Growth and Ripening Kinetics of Crystalline Polymorphs Giridhar Madras* Department of Chemical Engineering, Indian Institute of Science, Bangalore 560 012, India
CRYSTAL GROWTH & DESIGN 2003 VOL. 3, NO. 6 981-990
Benjamin J. McCoy Department of Chemical Engineering, Louisiana State University, Baton Rouge, Louisiana 70803 Received July 4, 2003;
Revised Manuscript Received July 30, 2003
ABSTRACT: Crystals that present different polymorphic forms during their preparation pose challenging questions for science and industry. Given their relevant physical and chemical properties, understanding how the size distributions of polymorphs evolve during crystallization is important fundamentally and practically. Here, we propose a distribution kinetics model for the essential phenomena associated with polymorph crystallization processes. Included in the model are temperature effects for growth and dissolution coefficients, Gibbs-Thomson effects of particle curvature on equilibrium solubility, phase-transition energies (heats of solidification or vaporization), critical nucleus sizes for denucleation during coarsening, and interfacial energies. Numerical solutions of the governing population dynamics equations show that interfacial and transition energies, but not activation energies, are significant in causing the less stable dimorph to vanish. Evaporative crystallization and cooling and heating programs are potential ways to control polymorph separation. I. Introduction Producing pure crystalline forms of active pharmaceutical ingredients is a huge challenge for the drug industry.1 A major difficulty is that crystals of identical chemical composition frequently have different forms, called polymorphs.2,3 Most importantly, different polymorphs may have a different solubility, dissolution rate, and chemical and physical stability, in addition to other properties. The solubility and stability have direct influences on drug bioavailability, effectiveness, and safety. Examples of polymorph transformation, either during or after preparation, have been well-documented.4-8 During simultaneous, or concomitant,4 crystallization, polymorphs of different growth rates may compete for substrate and cause one form to dominate over another. Even when crystals are fully formed, they may undergo Ostwald ripening, or coarsening, whereby smaller, less stable crystals dissolve (Gibbs-Thomson effect) and give up their mass to larger, more stable forms. Progesterone, for example, during drying, storage, and sampling, can undergo such transformations.5 A fundamental understanding of the underlying science of polymorph kinetics and dynamics, including nucleation, growth, and ripening, is needed to guide the research, development, manufacture, and storage of pharmaceuticals. In the present paper, we address the question: Given the relevant physical properties, what are the possible strategies for optimally producing a particular polymorph and how can they be evaluated? Cooling crystallization is a common method to prepare and separate polymorphs. Threlfall9 provided thermodynamic insight into how two polymorphs (dimorphs) respond during cooling at different concentrations when * Corresponding author. Ph.: 91-080-309-2321. Fax: 91-080-3600683. E-mail:
[email protected].
metastable zone limits and solubility curves (concentration vs temperature) cross. The lack of experimental reproducibility was attributed not only to differences in temperature, concentration, and solvent but also to seeding (intentional or accidental), cooling rate, solvent evaporation rate, and dynamic and kinetic features in general. Elaborate procedures10 for controlling the rate of cooling or solvent evaporation during seeding have been recommended to manage polymorph production. Solvent-mediated transformation10 to the more stable polymorph is akin to coarsening, whereby less stable particles dissolve and transfer their mass to more stable particles. Although such transformations can occur during sublimation (vapor-solid) transitions, they are less often observed in the dry solid state. Most studies of polymorph crystallization are qualitative or semiquantitative descriptions. The subject is complex, requiring an integration of thermodynamics, kinetics, reactor dynamics, population balances, and interfacial science. A recent quantitative study11 of process paths for controlling chiral and polymorph crystallization cited the importance of understanding coupled kinetics and phase equilibrium. Approximate models of nucleation, growth, and dissolution kinetics were proposed, but Gibbs-Thomson and ripening effects were neglected. The study demonstrated how to model several dynamic crystallization processes, including the case with evaporating solvent (evaporative crystallization). The work illustrated the value of quantitative descriptions of polymorph crystallization for evaluating operating strategies to produce the desired product at specified purity. Seeding, or heterogeneous nucleation, avoids the delay that frequently accompanies homogeneous nucleation and usually initiates precipitation of the desired polymorph. By rapidly decreasing the supersaturation,
10.1021/cg034117l CCC: $25.00 © 2003 American Chemical Society Published on Web 08/20/2003
