Ostwald Ripening - American Chemical Society

Article pubs.acs.org/crystal

Population Balance Modeling with Size-Dependent Solubility: Ostwald Ripening Martin Iggland and Marco Mazzotti* Institute of Process Engineering, ETH Zurich, Sonneggstrasse 3, 8092 Zurich, Swizerland ABSTRACT: In this work, we present a detailed population balance model for Ostwald ripening. The model is based on a size-dependent growth rate expression incorporating the Gibbs−Thomson relationship between particle size and solubility, and is solved numerically. The effect of parameters such as average initial particle size, initial width of the particle size distribution, and initial mass as well as solubility are investigated in simulations. This analysis focuses on understanding how the ripening phenomenon can be exploited in a crystallization process. The simulations are compared to the predictions of classical Lifshitz, Slyozov, Wagner (LSW) theory. Using our results, we assess the advantages and disadvantages of the full numerical simulation compared to the LSW model.

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INTRODUCTION In crystallization processes, a new population of particles forms and evolves through several steps, including nucleation and growth as well as secondary effects such as breakage and agglomeration. Nucleation and growth lead to a transfer of the solute from the liquid phase to the solid phase, whereas breakage and agglomeration lead to changes in the size and shape of solid particles, but do not affect the liquid phase concentration directly. The process is normally ended when the solute concentration has decreased to a level which practically corresponds to the bulk solubility at the process conditions. The development of the particle size distribution up until this stage can be described by established population balance models, finding applications in, for example, process design and control.1−7 Even though the supersaturation has been largely depleted at the end of a crystallization process, this does not mean that the population of crystals does not change any more, as the mechanism known as Ostwald ripening now takes control. Ostwald ripening, which is sometimes also referred to as coarsening, aging, or simply ripening, is a slow process which takes place over a time spanning from hours to days or even longer, depending on the materials and the conditions involved. The thermodynamic basis of ripening is the size-dependence of solubility, which can be explained as follows. When a crystal forms from a supersaturated solution, there is a Gibbs free energy gain proportional to the volume of the newly formed crystal and a Gibbs free energy loss associated with the creation of its surface. Since the former grows with size slower than the latter, the energy gain balances the loss at a critical size LC, beyond which new nuclei are stable and can grow.3,4 The critical size depends on the supersaturation of the solution S = c/c∞, according to LC(S) =

4γVm α = RT ln S ln S © 2012 American Chemical Society

where c is the solution concentration and c∞ is the bulk solubility, that is, the solubility of an infinitely large particle. The capillary length α combines all the physical parameters of the particles (surface tension γ, molar volume Vm); R and T are the universal gas constant and the absolute temperature. This equation is strictly valid only for spherical particles of nondissociating species with a homogeneous, flat surface and with sizes much larger than the size of the individual molecules.8 Furthermore, the interfacial tension should be independent of the particle size and the solution should be ideal, so that γ is independent of the solute concentration.9 These conditions are often not fulfilled, especially for faceted crystals. Wagner10 has modified the equation with a stoichiometric factor to account for species that dissociate. The equation has been used in modified form for crystals with nonspherical shapes, by including the area and volume shape factors ka and kv:11 LC(S) =

2ka γVm α = 3k vRT ln S ln S

(2)

In this case γ represents a weighted average of the values of the surface tension for all faces, and the capillary length α is defined accordingly. The fact that the stability of a crystal depends on its size implies that the solubility of a substance is dependent on the particle size, as it can be shown by using eq 1 to obtain the solubility c*(L) of a particle of size L (the Gibbs−Kelvin or Gibbs−Thomson equation): S*(L) =

⎛α⎞ c*(L) = exp⎜ ⎟ ⎝L⎠ c∞

(3)

Received: November 28, 2011 Revised: January 19, 2012 Published: February 23, 2012

(1) 1489

dx.doi.org/10.1021/cg201571n | Cryst. Growth Des. 2012, 12, 1489−1500

Crystal Growth & Design

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Equations 1 and 3 define the curve shown in Figure 1, which gives the critical length as a function of concentration, eq 1, or the solubility of a particle as a function of its size, eq 3. Every point in the plane represents a crystal of size L suspended in a solution having concentration c = Sc∞. Points underneath the curve represent crystals that dissolve either because they are small or because the supersaturation is low. Points above the curve correspond to crystals that grow because they are large or

dependent crystal growth, neglecting all other phenomena which affect crystal size or number:

∂(Gn) ∂n + =0 ∂t ∂L

(4)

Often, LSW theory, which is based on the above equation and was proposed by Lifshitz and Slyozov12 and Wagner10 in 1961, is used to predict and explain the behavior of crystal populations subject to Ostwald ripening. A discussion of the classical LSW theory, along with its most important conclusions, is given in the next section. The theory has been extended by several authors. Sarian and Weart13 and Ardell14 considered cases of high particle concentrations, where the assumption that the mean distance between particles is much larger than the particle size is no longer valid. They derived corrections to the growth terms and the time-dependence of the average particle radius, analogous to the classical LSW theory. Ardell additionally derived corrected expressions for the particle size distributions at the asymptotic limit. Kahlweit15 argued that some of the assumptions made by both Lifshitz and Slyozov and by Wagner are strictly valid only in open systems and concluded that the asymptotic rate of change of the mean radius (ripening rate) as predicted by LSW theory is not actually the asymptotic rate, but the maximum rate attainable. He also considered a secondorder growth rate expression, deriving expressions for the particle size distribution and for the mean size as a function of time. Marqusee and Ross16 carried out an analysis of the assumptions made about the mass conservation in LSW theory. They find that the assumption of constant total particle mass is not necessary in the derivation of the expressions for the asymptotic solutions. Furthermore, they argue that although the final state is independent of the initial conditions, the approach to this state is not independent, a conclusion contrary to the results of LSW theory. Venzl17 presented numerical solutions for the special case of diffusion-limited growth rate, the same as the one used by Lifshitz and Slyozov. The initial state is assumed to be the particles formed by nucleation, having a modal size slightly above the critical size, suspended in a solution with a solute concentration above the solubility. His results show that the same behavior is observed at long times for all simulations, independent of the initial state. Sugimoto18 has analyzed the influence of the solubility of a substance on the ripening behavior, thus concluding that the predictions obtained from LSW theory are only valid at low solubilities, whereas high solubilities lead to behavior which can not be predicted a priori. Other models have been presented to explain Ostwald ripening. De Smet et al.19 have used a Monte Carlo type model. They assume constant total particle mass, and as a consequence, that the critical radius is equal to the mean radius of a particle population and attempt to model the transition from an arbitrary initial state toward the stationary regime. The same model is applied by Hoang et al.20 and by Liu et al.21 to explain experimental results. McCoy22 has modified models used for reversible chain polymerization to apply to systems which have a size-dependent solubility. Using this model, he calculates the moments of the distributions, then recreates the distributions based on assumptions about their shape. When applying this model to ripening, McCoy assumes that the critical length is equal to the mean radius, as in other models, and introduces a denucleation rate to account for particle loss, the values for which are given explicitly, and not

