth behaviour under the influence of impurities
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Yang Zhang a, Jing Jing Liu a, Jian Wan a, Xue Z. Wang a,b,⇑
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a b
School of Chemistry and Chemical Engineering, South China University of Technology, 381 Wushan Road, Tianhe District, Guangzhou City 510641, China Institute of Particle Science & Engineering, School of Chemical and Process Engineering, University of Leeds, Leeds LS2 9JT, UK
a r t i c l e
i n f o
Article history: Received 6 October 2014 Received in revised form 21 December 2014 Accepted 5 January 2015 Available online xxxx Keywords: Impurity Additives Crystal shape distribution control Morphological population balance L-Glutamic acid
a b s t r a c t This short communication investigates the impact of impurity on crystal growth behaviour in terms of crystal shape distribution. A two dimensional population balance model is applied to study the crystal shape changing behaviour in cooling crystallization producing b-form L-glutamic acid crystals, in the presence and without the presence of an additive, L-phenylalanine. The simulation results were verified in experiments using a 1-litre crystallizer for crystallization of L-glutamic acid. Ó 2015 Published by Elsevier B.V. on behalf of The Society of Powder Technology Japan. All rights reserved.
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1. Introduction
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For crystalline particulate products obtained through crystallization, crystal morphology is an important property that affects the processability of the solid crystals and the end-use properties of the final products. Although much work has been done in order to understand the growth behaviour of single crystals, for quite some time, there has been a lack of knowledge on how to describe, model and ultimately control the morphology for a population of crystals growing in a crystallizer. As a result, at process level, modelling, simulation and control has focused on crystal size distribution where the size of a crystal is defined as the diameter of a sphere having the same volume of the crystal [1–7]. In recent years, there have been noticeable developments in modelling the dynamic evolution of crystal shape distribution using multi-dimensional and morphological population balance (PB) equations [8–15], as well as in on-line characterisation of crystal shape using imaging and image analysis [16–30]. These new developments paved the way for a practical solution to automatic control of crystal shape distribution. For example, Wan et al. [30] recently demonstrated that using a morphological PB model, an optimal supersaturation trajectory
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⇑ Corresponding author at: School of Chemistry and Chemical Engineering, South China University of Technology, 381 Wushan Road, Tianhe District, Guangzhou 510641, China. Tel.: +86 (0) 20 8711 4000. E-mail address: [email protected] (X.Z. Wang).
can be derived, and desired crystal shape distribution can be obtained by tracking the optimal trajectory. In addition to supersaturation, another effective means to influence crystal morphology is the use of additives or impurities, or additives [31,32]. An impurity can have strong influence on crystal growth, production throughput, yield and robustness of formulations. Quantitative knowledge on their influence can be used not only to develop strategies for their removal or moderating their effects, but also to possibly make positive use of an additive as a manipulated variable for optimization and control of crystal growth processes [33]. The effect of impurity is known to be highly selective: on the one hand, some impurity may inhibit the nucleation or growth of the crystals of one compound while promoting the nucleation or growth of the crystals of the other compound; on the other hand, the influence of the impurities on a single crystal may only demonstrate on certain crystallographic faces. Furthermore, the influence of impurities on crystallization is also closely correlated with process environments such as supersaturation, cooling rate and pH [34]. The mechanism and kinetics for the effect of the impurities on crystallization were studied intensively in the literature. Weissbuch et al. [35] investigated the effect of impurities on nucleation, growth and dissolution of crystals at the molecular level, and developed the tailor-made additives for blocking, docking and disrupting molecular packing arrangements during crystallization. Molecular modelling was thereafter widely applied to explain the mechanism for the effect of the impurities on crystallization of various compounds [36–39]. Davey et al. [40]
http://dx.doi.org/10.1016/j.apt.2015.01.001 0921-8831/Ó 2015 Published by Elsevier B.V. on behalf of The Society of Powder Technology Japan. All rights reserved.
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proposed guidelines for selecting suitable additive molecules for preventing the appearance of undesirable polymorphs by conformational mimicry. It was concluded that any successful additive molecule should have the appropriate conformation for taking part in the bonding network with minimum disruption. Along with the success of using molecular modelling to explain the mechanism for the effect of impurities on crystallization, kinetics relating the concentration of the added impurity as well as the supersaturation and faceted growth rates of crystals was also studied in literature. Kubota and Mullin proposed a kinetic model, the Kubota–Mullin model, to describe the faceted crystal growth rate as a function of impurity concentration [41]. Kubota–Mullin model or similar kinetic models were further applied to quantify the macro-scale effect of the added impurities on faceted growths of various crystals such as sodium chloride, hydroquinone, sucrose and L-glutamic acid [42–45]. An earlier experimental study on the crystallization of benzamide in the presence of impurities demonstrated that controlled modification of crystal habit was feasible by adding tailor-made impurities [46]. The influence of added impurities was mainly studied for single crystals to quantify the kinetic effect individually rather than for the population of crystals in the crystallizer. Patience and Rawlings designed a closed-loop feedback control system for shape control in which the shape was monitored by an imaging system, and the control was achieved by manipulating the flow rate of a habit modifier stream [19]. The purpose of the current work is to use a two-dimensional population balance model to study theimpact of impurity on crystal growth behaviour, in particular the crystal shape distribution. The work is different from previous work that either was restricted to a single crystal or a few crystals, or treat each crystal as a sphere for a population of crystals. A two population balance equations were built to model the effect of L-phenylalanine, as the impurity, on the shape changing behaviour of a growing crystal population during seeded crystallization of b-form L-glutamic acid (L-GA). Faceted growth kinetics of b-form L-GA with and without the added impurity L-phenylalanine is discussed in Section 2. The two dimensional PB model of seeded crystallization of b-form L-GA with the consideration of the added impurity L-phenylalanine is illustrated in Section 3. The experimental apparatus for validating simulation results is introduced in Section 4. The simulation results as well as its experimental validation are discussed in Section 5, which is followed by final remarks in Section 6. 2. Faceted growth kinetics of b-form L-GA with and without the presence of impurity L-phenylalanine The growth kinetics of both a-form and b-form L-GA was investigated in the literature [45,47]. Various methods were used to measure faceted growth kinetics relating to the operating conditions such as the relative supersaturation. For direct measurement, a single crystal can be mounted in the growth cell for online imagebased measurement using a video system [45]. An indirect approach was to estimate crystal growth rates using a predetermined growth pattern, where the initial and final crystal size distributions were measured offline [48]. The faceted growth rates of crystal faces during crystallization can also be measured in a statistical way using real-time in-process imaging techniques, where multiple crystals were snapshot from the reactor and a multi-scale segmentation algorithm was used to effectively extract objects from their backgrounds [18,22]. For example, the shape of b-form L-GA is shown in Fig. 1(a), which can be further simplified as a rod-like parallelepiped with the length L and the width W depicted in Fig. 1(b). The obtained faceted growth rates of b-form L-GA in the length and the width directions as a function of the relative supersaturation r were as follows [11]:
Fig. 1. The needle-shaped b-form L-glutamic acid (a), can be simplified as a rod-like structure (b).