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seeding can prevent unwanted homogeneous nucleation. Heterogeneous nuclei have accidental as well as intended origins. Examples have been cited of undesirable seeds inadvertently introduced by contamination,12 thus rendering the desired polymorph precipitation difficult or impossible. Such heterogeneous nuclei may be attached to surfaces of process equipment and can be longlived. A possible explanation for nucleation by ultrasound is that imperfections on vessel walls act as heterogeneous nucleation sites such that sonic vibrations dislodge the nanosized nuclei. Crystal habit and size distribution can be controlled by such sonocrystallization, in which ultrasonic vibrations are delivered to the outside walls of the crystallization vessel.1 Yu13 has lately reported evidence that classical concepts of crystallization are insufficient to describe polymorph nucleation phenomena. Cooling a crystal melt can nucleate a less stable polymorph on which a more stable polymorph can grow, and seeding with a particular polymorph can promote growth of another polymorph. The ability of a polymorph to grow on an unlike seed or nucleus suggests that polymorphs will sometimes crystallize opportunistically on any available and compatible surface. Moreover, the Ostwald law of stages (OLS) is not always followed. The OLS, a principle based on nonequilibrium thermodynamics stating that less stable polymorphs will crystallize prior to more stable polymorphs, is important in mineral crystallization.14 Differences in surface energy are known to favor the formation of a particular mineral polymorph.15 The overall results and experience support the conclusion that kinetic influences based on crystalline properties are crucial in interpreting observations of polymorph crystallization. In this paper, we expand upon the concept that crystal nucleation, growth, and ripening for polymorphs are competing processes determined by fundamental molecular and thermodynamic properties of crystalline particles. The approach described next is constructed along the lines of previous studies.16-19 Whenever a distribution of particle sizes exists, Ostwald ripening caused by the varying curvature of different interfaces can be crucially important. We have shown16 how nucleation, growth, and ripening for a single crystalline form can be combined into one distribution-kinetics theory. The mathematical model showed how homogeneous or heterogeneous nucleation preceded growth by the deposition of solute and was followed by Ostwald ripening. Similar ideas were applied to competitive crystal growth17 and ripening,18 specifically for polymorphs of interest in materials science. We proposed that crystal nucleation, growth, and ripening for polymorphs are competing processes determined by fundamental molecular and thermodynamic properties of crystalline particles. Computations showed how two polymorphs with slightly different properties respond differently to temperature varying with time.19 One polymorph may be more stable at a given temperature than another; thus, the more stable form would grow faster, whereas the less stable form would grow slower. Applying a temperature program can potentially optimize the particle size distributions. Our present aim is to examine the temperature dependence of the parameters that might influence
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crystal growth and ripening and thus polymorph separation. Homogeneous nucleation from metastable states is deferred to a future study, although for Ostwald ripening the critical nucleus size determines the denucleation rate and is a key feature of coarsening. We ignore temperature gradients within particles and fluid, an assumption that is realistic for pharmaceutical liquids and solids when temperature changes are slow. In the presence of an imposed temperature gradient, heat conduction effects can be significant.20 The temperature effects considered in the current model include the diffusion-influenced growth coefficient, GibbsThomson effect of particle curvature on equilibrium solubility, phase-transition energy (heat of solidification or vaporization), critical nucleus size for denucleation, and interfacial energy (surface tension). The dissolution rate coefficient is related to the growth rate coefficient by microscopic reversibility, thereby establishing its temperature dependence. The effect of the interfacial energy coefficient is a property of particular interest, even though for two polymorphs, any of the kinetic or thermodynamic parameters might have different values. Most previous discussions of polymorph crystallization emphasized solubility and its temperature dependence. Through the Gibbs-Thomson effect, the temperaturesensitive interfacial energy also offers an opportunity to control crystal or grain size by temperature programming. The paper is organized as follows. Concepts of distribution kinetics are introduced in Section II, including issues of solubility, interfacial effects, and temperature dependence. The governing equations for growth and ripening are presented in dimensionless form for dimorphs A and B. In Section III, we describe the numerical solutions of the governing equations for given parametric values and initial conditions. Principles illustrated with dimorphs A and B can be readily extended to a larger number of polymorphs. In the Conclusion (Section IV), we summarize the results and assess our approach to crystallization kinetics and dynamics. II. Distribution Kinetics Much of the following commentary on distribution kinetics and population dynamics for crystallization was discussed in detail in earlier work.17-19 The concentration of crystals at time t in the differential mass range (x, x + dx) is defined as c(x,t)dx. The moments are integrals over the mass
c(n)(t) )
∫0∞c(x,t)xn dx
(2.1)
Leading moments serve to characterize the crystal size distribution. The zeroth moment, c(0)(t), and the first moment, c(1)(t), are the time dependent molar (or number) concentration of crystals and the crystal mass concentration (mass/volume), respectively. The ratio of the two is the average crystal mass, cavg ) c(1)/c(0). The variance, cvar ) c(2)/c(0) - [cavg]2, and the polydispersity index, cpd ) c(2)c(0)/(c(1))2, are measures of the polydispersity. The molar concentration, m(0)(t), of solute monomer of molecular weight xm is the zeroth moment of the monomer distribution, m(x,t) ) m(0)(t)δ(x - xm).
