Theory of Conglomerate Crystallization in the Presence of Chiral

Sep 1, 2005 - this paper, we present a general theoretical approach to describe this process. Our formulation is founded on basic theories of nucleati...
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CRYSTAL GROWTH & DESIGN

Theory of Conglomerate Crystallization in the Presence of Chiral Impurities

2005 VOL. 5, NO. 6 2173-2179

Dilip K. Kondepudi* and Kenneth E. Crook Department of Chemistry, Wake Forest University, Winston-Salem, North Carolina 27103 Received April 26, 2005;

Revised Manuscript Received June 24, 2005

ABSTRACT: In the presence of an appropriate chiral impurity, the enantiomers of compounds that crystallize as conglomerates crystallize at different rates. During this process, the optical activity of the solution and the solid phase increases and then falls to zero. This transient optical activity happens on a time scale of about 15 min. In this paper, we present a general theoretical approach to describe this process. Our formulation is founded on basic theories of nucleation, crystal growth, and Langmuir adsorption. A computer code based on this theory is capable of describing both the time evolution of a single run of crystallization and the stochastic behavior observed in many crystallization runs; it also gives the crystal size distribution. Introduction The effects of impurities on nucleation and crystal growth is of fundamental importance to the study of kinetics of crystallization for various reasons.1,2 When we consider the crystallization of chiral compounds whose enantiomers crystallize separately as conglomerates of R and S, the stereoselective nature of the impurity’s effect has interesting consequences. If the impurity is chiral, the enantiomers of the crystallizing compound may nucleate and crystallize at different rates. If this happens, it would result in a transient enantiomeric enrichment of the solution and the crystallizing solid phase, and, as we describe below, a solid phase separated from the solution in an appropriate time window (which is about 15 min) would be nearly enantiopure. We briefly discuss how one can observe such an effect of a chiral impurity in the crystallization of amino acids such as glutamic acid. Our principal aim, however, is to formulate a theoretical framework for the kinetics of conglomerate crystallization in the presence of a chiral impurity. The processes described by the theory, which contains random generation of crystals, are simulated using a computer code written in Mathematica and C++. We also outline a procedure by which the numerical values of the parameters in the computer code can be set through comparison with experimental data. The code is capable of generating time evolution of the amount of each enantiomer in the solution and the solid phase, the crystal size distribution, and the stochastic nature of the variables in the system. In crystallization that takes place in a continuously stirred solution, crystal nuclei are generated through different mechanisms. One is the generation of crystal nuclei directly from the solution, called primary nucleation, and the other is the generation of nuclei from the surfaces of crystals that have already formed, called secondary nucleation. A theory of impurity effects on crystallization in a stirred solution must describe how * To whom correspondence should be addressed. Fax: 1-336-7584656. E-mail: [email protected].

an impurity changes the nucleation processes and crystal growth. The rate of primary nucleation depends on the difference in the chemical potentials of the solid and the solution phases, and the interfacial energy. Hence, the effect of the impurity on nucleation depends on how it alters the chemical potential of the solid phase and the interfacial energy. The processes of secondary nucleation and crystal growth are altered by the adsorption of impurities on the crystal surface. Among the several theories that one can find in the literature3-7 is a recent theory formulated by Kubota and Mullin8 based on Langmuir adsorption of the impurity on the solid surface and the consequent retardation of the crystal growth rate. This theory has been quite successful in describing the variation of growth rates of crystals as a function of impurity concentration.9,10 In fact it has been the basis of determining the adsorption isotherm of lead(II) ion onto the surface of a NaCl crystal.11 Theory As noted above, primary crystal nuclei are formed directly from the supersaturated solution, while secondary nuclei are created from the surfaces of crystals. If the crystals are chiral, they generate secondary nuclei with a high degree of chiral selectivity. Such chiral autocatalysis can cause spontaneous symmetry breaking;12,13 the extraordinary amplification of chiral asymmetry by this mechanism has been noted by McBride and Carter.14 For the crystallization considered in this paper, viz. conglomerate crystallization of chiral compounds, secondary nuclei generated from R-crystals are R nuclei and those from S-crystals are S nuclei. The processes of primary and secondary nucleation are distinct with different rate laws; the effect of the impurity on each process must be analyzed separately. Primary Nucleation. The rate of primary nucleation (nuclei/unit volume/unit time) as given by the classical nucleation theory is

