Process Paths of Kinetically Controlled Crystallization: Enantiomers

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Ind. Eng. Chem. Res. 2003, 42, 2230-2244

Process Paths of Kinetically Controlled Crystallization: Enantiomers and Polymorphs Joseph W. Schroer† Department of Chemical Engineering, University of Massachusetts, Amherst, Massachusetts 01003

Ka M. Ng* Department of Chemical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong

Reaction and crystallization kinetics play an important role in the recovery of enantiomers and polymorphs by crystallization. An approach that couples equilibrium phase behavior and kinetics is proposed for the design of such crystallization processes. A generic model that accounts for reaction, as well as crystal nucleation, growth, and dissolution, kinetics is developed. It allows the process paths to be tracked on phase diagrams and shows the effect of operating policies, design parameters, and kinetic constants on product purity and process productivity. Examples are provided to demonstrate this approach. Introduction

Model Development

Crystallization-based separations are an important problem for the chemical processing industries. With the current shift to high-value-added products, most of which are sold in solid form,1 the need for crystallization separations is bound to increase. Recently, systematic methods have been developed for the design of crystallization processes for molecular and ionic mixtures.2,3 Of particular interest are enantiomers and polymorphs, which are frequently encountered in the manufacture of pharmaceuticals. By using solid-liquid equilibrium phase diagrams, flowsheet alternatives can be identified for the resolution of racemic mixtures.4 The nature of phase diagrams involving polymorphs has also been clarified.5 However, crystallization is a rate-dependent process, and various transport phenomena contribute in determining the process outcome. Matsuoka and Garside6 highlighted some of the important heat- and masstransfer effects on crystallization. Kelkar and Ng7 considered reactive crystallization influenced by kinetics and mass-transfer effects. The recovery of enantiomers and polymorphs in chiral and polymorph crystallization, respectively, is known to depend on rate processes.8 Clearly, the development of operating policies and strategies for kinetically controlled crystallization of these compounds is highly desirable. To this end, a generic model is developed that accounts for chemical reaction, crystal growth, nucleation, and dissolution. It can be used for identifying the controlling mechanisms and developing operating strategies to yield the desired product with a specified purity. This approach is illustrated with select examples in racemate resolution and polymorph crystallization. Specifically, the asymmetric transformation of enantiomers and the solution-mediated transformation of polymorphs are considered.

We consider a mixture of two solutes, R and S, and two miscible solvents, W and E. Solute R can crystallize as only one polymorph, whereas solute S can crystallize into two different polymorphic forms, S1 and S2. Chemical reactions between the solutes occur only in the liquid phase. The balances for a well-mixed liquid phase are

* To whom correspondence should be addressed. Tel.: 852 2358 7238. Fax: 852 2358 0054. E-mail: [email protected]. † Current address: CWB Technology, 20311 Valley Blvd., Suite C, Walnut, CA 91789.

dW ) FFxFWF - FxPWP - vW dt

(1)

dE ) FFxFEF - FxPEP - vE dt

(2)

The first and second terms on the RHS of eqs 1 and 2 represent the feed and product removal rates, respectively. The terms vW (mol/s) and vE (mol/s) represent the evaporation rates (or other selective solvent removal) of the two solvents. We consider both a constant vapor removal rate and the case where the vapor rate is proportional to the amount of liquid in the crystallizer. A general expression for the overall solvent removal rate is

v ) v1 + Hv2

(3)

Here, H is the total molar liquid holdup of the crystallizer. If a constant relative volatility, REW, between E and W is assumed, the expressions for selective solvent removal are as follows

REWxE vE ) [v1 + (E + W + Rf + Sf)v2] xW + REWxE

(4)

xW vW ) [v1 + (E + W + Rf + Sf)v2] (5) xW + REWxE where Rf and Sf are the numbers of moles of R and S, respectively, in the liquid mixture. The balances on the solutes are as follows

10.1021/ie020722u CCC: $25.00 © 2003 American Chemical Society Published on Web 04/16/2003

Ind. Eng. Chem. Res., Vol. 42, No. 10, 2003 2231

dRf dt

r

∑ j)1

) FFxFRF - FPxPRP +

υRjΦjV + FRAR(DR/2) -

FRAR(GR/2) - FRkv(L0G,R)3VB0n,R (6) dSf

Φ ) kr(CR - CS/Keq)

r

FFxFSF

∑

FPxPSP

+ υSjΦjV + FS1AS1(DS1/2) ) dt j)1 0 0 )3VBn,S1 + FS2AS2(DS2/2) FS1AS1(GS1/2) - FS1kv(LG,S1 0 0 )3VBn,S2 (7) FS2AS2(GS2/2) - FS2kv(LG,S2

The first two terms on the RHS of eqs 6 and 7 represent the liquid-phase feed to and withdrawal from the crystallizer, respectively. The third term represents the change in number of moles of R or S due to r chemical reactions. The fourth, fifth, and sixth terms represent crystal dissolution, growth, and nucleation, respectively. The last three terms in eq 7 represent the crystal dissolution, growth, and nucleation of a second polymorph of S, S2. These equations can be generalized for any number of solutes, solvents, and polymorphs, if necessary. It is assumed that the crystals are geometrically similar. The characteristic crystal length, L, is chosen such that the relationship between the area and volume shape factors is kA ) 6kv. The equation for solid phase k is a population balance on the crystals. The population balance includes crystal growth, dissolution, and nucleation; input feed; and product slurry removal. Agglomeration and breakage are not considered.

