Temperature effects on the transition from nucleation and growth to Ostwald ripening

Chemical Engineering Science 59 (2004) 2753 – 2765

www.elsevier.com/locate/ces

Temperature e#ects on the transition from nucleation and growth to Ostwald ripening Giridhar Madrasa;∗ , Benjamin J. McCoyb a Department

b Department

of Chemical Engineering, Indian Institute of Science, Bangalore 560 012, India of Chemical Engineering, Louisiana State University, Baton Rouge, LA 70803, USA

Received 27 August 2003; received in revised form 17 March 2004; accepted 25 March 2004

Abstract Condensation, involving nucleation, growth, and ripening from a metastable state, is an important but complex phase transition process. The e#ect of physical parameters, including temperature, on condensation dynamics, the competition between homogeneous and heterogeneous (seeding) nucleation, and the separation of polymorphs are among several issues of practical interest. We present a model based on population dynamics that describes the time evolution of the particle size distributions for condensation of the 6uid phase and consequent decline in supersaturation. The crucial e#ect of interfacial curvature on energy, and hence on particle size (Gibbs–Thomson e#ect), causes larger particles to be less soluble, so that smaller particles dissolve and eventually vanish (denucleate). Numerical solutions of the governing equations show the transition from nucleation and growth to ripening occurs over a relatively long time period. The in6uence of temperature on these phenomena is primarily through its e#ect on interfacial energy, growth rate coe9cients, and equilibrium solubility. Temperature programming is proposed as a potential method to control the size distribution during the phase transition. The model suggests conditions to suppress homogeneous nucleation by seeding. We also explore how a temperature program for cooling crystallization based on di#erent properties of the crystal forms can separate two crystal polymorphs. ? 2004 Elsevier Ltd. All rights reserved. Keywords: Crystal size distributions; Crystal growth; Denucleation; Distribution kinetics; Nucleation; Ostwald ripening

1. Introduction Nucleation accompanied by growth and followed by Ostwald ripening constitute the processes during the formation of a condensed phase from a metastable phase (Stanley, 1971; Marqusee and Ross, 1983). In a closed system, clusters nucleate (either homo- or heterogeneously) and then grow while supersaturation correspondingly decreases. The condensation of the 6uid phase and consequent decline in supersaturation leads to an increase in the critical nucleus size, below which clusters are unstable and tend to disintegrate spontaneously. The e#ect of interfacial curvature on energy, and hence of particle size, causes larger particles to be less soluble, so that smaller particles dissolve and eventually vanish (denucleate). The mass of these dissipating particles is available to grow larger particles and coarsen

∗ Corresponding author. Tel.: +91-80-309-2321; fax: +91-80-360-0683. E-mail address: [email protected] (G. Madras).

0009-2509/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2004.03.022

the size distribution—the phenomenon known as Ostwald ripening. These processes often overlap during the evolution from a supersaturated phase to the Dnal state wherein a single condensed ripened particle remains in equilibrium with the noncondensed phase. A metastable supersaturated phase will not only generate nuclei (McClurg and Flagan, 1998) but will also cause deposition on the generated nuclei and consequent growth of the clusters (McCoy, 2001a, b). By this interpretation, nucleation without accompanying growth is impossible (McCoy, 2000). Due to growth, the supersaturation decreases, the critical nucleus size increases, and coarsening must also occur. The nucleation rate and the driving force for growth (given by the di#erence between the supersaturation and its equilibrium value) decrease. As growth and nucleation rates decline, the Gibbs–Thomson e#ect becomes signiDcant so that smaller clusters dissolve and transfer their mass to larger clusters that grow. When smaller clusters shrink to their critical nucleus size, they become thermodynamically unstable, and spontaneously disintegrate (Madras and McCoy, 2001, 2002a–c). As smaller clusters continue to shrink and disappear, the cluster number decreases, the

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G. Madras, B.J. McCoy / Chemical Engineering Science 59 (2004) 2753 – 2765

cluster mass shows a power law increase, and the cluster distribution narrows and approaches a single remaining particle at equilibrium (Madras and McCoy, 2001). We (Madras and McCoy, 2003d) have shown how the classic LSW model and subsequent enhancements of the model correctly depict the asymptotic time dependence of particle number concentration and average particle size, but often approximate the higher moments of the particle size distribution. The time evolutions of the average particle sizes determined by our theory match with that of the well-accepted asymptotic results of LSW at long times but are di#erent at short times (Madras and McCoy, 2003d). Though the topic of nucleation, growth, and ripening has been studied at length (Ratke and Voorhees, 2002), temperature e#ects are often not considered in detail. The e#ect of temperature on concurrent nucleation, growth, and coarsening was investigated for the spatial decomposition of immiscible alloys by phase separation during cooling (Ratke and Diefenbach, 1995), and for the microstructure evolution under solidiDcation conditions of various alloys (Zhao et al., 2000; Zhao and Ratke, 2004). The e#ect of di#erent thermal treatments on carbon containing two-phase titanium aluminide alloys to obtain precipitation hardening e#ects has also been investigated (Christoph et al., 1997; Appel and Wagner, 1998). In the current work, we present the full temperature dependence for crystallization dynamics. The new model provides a simple and e#ective technique to calculate important aspects of temperature processing, such as continuous cooling, which would be more di9cult to determine by the previous approaches mentioned above. The temperature a#ects all the participating and competing processes as the system proceeds toward its asymptotic (equilibrium) behavior. Knowing the temperature in6uence is of serious practical interest in industrial applications of phase condensation. Advanced materials synthesis frequently involves homogeneous nucleation from solution and consequent growth of crystals (Mintova et al., 1999). Avoiding homogeneous nucleation, however, is usually a requirement in industrial crystallization (Genck, 2003), where seeding (heterogeneous nucleation) is practiced. Understanding the e#ect of temperature, the key parameter in6uencing these processes, is essential for their rational design, construction, and operation. Exploiting the competition between homoand heterogeneous nucleation to control the size of crystals is a frequently used strategy. Control of ripening can also facilitate the development of self-assembling structures in materials and pharmaceutical synthesis (Medeiros-Ribeiro et al., 1998). The aim of this study is to explore the possibility that temperature programming can provide a way to tailor the particle distribution during nucleation, growth, and ripening. Our approach is to represent the dynamics of these processes by population balance equations (PBEs) for the crystal size distribution (CSD) and monomer mass distributions. The PBEs have an accumulation (time derivative) term, addition and dissociation terms, as well as source (or sink)