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The deposition or condensation process by which monomers of mass x′ ) xm are reversibly added to or dissociated from a crystal of mass x can be written as21,22
where the crystal radius is related to its mass x by r ) (3x/4πF)1/3, in terms of the crystal mass density F, which we assume to be constant with temperature. As usual in kinetics, we assume that the temperature dependence of the growth (addition, aggregation) rate is weak relative to the dissolution rate. When diffusion effects are significant, we assume an activation energy for the growth coefficient to account for activated diffusion, Dm ) Do exp(-E/RT); thus,
kg(x)
C(x) + M(x′) {\ } C(x + x′) k (x)
(2.2)
d
where C(x) is the crystal of mass x, and M(x′ ) xm) is the monomer. We allow for a process of solvent removal (e.g., by evaporation) with a time dependent volume, V(t), in the accumulation terms. The mass balance equations governing the crystal distribution, c(x,t), and the monomer distribution, m(x,t), are18
∫0 m(x′,t) dx′ +
(1/V)∂[Vc(x,t)]/∂t ) -kg(x)c(x,t)
∞
∫0 kg(x - x′)c(x - x′,t)m(x′,t) dx′ - kd(x)c(x,t) + ∫x∞kd(x′)c(x′,t)δ(x - (x′ - xm)) dx' - Iδ(x - x*) (2.3) x
and
(1/V)∂[Vm(x,t)]/∂t ) -m(x,t)
∫0∞kg(x′)c(x′,t) dx′ +
∫x∞kd(x′)c(x′,t)δ(x - xm) dx′ + Iδ(x - x*)x*/xm
(2.4)
kg(x) ) γxλ exp(-E/RT)
(2.8)
where E is the activation energy, R is the gas constant, and if λ ) 1/3, then γ ) 4πDo(3/4πF)1/3. The 1/3 power on x thus represents diffusion-controlled ripening.18 When growth is limited by monomer attachment and dissociation at the crystal surface, the rate coefficient is proportional to the crystal surface area, kg ∝ r2, so that kg is proportional to x2/3; thus, in eq 2.11, λ ) 2/3 for surface-controlled ripening.18 If the deposition is independent of the surface area, then kg varies as x0. The temperature dependence for growth and ripening is influenced by the thermodynamic properties. The interfacial curvature effect is prescribed by the GibbsThomson equation expressed in terms of m∞(0), the equilibrium solubility of a plane surface
Nucleation of crystals of mass x* at rate I are source terms, or in the case of ripening, sink terms for denucleation, which occurs when crystals shrink to their critical size, x*, and then spontaneously vanish. We thus distinguish between ordinary dissolution due to concentration driving forces and total disintegration due to thermodynamic instability. For ordinary particle growth or dissolution without nucleation, I ) 0. Initial conditions for eqs 2.3 and 2.4 are c(x,t ) 0) ) c0(x) and m(x,t ) 0) ) m0(0)δ(x - xm). The mass balance follows from eqs 2.3 and 2.4 and can be expressed in terms of mass concentrations
where xm/F is the monomer molar volume, σ is the interfacial energy, kB is Boltzmann’s constant, and T is temperature. The critical nucleus radius at a given solute concentration m(0) is
xmm0(0) + c0(1) ) xmm(0)(t) + c(1)(t)
r* ) 2σxm/[FkBT ln(m(0)/m∞(0))]
(2.4a)
The size distribution changes according to eq 2.3, which becomes, when the integrations over the Dirac distributions are performed, the finite-difference differential equation
∂c(x,t)/∂t ) -kg(x)c(x,t)m(0)(t) + kg(x - xm)c(x - xm,t)m(0)(t) - kd(x)c(x,t) + kd(x + xm)c(x + xm,t) - Iδ(x - x*) (2.5) Eq 2.5 can be expanded for xm , x to convert the differences into differentials, yielding the customary (approximate) continuity equation applied to particle growth and ripening.23 Microscopic reversibility (detailed balance) implies
kd(x) )
meq(0)kg(x)
(2.6)
Given an expression for kg(x), therefore, one can calculate kd(x). A monomer that attaches to a crystal may diffuse through the solution to react at the crystal surface. Such diffusion-controlled reactions have a rate coefficient represented24 by