Figure 1. The solubility curve of a generic substance as a function of the particle size. The red lines show how the curve can be used to obtain either the critical length from the concentration (according to eq 1) or the solubility of a particle from its size (using eq 3). The gray areas schematically represent the three different phases of a crystallization process.

because the solution is highly supersaturated. Points on the line correspond to crystals having exactly the critical size and therefore being stable. The three different phases of a crystallization process can be schematically shown in this figure. At high supersaturations nucleation and growth occur, and the only particles which are subject to dissolution are subcritical clusters. In the growth phase, nucleation is virtually nonexistent due to the low supersaturation, but most crystals grow because they are large enough to be above the critical size. Ostwald ripening becomes an important process when the critical size increases as the supersaturation approaches 1. At this stage, a large amount of particles end up being below the critical size. These will dissolve and the particles above the critical size will grow, ultimately leading to all the mass available in the system being contained in one particle. In an open system, this would in principle lead to one infinitely large particle. Thus Ostwald ripening is important at low supersaturation, where the critical size is large and the solution is undersaturated with respect to small particles, with sizes on the order of the capillary length. As a consequence, Ostwald ripening takes a long time (on the order of days to months), since at low supersaturations driving forces are small and hence crystal growth and dissolution are very slow. In order to model Ostwald ripening, a population balance equation (PBE) can be used. The growth rate is expressed in a size-dependent manner using the above definition of sizedependent solubility, that is, G(L) ∝ c − c*(L). The PBE for Ostwald ripening is then given by the standard PBE for size1490



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theory. We then present the model used, along with a dimensionless formulation. In the third section, we will compare our model to the simplified expressions for the short time range presented by Wagner10 and to the predictions of the asymptotic behavior predicted by classical LSW theory. We will then investigate the effect of various parameters on the ripening behavior, using different growth rate expressions, and comparing the results of our simulations to the predictions of classical LSW theory. Following this general investigation, we will consider an example which is more specific but potentially important for applications of Ostwald ripening, namely, the case of an initially bimodal particle size distribution. In an industrial application, a detailed understanding of ripening can be very valuable for process design and scale-up. Thus, our discussion of the simulations will focus on how the particle size distribution of a population of crystals can be affected, and on how the process time can be affected by the choice of conditions. It is clear that if ripening is to be included in a process model alongside other mechanisms, such as breakage or agglomeration, the asymptotic analysis of LSW theory is not sufficient. For these cases, the complete population balance equation is necessary.

derived from the physical basis of Ostwald ripening. Shukla et al.23 have used a population balance model to describe experimental data on ripening of silver halide particles, but they also make the assumption of constant mass and thus equality of critical size and mean particle size. Ludwig et al.24 presented an extension of the PBE for Ostwald ripening, which includes stochastic effects caused by temperature and concentration fluctuations. They argue that these stochastic effects can, at least partially, explain the discrepancy between the width of the predicted and experimentally observed particle size distributions at long observation times. Ozkan and Ortoleva25 use a model which describes nucleation and ripening based on the adsorption and desorption of monomer units on clusters. To simplify calculations, they split their model into a discrete and a continuous formulation, and thus they are able to describe the evolution in time of particle populations over a large size range. The results of simulations using their models agree with the results of LSW theory. Tavare26 and Ståhl et al.11 have presented PBE models for Ostwald ripening, based on eq 4. Neither work makes the assumption of constant mass, and both models have been solved numerically. Tavare separates dissolution from growth by specifying slightly different expressions for dissolution and growth. In the model, two compounds react to form a third product, whose concentration increases above its solubility. Once the solution is supersaturated with respect to this compound, nucleation leads to formation of particles at the critical size, and dissolution and growth lead to changes in the particle size distribution. The influence of the ratio of the dissolution rate to the growth rate is investigated, and compared to the case of only growth. Tavare concludes that while the ratio has only a small influence on the final particle size distribution, the result when assuming only growth is completely different. Ståhl et al. assume diffusion limited growth rates, and they estimate rate parameters from mass-transfer correlations. During ripening, the shape of the crystals is assumed to change, which is reflected in an exponentially decaying nonconstant shape factor. The rate parameters for such a decay are obtained by fitting the model to experimental data obtained from ripening of benzoic acid produced by reactive crystallization in an aqueous solution. There is a wealth of literature on experimental observation of Ostwald ripening,9,11,21,27−36 thus proving that Ostwald ripening plays an important role in several phenomena and processes. To give a complete overview of all of these is outside the scope of this paper; nevertheless, we would like to mention some examples and later compare some of the experimental findings to our modeling results. For example, the effects of Ostwald ripening are important to be able to estimate and improve the long-term stability of nanoparticles. Recently, the sizedependence of solubility has been cited as one of the mechanisms involved in the symmetry-breaking of enantiomers by the so-called Viedma ripening or attrition-enhanced deracemisation process.37−41 Ripening is also exploited industrially, for example, in the production of some pigments, although often Ostwald ripening is here only one of several mechanisms at work, since deagglomeration, grinding, polymorph transition, and the removal of crystal defects also play important roles.42,43 In this paper, we simulate Ostwald ripening using a PBE model similar to the one used by Ståhl et al.,11 thus avoiding the assumptions made in LSW theory. The aim is to demonstrate the use of the PBE model to simulate the dynamic behavior of a particle population at all stages of the Ostwald ripening process, including the beginning. In the first part of this paper, we review the most important results of LSW

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LSW THEORY Lifshitz and Slyozov12 and Wagner10 considered eq 4 in the case of diffusion-limited (GD) or reaction limited (GR) growth, using the following two growth rate expressions: GD(L) =

4+ (c − c*(L)) ρL

(5)

G R (L ) =

2k (c − c*(L)) ρ

(6)

where + is the molecular diffusivity, k is the rate constant of growth or dissolution, ρ is the density of the crystals, and the size-dependence of the solubility is simplified by retaining only the first two terms of the Taylor expansion in α/L of eq 3: ⎛ α⎞ c*(L) = c∞⎜1 + ⎟ ⎝ L⎠