[

10.1021/cg0501837 CCC: $30.25 © 2005 American Chemical Society Published on Web 09/01/2005

]

∆Gnuc kT

J ) J0 exp -

(1)

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in which the Gibbs energy of the critical crystal nucleus is 3

∆Gnuc )

16πγ Vm

2

(2)

3(∆µ)2

In the above expression, ∆µ ) µsolution - µsolid is the chemical potential difference between the solution and the solid phases, γ is the solution-solid interfacial energy, and Vm is the molar volume of the solid phase. For nucleation from a solution, in the simplest approximation, ∆µ ) RT ln(C/Cs), in which C is the solution concentration and CS is the concentration at saturation. Although the above expression is generally used to describe homogeneous nucleation, it can be adapted to describe heterogeneous nucleation that more realistically describes a typical laboratory situation. In a solution, crystal nuclei are generally formed on small particles, “nucleating agents”, which decrease the Gibbs energy barrier ∆Gnuc (by decreasing the interfacial energy). That heterogeneous nucleation is the major cause of nucleation from solution is clear from the fact that the rate of nucleation depends on the purity of the solvent; in highly filtered solvents the nucleation rate is lower. We already noted that the nucleation rate is altered by the impurity because it alters the interfacial energy γ, the chemical potential difference ∆µ, and the molar volume Vm. We may then assume that the function F ≡ γ3Vm2/(∆µ)2 is an analytical function of the impurity mole fraction Ximp. Consequently, we may express F as power series in Ximp and write:

F(Ximp) ≡

γ03Vm02 (1 + AimpXimp + ...) (∆µ0)2

(3)

[

2

16πγ0 Vm0

J ) J0 exp -

3kT(∆µ0)2

(1 + AimpXimp)

]

(4)

If we use the approximation ∆µ ) RT ln(C/Cs), the above expression becomes

[

16πNAγ03Vm02 (1 + AimpXimp) C 2 3RT3 ln Cs

J ) J0 exp -

( ( ))

[

16πg0 J ) J0 exp C 3RT3 ln Cs

( ( ))

(1 + AimpXimp)

2

]

(5)

As is discussed above, in practice we find that heterogeneous nucleation is dominant; however, the expression for the rate of nucleation still has the form shown in eq 5, except that the preexponential term J0 and the interfacial energy γ depend on the “nucleating

]

(6)

in which g0 and J0 depend on solvent purity, i.e., the number and nature of the nucleating agents. Since these are not directly measurable quantities, we take J0 and g0 to be parameters in the theory whose value is to be set by comparison with the experiment. We use eq 6 in our computer simulation of the crystallization process. Crystal Growth Secondary Nucleation. The effect of impurity on the rates of growth and secondary nucleation are closely related. They are both altered by the impurity adsorption on the crystal surface. Using the Langmuir adsorption theory, Kubota and Mullin have formulated a theory to describe the retardation of the growth rate by the adsorbed impurity.8 This theory is based on the idea that the adsorbed impurity interferes with the orderly growth process of the crystal surface. The well-established Langmuir theory relates the fraction, θ(Ximp, T), of adsorption sites occupied by the impurity at a given temperature to the impurity mole fraction. Data collected on the relationship between θ and Ximp show that, in some cases, the crystal growth rate reaches zero before all the available sites are occupied, i.e., when θ < 1, while, in other cases, even when the impurity occupies all the available sites, that is, when θ ) 1, the crystals continue to grow at a nonzero rate.8,9 Taking note of these observations, Kubota and Mullin8 introduced an impurity-effectiveness factor R in their theory and proposed the following equation for the growth rate (the rate at which the crystal size (“radius r”) increases in the presence of an impurity:

