Optimal Control of Crystal Shapes in Batch Crystallization Experiments

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Optimal Control of Crystal Shapes in Batch Crystallization Experiments by Growth-Dissolution Cycles Holger Eisenschmidt, Naim Bajcinca, and Kai Sundmacher Cryst. Growth Des., Just Accepted Manuscript • DOI: 10.1021/acs.cgd.6b00288 • Publication Date (Web): 27 Apr 2016 Downloaded from http://pubs.acs.org on May 2, 2016

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Crystal Growth & Design

Optimal Control of Crystal Shapes in Batch Crystallization Experiments by Growth-Dissolution Cycles Holger Eisenschmidt,a Naim Bajcinca,b Kai Sundmacher,a,b* a b

Otto-von-Guericke- University Magdeburg, Department Process Systems Engineering, Universitätsplatz 2, D-39106 Magdeburg, Germany Max Planck Institute for Dynamics of Complex Technical Systems, Department Process Systems Engineering, Sandtorstr.1, D-39106 Magdeburg, Germany

Abstract: The control of the evolution of the crystal size and shape distribution (CSSD) during crystallization processes is an important task in crystallization, as the final CSSD decisively influences the physical - and solid state properties of crystalline material. This work utilizes sequential growth and dissolution cycles, which turn out to result in an essentially enlarged region of attainable crystal sizes and shapes. Using potassium dihydrogen phosphate (KDP) as a model substance, such a cyclic crystallization process is realized in a batch scale and in a novel, fully automated and controlled manner. To this end, a novel observer setup is presented, which is based on video microscopy, and facilitates the real-time monitoring of the evolution of the crystal size and shape distribution. Given this information, optimal strategies for the control of supersaturation profiles, as well as CSSD are experimentally successfully implemented, proving a reliable and high-precision generic control scheme for crystal shape manipulation. The proposed theoretical and experimental research results are expected to be of use in targeted crystal size and shape manipulation in chemical and pharmaceutical industries.

Corresponding author: Kai Sundmacher Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstraße 1, 39106 Magdeburg, Germany Tel: 0049 391 6110351 Fax: 0049 391 6110523 Email: [email protected]

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Optimal Control of Crystal Shapes in Batch Crystallization Experiments by Growth-Dissolution Cycles Holger Eisenschmidt,a Naim Bajcinca,b Kai Sundmacher,a,b* a b

Otto-von-Guericke- University Magdeburg, Department Process Systems Engineering, Universitätsplatz 2, D-39106 Magdeburg, Germany Max Planck Institute for Dynamics of Complex Technical Systems, Department Process Systems Engineering, Sandtorstr.1, D-39106 Magdeburg, Germany

* Corresponding author: [email protected] Abstract The control of the evolution of the crystal size and shape distribution (CSSD) during crystallization processes is an important task in crystallization, as the final CSSD decisively influences the physical - and solid state properties of crystalline material. This work utilizes sequential growth and dissolution cycles, which turn out to result in an essentially enlarged region of attainable crystal sizes and shapes. Using potassium dihydrogen phosphate (KDP) as a model substance, such a cyclic crystallization process is realized in a batch scale and in a novel, fully automated and controlled manner. To this end, a novel observer setup is presented, which is based on video microscopy, and facilitates the real-time monitoring of the evolution of the crystal size and shape distribution. Given this information, optimal strategies for the control of supersaturation profiles, as well as CSSD are experimentally successfully implemented, proving a reliable and high-precision generic control scheme for crystal shape manipulation. The proposed theoretical and experimental research results are expected to be of use in targeted crystal size and shape manipulation in chemical and pharmaceutical industries. Keywords: Crystallization, Growth-Dissolution Cycles, Crystal Shape Control, Optimal Control

1. Introduction Crystallization is an important unit operation in chemical production systems. For instance, it is frequently used for the purification of specific substances and for the final product preparation. It is well-known that the final crystal size and shape distribution (CSSD) of crystallization processes significantly affects the physical and chemical product properties1. For example the massive occurrence of fines, which is typically a result of nucleation, can cause severe problems during further downstream processes such as filtration and drying2. Furthermore, uncontrolled nucleation can decrease the mean crystal size which is usually undesired in batch crystallization processes and is a main source of batch-to-batch variations. Beside the final crystal size distribution also the final crystal shape distribution is significantly influencing the final product properties. Platelet-like or rod-like crystals are known to cause difficulties during the downstream processes. However, due to the high surface to volume ratio of such shapes, they might be desirable in catalytic applications3 or solar cell applications4. In industrial practice, control of crystal shapes is frequently done by the usage of additives, which serve to decrease the growth rates of specific faces5. Since crystal growth is decreased by this approach, longer crystallization processes are expected, and, moreover, as ACS Paragon Plus Environment