(7)

Note that in the original papers all equations are written for spherical particles using the radius as a measure of the particle size. In this section, however, we choose a notation consistent with the equations used in elsewhere in the manuscript, that is, based on a characteristic length L, and shape factors ka and kv. For spherical particles, the characteristic length is the diameter, ka = π and kv = π/6. Short Process Times. Wagner found simplified solutions at short times for both types of growth rate expressions, making some important assumptions. He assumed that particle sizes are initially normally distributed, with a small standard deviation, and that the concentration change and decrease in number of particles at the beginning of the process could be neglected. Under these assumptions, the mean particle length is constant and equal to the critical length, and the number of crystals is constant. The coefficient of variation (defined as the standard deviation divided by the mean size) should obey the relation: ⎛t ⎞ ε(t ) = ε0 exp⎜ ⎟ ⎝ tj ⎠ 1491

(8)



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where tj equals tD or tR for diffusion-limited growth or for reaction-limited growth, respectively, where ρL̅ 03 , tD = 4α+c∞

ρL̅ 02 tR = 2αkc∞

(9)

L*(t ) =

9 L̅ (t ) 8

t ′R =

⎛ 9 ⎞2 ρL̅ 02 81 ⎜ ⎟ tR = ⎝ 8 ⎠ αkc∞ 32

(12)

Z0 1 + t /t ′D

(13) (14)

where *3 9 ρ(L 0 ) 9 = tD 16 α+c∞ 4

(15) 12

Equations 10−14 are, according to Lifshitz and Slyozov, valid for times where 2 ⎡ ⎛ ⎞⎤ ( ) L t ̅ ⎟⎥ ≫ 1 9⎢ln⎜⎜ ⎢⎣ ⎝ L * ⎟⎠⎥⎦ 0

dμ dc = − k vρ 3 dt dt

(17) 10

For the reaction-limited case, the following relationships apply:

(24)

(25)

where μ3 is the third moment of the distribution n, defined as (j = 3)

⎞ ⎟ 9 ⎟ if X < 4 ⎟ 1 ⎠ if X ≥

(23)

PBE MODEL The model equations are constituted of the population balance eq 4, together with the following mass balance for the solute:

(where L*0 is the critical size at time 0), or according to Wagner:

⎧ ⎛ ⎞5 ⎛ 4 ⎪ X ⎜ ⎪ *R X ⎜ 1 ⎟⎟ exp⎜⎜ 3 ⎪ ⎜1 − 4 X ⎟ ⎜ 4X − pR (X ) = ⎨ ⎝ ⎠ ⎝9 9 ⎪ ⎪ ⎪0 ⎩

(22)

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(16)

t ≫ t ′D or L̅ (t ) ≫ L̅ 0

(21)

It is worth noting that the predictions of LSW theory are largely independent of the initial conditions. Wagner’s predictions of the early stages reflect some influence of the initial mean particle size and coefficient of variation, whereas in the asymptotic state only the number of crystals depends on the initial state. An interesting consequence of the predictions of LSW theory is that the coefficient of variation of the particle size distribution approaches a constant value. The value can be derived by using the distributions pD and pR along with the definition of X in the equation for the coefficient of variation. This results in asymptotic values of the coefficient of variation of CVD ≈ 0.21 and CVR ≈ 0.35 for the diffusion-limited and reaction-limited case, respectively. Similarly, the skewness (γ1) and kurtosis (γ2) of the particle size distributions (see Appendix B for their exact definition) attain the values γ1D ≈ −0.92, γ1R ≈ −1.17, γ2D ≈ 0.68, γ2R ≈ −0.11.

(11)

t 16 +αtc∞ =3 ρ t ′D 9

(20)

and φ is the total particle volume per unit volume of solution, which is assumed to remain constant. The expressions in eqs 12 and 20 are obtained by assuming that eqs 14 and 22 are valid also at t = 0. Equation 19 is not given explicitly in Wagner’s work but can be derived from the equations that he reports. The constant * R is chosen so that the distribution pR is normalized. Equations 18 to 22 are valid when t ≫ t ′R or L̅ (t ) ≫ L̅ 0

9ρα2 16+c∞t

L*(t ) = L̅ (t )

t ′D =

αkc∞t ρ

where

(10)

Z (t ) =

(19)

3/2 φ ⎛ 2ρ ⎞ Z (t ) = ⎜ ⎟ 1.27π ⎝ αkc∞t ⎠

⎧ −11/3 ⎛ 2X ⎞ 3 ⎪ 81 ·2−5/3X 2(X + 3)−7/3 ⎛⎜ 3 − X ⎞⎟ ⎟ if X < exp⎜ ⎪ ⎝2 ⎠ ⎝ 2X − 3 ⎠ 2 pD (X ) = ⎨ ⎪ 3 if X ≥ ⎪0 ⎩ 2

L̅ (t ) = L̅ 0 3

t 8 = t ′R 9

L̅ (t ) = L̅ 0

The solution given by eq 8 applies as long as t is less than or equal to tj. Long Process Times. For longer times, LSW theory predicts that a quasi-stationary growth regime is attained, during which the total particle mass remains constant, and the particle size distribution, supersaturation, and mean particle size are independent of the initial conditions. The initial conditions define the state of the system at the start of the experiment or simulation. Such a state is described by the initial particle size distribution and the initial solute concentration in the continuous phase. In the quasi-stationary regime the particle size distribution p (using the scaled parameter X = L/L*), as well as the supersaturation S = c/c∞, the mean length L̅, the number of crystals Z (related to the initial number of crystals Z0), and the critical size L* are given by the following equations for diffusion-limited growth:10,12

SD(t ) = 1 + 3

ρα kc∞t

SR (t ) = 1 +

μj =

∫0

∞ j

L n dL

(26)

The time derivative of the third moment is given by

9 4

dμ 3 dt

(18) 1492

=3

∫0

∞

G(L)nL2 dL

(27)



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and the initial conditions at t = 0 for the system of eqs 4 and 25 are n(L , 0) = n0(L)

(28)

c(0) = c 0

(29)

The initial conditions for eqs 34 and 37 are f (y , τ = 0) = f0 (y) = L 0n0(yL 0) S(τ = 0) = S0 =