kg ) 4πDmr
(2.7)
meq(0) ) m∞(0) exp(Ω)
(2.9)
Ω ) 2σxm/(rFkBT)
(2.10)
with
(2.11)
Similar to a gas-liquid interface, we assume that the interfacial energy decreases linearly with temperature,25 σ ) σo(1 - T/Tc), where Tc is the critical (or reference) temperature. The temperature dependence of the equilibrium solubility is given by
m∞(0) ) µ∞ exp(-∆H/RT)
(2.12)
where ∆H is the molar energy of the crystallization, and µ∞ is the flat-surface equilibrium solubility at large T. We define scaled dimensionless quantities for the mass and temperature relationships
C ) cxm/µ∞, C(n) ) c(n)/µ∞xmn, ξ ) x/xm, θ ) tγµ∞xmλ, S ) m(0)/µ∞, Seq ) S exp(h/Θ - Ω), Ω ) w(Θ-1 - 1)/ξ1/3, w ) (3xm/4πF)-1/32σoxm/FkBTc, Θ ) T/Tc, J ) I/ (γµ∞2xmλ), h ) ∆H/RTc, ) E/RTc, v ) V/V0 (2.13) By the definition in eq 2.1, the scaled moments are
C(n)(θ) )
∫0∞C(ξ,θ)ξn dξ
(2.14)
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Note that ξ is the number of monomers in the crystal, and Θ is the reduced temperature (0 < Θ < 1). The zeroth and first moments, C(0) and C(1), represent the number and mass of crystals, respectively. The ratio S is defined19 relative to the high-temperature solubility µ∞ (here assumed identical for polymorphs A and B) rather than to the plane-surface solubility m∞(0). According to eq 2.12, if ∆H is different for A and B, then mA∞(0) is not equal to mB∞(0). The supersaturation ratio defined as Seq ) m(0)/meq(0) for either A or B evolves to unity at thermodynamic equilibrium. The scaled number (or moles) of particles, C(0) ) c(0)/µ∞, is also in units of the solubility µ∞. The Gibbs-Thomson factor Ω , eq 2.13, is expressed in terms of a scaled interfacial energy, w, which generally differs for A and B. Substituting the dimensionless quantities of eq 2.13 into the equations for balances on A, B, and solute yields the dimensionless equations
[1/v(θ)] ∂[v(θ)CA(ξ,θ)]/∂θ ) S(θ) exp(-A/Θ)[-ξλCA(ξ,θ) + (ξ - 1)λ CA(ξ - 1,θ)] -1
ξ exp[-(hA + A)/Θ] exp[wA(Θ λ
-1/3
- 1)ξ
]CA(ξ,θ) +
(ξ + 1) exp[-(hA + A)/Θ] λ
exp[wA(Θ-1 - 1)(ξ + 1)-1/3]CA(ξ + 1,θ) JAδ(ξ - ξA*) (2.15) and
Madras and McCoy
and
∂CB(ξ,θ)/∂θ ) S(θ) exp(-B/Θ)[-ξλCB(ξ,θ) + (ξ - 1)λCB(ξ - 1,θ)] ξλ exp[-(hB + )/Θ] exp[wB(Θ-1 - 1)ξ-1/3]CB(ξ,θ) + (ξ + 1)λ exp[-(hB + B)/Θ] exp[wB(Θ-1 - 1)(ξ + 1)-1/3]CB(ξ + 1,θ) + ψCB(ξ,θ)/(1 - ψθ) - JBδ(ξ - ξB*) (2.19) with
∂S(θ)/dθ ) -S(θ)[exp(-A/Θ)CA(λ) + exp(-B/Θ)CB(λ)] + exp[-(hA + A)/Θ] exp[wA(Θ-1 - 1)(CAavg)-1/3]CA(λ)+ exp[-(hB + B)/Θ] exp[wB(Θ-1 - 1)(CBavg)-1/3]CB(λ) + ψS(θ)/(1 - ψθ) + JAξA* + JBξB* (2.20) The initial conditions are S(θ ) 0) ) S0, CA(ξ,θ ) 0) ) CA0(ξ), and CB(ξ,θ ) 0) ) CB0(ξ). The terms JAξA* + JBξB* in eqs 2.17 and 2.20 account for the mass added to the solution as dimorphs A and B denucleate. As we are not considering homogeneous nucleation, the change (decrease) in crystal numbers is due solely to denucleation. The same solute produces the two polymorphs, so the mass balance is
CA(1)(θ) + CB(1)(θ) + S(θ) ) CA0(1) + CB0(1) + S0 (2.21)
[1/v(θ)] ∂[v(θ)CB(ξ,θ)]/∂θ ) S(θ) exp(-B/Θ)[-ξλCB(ξ,θ) + (ξ - 1)λCB(ξ - 1,θ)] ξ exp[-(hB + B)/Θ] exp[wB(Θ λ
-1
-1/3
- 1)ξ
]CB
The expressions19 for the number of monomers in the critical nucleus are
(ξ,θ) + (ξ + 1) exp[-(hB + B)/Θ] λ
exp[wB(Θ
-1
-1/3
- 1)(ξ + 1)
]CB(ξ + 1,θ) JBδ(ξ - ξB*) (2.16)
with
[1/v(θ)] ∂[v(θ)S(θ)]/dθ ) -S(θ)[exp(-A/Θ)CA(λ) + exp(-B/Θ)CB(λ)] + exp[-(hA + A)/Θ] exp[wA(Θ-1 - 1)(CAavg)-1/3]CA(λ) + exp[-(hB + B)/Θ] exp[wB(Θ-1 - 1) (CBavg)-1/3]CB(λ)} + JAξA* + JBξB* (2.17) The independent variables are time θ and crystal size ξ and thus need no subscripts. Supersaturation S(θ), temperature Θ(θ), and volume v(θ) are time dependent variables common to the two polymorphs and are also unsubscripted. For a linear volume decrease, v(θ) ) 1 - ψθ, differentiating with respect to θ in eqs 2.152.17 gives