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supersaturation can only slowly be depleted by the inhibited crystal growth, nucleation may occur during the process which results in an undesired final CSSD. Besides the usage of additives, also a change of the solvents may lead to different crystal shapes6,7. However, this approach also influences the entire production process, and hence may not always be applicable. Furthermore, the occurrence of solute inclusions within the crystals may violate purity specifications, especially in the pharmaceutical and food industry, and hence, the range of applicable solvents may be rather limited. Alternatively, the fact that the supersaturation dependency of the face specific growth rates differs between the individual face groups can be used to control crystal shapes8,9. With this approach control of crystal shapes is feasible via direct supersaturation – and thus temperature control. However, the range of crystal shapes which are attainable by pure growth process may be confined in practice. Therefore, sequential growth-dissolution cycles have been proposed by several authors10-13. In fact, the application of growth-dissolution cycles – or temperature cycles – is a popular technique in crystallization processes aiming at a reduction of the amount of fines and to obtain a narrower final crystal size distribution14-16. It was demonstrated both, theoretically and experimentally for single crystals, that this approach is also applicable to directly manipulate the final crystal shape and that the region of attainable crystal shapes can be extended as compared to pure growth processes11. In batch crystallization experiments, growth dissolution cycles were used for shape modifications of therephthalic acid17, monosodium glutamate13 and potassium dihydrogen phosphate18. In the later two examples, predefined temperature profiles were applied to the crystallization vessel in order to generate super- and undersaturated crystallization conditions. However, due to the uncertain crystallization kinetics, the realization of pure feedforward control turned out to be complex. Thus, it was suggested, that the underlying kinetics need to be re-estimated on the basis of the current process data, allowing for more accurate model predictions and, hence, better controllability of the entire crystallization process. Feedback control of crystallization processes however, requires the ability to observe the state of the crystalline phase in real-time. Although this is a challenging task even if the crystal population is characterized by one property, e.g. size, alone, major advances have been made in the last decades; see for example Nagy and Braatz19 for a recent review. To allow for the simultaneous observation of crystal size and shape, and therefore, to enable crystal shape control, video microscopy has been shown to be a promising measurement technique. Several approaches for reconstructing the crystal shape from the recorded video frames exist, including the application of wire frame models20, axis length distributions21, generic shape models22 or the application of Fourier descriptors23. In this work, an approach, which is based on the comparison of measured boundary curves of crystal projections to a pre-computed database24, is used to monitor the evolution of the CSSD in real-time. The obtained measurements are further used as feedback in a control loop for the experimental realization of a cyclic growth-dissolution process, and hence, a closed loop control of the crystal size and shape distribution is successfully established in this work. The remainder of the article is organized as follows. In Section 2, the experimental setup is presented, together with the model substance – potassium dihydrogen phosphate (KDP) – which is used throughout this work as a reference substance. In Section 3, the implementation of the crystal shape observer is briefly sketched. Furthermore, a strategy is presented that allows for supersaturation control during the growth and dissolution phases and further details of the necessary control laws are given in the supplementary material. Finally, the experimental results related to controlled growth-dissolution cycles are presented and discussed. Section 4 concludes this article.



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2. Experimental 2.1. Potassium Dihydrogen Phosphate

Potassium dihydrogen phosphate (KDP) represents the model substance in this work. KDP crystals exhibit prismatic {100} faces and pyramidal {101} faces on the outer crystal surface, as shown in Figure 1. Since KDP crystallizes in the tetragonal space group I-42d (a = 7.460 Å and c = 6.982 Å)25, the possible shapes of KDP range from elongated crystals with a high prominence of prismatic {100} faces to compact crystal shapes, and in extreme cases, to the octahedral crystal shapes exhibiting only {101} faces on the outer crystal surface, Figure 1. These possible crystal geometries are described by the h-representation26,27, where h = [h1, h2]T is the vector of perpendicular distances of the prismatic {100} faces (index 1) and the pyramidal {101} faces (index 2) to the crystal center. Within the framework of the hrepresentation it is possible that crystal faces detach from the outer crystal surface, as for example at high h1/h2 ratios in the case of KDP. In such cases, these crystal faces become virtual and their displacement from the crystal center is determined by the face distances of the neighboring faces24,26,27. Hence regions in h-space exists which are geometrically infeasible. In this work, such a region is marked by a dark grey area in the lower right part of the h1-h2-space, see Figures 3, 7 and 9. KDP was purchased from Carl Roth GmbH + Co. KG, purity of 98 % and pH 4.12 at a saturation temperature of 35 °C, and used without further purification. Seed material was purchased from Grüssing GmbH Analytica (purity 99.5%) and dry sieved. For each experiment, 0.8 g were taken from the sieve fraction of 212 µm to 300 µm. The seed crystals had a mean shape of h seed,0 ≈ (123 µm, 138 µm )T , which corresponds to rather compact crystals, see also Figure 8 a), and were added dry to the solution.