By solving this model, the entire evolution of a population of particles during Ostwald ripening can be simulated. Such simulations can be performed using different growth rate expressions, including but not limited to those considered within LSW theory. A general growth rate expression can be written as c λ kLβ G(L) = ∞ (S − S*(L))λ ρ

f (y = 0, τ) = 0

L , L0

τ=

t , t0

α* =

α L0

dφ0 dτ

(32)

ρL 01 −β λ kc∞

(33)

and L0 is defined to be close to the initial mean particle size of the distribution, L̅ 0 ≈ μ10/μ00; its exact value is specified for each simulation in the following. Note that, due to the very large variability of the solubility c∞, the range of possible values of t0 is likewise very large. With these definitions, eq 4 can be cast as ∂f ∂(G*f ) + =0 ∂τ ∂y

(34)

where G*(y) = yβ (S − S*(y))

B = ηφ A f0B (y) = ηf0A (y) → φ3,0 3,0

(35)

dτ

= 3L 03

∫0

∞ 2 dφ y G*(y)f dy = L 03 3

dτ

(36)

where φ3 is the third moment of the distribution f. The mass balance eq 25 can therefore be written as k ρL 3 dφ3 dS =− v 0 dτ c∞ dτ

(41)

(42)

The two populations can in principle be of different crystals, that is, having different shape factors kv and different crystal densities ρ, and the solubility c∞ can be different. All other parameters are the same. The mass balance equations can be written as

The derivative of the third moment of the size distribution is given by dμ 3

= [Gf ]∞ 0 = G(0)f (0, τ) − G(∞)f (∞ , τ)

The number of particles φ0 can only decrease during Ostwald ripening due to dissolution (nucleation cannot occur at such low supersaturations); hence, its rate of change must be negative and finite. Since f approaches zero for infinitely large particles, and G becomes infinitely large (negative) only for particles of size approaching zero, then f(0,τ) must be zero, and this is therefore the needed boundary condition. This model can be solved numerically, as described in Appendix A. Once the distributions are known, characteristic parameters can be calculated to allow for the comparison of different simulations. These are defined in Appendix B. Scaling Behavior: Effect of Mass and Solubility. The solubility c∞ appears only in the definition of the dimensionless time t0, eq 33, and in the mass balance, eq 37. The effect that the solubility has through t0 is straightforward: an increase in c∞ leads to a decrease in t0, thus implying that an increased solubility speeds up the process of ripening. The presence of c∞ in eq 37 suggests that increasing the solubility affects the solution of the equation in a manner similar to reducing the mass of crystals; that is, the ratio of the initial total volume density to the solubility is the important factor. Consider two populations of particles A and B whose initial particle size distributions have the same shape, but the particle concentrations in the suspension are different:

(31)

where the values of t0 and L0 can be chosen freely. A scaled distribution f is defined in terms of the dimensionless particle size y. Since f dy = n dL must hold, then f = L0n. The parameter t0 is chosen in such a way that the growth rate is simplified, namely as t0 =

(40)

This can be justified by considering the particle size distribution at size zero. Since the solubility for particles of zero size approaches infinity, the dissolution rate for these particles (using eq 30) also approaches infinity. A numerical integration with infinite values is not possible. Ståhl et al.11 and Shukla et al.23 have solved this problem by assuming that crystals below a certain size are completely dissolved. This assumption can in fact be justified by integrating both sides of the PBE (eq 34) from y = 0 to ∞. One obtains the following equation for the zeroth moment of the distribution, which is proportional to the number of particles in the distribution:

where S = c/c∞ and S*(L) is given by eq 3. The last equation results in the standard empirical growth rate expression for β = 0 and size-independent solubility.3 In this work, we consider only the case λ = 1. For β = −1, eq 5 is obtained, and by setting β = −1/2 an expression for the growth rate in a diffusionlimited system with large particles is obtained.44 In order to reduce the number of parameters in the model, the following dimensionless variables are introduced:

y=

(39)

The boundary condition for eq 34 is

(30)

⎛ α ⎞ ⎛ α* ⎞ c*(yL 0) S*(y) = = exp⎜ ⎟ = exp⎜ ⎟ c∞ ⎝ y ⎠ ⎝ L 0y ⎠

c0 c∞

(38)

(37) 1493

S0A = S0B

(43)

A k AρAL 3 dφ3 dS A =− v A 0 dτ dτ c∞

(44)


Crystal Growth & Design B k vBρBL 03 dφ3 dS B =− B dτ dτ c∞

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The approximation exp(α*/y) ≈ 1 + α*/y made in the LSW theory is certainly valid for small values of α*/y (large values of y), but the most interesting part of the growth rate expressions is the small size range. The values of G*R and G*P are very similar even for very small values of y, while G*P decreases only slightly faster for decreasing sizes. Since the growth rates G*R and G*P are so similar, no appreciable difference would be seen in the simulations, and hence G*P , which is the nonapproximated expression of the growth rate under reaction-limited conditions, will not be considered any further.

(45)

Since f(y,τ) appears linearly in the PBE, if the concentration in both solutions remains the same, the scaling of the initial distribution has an effect on the scaling of the solution of the PDE only, not on the shape of the distributions; that is f B (y , τ) = ηf A (y , τ) if S B(τ) = S A(τ)

(46)

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This is the case if A k vAρAL 03φ3,0 A c∞

=

SIMULATIONS AND PARAMETER ANALYSIS In this section, we investigate the effect of three key model parameters on the Ostwald ripening behavior, considering both the diffusion-limited (G*D) and the reaction-limited (G*R ) growth rate expressions. In particular, we vary the mean particle size of the initial distribution and its width (i.e., standard deviation) as well as the initial holdup ratio ν, which is defined as

B k vBρBL 03φ3,0 B c∞

(47)

This factor represents the ratio of the initial hold-ups in the solid phase and in the solution (at saturation, which is the initial condition for all simulations carried out in this work). Growth Rates and Model Parameters. In this work, three different growth rate expressions have been considered: ⎛ ⎞ * (y) = 1 S − 1 − α* GD ⎜ ⎟ y⎝ y ⎠

(48)

⎛ α* ⎞ GR* (y) = ⎜S − 1 − ⎟ y ⎠ ⎝

(49)

⎛ ⎛ α* ⎞⎞ GP*(y) = ⎜⎜S − exp⎜ ⎟⎟⎟ ⎝ y ⎠⎠ ⎝

ν=

k v ρμ3,0 c∞

(51)

As shown in Table 1, the ranges of variation of the three parameters are different, but in all cases they are broad enough Table 1. Initial Parameters for All Simulationsa ID #1a-d #2a-d #3a-c #4a-j #5