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2 GL ¼ 3:44 107 eðr0:49Þ=0:02 = 1 þ eðr0:49Þ=0:02
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GW ¼ ð0:51 þ 2:15r 2:22r2 Þ 106
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where the corresponding experiment was performed around a relative supersaturation of 0.5 in the solution and GL, GW are the growth rates in the length and the width directions respectively. It is worth noting that the obtained growth kinetics fit better with the corresponding experimental data than those using traditional Kubota-Mullin models for this case. Considering the effect of the added impurity L-phenylalanine on the growth of b-form L-GA, the growth kinetics was also measured using the single-crystal method [45]. It was observed that the added impurity only inhibited the growth in the length direction and the degree of the inhibition was closely related to the concentration of the added impurity as well as the solution supersaturation. Then the modified overall growth rate in the length direction can be described as follows:
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12 ; GL ¼ GL 1 ‘ KC np
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ð3Þ
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where ‘; K; n are parameters related to the corresponding molecular adsorption mechanism, C p denotes the concentration of L-phenylalanine, and GL is defined by Eq. (1) [45]. It can be seen that the modified growth rate GL depends on both the relative supersaturation of the solution and the concentration of the added impurity. It needs to pint out here that while studying the facet growth kinetics by observing a few single crystals, as Kitamura and Ishizu did [45,47], is fundamental, just like measuring hear transfer or mass transfer, it cannot reflect the growth behaviour for a population of crystals. To simulate the growth behaviour of a population of crystals, the measured facet growth kinetics for single crystals needs to be incorporated into a morphological population balance model. Therefore the work presented in this paper represents a step from the single crystal work of Kitamura and Ishizu [45] and is the main thrust of this contribution. Facet growth kinetics is fundamental data in morphological population balance modelling.
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3. Population balance model
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A two dimensional PB model for seeded cooling crystallisation that generate b-form L-glutamic acid crystals was built previously
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(Ma et al. [11]). It was a simplified PB model that used two characteristic sizes, i.e. the length L and width W, as illustrated in Fig. 1. In addition, agglomeration and breakage as well as primary and secondary nucleation are assumed that can be ignored without causing much errors. In this work, the assumption that these phenomena were negligible is acceptable since we had an on-line imaging system installed on the reactor and did not observe obvious breakage or aggregation of crystals. For the same material in a previous study [11] in which on-line imaging was also used and did not find crystal breakage and aggregation either. Under the above assumptions, the 2-D PB model can be written as follows:
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@ @ @ ½WðL; W; tÞ þ ½GL WðL; W; tÞ þ ½GW WðL; W; tÞ ¼ 0 @t @t @t
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ð4Þ
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where GL and GW are the growth rates in the length and width directions, respectively; and W is the number population density function of crystals. To include the effect of impurity on the growth rate of the length direction, GL in Eq. (4) is replaced by GL :
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@ @ @ ½WðL; W; tÞ þ ½GL WðL; W; tÞ þ ½GW WðL; W; tÞ ¼ 0 @t @t @t
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ð5Þ
where GL is the growth rate in the length direction, as a function of both supersaturation and the concentration of the added impurity. Below, simulation results using Eq. (4) without impurity, and Eq. (5) with the presence of impurity will be provided. During simulation, Eq. (4) or (5) is combined with the mass balance equations to update the relative supersaturation in the reactor [11]. Due to the fact that analytical solutions for PB equations can only be obtained for very few simple cases, solution for PB equations requires numerical methods such as the method of moments, discretization techniques, finite element methods and high resolution algorithms [13]. According to the structure of the PB Eqs. (4) and (5), which is homogeneous, a 2-D high resolution algorithm used in the literature [13,30,49] was applied here for solving them with the size distribution of the added seeds as the initial condition.
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4. Experiments
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Experiments were conducted in a 1-litre reactor, as shown in Fig. 2. The process was monitored by an online imaging system developed by GlaxoSmithKline. The system uses a Sony XC-55 progressive scan monochrome camera with a microscopic lens. The video camera has a frequency up to thirty images per second with a pixel resolution of 640 480 and a field of view varying from 140 105 lm to 16 12 mm depending on the calibration lens used. A xenon stroboscopic light source is also used to provide illumination with a fibre optic guide to conduct the light. More details about the imaging system and its use in monitoring crystallization processes can be found from the literature [16–18,24,50]. The heating and the cooling processes for the reactor are controlled by a Julabo FP50-HD thermostated bath, where the temperature in the reactor is measured by platinum resistance thermometers (PT100). A turbidity probe is also inserted into the reactor to monitor the crystallization process especially for any potential onset of nucleation before adding the seeds. The shape and size distributions of the added seeds and the dried crystals of b-form L-GA obtained at the end of the crystallization process were measured by Malvern PharmaVision 830. This is an image-based particle analyzer that allows the collection of image data of particulate materials dispersed as dry powders. The dry crystals of b-form L-GA were firstly dispersed evenly on a glass plate using the attached sample preparation device. Then shape distributions of the samples can be measured by Malvern PharmaVision 830 through its self-calibrated imaging system and the corresponding software system. Malvern PharmaVision 830
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Fig. 2. The 1-litre crystalliser.