terms to represent nucleation (or denucleation). The PBE can be converted to a Fokker–Planck equation by an expansion valid for a small ratio of monomer to crystal mass. We have shown (Madras and McCoy, 2002c) how nucleation, growth, and ripening for a single crystalline form at constant temperature can be combined into one distribution kinetics theory. The mathematical model showed how homogeneous or heterogeneous nucleation preceded growth by deposition of solute, and was followed by Ostwald ripening. Although a primary interest is the liquid–solid condensation, the model can be readily adapted to the vapor–liquid phase transition. Temperature need not be constant when condensation occurs, for example, during cooling crystallization (Beckmann, 2000; Mohan et al., 2001). Heating has been proposed (Lewiner et al., 2002) as a method to dissolve small particles. Describing the quantitative e#ects of such temperature variations requires knowledge of the temperature dependence of all parameters. The temperature e#ects incorporated in the present model include the di#usion-in6uenced growth coe9cient, the Gibbs–Thomson e#ect of particle curvature on equilibrium solubility, the phase transition energy (heat of solidiDcation or vaporization), the critical nucleus size, and interfacial energy (surface tension). The dissolution rate coe9cient is related to the growth rate coe9cient by microscopic reversibility, thereby determining its temperature dependence. In earlier papers, we have presented the e#ect of temperature on crystal growth (Madras and McCoy, 2003a) and Ostwald ripening (Madras and McCoy, 2003b), whereas in the present work we focus on describing the temperature e#ects during the complete time range for transition from a homogeneous supersaturated phase through the Drst appearance of nuclei, subsequent monomer deposition and growth, and eventual ripening to a single large remaining particle. We examine the e#ect of temperature on the evolution of polymorphs. Producing pure crystalline forms of active pharmaceutical ingredients (Rouhi, 2003) is an enormous challenge because crystals of identical chemical composition frequently have di#erent forms called polymorphs (Mohan et al., 2001; Laird, 2000). Polymorphs di#er in their properties such as solubility, dissolution rate, and chemical and physical stability. During competitive crystallization, wherein both the polymorphs compete for the same substrate, one form of the polymorph may be preferred over the other (Bernstein et al., 1999). Even when crystals are fully formed they may undergo Ostwald ripening owing to the Gibbs–Thomson e#ect. 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. Many studies of polymorph crystallization, however, are qualitative or semi-quantitative descriptions. This is primarily because the subject is complex, requiring an integration of thermodynamics, kinetics, reactor dynamics, population balances, and interfacial science

G. Madras, B.J. McCoy / Chemical Engineering Science 59 (2004) 2753 – 2765

(Madras and McCoy, 2003c). Approximate models focusing on nucleation, growth, and dissolution kinetics, but ignoring Gibbs–Thomson and ripening e#ects, have been proposed. Cooling crystallization is a common method to prepare and separate polymorphs, and elaborate procedures for controlling the rate of cooling (Beckmann, 2000) during seeding have been recommended to manage polymorph production. Thus, temperature plays a crucial role in the growth and separation of polymorphs. Here, we report how cooling can in6uence the growth and ripening of dimorphs and suggest strategies for optimally producing a particular polymorph when homogeneous nucleation occurs. The current paper develops and evaluates these concepts in the following sections. The elements of distribution kinetics through population balance equations are introduced in Section 2. We extend a previous paper (Madras and McCoy, 2002c) by incorporating temperature e#ects, and thus utilize dimensionless governing equations already derived. Changes are presented to include the temperature dependences of the participating phenomena. The theory so outlined in Section 2 is applicable to vapor–liquid or liquid –solid transitions. We present the numerical analysis of the population dynamics equation (Section 3), and discuss the results and provide comparisons with experimental observations (Section 4) along with conclusions (Section 5). 2. Theory Classical homogeneous nucleation theory in the capillarity approximation (McClurg and Flagan, 1998) utilizes the sum of surface energy and formation free energy for a spherical cluster of radius r, W (x) = 4r 2  − (4=3)r 3 (=xm )RT ln S:

(2.1)

Here,  is the cluster interfacial energy and NG = −RT ln S is the chemical potential di#erence between the two phases in terms of supersaturation, S. While classical nucleation theory is reasonably accurate in determining the dependence of the nucleation rate on supersaturation, it can be improved in determining the dependence of the nucleation rate on temperature (Wolk et al., 2002). The cluster mass x is related to the condensed-phase mass density  and cluster radius r by x =(4=3)r 3 . For a spherical particle, the local-equilibrium interfacial concentration at the particle surface is given by the Gibbs–Thomson equation in terms of the solubility of a 6at surface, m(0) ∞, (0) m(0) eq = m∞ exp():

(2.2)

cluster of radius r; W (r), reaches a maximum value, W ∗ , at the critical cluster radius, r ∗ , r ∗ = 2xm =(RT ln S):

(2.4)

Thus by Eq. (2.1) the maximum is 2 W ∗ = (16=3)xm2 3 =[RT ln (m(0) =m(0) ∞ )] :

(2.5)

The classical expression (Adamson, 1990; McClurg and Flagan, 1998) for the nucleation rate (moles of nuclei/vol time) is the 6ux over the maximum energy barrier (at r=r ∗ ), Inuc = knuc exp(−W ∗ =RT )

(2.6)

with the prefactor knuc = (m(0) )2 (2xm =NA )1=2 =

(2.7)

expressed in terms of Avogadro’s number, NA . Classical nucleation theory prescribes that nuclei of identical size, depending on the parameters in Eqs. (2.5)–(2.7), form in the system, and are then free to grow as equilibrium is approached. The deposition or condensation process by which monomers of mass x = xm are reversibly added to or dissociated from an established particle of mass x can be written as the reaction-like process (Madras and McCoy, 2001), kg (x)

