Secondary Nucleation by Interparticle Energies. II. Kinetics

Abstract

This work presents a mathematical model that describes growth, homogeneous nucleation, and secondary nucleation that is caused by interparticle interactions between seed crystals and molecular clusters in suspension. The model is developed by incorporating the role of interparticle energies into a kinetic rate equation model, which yields the time evolution of nucleus and seed crystal populations, as in a population balance equation model, and additionally that of subcritical molecular clusters, thus revealing an important role of each population in crystallization. Seeded batch crystallization at a constant temperature has been simulated to demonstrate that the interparticle interactions increase the concentration of the critical clusters by several orders of magnitude, thus causing secondary nucleation. This explains how secondary nucleation can occur at a low supersaturation that is insufficient to trigger primary nucleation. Moreover, a sensitivity analysis has shown that the intensity of the interparticle energies has a major effect on secondary nucleation, while its effective distance has a minor effect. Finally, the simulation results are qualitatively compared with experimental observations in the literature, thus showing that the model can identify operating conditions at which primary or secondary nucleation is more prone to occur, which can be used as a useful tool for process design.

Short abstract

For describing secondary nucleation caused by interparticle interactions between seed crystals and molecular clusters in suspension, a kinetic model is developed. The model demonstrates that the interparticle interactions increase the concentration of the critical clusters by several orders of magnitude, thus causing secondary nucleation at a low super saturation that is insufficient to trigger primary nucleation.

1. Introduction

Crystallization is an essential unit operation used as a purification technique in various industries for the production of chemicals, agrochemicals, nutritionals, and pharmaceuticals. Crystallization process is often initiated by the addition of seed crystals (seeds) of the target product in a supersaturated solution. While these seeds grow in the solution, they can generate new crystals (i.e., nuclei), through the phenomenon called secondary nucleation, which is widespread in industrial crystallization,¹ in particular, in the continuous crystallization of active pharmaceutical ingredients.² Hence, to operate and design continuous crystallization efficiently, we need a sound understanding of secondary nucleation mechanisms.

Traditionally, secondary nucleation has been largely attributed to microbreakages of seed crystals caused by crystal-impeller collisions,³ that is, attrition.^1,4,5 However, recent research has noted that this mechanism cannot describe secondary nucleation completely,^3,6−13 based on two main reasons. First, secondary nucleation can occur even when the crystal microbreakages are prevented by precluding any crystal-impeller collision.¹¹⁻¹³ Second, the attrition mechanism cannot explain some crucial experimental observations according to which secondary nucleation can produce nuclei having a polymorphic¹⁴ or chiral¹⁵⁻¹⁷ form different from that of the seeds. In contrast with these observations, secondary nuclei generated by the attrition mechanism must exhibit the same polymorphic form and the same handedness in the case of chiral molecules of the seeds because these nuclei are the broken pieces of the seeds. These reasons clearly highlight that when describing secondary nucleation, we need to consider not only the attrition mechanism but also other secondary nucleation mechanisms.

Another perspective for rationalizing secondary nucleation suggests that secondary nucleation is induced by interparticle interactions,¹⁸ in particular, the interaction energies between seed crystals and nearby molecular clusters. From this perspective, the interactions facilitate nucleation by reducing the energy barrier for nucleation in the vicinity of the seeds. This type of secondary nucleation can be called secondary nucleation caused by interparticle energies (SNIPE). As opposed to secondary nucleation by attrition, the SNIPE mechanism can explain why some secondary nuclei exhibit a polymorphic/chiral crystal structure different from that of the seeds; this is because the interparticle energies decrease the energy barrier for nucleation independent of the crystal structure of the relevant cluster,^15,18 thus causing secondary nucleation of any crystal structures.

A simple version of the SNIPE mechanism is the embryo-coagulation secondary nucleation (ECSN),¹⁸ a key mechanism according to recent reviews on secondary nucleation.^1,4,5 However, despite its important contribution to our understanding of the SNIPE mechanism, the ECSN model has three critical limitations. First, the ECSN model assumes that subcritical molecular clusters are monodispersed. This assumption contradicts the classical nucleation theory¹⁹ (CNT) whereby a cluster population develops in solution following a Boltzmann distribution. Second, the ECSN model further assumes that the subcritical molecular clusters grow by aggregation only, thus being inconsistent with the CNT where the clusters grow via a sequence of molecule attachments. Third, the ECSN model cannot describe the temporal evolution of any of the three key populations in crystallization—the (subcritical) molecular clusters, nuclei, and seed crystals—thus limiting a deeper understanding of how the SNIPE mechanism works.

The goal of this work is to develop a mathematical model that simultaneously describes homogeneous nucleation, growth, and the SNIPE phenomenon while overcoming the three limitations of the ECSN model. To this aim, the present work extends the conventional kinetic rate equation (KRE) model of nucleation by including a description of the SNIPE mechanism. The structure of this article is as follows. In Section 2, we provide an overview of the conventional KRE model that describes homogeneous nucleation and growth while being consistent with the CNT. In Section 3, we incorporate the description of the SNIPE mechanism into the conventional KRE model. In Section 4, using the developed model, we demonstrate how the SNIPE mechanism works, show the results of a sensitivity analysis, and compare the simulation results qualitatively with available experimental observations.

2. Conventional KRE Model of Nucleation

To describe the SNIPE mechanism, we extend the conventional KRE model of nucleation (i.e., the Szilard model^19,20) by integrating the effect of the interparticle energies. Hence, knowing the working principle of the Szilard model is essential for understanding how our SNIPE model works. To give a brief overview, the basic principle of the model and its governing equations are presented in this section, followed by equations for the attachment and detachment rate constants that are necessary for solving the governing equations.