automatically scan and take images for a population of particles dispersed on a plate; then individual particles are segmented from the background and their morphological information such as the length and the width are measured. The measured length and width of a particle corresponds to the minimal rectangle containing the particle, which is consistent with the definition in Fig. 1(b). A slurry was prepared by adding 26.96 g L-GA b-form crystals of L-GA purchased from Van Waters and Rogers (VWR) International Ltd. into one litre distilled water in a 1-litre jacketed glass crystallizer. The mixture was then heated quickly to 70 °C while it was stirred by a pitched blade stirrer at a constant speed of 200 rpm and the mixture was held for about 100 min at 70 °C under constant stirring, to allow the crystals to completely dissolve (see Fig. 3 for the temperature and turbidity profiles). It is also worth noting that the value of turbidity is opposite to its physical meaning, i.e., the lower the value is, the higher the turbidity of the slurry is. The solution was then cooled at a speed of 1 °C/min to 45 °C. According to the solubility estimated from b-form L-GA solubility equation, Eq. (6) [51], the supersaturation at 45 °C is 0.5:
C b
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¼ ð2:204 0:07322 T þ 0:00893 T 0:000148183 T þ 0:00000134069 T 4 Þ:
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3
ð6Þ
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At 45 °C, 2% additional b-form L-GA, i.e. 545.2 mg, was added into the solution as seeds. A slow cooling rate, 0.1 °C/min, was maintained, which according to previous study [18] leads to b-form L-GA instead of a form. In addition, slow cooling also generates low supersaturation, a condition that minimizes nucleation, ensuring the major process is the growth of the seeded crystals. The window for simulation is 12 min from the time that seeds were added in. The experiment was conducted for the same time length of 12 min for the purpose of comparison. At the end of 12 min, samples were taken from the crystallizer, filtered, dried and analysed.
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Fig. 3. The temperature and turbidity profiles for the crystallization experiment without the presence of impurity.
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5. Results and discussions
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Fig. 3 shows the temperature and the turbidity profiles for the case of seeded crystallization without adding impurity. It can be seen that there was no sign of sharp changes for turbidity before adding the seeds, which was consistent with the assumption for the simulation that no nucleation happened before adding the seeds. After seeds were added at 45 °C, the turbidity is seen increasing quickly according to the physical meaning of the measured value. The grown b-form L-GA crystals were then filtered from the reactor and dried in an oven. The dried crystals were analysed using Malvern PharmaVision 830 for shape and size distributions. Fig. 4 is an image of the seeds used, confirming that the seeds are b-form L-GA. The shape distribution obtained from Malvern PharmaVision 830 for a sample of 3961 particles turned out to be that approximately normal and the mean values for the length and the width are 95.26 lm and 40.52 lm respectively, with the standard deviation being 27.68 lm and 9.98 lm. Assuming that all the seeds have the same shape distribution to the measured sample, the number density function for all the seeds was then deduced from their overall weight. The deduced number density function for the added seeds was shown in Fig. 5. This was also used as the initial shape distribution of seeds in the simulation. The same method was used for all simulations, either in the presence of added impurity or without the presence of impurity. The simulation and experiments were conducted under two conditions: seeded crystallization without the presence of impurity, and seeded crystallization with added impurity L-phenylalanine of
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Fig. 4. A snapshot of the seeds by the microscope.
82.595 mg. The corresponding simulation results for the seeded crystallization processes without and with adding the impurity Lphenylalanine are shown in Figs. 6 and 7, respectively. The number of the size classes for the high resolution algorithm used in the simulation is 2000 1000 and the corresponding width is 0.2 lm 0.2 lm. In Figs. 6 and 7, only the shape distributions at 720th seconds and for the seeds were given. Comparing Figs. 6 and 7, it can be seen that the crystals at t = 720 s obtained from the experiment without adding impurity is larger in the mean length than those crystals obtained from the experiment with adding impurity also at 720th second. For the simulation without adding impurity, the mean length and width for the crystals obtained are 131.9 lm and 45.3 lm respectively with the standard deviation of 27.68 lm and 9.98 lm. For the simulation with adding impurity; the mean length and width are 120.3 lm and 44.9 lm respectively with the standard deviation of 27.68 lm and 9.98 lm. For the corresponding experiments, the crystals in the crystalliser at the time of t = 720 s were filtered and then dried before measurement of shape distribution using Malvern PharmaVision 830. For the experiment without adding impurity, the mean values for the length and the width are 124.43 lm and 46.37 lm respectively with the standard deviation of 38.08 lm and 11.81 lm according to a measured sample of 3135 particles. For the experiment with adding impurity, the mean length and width are 116.36 lm and 44.59 lm respectively with the standard deviation of 35.42 lm and 10.92 lm according to a measured sample of 3174 particles. Snapshots of the samples from the obtained dry crystals without and with adding impurity by a microscope are shown in Figs. 8 and 9, respectively. The overall data for the experiments and the corresponding simulation results are listed in Table 1 and plotted in Fig. 10. For a fixed width W ¼ 45:1 l m, Fig. 10 shows the simulated and experimentally obtained length and width distributions of crystals, under two conditions, with and without adding impurity. Below are some observations made in Fig. 10. Firstly, it is found that the simulated results have narrower size distributions, i.e. smaller standard deviations than experimentally obtained. The discrepancies between the simulation and experimental results, and the large standard deviations for experimental data can be due to various factors, including:
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(1) Errors caused by the characterisation method. The seeds and the final crystals were characterised by the instrument Malvern PharmaVision 830. The instrument is based on analysis of images taken for crystal particles dispersed on a plate, either due to poor dispersion (or overlapped particles) or as a result of inaccuracies in image analysis, errors can be brought into the final result.
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Fig. 5. Number density function of the seed crystals.
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Fig. 6. The evolution of shape distribution for the seeded crystallization without the presence of impurity. Fig. 9. A snapshot of the dry crystals from the seeded crystallization experiment with the presence of impurity. 367
Fig. 7. The evolution of shape distribution for the seeded crystallization with the presence of impurity.
Fig. 8. A snapshot of the dry crystals from the seeded crystallization experiment without the presence of impurity.
