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Crystallization from Solutions Containing Multiple Conformers. 1. Modeling of Crystal Growth and Supersaturation L. Derdour*,† and D. Skliar‡ †

Drug Product Science and Technology, Material Science, and ‡Chemical Development, Bristol-Myers Squibb, 1 Squibb Drive, New Brunswick, New Jersey 08901, United States ABSTRACT: A mechanistic model for supersaturation and crystal growth during batch crystallization of organic solids from solutions containing multiple conformers is presented. The model is based on the approach of the right conformer (RC) which assumes that only one conformer participates in the surface integration step. The model is concerned with systems characterized by slow crystal growth and low supersaturation, a behavior favored when the RC is the minority species in solution. Crystal growth is assumed to occur via a step advance mechanism with a variable step advance velocity (VSAV). Model derivation indicated that when the approach of the RC applies, crystal growth is inversely proportional to crystal size. Lastly, model simulations predicted an exponential increase of maximum relative supersaturation with linear antisolvent addition rate. The simulation also allowed estimation of maximum addition rate below which secondary nucleation is minimized.

1. INTRODUCTION A common characteristic of organic molecules is their ability to adopt multiple conformations due to their inherently high degrees of freedom compared with inorganic substances. With respect to crystallization, a frequently asked question is: How do multiple conformations affect nucleation and crystal growth of organic molecules? To date, there is no consensus in the literature regarding the effect of the existence of several conformers in solution of a given molecule on its crystallization behavior. However three general ideas were proposed from the literature: namely, (1) the presence of multiple conformations in solution can inhibit nucleation and crystal growth, (2) it can enhance those phenomena or (3) it can have a negligible effect on them. 1. Yu et al.1 presented a general rule of thumb for predicting which conformer in solution is expected to crystallize depending on the crystal structure. These authors also stipulated that systems of higher degrees of freedom (i.e., high number of conformers) are likely to be more difficult to crystallize. This was attributed to the lower effective concentration of the conformer that crystallizes. Because of the presence of all other conformers, the “right” conformer (RC) is diluted, which in turn gives a low supersaturation. Furthermore, the other conformers can act as “impurities” and inhibit the integration of the right conformer on the crystal surface. This rationale leads to the expectation that systems with multiple conformations have a decreased crystal growth/nucleation rate. This effect of multiple conformations in impeding crystallization kinetics was recently reported by Derdour et al.2 2. On the other hand, as noted by Buttar et al.,3 increased molecular flexibility and hence the presence of multiple © 2012 American Chemical Society

conformers in solution can result in increased possibilities of self-assembly, which can lead to higher probability for nucleation. The existence of multiple conformers increasing the number of polymorphs is exemplified by Yu,4 who attributes the finding of 6 polymorphs of 5-methyl-2-[(2-nitrophenyl)amino]-3-thiophenecarbonitrile to the crystallization of 5 different conformers into different crystals. It was reported by Yu et al.5 that high energy conformers can form crystals if they can pack into a stable crystal lattice. They concluded that crystallization has a stabilizing effect for high energy conformers if they are not prone to H-bonding. Yu4 and Yu et al.5 also reported that dipole−dipole interactions stabilize the high dipole conformers in solution, thus enhance the possibility of crystallization. 3. The literature also suggests that in most cases the crystal surface influences the conformation of the crystallizing species, which results in a negligible effect of the presence of multiple conformations in solution on crystallization behavior. This probably explains the few reports in the literature of crystallization kinetics dependent on conformations in solution. Yu et al.5 found that the crystal lattice can actually influence the conformation of the molecule that crystallizes and implicitly stipulated that the crystallization has an effect on the proportion of the conformers in fluid layers adjacent to the crystal surface. The crystal surface influence on the conformation of the crystallization species was also reported by many authors such as McPherson6 for crystallization of macromolecules and Bernstein and Hagler,7 Hagler and Received: July 13, 2012 Revised: September 6, 2012 Published: October 9, 2012 5180