G ) G0(1 - Rθ)

in which Aimp is an appropriate coefficient and subscript “0” denotes the value of quantities in the absence of the impurity. Since Ximp is on the order of 10-5 in experimental conditions of interest, we may ignore second and higher order terms in Ximp in eq 3. Treating Aimp as an empirical parameter, we can combine eqs 1-3 and write an expression for the nucleation rate as a function of the impurity mole fraction Ximp: 3

agent”. So we may write the nucleation rate as

(7)

in which G0 is the growth rate when the impurity is absent. The well-known Langmuir expression relates the impurity concentration, which we express as mole fraction Ximp, to the fraction of the occupied sites θ on the crystal surface:

θ)

XimpKl 1 + XimpKl

(8)

Kl in the above expression is the Langmuir adsorptiondesorption equilibrium constant. In using this expression, we tacitly assume that the adsorption equilibrium is reached during the crystal growth, an assumption that may be adequate as a first approximation but whose justification needs further study. The Burton-Cabrera-Frank theory of crystal growth15 gives the dependence of G0 on supersaturation:

G0 ) A(C - Cs)2 tanh

(

B C - Cs

)

(9)

in which A and B are empirical parameters. For small values of (C - Cs), the growth rate G0 is proportional to (C - Cs)2, but it is linear in (C - Cs) for higher values of the solute concentration C. Equations 7-9 tell us how

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the rate of crystal growth is retarded by the impurity. The parameter A is larger in stirred systems. Next, we turn to impurity effects on secondary nucleation. Various mechanisms have been proposed for the generation of secondary nuclei from the surface of an existing crystal when the crystallizing solution is stirred, but it remains a process whose rate is largely described from empirical data fitting.16,17 Among the more successful expression for the rate of secondary nucleation, S0, is the following expression16 in which σ is the total area of the crystal surface on which secondary nuclei are generated, s is a parameter proportional to the stirring rpm, and Ksec is a temperature-dependent parameter:

S0 ) Ksecσs(C - Cs)lΘ(r - rmin)

(10)

Because of the lack of an accepted fundamental theory, the dependence of S on supersaturation (C Cs) is given the form of a power law, with exponent l, which is often given a noninteger value to fit the experimental data.16,17,19,27 The temperature dependence of Ksec is yet to be established. It was also found that the secondary nucleation is associated with a minimum size (radius rmin) that the parent crystal must reach to generate an observable number of secondary nuclei,18,19 hence the inclusion of the theta function, Θ(r - rmin), which equals 0 when the argument is negative and 1 when the argument is positive; here r is the crystal radius and rmin is the radius below which no secondary nuclei are generated. (To distinguish it from the fraction of occupied sites, θ, defined in eq 8, upper case “Θ” is used to represent the theta function). If secondary nucleation is a surface phenomenon, the adsorption of the impurity must influence it as it does the growth rate. Accordingly, we propose the following expression for the effect of impurity on the rate of secondary nucleation:

S ) S0(1 - βθ)

(11)

That is to say, just as the adsorption of the impurity in the crystal surface interferes with the growth process, the generation of secondary nuclei, which in some manner involves attachment of solute molecules to the surface and subsequent detachment as crystal nuclei, must also be inhibited due to impurity adsorption. In addition, we assume the effectiveness factor β, for secondary nucleation is, in general, different from the effectiveness factor for growth retardation, R. Equations 6, 7, and 11 based on classical theory of nucleation and Langmuir theory of adsorption quantify the retardation of nucleation, primary and secondary, and the growth rates due to the impurity. We apply this theory to the conglomerate crystallization of chiral compound in the presence of a chiral impurity. The parameters in this theory can be determined by comparison with experiments as described in the subsequent sections. As a particular example, we choose the crystallization of DL-glutamic acid in the presence of Lor D-lysine, which has been studied experimentally.20 Crystallization of DL-Glutamic Acid in the Presence of L-Lysine The crystallization of glutamic acid (Glu) is quite sensitive to the presence of lysine (Lys). Both nucleation

