A Comprehensive Approach to Predicting Crystal Morphology Distributions with Population Balances

A modeling framework is developed for predicting crystal morphology distributions with a goal toward their control in the manufacture of crystalline products. This work distinguishes itself from prior efforts in this direction by its comprehensive coverage of all possible morphologies based only on fundamental molecular information on the material. The morphology of growing crystals is composed of a finite number of low-energy faces characterized by their Miller indices and perpendicular distances. The symmetry of crystals allows the classification of kinetically and geometrically similar faces into different groups identified by their perpendicular distances (h-vector). A set of different kinds of morphologies obtained by combining different groups of faces is referred to as a morphology set. Further, a morphology graph is constituted with vertices as elements of morphology set and edges specifying conditions for transformations between morphologies. These conditions form a polyhedral cone, which will be called a morphology domain, in the space of h-vectors. The dynamics of crystal morphology is given by the trajectories of the h-vector inside this morphology domain. The evolution of morphology distributions due to crystal growth inside the morphology domain is described by a morphological-population balance model (M-PBM) which readily submits to solution by the method of characteristics. The methodology, illustrated in controlling crystal morphologies of potassium acid phthalate using additives, paves the way for model-based control of shape control of crystallization processes.