dx.doi.org/10.1021/cg300974u | Cryst. Growth Des. 2012, 12, 5180−5187

Crystal Growth & Design

Article

Bernstein8 and Bernstein et al.9 for small organic molecules. In this case, the crystal can be considered as a “catalyst” to the crystallization and nucleation and growth kinetics are independent of the conformers’ concentrations in solution. In a recent study we reported that the effect of crystal surface on the conformation of the crystallizing species diminishes for energy barriers of transition between conformers higher than 10 kcal/mol.2 In solution, population of conformers and atropisomers depends on conformational energies, energy barriers of transition between conformers/atropisomers, temperature and solvent/ solute interactions. As mentioned above, if molecules already crystallized on the crystal surface do not influence the conformation of the approaching molecules, one can expect an impeded growth rate from a solution with multiple conformers. Therefore we assume that if the conformer that crystallizes is a high energy conformer (i.e., in low proportion in solution), crystal growth is expected to be slow if crystal surface does not act as a conformation modifier on upcoming molecules. In addition, the conformer that does not participate in the surface integration mechanism, i.e., the wrong conformer (WC), can block crystallization sites and inhibit crystal growth. The low nucleation and growth rates encountered in multiple conformation systems are also expected to deteriorate if the equilibration kinetics between conformers is slow. From this perspective, increasing the temperature can be a way to facilitate crystallization since high temperatures are expected to increase the population of the high energy conformers in solution. Elevating the temperature also leads to a faster interconversion between conformers especially beyond the NMR peak coalescence temperature where rotamers’ lifetimes are shorter than typical NMR time scales, which can be as low as the millisecond−microsecond range.10−14 This indicates that the equilibration rates between conformers at temperatures above the coalescence temperature are high and are not expected to limit crystallization kinetics. On the other hand, faster equilibration rates can also lead to shorter RC mean lifetime. Therefore, it is expected that very high conformation interchange kinetics can also impede crystallization for systems where the approach of the RC is applicable if the RC mean lifetime at the interface crystal/film is shorter than the time required for solute surface integration. In a previous study15 we presented a model for crystal growth and supersaturation based on a constant step advance velocity (CSAV). That model was found to provide a good prediction of supersaturation for ca. 70% of the crystallizations. Its limitations were attributed to the nonconstancy of the step advance velocity during crystallization. In this paper, the model is modified to account for the variation of the step advance velocity during crystallization. The new model was derived for crystallization of solids growing from solutions containing multiple conformers or atropisomers where the solute has to transit to a single conformation (RC) before it integrates onto the crystal surface. It addresses the cases where the approach of the right conformer is applicable (i.e., crystal surface does not influence the conformation of the crystallizing species) and when crystal growth is the dominating phenomenon during crystallization.

as the concentration of the RC at equilibrium and the concentration of the RC minus the intrinsic solubility.2 The measurable supersaturation (MSS) corresponds to the classical definition of supersaturation and for a two-conformer system was found to be related to the ISS by2 ISS = MSS Keq /(Keq + 1)

(1)

where MSS = C − C*

(2)

Keq is the equilibrium constant between conformers, defined as Keq = C RC/C WC = CCR /(C − CCR )

(3)

The integration mechanism presented in this study is based on the step advance theory (Kossel16 and Bennema and Gilmer17). Non-Kossel models are not considered since they deal with the rare cases of melts where molecules occupy nonequivalent positions within the unit cell (Chernov18,19). The main assumptions of the following formulation are as follows: • Diffusion is not limiting crystallization kinetics. • Two main conformers are considered, namely, the RC and the wrong conformer, noted WC. • Concentration is uniform in the crystallizer. • Crystal growth is isotropic. • The crystal surface does not affect the conformation of crystallizing molecules. • Supersaturation is low since crystallization is driven by the ISS. • The system is characterized by slow crystal growth. • Step bunching is neglected: Steps are stable, and step height is constant. • The rate of equilibration between conformers is not limiting crystallization kinetics: RC depletion by surface integration is instantaneously compensated for by conversion of WC to RC to reestablish equilibrium between conformers. • At the interface crystal/film, the mean lifetime of the RC is higher than the time required for solute surface integration. 2.1. Derivation of the Relationships for C, MSS, G and Li. The model developed in this paper applies to slow kinetics, crystal growth dominated crystallizations (i.e., other phenomena such as nucleation, agglomeration and breakage are negligible). In this paper, the modeling approach was applied to an isothermal antisolvent batch crystallization. However, the approach is also applicable to cooling crystallization. In the subsequent model formulation, we will consider the evolution of n classes of crystal sizes starting with a given initial size distribution of the seeds. The number of size classes will be maintained constant and equal to the number set for the initial PSD. During crystallization, the total number of growing crystals in each class will also be maintained constant and equal to the initial number of particles, i.e., negligible nucleation throughout crystallization. Our approach is to determine the concentration variation based on the crystal growth of each individual class of crystals starting from the initial PSD. The mass of the dissolved solid is determined by subtracting the sum of the weight gain of all crystals from the initial mass of the dissolved solid. The temporal weight change of a single crystal of class i is