(50)

where G*D corresponds to diffusion-limited growth,10,12,44 and G*R and G*P correspond to reaction-limited (or surfaceintegration limited) growth.44 The different growth rate expressions are plotted in Figure 2. It can readily be seen that the dissolution behavior of the

α* −2

10 10−2 10−2 10−2 10−2

L0 [m] −5

10 10−5 10−5 10−5 10−5

S0

ν

y̅ 0

σ0

G*

1 1 1 1 1

1.5 1.5 1.5 10−3 − 15 1.5

0.5−1.2 0.5−1.2 1 1 0.23

0.1 0.1 0.05−0.2 0.1 0.1

G*D G*R G*D G*D G*D

a

G* denotes which growth rate was used for the simulation, see eqs 48 and 49 for the corresponding expressions. All of the parameters, except L0, are dimensionless.

to appreciate the trends caused by the change in the relevant parameter. The simulations have been carried out with the same initial conditions for the two different growth rate expressions. The values of all parameters for all the simulations presented in this paper are reported in Table 1. By initial conditions, we mean those defined in eqs 38 and 39. The value of f 0 affects other parameters, such as the mean particle size, standard deviation, and the holdup ratio. In Table 1, we show the values of ν, y0̅ and σ0 as statistical measures of the particle size distribution f 0. Since α = α*L0, the value of the capillary length corresponding to the parameters chosen in this work is α = 100 nm. Such values have been chosen for the sake of illustration but without losing generality. A smaller, physically more realistic value of the capillary length17,21 would be obtained using for instance L0 = 10−7 m. By looking at the dimensionless model equations, one can readily observe that this choice would lead to the same results as below, except for the scaling of the PSD f and for the value of t0, which is proportional to L0 (reactionlimited growth) or to L0 squared (diffusion-limited growth). Therefore, reducing L0 from 10−5 to 10−7 m would yield a reduction of t0 by a factor 102 or 104, while leaving the dimensionless time unaltered; however, each dimensionless time unit would correspond to a 102 or 104 times shorter real time unit, thus making the process much faster.

Figure 2. Growth rates as a function of particle size, at S = 1.01, α* = 1 × 10−2. Note that the curves for G*R and G*P overlap.

diffusion and reaction-limited growth rates is similar, while the growth behavior becomes different for larger sizes. In the reaction-limited case, the growth rate increases with size, while in the diffusion-limited case, particles with sizes slightly above the critical size will grow with the highest rate. 1494



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Contrary to classical LSW theory and to work by Tavare26 and Ståhl et al.,11 the initial supersaturation is equal to 1 in all simulations. Such an initial state corresponds to the case of crystals being transferred to a saturated solution after growth (quenching), or crystals being suspended in a saturated solution after precipitation in order to be subjected to ripening. The latter case is typical for example in the synthesis of pigments and occurs also in the case of Viedma ripening.41 Effect of Initial Size. The difference between the capillary length and the initial mean size determines the amount of crystals in the initial PSD which are on either side of the critical size. To investigate the effect of this on the ripening behavior, we have varied the initial mean particle size y ̅ 0 between 0.5 and 1.2. The first simulation set, #1a-d, uses the diffusion-limited growth rate expression (eq 48), while the second set, #2a-d, uses the reaction-limited growth rate (eq 49). The results for these simulations are illustrated in Figure 3, showing the mean

For both mechanisms, changes to the particle population and the solution begin earlier for smaller initial mean sizes. Similarly, Greenwood9 observed slower ripening in lead melts for large UPb3 particles than for small ones, and Kumar et al.35 have experimentally observed increased stability of large nanoparticles compared to smaller nanoparticles. The supersaturation quickly attains a constant value, the level of which is independent of the initial supersaturation but strongly dependent on the initial particle size. The value of the supersaturation is related to the critical size, which is in turn determined by the relative rates of dissolution and growth. After the initial, fast dissolution, the system very quickly reaches a state in which approximately the same amount of substance enters the solution per unit time through dissolution as is consumed by growth. The distribution of particles above and below the critical size in this state depends on the growth mechanism. In both cases considered in this work, the critical size turns out to be close the mean particle size. As Wagner10 predicted, the mean particle size and the number of crystals remains constant at the beginning of the process. The coefficient of variation slowly increases initially, following the evolution predicted by eq 8. Furthermore, Wagner assumed that the shape of the distribution (a normal, or Gaussian, PSD) does not change until the point in time where the first particles disappear. Here, we use the skewness and kurtosis as measures of the shape of the PSD. As can be seen in Figure 3e,f, the shape change starts earlier than Wagner assumed. The skewness and kurtosis of a normal distribution are both 0; in all cases the value for the simulated PSDs has changed substantially before the number of crystals starts decreasing. At longer times, the coefficient of variation as well as the skewness and curtosis approach the values predicted by LSW theory, as given above. All characteristic parameters of the system also evolve according to the predictions of LSW theory, including the critical size. In the experimental study by Cabane et al.,34 the coefficient of variation approaches a constant value of about 0.5 and 0.4 for two different substances studied. These values are different to the ones mentioned here since different growth mechanisms are active in these cases. Effect of Initial Distribution Width. Another way to influence the distribution of particles above and below the critical size is by varying the width of the initial particle size distribution. Here, we measure the width using the standard deviation of the particle size distributions, as defined in Appendix B. We have carried out simulations with initial distributions having standard deviations in the range 0.05−0.2. As before, we have used two growth rate expressions. For the sake of brevity, only the simulations using G*D (simulations #3a-c) are illustrated and discussed, since the simulations using G*R follow the same trends. A summary of the results of these simulations is shown in Figure 4. The simulations clearly show that the broader particle size distributions experience changes due to ripening earlier than the narrower distributions; the mean particle size and the number of particles start changing at an earlier time for the higher initial widths. This behavior has been observed experimentally by others, leading to the conclusion that crystal populations that have a large variation in size are more affected by ripening than populations that have a narrow size distribution.21 This is because net growth is achieved earlier in time, as it can be seen by the earlier decrease in supersaturation. With the wider particle size distributions, there are more large particles

Figure 3. Summary of simulations with varying initial length. Profiles of (a) mean particle size, (b) supersaturation, (c) coefficient of variation, (d) number of crystals, (e) skewness, and (f) kurtosis. Solid lines represent simulations #1a-d, which use the diffusion-limited growth rate, and the dashed lines represent simulations #2a-d, which use the reaction-limited growth rate.