2.1. Principle

The Szilard model is a well-established theoretical framework that describes homogeneous nucleation and growth via a series of molecule attachments to and detachments from the molecular clusters,^19,21 that is, a description consistent with the CNT.¹⁹ In this approach, a molecular cluster consisting of n solute molecules (i.e., a n-sized cluster) represents a solid particle (i.e., a crystal), when its size n is larger than the critical nucleus size n* (i.e., n > n*) defined in the CNT.¹⁹ Otherwise (i.e., when n ≤ n*), the n-sized cluster is considered as part of the liquid phase; hence, its volume is excluded from the calculation of the solid-phase volume as in the literature.¹⁹ The n-sized clusters can grow to (n+1)-sized clusters via molecule attachments, while (n+1)-sized clusters decay to the n-sized clusters through molecule detachments. This mechanism can be viewed as the following pseudo reaction^19,21

1

where W_n represents a n-sized cluster, k_n^a is the rate constant of monomer attachment to the n-sized clusters, and k_n+1 is the rate constant of monomer detachment from the (n+1)-sized clusters. This formula suggests that the temporal evolution of the cluster concentrations can be described using rate equations once the rate constants k_n^a and k_n+1 are known.

2.2. Governing Equations

On the basis of eq 1, we can formulate a system of KREs for a closed system, each characterizing the rate at which the concentration of the n-sized clusters evolves upon molecule attachments and detachments:¹⁹

2a

2b

where Z_n(t) is the concentration of the n-sized clusters (i.e., their number per unit suspension volume) and Z_nmax+1(t) ≡ 0, with n_max being the upper bound of the cluster size, which must be sufficiently large to avoid any effect on the numerical solution.²²Equation 2b is the mass balance for the solute, ensuring that the total number of solute molecules is conserved. The initial condition is as follows

3

where Z_n,0 is the initial cluster size distribution. The system of eqs 2a and 2b can be solved with the initial condition (eq 3) to obtain the temporal evolution of any initial cluster size distribution for a given set of rate constants (possibly evolving in time, for instance, due to temperature changes).

2.3. Attachment and Detachment Rate Constants

The attachment rate constant k_n^a is determined by the rate-limiting mechanism of the monomer transport to the clusters. It is common that the monomer transfer is limited by the monomer integration step on a cluster surface, for example, in the crystallization of l-glutamic acid in an aqueous solution,²³ lactose in an aqueous solution,²⁴ and paracetamol in an ethanol solution.²⁵ Under this surface-integration limited mechanism, k_n is given by^19,26

4

where α is the sticking coefficient, k_s is the surface area shape factor (e.g., k_s = π for a sphere), k_v is the volume shape factor (e.g., k_v = π/6 for a sphere), V₁ is the volume of a solute molecule, and D is the diffusion coefficient of the solute molecule in the solution. Equation 4 shows that k_n^a is a purely kinetic quantity,¹⁹ as it does not contain any thermodynamic parameter. The detachment rate constant k_n can be derived based on thermodynamic considerations about the concentration of clusters.¹⁹ For instance, at equilibrium, the monomer attachments to the n-sized clusters occur with the same frequency as the monomer detachments from (n+1)-sized clusters, thus kinetically ensuring the microscopic balance of the pseudo reaction (eq 1); this implies

5

where C_n is the equilibrium concentration of n-sized clusters at the actual level of Δμ, that is, the thermodynamic driving force for crystallization;¹⁹ therefore, we write C_n(Δμ). The equilibrium cluster concentration C_n can be expressed as¹⁹

6

where C₀ is the concentration of nucleation sites in the system, ΔG_n(Δμ) is the Gibbs free energy for the formation of a n-sized cluster from n monomers at a given driving force Δμ, k_B is Boltzmann constant, and T is the absolute temperature of the system. The nucleation site concentration C₀ is typically defined as V₁^–1 in the literature,^19,26 but a different value of C₀ calculated from a mass balance was used in this work to consider the bulk solubility c_e of a system, as explained in Appendix A. It is worth highlighting that C₀ is cancelled out in the nondimensional Szilard model (see eqs A.5a–A.8 with E_st = 1), so the general behavior of the model is not affected by the choice of a specific value of C₀. We further note that the probability C_n/C₀ of finding a n-sized cluster at a given nucleation site is exponentially proportional to −ΔG_n/k_BT,¹⁹ thus indicating that the equilibrium cluster size distribution C_n is a Boltzmann-type distribution. The driving force Δμ is defined as²³

7

where C_1,e≡ C₁(0) is the monomer concentration at phase equilibrium (Δμ = 0) and s = Z₁/C_1,e is the supersaturation (ratio) based on the actual monomer concentrations. According to the CNT, ΔG_n is given by¹⁹

8

where γ is the specific surface energy of a cluster and b = k_s(V₁/k_v)^2/3 is the surface area of a solute molecule. Combining eqs 5 and 6, with a consideration that C₁ is very close or equal to Z₁,¹⁹ leads to

9

or with the help of eqs 7 and 8

10

where k₀^a = 0 and k₁ = 0 by definition.¹⁹ Note that eq 9 shows that k^d is determined by both kinetics (i.e., k^a) and thermodynamics (i.e., ΔG).¹⁹

3. Modeling SNIPE

In this section, we extend the Szilard model to describe the SNIPE mechanism by incorporating the assumption that interparticle interactions stabilize cluster formation. This stabilization effect alters the thermodynamics of nucleation (i.e., ΔG) near seed crystals such that the effective detachment rate constants k^d,eff(t) become smaller than those without stabilization, thus accelerating nucleation. For implementation, the identical governing equations of the Szilard model (eqs 2a and 2b) can be reused after replacing the original detachment rate constants k^d with k^d,eff whose formulation is described in the following by considering that only part of a cluster population close to seed crystals gets stabilized by the interparticle energies.