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(2) Complexity in the crystallisation process, leading to size distribution of large standard deviation. Assumptions were made that nucleation and breakage of crystals are negligible. Despite the assumptions seem reasonable under the operational condition for this material, they after all have simplified the real process.
These need future more detailed investigation. Nevertheless, both the results of simulation and experiments do show the correct trend with regard to the impact of the impurity. This is clearer in Fig. 10. Both simulation and experiments showed that presence of impurity in crystallization have led to smaller mean size in the length direction, though no obvious difference was observed in the shape of the distribution curves. This is clear indication that the added impurity L-phenylalanine did inhibit the growth of bform L-GA in the length direction. Similar observation can be made by inspecting the aspect ratio. Table 1 shows the changes in aspect ratios. As can be seen from the table that for all the crystallisation experiments and simulations the product crystals have larger aspect ratios than the seeds, indicating crystals have grown faster in length directions than in the width direction. In addition, from the aspect ratios, it is also clear that impurity has inhibited growth of crystals in the length direction since crystals produced in the presence of impurity showed lower aspect ratios than that produced without the addition of impurity.
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6. Final remarks
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Impurities or additives, can have major influence on crystal growth. The presence of impurity can change the surface chemistry of crystals growing from solution, resulting in changes of the relative growth rates of the individual crystal surfaces, leading to changes in crystal shape as well as size distribution of final crystalline products. This will have profound impact on down-stream manufacturing processes such as filtration. Quantitative understanding of the influence of impurity on crystal growth is therefore critical to the competitiveness of manufacturers of high value crystalline products. The knowledge can be used to develop strategies for their removal or moderating their effects, or to make positive use of impurity as a manipulated variable for optimisation and control of crystal growth processes. The effect of impurity on crystal growth, including the effects on the growth kinetics of individual faces is a very complicated process. Being able to model such effect at process level for a population of crystals in a crystallizer is no doubt of great importance. In this paper, the dynamic behaviour of shape distribution of bform L-GA crystals growing from solution with the presence of impurity, L-phenylalanine, is studied computationally and experimentally, and compared with simulation and experimental results when no impurity is added. The simulation study was conducted using two dimensional population balance model where the growth
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Table 1 Experimental and simulation data.
Fig. 10. Comparison between simulation and experimental results.
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rate in the length direction is a function of impurity concentration as well as of the supersaturation, while the growth rate in the width direction only depends on the supersaturation. The experiments were carried out in a 1-litre crystallizer. b form L-GA seed crystals and the crystallized crystals were characterized for shape distribution using an imaging instrument. Both simulation and experiments demonstrated that the presence of the impurity, L-phenylalanine, inhibits the growth in the length direction, but has little effect on the width. The experimentally obtained crystals have larger standard deviation compared to simulation. This is considered as logical and acceptable given the various sources of errors that could be introduced in crystal characterisation as well as in simulation. This preliminary study provides a proof of concept of applying morphological or multi-dimensional population balance models for the assessment of impurities, and for modifying the morphology of products in relation to their commercial effect.
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Acknowledgements
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Financial support from the China Thousand Talents scheme is acknowledged. The work has also benefited early research funded Q4 by the UK Engineering and Physical Sciences Research Council Q5 (EPSRC) for the projects of Shape (EP/C009541) and Stereo Vision (EP/E045707). References [1] M.A. Lovette, A.R. Browning, D.W. Griffin, J.P. Sizemore, R.C. Snyder, M.F. Doherty, Crystal shape engineering, Ind. Eng. Chem. Res. 47 (2008) 9812–9833. [2] S.B. Gadewar, M.F. Doherty, A dynamic model for evolution of crystal shape, J. Cryst. Growth 267 (2004) 239–250. [3] Z.Q. Yu, J.W. Chew, P.S. Chow, R.B.H. Tan, Recent advances in crystallization control: an industrial perspective, Chem. Eng. Res. Des. 85 (2007) 893–905. [4] P.A. Larsen, D.B. Patience, J.B. Rawlings, Industrial crystallization process control, Control Syst. Mag., IEEE 26 (2006) 70–80. [5] M. Fujiwara, Z.K. Nagy, J.W. Chew, R.D. Braatz, First-principles and direct design approaches for the control of pharmaceutical crystallization, J. Process Control 15 (2005) 493–504.
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[29] J. Eggers, M. Kempkes, J. Cornel, M. Mazzotti, I. Koschinski, E. Verdurand, Monitoring size and shape during cooling crystallization of ascorbic acid, Chem. Eng. Sci. 64 (2009) 163–171. [30] J. Wan, X.Z. Wang, C.Y. Ma, Particle shape manipulation and optimization in cooling crystallization involving multiple crystal morphological forms, AIChE J. 55 (2009) 2049–2061. [31] C. Thompson, M.C. Davies, C.J. Roberts, S.J.B. Tendler, M.J. Wilkinson, The effects of additives on the growth and morphology of paracetamol (acetaminophen) crystals, Int. J. Pharm. 280 (2004) 137–150. [32] T.R. Keel, C. Thompson, M.C. Davies, S.J.B. Tendler, C.J. Roberts, AFM studies of the crystallization and habit modification of an excipient material, adipic acid, Int. J. Pharm. 280 (2004) 185–198. [33] I. Weissbuch, M. Lahav, L. Leiserowitz, Toward Stereochemical Control, Monitoring, and Understanding of Crystal Nucleation, Cryst. Growth Des. 3 (2003) 125–150. [34] K. Kadota, K. Takase, A. Shimosaka, Y. Shirakawa, J. Hidaka, Influence of habit modifiers on particle shape in a crystallization process, KONA 24 (2006) 203–212. [35] I. Weissbuch, L. Addadi, L. Leiserowitz, Molecular recognition at crystal interfaces, Science 253 (1991) 637–645. [36] G. Clydesdale, K.J. Roberts, R. Docherty, Modelling the morphology of molecular crystals in the presence of disruptive tailor-made additives, J. Cryst. Growth 135 (1994) 331–340. [37] G. Clydesdale, K.J. Roberts, K. Lewtas, R. Docherty, Modelling the morphology of molecular crystals in the presence of blocking tailor-made additives, J. Cryst. Growth 141 (1994) 443–450. [38] S.K. Poornachary, P.S. Chow, R.B.H. Tan, R.J. Davey, Molecular speciation controlling stereoselectivity of additives: impact on the habit modification in a-glycine crystals, Cryst. Growth Des. 7 (2007) 254–261. [39] S.K. Poornachary, P.S. Chow, R.B.H. Tan, Influence of solution speciation of impurities on polymorphic nucleation in glycine, Cryst. Growth Des. 8 (2008) 179–185.