2. MODEL FORMULATION In a previous study the intrinsic solubility and intrinsic supersaturation (ISS) were introduced and defined respectively 5181



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d(ρcr (Vcr)i ) d(mcr )i = dt dt

(5.1), which may explain the quasi-independence of Keq upon solvent composition observed experimentally. On the other hand, the effect of the solvent composition change during crystallization on the change in energy barrier can be minimal if the latter is high, i.e., the relative change in energy barrier due to solvent composition variation is negligible. On the basis of the considerations above and since the operation is carried out at constant temperature, Keq is considered constant throughout the crystallization. Combination of eqs 1−3, 8 and 11 leads to the relationships of C, CCR, MSS and ISS, which are summarized in Table 1. In Table 1, the solubility (C*) is determined by the following polynomial expression, determined from experimental data (coined as total solubility in Derdour et al.2):

(4)

We introduce the following cube based volume shape factor:

kv = Lci /Li

(5)

where Lci is the characteristic length of crystals of class i and L is the edge length of a cubic crystal of the same volume as crystals of class i. Assuming a constant kv during crystallization, eq 4 becomes

d(mcr )i dL 3 = ρcr kv i dt dt

(6)

Resolution of eq 6 in terms of Li yields

P

Δ(mcr )i = ρcr kv(Li 3 − Li0 3)

C* =

(7)

2

and C*CR is the intrinsic solubility (reported to the RC). The temporal variation of particle size (Li(t)) in eqs 10−13 is unknown. In the following section, an approach to estimate the evolution of particle size is described. The creation of steps is related to the density of kinks on the crystal surface and is believed to increase dramatically with crystal growth due to the likelihood of surface defect formation under these conditions (Myerson and Ginde25). However, for slow crystallizing systems at low supersaturations, the density of kinks and defects on the crystal surface can be considered low, hence there is a reduced probability for surface nucleation. If the concentration of the RC is low leading to a low ISS, one can expect crystal growth to be slow, which translates into a low density of kinks and defects. Mullin26 suggested that the behavior of a face containing multiple steps is practically the same as that of a face containing only one step. On the other hand, for systems characterized by slow growth rates and low supersaturation like the system at hand, step stability is expected to increase the propensity for step bunching is low as noted by Chernov and Komatsu.27 Based on these considerations, step bunching and macrosteps are neglected and only one moving step per face is considered for simplicity. In addition, under conditions of slow growth and low supersaturation, new steps are assumed to be created preferentially at the surface edges once the step has finished covering the whole face as reported by Kossel16 and Mullin.26 This behavior is consequent to the surface energy minimization principle: If a new layer is created on a flat surface, two additional steps are created while only one step is created if an additional layer appears at the edge of a surface (cf. Figure 1). Steps are more energetic than flat surfaces according to the periodic bond

n

(8)

i=1

The mass of dissolved RC can be easily determined from eq 3 and found to be mRC = mKeq /(1 + Keq) = f (T , xAS)

(14)

p=0

Equation 7 is the relationship between the decrease of the mass of dissolved solute, due to crystallization on a single crystal of class i, and crystal size Li. Knowing the population of each class of particles Ni, the total mass of the dissolved solute can be determined by the following equation: m = m0 − ρcr kv ∑ Ni(Li 3 − Li0 3)