C(x) + M(x )
C(x + x ); kd (x)

(2.8)

where C(x) represents the particle of mass x and M(x =xm ) is the monomer with a distribution written as m(x; t) = m(0) (t)(x − xm ). An activation energy for the growth coefDcient is assumed to account for activated di#usion, kg (x) = x exp(−E=RT ) which is simply related to kd (x) by microscopic reversibility (Madras and McCoy, 2002c). Values of  = 0; 1=3, and 2=3 (Madras and McCoy, 2002a) represent surface-area-independent ripening, di#usion-controlled ripening, and surface-controlled ripening, respectively. The temperature dependence for growth and ripening is in6uenced by the thermodynamic properties. The interfacial curvature e#ect is prescribed by the Gibbs–Thomson equation (Eq. (2.2)). The temperature dependence of the equilibrium solubility in Eq. (2.2) is given by m(0) ∞ = ∞ exp(−NH=RT );

(2.9)

where NH is the molar energy of the phase transition and ∞ is the 6at-surface equilibrium solubility at large T . Following previous work (Madras and McCoy, 2002c, 2003c), we deDne the dimensionless quantities as follows: C = cxm =∞ ; C (n) = c(n) =∞ xmn ; ! = x=xm ; " = t∞ xm ; S = m(0) =∞ ; Seq = S exp(h=$ − );  = !=!1=3 ; w = (3xm =4c )−1=3 20 xm =kB Tc ; $ = T=Tc ;

Consistent with Eq. (2.1), we write  = 2xm =rRT

2755

(2.3)

in terms of monomer molar volume v = xm =, interfacial energy , the gas constant R, and absolute temperature T . Thus smaller particles are more soluble than larger particles. For a supersaturated (metastable) system, the energy of a

2  J = I=(∞ xm ); h = NH=RTc ; ) = E=RTc ;

(2.10)

where c(x; t) is the distribution of particles of mass x at time t. The scaled moments of the distribution are deDned as
∞ C (n) (") = C(!; ")!n d!; (2.11) 0

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G. Madras, B.J. McCoy / Chemical Engineering Science 59 (2004) 2753 – 2765

where ! as deDned in Eq. (2.10) is the number of monomers in the particle. The zeroth moment, C (0) ("), and the Drst moment, C (1) ("), are the time-dependent (scaled) molar (or number) concentration of particles and the particle mass concentration, respectively. The ratio of the two is the average particle mass, C avg = C (1) =C (0) . If  is assumed to vary linearly with temperature (Reid et al., 1977),  = 0 (1 − $), causing  to vanish at the reference temperature, then ! = w($−1 − 1). Some studies (Lancia et al., 1999) report the interfacial energy to be temperature independent, in which case, ! = w$−1 . In Eq. (2.10), ! represents the number of monomers in the particle and $ is the reduced temperature (0 ¡ $ ¡ 1). The ratio S is deDned relative to the high-temperature solubility ∞ , rather than to the plane-surface solubility m(0) ∞ as in earlier studies (Madras and McCoy, 2001, 2002a). The supersaturation ratio deDned as Seq = m(0) =m(0) eq evolves to unity at thermodynamic equilibrium. The scaled number (or moles) of particles, C (0) = c(0) =∞ , is also in units of the solubility ∞ . The Gibbs–Thomson factor , Eq. (2.10), is expressed in terms of the scaled interfacial energy !, which depends on $. The dimensionless equations for crystal growth and nucleation (Madras and McCoy, 2003c) are @C(!; ")=@" =S(")exp(−)=$)[ − ! C(!; ") + (! − 1) C(! − 1; ")] − ! exp[ − (h + ))=$] exp[!!−1=3 ]C(!; ") + (! + 1) exp[ − (h + ))=$] ×exp[!(!+1)−1=3 ]C(!+1; ")+J(!−!∗ )

(2.12)

and dS(")=d" = exp(−)=$)[ − S(") + exp(−h=$) ×exp[!(C avg )−1=3 ]]C () + J!∗ :

(2.12a)

The initial conditions are S(" = 0) = S0 and C(!; " = 0) = C0 (!). The term J(! − !∗ ) in Eq. (2.12) represents the rate of nucleation of particles of mass !∗ and is a source term. When ripening occurs, it is a sink term representing the rate of denucleation when particles shrink to their critical size, !∗ , and then spontaneously vanish. Because the rate coefDcients are related by microscopic reversibility, Eq. (2.12) provides the required thermodynamic equilibrium, Seq = 1 when dS=d" = 0 and J = 0. The number of monomers in the critical nucleus (Madras and McCoy, 2003b) is !∗ = [!=(ln S + h=$)]3 ;

(2.13)

which varies with time because of the time dependence of supersaturation, S. During cooling or heating, $ is a function of time and also in6uences the critical nuclei size. As explained before, depending on whether the interfacial energy is linearly dependent on temperature or independent of temperature, ! will be equal to w($−1 − 1) or w$−1 ,

respectively. The scaled mass balance for a closed (batch) system follows from Eqs. (2.12) and (2.12a), C (1) (") + S(") = C0(1) + S0 ;

(2.14)

where C0(1) is the initial mass of particles, representing heterogeneous nuclei or seeds. If we deDne the prefactor by Eq. (2.7), then 2  J0 = knuc =(∞ xm ):

(2.15)

Substituting knuc from Eq. (2.7), the resulting equation for Eq. (2.8) is J0 = J00 (1 − $)1=2 S 2 ;

(2.16)

where J00 = (20 xm1−2 =NA 2 2 )1=2 :

(2.17)