Figure 1. The decrease of excess concentration of DL-glutamic acid in the presence of L-Lys. To a solution of 1.6 g of DL-Glu in 80 mL of water, 120 mL of 2-propanol is added with rapid stirring to drive the system to supersaturation. The solute begins to precipitate immediately. Data collection begins when the solution becomes homogeneous. The concentration which drops from about 0.008 g/mL to about 0.003 g/mL is normalized as excess concentration that varies from 1 to 0. The total amount of L-Lys impurity added to the 200 mL DL-Glu solution is 0.02 and 0.08 g for the two curves shown.

and growth rates of glutamic acid are retarded by the presence of lysine. The effect is highly stereoselective: L-Lys retards the crystallization rate of L-Glu, but its effect on D-Glu, if any, is hardly noticeable. Similarly, the crystallization of DL-threonine is influenced by the presence of alanine.21 The stereoselectivity follows a general principle noted by Addadi et al.,22 which states that an impurity with a structure similar to that of the crystallizing compound will retard nucleation and growth rates. Through a systematic study Addai et al.23 arrived at a stereochemical explanation for the impurity effects on crystallization. In a recent review,24 Weissbuch et al. state their inference that “the tailor-made additive is adsorbed on the growing crystal, but only at certain surfaces and then with the part of the adsorbate that differs from that of the substrate emerging from the crystal”. Thus, we might expect that amino acids such as Lys might stereoselctively adsorb on the surface of Glu crystals and retard its growth. This expectation is well confirmed by experiments.20,24-26 Indeed, the steroselectivity is so good, it could be used as an analytical method for the detection of L-Lys.25,26 When DL-Glu is crystallized in the presence of L-Lys, D-Glu crystallizes first followed by L-Glu, thus causing a transient separation of Glu enantiomers. In the initial stages of crystallization, the solid-phase enantiomeric excess (EE) is nearly 100%. Our studies have shown that transient separation of enantiomers of DL-Glu in the presence of L- or D-lysine can occur in about 20 min,20 a time in which the solid phase can easily be separated. More recent experiments with highly purified water show clearly the well-separated crystallization of L- and D-Glu; the stepwise graph in Figure 1 shows crystalyzation of D-Glu followed by the crystallization of L-Glu. In these experiments, an aqueous solution of DL-Glu is driven to supersaturation by the addition of

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Figure 2. Crystallization of 1.6 g of DL-Glu in the presence of 0.04 g of L-Lys. Repeated runs with identical initial conditions, each represented by a different symbol (crosses, filled circles, etc.), give different concentration-vs-time curves, indicating the stochastic nature of the crystallization process. The stochastic behavior originates from primary and secondary nucleation.

2-propanol; the concentration of Glu is monitored using a conductivity meter. More details of data collection can be found in ref 20; here we focus on a theoretical description of the impurity effect. In stirred crystallization, nucleation is autocatalytic due to the process of secondary nucleation. An initial nucleation, which is random, gives rise to rapid conversion of the solute-to-solid phase through secondary nucleation and subsequent crystal growth. In other words, this process amplifies an initial random nucleation process. There is thus an essential stochastic nature to this process: repeated runs of crystallization with identical initial conditions give different curves for the decrease of solute concentration.27 Figure 2 shows experimental data when 0.04 g of L-Lys impurity is in a solution containing 1.6 g of DL-Glu. The different curves obtained for identical initial conditions are initiated through primary nucleation, which is random.

Figure 3. Crystallization of 1.6 g of DL-Glu in the presence of 0.04 g of L-Lys. (a) Comparison between the concentrationvs-time curves generated by theory and experiment. Experimental curves, marked by X’s, are the same as in Figure 2. Filled circles are generated by a theory-based computer code with parameter values specified in the text. (b) Concentration of Glu-vs-time curves generated by the computer code for various amounts of the impurity L-Lys; the amounts vary from 0.02 to 0.08 g of Lys per 1.6 g of DL-Glu. The two “steps” correspond to crystallization of D-Glu followed by L-Glu.