∑ apxas p

(9)

where xAS is the mass fraction of the antisolvent in the solvent system (see Appendix). It is generally believed that the equilibrium constant is dependent on temperature and solvent properties. High temperatures are known to shift the equilibrium toward the high energy conformers because of the Boltzmann distribution law, and solvent can affect the equilibrium between conformers by polar interactions such as solvation and/or H-bonding. However, for the system at hand, Keq was found to be practically independent of solvent composition.2 At a given temperature, the energy barrier of conformational change and the proportion of conformers at equilibrium depend upon solvent polarity and proticity with more polar atropisomers being better stabilized in more polar and/or protic solvents.20−24 In addition, the proportion of atropisomers/conformers at equilibrium and the energy barrier of conformational change are expected to be less solvent-dependent in polar solvents.24 For our system, polar solvents (methanol as a solvent and acetone as an antisolvent) were used and both have the same polarity index Table 1. Relationships Obtained for C, CCR, MSS and ISS variable C

relationship n

C = [m0 − ρcr kv ∑ Ni(Li 3 − Li03)]/(Vsol Mwslt) i=1

CCR

C RC = Keq /(1 + Keq)[m0 − ρcr kv ∑ Ni(Li 3 − Li03)]/(Vsol Mwslt) i=1

(11)

n

MSS

MSS = ISS

(10)

n

m0 − ρcr kv ∑i = 1 Ni(Li 3 − Li03) Mwslt((Vsol)0 + AR m0t )

− C*

(12)

n

ISS = Keq /(1 + Keq)[m0 − ρcr kv ∑ Ni(Li 3 − Li03)]/(Vsol Mwslt) − C*CR i=1

5182

(13)



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Figure 1. Comparison between a step formation from a flat surface and an edge.

chains (PBC) theory (Hartman and Perdok28−30). Since the growing crystal has a tendency to minimize its surface energy, new layers are expected to be created preferentially at the edges of the faces once the step had finished forming an additional layer of molecules. One of the bases of the model presented herein is the following expression of the step advance velocity on a given face A(hkl) reported by Bennema and Gilmer:17 ⎡V L 2 ⎤ + vsa = ⎢ m m ν e−E /(RT )⎥(CA(hkl) − C*) ⎢⎣ λA(hkl) ⎥⎦

Figure 2. Gradient of concentration in continuous phase. Ci: Concentration of all conformers at interface diffusion film/integration film. CRC: Concentration of right conformer. C*RC: Solubility of the right conformer (intrinsic solubility).

(15)

where CA(hkl) is the concentration at the interface integration film/diffusion film, on face A(hkl), C* is the solubility, Lm is the molecular length, Vm is the molecular volume, λA(hkl) is the distance between kinks of a hypothetical crystal face A(hkl), ν is the molecular vibration frequency and E+ is the activation energy for integration. This relationship was obtained for anisotropic crystal growth using the classical definition of concentration (concentration of all conformers in the case of a multiconformation system). It shows that, for a given crystal face A(hkl), the step advance velocity is directly proportional to the supersaturation via a kinetic constant depending on molecular characteristics, density of kinks, and temperature. For a given solute, molecular characteristics are constant and, if crystal growth is slow, surface nucleation and crystal defects can be considered minimal. Consequently, for an isotropic crystal growth, the step advance velocity is directly proportional to the supersaturation at the interface crystal−film. For our system, the ISS is the driving force for crystallization. We also introduce a WC crystal growth inhibition term (Ψ) to account for kink poisoning by the WC, and hence eq 16 can be modified to account for the relationship between ISS and MSS (eq 1): ⎡ ⎛V L 2 ⎞ ⎤ + vsa = ⎢K Ψ⎜⎜ m m ν⎟⎟ e−E /(RT)⎥MSS ⎢⎣ ⎝ λA(hkl) ⎠ ⎥⎦

Figure 3. Increase of crystal size due to step advance.