3.1. Interparticle Energies

In part I of this series,²⁷ the thermodynamic aspects of the SNIPE mechanism are examined in detail by considering two classical interparticle forces, that is, van der Waals forces and Born repulsion forces. This contribution, as part II of this series, examines the kinetic aspect of the SNIPE mechanism by including the role of these interparticle interactions in the Szilard model under the following simplifying assumptions:

• As illustrated in Scheme 1a, a seed crystal is surrounded by a stabilization volume V_st(t) consisting of a thin shell with thickness d₁l_st, where d₁ is the characteristic length of a solute molecule and l_st(≥0) is an adjustable nondimensional parameter.

• A uniform field of interparticle energies is applied within V_st, which vanishes outside V_st; this external field stabilizes the cluster formation by decreasing ΔG_n.

Scheme 1. Schematic representation of a seed crystal in a solution when a crystal is spherically shaped; not drawn to scale.

(a) A seed crystal consisting of n solute molecules is surrounded by the stabilization volume, consisting of a thin shell with thickness d₁l_st, where d₁ is the characteristic length of a solute molecule and l_st is an adjustable parameter. (b) A suspension volume consists of a solid phase volume V_s, a stabilization volume V_st, and a bulk solution volume V_b. Cluster formation occurring in V_b is governed by the Gibbs free energy ΔG^b, whereas the cluster formation in V_st is controlled by the Gibbs free energy ΔG^st in that volume. The concentrations Z^b and Z^st represent the cluster concentrations in V_b and V_st, respectively. A flux of molecular clusters f^st→b from V_st to V_b through the interfacial area A is represented by an arrow.

Under these assumptions, we can define the Gibbs free energy ΔG_n^st(Δμ) for the formation of a n-sized cluster in V_st at a given Δμ as follows (see also eq 11 in part I of this series)

11

where E_st(≥1) is a nondimensional parameter that accounts for the intensity of the interparticle energies. By substituting ΔG in eq 9 with ΔG^st, the detachment rate constant k_n^d,st in V_st can be written as

12

Here, k_n^d,b(=k_n) is the detachment rate constant in the bulk solution volume V_b(t), with the corresponding Gibbs free energy ΔG_n^b(=ΔG_n). Equation 12 shows that when the interparticle energies are present (i.e., E_st > 1), k_n is smaller than k_n^d,b, thus implying that clusters grow faster in V_st than they do in V_b. Obviously, when the interparticle energies are absent (i.e., E_st = 1), k_n is equivalent to k_n^d, thus yielding no stabilization effect. To consider different kinetics in V_b and V_st and to describe the evolution of an entire cluster population, the system can be divided into multiple compartments, each described by a mass balance with a corresponding kinetic description. In this work, the system/suspension volume V_susp is partitioned into three compartments, as illustrated in Scheme 1b: the bulk solution volume V_b, the stabilization volume V_st, and the volume of the solid phase V_s, with corresponding time-dependent volume fractions defined as

13

with ϕ_b(t) + ϕ_st(t) + ϕ_s(t) = 1. The solid volume fraction ϕ_s can be calculated by considering that the solid phase consists of supercritical clusters¹⁹

14

where V₁n approximates the volume of a n-sized cluster and n* = (2γb/3Δμ)³ is the critical size.¹⁹ Likewise, ϕ_st can be calculated by considering all stabilization (shell) volumes in the suspension

15a

15b

where n_st is the minimum cluster size that enables the existence of the stabilization volume and d_n is the characteristic length of a n-sized cluster.

For simplicity, n_st is set to be the minimum size of the initial seed crystals, being on the order of 10¹³ in the present work (corresponding to 25–50 μm for small organic molecules). The approximation in eq 15b is valid, provided that the size of seed crystals (i.e., d_n = d₁n^1/3) is much larger than the thickness of the stabilization volume d₁l_st, that is, n^1/3 ≫ l_st, where l_st is varied from 1 to 100 in this study. Note that eq 15b shows that ϕ_st is linearly proportional to the parameter l_st and the total surface area of seed crystals. The concentrations of n-sized clusters in V_b and V_st, denoted by Z^b and Z^st, respectively, can be determined by considering that the total number of n-sized clusters is conserved, that is,

16

and by assuming that their ratio Z_n^b/Z_n is well approximated by that at equilibrium, C_n^b/C_n, given by

17

where C_n^st(Δμ) and C_n(Δμ), likewise in eq 6, are the equilibrium concentrations of the n-sized clusters in V_st and V_b, respectively. Combining eqs 8, 11, 13, 16, and 17 yields

18

19

The mass balances for Z^b and Z^st can be written as

20

21

where f_n^st→b is the flux of the n-sized clusters from V_st to V_b through the interfacial area A, illustrated by an arrow in Scheme 1b, and the monomer concentration in each volume is approximated by Z₁, as in part I of this series.²⁷ Summing up these two equations and using eqs 12, 13, 16, 18, and 19 lead to a system of governing equations for Z_n (n = 2, 3, ..., n_max), analogous to eq 2a, that can be cast as follows

22

where the effective detachment rate constant k_n^d,eff is

23

Equation 23 implicitly assumes that all clusters including the supercritical (n > n*) clusters experience the stabilization effect; it has been used in all simulations reported in this paper. This is because the influence of k_n>n*^d,eff on the simulation results Z_n(t) is negligible compared to that of k_n≤n*, and setting k_n>n*^d,eff ≈ k_n>n* increases the computational time by a factor of 20. Summarizing, a system of the governing equations (eqs 2b and 22) can be solved with the initial condition (eq 3) for simulating a crystallization process in the presence of the stabilization effect that is described by eqs 4, 10, 13–15, and 23. The developed model contains 11 parameters: α_s, k_s, k_v, V₁, D, T, γ, C_1,e(or c_e), Z_n,0, E_st, and l_st. As elaborated in Appendix B, the model can be made dimensionless by introducing two dimensionless variables

24a

24b

where Y_n and τ are the nondimensional concentration and time, respectively. Consequently, the nondimensional model contains five dimensionless parameters: β, Ω, Y_n,0 = Z_n,0/C_1,e, E_st, and l_st. Here, β represents the volume fraction of monomers at phase equilibrium, defined as

25

and Ω is the dimensionless surface energy, defined as

26

The nondimensional model has been solved numerically as explained in Appendix C.