∂(nkV) ∂(nkGkV) ∂(nkDkV) + - Bn,kV ∂t ∂L ∂L nF,kF + nkP ) 0 (8) Here, nk(L,t) dL is the number of particles of solid k of length L + dL per unit volume of solids-free liquid. To simplify the model analysis, we will use the moment method by transforming the population balance equation into a set of ordinary differential equations consisting of moments

∫0∞LjnV dL

mj )

(9)

For solid k, eq 8 becomes

dmj P 0 0 ) j(Gk - Dk)mj-1 - Bn,k (LG,k )3V + FmFj - mj dt V (10) where mFj is defined as

mFj )

∫0∞LjnF,kF dL

1 F

of parameter regression for a particular process, provided that the functional form is adequate. For racemization of the R and S enantiomers, we have a reversible first-order reaction

(11)

In our analysis, we use moments zero through three to calculate the number, the total length, the total area, and the mass of the crystals. The following rate laws are used with the model. The rate laws are most appropriate for the examples given in this article. Thus, although they are commonly used, they are not the most general rate laws possible. The modeling of other specific cases might warrant the use of other functional forms or correlations among additional variables. However, the simplicity of a model with fewer unknown constants is advantageous in cases

(12)

The rate constant kr has units of s-1, and the equilibrium constant Keq has been shown to be unity for a number of enantiomers with a single chiral center.9 For the rate laws of crystal growth, nucleation, and dissolution, we use conventional power-law expressions

Gk ) kG,kσkg

(13)

0 ) kn,kσkn Bn,k

(14)

The supersaturation, σ, is defined as sat σk ) (xi - xsat k )/xk

(15)

where xi is the mole fraction of the crystallizing component in the liquid phase and xsat k is the mole fraction of the crystallizing component at saturation of solid k. The growth order, g, can vary depending on the crystal growth mechanism, whether surface integration, 2dimensional nucleation, or mass-transfer-controlled growth is taking place.10 The rate law for dissolution is defined as

Dk ) kD,kuk

(16)

with the undersaturation defined by sat uk ) (xsat k - xi)/xk

(17)

Because of the variety of phenomena captured by this model, there are several characteristic time scales: reaction, (kr)-1; crystal growth, (kG,k A ˆ k)-1; nucleation, 0 3 -1 -1 [kv(Ln,k) kn.k] ; dissolution, (kD,k A ˆ k) ; feed time, V/F; and product removal time, V/P. Ratios of these time scales form various dimensionless numbers that can be used to determine the dominant mechanism. Range of Parameter Values. Data for growth and nucleation rate parameters for organic systems are scarce.11 We summarize below such information on a number of organic solutes in the literature. Mahajan et al.12 measured the growth and nucleation kinetics for crystallization of L-asparagine from 1- and 2-propanol in a batch crystallizer. Zipp and Rodrı´guez-Hornedo13 investigated the effects of pH on the crystallization of a monoprotic organic acid from a buffered aqueous solution. Saska and Garandet14 measured the crystallization kinetics of sucrose, and Johns et al.15 measured the crystallization kinetics of fructose from mixed solvents of various ethanol/water ratios. Yokota et al.16 measured the growth rate of (S)-carboxymethyl-D-cysteine and investigated the causes of impurity in its optical resolution. Kitamura et al.17 measured the growth rate for two different polymorphs of L-histidine in mixed ethanol/ water solvents of different proportions. They found that high concentrations of ethanol could suppress the crystallization of one of the polymorphs and showed the difference in growth rates of the two polymorphs at different solvent compositions. Kitamura and Sumi18 measured the growth rate of glutamic acid and investigated the effect of L-phenylalanine on its morphological

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Ind. Eng. Chem. Res., Vol. 42, No. 10, 2003

Table 1. Examples of Asymmetric Transformation from the Literature molecule L-768,673 N-benzyl-3-(S)-(+)-(4-fluorophenyl)-1,4-oxazin-2-one (-)-galanthamine naproxen N-phthalyl-(S)-3-amino2-phenyl propanoic acid (R)-4-chlorophenylalanine D-p-hydroxyphenylglycine (R)-phenylglycine N-acetyl-2-(4-hydroxyphenyl) glycine N-alkyl-2-phenylglycines L-histidine L-proline N-acetyl-D,L-leucine oxazaborolidinones

use

ref

antiarrythmia drug candidate drug intermediate

52 53

Alzheimer’s drug analgesic β-amino acid

54, 55 56 57

drug candidate antibiotics antibiotics anitbiotics

58 59 29 32

antibiotics chiral intermediate chiral intermediate

33 31 30 60 61

change. From the data of these studies, the range of the crystal growth rate parameter kG is from 1 × 10-9 to 1 × 10-6, with the most typical value being between 1 × 10-8 and 1 × 10-7. Growth orders range from 1 to 4. Calculation Method. The model equations were integrated using Matlab (Mathworks, Inc., Natick, MA), a software package for numerical computation and analysis. Consistency checks were made by integrating the equations using more than one integration algorithm given in the ODE solver package and by making comparisons after adjusting integrator parameters. Chiral Crystallization and Asymmetric Transformation Asymmetric transformation is a means for separating enantiomers from their racemic mixtures.19 The racemization reaction proceeds simultaneously with crystallization to improve the yield and purity of the product. Asymmetric transformation has been used to obtain pure single enantiomers starting from racemic mixtures, as well as to convert one enantiomer to the other. Selected examples are listed in Table 1. Most are for current drugs, drug candidates, amino acids, or amino acid derivatives. Because racemization and crystallization take place simultaneously, the relative rates between crystal growth and racemization are likely to be important. Ebbers et al.20 pointed out that racemization and resolution usually require conditions (i.e., temperature, concentration, pH) that are often mutually incompatible. The effects of seeding and the operating policy should also be taken into account. Many reviews of nucleation and growth in crystallization processes21-24 and studies on the effects of optimal seeding25 are available. In particular, Chung et al.25 studied using the particle size distribution of the seed to accomplish specified objectives, either maximizing the weight mean size, minimizing the coefficient of variation, or minimizing the ratio of nucleated crystal to seed crystal mass. Racemization. Much of the information known about racemization comes from studies of the racemization of amino acids. In fact, a recent review notes that almost half of the papers on racemization deal with amino acids or their derivatives, which are intermediates in many modern drugs.20,26 A wide range of amino acids are racemized by heating to 80-100 °C in the presence of aldehydes in acetic acid.27 There is the