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[40] R.J. Davey, N. Blagden, G.D. Potts, R. Docherty, Polymorphism in molecular crystals: stabilization of a metastable form by conformational mimicry, J. Am. Chem. Soc. 119 (1997) 1767–1772. [41] N. Kubota, J.W. Mullin, A kinetic model for crystal growth from aqueous solution in the presence of impurity, J. Cryst. Growth 152 (1995) 203–208. [42] S. Al-Jibbouri, J. Ulrich, The influence of impurities on crystallization kinetics of sodium chloride, Cryst. Res. Technol. 36 (2001) 1365–1375. [43] A. Chianese, M. Di Luozzo, N. Kubota, Effect of pyrogallol additive on the growth rate and the habit of hydroquinone crystals, Cryst. Growth Des. 3 (2003) 425–430. [44] G. Sgualdino, D. Aquilano, L. Pastero, G. Vaccari, Face-by-face growth of sucrose crystals from aqueous solutions in the presence of raffinose–II: Growth morphology and segregation, J. Cryst. Growth 308 (2007) 141–150. [45] M. Kitamura, T. Ishizu, Kinetic effect of L-phenylalanine on growth process of Lglutamic acid polymorph, J. Cryst. Growth 192 (1998) 225–235. [46] Z. Berkovitch-Yellin, L. Addadi, M. Idelson, M. Lahav, L. Leiserowitz, Controlled modification of crystal habit by tailor-made impurities: application to benzamide, Angew. Chem., Int. Ed. Engl. 21 (1982) 631–632. [47] M. Kitamura, T. Ishizu, Growth kinetics and morphological change of polymorphs of L-glutamic acid, J. Cryst. Growth 209 (2000) 138–145. [48] Q. Hu, S. Rohani, D.X. Wang, A. Jutan, Nonlinear kinetic parameter estimation for batch cooling seeded crystallization, AIChE J. 50 (2004) 1786–1794. [49] D.L. Ma, D.K. Tafti, R.D. Braatz, High-resolution simulation of multidimensional crystal growth, Ind. Eng. Chem. Res. 41 (2002) 6217–6223. [50] M.J. Wilkinson, K.H. Jennings, M. Hardy, Non-invasive video imaging for interrogating pharmaceutical crystallization processes, Microsc. Microanal. 6 (2000) 996–997. [51] R.F. Li, X.Z. Wang, S.B. Abebe, Monitoring batch cooling crystallization using NIR: development of calibration models using genetic algorithm and PLS, Part. Part. Syst. Charact. 25 (2008) 314–327.
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Contents lists available at ScienceDirect
Advanced Powder Technology journal homepage: www.elsevier.com/locate/apt
Rapid Communication
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Two dimensional population balance modelling of crystal growth behaviour under the influence of impurities
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Yang Zhang a, Jing Jing Liu a, Jian Wan a, Xue Z. Wang a,b,⇑
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a b
School of Chemistry and Chemical Engineering, South China University of Technology, 381 Wushan Road, Tianhe District, Guangzhou City 510641, China Institute of Particle Science & Engineering, School of Chemical and Process Engineering, University of Leeds, Leeds LS2 9JT, UK
a r t i c l e
i n f o
Article history: Received 6 October 2014 Received in revised form 21 December 2014 Accepted 5 January 2015 Available online xxxx Keywords: Impurity Additives Crystal shape distribution control Morphological population balance L-Glutamic acid
a b s t r a c t This short communication investigates the impact of impurity on crystal growth behaviour in terms of crystal shape distribution. A two dimensional population balance model is applied to study the crystal shape changing behaviour in cooling crystallization producing b-form L-glutamic acid crystals, in the presence and without the presence of an additive, L-phenylalanine. The simulation results were verified in experiments using a 1-litre crystallizer for crystallization of L-glutamic acid. Ó 2015 Published by Elsevier B.V. on behalf of The Society of Powder Technology Japan. All rights reserved.
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1. Introduction
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For crystalline particulate products obtained through crystallization, crystal morphology is an important property that affects the processability of the solid crystals and the end-use properties of the final products. Although much work has been done in order to understand the growth behaviour of single crystals, for quite some time, there has been a lack of knowledge on how to describe, model and ultimately control the morphology for a population of crystals growing in a crystallizer. As a result, at process level, modelling, simulation and control has focused on crystal size distribution where the size of a crystal is defined as the diameter of a sphere having the same volume of the crystal [1–7]. In recent years, there have been noticeable developments in modelling the dynamic evolution of crystal shape distribution using multi-dimensional and morphological population balance (PB) equations [8–15], as well as in on-line characterisation of crystal shape using imaging and image analysis [16–30]. These new developments paved the way for a practical solution to automatic control of crystal shape distribution. For example, Wan et al. [30] recently demonstrated that using a morphological PB model, an optimal supersaturation trajectory
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⇑ Corresponding author at: School of Chemistry and Chemical Engineering, South China University of Technology, 381 Wushan Road, Tianhe District, Guangzhou 510641, China. Tel.: +86 (0) 20 8711 4000. E-mail address: [email protected] (X.Z. Wang).
can be derived, and desired crystal shape distribution can be obtained by tracking the optimal trajectory. In addition to supersaturation, another effective means to influence crystal morphology is the use of additives or impurities, or additives [31,32]. An impurity can have strong influence on crystal growth, production throughput, yield and robustness of formulations. Quantitative knowledge on their influence can be used not only to develop strategies for their removal or moderating their effects, but also to possibly make positive use of an additive as a manipulated variable for optimization and control of crystal growth processes [33]. The effect of impurity is known to be highly selective: on the one hand, some impurity may inhibit the nucleation or growth of the crystals of one compound while promoting the nucleation or growth of the crystals of the other compound; on the other hand, the influence of the impurities on a single crystal may only demonstrate on certain crystallographic faces. Furthermore, the influence of impurities on crystallization is also closely correlated with process environments such as supersaturation, cooling rate and pH [34]. The mechanism and kinetics for the effect of the impurities on crystallization were studied intensively in the literature. Weissbuch et al. [35] investigated the effect of impurities on nucleation, growth and dissolution of crystals at the molecular level, and developed the tailor-made additives for blocking, docking and disrupting molecular packing arrangements during crystallization. Molecular modelling was thereafter widely applied to explain the mechanism for the effect of the impurities on crystallization of various compounds [36–39]. Davey et al. [40]
http://dx.doi.org/10.1016/j.apt.2015.01.001 0921-8831/Ó 2015 Published by Elsevier B.V. on behalf of The Society of Powder Technology Japan. All rights reserved.
