Secondary nucleation is ubiquitous in nature and of fundamental importance for both batch and continuous crystallization processes. Attrition is the mechanism through which fragments are formed after the collision of a crystal with a stirrer. Those fine fragments, if small enough, are considered secondary nuclei. In this work, starting from the mechanistic description of attrition by Gahn and Mersmann (Crystallization Technology HandbookCRC Press2001), two population balance equation models to simulate secondary nucleation processes have been derived. The first simulates attrition as a breakage term, and growth rate is the result of size-dependent solubility. The second model considers attrition as a boundary condition at zero crystal size, where the expression for secondary nucleation rate already takes into account the effect of supersaturation, while the growth rate is size-independent. The two models are proven equivalent in the growth regime, thus where secondary nucleation and growth are the dominant phenomena. At extremely low values of supersaturation, thanks to size-dependent solubility, the first model yields to further development of the crystal population, e.g., ripening and aging. The main result is that secondary nucleation by attrition can be described as a birth/death term or, alternatively, as a source term according to the final application of the model. Since the two approaches have very different computational intensities, one can choose the right model based on the objective of the simulation study. The evolution of the crystal population for high values of supersaturation will be the same in both cases.

Primary nucleation is a stochastic process; hence, detection times are statistically distributed even though experiments are repeated at the same conditions. This contribution aims at defining and discussing a method to perform an accurate statistical analysis of detection times; it focuses on three main aspects. First, we develop an accurate experimental protocol and set criteria on temperature, T, and supersaturation, S, to accept measurements as part of the experimental series. Applying such protocol to the isonicotinamide–ethanol system at several different supersaturations, we perform multiple series of isothermal nucleation experiments, calculating for each series the associated empirical cumulative distribution function (eCDF) of the detection times. Second, exploiting a set of statistical tools, we investigate whether the repetitions of the experiments at the same supersaturation conditions belong to the same stochastic process, assessing also the possibility to combine them in one single distribution. Third, we estimate for the eCDFs the values of its associated nucleation rate and derive an analytical expression to calculate the propagation of the uncertainty from the eCDF to the nucleation rate estimate.

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, in the absence of attrition, is known to be dependent on external fields, such as contact forces, shear, or interparticle forces. In this contribution, the thermodynamic effect of the presence of the seed crystal surface on secondary nucleation is derived in the context of the classical nucleation theory. The Gibbs free energy for the formation of a cluster close to a seed crystal is calculated with the addition of interparticle energies, namely, van der Waals attractive forces and Born repulsive forces. This results in the stabilization of a subcritical cluster close to the seed surface that can become a secondary nucleus more easily than under homogeneous nucleation conditions. Far from the seed surface, the developed model is reduced to the homogeneous nucleation described by the classical nucleation theory. The crystallization of paracetamol from an ethanol solution is taken as a case study, and the stabilization effect, given by the presence of interparticle energies, can be observed at different values of supersaturation. Three key indicators have been defined and calculated to describe the intensity of the stabilization effect, two of which, namely, the distance from the seed surface where the stabilization is active and the enhancement factor for supersaturation, are used in Part II of this series to describe the kinetics of secondary nucleation by interparticle energies.

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.