∂CA(ξ,θ)/∂θ ) S(θ) exp(-A/Θ)[-ξλCA(ξ,θ) + (ξ - 1)λCA(ξ - 1,θ)] ξλ exp[-(hA + A)/Θ] exp[wA(Θ-1 - 1)ξ-1/3]CA(ξ,θ) + (ξ + 1)λ exp[-(hA + A)/Θ] exp[wA(Θ-1 - 1)(ξ + 1)-1/3]CA(ξ + 1,θ) + ψCA(ξ,θ)/(1 - ψθ) - JAδ(ξ - ξA*) (2.18)
ξA* ) [wA(Θ-1 - 1)/(ln S + hA/Θ)]3
(2.22)
ξB* ) [wB(Θ-1 - 1)/(ln S + hB/Θ)]3
(2.23)
and
Because the critical nucleus size increases with the cube of w, larger values of w imply much greater instability (increased denucleation) during growth and coarsening. The effect of h in the denominator is also significant in modulating denucleation when hB/Θ is larger than ln S. III. Computations and Discussion of Results The governing equations, eqs 2.18-2.20, are simultaneous differential equations that we solved by a Runge-Kutta technique with an adaptive time step.19 The dimorph distributions, CA(ξ,θ) and CB(ξ,θ), were evaluated at each time step sequentially. The mass variable (ξ) was divided into 5000 intervals, and the adaptive time (θ) step varied from 0.001 to 0.1. The mass balance was confirmed at every step by comparing the computed value of S to its value from the mass balance, eq 2.21. If the two values are within a tolerance of 0.01, the iteration is permitted to continue. As discussed in earlier work,19 the ratio S ) m(0)/µ∞ does not decrease to unity, but the temperature dependent supersaturation ratio (Seq), defined in eq 2.13, does, thus satisfying the thermodynamic condition for equilibrium. For the less stable polymorph, the scaled particle number and
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mass concentrations, C(0) and C(1), decrease, and the average mass, C(1)/C(0) ) Cavg, increases during growth and ripening. The polydispersity, Cpd, of the coarsening distribution evolves to unity, approaching a single particle after a very long time.18,19 For polymorphs A and B, the initial exponential distributions with smallest cluster mass, ξA0* or ξB0*, are (with appropriate subscripts)
CA0(ξ) ) [CA0(0)/βA0] exp[-(ξ - ξA0*)/βA0] (3.1a) and
CB0(ξ) ) [CB0(0)/β B0] exp[-(ξ - ξB0*)/βB0] (3.1b) The moments (eq 2.14) can be readily evaluated n
CA0(n) ) CA0(0)
(nj)j!ξA0*n-jβA0j ∑ j)0
(3.2)
and a similar expression for polymorph B is found, so that
CA0avg ) βA0 + ξA0*, CA0var ) βA02, CB0avg ) βB0 + ξB0*, and CB0var ) βB02 (3.3) The computations are for the case with dimorph A growing exclusively on A seeds and B growing on B seeds. Initial seeding conditions were chosen to emphasize how the less stable B (e.g., wB > wA) loses in the growth and coarsening competition. With CA0(0) ) 1 and CB0(0) ) 2, the number concentration of seeds of B is double that of A. If we also take CA0avg ) 100 and CB0avg ) 50, then the initial masses of A and B are equal, CA0(1) ) CB0(1) ) 100. Of course, the dynamic details depend on the specific values of parameters and initial values, but the general behavior is the same. For the three properties w, , and h, when one B value is larger than the A value, the B number concentration declines to zero approximately with the logarithm of time for the above initial conditions. We are primarily interested in examining how the parameters, temperature (Θ), transition energy (h), activation energy (), and interfacial energy (w), influence the growth of the polymorphs. A reference temperature scales the absolute temperature so that the reduced temperature obeys 0 < Θ < 1. The molar energy of the phase transition, ∆H, similar to a heat of crystallization, is usually in the range of 1-3 kcal/mol;26 therefore, we have chosen h ()∆H/RTc) to be about 1 for both the polymorphs. The