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proposed guidelines for selecting suitable additive molecules for preventing the appearance of undesirable polymorphs by conformational mimicry. It was concluded that any successful additive molecule should have the appropriate conformation for taking part in the bonding network with minimum disruption. Along with the success of using molecular modelling to explain the mechanism for the effect of impurities on crystallization, kinetics relating the concentration of the added impurity as well as the supersaturation and faceted growth rates of crystals was also studied in literature. Kubota and Mullin proposed a kinetic model, the Kubota–Mullin model, to describe the faceted crystal growth rate as a function of impurity concentration [41]. Kubota–Mullin model or similar kinetic models were further applied to quantify the macro-scale effect of the added impurities on faceted growths of various crystals such as sodium chloride, hydroquinone, sucrose and L-glutamic acid [42–45]. An earlier experimental study on the crystallization of benzamide in the presence of impurities demonstrated that controlled modification of crystal habit was feasible by adding tailor-made impurities [46]. The influence of added impurities was mainly studied for single crystals to quantify the kinetic effect individually rather than for the population of crystals in the crystallizer. Patience and Rawlings designed a closed-loop feedback control system for shape control in which the shape was monitored by an imaging system, and the control was achieved by manipulating the flow rate of a habit modifier stream [19]. The purpose of the current work is to use a two-dimensional population balance model to study theimpact of impurity on crystal growth behaviour, in particular the crystal shape distribution. The work is different from previous work that either was restricted to a single crystal or a few crystals, or treat each crystal as a sphere for a population of crystals. A two population balance equations were built to model the effect of L-phenylalanine, as the impurity, on the shape changing behaviour of a growing crystal population during seeded crystallization of b-form L-glutamic acid (L-GA). Faceted growth kinetics of b-form L-GA with and without the added impurity L-phenylalanine is discussed in Section 2. The two dimensional PB model of seeded crystallization of b-form L-GA with the consideration of the added impurity L-phenylalanine is illustrated in Section 3. The experimental apparatus for validating simulation results is introduced in Section 4. The simulation results as well as its experimental validation are discussed in Section 5, which is followed by final remarks in Section 6. 2. Faceted growth kinetics of b-form L-GA with and without the presence of impurity L-phenylalanine The growth kinetics of both a-form and b-form L-GA was investigated in the literature [45,47]. Various methods were used to measure faceted growth kinetics relating to the operating conditions such as the relative supersaturation. For direct measurement, a single crystal can be mounted in the growth cell for online imagebased measurement using a video system [45]. An indirect approach was to estimate crystal growth rates using a predetermined growth pattern, where the initial and final crystal size distributions were measured offline [48]. The faceted growth rates of crystal faces during crystallization can also be measured in a statistical way using real-time in-process imaging techniques, where multiple crystals were snapshot from the reactor and a multi-scale segmentation algorithm was used to effectively extract objects from their backgrounds [18,22]. For example, the shape of b-form L-GA is shown in Fig. 1(a), which can be further simplified as a rod-like parallelepiped with the length L and the width W depicted in Fig. 1(b). The obtained faceted growth rates of b-form L-GA in the length and the width directions as a function of the relative supersaturation r were as follows [11]:
Fig. 1. The needle-shaped b-form L-glutamic acid (a), can be simplified as a rod-like structure (b).
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2 GL ¼ 3:44 107 eðr0:49Þ=0:02 = 1 þ eðr0:49Þ=0:02
ð1Þ
GW ¼ ð0:51 þ 2:15r 2:22r2 Þ 106
ð2Þ
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where the corresponding experiment was performed around a relative supersaturation of 0.5 in the solution and GL, GW are the growth rates in the length and the width directions respectively. It is worth noting that the obtained growth kinetics fit better with the corresponding experimental data than those using traditional Kubota-Mullin models for this case. Considering the effect of the added impurity L-phenylalanine on the growth of b-form L-GA, the growth kinetics was also measured using the single-crystal method [45]. It was observed that the added impurity only inhibited the growth in the length direction and the degree of the inhibition was closely related to the concentration of the added impurity as well as the solution supersaturation. Then the modified overall growth rate in the length direction can be described as follows:
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12 ; GL ¼ GL 1 ‘ KC np
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ð3Þ
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where ‘; K; n are parameters related to the corresponding molecular adsorption mechanism, C p denotes the concentration of L-phenylalanine, and GL is defined by Eq. (1) [45]. It can be seen that the modified growth rate GL depends on both the relative supersaturation of the solution and the concentration of the added impurity. It needs to pint out here that while studying the facet growth kinetics by observing a few single crystals, as Kitamura and Ishizu did [45,47], is fundamental, just like measuring hear transfer or mass transfer, it cannot reflect the growth behaviour for a population of crystals. To simulate the growth behaviour of a population of crystals, the measured facet growth kinetics for single crystals needs to be incorporated into a morphological population balance model. Therefore the work presented in this paper represents a step from the single crystal work of Kitamura and Ishizu [45] and is the main thrust of this contribution. Facet growth kinetics is fundamental data in morphological population balance modelling.
