Modeling Nucleation, Growth, and Ostwald Ripening in Crystallization Processes: A Comparison between Population Balance and Kinetic Rate Equation

Abstract

In this work, we investigate a comprehensive model describing nucleation, growth and Ostwald ripening based on the kinetic rate equation and compare it to commonly used population balance equation models that either describe nucleation and crystal growth or crystal growth and Ostwald ripening. The kinetic rate equation gives a microscopic description of crystallization, i.e., the process is seen as an attachment and detachment of crystals of different sizes to and from each other, thereby changing their size. A hybrid model is employed in which the discrete kinetic rate equation is used to describe the smallest particle sizes while a Fokker–Planck equation is used to approximate the kinetic rate equation at larger particle sizes. This allows us to cover crystals in a size range starting from a single molecule up to macroscopic particle sizes and to solve the model numerically with reasonable computational effort and great accuracy. We show that the model based on the kinetic rate equation describes the processes of nucleation, crystal growth, and Ostwald ripening accurately in a single, continuous model. This is set in contrast with classical population balance equation models that require, due to their underlying assumptions, separation of the process of nucleation from the process of Ostwald ripening. We compare the results of the two models for different sets of parameters (such as different solubilities, surface tensions, initial supersaturations, and seed distributions). Using these results, we assess the advantages and disadvantages of models based on the kinetic rate equation in comparison to models employing a population balance equation.