possibility of side reactions that form undesired byproducts. An example is the racemization of alanine at 275 °C, where two alanine molecules can condense to form alanine anhydride.28 Investigations of the reaction kinetics of free amino acids in buffered aqueous solutions indicate that reversible first-order rate laws are valid; however, some peptides and proteins have more complex kinetics.9 Rate constants for racemization have been measured for several molecules,28-33 with additional values cited in the review by Bada.9 Typical values of the rate constant kr range from 1 × 10-4 to 1 × 10-1 s-1. Rates can vary greatly with temperature, as well as with pH. Regimes of racemization are acid-catalyzed racemization, basecatalyzed racemization, and racemization of the zwitterion. Racemization kinetics accounting for changes in pH can be accurately modeled using rate constants for hydronium- and hydroxide-catalyzed reactions for each ionic species (six constants total for a neutral amino acid).34 However, because pH swing crystallization is not emphasized in this work, a first-order rate law is sufficient (eq 12). Although the equilibrium constant is usually unity, some examples of molecules with uneven isomeric distributions are isoleucine and 4-hydroxyproline, which have Keq values of 1.3 and 0.8, respectively, at a pH of 7.6 (eq 12). Model for Asymmetric Transformation. From the generic model, we can simplify the equations to the form needed in modeling asymmetric transformation without dissolution. We consider only one solvent, W, and only one polymorphic form of each isomer. The balance equation on the liquid is then

vW 1 dmfluid F P ) (FF/F) - mfluid dt V V mfluid

()

ˆ RkG,RσRg - (FR/F)kv(L0n,R)3kn,RσRn 0.5(FR/F)A ˆ SkG,SσSg - (FS/F)kv(L0n,S)3kn,SσSn (18) 0.5(FS/F)A where mfluid is the number of moles of fluid in the crystallizer. The two enantiomers are denoted as R and S. The first two terms on the RHS are the feed and product removal rates, respectively, which are zero for a batch crystallizer. The third term is the rate of solvent removal by vaporization or by some other selective removal process such as membrane filtration. The next two terms are the rates of crystal growth and nucleation, respectively, for enantiomer R, and the final terms are the corresponding rates for enantiomer S. Because it is convenient to work with mole fractions, eqs 6, 7, and 1 from the generic model are rewritten as follows

dxR F P ) -kr(xR - xS/Keq) + xFR(FF/F) - xR dt V V 0.5(FR/F)A ˆ RkG,RσRg - (FR/F)kv(L0n,R)3kn,RσRn xR dmfluid (19) mfluid dt

()

dxS F P - xS ) kr(xR - xS/Keq) + xFS(FF/F) dt V V ˆ SkG,SσSg - (FS/F)kv(L0n,S)3kn,SσSn 0.5(FS/F)A xS dmfluid (20) mfluid dt

()


xW dmfluid dxW vW F P - xW ) xFW(FF/F) (21) dt V V mfluid mfluid dt

()

The solid-side equations are

dNR ) (mfluid/F)kn,RσRn dt

(22)

dLR ) (mfluid/F)kn,RσRn(L0n,R) + (kG,RσRg)NR (23) dt dAR ) kA[(mfluid/F)kn,RσRn(L0n,R)2 + 2(kG,RσRg)LR] (24) dt

Table 2. Input Data for Asymmetric Transformation Example Model Parameters liquid density (mol/m3) Fliq 3 solid density (mol/m ) Fs area shape factor kA volume shape factor kv size of nuclei (m) L0n,R, L0n,S reaction equilibrium constant Keq growth rate constant (m/s) kG nucleation rate constant kn [no./(s m3)] nucleation exponent n growth exponent g solubility of S xsat S solubility of R xsat R

17 000 17 000 π π/6 1 × 10-8 1 1 × 10-8 1 × 108 1 1 0.1 0.1

dMR ) FRkv[(mfluid/F)kn,RσRn(L0n,R)3 + 3(kG,RσRg)AR/kA] dt (25)

Initial Fluid Composition xS xR xW

dNS ) (mfluid/F)kn,SσSn dt

(26)

Seed Characteristics amount of solid R seed none shape of solid S seed uniform spherical particles Seed size (m) LS 1 × 10-5

dLS ) (mfluid/F)kn,SσSn(L0n,S) + (kG,SσSg)NS dt

(27)

0.25 0.25 0.5

dAS ) kA[(mfluid/F)kn,SσSn(L0n,S)2 + 2(kG,SσSg)LS] (28) dt dMS ) FSkv[(mfluid/F)kn,SσSn(L0n,S)3 + 3(kG,SσSg)AS/kA] dt (29) Example 1: Resolution of a Racemic Mixture by Asymmetric Transformation. In this example, we model asymmetric transformation processes for the resolution of a racemic mixture. The initial solute is a 50:50 mixture of the R and S enantiomers. This process is different from an asymmetric synthesis process where the two enantiomers are formed from some other intermediates by chemical or biological reaction. An asymmetric synthesis does not necessarily involve a racemization reaction for interconversion of the enantiomers and does not necessarily use crystallization as a means of producing a resolved or partially resolved enantiomeric product. We illustrate the effects of crystal nucleation and growth, seeding, and the racemization reaction independently by starting from a simple base case and building up to the full process models. The model parameters for the base case scenario are given in Table 2. The nucleation and crystal growth rate parameters in the rate laws are the same for both enantiomers. The inputs for the process are typical of the racemization kinetics and crystallization kinetics described earlier. Base Case (BC). The calculations in this example were for a batch crystallizer. Process paths for different cases and operating conditions are presented in Figure 1. This phase diagram is a ternary isothermal diagram. The solvent is at the vertex labeled by W, and the two enantiomers are at the vertices labeled R and S. The single-phase region is the rectangular region closest to the W vertex. The process path lies within the multiphase region where both enantiomers R and S are supersaturated. Because the two components are enantiomers, they exhibit similar solubilities in the solvent. Thus, the ratio of the two enantiomers at the double saturation point is 1:1, as it is in the racemate. The base

Figure 1. Isobaric isothermal ternary phase diagram containing process paths for an asymmetric transformation example. The two enantiomers are labeled R and S, and the solvent is labeled W. Process paths begin at point b and terminate at the double saturation point, ds.

case examines an unseeded crystallization without the effect of the racemization reaction. Both enantiomers nucleate and crystallize. The process path begins at point b and moves directly toward the double saturation point, ds, which is the equilibrium point. The process path for this case is labeled BC in Figure 1. Seeded Crystallization (SC). The second case considered is that of an unreactive, seeded crystallization. Seeds of the S crystal are added at the start of the simulation such that the amount of seed is 2.5% of the number of moles of S present in the fluid. The seeding enhances the crystallization rate of S, and initially, the process path moves away from the S vertex, with the product at short crystallization times significantly enriched in the S isomer. However, as R nucleates and

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Figure 2. Crystal product purity versus time. Figure 4. Average crystal sizes versus time.

