Model for Diffusion Limited Crystal Growth with Growth Rate Dispersion Yielding a System of Discrete Kinetic Modes

We show, in this report, how a population balance model differential equation describing batch crystal growth from solution can be solved in closed form for the case of diffusion limited growth while also modeling the effects of growth rate dispersion. By letting a constant growth rate dispersion (GRD) diffusivity coefficient be directly proportional to the supersaturation, a closed form solution can be found for the case of a separable distribution function of crystal sizes. The result leads to an infinite spectrum of possible GRD diffusivity coefficients, and thus growth rate, values. Therefore, a family of possible equilibrium size distributions exist. It is demonstrated how this result can be used to describe the equilibrium size distributions reported for lactose and sucrose crystal growth where the distribution was suggested to be composed of two crystal ensembles each with a distinct kinetic behavior. The model predicts that different types of crystals, each with distinct kinetic behavior growing from a pure solute, are most likely to be observed when the growth rate diffusivity coefficient is on the order of the bulk solute diffusivity coefficient. Assuming different crystal species forming from a pure solute are crystal polymorphs, and assuming that for polymorphs to appear conditions must be favorable both energetically and kinetically, the model leads to a qualitative description for why certain polymorphs appear while other do not.