4. Results

To demonstrate the SNIPE mechanism and the application of the developed model to experimental observations in the literature, seeded batch crystallization processes have been simulated at a constant temperature with varying initial supersaturation, initial seed size distribution, strength, and effective range of the stabilization effect. For simulating such conditions, two parameters (Ω and β) have been kept constant, while the other three [E_st, l_st, and Y₀(n)(≡Y_n,0)] have been varied. The values of these five parameters have been specified by adopting the physicochemical properties of a reference case, that is, paracetamol crystallization from an ethanol solution at 20 °C; the physicochemical properties and the corresponding values of the model parameters are reported in Tables 1 and 2, respectively.

Table 1. Physicochemical Properties of the Paracetamol–Ethanol System at 20 °C.

properties	value	description
molecular weight of the solute, Mw	0.151 kg mol–1
solid density, ρs	1263 kg m–3
solvent density, ρsolv	789 kg m–3
mass-based bulk solubility, c̅e	0.180 kg kg–1	from ref (28).a
bulk solubility, ce	5.65 × 1026 m–3	≡c̅eNAρsolv/Mwb
specific surface energy, γ	0.0111 J m–2	assumed value.c
molecular volume of the solute, V1	1.99 × 10–28 m3	≡Mw/ρsNAb
molecular surface area of the solute, b	1.65 × 10–18 m2	≡ks(V1/kv)2/3d
monomer solubility, C1,e	4.82 × 1026 m–3	≡C1(0)

^aThe solubility is based on solute mass per solvent mass.

^bN_A is the Avogadro number.

^cThe specific surface energy is within the range from an experimentally fitted value (0.0043 J m^–2 at 40–46 °C by ref (29)) to a theoretically predicted value (0.0134 J m^–2 at 20 °C by eq 12 in ref (26)).

^dSpherical shape is assumed: k_s = π and k_v = π/6.

Table 2. Model Parameters Used in the Simulations.

symbol	value			description
β	0.0958			monomer volume fraction at phase equilibrium
Ω	4.5			nondimensional surface energy
Est	1, 1.1, 1.2			strength of the stabilization effect
lst	0, 1, 10, 100			effective range of the stabilization effect
Y0(n)	n = 1	1.52–2.34		initial supersaturation s0
	2 ≤ n ≤ n*	0		initial subcritical and critical clusters
	n* < n	ϕs,0	0.0016	initial solid volume fraction
		q3(L)a	normal(50, 0.5) or log–normal(5.14, 0.51)	normal dist. in Sections 4.1 and 4.2; log–normal dist. in Section 4.3

^aVolume-weighted probability density function of the crystal size in micrometer .

The parameters β and Ω have been calculated using eqs 25 and 26, respectively, with the physicochemical properties in Table 1: β = 0.0958 and Ω = 4.5. The parameter E_st = {1, 1.1, 1.2} reflects the strength of interparticle interactions, and its values are conservative estimates of E_st, ranging between 1.04 and 1.26 (see Section 3 in part I of this series²⁷). The parameter l_st represents the range of the stabilization effect. A wide range of {0, 1, 10, 100} has been investigated; a realistic value has been estimated to be around 4 in part I.²⁷ By setting either E_st = 1 (zero strength) or l_st = 0 (zero thickness) or both, the stabilization effect can be eliminated, thus reducing the new SNIPE model to the Szilard model.

The last parameter Y₀(n) determines the initial condition: the initial supersaturation s₀ via Y₀(n = 1), the initial subcritical and critical cluster concentrations via Y₀(2 ≤ n ≤ n*), and the initial seed size distribution via Y₀(n > n*). The initial supersaturation s₀ is equivalent to Y₀(n = 1) (see eqs 7 and 24b), and the studied range is s₀ = 1.52–2.34. For the subcritical and critical clusters, it is typical to assume that they are initially absent,^21,22 thus setting Y₀(2 ≤ n ≤ n*) = 0. To define the initial seed crystal distribution Y₀(n > n*), it is required to set the initial amount of seeds and the initial probability density function (PDF) of the crystal size. The initial amount of seeds has been kept constant by fixing the initial solid volume fraction ϕ_s,0 = 0.0016. This condition corresponds to 1 g of seeds in a 500 mL solution, a reported experimental condition that enables secondary nucleation in the paracetamol–ethanol system at 20 °C.³⁰ For the initial PDF, two cases are considered. In Sections 4.1 and 4.2, we use a narrow Gaussian distribution (characterized by the mean size 50 μm and the standard deviation 0.5 μm) to discuss the general behavior of the developed model. In Section 4.3, to compare our simulation results with experimental observations qualitatively, we use a log–normal distribution that adequately characterizes an experimental seed size distribution (explained in detail in Section 4.3).

The simulation results are presented in three parts. In Section 4.1, we demonstrate the key features of the model in describing the time evolution of a cluster size distribution, including the formation of primary and secondary nuclei, thus leading to a bimodal crystal size distribution. Moreover, we verify that the stabilization effect can indeed cause secondary nucleation. In Section 4.2, we present the results of a sensitivity analysis that elucidates the effect of the simulation parameters on the final crystal size distribution. Finally, in Section 4.3, we make a qualitative comparison between simulation results obtained using the SNIPE model and experimental observations reported in the literature.