Figure 3. Cumulative amounts of each solid enantiomer versus time.

begins to crystallize, the path turns towards the double saturation point. This corresponds to a drop in purity of the solid product, as can be seen in Figure 2, which displays a plot of the fraction of the crystallized solid consisting of S crystals, zS. The left axis in Figure 2 is the mole fraction scale, and the right axis is the purity expressed in terms of the percent enantiomeric excess. Toyokura et al.35 found a similar result in experimental studies on the selective seeding crystallization of (S)carboxymethyl-L-cysteine (L-SCMC) from a solution containing its racemate. After a period of desupersaturation of the L-SCMC enantiomer, the concentration of D-SCMC would decrease as a result of its nucleation and crystal growth. A decrease in the purity of racemic mixtures with resolution time has also been discussed in detail by Matsuoka.36 The total amount of crystallized solid and the amounts of solids R and S versus time are given in Figure 3. In comparison to the unseeded case (BC), the amount of R enantiomer is less than the amount of S enantiomer in the seeded case (SC). Trajectories of the first moment of the crystal size distribution representing the average crystal size are given in Figure 4. The total surface areas of the different types of crystals are given in Figure 5. In the seeded unreactive case, the product purity obtained after a long period of time is determined by phase equilibrium limitations. Some enrichment in the S enantiomer can be obtained if the crystallization is stopped before equilibrium is reached. The product purity can be controlled by adjusting the time at which the crystallization is stopped. Seeded Crystallization with Racemization (SR). The third case considers the racemization reaction in

conjunction with seeding of the S enantiomer. Results are given for two values for the racemization constant, 1 × 10-4 s-1 (SR1) and 1 × 10-3 s-1 (SR2). As can be seen in Figure 1, the effect on the process path is to hold the path closer to a 50:50 mixture of enantiomers. In addition, the product purity at the end of the batch is significantly higher than that obtained in the other cases (Figure 2). Although some of the undesired R enantiomer crystallizes in this example, the ratio of the amounts of the two crystallizing solids has dramatically changed. This is because the racemization reaction, which consumes R, acts to reduce some of the supersaturation of this component. The particle size distribution is also affected. In the unseeded case (BC), the crystal sizes are identical for the two enantiomers (Figure 4). In the seeded case without reaction (SC), the R crystals are 48% larger in size than the S crystals, whereas in case SR1, the S crystals are 37% smaller than the R crystals. This is because, in seeding, a large number of small crystals were added to the mother liquor. The number of seed crystals in this case is greater than the number of nuclei formed by nucleation. However, this trend can be reversed with fast racemization from R to S. For case SR2, the S crystals are 36% larger than the R crystals. The relative effects of racemization and seeding are illustrated in Figure 6, which shows the product purity at the end of the batch. At this time, the supersaturation is depleted, and the process path is close to the equilibrium double saturation point on the phase diagram. The results are corrected for the amount of seed used. That is

z′S )

MfS - M0S MfR + MfS - M0S

(30)

For this system, a racemization rate constant on the order of 1 × 10-4 s-1 is needed before an appreciable effect of the chemical reaction on the product purity is produced. A racemization rate greater than 1 × 10-3 s-1 is needed to make a product with greater than 98% purity. The product purity increases with the amount of seeding used but encounters a plateau, and amounts of seeding in excess of 4 mol % of the amount of S in the mother liquor provide little improvement in the final product purity. It should be mentioned that these results are for a uniform seed size. Although the general trends will be the same, the results in Figure 6 are


Figure 5. Cumulative crystal areas versus time.

Figure 7. Process paths with decreasing evaporation rate policy. Higher solvent removal rates give larger loops in the process path.

Figure 6. Product purity at the end of the batch versus amount of seeding and racemization reaction rate.

affected by the particle size distribution of the seed crystals used. Effect of Reducing the Initial Supersaturation (LS). Reducing the amount of supersaturation is one way to improve the purity. In case SR1 discussed above, the purity is only 66.6%. By reducing the amount of supersaturation at the start of the batch from 0.15 to 0.01, the purity can be increased to 98.5%. However, this is not done without expense. Assuming that the batch size contains the same amount of solute, the fractional product recovery is much less. In terms of the productivity ratio, the result is 0.50 for case SR1 and 0.11 for this case. (The productivity ratio Rp is defined as the ratio of the amount of S recovered to the sum of the amounts of S and R in the feed.) In addition, the crystallization takes about 5 times as long to complete. Semibatch Evaporative Crystallization (BE). Now consider asymmetric transformation in a semibatch evaporative crystallizer. The vaporization rate is set in a decreasing fashion such that it is proportional to the total amount of liquor present in the crystallizer. The model parameter v1 (in eq 5) is set to zero, so that the parameter v2 controls the solvent removal rate. This solvent removal rate has the property that the liquid is completely evaporated only as the batch time becomes infinite, although the dynamics approaching this state are very slow and all but a very small percentage of the solute can be crystallized within realistic time scales. The input parameters are the same as in case SR2, except that the initial liquor is a saturated solution of

Figure 8. Cumulative amounts of each solid enantiomer versus time for the evaporation rate of v2 ) 3 × 10-4 mol/(mol s).

the racemate. Figure 7 shows process paths for this example. The process path loops were calculated for several values of the evaporation rate parameter: v2 ) 2 × 10-5, 5 × 10-5, 1 × 10-4, 2 × 10-4, and 3 × 10-4 mol/(mol s). The higher the evaporation rate, the larger the loop. Each loop begins at the double saturation point for the two enantiomers. Initially, the solvent is removed, and supersaturation is generated. The process path moves away from the solvent vertex (W). Because of seeding, the S enantiomer begins to crystallize first, although the solution is supersaturated in both R and S. Crystallization of S causes the process path to turn away from the S vertex. As the supersaturation of R increases, the R enantiomer begins to nucleate. Crystallization of R causes the process path to turn again, this time away from the R vertex. The supersaturation generation rate slows as the evaporation rate falls with decreasing liquid holdup. As a result, the process path returns to the double saturation point as the crystallizer operates closer and closer to equilibrium conditions. Eventually, the solvent is nearly completely evaporated, and nearly all of the solute is crystallized. Figure 8 shows plots of the amount of each crystallized solid versus time. The numbered points in this graph correspond to the numbered points in Figure 7. Figure 9

2236


Figure 9. Product purity and productivity ratio versus time for v2 ) 3 × 10-4 mol/(mol s).