Figure 1: Different geometrically possible crystal shapes of KDP exhibiting prismatic {100} faces (index 1) and pyramidal {101} faces (index 2) and the shape representation using the perpendicular face distances h1 ({100}-faces) and h2 ({101}-faces). 2.2. Experimental Setup

The details of the experimental setup are given elsewhere28 and are therefore only briefly sketched here. The experiments are performed in a flat bottomed 3 L crystallization vessel, whose cooling jacket is connected to two different thermostats by two three-way valves. This particular setup is chosen to enable abrupt temperature changes, which are necessary to induce the desired levels of supersaturation and undersaturation, see Section 3.3. The solution is agitated by a four blade pitch bladed impeller operated at a speed of 400 rpm. The temperature is monitored by a PT100 thermocouple and the solute concentration is measured by an ATR-FTIR probe (Nicolet iS10, Thermo Fisher). Absorption spectra are continuously collected every 20 seconds over a range of 700 to 1800 cm-1 with a resolution of 4 cm-1. Using previously collected calibration standards, the solute concentration is obtained from the recorded spectra by a partial least squares approach. To monitor the state of the dispersed phase during the crystallization runs, the experimental setup is equipped with an external sampling loop. The suspension is continuously withdrawn from the vessel and pumped, by a peristaltic pump (Heidolph PD 5206, Heidolph Instruments ACS Paragon Plus Environment

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GmbH & Co KG), operated at a speed of 150 rpm with an inner tube diameter of 5 mm, through a flow-through microscope (QicPic, Sympatec) back to the crystallization vessel. The flow rate in the external sampling loop is approximately 0.52 l/min. To avoid, that temperature losses in the sampling loop are affecting the crystallization behaviour, all tubing of this loop were tempered by an additional thermostat whose setpoint temperature was set to the current temperature in the crystallization vessel. The microscope continuously collects videos of the bypassing suspension which are subsequently used to estimate the current state of the CSSD. Prior to each experiment, a defined mass of KDP is dissolved in 1.8 kg of water to create a solution with a desired supersaturation level at the initial temperature of 35 °C. When all crystals are dissolved, the temperature is lowered to this starting temperature, and the seed crystals are added to the solution. Subsequently, the automated programs for process control, described in Section 3, are run.

3. Results and Discussion 3.1 Crystal Shape Observer

To control crystallization processes, reliable measurement techniques are required to observe the state of the suspension. While several different measurement techniques exist to monitor the state of the continuous phase, the observation of the state of the disperse phase remains challenging. This is in particular true if the size and shape distribution of the crystal population has to be measured in real-time. For this purpose, a video microscope is used. Videos of the bypassing suspension are continuously collected. Each video consists of 200 individual frames, as exemplarily shown in Figure 2, collected at a sampling rate of 20 frames per second with a resolution 5 µm per pixel and a field of view of 5000 µm x 5000 µm. All collected videos are subsequently used as a basis for the estimation of the CSSD.

Figure 2: Left: Example frame obtained by video microscopy, right: depiction of the estimated crystal shapes based on the recorded video frame. Since the collected video frames yield only the crystal projections on the image plane, the true 3 dimensional crystal shapes need to be reconstructed therefrom, as illustrated in Figure 2. For this purpose, an estimation algorithm was proposed24, which is based on the comparisons of an observed crystal projection to a pre-computed database. The application of this algorithm to all observed single crystals in a collected video leads to estimates for the CSSD as exemplarily shown in Figure 3. In both figures, two distinct subpopulations can be distinguished, stemming from the seed crystal population and nucleated crystals. In this work, we are only concerned with the monitoring and control of the seed crystal population, and ACS Paragon Plus Environment


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hence a classification between the seed crystals and nucleated crystals is required. To achieve this, a rectangular region is defined in the state space with a size of 200 µm x 200 µm, depicted by the dark green area in Figure 3 (right). Every crystal which is found to lie within this region is considered to belong to the seed crystal population. This region is initially centered at the mean seed crystal size of h seed,0 and subsequently moved through the state space according to the mean seed crystal evolution. As can be seen from Figure 3, the evolution of the seed crystal population can be tracked reliably by this approach. However, it is also apparent, that a large number of nuclei are observed, especially after prolonged growth times. To exclude the estimation of the nucleated crystals that would dominate the major estimating computation time, a second rectangular region, centered at the mean seed crystal state is introduced with a width of 350 µm x 350 µm, Figure 3 (light green region, right). This region serves to obtain a minimal and maximal projection area which is possible for seed crystals. By using these thresholds, the majority of nucleated crystals could be classified as nuclei, prior to the shape estimation. These nuclei are depicted as light red circles in Figure 3 right. Thus, the entire computation time is significantly reduced to approximately 14 seconds per video, each with a length of 10 seconds, which is found to be sufficiently fast, for the realtime monitoring and the subsequent control of the crystallization experiments described in Section 3.3.