particle size, the supersaturation, the coefficient of variation, the number of crystals, the skewness and the kurtosis as a function of time for simulations #1a-d and #2a-d (dashed lines). 1495



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In this section, we have varied ν in the range 0.001−15, using G* D (simulations #4a-j). It is worth noting that when varying the particle concentration in such a broad range the growth mechanism also might change. The diffusion-limited growth rate used in this work and in the classical LSW theory is in fact valid only for cases where the distance between particles is much larger than the particle size, that is, an assumption that is not necessarily met at very high particle concentrations.13,14,45 For the sake of simplicity and without loss of generality, we have chosen to work with the same growth mechanism through the whole range of holdup ratios investigated. A summary of the results for these simulations is illustrated in Figure 5. Simulations carried out with G*R show results similar to the ones presented here, and thus they are not included. Two curves, namely, those at very low particle concentrations, stop after a short time because then nearly all particles have dissolved. As readily observed in Figure 5b, the initial change in supersaturation is slower at low holdup ratios than at

Figure 4. Summary of simulations with varying initial standard deviation of the PSD. Profiles of (a) mean particle size, (b) supersaturation, (c) coefficient of variation, and (d) number of crystals. The plots are for simulations #3a-c, which use the diffusion-limited growth rate.

that are above the critical size. Because of their size, the amount of substance deposited on these particles through growth outweighs the amount of material transferred into the solution through dissolution of small particles. The larger number of small particles makes no difference in this context, as the total amount of material contained in these is quite small. The behavior of the critical size is similar to that described in the previous section. As before, these simulations follow the trend predicted by LSW theory at longer times. There are, however, some differences. For example, for both the diffusion and reaction-limited growth rates, the final particle size distributions have shapes similar to those predicted by LSW theory (see eqs 10 and 18), but are broader for the broader initial distributions than for the narrower ones. In general, when starting with a narrower initial distribution, all parameters (e.g., mean particle size, supersaturation, PSD shape) are closer to the predictions made by LSW theory than those obtained using broader initial distributions. This result agrees with simulation results by DeSmet et al,19 who concluded that broader distributions “evolve slower toward a limiting form”. Thus it seems that the wider PSD requires a longer time before it reaches the asymptotic growth stage assumed by LSW theory. While the prediction and simulation may become closer for longer times, it is clear that the effect of the width of the particle size distribution can not be captured completely in LSW theory. Effect of Initial Holdup Ratio. In this section we investigate the effect of the initial holdup ratio, that is, the ratio between the holdup in the solid phase to the holdup in the liquid phase, as defined in eq 51: ν = kvρμ3,0/c∞. A change in ν can be obtained by either a change in the solid or the liquid phase holdup, that is, the solubility. Note that if the solubility changes, then t0 changes (see eq 33) and thus the time scale does as well. A large value of the holdup parameter ν represents a large initial particle concentration or a low solubility, and a low value represents low particle concentration or high solubility.

Figure 5. Summary of simulations with varying initial holdup ratios. Profiles of (a) mean particle size, (b) supersaturation, (c) coefficient of variation, and (d) number of crystals (normalized with initial number), and (e) mean particle size (solid lines) and critical size (dashed lines) (for the sake of simplicity, only some curves are shown; the trend for the surrounding simulations is comparable to the data shown). (f) Maximum relative mass change as a function of initial holdup ratio. The plots are for simulations #4a-j, which use the diffusion-limited growth rate. The line in (f) is to guide the eye. 1496



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or attrition lead to the creation of fines, which is generally to be avoided since fines often cause problems in downstream processing. There are several ways to remove fines, for example, by dissolving them. Since Ostwald ripening also leads to the dissolution of small particles, it is interesting to investigate its effect on a population of crystals with a bimodal particle size distribution. In this section, we assume an initial distribution which is bimodal (simulation #5). This bimodal distribution is made up of two normal distributions, with their corresponding means at y = 0.2 (fines) and 1 (target crystals), and the smaller peak making up 20% of the total mass (the overall initial mean size is then y0̅ = 0.23 (number weighted)). The simulation results are summarized in Figure 6, showing mean particle size, supersaturation, coefficient of variation, and

high holdup ratios. This is expected, since at low holdup ratios either there is less surface area available for growth and dissolution, or the amount of dissolved material, which corresponds to a certain change in supersaturation, is larger, or both. This leads to a larger change in the mean particle size for low holdup ratios (Figure 5a) and, as shown in Figure 5d, a larger change in the number of crystals (note that the number of crystals in the plot is normalized with the initial number). The slower increase in supersaturation means that the critical size decreases more slowly for low values of the holdup ratio than for high values (see Figure 5e). As a consequence, the time taken until growth and dissolution are in equilibrium increases for decreasing holdup ratios; however, the maximal supersaturation and correspondingly the minimum critical size reached do not follow exactly the same trend. Dissolution will continue until the critical size reaches a value small enough to allow for net growth, which depends on several factors, such as the particle size distribution, the particle concentration, and the growth rate. As can be seen in Figure 5c, the PSD widens more for low holdup ratios than for high holdup ratios. Thus, there are more large particles, which means that the critical size does not have to decrease so much in order to allow for net growth. The shape of the particle size distributions is not affected strongly by the initial holdup ratio and is as predicted by LSW theory. Figure 5f shows the maximum relative mass change for different initial holdup ratios. For very small holdup ratios, all particles dissolve before the supersaturation has reached a level high enough to allow for growth to become more important than dissolution. For very high holdup ratios, only a very small fraction of the initial particles dissolve before the growth regime is reached. The fact that an increased solubility leads to faster ripening has been observed experimentally by Liu et al.21 and by Kumar et al.35 This effect is obvious when considering t0, as given by eq 33. Except for the extreme values, the holdup ratio ν has only a minor effect on the dimensionless time at which the asymptotic growth regime is reached. In other words, the effect of c∞ on t0 is what makes a difference. If the change in ν comes from a change in the particle concentration, then no difference is observed. Experiments have shown that, at least in the investigated range, the rate of ripening is independent of the particle concentration.32 When extremely low values of ν are considered, a very different behavior to that predicted by LSW theory is observed. In these simulations, the asymptotic growth regime is not reached in the time considered. Furthermore, the profiles of supersaturation and coefficient of variation are very different. Thus, the conclusion of LSW theory that Ostwald ripening is independent of the particle concentration is not valid in all cases. Sugimoto18 has also considered the effect of solubility on ripening, but in a theoretical model. He concludes that the assumptions of LSW theory are not valid for high solubilities and thus the predictions are not generally valid. According to his work, the initial conditions should be important at high solubilities, an argument which holds also for our simulations. Marqusee and Ross16 argued that while the approach to the final state is not given, the final state itself is independent of the initial conditions and is the same as that given by classical LSW theory. However, this does not hold for very low holdup ratios, at which complete dissolution is observed. This behavior cannot be explained at all using LSW theory. Ostwald Ripening with Bimodal Distributions. In crystallization processes, mechanisms such as secondary nucleation