Figure 11. Process paths for batch asymmetric transformations with constant vapor removal rate policies.

Figure 10. Product purity versus productivity ratio for several initial solvent removal rates in a crystallizer with a decreasing vaporization rate.

shows plots of the product purity and the productivity ratio versus time for the evaporation parameter v2 ) 3 × 10-4 mol/(mol s). The productivity ratio and product purity are defined as

Rp )

MS M0S

+

M0R

+ R0f + S0f

(31)

and

zS )

MS M S + MR

(32)

respectively. As the batch processing time increases, the productivity increases while the product purity decreases. At the point of complete solvent evaporation (infinite time for the current vapor rate policy), the productivity ratio equals the product purity because of the mass balance. The effect of the initial evaporation rate on the production rate and product quality was also investigated. Figure 10 shows the product purity versus the productivity ratio for several solvent removal rates. This type of plot can be considered a feasibility plot for asymmetric transformation processes because the productivity and the product purity are relevant measures.

Figure 12. Product purity versus productivity ratio for several solvent removal rates at a seeding policy of 0.36%.

The goal, of course, is to achieve a productivity ratio higher than 0.5 beyond which, because of the racemization, an amount of S that exceeds the initial amount of S in the feed is recovered. In this example, improved purities are obtained with lower solvent removal rates for a given productivity ratio. At some point, the vapor rate is sufficiently low that it is no longer the limiting step in the crystallization process. At this rate, a further lowering of the solvent removal rate is of limited benefit. Process paths on the phase diagram for constant evaporation rate policies are given in Figure 11. The batch becomes completely dry at a finite time, and the process path does not return to the double saturation point. As can be seen in Figure 10, the overall system performance, neglecting the batch time, is relatively insensitive to the evaporation rate. The slope of the curves in Figure 10 can be changed by varying the amount of seeding used and the racemization kinetics. For example, Figures 12 and 13 show the feasibility plot and the process paths when less seeding is used. In these figures, the size of the seed is the same as in the above examples, but the seed amount is 0.36% of the S in the initial batch liquor. Application to Process Synthesis. We now use the feasibility plot to synthesize an asymmetric transforma-

Ind. Eng. Chem. Res., Vol. 42, No. 10, 2003 2237 Table 3. Some Examples of Industrial Compounds Exhibiting Polymorphism

Figure 13. Process paths with a seeding policy of 0.36%.

molecule

use

monosodium glutamate paclobutrazol stearic acid taltirelin L-histidine abecarnil 5-methyl-2-[(2-nitrophenyl)amino]-3-thiophenecarbonitrile progesterone ranitidinium chloride (unmentioned) (unmentioned)

food additive plant growth regulator cosmetics pharmaceutical nutrition supplement pharmaceutical pharmaceutical

41 49 47 50 17 44 62

ref

pharmaceutical pharmaceutical photographic couplers dyestuff

63 39 40 48, 49

improvement in performance by the line Vm-S1. By decreasing the vapor rate to 4 × 10-5 mol/(mol s) and increasing the amount of seeding to 7.1%, we obtain a productivity ratio of only 61%, but a purity of 99.6% (line Vi-S2). This result is within the design specification. Polymorph Crystallization and Solution-Mediated Transformation

Figure 14. Feasibility plot showing design specifications and effects of varying percent seeding and evaporation rate for a 5.6-h batch time.

tion process for a fixed batch time of 5.6 h for a system with a decreasing vapor removal rate. The system characteristics are the same as in case SR2 discussed above. The specifications for the design are that the product purity must be above 98% and the productivity ratio must be above 60%. The curve in Figure 14 shows the purity and productivity that can be achieved with increasing initial vaporization rate. The amount of seeding used is 0.36%. At point Vl, v2 ) 1 × 10-5 mol/(mol s), and at point Vh, v2 ) 2 × 10-4 mol/(mol s). Thus, the productivity increases with increasing initial evaporation rate, whereas the purity decreases. There is a maximum initial vaporization rate [v2 )1.5 × 10-4 mol/(mol s)] beyond which both the purity and productivity decline. It can be seen in Figure 14 that, to obtain a productivity ratio greater than 50%, the product purity must be less than 85%. We can increase the amount of seeding to achieve the process specifications indicated in the shaded region of the figure. An increase in the amount of seeding from 0.36% of the initial amount of S in the crystallizer to 7.1% gives a purity of 95.7%. Figure 14 shows the

Different polymorphs of the same molecule can have different physical properties, such as color, texture, transparency, crystal shape, and solubility.37-39 For this reason, polymorphism has many implications in product and process development. For example, polymorphism is an important issue for pharmaceutical products, where the regulatory demands for product characterization can include the polymorphic form. Polymorphism in photographic image couplers can diminish photographic performance.40 In the production of monosodium glutamate, crystallization of the R-form glutamic acid crystals and subsequent solvent-mediated transformation to β-form crystals are important process steps.41 Table 3 lists examples of commercially relevant compounds for which polymorphism is observed. When multiple polymorphs can be crystallized from the same solvent, normally, the polymorph with the lower solubility, and thus the greater thermodynamic stability, is produced. However, the approximately 5-20 kJ/mol difference in lattice energy between different polymorphs is small compared to the lattice energy of the crystal.39 The ability to observe the thermodynamically unstable polymorph has been attributed to kinetic effects, such as nucleation. An explanation for this could be that the driving forces of the chemical potential difference and the interfacial energy could be in opposition to each other.42 In this case, the polymorph with the higher nucleation rate could actually be the less thermodynamically stable one. Control of nucleation by seeding has been promoted by several authors as a method of obtaining the desired polymorph, especially if it is the thermodynamically less stable one.40,43-44 Solvent selection for the crystallization of polymorphs is often explored as a means of obtaining the desired polymorph or optimizing the rates of the process. For example, phenylbutazone can be produced in four polymorphic forms depending on the solvent.45 Spruijtenburg8 discussed an example in which a polymorph of a pharmaceutical active ingredient was initially thought to be crystallizable only from butanol, but after some effort, a process was developed to obtain the desired form by crystallization from ethanol. Solution-mediated transformation is a process used to convert one polymorphic form to another. In this

