Modeling of crystal morphology distributions. Towards crystals with preferred asymmetry

Exploitation of crystal symmetry is very important in formulation and efficient simulation of population balance models for crystal morphology. This work presents the first population balance model for morphology distribution considering the diversity of symmetry. In this model, we analyze the symmetry of a population of crystals using group theory and divide the population into various symmetry classes, which, in turn, is subdivided into various morphological forms. The internal coordinate vector for any given crystal can be symmetry reduced and can be grouped into various sets characterized by identical growth rates. It has been shown that the internal coordinate vector can be represented in such a way that only one internal coordinate needs to be treated dynamically for each set while all other coordinates remains invariant during growth. This leads to a very small number of dynamic internal coordinates and the effective dimensionality of the problem becomes very small, allowing simulation of a population of asymmetric crystals with minimal computational effort. It has been shown using this model that the concentration of more symmetric crystals invariably increases during the growth process. However, this natural gravitation of the crystal population towards more symmetric forms can be controlled by manipulating the supersaturation which has been shown using numerical examples.