concentration of all conformers between the bulk and the diffusion/integration films. It is assumed that, during the crystallization of a solid containing a single high energy conformer, the integration step is limiting due to the low amount of the RC available which results in a low ISS. For the system at hand, it was demonstrated experimentally that hydrodynamics (i.e., diffusion) has no effect on desupersaturation rates.2 For the sake of simplicity, we will write the model for the case of a growing cube and we will use the cube-based shape factor (eq 5) in order to obtain the relationship for our particular system (cf. Figure 3). In this case, the step advance velocity is defined as

(16)

vsa = (L + ΔL)/tlay

where K = Keq /(Keq + 1)

(18)

with ΔL = 2σs

(17)

In eq 16, Ψ value ranges from 0 for a complete inhibition of crystal growth to 1 for a negligible growth inhibition. Figure 2 summarizes the main mechanisms involved during the crystallization. A double film is assumed to exist between the surface of the crystal and the bulk solution. Diffusion occurs in the film adjacent to the bulk, and integration occurs in a narrower film located between the diffusion film and the crystal surface. As these mechanisms occur simultaneously, the slowest step will dictate the kinetics of the process. The equilibrium between the two main conformations occurs throughout the solution (in the bulk, in the diffusion film, in the interface diffusion/integration and in the integration film). Integration is driven by the ISS while diffusion is driven by the difference in

(19)

where tlay is the time needed for a step to create an additional layer per face and σs is the thickness of a step. For the case of a cubic shape, one can write the following equations related to the addition of consecutive layers of solute molecules on the crystal surface of class i and the durations to add a complete layer: ⎧ ΔLi(t Lay _1)/t Lay _1 = 2σsvsa /Li(t Lay _1) ⎪ ⎪ ΔL (t = 2σsvsa /Li(tlay _2) )/t ⎨ i lay _2 lay _2 ⎪············ ⎪ ⎩ ΔLi(tlay _H)/tlay _H = 2σsvsa /Li(tlay _H) 5183

(20)



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The incremental increase in size of crystals of class i ΔLi(tlay _h)| i = 1 → n can be approximated to be of the order of h=1→H

twice a molecular diameter and therefore can be considered infinitesimal. On the other hand, for crystallization to be achieved in reasonable time scale (few hours), the time needed to complete an additional layer (th|h=1→H) has to be extremely short for most of the crystallization. Therefore, the preceding equations can be approximated to give the expression of the temporal variation of the characteristic length, i.e., the linear growth rate. Substitution of eqs 16 in the series of eqs 20 yields the following expression for the crystal growth rate:

Gi =

dLi Φ MSS = dt Li

(21)

where Φ is an intrinsic solute integration coefficient defined as Φ = αK Ψ e − E

+

/(RT)

(22)

where α = 2σsν

VmLm 2 λA(hkl)

(23)

Discretization of eq 21 allows estimating crystal size at time tH from previous size estimations (with the size at initial time being the size of seeds of class i): H

Li = [(Li0)2 + 2Φ ∑ MSS(th)Δt ]1/2 h=1

(24)

31

In another study, crystal growth rate and supersaturation were determined experimentally for the crystallization of a Bristol-Myers Squibb proprietary substance for which data showed that the approach of the RC is applicable. Crystallization conditions were optimized so that the operation is growth dominated, and the data obtained was compared to model predictions to check its validity. The solute integration coefficient Φ is determined experimentally. Once Φ is identified, and if it is confirmed to be independent of crystallization conditions within the range investigated, it can be used in solving eqs 10, 12, 21 and 24 numerically to determine concentration, MSS, crystal growth rate and crystal size respectively. A script was written in Matlab to perform the resolution, and the corresponding resolution flow diagram is shown in Appendix. 2.2. Estimation of Crystal Growth Inhibition by the WC. For two isothermal antisolvent crystallizations conducted at different temperatures T1 and T2, eq 22 can be written as ⎧ −E + /(RT1) ⎪ Φ1 = αK (T1) Ψ(T1) e ⎨ ⎪ Φ = αK (T ) Ψ(T ) e−E+ /(RT2) ⎩ 2 2 2

Figure 4. Simulation of RSS (a) and ISS (b) for different antisolvent addition rates.

(25)

Combination of the equations above yields the ratio between ̀ the WC inhibition factor at different temperatures: Figure 5. Estimation of maximum antisolvent addition rate tolerable to minimize secondary nucleation.