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Table 4. Process Considerations for Polymorphic Compounds • Complex solid-liquid equilibrium behavior is captured in the phase diagram that can be used to target operating temperatures and solvent composition. • Polymorphs can be characterized using a variety of experimental techniques, including single-crystal X-ray crystallography, powder X-ray diffraction, solid-state spectroscopy (NMR, IR, and Raman), microscopy, and thermal techniques.38,64 • Solvent composition might affect not only polymorph solubility but also crystallization kinetics.17 • Solvent in which the compound has limited solubility favors formation of the corresponding polymorph. • Temperature is important, particularly in enantiotropic systems, in determining the polymorphic form of the product. • To obtain the desired polymorph, seeding is recommended.44 • Cooling profile should be controlled to prevent nucleation of the undesired polymorph.40 • Polymorphs can undergo interconversion during subsequent processing steps, particularly if downstream processes operate at different temperatures than the crystallization processes.8 • Crystallization path should stay within the boundary of the saturation variety of the desired polymorph and avoid the saturation variety of undesired polymorphs as much as possible. [A saturation variety is the region of space (T, P, x) on the phase diagram where a particular phase forms.] • If the metastable polymorph has desirable solids processing properties, consider crystallizing the metastable polymorph and using solution-mediated transformation to convert the product to the stable polymorph at a later stage. • Solvent trapped in solids could mediate transformation to the stable polymorph during drying.

process, a solvent is used as a medium to dissolve the more soluble and less stable form and to recrystallize the more stable and less soluble polymorphic form. In addition to the above-mentioned solution-mediated transformation of glutamic acid, several other studies have been documented.46-50 In particular, Cardew and Davey46 provided an analysis of solution-mediated transformation processes that incorporated the effects of crystal dissolution, growth, relative supersaturation of the polymorphs, and seeding. Their analysis was repeated by Nass.40 Our generic model can be used to represent solutionmediated transformation as in the Cardew and Davey model. However, it is more general and can be used for other operating policies of polymorph crystallization that include drowning-out and evaporative crystallization. The model is expected to be useful in process data analysis and optimization. General process considerations for polymorphic compounds are summarized in Table 4. We consider only one solute in a mixed solvent with no chemical reactions. Selective solvent removal by evaporation from the crystallizer is permitted. Dissolution of polymorph solid S1 and nucleation and growth of polymorph solid S2 occur. Discussion of the equations derived from the generic models is omitted. Example 2: Solution-Mediated Transformation in a Single Solvent. This example considers a batch process for the dissolution of polymorph S1 and the nucleation and growth of polymorph S2. The input data are given in Table 5. We have a solution with an initial concentration of S at the solubility of solid S1. The initial charge of solid S1 is composed of particles of spherical shape and uniform size. Simulations were performed for both unseeded and seeded crystallization of solid S2. In the case of seeding with S2, the seed particles were also assumed to be of spherical shape and uniform size.

Table 5. Input Data for Solution Mediated Transformation Example Model Parameters liquid density (mol/m3) solid S1 density (mol/m3) solid S2 density (mol/m3) area shape factor volume shape factor size of nuclei (m) dissolution rate constant (m/s) growth to dissolution rate ratio nucleation rate constant [no./(s m3)] nucleation exponent growth exponent solubility of polymorph S1 solubility of polymorph S2

F FS1 FS2 kA kv L0n,S2 kD kG/kD kn n g xsat S1 xsat S2

Initial Fluid Composition mfluid xS xW Seed Characteristics amount of solid S1 (mol) MS1 S1 particle size (m) LS1 amount of solid S2 seed (mol) MS2 S2 particle size (m) LS2

17 000 17 000 17 000 π π/6 1 × 10-8 1 × 10-8 0.5, 1, 10 1 × 108 1 1 0.12 0.1 10 0.12 0.88 0.89 1 × 10-4 0.0089 1 × 10-5

Figure 15. Supersaturation versus time for unseeded solventmediated transformation.

Calculations were performed for several values of the relative rates of growth and dissolution, kG and kD, respectively. Figure 15 shows the supersaturation versus time profiles for the unseeded case. When the growth rate of S2 is much faster than the dissolution rate of S1, there is a rapid drop in supersaturation, followed by a slower approach to equilibrium as solid S2 crystallizes. In the opposite extreme where crystal growth of S2 is the rate-limiting step, the supersaturation is maintained at a high level until it is depleted by crystal growth of solid S2. Figure 16a and b shows profiles of the amount of each solid versus time for this example. Figures 17 and 18 show results in the case where the solid S2 is seeded by an amount of 1% of the initial charge of S1 solid. The seeding increases the crystal growth rate, and the supersaturation is depleted more rapidly than in the unseeded case. Example 3: Selective Polymorph Crystallization by Evaporation of a Mixed Solvent. Polymorph S2, the more soluble polymorph in this example, is the desired product form to be crystallized from a solvent of mixed composition. The polythermal solid-liquid phase diagram for the system of interest is given in Figure 19. Polymorph S1 is the stable polymorph and remains so with changes in temperature (monotropic


Figure 16. Number of moles of solid versus time for unseeded solvent-mediated transformation: (a) solid S1 and (b) solid S2. Figure 18. Number of moles of solid versus time for seeded solvent-mediated transformation.