Figure 3: Estimated crystal size and shape distributions, left: original output of the shape estimation routines, right: classification of seed – and nucleated crystals by two rectangular regions (light and dark green) and observed mean seed crystal evolution (blue dots). For the control of the crystallization experiments, only the mean seed crystal evolution is tracked, as depicted by a series of blue dots in Figure 3. Note that there is an increase in the measurement error after prolonged growth phases, i.e. at time instances where a controlled switching to a dissolution phase is performed, see Section 3.3. In order to reduce this measurement noise and therefore to allow for more precise controller actions, we recently proposed the application of a Kalman filter18. This filter has been constructed to drop the bias between the measured and filtered mean crystal shape observations, even in case of uncertain crystallization kinetics. Furthermore, it turns out that a useful filtering performance in terms of smoothness and convergence is obtained thereby. The application of the Kalman filter to measurements of a cyclic growth-dissolution process is depicted in Figure 4. As can be seen, the measurement noise is significantly reduced by this approach, while no bias between original and filtered measurements is observable. With high accuracy observations obtained at reasonable computation times, this observer is well-suited for the control of the cyclic growthdissolution experiments and is therefore applied for the algorithms for supersaturation control,


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presented in section 3.2, and for the control of the entire cyclic growth-dissolution process as discussed in Section 3.3.

Figure 4: Observations of the mean seed crystal evolution over time. The measured evolution of the {100} faces is depicted with red squares, whereas the evolution of the {101} faces is depicted with green diamonds. The filtered measurements are shown as solid lines.

3.2 Supersaturation Control

A crystallization process that consists of several growth and dissolution phases will naturally result in prolonged process times, as compared to the pure growth processes. It is therefore convenient to design the process conditions such that the total process time is minimized, this introducing an optimal control problem. Recently, it has been shown12 that the time optimal process route starting from an initial shape h0 with a target shape hf requires only constant supersaturation profiles during the entire process. As a consequence, if the target crystal shape is attainable by a pure growth scenario solely, then a proper unique optimal supersaturation level solves the optimal control problem. If, however, the target crystal shape can only be attained by applying additional dissolution phases, then the time-minimal control process consists of growth and dissolution phases with constant levels of supersaturation and undersaturation, respectively. In other words, the realization of constant supersaturation and undersaturation profiles not only prevents the occurrence of uncontrolled nucleation, but much more, it is a prerequisite for the realization of time- minimal cyclic growth-dissolution processes. A simple way to control supersaturation in batch crystallization is to use a PI controller based on online measurements of the concentration and temperature of the suspension. Such a controller does not require any a priori knowledge about the crystallization process and is therefore easy to implement in practice. In this work however, we present an alternative controller, which uses the knowledge about the kinetics of crystallization and heat transfer to derive a setpoint value for the thermostat temperature that has to be applied. By exploiting this additional information, the resulting controller shows, according to preliminary simulation studies, a better performance compared to a simple model free PI controller, and is derived in detail in this section. Since solute mass is transferred to the crystalline phase during a growth process, the temperature of the crystallization vessel has to be adjusted, typically lowered, such that, the supersaturation level remains constant over time. For this purpose, cooling policy was derived29, which was adapted to account for changes in the crystal shape over time28 and is used in this work to construct a supersaturation controller. The supersaturation S is defined as ACS Paragon Plus Environment


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the ratio of the concentration w, given in kg solute per kg solvent, and the equilibrium concentration weq: w S= (1) weq (T ) The temperature dependence of the equilibrium concentration is described by the empirical equation: (2) weq (TC ) = aTC2 + bTC + c = 4.6479 ⋅ 10 −5 TC2 − 0.022596TC + 2.8535 Using these two definitions, the following differential equation for the crystallizer temperature TC can be derived28, which has to be fulfilled for a constant supersaturation profile: 1 dw 1 dw dTC S dt = = S dt (3) dw dt 2aTC + b eq dTC The temperature profile resulting from eq. (3) has to be realized by applying an appropriate temperature profile of the thermostat which is connected to the cooling jacket of the crystallizer. To calculate this temperature profile, the crystallization vessel is modeled as ideally mixed and the cooling jacket is modeled as a cascade of two CSTRs consisting of the bottom element of the cooling jacket (index G) and the cylindrical side segment of the cooling jacket (index S), see Figure 10 in the supporting information. Starting from the energy balances, the following set of differential equations describing temperature profiles in the crystallizer and the cooling jacket can be obtained: dTC dmKDP = a1TG + a2TS − a3TC + a4 (4 a) dt dt dTS = b1TG − b2TS + b3TC + b4 (4 b) dt dTG (4 c) = c1TTh − c2TG + c3TC + c4 dt A detailed derivation of these balance equations can be found in the supporting information of this manuscript. As can be seen from eq. (4 c), the thermostat temperature TTh appears explicitly on the right hand side of the equation. Eq.s (4 a) to (4 c) constitute a system of equation with four unknown variable, namely the three derivatives on the left hand side as well as the thermostat temperature TTh. Hence, an additional independent equation is required to calculate TTh, which is obtained by taking the second derivative of the crystallizer temperature TC with respect to time using eqs. (3) and (4 a). d 2 w dweq dw d  dweq  d 2w dw dTC   − ( 2aTC + b ) − 2a 2 2 2 dT dt dt dT dt d TC 1 C  C  = 1 dt dt dt (5 a) = 2 2 2 S S dt (2aTC + b)  dweq    dT  C