4.1. Demonstration of the Stabilization Effect

The developed model can provide a comprehensive description of batch crystallization processes in which the growth of seeds can occur together with secondary nucleation induced by the interparticle energies. These processes are simulated in the model such that the initial cluster size distribution evolves over time with or without the stabilization effect while experiencing a series of molecule attachments (i.e., growth) and detachments (i.e., dissolution). The model yields the time evolution of a cluster size distribution Y_n(τ) that reveals insightful dynamics of the three key populations in the crystallization processes, namely, the populations of subcritical clusters, nuclei, and seed crystals. We emphasize that this type of insightful information cannot be attained using a (classical) population balance model.²¹

In this section, we demonstrate how the stabilization effect can cause secondary nucleation by examining three simulation results. Two results are obtained from the simulations with no stabilization effect (i.e., E_st = 1 and l_st = 0) at two levels of the initial supersaturation s₀: high level s₀ = 2.22 (Figure 1a) and low level s₀ = 1.93 (Figure 1b). The other simulation result is obtained when including the stabilization effect (corresponding to E_st = 1.2 and l_st = 10) and at the low supersaturation s₀ = 1.93 (Figure 1c). All three simulations are illustrated in Figure 1 in terms of the volume-weighted size distributions nY_n (solid lines) and the critical sizes n* (dashed lines) at three different times. Note that nY_n is called volume-weighted distribution in the literature,²¹ though the distribution is weighed by the number of molecules because it is assumed quite rightly that the volume of a n-sized cluster is proportional to the number of molecules that comprise the cluster.¹⁹

Figure 1.

Comparison of three simulations conducted with two levels of the initial supersaturation s₀ and in the absence (a,b) or the presence (c) of the stabilization effect. For each simulation, the volume-weighted size distributions (nY_n(τ) vs n) at three times τ are depicted by the solid lines, and the corresponding critical sizes (n*(τ)) are depicted by the dashed lines. In all relevant subplots, the point where the solid line intersects the dashed vertical line corresponds to the critical clusters, and its vertical coordinate is an indication of their concentration. A set of parameters for each simulation is as follows: (a) E_st = 1; l_st = 0; s₀ = 2.22. (b) E_st = 1; l_st = 0; s₀ = 1.93. (c) E_st = 1.2; l_st = 10; s₀ = 1.93.

The simulation without the stabilization effect at high supersaturation is analyzed first. The earliest snapshot [Figure 1(a1)] reveals the fast dynamics of a subcritical cluster population; a large number of subcritical clusters form rapidly. A small part of this population grows larger than the critical size, thus forming a population of primary nuclei through homogeneous nucleation. On the right-hand side of the figure, one can observe a seed crystal population in the form of a narrow Gaussian distribution. In the next snapshot [Figure 1(a2)], a growing nucleus population dramatically changes the cluster size distribution by increasing both the number and size of the nuclei. The third snapshot [Figure 1(a3)] is mainly characterized by a bimodal crystal size distribution consisting of both the seeds and the population of primary nuclei. All snapshots at the bottom row of Figure 1 represent the results at the final process times τ_f at which the supersaturation is close to unity and the number of crystals is at its maximum (see Section S2 in the Supporting Information for more details).

The next simulation, illustrated in Figure 1b, is performed with no stabilization effect and a low level of the initial supersaturation s₀ = 1.93. In this simulation, a nucleus population does not form (see Figure 1(b1,b2)); consequently, the seed crystals completely dominate the crystallization process through their growth, thus resulting in a unimodal crystal size distribution, as presented in Figure 1(b3). A comparison of the two simulations discussed so far (Figure 1a,b) indicates that the low initial supersaturation s₀ = 1.93 is insufficient to trigger primary nucleation in the bulk solution for the system under consideration here.The third simulation is illustrated in Figure 1c. Although this simulation begins with the low initial supersaturation, a substantial amount of nuclei appear [see Figure 1(c1,c2)] because of the stabilization effect. This stabilization effect enhances the overall driving force for crystallization and thus increases the concentration of the critical clusters by several orders of magnitude, as shown in Figure 1(c1,c2). Their subsequent growth and that of the seeds yield a bimodal crystal size distribution [see Figure 1(c3)]. The observed nucleation can be regarded as secondary nucleation caused by the stabilization effect, in other words, SNIPE. This result, obtained using the SNIPE model developed here, can explain why secondary nucleation can occur even when the initial supersaturation is at such a low level that primary nucleation does not occur.

4.2. Sensitivity Analysis

A sensitivity analysis has been conducted in two steps. First, various simulations have been carried out by altering three model parameters: the intensity of the stabilization effect E_st = {1, 1.1, 1.2}, the effective range of the stabilization effect l_st = {0, 1, 10, 100}, and the initial supersaturation s₀ = {1.52, 1.64, 1.76, 1.87, 1.93, 1.99, 2.11, 2.17, 2.22, 2.34}. Second, simulation results have been compared with respect to two characteristic quantities: the degree of nucleation, ψ_N, which indicates how many new crystals have formed with respect to their initial count, and the degree of growth, ψ_G, which specifies how much seed crystals have grown with respect to their initial size.

4.2.1. Degree of Nucleation and Growth

These two characteristic quantities can be readily calculated from a cluster size distribution, Y_n(τ), by using its moments. Hence, we first express the j-th moment m_j(p,τ) of the distribution Y_n(τ) in which the considered cluster size n is larger than a given size p

27

Based on this definition, the degree of nucleation ψ_N can be defined by dividing the zeroth moment of the final crystal size distribution, m_0,f^*, by the initial one, m_0,i

28

where n*(τ) is the critical size at time τ. This equation gives ψ_N ≈ 1 when nucleation is negligible, simply because the final number of crystals is close to the initial one (i.e., m_0,f^* ≈ m_0,i). On the contrary, this definition yields ψ_N ≫ 1 when nucleation occurs because the final number of crystals is much larger than the initial one (i.e., m_0,f^* ≫ m_0,i). Similarly, the degree of growth ψ_G can be defined by dividing the final volume-weighted average size of the seed crystals, n̅_seed,f, by the initial one, n̅_seed,i

29

where n_seed,min(τ) is the smallest size of a seed crystal size distribution at time τ. This equation gives ψ_G ≈ 1 when the growth of seed crystals is negligible (n̅_seed,f ≈ n̅_seed,i), whereas it gives ψ_G > 1 when the seeds grow noticeably (n̅_seed, f > n̅_seed,i).