Computer Simulation and Results Since heterogeneous primary nucleation depends on the particular experimental conditions, such as the purity of the solvent, some parameters of the theory must be set for each experimental condition. To be able to discuss the implications of the theory meaningfully, the numerical values of the parameters in the theory must reflect, at least approximately, the experimental situation. As shown in Figure 3, we used preliminary experimental data to set the parameters of the theory. In setting the theory parameters, one must look at the different stages of crystallization. The initial rapid drop in the concentration, for example, depends on the secondary nucleation rate; hence, the secondary nucleation rate must be adjusted to fit this part of the curve. As the supersaturation drops, the rate of nucleation decreases rapidly so it does not contribute to the decrease of concentration at lower supersaturation; the decrease of concentration at lower supersaturation is almost entirely due to the growth of nucleated crystals.

Accordingly, the growth rate parameters are adjusted to fit these parts of the curve. The random variation in the concentration-vs-time trajectories comes from the stochastic nature of the primary nucleation process. The number of crystals generated through secondary nucleation far exceeds number of crystals generated by primary nucleation. Hence, secondary nucleation rate essentially determines the total number of crystals generated. By measuring the total amount of crystallized solute and the approximate crystal size, we were able to estimate that the total number of crystals generated in our experiments is in the range 107-108; the secondary nucleation parameters of the theory were then adjusted to generate numbers of crystals that are comparable to these experimental estimates (as can be seen in Figure 6). And, of course, the retardation of the crystallization process must be used to determine the parameters such as the Langmuir equilibrium constant Kl and the effectiveness factors R and β. In our theory, solute concentration, C, is expressed as moles of solute

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per moles of solvent and impurity concentration, Ximp, is expressed as a mole fraction in the solution phase. These concentrations are independent of temperature. The theoretical curve shown in Figure 3a is obtained by setting the numerical values of the parameters as explained above. The parameter values are

Primary nucleation: J0 ) 8.43 × 10-2 s-1; g0 ) 1.0 × 10-23 J K2/mol; Aimp ) 0.8 × 109; Cs ) 2.85 × 10-4 Langmuir adsorption: Kl ) 2.0 × 103; R ) 4.9 × 103; β ) 3.0 Secondary nucleation: l ) 2.5; s ) 2.0; Ks ) 0.8 × 1012 cm-2 s-1; Rmin ) 15.0 × 10-4 cm Burton-Cabrera-Frank growth rate: A ) 6.67 × 102 (1 + s) cm/s; B ) 5.2 × 104 The (1 + s) factor, in which s depends on the stirring rpm, is included in A to have a growth rate that increases with stirring rpm. With these parameter values that correspond approximately to the actual time scales and concentrations of the experiment, we are in a position to theoretically analyze the process in some detail. Using the rates discussed above for primary and secondary nucleation, and the subsequent growth of the crystals, computer codes were written in Mathematica and C++. The codes use random number generators for the generation of crystal nuclei. In the computer code, each iteration, which is also a time step, goes through the following cycle: (i) Computes the solute concentration (ii) Creates primary nuclei using a Poisson random number generator with an average equal to J in eq 6 (iii) Grows each crystal with a growth rate given by eq 7. Here the crystals are assumed to be spherical with radius r, and dr/dt ) G. (iv) Computes the total surface area of all of the crystal, σ, and, depending on the stirring rate s and concentration C, generates secondary nuclei using a Poisson random number generator with an average equal to S in eq 10. (v) Computes the amount of L- and D-enantiomers of the solute that has crystallized and updates the solution concentration, thus returning to the first step in the cycle. During each cycle, the concentration, the number of primary and secondary nuclei, and their radii are recorded; the crystal size distribution could also be obtained at any stage of the process. This is one more aspect of the theory that could be compared with the experimental data. As shown in Figure 3a, experimentally meaningful values of the parameters could be obtained by fitting the theory to preliminary experimental data. The data in Figures 1 and the computer simulation shown in Figure 3b have reasonable quantitative similarity. For a good fit, that including stochastic variation, a large amount of experimental data are needed. To obtain information regarding stochastic variation of the tra-