Figure 6. Summary of simulation #5 with initially bimodal distribution. Profiles of (a) mean particle size (solid, black line) with one standard deviation marked by a dashed line above and below and critical size (gray line), (b) supersaturation, (c) coefficient of variation, and (d) number of crystals.

number of crystals as a function of time, and in Figure 7, showing the PSD as a function of time. It can be clearly seen that the

Figure 7. Volume-based particle size distribution as a function of size for simulation #5.

smaller peak dissolves quickly, while the larger peak remains largely unchanged. The dissolution of the smaller peak leads to an increase in mean particle size and supersaturation, and a 1497



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between the initial state to the asymptotic growth stage and on the effect of the initial conditions. Toward the end of all our simulations, the mean particle size increases with time. The rate of increase is independent of the initial conditions at this stage, except at very low holdup ratios, and is determined by the growth mechanism. Furthermore, the width of the particle size distributions is affected in such a way that a constant coefficient of variation is approached in all cases. The value is again determined by the growth mechanism. The same value (0.21 and 0.35 for the diffusion-limited and reaction-limited growth rates, respectively) is approached, regardless of whether the initial coefficient of variation is above or below the final value. These values can be compared to experimentally measured particle size distributions; for example, for L-glutamic acid particle size distributions with a coefficient of variation around 0.33 have been reported,46 and for acetylsalicylic acid the corresponding values range from 0.34 to 0.65.7 The initial conditions have a strong influence on the initial stages of the ripening process. We have shown that the time until ripening causes changes to a population of particles is shortened if the particles are small, if they have a large variation in size, if the mass of particles is low, or if their solubility in the continuous phase is high. These considerations are important for the design of an industrial process exploiting Ostwald ripening. As our simulations show, ripening can lead to an increase in particle size, a focusing of wide particle size distributions or a widening of narrow particle size distributions. When designing a process, the aim could be to speed up the ripening process and to choose the correct moment at which to end the process in order to obtain the desired particle size distribution. As an example, we have presented a simulation which shows the effect of Ostwald ripening on a population of particles with a bimodal particle size distribution, a situation which is not covered by LSW theory. This simulation suggests that Ostwald ripening can be used in order to remove fines. Thus, a detailed population balance model for Ostwald ripening, such as the one presented here, can give insights not available from LSW theory. Using this approach is unavoidable if one wishes to construct a model of a process which combines the effects of a size-dependent solubility with secondary effects such as breakage or agglomeration. This is, for example, the case when modeling Viedma ripening.41

decrease in coefficient of variation and number of crystals. After this initial dissolution, the particle size distribution is unimodal, with only the target particles remaining. These will continue to be affected by ripening, according to the general behavior discussed above. If the process is stopped at this stage, Ostwald ripening has effectively been exploited for the removal of fines. If the process is allowed to continue, the system evolves toward the asymptotic behavior described by LSW theory. Pesika et al.31 have experimentally observed the ripening of a bimodal distribution, and reported the same behavior as in the simulations presented here.

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DISCUSSION An important aspect when considering Ostwald ripening is how to influence the time required for substantial changes to the particle size distribution to occur. As shown above, there is a period of time at the beginning of a ripening process during which the supersaturation, the mean particle size, and the number of crystals are constant. The first parameter which changes during ripening is the coefficient of variation, that is, the width of the PSD. As suggested by Wagner, in the early stages of Ostwald ripening, the evolution of the coefficient of variation can be described using eq 8. The faster the coefficient of variation increases at the beginning of the process, the earlier particles start to dissolve completely and the earlier substantial changes to the PSD occur. The rate of change of the coefficient of variation is given by ⎛t ⎞ ε dε = 0 exp⎜ ⎟ dt tj ⎝ tj ⎠

(52)

where ε0 is its initial value. It is clear that either a large value of ε0 or a small value of tj lead to a faster change in ε. In other words, a small particle size (which appears in tj as L̅0 to the power of either 2 or 3, depending on the growth mechanism), or a wide particle size distribution or a high solubility (which appears in the denominator of tj as given by eq 9) will lead to faster changes in the particle size distribution. This is exactly what we have observed in the simulations described above. In processes where the precipitation produces particles of the desired size and shape, a large value of tj is required, in order to avoid changes as long as possible. However, if ripening is used to fine-tune the shape and size of particles, as is the case in pigment production, a small value of tj is required, since then the process time can be minimized.

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APPENDIX A: NUMERICAL SOLUTION To solve the PBE the method of pivots described by Ramkrishna47 is used. The distribution function f is discretized onto the points yi using M gridpoints, assuming the population is concentrated at the gridpoints:48

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CONCLUSIONS In this work, we investigate the dynamics of the population balance model used to describe Ostwald ripening. While the basic PBE is the same as that used in the classical LSW theory, the numerical simulations carried out as part of this work have the advantage that they avoid some of the strong assumptions made in other models. The disadvantage is the computational effort required for the full numerical solution. We have carried out a parameter analysis to investigate the effect of the initial mean particle size, the width of the initial particle size distribution, and the initial holdup ratio (a function of the solid phase density, the initial particle mass and their solubility in the continuous phase) on the ripening behavior of particles dispersed in a continuous phase. Some of our results can be explained by LSW theory, but a numerical simulation gives extra information on the transition

M

f (y , t ) =

∑ Fi δ(y − yi ) i=1

(53)

Since the distribution at y = 0 is zero, this point is not included in the sum. The first pivot is located at y1 = ymax/M. The lengthderivative in eq 34 is then approximated using a central difference scheme (forward and backward difference at the grid boundaries). The resulting ODEs in time are solved in MATLAB, using the built-in ODE15s solver. The number of gridpoints used for the simulations varies, but is chosen sufficiently large to give a good agreement between the third moments of the particles calculated using eq 36 and 1498