scaled activation energy for diffusion, , is usually smaller than the molar energy of phase transition, h (e.g., for the ripening of precipitated amorphous alumina gel).27 However, can be comparable or greater than h (e.g., for ripening of metallic grains).28 The values of w can be directly calculated from the fundamental parameters given by eq 2.13, which for vapor-liquid systems29,30 range from 2 (methanol at 350 K) to 33 (mercury at 290 K). For solids, we have chosen values of w ) 1-3. When the exponent λ for the rate coefficients is changed from 0 to 1.0, the profiles have the same appearance, and only the time scale is changed; hence, these results are not shown. For the computational results presented below,
Figure 1. Effect of (a) wB ) 2 and (b) wB ) 3 at temperature Θ ) 0.5 with wA ) 1, hA ) hB ) 1.0. For all graphs, the scaled mass concentrations, CA(1) and CB(1), are solid lines; number concentrations, CA(0) and CB(0), are dotted lines. Unless otherwise stated, the initial conditions are an exponential distribution (eq 3.1) with CA0(0) ) 1, CB0(0) ) 2, CA0avg ) 100, CB0avg ) 50, and S0 ) 5 with parameters λ ) 0 and Α ) Β ) 0.01.
the reduced time θ varies from 1 to 1000 (i.e., over 3 orders of magnitude). Figures 1-5 show the influence of the parameters for constant (isothermal) systems; Figures 6-10 deal with programmed (evaporative or heated or cooled) systems. For all graphs, the scaled mass concentrations, CA(1) and CB(1), are solid lines; number concentrations, CA(0) and CB(0), are dotted lines. Unless otherwise stated, the initial conditions are an exponential distribution (eq 3.1a,b) with CA0(0) ) 1, CB0(0) ) 2, CA0(1) ) 100, CB0(1) ) 100, and S0 ) 5 with parameters λ ) 0 and Α ) Β ) 0.01. The effect of interfacial energy on the evolution of A and B number and mass concentrations is displayed in Figure 1. The concentration of the less stable B (wB > wA) decreases almost with the logarithm of time, while the concentration of A is nearly constant. The mass of A increases at the expense of B. The average size of A and B (the ratio of mass to number concentration) both increase, approaching the well-known asymptotic power law time dependence.18,19 Because the B polymorph has a larger critical nucleus than A, and is more soluble for a given size, the B crystals eventually vanish, as shown in Figure 1b. The average size of B diverges to infinity because the number concentration is less than the mass concentration. The effect of transition energy (heat of crystallization) and thus solubility is shown in Figure 2. Smaller values
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Figure 2. Effect of (a) hB ) 0.75 and (b) hB ) 0.9 at temperature Θ ) 0.5 with wA ) wB ) 1.0, A ) 0.01, and hA ) 1.0.
Figure 3. Effect of Α . Β at temperature Θ ) 0.5 with wA ) wB ) 1.0, Α ) 100, Β ) 0.01, and hA ) hB ) 1.0.
of h imply higher solubility. In Figure 2a,b, hB < hA, so that B is more soluble than A. Again, the decline of B concentration is nearly proportional to log θ, eventually falling to zero. The computations corroborate the wellknown principle11 that solubility is significant in determining polymorph separation. The effect of activation energy is shown in Figure 3. Despite the widely different values of B ()0.01) , A ()100), the influence is minimal. This implies that the activation energy of rate coefficients may not play an important role in crystallization kinetics for polymorph separation. Because the values of and h for A and B are equal in Figure 3, number concentrations for both A and B decrease. In all the other figures, parameters w and h are chosen so that B is less stable; hence, its number concentration decreases, while A is nearly constant.