Figure 17. Supersaturation versus time for solvent-mediated transformation with 1% seeding of solid S2.

behavior). Points E, W, S1, and S2 in Figure 19 are the pure-component melting points for E, W, and the two polymorphs S1 and S2, respectively. The melting surfaces for the polymorphs are shown as transparent to each other whereas the surfaces for E and W are shaded. The shading changes where the melting surfaces of E and W undercut the S1 melting surface to reach the S2 surface. Point s1e is the eutectic point between E and S1. Point s2e is the point representing the metastable eutectic between E and S2. Point s1w is the eutectic point between W and S1, and the point s2w is the point representing the metastable eutectic between W and S2. Similarly, points s1ew and s2ew are the ternary eutectic points. The dashed line shows an isothermal cut of this phase diagram at temperature T1. At this temperature, the solubility of polymorph S1 is shown by the line a-b. The solubility of the polymorph S2 at T1 is shown by the line c-d (dotted). The equations describing lines a-b and c-d giving the solubilities of the two polymorphs

Figure 19. Isobaric phase diagram for the system in example 3.

in the mixed solvent at T1 are

xsat S1 ) 0.29 - 0.09RE

(33)

xsat S2 ) 0.35 - 0.13RE

(34)

In the above equations, RE is the molar fraction of solvent that is component E. A summary of the input parameters is given in Table 6. The nucleation rate constant of polymorph S2, kn,S2, is 5 times the nucleation

2240


Table 6. Input Data for Evaporative Polymorph Crystallization Example Model Parameters liquid density (mol/m3) F solid S1 density (mol/m3) FS1 solid S2 density (mol/m3) FS2 area shape factor kA volume shape factor kv size of nuclei (m) L0n,S2 S1 dissolution rate constant (m/s) kD,S1 S2 dissolution rate constant (m/s) kD,S2 S1 growth to dissolution rate ratio kG,S1/kD,S1 S2 growth to dissolution rate ratio kG,S2/kD,S2 S1 nucleation rate constant kn,S1 [no./(s m3)] S2 nucleation rate constant kn,S2 [no./(s m3)] S1 and S2 nucleation exponent n S1 and S2 growth exponent g solvent evaporation rate (mol/s) v1 solvent relative volatility REW Initial Fluid Composition mfluid xS RE Seed Characteristics amount of solid S1 (mol) MS1 amount of solid S2 seed (mol) MS2 particle size (m) LS2

17 000 17 000 17 000 π π/6 1 × 10-8 1 × 10-7 1 × 10-7 0.5 0.5 1 × 108 5 × 108 1 1 1.563 × 10-3 2

Figure 20. Process path for the selective crystallization of S2: (O) unseeded, (0) 3% seeding.

10 0.218 0.8 0 0, 0.0654 1 × 10-5

rate constant of polymorph S1. This difference somewhat offsets the disadvantage of S1 having a higher supersaturation. The strategy for crystallization of S2 is to evaporate the solvent quickly to minimize the time that the process path lies on the S1 solubility surface but outside of the S2 solubility surface. The process paths for two cases, crystallization without seeding and crystallization with seeding of S2 crystals to the amount of 3% of the solute, are illustrated in the isothermal cut of the phase diagram in Figure 20. In the seeded case, the seed is added at the start (t ) 0). Initially, the solvent is 80% E (point 1 in Figure 20). The solution is a saturated solution with S1. The solvent mixture is evaporated isothermally at a constant rate. The evaporation rates of the two solvents are described by eqs 4 and 5. The relative volatility between the two solvents, REW, is 2. As the solvent evaporates, the concentration of E in the liquid is depleted. After 1.38 h, the evaporation is stopped (point 6). The amounts and average crystal sizes of the two solids are plotted as functions of time in Figures 21 and 22, respectively. The numbered points in these two figures correspond to the numbered points in the isothermal phase diagram. In comparison to the seeded case, the unseeded situation yields nearly an equal mixture of polymorphs. The average crystal size of S1 crystals is greater than the size of the S2 crystals because S1 is supersaturated for a longer period of time. The seeded operation gives more desirable results, with most of the crystals being S2. The key to obtaining S2 in this case is (1) selective seeding of the desired polymorph and (2) maximization of the time during which the process path is inside the supersaturation region for S2 and minimization of the time during which the process path is in the supersaturation region for S1 only. In addition, it is important to filter and dry the crystals as soon as possible to prevent solution-mediated transformation to the stable polymorph. If the objective were to obtain solid S1 as the product, insights from the phase behavior of this example are

Figure 21. Amount of solid polymorphs versus time for the selective crystallization of S2: (O) unseeded, (0) 3% seeding.

Figure 22. Average crystal size for the selective crystallization of S2: (O) unseeded, (0) 3% seeding.

helpful. One heuristic is to operate at low supersaturations so that the process path lies outside the saturation variety of S2. If operation is such that some S2 is crystallized, then allowing the crystals to equilibrate in the mother liquor, before filtering, for sufficient time for the solution-mediated transformation to the stable polymorph to occur is recommended. Processing in this manner might allow for a reduction in the batch operation time. Thus, although thermodynamics favors crystallization of polymorph S1, polymorph S2 can be obtained through a carefully planned kinetically controlled operation.


Figure 24. Process paths for the base case (BC) and for faster S2 nucleation (FN). Figure 23. Isobaric phase diagram for the system in example 4 showing saturation surfaces with a crossover temperature. The solubility surface of polymorph S1 is outlined by the solid line; S2, dotted line.

Example 4: Drowning-Out for Polymorph Crystallization. Crystallization of the desired solute can be achieved by the addition of an antisolvent. This approach is commonly used in batch crystallization of high-value-added products that are sensitive to temperature. Drowning-out crystallization is viewed as a difficult separation technique in that care must be taken to control the level of supersaturation.51 This example explores some of the kinetics and control issues involved in the drowning-out crystallization of polymorphs. The isobaric phase diagram for the system of interest is shown in Figure 23. The solubility surfaces for the two polymorphs are shown. The solubility behavior of solute S is such that it is soluble in solvent E but only sparingly soluble in solvent W. In this system, there is a crossover temperature Tc at which the relative stability of the polymorphs is reversed (enantiotropic behavior). At temperatures above Tc, polymorph S1 is the stable form, whereas below Tc, S2 is the stable solid. Line a-b (solid line) follows the solubility of the polymorphs at the crossover temperature. The objective is to obtain polymorph S2. Crystallization at a temperature below Tc, where S2 is the less soluble polymorph, is recommended (Table 4). Otherwise, a polymorph transformation to the undesired form might occur, as recounted by Spruijtenburg8 in an example of the drowning-out crystallization of a food colorant. We consider operation for which the isothermal solubility relationships for S1 and S2 in the solvent mixture are given by -4 xsat exp(7.9RE) S1 ) 1 × 10