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3. Population balance model
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A two dimensional PB model for seeded cooling crystallisation that generate b-form L-glutamic acid crystals was built previously
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(Ma et al. [11]). It was a simplified PB model that used two characteristic sizes, i.e. the length L and width W, as illustrated in Fig. 1. In addition, agglomeration and breakage as well as primary and secondary nucleation are assumed that can be ignored without causing much errors. In this work, the assumption that these phenomena were negligible is acceptable since we had an on-line imaging system installed on the reactor and did not observe obvious breakage or aggregation of crystals. For the same material in a previous study [11] in which on-line imaging was also used and did not find crystal breakage and aggregation either. Under the above assumptions, the 2-D PB model can be written as follows:
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@ @ @ ½WðL; W; tÞ þ ½GL WðL; W; tÞ þ ½GW WðL; W; tÞ ¼ 0 @t @t @t
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ð4Þ
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where GL and GW are the growth rates in the length and width directions, respectively; and W is the number population density function of crystals. To include the effect of impurity on the growth rate of the length direction, GL in Eq. (4) is replaced by GL :
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@ @ @ ½WðL; W; tÞ þ ½GL WðL; W; tÞ þ ½GW WðL; W; tÞ ¼ 0 @t @t @t
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ð5Þ
where GL is the growth rate in the length direction, as a function of both supersaturation and the concentration of the added impurity. Below, simulation results using Eq. (4) without impurity, and Eq. (5) with the presence of impurity will be provided. During simulation, Eq. (4) or (5) is combined with the mass balance equations to update the relative supersaturation in the reactor [11]. Due to the fact that analytical solutions for PB equations can only be obtained for very few simple cases, solution for PB equations requires numerical methods such as the method of moments, discretization techniques, finite element methods and high resolution algorithms [13]. According to the structure of the PB Eqs. (4) and (5), which is homogeneous, a 2-D high resolution algorithm used in the literature [13,30,49] was applied here for solving them with the size distribution of the added seeds as the initial condition.
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4. Experiments
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Experiments were conducted in a 1-litre reactor, as shown in Fig. 2. The process was monitored by an online imaging system developed by GlaxoSmithKline. The system uses a Sony XC-55 progressive scan monochrome camera with a microscopic lens. The video camera has a frequency up to thirty images per second with a pixel resolution of 640 480 and a field of view varying from 140 105 lm to 16 12 mm depending on the calibration lens used. A xenon stroboscopic light source is also used to provide illumination with a fibre optic guide to conduct the light. More details about the imaging system and its use in monitoring crystallization processes can be found from the literature [16–18,24,50]. The heating and the cooling processes for the reactor are controlled by a Julabo FP50-HD thermostated bath, where the temperature in the reactor is measured by platinum resistance thermometers (PT100). A turbidity probe is also inserted into the reactor to monitor the crystallization process especially for any potential onset of nucleation before adding the seeds. The shape and size distributions of the added seeds and the dried crystals of b-form L-GA obtained at the end of the crystallization process were measured by Malvern PharmaVision 830. This is an image-based particle analyzer that allows the collection of image data of particulate materials dispersed as dry powders. The dry crystals of b-form L-GA were firstly dispersed evenly on a glass plate using the attached sample preparation device. Then shape distributions of the samples can be measured by Malvern PharmaVision 830 through its self-calibrated imaging system and the corresponding software system. Malvern PharmaVision 830
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Fig. 2. The 1-litre crystalliser.
automatically scan and take images for a population of particles dispersed on a plate; then individual particles are segmented from the background and their morphological information such as the length and the width are measured. The measured length and width of a particle corresponds to the minimal rectangle containing the particle, which is consistent with the definition in Fig. 1(b). A slurry was prepared by adding 26.96 g L-GA b-form crystals of L-GA purchased from Van Waters and Rogers (VWR) International Ltd. into one litre distilled water in a 1-litre jacketed glass crystallizer. The mixture was then heated quickly to 70 °C while it was stirred by a pitched blade stirrer at a constant speed of 200 rpm and the mixture was held for about 100 min at 70 °C under constant stirring, to allow the crystals to completely dissolve (see Fig. 3 for the temperature and turbidity profiles). It is also worth noting that the value of turbidity is opposite to its physical meaning, i.e., the lower the value is, the higher the turbidity of the slurry is. The solution was then cooled at a speed of 1 °C/min to 45 °C. According to the solubility estimated from b-form L-GA solubility equation, Eq. (6) [51], the supersaturation at 45 °C is 0.5:
C b
2
¼ ð2:204 0:07322 T þ 0:00893 T 0:000148183 T þ 0:00000134069 T 4 Þ:
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3
ð6Þ
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At 45 °C, 2% additional b-form L-GA, i.e. 545.2 mg, was added into the solution as seeds. A slow cooling rate, 0.1 °C/min, was maintained, which according to previous study [18] leads to b-form L-GA instead of a form. In addition, slow cooling also generates low supersaturation, a condition that minimizes nucleation, ensuring the major process is the growth of the seeded crystals. The window for simulation is 12 min from the time that seeds were added in. The experiment was conducted for the same time length of 12 min for the purpose of comparison. At the end of 12 min, samples were taken from the crystallizer, filtered, dried and analysed.
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Fig. 3. The temperature and turbidity profiles for the crystallization experiment without the presence of impurity.
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5. Results and discussions
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Fig. 3 shows the temperature and the turbidity profiles for the case of seeded crystallization without adding impurity. It can be seen that there was no sign of sharp changes for turbidity before adding the seeds, which was consistent with the assumption for the simulation that no nucleation happened before adding the seeds. After seeds were added at 45 °C, the turbidity is seen increasing quickly according to the physical meaning of the measured value. The grown b-form L-GA crystals were then filtered from the reactor and dried in an oven. The dried crystals were analysed using Malvern PharmaVision 830 for shape and size distributions. Fig. 4 is an image of the seeds used, confirming that the seeds are b-form L-GA. The shape distribution obtained from Malvern PharmaVision 830 for a sample of 3961 particles turned out to be that approximately normal and the mean values for the length and the width are 95.26 lm and 40.52 lm respectively, with the standard deviation being 27.68 lm and 9.98 lm. Assuming that all the seeds have the same shape distribution to the measured sample, the number density function for all the seeds was then deduced from their overall weight. The deduced number density function for the added seeds was shown in Fig. 5. This was also used as the initial shape distribution of seeds in the simulation. The same method was used for all simulations, either in the presence of added impurity or without the presence of impurity. The simulation and experiments were conducted under two conditions: seeded crystallization without the presence of impurity, and seeded crystallization with added impurity L-phenylalanine of
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Fig. 4. A snapshot of the seeds by the microscope.