4.2.2. Results

The effect of the initial supersaturation s₀ on ψ_N and ψ_G in the absence of the stabilization effect (E_st = 1 and l_st = 10) is illustrated in Figure 2 by the circle markers. When the initial supersaturation s₀ is below the threshold value s_th(E_st = 1) = 2.17 (the left side of the vertical dotted line in Figure 2), ψ_N remains around 1 (see Figure 2a), while ψ_G increases linearly with s₀ (see Figure 2b). By increasing s₀ beyond the threshold value s_th(E_st = 1), ψ_N increases by orders of magnitude, whereas ψ_G drops to 1. This effect of s₀ on ψ_N and ψ_G is also present in the simulations with the stabilization effect (E_st = {1.1, 1.2}) but with the lower threshold values, s_th(E_st = 1.1) = 2.10 and s_th(E_st = 1.2) = 1.87, as illustrated in Figure 2 by the square markers for E_st = 1.1 and the triangle markers for E_st = 1.2. In all three cases, if s₀ < s_th, the growth of the seeds is the only dominant crystallization mechanism, indicated by ψ_N ≈ 1 and ψ_G > 1. By increasing s₀ beyond s_th, the dominant crystallization mechanism continuously shifts from growth to primary nucleation (in the case of E_st = 1) or secondary nucleation (in the case of E_st > 1), indicated by ψ_N ≫ 1 and ψ_G → 1. Furthermore, the threshold initial supersaturation s_th decreases with increasing strength of the stabilization effect E_st. This suggests that secondary nucleation by interparticle energies can occur with a smaller value of s₀ if a system exhibits strong interparticle interactions between seeds and molecular clusters.

Figure 2.

Degree of nucleation ψ_N is shown in plot (a) and degree of growth ψ_G is presented in plot (b) corresponding to the simulations conducted with varying values of the initial supersaturation s₀ and the stabilization effect strength E_st and a fixed value of the stabilization effect range l_st = 10. A vertical dotted line indicates the threshold initial supersaturation for E_st = 1: s_th(E_st = 1). Other lines are just a guide for the eye.

Next, we shift our attention to another group of simulations for studying the influence of s₀ and l_st = {1, 10, 100} at a fixed value of E_st = 1.2. In the following, the degree of nucleation ψ_N alone is sufficient to demonstrate the impact of the model parameters. A higher value of l_st means a longer effective range of the interparticle energies. As illustrated in Figure 3, although we increase l_st by 1 or 2 orders of magnitude, the threshold supersaturation barely decreases. This minor effect of l_st on the degree of nucleation ψ_st sharply contrasts with the major effect of E_st.

Figure 3.

Degree of nucleation ψ_N of the simulations conducted with varying values of the initial supersaturation s₀ and the stabilization effect range l_st and a fixed value of the stabilization effect strength E_st = 1.2. Lines are just a guide for the eye.

To further understand the relative importance of these two parameters, a set of simulations have been executed with varying values of E_st and l_st at a constant value of s₀ = 2.22, as illustrated in Figure 4. This figure clearly shows that the degree of nucleation ψ_N changes more dramatically when we change E_st than when l_st is varied. This result can be explained by considering the functional form of eqs 15b and 23, where E_st determines the effective detachment rate constant k^d,eff more dominantly than l_st does.

Figure 4.

Degree of nucleation ψ_N of the simulations conducted with varying values of the stabilization effect strength E_st and the stabilization effect range l_st and a fixed value of the initial supersaturation s₀ = 2.22.

4.3. Comparison with Experimental Observations: Paracetamol Crystallization from an Ethanol Solution

In this section, we show the relevance of SNIPE-based simulations to experimental crystallization processes by comparing our simulation results with available experimental observations qualitatively. For the comparison, simulation results [i.e., Y_n(τ)] were converted into experimentally relevant quantities, namely, the bulk supersaturation S and the volume-weighted PDF of the crystal size q₃(L), as explained in Appendix A and Appendix D, respectively. The conditions of the reference experiments were set up in the model as detailed in the following.

4.3.1. Setting up Experimental Conditions in the Model

References (30) and (31) are selected as benchmark experimental studies for two main reasons. One reason is that these experiments were conducted with the paracetamol–ethanol system, a widely studied system from various perspectives.^{25,28,30−34} Another reason is that the chosen works were carried out systematically as a series; prior to their investigation of secondary nucleation,³⁰ the authors extensively studied primary nucleation^33,34 and growth³¹ for the same system, thus offering various types of experimental data sets obtained by the same researchers in a consistent manner.

In the following, we summarize the experimental conditions in refs (30) and (31) and explain how these conditions are reflected in the model parameters summarized in Table 2. The reference experiments consist of seeded batch crystallization of paracetamol from an ethanol solution at 20 °C; these conditions were implemented in the model by setting β = 0.0958 and Ω = 4.5, calculated using eqs 25 and 26, respectively, with the properties reported in Table 1. The experiments are started by seeding 1 g of crystals in a 500 mL supersaturated solution; the corresponding seed size distribution is shown in Figure 5 in terms of the volume-weighted PDF q₃(L) of the crystal size L; these conditions were reproduced in the model by setting the initial solid volume fraction ϕ_s,0 = 0.0016 and by approximating the experimental seed size distribution with a log–normal distribution (see the dash-dotted line in Figure 5), characterized by the mean size 5.14 and standard deviation 0.51 of ln L.