⎡ ⎛ Φ(T )/Φ(T ) ⎞ Ψ E+ ⎛ 1 1 ⎞⎤ 1 2 Θ = 1 = exp⎢Ln⎜ ⎜ − ⎟⎥ ⎟+ ⎢⎣ ⎝ K (T1)/K (T2) ⎠ R ⎝ T1 T2 ⎠⎥⎦ Ψ2 (26)

via supersaturation and crystal growth rate measurements.31 In addition the variation of K with temperature can also be estimated experimentally through solubility measurements and NMR analysis.2 That data in combination with the knowledge of the solute integration activation energy can allow estimating Θ. Crystal growth inhibition by the WC is likely to occur if Θ is found to

For systems where the WC is the low energy conformer, the population of the WC increases with decreasing temperature. Therefore if the WC actually inhibits crystal growth, Θ should decrease with temperature (i.e., if T1 > T2, then Θ > 1). The variation of Φ with temperature can be determined experimentally 5184



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Figure 6. Parameter estimation and model resolution flowcharts. Blue section: Common to both scripts. Green section: Specific to model resolution script. Red section: Specific to parameter estimation script.

decrease with temperature. In another study,31 estimation of Θ is provided to assess the validity of the assumption of crystal growth inhibition by the wrong conformer.

The initial increase in RSS can be explained by a higher rate of generation of RSS by antisolvent addition versus the rate of RSS depletion by crystal growth. As crystals grow, their surface growths allow for an increasing area for solute integration. As a result, the total crystal surface becomes sufficient to consume the RSS created. The decreasing part of the RSS curves corresponds to the period where total crystal surface leads to a RSS depletion higher than the RSS creation. Figure 5 shows that the model predicts an exponential increase of the RSS maximum with the antisolvent addition rate. This behavior is expected because at high antisolvent addition rates, during the initial period of antisolvent addition, RSS creation rate becomes exceedingly higher than supersaturation consumption rate by crystal growth. As a result the maximal supersaturation reached during crystallization increases with antisolvent addition rate. High supersaturations are known to increase the risk of causing primary heterogeneous nucleation and increasing secondary nucleation, both of which can be detrimental for product quality. For our system, contact secondary nucleation was negligible throughout the operation. Based on FBRM measurements, secondary nucleation was observed if the RSS exceeds 60%. This nucleation event maybe due to surface secondary nucleation since it is an activated mechanism. According to Figure 5, maintaining the antisolvent addition rate below 0.094 mL/(g·mn) will keep the RSS below 60% and hence should prevent the occurrence of significant secondary nucleation. This computation can be useful in selecting ranges for antisolvent addition rate that allow minimizing secondary nucleation hence decreasing number of experiments to be conducted, which can have a great return in terms of resource savings and shortening of crystallization development and optimization efforts.

3. SIMULATION OF SUPERSATURATION DURING ANTISOLVENT ADDITION In situ and in-line measurement of ISS are impossible due to the lack of conformer-specific probes (Derdour et al). In addition, conducting crystallization at temperatures higher than the rotamers’ peak coalescence makes off-line NMR analysis an unreliable tool to evaluate the concentration of the RC (and hence ISS). To alleviate this problem, we choose to simulate the relative supersaturation (RSS) along with ISS because experimental values of RSS can be obtained. The RSS is defined as RSS = 100 MSS/C*

(27)

Different linear antisolvent additions were simulated using the VSAV approach for seeded crystallization presented in the previous section. For the purpose of simulation, solute integration coefficient identified for a constant step velocity approach was employed.15 The seed loading (SL) was kept constant at 1% of the initial dissolved solute, and the solvent and antisolvent levels at seeding were kept at 5.7 l/(kgdrug) and 2.1 l/(kgdrug) (liter of solvent per unit mass of initially dissolved drug) respectively. Computation of RSS and ISS for different antisolvent addition rates after seeding was performed. Figure 4 shows the computation of the RSS for different antisolvent addition rates during the antisolvent addition period. As shown on the curves, the VSAV approach predicts a maximum in RSS at some point during the linear antisolvent additions. 5185