Madras and McCoy
The influence of the ratio CB0(0)/CA0(0) is shown in Figure 4 for equal initial mass concentrations of A and B. As the ratio increases, the time dependence for the decline of the B number concentration is still logarithmic, but the time when B vanishes is smaller. The mass difference between A and B increases significantly until B is depleted. As before, the reason for the behavior is that B crystals are less stable and thus denucleate, while A crystals grow. To further illustrate the seeding effect, we examine the evolution of the polymorphs with the initial conditions such that the average sizes of both A and B are equal ()25) but have a different number and mass concentration (Figure 5). When B is initially in excess, its number and mass concentrations decrease. For example, the initial number concentration and average size of polymorph B in Figure 5c are 10 and 25, and the number concentration and the average size of polymorph B at θ ) 1000 are nearly 2 and 100, respectively. Thus, even when A is more stable (wA < wB), the polymorph B is favored because of the excess seeding. Similar to Figure 4, when the ratio of CB0(0)/CA0(0) increases, the time dependence for the decline of B is still logarithmic, but more B is present, and the time when it vanishes is larger. As observed in practice, heterogeneous nuclei introduced accidentally are extremely small and therefore potentially in enormous abundance, even when careful measures are imposed to avoid such contamination. Yu observed13 that one polymorphic form may deposit on the nuclei of its associated form. We can simulate the effect of indiscriminant deposition of a polymorphic form (B) on seeds composed of its associated form (A) simply by changing the subscript A to B in eqs 2.182.20. Or if both A and B deposit opportunistically on A, then for computational purposes we would let the initial distribution of A equal that of B. The method for selective polymorph crystallization by evaporation of the solvent is commonly applied.11 This process of evaporative crystallization is illustrated in Figure 6 with wA ) 1 and wB ) 2. The scaled evaporation rate, ψ, is larger by a multiple of 10 in Figure 6b as compared to Figure 6a. When all other parameters remain constant, increasing the evaporation rate causes the less stable B to disappear more rapidly. A similar effect is observed if wA ) wB and hB < hA. Temperature can be varied to control crystallization. We propose temperature programs that either decrease or increase with time according to
Θ ) 0.95 - 0.9[1 - exp(-Rθ)]
(3.4)
Θ ) 0.05 + 0.9[1 - exp(-Rθ)]
(3.5)
or
With R in the range of 0.01-0.1, most of the temperature change occurs in the time range 10-100. Figures 7 and 8 show the effect of cooling and heating, respectively, for polymorphs of different interfacial energies, wA < wB. For both heating and cooling programs, the B form rapidly decreases in number concentration, while the A form is relatively constant. The mass concentrations for A and B, respectively, increase and decrease. An interesting observation is that
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Figure 4. Effect of CB0(0)/CA0(0) with (a) CB0(0) ) 2; (b) CB0(0) ) 5; and (c) CB0(0) ) 10 at temperature Θ ) 0.5 with wA ) 1, wB ) 2, and hA ) hB ) 1.0. The initial condition is an exponential distribution (eq 3.1) with CA0(0) ) 1 and CA0(1) ) CB0(1) ) 100.
Figure 5. Effect of CB0(0)/CA0(0) with (a) CB0(0) ) 2; (b) CB0(0) ) 5; and (c) CB0(0) ) 10 at temperature Θ ) 0.5 with wA ) 1, wB ) 2, and hA ) hB ) 1.0. The initial condition is an exponential distribution (eq 3.1) with CA0(0) ) 1 and CA0avg ) CB0avg ) 25.
the slower heating and cooling rates are more effective in separating the dimorphs.
Cooling crystallization effects are usually explained by the relative solubilities, whose temperature influ-
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Figure 6. Effect of evaporation rate (a) ψ ) 10-4 and (b) ψ ) 10-3 at temperature Θ ) 0.5 with wA ) 1, wB ) 2, and hA ) hB ) 1.0.
Madras and McCoy
Figure 8. Effect of heating, Θ ) 0.05 + 0.9[1 - exp(-Rθ)] with (a) R ) 0.01 and (b) R ) 0.1 with wA ) 1, wB ) 2, and hA ) hB ) 1.0.
