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.

Secondary Nucleation by Interparticle Energies. III. Nucleation Rate Model

“…To account for the localized stabilization effect and its influence on the kinetic of the SNIPE mechanism, three compartments and the corresponding volume fractions were introduced in part II, 18 as illustrated in Figure 2 b: the solid volume fraction, ϕ s , the stabilization volume fraction, ϕ st , and the bulk solution volume fraction, ϕ b , with ϕ s + ϕ st + ϕ b = 1. The solid volume fraction ϕ s and the stabilization volume fraction ϕ st are given by 18 respectively, where l st (≥0) is the model parameter determining the thickness of the stabilization volume and m 2 seed is the second moment of the PSD of seeds, which measures their total surface area. Note that, in eqs 17 and 18 , the unit conversion factor κ is used for the same reason as in eq 4 and that ϕ st given by eq 18 is an approximation of its exact geometric value, which is justified as discussed in part II.…”

“…Due to the localized stabilization effect, nucleation in the stabilization volume is thermodynamically more favorable than in the bulk solution. This can be described using a different Gibbs free energy for the formation of an n -sized cluster in the different compartments, 17 , 18 denoted by F n st and F n b for the stabilization volume and the bulk solution volume, respectively; 17 , 18 these are defined as where E st (≥1) is the model parameter determining the intensity of the stabilization effect. Replacing F n in eq 12 with Gibbs free energies in each volume ( eq 19 ) results in different equilibrium concentrations of the n -sized clusters in the stabilization volume and the bulk solution volume (denoted by C n st and C n b , respectively), thus yielding the effective equilibrium concentration of the n -sized clusters in the system, C n eff , as a volume-weighted average of C n st and C n b or with the help of eqs 12 and 19 where the effective Gibbs energy F n eff , which is not a thermodynamic quantity, is defined as …”

“…The first is to derive the SNIPE-based nucleation rate model (called SNIPE rate model for brevity). The second is to verify the SNIPE rate model by showing that its output matches the nucleation kinetics described by the KRE model developed in part II 18 and by conducting a sensitivity analysis of the SNIPE rate model. The third objective is to validate the SNIPE rate model by using it for fitting experimental data and to assess its goodness-of-fit and theoretical consistency in comparison with those of other secondary nucleation models.…”

Accounting for the Presence of Molecular Clusters in Modeling and Interpreting Nucleation and Growth

“…The KRE modeling framework has been successfully applied to describe several systems, e.g., the polymorphic system of silica, 32 that of l -glutamic acid, 10 and a system for the synthesis of semiconductor nanocrystals. 33 Moreover, the model has been extended to describe secondary nucleation caused by interparticle energies between seed crystals and molecular clusters, 34 with its description of underlying interparticle interactions built on a thermodynamic analysis. 35 In this work, the Szilard model was used for generating kinetic data.…”

“…According to the assumption on which eq 5 is based, the clusters grow through the mechanism of rough growth. 1 , 37 − 39 The expression for k n d is 1 , 34 where ΔG n ( Δμ , γ) is the Gibbs free energy for the formation of a n -sized cluster at a given driving force for crystallization, Δμ , k B is the Boltzmann constant, and T is the absolute temperature of the system. With the initial condition ( eq 4 ) and the rate constants ( eqs 5 and 6 ), the system of the governing equations (eqs 3) can be solved to describe the temporal evolution of Z n ,0 .…”