Thus the homogeneous nucleation rate is J = J00 (1 − $)1=2 S 2 exp[ − !3 ={2(ln S + (h=$))2 }]: (2.18) Di#erent polymorphs may have di#erent solubility, interfacial energies, dissolution rate, and chemical and physical stability (Madras and McCoy, 2003c). For A and B polymorphs we assume that di#erences are determined by the interfacial energies, wA and wB , or activation energies, )A and )A , or phase transition energies, hA and hB . Because the same solute produces the two polymorphs, and the quantity S is a dimensionless solute concentration, we have deDned S to be the same for both polymorphs (Madras and McCoy, 2003c). Thus, in Eq. (2.10), we deDne S as the ratio of solute concentration to ∞ , which is related in Eq. (2.9) to the solubility. Clearly, ∞ is the solubility of either polymorph when the temperature is very large. We have assumed that ∞ is identical for the two polymorphs, even though the low-temperature supersaturation ratio, Seq , which is also deDned in Eq. (2.10), does indeed depend on the interfacial and phase transition energies. The quantity S is written in the equations for the polymorphs rather than writing SeqA and SeqB . With appropriate subscripts we distinguish between the two distributions, CA (!; ") and CB (!; "), which have moments deDned as in Eq. (2.11). The mass balance is (1) (1) + CB0 + S0 : CA(1) (") + CB(1) (") + S(") = CA0

(2.19)

The governing population balance equations are written as @CA (!; ")=@" =S(") exp(−)A =$)[ − ! CA (!; ") + (! − 1) CA (! − 1; ")] − ! exp[ − (hA + )A )=$] exp[!A !−1=3 ]CA (!; ") + (! + 1) exp[ − (hA + )A )=$] exp[!A (! + 1)−1=3 ] ×CA (! + 1; ") − JA (! − !∗A )

(2.20)

G. Madras, B.J. McCoy / Chemical Engineering Science 59 (2004) 2753 – 2765

and @CB (!; ")=@" =S(") exp(−)B =$)[ − ! CB (!; ") + (! − 1) CB (! − 1; ")] − ! exp[ − (hB + )B )=$] exp[!B !−1=3 ]CB (!; ") + (! + 1) exp[ − (hB + )B )=$] exp[!B (! + 1)−1=3 ] ×CB (! + 1; ") − JB (! − !∗B )

(2.21)

with @S(")=d" = −S(")[exp(−)A =$)CA() + exp(−)B =$)CB() ] + exp[ − (hA + )A )=$] exp[wA ($−1 − 1) ×(CAavg )−1=3 ]CA() + exp[ − (hB + )B )=$] exp[wB ($−1 − 1) ×(CBavg )−1=3 ]CB() + JA !∗A + JB !∗B :

(2.22)

The initial conditions are S(" = 0) = S0 ; CA (!; " = 0) = CA0 (!), and CB (!; " = 0) = CB0 (!). The terms JA !∗A + JB !∗B in Eq. (2.22) account for the mass added to the solution as polymorphs A and B denucleate.

3. Numerical solution The general problem of unsteady-state, isothermal nucleation and growth with subsequent ripening in a closed vessel requires a numerical solution. The numerical scheme is similar to that of the ripening problem (Madras and McCoy, 2001), except that initially we have homogeneous nucleation with an explicit expression for I (or J ) given by Eq. (2.18). No particles are present initially, unlike the case of ripening alone, for which an initial particle distribution is present (Madras and McCoy, 2001). The di#erential equation (2.12) is solved by a Runge–Kutta technique with an adaptive time step with C(!; ") evaluated sequentially at each time step. The particle moments were calculated by integration of the nonzero CSD from !∗ to ∞. Because C(!; ") lies in the semi-inDnite domain, it was converted to a bounded range (0,1) by the mapping function, !=!∗ +(C avg −!∗ )y=(1−y) with 0 6 y 6 1. This causes y to vary from 0 to 1 when ! varies from !∗ to ∞. By choosing this mapping, we ensure that when y is centered at 0.5, the distribution is centered around C avg (") and is bounded at the lower end by !∗ , which increases as S decreases, according to Eq. (2.22). Choosing this grid provides that the mapping is Dne in the range of prevalent sizes and coarse at very high and very low sizes. It is, therefore, possible to consider a narrow CSD with a few hundred intervals to do the numerical analysis. The mass variable (!) was divided into 1000 intervals and the

2757

adaptive time (") step varied from 0.001 to 0.1 ensuring stability and accuracy at all values of the parameters. At every time step, the mass balance, given by Eq. (2.14), is veriDed. Typically, 30 min were required to run the program on a Compaq AlphaServer. An advantage of the model-based, di#erential equation approach over Monte Carlo or molecular dynamics method is the e9ciency of the computations, and also the clear e#ect of parameters displayed by the governing equations. In the Drst step of the numerical routine, the CSD is zero and nuclei are generated at their critical size according to Eq. (2.18). The Drst terms on the right-hand sides of Eqs. (2.12) and (2.12a) are zero, so that the decrease of S and the increase of C(! = !∗ ; ") can be determined. The moments of the CSD are computed at each iteration, and at the Drst step yield moments of the delta distribution. At the second step, the critical nucleus size has grown owing to the reduced value of S, but the crystals evolving from the initial nuclei have also grown because of monomer deposition. All terms in Eqs. (2.12) contribute to the computation. At each step of the iteration, particles smaller than the critical size dissolve (denucleate) and give back their mass to the uncondensed phase. The denucleation is assumed instantaneous (Madras and McCoy, 2002b), compared to the relatively slow reversible deposition (growth) process based on rate coe9cients, kg and kd . Our choice of parameters is based on published values, although methods for measuring the free interfacial energy between a solid and a liquid have been questioned (Washburn, 1930). For some vapor–liquid and liquid–solid systems, the interfacial energies and ratio w are listed in Table 1. The interfacial energy for vapor–liquid systems is approximately linear with temperature (Reid et al., 1977). The interfacial energy () of a solid–liquid system CaSO4 · 2H2 O does not vary with temperature in the range 298–363 K (Lancia et al., 1999). We have, therefore, discussed the time evolution of the crystal size distribution for both temperature dependencies. The molar energy of the phase transition, NH , is similar to a heat of crystallization and is usually in the range 1–3 kcal=mol (Perry and Green, 1997). We have, therefore, chosen h(=NH=RTc ) to span two orders of magnitude, 0.01–1. The scaled activation energy for di#usion, ), is usually smaller than the molar energy of phase transition, h, e.g., for the ripening of precipitated amorphous alumina gel (Rousseaux et al., 2002). However, ) can be comparable or greater than h for ripening of metallic grains (Baldan, 2002). We have, therefore, chosen ) to span two orders of magnitude, 0.001–0.1. Because the homogeneous nucleation rate is so sensitive to supersaturation S in the exponential, the e#ect of prefactor J00 is much less and is chosen to span two orders of magnitude, 0.001–0.1. A slight increase in S causes a huge increase in the classical nucleation rate (Adamson, 1990), whereas by contrast, the value of J00 in the prefactor in6uences J only linearly.