Figure 4. Computer simulation of crystallization of 1.6 g DLGlu in the presence of 0.04 g of L-Lys. (a) The curves show the decrease in the amount of D- and L-Glu in the solution phase during crystallization. The presence of the impurity L-Lys retards the crystallization of L-Glu. Most of the D-Glu crystallizes before L-Glu begins to crystallize. (b) The change of EE in the solution and solid phases during the crystallization. In the shaded region, the solid phase has an EE close to 100%.

jectories shown in Figure 2, that experiment must be repeated with the same initial conditions 10-15 times for each concentration of the impurity. At this time such extensive data are not available; nevertheless, with the type of agreement between theory and experiment shown in Figure 3, one can make realistic predictions about this system. The first point to note is that the chirally selective effect of the impurity is a viable means for separation of Glu enantiomers. Figure 4a shows the concentration of the two enantiomers in the solution as a function of time. From this graph, it is clear for a significant time (15-20 min) that only D-Glu is in the solid phase. This solid phase, whose EE is nearly 100%, can be separated; thus, a 5% impurity could generate a much larger amount of enantiopure D-Glu. Figure 4b shows the EE in the solution and the solid phases as functions of time. The second point is the possibility of amplification of EE in the following sense. In the above process, we assumed that the EE of the impurity (L-Lys) is 100%; hence obtaining solid-phase D-Glu with 100% EE is not “amplification” of EE. However, obtaining D-Glu with

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Concluding Remarks Our formulation of crystallization kinetics in the presence of a chiral impurity gives us a means to extract theoretical parameters from the experimental data and make predictions. The theory gives the time that separates the crystallization of the enantiomers; it predicts the possibility of using an impurity with less than 100% EE to produce a solid product with 100% EE, and it gives such details as crystal size distribution. This theory could easily be extended to solutes that racemize at a rate comparable to its crystallization rate. In that case, the enantioseparation using a chirally selective impurity would be even more effective.

Figure 5. Computer simulation of crystallization of 1.6 g DLGlu in the presence of 0.08 g of L-Lys and 0.02 g of D-Lys. Even when the EE of the impurity is less than 100%, the EE of the solid phase during the crystallization can reach 100%.

Acknowledgment. We thank Wake Forest University for providing support for this work. We also acknowledge a grant from Japan Space Forum (15JSF14026), Grant-in-Aid for Scientific Research (C) from Japan Society for Promotion of Science (4830). Notation A B Aimp C Cs EE J0 g0 G0 G ∆G(r)nuc Kl Ksec

Figure 6. Crystal size distribution of L- and D-Glu crystals obtained through computer simulation. As in Figure 4, the simulation corresponds to crystallization of 1.6 g DL-Glu in the presence of 0.04 g of L-Lys. The distribution shows larger number of small L-Glu crystals because the impurity, L-Lys, retards the growth of L-Glu. In the size range 10-2-10-4 cm, there are more D-Glu crystals.

100% EE using an impurity with EE < 100% would be amplification of EE. This possibility is shown in Figure 5; it shows crystallization in the presence of 0.08 g of L-Lys and 0.02 D-Lys. From this curve, it is clear that the impurity with an EE equal to 60% can slow the crystallization of L-Glu sufficiently to produce D-Glu in the solid phase with 100% EE, a clear case of chiral amplification. Finally, Figure 6 shows the crystal size distribution at the end of the run. This carries much information about the rates of secondary nucleation and crystal growth in the presence of impurity. As is clear from this plot, L-crystals are smaller and larger in numbers due to the presence of L-Lys. A total of 1.4 × 108 L-crystals and 1.2 × 108 D-crystals are generated during the process. The estimate of the number of crystals generated in the experiment is in the range 107-108. With particle size counters, one could check the theoretical predictions of the crystal size distributions more accurately.

