Online Monitoring of the Concentrations of Amorphous and Crystalline Mesoscopic Species Present in Solution.

Despite the growing evidence for the existence of amorphous mesoscopic species in a solution and their crucial roles in crystallization, there has been the lack of a suitable method to measure the time-resolved concentrations of amorphous and crystalline mesospecies in a lab-scale stirred reactor. This has limited experimental investigations to understand the kinetics of amorphous and crystalline mesospecies formation in stirred solutions and made it challenging to measure the crystal nucleation rate directly. Here, we used depolarized light sheet microscopy to achieve time-resolved measurements of amorphous and crystalline mesospecies concentrations in solutions at varying temperatures. After demonstrating that the concentration measurement method is reasonably accurate, precise, and sensitive, we utilized this method to examine mesospecies formation both in a mixture of two miscible liquids and in an undersaturated solution of dl-valine, thus revealing the importance of a temperature change in the formation of metastable and amorphous mesospecies as well as the reproducibility of the measurements. Moreover, we used the presented method to monitor both mesospecies formation and crystal nucleation in dl-valine solutions at four different levels of supersaturation, while achieving the direct measurement of the crystal nucleation rates in stirred solutions. Our results show that, as expected, the inherent variability in nucleation originating from its stochastic nature reduces with increasing supersaturation, and the dependence of the measured nucleation rate on supersaturation is in reasonable agreement with that predicted by the classical nucleation theory.

Secondary Nucleation by Interparticle Energies. III. Nucleation Rate Model.

A nucleation rate model for describing the kinetics of secondary nucleation caused by interparticle energies (SNIPEs) is derived theoretically, verified numerically, and validated experimentally. The theoretical derivation reveals that the SNIPE mechanism can be viewed as enhanced primary nucleation, i.e., primary nucleation with a lower thermodynamic energy barrier (for nucleation) and a smaller critical nucleus size, both caused by the interparticle interactions and the associated energy between the surface of a seed crystal and a molecular cluster in solution, as shown in part I of this series. In the case of a sufficiently agitated suspension, the model depends on four parameters: two reflecting primary nucleation kinetics and the other two accounting for the intensity and effective spatial range of the interparticle interactions. As a numerical verification of the model, we show that the nucleation kinetics described by the SNIPE rate model is in quantitative agreement with those given by the kinetic rate equation model developed in part II of this series. A sensitivity analysis of the SNIPE rate model is conducted to present the effect of key model parameters on the nucleation kinetics. Moreover, the SNIPE rate model is validated by fitting the model to the time-resolved data of secondary nucleation experiments as well as to two other, well-known secondary nucleation rate models. Importantly, all of the estimated parameter values for the SNIPE model were consistent with the theoretical estimates, while some of the estimated parameter values for one of the well-known secondary nucleation models deviated from the corresponding theoretical values significantly.

Secondary Nucleation by Interparticle Energies. II. Kinetics.

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.

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

The effect of molecular cluster formation on the estimation of kinetic parameters for primary nucleation and growth in different systems has been studied using computationally generated data and three sets of experimental data in the literature. It is shown that the formation of molecular clusters decreases the concentration of monomers and hence the thermodynamic driving force for crystallization, which consequently affects the crystallization kinetics. For a system exhibiting a strong tendency to form molecular clusters, accounting for cluster formation in a kinetic model is critical to interpret kinetic data accurately, for instance, to estimate the specific surface energy γ from a set of primary nucleation rates. On the contrary, for a system with negligible cluster formation, a consideration of cluster formation does not affect parameter estimation outcomes. Moreover, it is demonstrated that using a growth kinetic model that accounts for cluster formation allows the estimation of γ from typical growth kinetic data (i.e., de-supersaturation profiles of seeded batch crystallization), which is a novel method of estimating γ developed in this work. The applicability of the novel method to different systems is proven by showing that the estimated values of γ are closely comparable to the actual values used for generating the kinetic data or the corresponding estimates reported in the literature.