Figure 5.

Experimental data of the volume-weighted PDF of the initial crystal size (q₃(L) vs L) shown by a bar graph and a fitted log–normal distribution (characterized by the mean 5.14 and standard deviation 0.51 of ln L) by a dash-dotted line. The displayed experimental data are adapted with permission from Figure 5c in ref (31). Copyright Elsevier (2011).

4.3.2. Comparison

Two sets of simulations were performed, namely, either in the absence or presence of the stabilization effect (E_st = 1 and l_st = 0 and E_st = 1.2 and l_st = 10, respectively), each for estimating a threshold initial bulk supersaturation for primary nucleation and secondary nucleation, respectively. These estimates were compared with the values inferred from the experiments.

The results of the first set are presented in Figure 6a in terms of q₃(L). When S₀ is larger than around 1.8, the simulation yields a bimodal size distribution, thus indicating the occurrence of primary nucleation. This suggests that the threshold initial bulk supersaturation is around S_th(E_st = 1) ≈ 1.8. It is important to note that S_th(E_st = 1) varied around ±5% when the specific surface energy γ in Table 1 was varied around ±10%. A corresponding experimental value for the threshold supersaturation can be inferred from the experimental data in ref (31) that characterized the metastable zone width, reproduced in Figure 6c. This figure shows that, at 20 °C, primary nucleation was detected when the initial bulk supersaturation was larger than 1.84 (corresponding to the point B in the figure), which is close to the value estimated by the model. This comparison demonstrates that the model can estimate a threshold supersaturation for primary nucleation adequately.

Figure 6.

(a) Simulations conducted with no stabilization effect (E_st = 1 and l_st = 0) at two values of the initial bulk supersaturation S₀ = {1.78, 1.83}. (b) Simulations performed with the stabilization effect (given by E_st = 1.2 and l_st = 10) at two values of the initial bulk supersaturation S₀ = {1.5, 1.6}. (c) Solubility and metastable zone width data. [Reproduced with permission from ref (31). Copyright Elsevier (2011)]. (d) SEM image of the final crystals showing agglomerates. [Reprinted with permission from ref (30). Copyright Elsevier (2012.)].

The outcome of the second simulation set is illustrated in Figure 6b, indicating that the threshold supersaturation for secondary nucleation falls in the range between 1.5 and 1.6. This range agrees well with the experimental observations in ref (30) where a value of the initial supersaturation in the range S₀ = 1.42–1.71 indeed caused secondary nucleation. This again shows that the developed model could be used to support future experiments, for instance, when selecting the initial supersaturation for a crystallization process.

We do acknowledge that the final PDFs in our simulations may not quantitatively match the experimental ones because of two main reasons. One reason is that, in experiments, secondary nucleation is caused by a combination of multiple mechanisms (e.g., attrition and SNIPE), but the developed model considers only the SNIPE mechanism. The second reason is that the final PDF can be critically modified by other crystallization phenomena (e.g., crystal breakage³⁵⁻³⁷ and agglomeration^38,39), and these phenomena are not always predictable. In the reference experiment (ref (30)), agglomeration clearly occurred, as evident from the reported SEM image of the final crystal population (see Figure 6d). Since the developed model describes neither agglomeration nor breakage, the simulation outcome cannot quantitatively agree with the experiments where any of these phenomena may occur. Though modeling crystal agglomeration and/or crystal breakage in a KRE model is possible, this is nevertheless beyond the scope of this work.

5. Conclusions

We have developed a mathematical model that simultaneously describes growth, homogeneous nucleation, and secondary nucleation caused by the interparticle energies, that is, due to the stabilization effect caused by the interactions between seeds and clusters in suspension. The key feature of the model is the incorporation of the stabilization effect into the conventional KRE model of nucleation. By simulating seeded batch crystallization of paracetamol from an ethanol solution at 20 °C, this work demonstrates how the stabilization effect causes secondary nucleation at a low initial supersaturation by increasing the concentration of the critical clusters by orders of magnitude. Moreover, a sensitivity analysis has been carried out to understand the relative importance of three key simulation parameters: the initial supersaturation, s₀, the strength of the stabilization effect, E_st, and the effective range thereof, l_st. The analysis shows that primary and secondary nucleation are enhanced to varying degrees on increasing any of the three parameters, and the parameter E_st has a major effect on the stabilization effect, while the parameter l_st has a minor effect. The simulation results are compared with the experimental observations in the literature qualitatively. The comparison verified a capability of the model to estimate a threshold initial bulk supersaturation at which primary or secondary nucleation occurs as the estimated values adequately agree with the values inferred from the reference experiments.

The model can be employed to obtain qualitative information on threshold supersaturation for nucleation, which is a useful tool for designing crystallization processes. Moreover, the developed model presents a new perspective on secondary nucleation mechanisms through which one can rationalize some experimental observations that cannot be explained by the attrition mechanism, for instance, the formation of secondary nuclei whose polymorphic form or chiral structure is different from that of a seed,¹⁴⁻¹⁷ the occurrence of secondary nucleation caused by a seed crystal that was held stationary,¹¹⁻¹³ and the appearance of secondary nuclei in a no-impeller crystallizer (e.g., airlift crystallizer^40,41).

Finally, demonstrating that the theory developed in part I and II of this series has potential for application and experimental validation remains a challenge. To address this, the complex KRE model used here needs to be transformed into a nucleation rate model, which can be used in a standard population balance modeling framework. This would prove that the SNIPE theory can describe experiments and would allow for a comparison of the SNIPE rate model with other well-recognized secondary nucleation models. These aspects will be addressed in part III of this series,⁴² where the practical relevance of secondary nucleation rate by interparticle energies will be proved.