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G. Madras, B.J. McCoy / Chemical Engineering Science 59 (2004) 2753 – 2765

Table 1 Values of the interfacial energies for various systems  (mJ=m2 )

xm

 (g=cm3 )

Tc (K)

Vapor–liquid Mercury Water Ethanol Benzene

485 72 25.4 37.4

200.6 18 46 78

13.6 1.0 0.79 0.876

1750 647 513 562

5.5 2.5 2.4 4.3

Liquid–crystal Ampicillin CaSO4 · 2H2 O Benzoic acid SrSO4 NaCl in alcohol NaCl in water

5.83 37 15 85 171 276

349 172 122 183.7 58.5 58.5

2.17 2.96 1.32 3.95 1.14 1.14

373 373 373 373 373 373

1.5 4.9 2.7 9.7 20.9 33.8

w

Reference Girshick (1990) Adamson (1990) Yang and Qiu (1986) Yang and Qiu (1986) Ottens et al. (2001) Lancia et al. (1999) Stahl et al. (2001) Adamson (1990) Adamson (1990) Adamson (1990)

As deDned in Eq. (2.10), w = (3xm =4)−1=3 2xm =kB Tc , where Tc is the reference temperature for liquid–crystal systems and critical temperature for vapor–liquid systems.

Evolution of C(0) and C(1)

Evolution of S

0.4 w=5 w=5 6

50

40

40

30

30 C(1)

C(0)

0.3

50

7

0.2

S

0.5

20

20

10

10

w=5 6

7

6 0.1

7

0 1000

0.0 0.1

1

10

100

θ

(a)

0 0.01

Evolution of (de)nucleation rate, J, on log-linear coordinates 0.30

1

w=5 10

10

1000

J

6

-3

10

0.05 7

0.00

100

7 Jnuc

10-2

0.15 J

θ

w=5 6

-1

0.20

Abs(Jdenuc)

-4

10

-0.05

-5

10

-0.10 0.01

(c)

1

Evolution of (de)nucleation rate, J, on log-log coordinates

0.25

0.10

0.1

(b)

0.1

1

θ

10

100

1000

0.01

(d)

0.1

1

θ

10

100

1000

Fig. 1. E#ect of interfacial energy, w, at temperature $ = 0:5 with hA = 1:0 on (a) evolution of C (0) (scaled number of clusters) and C (1) (scaled cluster mass), (b) evolution of scaled solute concentration, S, (c) evolution of (de)nucleation rate, J , on log-linear coordinate, and (d) evolution of (de)nucleation rate, J , on log-log coordinates.

4. Results We are primarily concerned with the e#ect of temperature on the evolution of the CSD. We initially examine how the factors, w; S0 ; J00 ; ), and h in6uence the time evolution of the crystal number and crystal size at a constant temperature of $ = 0:5 (Figs. 1–5). We then investigate the e#ect of seeding (Fig. 6) and temperature on the evolution of the

crystals (Figs. 7–9). After ascertaining that the e#ects of w and h are critical in determining how crystals grow and ripen, we examine the evolution of polymorphs with di#erent w (Fig. 10) and h (Fig. 11) and the e#ect of temperature on the evolution of polymorphs (Figs. 12 and 13). For all graphs, the scaled mass concentrations, CA(1) and CB(1) , are solid lines, and number concentrations, CA(0) and CB(0) , are dotted lines. Unless otherwise stated, the initial conditions are

G. Madras, B.J. McCoy / Chemical Engineering Science 59 (2004) 2753 – 2765

5

50

2759

100

1.2 S0 = 100

4 0.1

70

S0 = 100

0.8

J00 = 0.001

60

50

0.6

50

20

2

40 0.4

10

0

0 1

10

Fig. 2. E#ect of nucleation rate prefactor, J00 , at $ = 0:5 with hA = 1:0 and wA = 5:0.

0.5

50

0.4

40

0 1000

0.5

50

0.4

40 seeded

0.3

30

C(0)

C(0)

100

C(1)

20

0.2

0.1

0.1

10 θ

30

0.1 0.01

0.2

1

Fig. 5. E#ect of initial supersaturation, S0 , at temperature $ = 0:5 with w = 5:0 and h = 1:0.

C(1)

h=1

10 10

0.1

θ

h=1

10

0.0

100

0.3

20

0.2

J00 = 0.001

0.1

30

50

0.01

1

C(1)

C(1)

C(0)

30

C(0)

3

80

0.01

0.1

90

1.0

40

20

seeded

10 0.01

0.0 0.1

1

10 θ

100

0.1 0 1000

0.0 0.1

Fig. 3. E#ect of scaled heat of condensation, h, at temperature $ = 0:5 with w = 5:0.

50

0.8

40

C(0)

50

0.4

40

20 10

0.1

0.1

10

0.0 0.1

0.0 0.1

1

10 θ

100

0 1000

Fig. 4. E#ect of scaled activation energy, ), at temperature $ = 0:5 with w = 5:0 and h = 1:0. (0) (0) avg avg CA0 = CB0 = 0; CA0 = CB0 = 0, and S0 = 50. Also, we take J00 = 0:001 and use parameters  = 0 and )A = )B = 0:01, unless stated otherwise.