(35)

-5 exp(7.85RE) xsat S2 ) 9 × 10

(36)

The model input parameters are given in Table 7. At the start of the batch, the mother liquor is a saturated solution of the solute in pure solvent E. The antisolvent W is added isothermally at a constant rate to induce crystallization. Figure 24 shows the process paths on

Table 7. Input Data for Drowning-out Polymorph Crystallization Example Model Parameters liquid density (mol/m3) F solid S1 density (mol/m3) FS1 solid S2 density (mol/m3) FS2 area shape factor kA volume shape factor kv size of nuclei (m) L0n,S2 S1 dissolution rate constant (m/s) kD,S1 S2 dissolution rate constant (m/s) kD,S2 S1 growth to dissolution rate ratio kG,S1/kD,S1 S2 growth to dissolution rate ratio kG,S2/kD,S2 S1 nucleation rate constant kn,S1 [no./(s m3)] S2 nucleation rate constant kn,S2 [no./(s m3)] S1 and S2 nucleation exponent n S1 and S2 growth exponent g antisolvent addition rate (m3/s) F Initial Fluid Composition mfluid xS RE

17 000 17 000 17 000 π π/6 1 × 10-8 1 × 10-7 1 × 10-7 0.1 0.1 1 × 108, 5 × 108 1 × 108 1 1 4 × 10-8 10 0.231 1.0

the phase diagram for two cases. The difference in the base case (BC, circles) and the process path for faster nucleation (FN, squares) is the nucleation rate constant for S1, kn,S1, which is 5 × 108 no./(s m3). This is 5 times the value of the base case, where the nucleation constants for the polymorphs are equal. Initially, the process path follows the material balance line for the addition of W. Later, crystal nucleation and growth cause the process path to turn towards the solid-liquid equilibrium curve. The amounts of each of the crystallized solids and the total amount of solid crystals present are plotted as functions of time in Figure 25. The squares in this figure refer to case FN, whereas circles mark points on the base case curves. Numbered points correspond to the numbered points on the isothermal phase diagram. Some of the undesired polymorph S1 crystallizes but redissolves after the process path leaves the saturation variety of S1. It is interesting to note that the bulk of the solute is precipitated during the initial 1.5 h. The remaining processing time involves the solution-mediated transformation of polymorph S1 into polymorph S2.

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Acknowledgment The support of the National Science Foundation (Grant CTS-9908667) for this work is gratefully acknowledged. Notation

Figure 25. Amounts of solid polymorphs versus time for the drowning-out crystallization of S2: base case (BC), faster S2 nucleation (FN).

Figure 26. Average crystal size for the drowning-out crystallization of S2: base case (BC), faster S2 nucleation (FN).

In case FN, where the nucleation rate of S1 is greater, there are more S1 crystals in the vessel at this time. A longer batch time is required to allow for the solutionmediated transformation in this case. If it is desirable to conserve solvent, then the addition of W could be stopped after 1.5 h, and only a marginal amount of product would be lost. Figure 26 gives the average crystal sizes of the two polymorphs as functions of time. The FN case gives smaller average crystal sizes. Conclusions A generic model was developed for kinetically controlled crystallization processes that accounts for chemical reaction, crystal growth, nucleation, and dissolution. The model was used in the study of asymmetric transformation and solution-mediated transformation. Several examples were used to provide insight into the role of phase equilibrium versus reaction and crystallization kinetics in these processes. The impact of operating policies, design parameters, and kinetic parameters on product purity and productivity is elucidated. This approach of coupling solid-liquid phase diagrams and kinetic effects is expected to help synthesize crystallization processes involving enantiomers and polymorphs. It is highly desirable to develop software tools to facilitate the generation and screening of different processing alternatives. Efforts in this direction are underway.

Ak ) total area of crystals of solid k, m2 A ˆ k ) specific area of crystals of solid k, m2/m3 0 Bn,k ) nucleation rate for solid k, no./(s m3) Ci ) concentration of component i, mol/m3 D ) solid dissolution rate, m/s E ) amount of solvent E in the fluid phase, mol F ) feed flow rate, m3/s g ) crystal growth exponent G ) crystal growth rate, m/s H ) liquid holdup of the crystallizer, mol kA ) area shape factor kD,k ) rate constant for dissolution of solid k, m/s Keq ) equilibrium constant for chemical reaction kG,k ) rate constant for crystal growth of solid k, m/s kn,k ) nucleation rate constant for solid k, no./(s m3) kr ) rate constant for racemization reaction, s-1 kv ) volumetric shape factor 0 LG,k ) critical size of nuclei for growth of solid k, m Lk ) total length of crystals of solid k, m mfluid ) amount of fluid in the crystallizer, mol mj ) moment of eq 8 from the moment transformation Mk ) total amount of crystals of solid k, mol n ) nucleation exponent nk ) population density of solid k, no./m4 Nk ) number of crystals of solid k P ) product flow rate, m3/s RE ) fraction of solvent that is solvent E Rf ) amount of solute R in the fluid, mol Rp ) productivity ratio Sf ) amount of solute S in the fluid, mol S1 ) polymorph of solute S S2 ) polymorph of solute S uk ) undersaturation with respect to solubility of solid k V ) volume of fluid in crystallizer, m3 W ) amount of solvent W in the fluid phase, mol xi ) mole fraction of component i xFi ) mole fraction of component i in the feed zS ) fraction of component S in the crystalline product Greek Letters and Symbols REW ) relative volatility between components E and W vi ) selective removal rate of component i from the fluid, mol/s v1 ) constant evaporation rate, mol/s v2 ) proportional evaporation rate, mol/(mol s) υSj ) stoichiometric coefficient for component S in reaction j Φj ) rate law for chemical reaction j, mol/(s m3) F ) fluid density, mol/m3 FF ) feed density, mol/m3 Fk ) density of solid k, mol/m3 σk ) supersaturation with respect to solubility of solid k

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Received for review September 12, 2002 Revised manuscript received February 27, 2003 Accepted March 11, 2003 IE020722U

Process Paths of Kinetically Controlled Crystallization: Enantiomers

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