Appendix A

Mass Balance in Liquid Phase

The solute concentration c(Δμ) at a given Δμ is equal to the total number of the solute molecules forming subcritical and critical clusters, which can be written as

A.1

where the critical size n*(Δμ) is defined according to the CNT¹⁹

A.2

Equation A.1 was applied to calculate the bulk supersaturation S = c/c_e, which was used only for comparing simulation results with the relevant experimental observations in Section 4.3. Note that the monomer-based supersaturation s was used to define the driving force Δμ (see eq 7). At phase equilibrium (i.e., Δμ = 0), Z_n can be approximated by the equilibrium concentration C_n (eq 6) and c(0) is equal to the bulk solubility c_e by definition; eq A.1 leads to

A.3

Finally, C₀ can be obtained by substituting eqs 6 and 8 into eq A.3

A.4

Appendix B

Nondimensionalization

Nondimensionalization of differential equations is a convenient technique that makes physical quantities in differential equations dimensionless through the chain rule and some algebra. This technique effectively decreases the number of model parameters by grouping them into a smaller set.⁴³ This can ease computational burden significantly because one simulation result in a nondimensional model can represent multiple results in the original model. For nondimensionalization, two dimensionless variables Y_n and τ (see eqs 24a and 24b) are introduced. Note that Y₁ is equivalent to s by definition (see eq 7) and that the time t is scaled by the characteristic time for monomer attachments at phase equilibrium, that is, (k₁^aC_1,e)⁻¹. Substituting these definitions into the governing equations (eqs 2b and 22) leads to

A.5a

A.5b

where η_n and ξ_n(τ) are the nondimensional rate constants defined as

A.6

A.7

The corresponding initial condition is

A.8

It is worth noting that one can recast eqs 14 and 15b using dimensionless quantities as

A.9

A.10

Appendix C

Solution of the SNIPE Model

The governing equations (eq A.5a) are a set of integro differential equations as the rate constants ξ(τ) depend on the moments of a cluster size distribution Y(τ). These equations can be approximated by a set of ordinary differential equations (ODEs) by treating ξ as constants at every time integration, for example, for the j-th integration, ξ_n(τ) = ξ_n(τ_j) for τ_j ≤ τ < τ_j+1. This approximation is reasonable because the cluster concentrations Y_n vary much faster than ξ_n, yielding a small integration time step Δτ_j ≡ τ_j+1 – τ_j, and during each time integration, the relative change in ξ_n is very small, satisfying the following condition

A.11

Accordingly, we can approximate ξ_n(τ) by ξ_n(τ_j) for τ_j ≤ τ < τ_j+1 as

A.12

A.13

Moreover, the governing equations (eqs A.5a and A.5b) are a set of n_max discrete rate equations, each equation describing the concentration of clusters, with the size n ranging from 1 to n_max. For describing the dynamics of large crystals, for instance, 800 μm crystals, the required n_max can be on the order of 10¹⁸. This turns the governing equations into a set of 10¹⁸ or more differential equations that definitely causes formidable computational burden. This issue can be adequately resolved by approximating the discrete rate equations with the Fokker–Planck partial differential equation (FPE).^21,22,44 The FPE can be readily derived by applying the second-order Taylor series expansion to the two terms of eq A.5a around n, namely, η_n–1Y₁Y_n–1 and ξ_n+1Y_n+1,¹⁹ yielding

A.14

A.15

A.16

where Ỹ(ñ,τ), the continuous counterpart of Y_n, is a function of the continuous size ñ, v(ñ,τ) is the growth rate (i.e., the net molecule attachment frequency) of the n-sized cluster, and H(ñ,τ) is the dispersion coefficient of the clusters along the size axis ñ, reflecting a fluctuating change in the cluster size ñ.¹⁹ However, because the approximation with the FPE is not sufficiently accurate for small ñ, it is common to use the original discrete equations for small size (i.e., n ≤ N) and the FPE for larger size (i.e., ñ > N),^21,22,44 yielding a hybrid size coordinate. In this work, N is set to 50 as it has been reported that increasing N beyond 50 does not improve the accuracy of the solutions.^21,22 To smoothly combine the FPE with the discrete equations, it is necessary to adapt eqs 27, A.5b, A.9, and A.10 to the hybrid size domain, spatially discretize the FPE by using a suitable discretization method, and impose two boundary conditions, as explained in Section S1 in the Supporting Information. In this work, the incorporation of the FPE reduces the total number of ODEs substantially from a value on the order of 10¹⁸ to 4000. The discrete rate equations and the spatially discretized (or semi-discrete) FPE are integrated over time using the MATLAB’s ode15s solver for stiff ODEs.

Appendix D

Conversion of Simulation Results

This section explains a procedure for converting a crystal size distribution Y_n>n*(τ) to a corresponding volume-weighted PDF q₃(L,τ) of the crystal size in micrometer, L. The size variable n is connected to the size L through the expression for a cluster volume as

A.17

where the number 10^–18 converts the unit [μm³] into [m³]. By rearranging eq A.17 to obtain L, we can derive a conversion function L = f(n)

A.18

A crystal size distribution Y_n>n*(τ) can be first converted to a volume-weighted PDF q̅₁(n > n*,τ) of the size n

A.19

Next, by using eq A.18 and applying the change-of-variable technique, q̃₁(n,τ) can be converted to q₃(L,τ)

A.20

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.cgd.1c00928.

• Adaptation of eqs 24b, A.5b, A.9, and A.10 to the hybrid size domain; ñ-space discretization of the FPE; two boundary conditions to smoothly combine the FPE with the original discrete equations; numerical method to solve FPE; time evolution of supersaturation; and the zeroth moment of a crystal population (PDF)

The authors declare no competing financial interest.