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4. CONCLUSION A mechanistic model based on the step advance theory is presented for slow growing crystals of substances that have high energy barriers of transition between conformers. The intrinsic supersaturation (ISS), defined as the concentration of the conformer that crystallizes minus the true solubility of the solid (intrinsic solubility), is assumed to be the driving force for crystallization. The model presented is based on Kossel’s theory which considers a variable step advance velocity (VSAV). The VSAV model presented in this paper (1) leads to the identification of the surface integration step as the limiting mechanism during crystallization and (2) indicates that when the approach of the right conformer applies, crystal growth can decrease with crystal size. Lastly, simulation of different linear antisolvent addition rates using the VSAV approach predicted an exponential increase of maximum supersaturation with antisolvent addition rate. This allows us to determine a suitable range of antisolvent addition rate within which secondary nucleation is minimized.

C*RC intrinsic solubility (mol/L) concentration of all conformers at interface diffusion film/integration film (mol/L) Cid E+ energy of activation of solute integration (J/mol) G crystal linear growth rate (m/s) H highest index of time for computation of crystal size i index of class of sizes ISS intrinsic supersaturation (mol/L) Keq constant of equilibrium between conformers kv cube based shape factor L crystal characteristic length (size) (m) Li0 initial crystal size (m) Lm molecular size (m) m0 mass of solute, all conformers included, dissolved before crystallization (kg) m mass (kg) MWslt molar mass of solute (kg/mol) MSS measurable supersaturation (mol/L) Ni number of particles of class i n number of classes of particles P highest coefficient index/polynomial order for solubility correlation R universal gas constant (J/(K·mol)) SL seed loading (SL = 100mseeds/m0) (%) RSS Relative supersaturation (%) T temperature (K) tlay time need to complete an additional solute layer per crystal face (s) tmax total duration of antisolvent addition (s) V volume (m3) Vm molecular volume (m3) vsa step advance velocity (m/s) xas mass fraction of antisolvent in solvent system

■

APPENDIX The solvent composition is expressed in mass fraction of the antisolvent, which is expressed xas =

(Vas0 + m0 AR t )ρas (Vas0 + m0 AR t )ρas + Vsρs

(A1)

The principle of parameter estimation of Φ is the following: An initial guess for the range of Φ is set. Starting with the minimal expected value of Φ, temporal variation of MSS is estimated by time discretization. In each time increment, MSS is estimated by minimization of the error between the computed and experimental values. The calculation is executed by iteration starting from the preceding value of MSS. The error calculation is performed for the entire duration of the crystallization. Φ is then increased incrementally, and the error estimation described above is repeated for each increment of Φ. The parameter estimation is halted once the Φ increment reaches its expected maximal value. The value of Φ that provides the minimal total error between the computation and experimental data is set as the estimated solute integration coefficient. The parameter estimation and model resolution are based on the flowchart shown in Figure 6.

■

Greek Letters

ρ Φ Δt ΔL λA(hkl) σs

density (m3/kg) intrinsic solute integration coefficient (m2/(mol/L)·s) increment of time for computation (s) increase of crystal size after completion of one solute layer per crystal face (m) density of kinks of hypothetical face, A(hkl) step thickness (m)

Subscripts

0 as cr C exp h i MSS mod p

at the start of antisolvent addition related to antisolvent related to crystal related to concentration related to experiment index of time for computation of crystal size index of class of particles related to MSS related to model coefficient index/polynomial order for solubility correlation RC related to the right conformer s related to solvent seeds related to seeds sol related to solution WC related to the wrong conformer

AUTHOR INFORMATION

Corresponding Author

*E-mail: lotfi[email protected]. Tel: 007-732-227-6702. Fax: 007-732-227-3002. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The Senior Leadership Team at Bristol-Myers Squibb Co. is kindly acknowledged for providing means and incentives that made this study possible.

■

NOMENCLATURE ap coefficient in polynomial expression of solubility versus time AR antisolvent addition rate reported to unit mass of dissolved solute before crystallization (m3/(kg·s) or mL/(g·mn)) C concentration (mol/L) C* solubility (mol/L)

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REFERENCES

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