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+ = Molecular diffusivity, used in diffusion-limited growth rate ε = Coefficient of variation ε0 = Coefficient of variation at time 0 η = Scalar factor, see section Scaling behavior f = Scaled particle size distribution for dimensionless model f 0 = Scaled distribution at time 0 Fi = Discretized PSD, see Appendix A γ = Surface tension γ1D = Skewness of asymptotic PSD, diffusion-limited growth, according to LSW theory γ1R = Skewness of asymptotic PSD, reaction-limited growth, according to LSW theory γ2D = Kurtosis of asymptotic PSD, diffusion-limited growth, according to LSW theory γ2R = Kurtosis of asymptotic PSD, reaction-limited growth, according to LSW theory G(L) = Growth rate of particles of size L G* = Growth rate in dimensionless model G*D = Diffusion-limited growth rate in dimensionless model G*P = Reaction-limited growth rate in dimensionless model, without simplification of supersaturation G*R = Reaction-limited growth rate in dimensionless model GD = Diffusion-limited growth rate GR = Reaction-limited growth rate i = Counter for gridpoints, see Appendix A k = Rate constant for growth or dissolution, used in reactionlimited growth rate ka = Area shape factor kv = Volume shape factor L̅ = Mean particle size λ = Exponent in expression for general growth rate L = Particle size L* = Critical size L0 = Parameter for scaling of dimensionless length LC = Critical length of a particle μj = jth moment of particle size distribution n μjC = jth central moment M = Number of gridpoints, see Appendix A v = Initial holdup ratio n = Particle size distribution n0 = Initial particle size distribution φj = jth moment of scaled PSD f PD(X) = Asymptotic PSD for diffusion-limited growth as function of scaled particle size X, according to LSW theory PR(X) = Asymptotic PSD for reaction-limited growth as function of scaled particle size X, according to LSW theory ρ = Crystal density R = Universal gas constant σ = Standard deviation S = Supersaturation of solute in continuous phase, S = c/c∞ S*(L) = Equilibrium supersaturation (solubility) of a particle of size L S0 = Supersaturation at time 0 SD = Supersaturation profile predicted by LSW theory for diffusion-limited growth SR = Supersaturation profile predicted by LSW theory for reaction-limited growth τ = Dimensionless time, τ = t/t0 T = Absolute temperature t = Time t′D = Alternative time constant for diffusion-limited growth t′R = Alternative time constant for reaction-limited growth

that calculated directly from the simulated particle size distributions.

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APPENDIX B: MEASURES OF THE SHAPE OF PARTICLE SIZE DISTRIBUTIONS In order to easily assess the change in shape of the particle size distributions during a simulation, the standard deviation, coefficient of variation, skewness, and kurtosis are used. All of these can be calculated from a combination of the moments of the distribution (μj represents the jth moment) or the central moments (μjC represents the jth central moment). The standard deviation σ is defined as the square root of the second central moment normalized with the zeroth moment of the distribution as follows: μC 2

σ=

μ0

(54)

and the coefficient of variation is the standard deviation divided by the mean particle size: σμ0 ε= μ1 (55) The skewness γ1 is defined as γ1 =

μC 3 μ0σ3

(56)

For a completely symmetric distribution, for example, a normal distribution, the skewness is 0. Distributions with a tail toward the left (i.e., small sizes in our case) have a negative skew, whereas distributions with a tail to the right (large sizes) have a positive skew. The kurtosis γ2 is defined as γ2 =

μC 4 μ0σ4

−3 (57)

μ4C

where is the third central moment. The kurtosis has a large, positive value for distributions with sharp peaks and long tails, and negative values for distributions with wide peaks and short tails. The normal distribution has a kurtosis of 0.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors would like to thank Dr. Lars Vicum, BASF AG, Ludwigshafen, Germany, for stimulating discussions. NOMENCLATURE α* = Dimensionless capillary length, α* = α/L0 α = Capillary length (combined parameter). α = (4γVm)/ (RT) β = Exponent in expression for general growth rate c = Concentration of solute in continuous phase c*(L) = Solubility of a particle of size L c0 = Initial concentration of solute in continuous phase c∞ = Bulk solubility of particles in continuous phase 1499



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(36) Segets, D.; Hartig, M. A.; Gradl, J.; Peukert, W. Chem. Eng. Sci. 2012, 70, 4−13. (37) Viedma, C. Phys. Rev. Lett. 2005, 94, 065504. (38) McBride, J. M.; Tully, J. C. Nature 2008, 452, 161−162. (39) Noorduin, W. L.; Meekes, H.; Bode, A. A. C.; van Enckevort, W. J. P.; Kaptein, B.; Kellogg, R. M.; Vlieg, E. Cryst. Growth Des. 2008, 8, 1675−1681. (40) Noorduin, W. L.; van Enckevort, W. J. P.; Meekes, H.; Kaptein, B.; Kellogg, R. M.; Tully, J. C.; McBride, J. M.; Vlieg, E. Angew. Chem., Int. Ed. 2010, 49, 8435−8438. (41) Iggland, M.; Mazzotti, M. Cryst. Growth Des. 2011, 11, 4611− 4622. (42) Löbbert, G. Ullmann’s Encyclopedia of Industrial Chemistry; Wiley-VCH Verlag GmbH & Co. KGaA: New York, 2000; Chapter Phthalocyanines. (43) Herbst, W.; Hunger, K.; Wilker, G.; Ohleier, H.; Winter, R. Industrial Organic Pigments; Wiley-VCH Verlag GmbH & Co. KGaA: New York, 2005. (44) LeBlanc, S. E.; Fogler, H. S. AIChE J. 1987, 33, 54−63. (45) Asimow, R. Acta Metall. 1963, 11, 72−73. (46) Lindenberg, C.; Schöll, J.; Vicum, L.; Mazzotti, M.; Brozio, J. Cryst. Growth Des. 2008, 8, 224−237. (47) Ramkrishna, D. Population Balances: Theory and Applications to Particulate Systems in Engineering; Academic Press: San Diego, CA, 2000. (48) Bäbler, M. U. Modelling of Aggregation and Breakage of Colloidal Aggregates in Turbulent Flows; ETH: Zürich, 2007.

t0 = Parameter for scaling of dimensionless time tD = Time constant for diffusion-limited growth tR = Time constant for reaction-limited growth tj = Time constant, tR or tD Vm = Molar volume X = Scaled size, X = L/L* y = Dimensionless length, y = L/L0 Z = Number of particles Z0 = Number of particles at time 0

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Ostwald Ripening - American Chemical Society

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