30

seeded

seeded

20

0.2

0 1000

0.5

0.2

C(1)

ε = 0.001

0.4

100

C(1)

30

10 θ

(a)

0.3

0.1

0.6

1

C(0)

1.0

ε = 0.001

10

(b)

1

10 θ

100

0 1000

Fig. 6. E#ect of heterogeneous seeding with (a) w = 6 and (b) w = 7. The line marked seeded represents the case when the system is heterogeneously avg (0) seeded with C0 of 30 and C0 of 0.15 and 0.1 for (a) and (b), respectively.

Fig. 1 shows the e#ect of w on crystal nucleation, growth, and ripening. The ratio of interfacial to thermal energy, w, through Eqs. (2.12) and (2.12a), in6uences

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G. Madras, B.J. McCoy / Chemical Engineering Science 59 (2004) 2753 – 2765

1.0

50

0.5

0.8

40

0.4

50 40 0.9

C(0)

C(1)

0.5 0.45 0.4

0.4

0.5

0.2

0.45

0.75

0.3

30

Θ = 0.5

30 C(1)

Θ = 0.6

Θ = 0.6

C(0)

0.6

20

0.2

20

10

0.1

10

0.4

0.0 0.1

1

10 θ

0 1000

0.0

0 1000

100

0.1

1

(a)

10 θ

100

4

50

Fig. 7. E#ect of scaled temperature $ with w = 5:0 and h = 1:0.

40

3 1.0

α is assumed to vary linearly with temperature

50 C(0)

0.5

0.8

40 1

C(0)

C(1)

0.1

1

0.1

1

10 θ

100

α is assumed to be independent of temperature.

0.08

C(0)

30 C(1)

α = 0.1

0.5

0.04

1 α = 0.1

0.02 1

0.00 0.1 (b)

50 40

0.06

1

20 10

0.5

10

100

1

10

100

0 1000

θ

Fig. 9. E#ect of heating with $ = 0:5 till (a) " = 10 (b) " = 100 and then $ is ramped to the value given in the Dgure. The other parameters are h = 1; w = 5 and (a) e = 0:01 and (b) e = 2.

0 1000

0.0

0.10

(b)

10

0.2

20 10

0

20

0.4

30

0.9

30

0.5

(a)

0.75

1

α = 0.1

0.6

Θ= 0.5

2

C(1)

α = 0.1

0 1000

θ

Fig. 8. E#ect of cooling, $ = 0:95 − 0:9[1 − exp(−-")] with w = 5:0 and h = 1:0 when (a)  is assumed to vary linearly with temperature and (b)  is assumed to be independent of temperature.

nucleation and growth and hence the time evolution of the crystal number concentration and mass concentration (Fig. 1a). Increasing w delays nucleation and the supersaturation decrease (Fig. 1b). According to the mass balance, Eq. (2.14), S(") and C (1) (") are mirror images. The scaled

particle-number concentration, C (0) , rises by nucleation and later decreases by denucleation. At smaller w, nucleation is greater, but denucleation sets in earlier and thus the average size of the crystals generated would be larger compared to large w. The nucleation rate Jnuc decreases as S decreases in time as indicated in Fig. 1c. As supersaturation declines from a large value, the critical nucleus size, given by Eq. (2.13), increases faster than the crystals grow, causing denucleation at rate Jdenec (Fig. 1c and d). The net nucleation rate, J = Jnuc + Jdenec , decreases to zero and becomes negative when denucleation dominates during ripening (Fig. 1c and d). As J is the time derivative of the crystal number concentration (Madras and McCoy, 2001), the transition time between nucleation growth and ripening occurs when C (0) is maximum. The decline in the crystal number concentration, C (0) , especially at larger values of w, demonstrates that denucleation is a signiDcant process. Figs. 2–4 show the e#ects of changing J00 ; h, and ), respectively. The time when crystal generation (nucleation) becomes crystal loss (denucleation) is the principal marker in the transition from nucleation growth to ripening. The turnaround time occurs when C (0) (") is maximum and

G. Madras, B.J. McCoy / Chemical Engineering Science 59 (2004) 2753 – 2765

0.5 0.4

2761

50

0.5

50

40

0.4

40

30

0.3

A

0.2

B

0.1

B

B

30

C(0)

A

C(1)

C(1)

C(0)

0.3

20

0.2

10

0.1

20

10 B

0.0 0.1

1

10

100

0.1

θ

(a)

A A

0.0

0

1

50

100

θ

(a)

0.5

0

10

50

0.5

z

40

0.4

0.4

40 A

B B

0.0 0.1

1

10

0.3 C(0)

30 C(1)

20

0.2 0.1

(b)

30

A

C(1)

C(0)

0.3

A

0.2

10

20

0.1

B

10

B

0 100

θ

Fig. 10. E#ect of (a) wB = 5:5 and (b) wB = 6 at temperature $ = 0:5 with wA = 5 and hA = hB = 1:0 on the evolution of polymorphs.

hence, J = 0. According to our computations here and in earlier work (Madras and McCoy, 2002c), the maximum can occur over a time interval separating the two stages of nucleation growth and ripening. The number-density maxima determine when maximum curvature appears for the average mass (Figs. 2–4). Increasing J00 speeds up the onset of nucleation, and with the larger number of small particles, the denucleation rate will also increase. As evident from Eq. (2.18), the dependence of the homogeneous nucleation rate, J , on J00 is linear. Since homogeneous nucleation is avoided in industrial crystallization (Genck, 2003), lower values of J00 should be preferred. The e#ect of solubility on crystal growth and ripening has been discussed in several studies (Schroer and Ng, 2003; Mohan et al., 2001). Fig. 3 shows the e#ect of phase transition energies (h) on the time evolution of crystal number and mass concentrations. Smaller values of h imply higher solubility. For comparable changes in h and w, the in6uence of h on the evolution of the mass and number concentrations is much smaller than the in6uence of w. The e#ect of activation energy is shown in Fig. 4. Despite a two order of magnitude change in the values of ) (0.001–0.1), the in6uence of this parameter on

0

0.0 0.1 (b)

1

10 θ

Fig. 11. E#ect of (a) hB = 0:1 and (b) hB = 0:5 with wA = wB = 5 and hA = 1:0 at temperature $ = 0:5 on the evolution of polymorphs.