d 2TC dTG dTS dTC d 2 mKDP = a + a − a + a 1 2 3 4 (5 b) dt dt dt dt 2 dt 2 It has to be mentioned, that both equations, (5a) and (5b), as well as eqs. (3) and (4 a), are not only dependent on the temperatures of the crystallizer, but as well on the change of crystalline mass mKDP and concentration w. A detailed model to express these derivatives as a function of the state of the CSSD is presented in the supporting information of this manuscript. By


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rearranging Eq. (5 b) and inserting eqs. (5 a), (3) and (4 b), the derivative of the ground element temperature of the cooling jacked dTG/dt can be obtained. dTG 1 d 2TC a 2 dTS a3 dTC a 4 d 2 m KDP = − + − (6) dt a1 dt 2 a1 dt a1 dt a1 dt 2 Finally, this information can be used, to solve eq. (4 c) for the thermostat temperature TTh to give: 1 dTG c2 c c TTh = + TG − 3 TC − 4 (7) c1 dt c1 c1 c1 This equation provides the required temperature profile for given dynamical states, guaranteeing a constant level of supersaturation. Figure 5 depicts the resulting profile. It can be seen, that the profiles of the crystallizer and thermostat temperatures resulting from eqs. (3) and (7) can be realized with high accuracy. Also the corresponding setpoint supersaturation profile indicates a reasonable maintenance. However, after prolonged growth phases, the supersaturation level is decreasing which can be attributed to uncertainties in the crystallization kinetics and heat transfer coefficients. Furthermore, also nucleation, which occurs even under rather moderate conditions (low temperature and intermediate supersaturation), and subsequent growth, which becomes more significant at prolonged growth times, leads to an additional depletion of the supersaturation.

Figure 5: Open loop supersaturation control using eq. (7), left: Temperature profiles of the crystallizer and thermostat (green dashed and yellow dashed) calculated by eqs. (3) and (7) respectively, and measured crystallizer and thermostat temperature profiles (green solid and yellow solid), right: Measured supersaturation profile (solid line) and setpoint supersaturation level (dotted line). To account for these model uncertainties, and to compensate for potential offset in the initial temperature, and therefore to achieve a better controller performance, which is capable of compensating the above mentioned phenomena even at higher supersaturations and temperatures, eq. (7) has been used as a basis for designing the underlying PI supersaturation controller. This controller is designed to adjust the thermostat setpoint temperature based on the difference between measured supersaturation and the desired level of supersaturation:   1 t TTh,PI = TTh,mod −  K p (S set − S ) + ∫ (S set − S (τ )) dτ  (8) TN 0   In this equation, TTh,mod denotes the thermostat temperature which is calculated by eq. (7). Based on previous simulation studies, the following values for the parameters Kp and TN of this controller were determined: Kp = 300 K and TN = 1.33 s/K. The resulting temperature and ACS Paragon Plus Environment


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supersaturation profiles are depicted in Figure 6. Despite some mild oscillations, the supersaturation level can be kept constant even after prolonged growth and under conditions which are resulting in faster crystallization kinetics, compared to Figure 5, which in turn are promoting the process uncertainties mentioned above. In the next section, we present experimental results related to control of growth-dissolution cycles, where the presented methods for the crystal shape observer and the PI supersaturation control are applied.

Figure 6: Closed loop supersaturation control, left: realized crystallizer temperature profile (green) together with the setpoint (dark yellow) and realized thermostat temperature profile (yellow), right: measured supersaturation profile (solid line) and setpoint supersaturation level (dotted line).