k l r rmin R T Vm Ximp R γ µsolid µsolution θ σ

constant used in growth rate (eq 9) constant used in growth rate (eq 9) empirical constant for the impurity effect on primary nucleation solution concentration solution concentration at saturation enantiomeric excess preexponential factor in nucleation rate (eq 6) Empirical parameter in primary nucleation rate (eq 6) growth rate constant in the absence of impurity growth rate in the presence of impurity Gibbs energy change for the formation of a solidphase nucleus of radius r equilibrium constant for Langmuir adsorption (eq 8) empirical constant used in secondary nucleation rate (eq 10) Boltzmann constant J/K empirical exponent used in secondary nucleation rate (eq 10) crystal radius crystal size below which secondary nucleation does not occur (eq 10) gas constant, J/(K mol) temperature, K molar volume of the solid-phase mole fraction of the impurity impurity effectiveness factor for growth rate solid-solution interfacial energy chemical potential of the solute in the solid phase chemical potential of the solute in the solution phase fractional coverage of the adsorption sites by impurity area of crystal surface that generates secondary nuclei

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Impurity Effects in Conglomerate Crystallization (7) Al-Jibbouri, S.; Ulrich, J. Cryst. Res. Technol. 2001, 36, 1365-1375. (8) Kubota, N.; Mullin, J. W. J. Cryst. Growth 1995, 152, 203208. (9) Kubota, N.; Yokota, M.; Mullin, J. W. J. Cryst. Growth 1997, 182, 86-94. (10) Kubota, N.; Yokota, M.; Mullin, J. W. J. Cryst. Growth 2000, 212, 480-488. (11) Kubota, N.; Shigeko, S.; Doki, N.; Yokota, M. Cryst. Growth Des. 2005, 5, 509-512. (12) Kondepudi, D. K.; Kaufman, R. J.; Singh, N. Science 1990, 250, 975-976. (13) Kondepudi, D. K.; Asakura, K. Acc. Chem. Res. 2001, 34, 946-954. (14) McBride, J. M.; Carter, R. L. Angew. Chem. 1991, 30, 293295. (15) Burton, W. K.; Cabrera, N.; Frank, F. C. Philos. Trans. R. Soc. (London) 1951, 243, 299-358. (16) Randolph, A. D.; Larson, M. D. Theory of Particulate Processes, 2nd ed.; Academic Press: San Diego, 1988. (17) Jancic, S. J.; Grootscholten, P. A. M. Industrial Crystallization; D. Reidel Publishing Co.: Boston, 1984.

Crystal Growth & Design, Vol. 5, No. 6, 2005 2179 (18) Kubota, N.; Fuliwara, M. J. Chem. Eng. Jpn. 1990, 23, 691696. (19) Kondepudi, D. K.; Bullock, K. L.; Digits, J. A.; Hall, J. K.; Miller, J. M. J. Am. Chem. Soc. 1993, 115, 10211-10216. (20) Buhse, T.; Kondepudi, D. K.; Hoskins, B. Chirality 1999, 11, 343-348. (21) Shiraiwa, T.; Kubo, M.; Fukuda, K.; Kurokawa, H. Biosci., Biotechnol., Biochem. 1999, 63, 2212-2215. (22) Addadi, L.; Weinstein, S.; Gati, E.; Weissbuch, I.; Lahav, M. J. Am. Chem. Soc. 1982, 104, 4610-4617. (23) Addadi, L.; Berkovitchyellin, Z.; Weissbuch, I.; Vanmil, J.; Shimon, L. J. W.; Lahav, M.; Leiserowitz, L. Angew. Chem. Int. Ed. Engl. 1985, 24, 466-485. (24) Weissbuch, I.; Lahav, M.; Leiserowitz, L. Cryst. Growth Des. 2003, 3, 125-150. (25) Ballesteros, E.; Gallego, M.; Valcarcel, M. Anal. Chem. 1996, 68, 322-326. (26) Hosse, M.; Ballesteros, E.; Gallego, M.; Valcarcel, M. Analyst 1996, 121, 1397-1400. (27) Kondepudi, D. K.; Culha, M. Chirality 1998, 10, 238-245.

CG0501837