the time evolution of the crystal concentrations is minimal. This implies that activation energy of rate coe9cients usually will not play an important role in crystallization kinetics. Fig. 5 shows the e#ect of initial supersaturation, S0 . Larger initial supersaturation increases homogeneous nucleation. If S is excessively large, the critical nucleus size may be less than a monomer, and activated nucleation concepts are not applicable. For small values of h and ! = 5, the critical nucleus size given by Eq. (2.13) is unity when S is 100. Such highly unstable systems undergo a spontaneous decomposition to phase condensation. Fig. 6 shows the e#ect of heterogeneous seeding. For larger values of w, decreased homogeneous nucleation is overwhelmed by seeding. For smaller values of w, solutions have a greater tendency to nucleate homogeneously leading to a larger number of smaller crystals. An appropriate mass of seeds (heterogeneous nuclei) can overpower this undesirable behavior to yield a more favorable product of fewer but larger crystals. Fig. 6a and b shows that with appropriate seeding (for C0(0) = 0:15 and 0.1 with w = 6 and 7, respectively), homogeneous nucleation can be suppressed.

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G. Madras, B.J. McCoy / Chemical Engineering Science 59 (2004) 2753 – 2765

1.0

50

0.010

40

0.008

50

1.0 Θ

0.8

0.8

0.6

40

0.4 0.2 10

100

A B

0.4

20

0.2

10

30

0.006

30

C(0)

θ

C(0)

1

C(1)

0.0 0.1

C(1)

A B

0.6

A

B

20

0.004

10

0.002 A B

0

0.000

0.0

0.1

0 0.1

1

(a)

10

10

100

θ

(a)

100

θ 0.5

1

0.0010

5

0.0008

4

50 A

1.0

0.4

B

Θ

0.8

40

0.6

0.0006

0.4 0.0 0.1

1

10

30

100

0.2

20 10

0.1

0 0.1

(b)

1

1 A B

0.0000

B

0.0 10

100

θ

Fig. 12. E#ect of cooling, $ = 0:95 − 0:9[1 − exp(−-")] with (a) - = 0:1 and (b) - = 1:0; wA = 5:0; wB = 5:5, and hA = hB = 1:0 when  is assumed to vary linearly with temperature in an unseeded solution. The insets of the Dgures show the variation of $ with ".

Thus, C (0) (") values are relatively constant compared to the unseeded cases, for which C (0) (") rises from zero by nucleation. Fig. 7 shows how changing temperature ($ = 0:4 − 0:6) in6uences the evolution of the crystal concentrations. The e#ect is similar to that of changing w, with higher temperature speeding the onset of both nucleation and denucleation. Homogeneous nucleation is strongly reduced at lower temperatures, leading to larger crystal sizes. The onset of denucleation is also delayed leading to a longer transition period between nucleation and ripening. Fig. 8a and b shows how cooling crystallization in6uences the crystal growth. As discussed earlier, the interfacial energies for liquids generally depend on temperature (Reid et al., 1977; Yang and Qiu, 1986; Girshick, 1990) while the interfacial energies for solids (Lancia et al., 1999) may not vary much with temperature. Since the theory applies to both vapor–liquid and solid–liquid transitions, we have shown how condensation is a#ected by cooling for both the cases. As seen from Fig. 8a and b, the in6uence of temperature is

2

B

0.0002

A

0.1

A

0.0004

C(1)

θ

C(0)

C(0)

0.3

3 C(1)

0.2

(b)

1

10

0 100

θ

Fig. 13. E#ect of cooling, $ = 0:95 − 0:9[1 − exp(−-")] with (a) - = 0:1 and (b) - = 1:0; wA = 5:0; wB = 5:5, and hA = hB = 1:0 when  is assumed to be independent of temperature in an unseeded solution.

striking, and homogeneous nucleation is nearly negligible when the interfacial energy is assumed to be independent of temperature. To illustrate the e#ect of heating after nucleation has occurred (Lewiner et al., 2002), we consider crystal growth and nucleation at $=0:5. At the point where the number concentration, C (0) , is maximum, the temperature $ is increased and the subsequent evolution of the crystal concentration is observed. Fig. 9a and b shows the evolution of the crystal concentration for these cases. When ) is changed from its value of 0.01 in Fig. 9a to 2.0 in Fig. 9b, the results reveal a reversal of temperature dependence of ripening rates. Fig. 9a shows that increasing the temperature decreases the ripening rate; both the concentration of crystals and the number of crystals evolve more slowly. When supersaturation becomes small, the expression for the critical nuclei size (Eq. (2.13)) reduces to !∗ =(w(1−$)=h)3 when the interfacial energy is linearly dependent on temperature. Thus, if denucleation depends on the critical size alone, ripening rates decrease with the increase in temperature. Fig. 9b shows that increasing T (or $) has the opposite behavior. This can be understood by recognizing that Eq. (2.12) with Eq. (2.12a) shows how J is directly in6uenced by exp(−)=$) in the range when