3.3. Control of Crystal Shapes by Growth-Dissolution Cycles

Obtaining a desired final crystal shape distribution by means of a cyclic growth-dissolution process requires applications of process control routines. In this work, we aim not only at controlling the final crystal shape distribution, but also at reaching the target shape in minimal time. As already mentioned in Section 3.2, it was shown12 that the time optimal supersaturation profile consists of growth and dissolution phases with constant levels of supersaturations. With this finding, the optimal control problem can be solved by solving a parameter optimization problem with two decision variables only, namely the level of the supersaturation (for the growth phases) SG and undersaturation (for the dissolution phases) SD respectively12. min (t G + t D ) [S G , S D ]

S G,min ≤ S G ≤ S G,max s.t. :

S D,min ≤ S D ≤ S D,max

(9)

h(t G + t D ) = h F Note that both are bounded by corresponding upper and lower bounds. The constraints for SG are chosen to prevent the onset of primary nucleation, as well as to avoid a dependence of the crystallization kinetics on impurities in the crystal lattice. At undersaturated conditions, a minimal dissolution rate is required, whereas an uncontrollable dissolution rate is prevented by SD,min. Due to the rather low suspension densities that are used in the experiments, the temperature changes during the individual growth and dissolution phases are small, see Figures 7 and 9, and can be neglected in the optimization problem. By this assumption, the evolution of the CSSD is decoupled from the state of the liquid phase, and no population balance model has to be employed. Therefore, such a numerical optimization problem can be ACS Paragon Plus Environment

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solved efficiently and hence, we suggest an online optimization of the values SG and SD during the crystallization process based on the current observation h(t). The optimization problem was implemented in MATLAB 2014a, and solved by the SQP algorithm provided by the routine fmincon of the optimization toolbox. The accuracy of the optimization results, in particular the resulting times for the growth phase and dissolution phase, are strongly dependent on the accuracy of the crystallization kinetics used in the optimization. In this work, we apply the estimated kinetics presented by the authors in a recent publication28. The growth kinetics can be described by:  E A , G ,i   S − S G* ,i Gi = k 0,G ,i exp − (10 a) RT   (10 b) S G* ,i = s1,G ,iϑ 2 + s 2,G ,iϑ + s 3,G ,i

(

)

Where EA,G,i denotes the face specific activation energies and S*G,i a threshold supersaturation at which no significant growth occurs, whose temperature dependence is parameterized by a second order polynomial. Note, that the temperature T in eq. (10 a) is given in K whereas the temperature ϑ in eq. (10 b) is given in °C. A similar expression is also given for the dissolution kinetics.  E A , D ,i   S − S D* ,i Di = k 0,D,i exp − (11 a)   RT  (11 b) S D* ,i = s1,D,iϑ 2 + s 2,D,iϑ + s 3,D,i

(

)

The parameter values of these crystallization kinetics are given in Table 1. To improve the accuracy of the kinetic parameters and therefore to account for possible impurities it was suggested to re-estimate the crystallization kinetics based on the actual observations of CSSD evolution13,18. In this work, this re-estimation is done during the first growth or dissolution phase respectively, to obtain initial estimated for the kinetics, and a final update of the kinetics is obtained after the first growth or dissolution phase is finalized. Table 1: Parameter values for the crystallization kinetics of eqs. (10) and (11). Values marked with * correspond to the initial parameter values. Growth {100} Growth {101} Dissolution {100} Dissolution {101} k0,i [m/s] 6.011 13.705 1.086* 0.346* EA,i [kJ/mol] 37.05 39.14 29.71 26.83 -5 -6 -5 s1,i [°C²] 1.764·10 -7.532·10 2.732·10 2.316·10-5 s2,i [°C] -1.934·10-3 -1.185·10-4 -1.356·10-3 -1.158·10-3 s3,i [-] 1.0943* 1.0469* 1.0175 1.0183 The mean shape estimates obtained by the methods introduced in Section 3.1 are not only used for the supersaturation control and re-estimation of the governing crystallization kinetics, but also for the planning of the crystal shape trajectory. Obviously, such planning policies are needed to reach the target crystal shape with a high precision. Furthermore, for a given optimal pair of super- and undersaturation levels [SG, SD], we also aim at minimizing the total number of switches between growth and dissolution phases. This is not just to avoid the related energy losses during the switching process, but also to reduce the required total crystallization time. If it is possible to grow crystals to an arbitrary size, one supersaturation switch would be sufficient to reach a desired final shape12. However, the maximal obtainable crystal size is typically limited by the solubility, the solute consumption or technical reasons. In this work, we assume that the mean seed crystal volume should not exceed a value of Vmax = 0.288 mm³ to prevent blockage of the flow-through microscope. Additionally, the minimal seed crystal volume during the dissolution phase is bounded to Vmin = 0.08 mm³ to prevent ACS Paragon Plus Environment


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complete dissolution of any seed crystals. Under these constraints, the total number of supersaturation switchings is minimized, if the whole span between Vmin and Vmax is used for the cyclic crystallization. This is ensured, e.g., by adjusting the duration of the growth and dissolution phases in such a manner that none of the two volume constraints is violated within the subsequent switching phases. These adjustments rely either on model predictions or prior observations of the same supersaturation switching scenario. Based on the model predictions, the thermostat temperature profiles are then automatically calculated to minimize the total time which is required for the remaining switching sequence. Finally, also the duration of the last growth or dissolution phases is similarly adjusted such that the desired final crystal shape can be reached. This is necessary since a complete growth- or dissolution phase between Vmax and Vmin would lead to an overshoot of the crystal shape evolution which would have to be corrected by further control actions, see Figure 7 and Figure 9. The entire process is stopped as soon as the mean crystal shape reaches a minimal distance to the desired target crystal shape.