Figure 7. Effect of cooling, Θ ) 0.95 - 0.9[1 - exp(-Rθ)] with R ) 0.01, wA ) 1, wB ) 2, and hA ) hB ) 1.0.
ences depend on hA and hB. Figure 9 shows the effect of cooling for polymorphs with hA ) 1.0 and hB ) 0.75. Both mass and number concentrations of B decrease during the temperature change. Figure 10 shows the effect of heating with hA ) 1.0 and hB ) 0.75. In this case, both mass and number concentrations of B vanish at definite times less than θ ) 1000. For both heating and cooling programs, the number concentration of A is nearly constant, while its mass concentration increases. In both cases, the slower heating or cooling rate allows more time for the rate processes to take place and is more effective in separating polymorphs. We conclude that differences in either solubility or interfacial parameters allow separation of polymorphs and that slower processes are optimal. The dynamic temperature computations are useful in interpreting pro-
Figure 9. Effect of cooling, Θ ) 0.95 - 0.9[1 - exp(-Rθ)] with (a) R ) 0.01 and (b) R ) 0.1 with wA ) wB ) 1, hA ) 1.0, and hB ) 0.75.
cesses such as cooling crystallization. The effect of temperature changes on nucleation will be investigated in future work.
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included in the computations to explore heating or cooling as process control methods. The rate of solvent evaporation during crystallization was also incorporated as a possible method to control polymorph preparation. The results may be useful not only in explaining observations but also in guiding manufacturing strategies. Acknowledgment. We thank Darren Oufnac for helpful discussions. Notation c C Figure 10. Effect of heating, Θ ) 0.05 + 0.9[1 - exp(-Rθ)] with R ) 0.01, wA ) wB ) 1, hA ) 1.0, and hB ) 0.75.
IV. Conclusion A consistent theme in our studies of particulate systems is the foundation in distribution kinetics and dynamics. The distribution of sizes is an obvious characteristic of particulates, which if ignored glosses over fundamental principles, such as the Gibbs-Thomson effect and instability of subcritical clusters. Crystal growth and coarsening are crucial in the evolution of a crystallizing solution or melt. The time dependence of a particulate size distribution is governed by a population dynamics equation that describes these essential processes. The integro-differential equation (for continuous distributions) can assume other guises, including moment, finite difference, or partial-differential Fokker-Planck (convective) forms. Each form provides insight for specific cases into the kinetics and dynamics of the evolving distribution. Most analyses of polymorph crystallization are based on equilibrium thermodynamic relationships and kinetic rate effects growing out of classical nucleation theory and conventional crystal growth kinetics. The kinetics and dynamics of particle size distributions as influenced by surface curvature (Gibbs-Thomson effect) are typically avoided, except in classical Ostwald ripening studies. Our goal has been to provide a fundamental explanation for the range of polymorph crystallization observations through a distribution kinetics and population dynamics theory. The model accounts for the Gibbs-Thomson influence that favors growth of larger particles and the dissolution of unstable particles smaller than critical nucleus size. Denucleation can eliminate unstable forms and is therefore particularly important in polymorph crystallization. The approach advocated here allows results to be computed readily for a wide range of physicochemical and operating parameters. Computations of this kind show a variety of behavior for polymorph crystallization. The interfacial and crystallization energies, but not the activation energies, are key parameters in the separation of polymorphs. According to the results discussed in this paper, the less stable form will decline in number and mass concentration, while the more stable forms will grow. The critical nucleation size not only determines which polymorph will denucleate first (the least stable) but is expected also to determine which polymorph will nucleate first (the most stable). The temperature dependence was
E h ) ∆H/RTc ∆H I J kB m(0)(t) m∞(0) r r* S ) m(0)/µ∞ Seq ) m(0)/meq(0) T Tc v w ) (3xm/4πFc)-1/32σo xm/FckBTc x xm x* Greek Symbols ) E/RTc θ Ω ) w(Θ-1 - 1)/ξ1/3 σ λ F ) r/r* Fc ξ ) x/xm µ∞ Θ ψ
cluster size distribution dimensionless cluster size distribution activation energy for diffusion dimensionless transition energy molar energy of the phase transition denucleation rate dimensionless denucleation rate Boltzmann’s constant molar concentration of solute as a function of time molar concentration of solute in equilibrium with plane cluster surface radius of the cluster radius of critical nucleus supersaturation supersaturation absolute temperature reference temperature volume Gibbs-Thomson ratio of interfacial to thermal energy for a cluster of monomer size cluster mass monomer mass mass of critical nucleus dimensionless activation energy dimensionless time Gibbs-Thomson factor interfacial free energy exponent on mass in rate coefficient expression dimensionless cluster radius cluster mass density dimensionless cluster mass high-temperature solubility T/Tc, dimensionless temperature parameter for linear volume decrease
Subscript and Superscripts n superscript indicating nth mass moment of cluster size distribution 0 subscript indicating initial condition A subscript indicating polymorph A B subscript indicating polymorph B
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