G. Madras, B.J. McCoy / Chemical Engineering Science 59 (2004) 2753 – 2765

S is changing only slowly. The denucleation rate J is thus smaller for larger ) and vice versa. As expected, the larger activation energy has a greater e#ect on the ripening rate, actually reversing the weaker temperature dependence of smaller ). The reduction of Dnes with the increase in temperature is usually attributed to the increase of solubility with temperature (Lewiner et al., 2002). However, our results show that it is not just solubility but the large activation energy values that allow the heating to increase the average crystal size. The interpretation of Figs. 1–9 indicates that the interfacial energies, solubilities, and temperature play important roles in determining the time evolution of the condensed phase concentrations. Therefore, we next examine the evolution of two polymorphs with di#erent interfacial energies or solubilities and the in6uence of temperature on their evolution. The e#ect of interfacial energy on the evolution of A and B number and mass concentrations is displayed in Fig. 10a and b. The graphs show both number and mass concentrations rising from zero owing to homogeneous nucleation. The concentration of the less stable B(wB ¿ wA ) increases less steeply (versus the logarithm of time) while concentration of A increases sharply. The di#erence in mass concentration of A and B is a measure of separation of the dimorphs. As expected, the separation is better when the di#erence in the interfacial energies of A and B is signiDcant. The e#ect of transition energy (heat of crystallization) and thus solubility is shown in Fig. 11a and b. Smaller values of h imply higher solubility. In both Fig. 7a and b, hB ¡ hA , so that B is more soluble than A. An order of magnitude di#erence in the transition energies (hA = 1:0; hB = 0:1) ensures that the homogeneous nucleation of A is completely suppressed compared to the homogeneous nucleation of B. This indicates the importance of solubility (Threlfall, 2000) in determining the separation of polymorphs in the presence of homogeneous nucleation. As discussed earlier and in previous studies (Beckmann, 2000; Mohan et al., 2001; Lewiner et al., 2002), temperature is often used to control crystallization and polymorph separation. We propose temperature programs that decrease with time according to $ = 0:95 − 0:9[1 − exp(−-")]. With - in the range 0.1–1.0, most of the temperature change occurs in the time range 1–10. Figs. 12 and 13 show the e#ect of cooling for polymorphs of di#erent interfacial energies, wA ¡ wB , for two temperature dependencies. Fig. 11 indicates that the separation of the dimorphs is negligible when the cooling rate is slow. Faster cooling (Fig. 11b) allows stronger growth of A when denucleation ceases (C (0) becomes constant) and therefore a larger mass concentration of A. Homogeneous nucleation is nearly negligible when the interfacial energy is assumed to be independent of temperature, as shown in Fig. 13 where the C (0) coordinate is 1/100 that of Fig. 12a. Figs. 12 and 13 conDrm the importance of temperature control (Beckmann, 2000) for polymorph separation.

2763

5. Conclusions Nucleation, crystal growth, and ripening are generally necessary in describing the evolution of a crystallizing solution or melt. A kinetics model based on population balance dynamics describes the time evolution of the crystal size distribution for condensation of the 6uid phase and consequent decline in supersaturation. Numerical solutions of the governing equations show that the transition from nucleation and growth to ripening occurs over a relatively long time period. It is shown that interfacial energies, compared to solubilities and activation energies, play an important role in determining the time evolution of the crystal size distribution. The temperature dependence was introduced using cooling as a potential process technique to control the size distribution during the phase transition. Separation of two crystal dimorphs that di#er in crystal properties by such a temperature program was examined. The work shows that including the in6uence of the interfacial energy (Gibbs–Thomson e#ect) is critical for understanding crystal growth and is also important for continuous cooling conditions. It has long been recognized that crystallization processes always yield distributions of crystal sizes that must be incorporated into rate models. Less acknowledged is the solubility increase for smaller particles owing to the Gibbs–Thomson e#ect, which has usually been considered explicitly only in investigations of Ostwald ripening. Our proposal is that the combined processes of nucleation, crystal growth, and ripening can be handled by a distribution kinetics, population dynamics approach. The competition between homogeneous and heterogeneous nucleation in seeded and unseeded processes can be treated by this method. The e#ect of the temperature dependency of interfacial energy is signiDcant for quantitative descriptions of homogeneous and heterogeneous nucleation and for polymorph separation. Di#erent rates of cooling during crystallization a#ect the system in ways that depend on the physical property values. The complexities of phase condensation phenomena are caused by the interactions of molecular processes at di#erent scales in time and space. Only a theory that incorporates simultaneously the e#ects of crystal size distribution, interfacial energy, and particle solubility can be expected to portray the phenomena realistically.

Notation c C E h NH I J J00

crystal size distribution dimensionless crystal size distribution activation energy for di#usion (=NH=RTc ), dimensionless transition energy molar energy of the phase transition denucleation rate dimensionless denucleation rate prefactor for nucleation rate in (Eq. (2.16))

2764

kB m(0) (t) m(0) ∞ r r∗ S Seq T Tc v w x xm x∗

G. Madras, B.J. McCoy / Chemical Engineering Science 59 (2004) 2753 – 2765

Boltzmann’s constant molar concentration of solute as a function of time molar concentration of solute in equilibrium with plane crystal surface radius of the crystal radius of critical nucleus =m(0) =∞ , supersaturation ratio =m(0) =m(0) eq , supersaturation absolute temperature reference temperature volume (=(3xm =4c )−1=3 20 xm =c kB Tc ), Gibbs– Thomson ratio of interfacial to thermal energy for a crystal of monomer size crystal mass monomer mass mass of critical nucleus

Greek letters ) " $  ∞ ! c  ! 

(=E=RTc ), dimensionless activation energy dimensionless time (=T=Tc ), dimensionless temperature exponent on mass in rate coe9cient expression high-temperature solubility (=x=xm ), dimensionless crystal mass crystal mass density interfacial free energy (=w($−1 −1) or w$−1 ), depending on the temperature dependency of  (=!=!1=3 ), Gibbs–Thomson factor

Superscript (n)

nth mass moment of crystal size distribution

Subscripts 0

A B

initial condition polymorph A polymorph B

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Threlfall, T., 2000. Crystallisation of polymorphs: thermodynamic insight into the role of solvent. Organic Process Research and Development 4, 384. Washburn, E.W. (Ed.), 1930. International Critical Tables of Numerical Data, Physics, Chemistry and Technology, Vols. 1–7. McGraw-Hill, New York. Wolk, J., Strey, R., Heath, C.H., Wyslouzil, B.E., 2002. Empirical function for homogeneous water nucleation rates. Journal of Chemical Physics 117, 4954. Yang, C.H., Qiu, H., 1986. Theory of homogenous nucleation. A chemical kinetic view. Journal of Chemical Physics 84, 416. Zhao, J.Z., Ratke, L., 2004. A model describing the microstructure evolution during a cooling of immiscible alloys in the miscibility gap. Scripta Materialia 50, 543. Zhao, J.Z., Drees, S., Ratke, L., 2000. Strip casting of Al–Pb alloys—a numerical analysis. Materials Science and Engineering A 282, 262.