Figure 7: Experimental realization of controlled growth dissolution cycles using a minimal supersaturation level of SG,min = 1.10. a) Observed mean crystal shape evolution during growth and dissolution phases (dark blue dots) and during the switching phases (light blue dots) together with the mean crystal shapes at the end of each growth - or dissolution phase; b) Measured temperature profiles of the thermostats (red and blue) and the crystallizer temperature (green); c) Detailed depiction of the measured crystallizer temperature; d) Measured concentration profile; e) Measured supersaturation profile (solid line) during the growth and dissolution phases (dark blue) as well as during the switching phases (light blue), applicable super- and undersaturation levels (light blue areas) and optimal supersaturation levels (dashed line). The experimental realization of crystal shape control by growth-dissolution cycles is depicted in Figure 7. In this experiment, the supersaturation ranges were set to 1.10 ≤ SG ≤ 1.12 and 0.975 ≤ SD ≤ 0.995, respectively. The target crystal shape was set to h F = (200 µm, 440 µm )T and thus to rather elongated crystals shapes, see Figure 7 a) and Figure 8 i). After preparation of the solution with a supersaturation level of 1.10 at 35°C, the seed crystals were added to the solution and the control routines were started together with the video collection. The resulting observed mean crystal shape evolution is depicted in the subfigure (a). It is obvious that the region which is attainable by a growth phase under these supersaturation constraints indicated by a gray shaded region originating at the initial seed shape - can be left by a subsequent dissolution phase. As a consequence, switching trajectories with a number of subsequent growth and dissolution phases may be utilized to achieve crystal morphologies which do not result directly from a pure growth process only. As depicted by the micrographs ACS Paragon Plus Environment

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in Figure 8, the mean crystal shape evolves from compact shapes to more elongated crystals under such experimental conditions. Due to the ability to observe and control the CSSD evolution over time, also a cyclic crystallization between the minimal and maximal seed crystal volume can be achieved reliably and the target crystal shape, indicated by the yellow point, can be reached with high precision. Interestingly, the slopes of the mean shape evolution during different growth and dissolution phases respectively are essentially parallel, indicating that the underlying growth and dissolution kinetics are indeed independent on the crystal size and shape. In Figure 7 b) and c) the measured temperature profiles of the thermostats and the crystallizer temperature are shown. Due to the use of two thermostats, the temperature changes which are necessary to realize the supersaturation switchings can be realized reasonably fast. It has to be mentioned, that the times which is required for the individual growth – and dissolution phases are similar in this experiment. This is mainly due to the rather fast growth rates of approximately 150 nm/s and 200 nm/s as well as to the limited driving forces during the dissolution phases. The measured concentration profile is shown in subfigure d). The concentration level is reliably cycling between a maximal and minimal value. This indicates that a cyclic crystallization process bounded by the maximal and minimal crystal volume Vmin and Vmax, can indeed be controlled using the concentration measurements as proposed in ref 12, provided that the solute consumption due to the growth of nucleated crystals can be neglected. Figure 7 e) depicts the measured supersaturation profile over time. It can be seen, that the level of the supersaturation can be well controlled with the methods proposed in Section 3.2. Overshoots in the supersaturation level, e.g. during the second and third growth phases, are compensated reasonably fast, maintaining the setpoint supersaturation level. The dashed lines in this figure indicate the optimized supersaturation levels over time. As can be seen, the lower bound for SG and SD are active throughout the entire crystallization time. For undersaturated conditions this lower bound corresponds to the maximal dissolution velocity, while for supersaturated conditions, the application of the minimal growth velocities were found to be optimal. This can be explained by the growth kinetics in the chosen supersaturation interval. At lower supersaturation, KDP crystals grow toward more elongated crystals – and thus more towards the final crystal shape – as compared to higher supersaturations. Therefore, an even further decrease in the total crystallization time can be expected if the value for the minimal applicable supersaturation is decreased.

Figure 8: Example photographs of the KDP crystals during different time instances: a) Initial seed crystals; b) Crystals after the first growth phase; c) After first dissolution phase; d) After the second growth phase; e) After the second dissolution phase; f) After the third growth phase; g) After the third dissolution phase; h) After the final growth phase; i) Final crystal shape. Bars in the lower right corners correspond to a length of 400 µm.