82.595 mg. The corresponding simulation results for the seeded crystallization processes without and with adding the impurity Lphenylalanine are shown in Figs. 6 and 7, respectively. The number of the size classes for the high resolution algorithm used in the simulation is 2000 1000 and the corresponding width is 0.2 lm 0.2 lm. In Figs. 6 and 7, only the shape distributions at 720th seconds and for the seeds were given. Comparing Figs. 6 and 7, it can be seen that the crystals at t = 720 s obtained from the experiment without adding impurity is larger in the mean length than those crystals obtained from the experiment with adding impurity also at 720th second. For the simulation without adding impurity, the mean length and width for the crystals obtained are 131.9 lm and 45.3 lm respectively with the standard deviation of 27.68 lm and 9.98 lm. For the simulation with adding impurity; the mean length and width are 120.3 lm and 44.9 lm respectively with the standard deviation of 27.68 lm and 9.98 lm. For the corresponding experiments, the crystals in the crystalliser at the time of t = 720 s were filtered and then dried before measurement of shape distribution using Malvern PharmaVision 830. For the experiment without adding impurity, the mean values for the length and the width are 124.43 lm and 46.37 lm respectively with the standard deviation of 38.08 lm and 11.81 lm according to a measured sample of 3135 particles. For the experiment with adding impurity, the mean length and width are 116.36 lm and 44.59 lm respectively with the standard deviation of 35.42 lm and 10.92 lm according to a measured sample of 3174 particles. Snapshots of the samples from the obtained dry crystals without and with adding impurity by a microscope are shown in Figs. 8 and 9, respectively. The overall data for the experiments and the corresponding simulation results are listed in Table 1 and plotted in Fig. 10. For a fixed width W ¼ 45:1 l m, Fig. 10 shows the simulated and experimentally obtained length and width distributions of crystals, under two conditions, with and without adding impurity. Below are some observations made in Fig. 10. Firstly, it is found that the simulated results have narrower size distributions, i.e. smaller standard deviations than experimentally obtained. The discrepancies between the simulation and experimental results, and the large standard deviations for experimental data can be due to various factors, including:
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(1) Errors caused by the characterisation method. The seeds and the final crystals were characterised by the instrument Malvern PharmaVision 830. The instrument is based on analysis of images taken for crystal particles dispersed on a plate, either due to poor dispersion (or overlapped particles) or as a result of inaccuracies in image analysis, errors can be brought into the final result.
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Fig. 5. Number density function of the seed crystals.
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Fig. 6. The evolution of shape distribution for the seeded crystallization without the presence of impurity. Fig. 9. A snapshot of the dry crystals from the seeded crystallization experiment with the presence of impurity. 367
Fig. 7. The evolution of shape distribution for the seeded crystallization with the presence of impurity.
Fig. 8. A snapshot of the dry crystals from the seeded crystallization experiment without the presence of impurity.
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(2) Complexity in the crystallisation process, leading to size distribution of large standard deviation. Assumptions were made that nucleation and breakage of crystals are negligible. Despite the assumptions seem reasonable under the operational condition for this material, they after all have simplified the real process.
These need future more detailed investigation. Nevertheless, both the results of simulation and experiments do show the correct trend with regard to the impact of the impurity. This is clearer in Fig. 10. Both simulation and experiments showed that presence of impurity in crystallization have led to smaller mean size in the length direction, though no obvious difference was observed in the shape of the distribution curves. This is clear indication that the added impurity L-phenylalanine did inhibit the growth of bform L-GA in the length direction. Similar observation can be made by inspecting the aspect ratio. Table 1 shows the changes in aspect ratios. As can be seen from the table that for all the crystallisation experiments and simulations the product crystals have larger aspect ratios than the seeds, indicating crystals have grown faster in length directions than in the width direction. In addition, from the aspect ratios, it is also clear that impurity has inhibited growth of crystals in the length direction since crystals produced in the presence of impurity showed lower aspect ratios than that produced without the addition of impurity.
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6. Final remarks
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Impurities or additives, can have major influence on crystal growth. The presence of impurity can change the surface chemistry of crystals growing from solution, resulting in changes of the relative growth rates of the individual crystal surfaces, leading to changes in crystal shape as well as size distribution of final crystalline products. This will have profound impact on down-stream manufacturing processes such as filtration. Quantitative understanding of the influence of impurity on crystal growth is therefore critical to the competitiveness of manufacturers of high value crystalline products. The knowledge can be used to develop strategies for their removal or moderating their effects, or to make positive use of impurity as a manipulated variable for optimisation and control of crystal growth processes. The effect of impurity on crystal growth, including the effects on the growth kinetics of individual faces is a very complicated process. Being able to model such effect at process level for a population of crystals in a crystallizer is no doubt of great importance. In this paper, the dynamic behaviour of shape distribution of bform L-GA crystals growing from solution with the presence of impurity, L-phenylalanine, is studied computationally and experimentally, and compared with simulation and experimental results when no impurity is added. The simulation study was conducted using two dimensional population balance model where the growth
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Table 1 Experimental and simulation data.
Fig. 10. Comparison between simulation and experimental results.
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rate in the length direction is a function of impurity concentration as well as of the supersaturation, while the growth rate in the width direction only depends on the supersaturation. The experiments were carried out in a 1-litre crystallizer. b form L-GA seed crystals and the crystallized crystals were characterized for shape distribution using an imaging instrument. Both simulation and experiments demonstrated that the presence of the impurity, L-phenylalanine, inhibits the growth in the length direction, but has little effect on the width. The experimentally obtained crystals have larger standard deviation compared to simulation. This is considered as logical and acceptable given the various sources of errors that could be introduced in crystal characterisation as well as in simulation. This preliminary study provides a proof of concept of applying morphological or multi-dimensional population balance models for the assessment of impurities, and for modifying the morphology of products in relation to their commercial effect.
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Acknowledgements
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