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In a second experiment, the lower supersaturation bound for growth conditions was therefore relaxed to SG,min = 1.09. The corresponding experimental results are shown in Figure 9. Again, the evolution of the mean seed crystal shape as well as the supersaturation levels are controlled extremely well, and the target crystal shape can be obtained almost exactly. As expected, the total time that was required to reach the target crystal shape is reduced to approximately 7000 seconds, as compared to 8000 seconds for the experiment depicted in Figure 7. Notably, also the number of supersaturation switches could be reduced by two. As can be seen, from Figure 9 e), the lower bounds on SG and SD are again active throughout the experiments which indicates that a further relaxation of SG,min will result in even shorter process times. However, particularly the growth of the {100} faces of KDP is massively influenced by impurities present in the solution at lower levels of supersaturation5,28-31. Therefore, the controllability of the entire process will be affected as the relative growth rate Grel = G1/G2, which dictates the direction in which the crystal population will evolve, will be extremely sensitive to even small changes in supersaturation and purity of the initial material. Furthermore, the {100} faces of KDP are known to exhibit the phenomenon of growth rate hysteresis, that is, the growth rate G1 is dependent of the history of the solution31 at lower levels of supersaturation. Due to the dissolution phases as well as the switchings between undersaturated and supersaturated conditions, all growth phases after the first one might be affected by this phenomenon, which would lead to an undesired evolution of the CSSD and is not captured by the kinetics and models used in this work. However, growth rate hysteresis is known to diminish at higher supersaturation levels31,32 and hence, the applicable lower supersaturation levels SG,min were kept at rather high values in this work. Finally, there is strong evidence given by Zaitseva et al.30 that the incorporation of additives, in particular metal ions, of the {100} faces of KDP is more pronounced at lower supersaturations.

Figure 9: Experimental realization of controlled growth dissolution cycles using a minimal supersaturation level of SG,min = 1.09. a) Observed mean crystal shape evolution during growth and dissolution phases (dark blue dots) and during the switching phases (light blue dots) together with the mean crystal shapes at the end of each growth - or dissolution phase; b) Measured temperature profiles of the thermostats (red and blue) and the crystallizer temperature (green); c) Detailed depiction of the measured crystallizer temperature; d) Measured concentration profile; e) Measured supersaturation profile (solid line) during the growth and dissolution phases (dark blue) as well as during the switching phases (light blue), applicable super- and undersaturation levels (light blue areas) and optimal supersaturation levels (dashed line).


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4. Conclusions In this work, we demonstrated the applicability of growth-dissolution cycles to modify crystal shapes in a fully controlled manner. The control algorithms that were employed in this work are based on an observer technique to monitor the evolution of the CSSD in real time. The ability to observe the disperse phase during crystallization processes is of value for its own, as it can lead to more detailed insights in the governing phenomena, better estimates of the crystallization kinetics and, of course, it provides the opportunity to control crystallization processes in a closed loop form, as employed in this work. Additionally, a strategy to control the level of the supersaturation during a growth or dissolution scenario is presented in this manuscript. Although we were concerned with the rather simple task of maintaining constant levels of supersaturation or undersaturation in this work, we emphasize, that the derived equations can be easily adjusted to control any differentiable supersaturation profile over time. Hence, other process concepts can be realized and controlled by the presented approach such as, e.g., nucleation control, as well. Finally, the application of growth-dissolution cycles has been shown to be a promising technique to tailor crystal shapes towards a desired final shape distribution. We demonstrated that crystal shapes can indeed be obtained by this process concept, which would not have been attainable by pure growth processes. By applying the proposed control concepts, we were able to obtain a desired target crystal shape distribution reliably with a high precision, while minimizing the required total process time.

Acknowledgements The financial support of this work by the German Research Foundation (DFG) under the grant SU 189/5-1 is gratefully acknowledged.

Supporting Information A detailed derivation of the model equations, including the energy balances and the equations derived from the morphological population balance, that were used to derive the equation in section 3.2, can be found in the supporting information of this manuscript.

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For Table of Contents Use Only Manuscript title: Optimal Control of Crystal Shapes in Batch Crystallization Experiments by GrowthDissolution Cycles Author list: Holger Eisenschmidt, Naim Bajcinca, Kai Sundmacher TOC graphic

Synopsis: A closed loop control of crystal shape distributions during cyclic growth-dissolution crystallization processes is realized. Using video microscopy, the evolution of the crystal shape distribution is monitored in real time. Hence, feedback for subsequent control actions is obtained. Furthermore, a supersaturation control strategy is presented which ensures timeoptimal cyclic growth-dissolution processes to obtain a desired final crystal shape distribution.


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Optimal Control of Crystal Shapes in Batch Crystallization Experiments

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