Formulation of partition coefficient of impurities in a growing crystal based on the Burton-Cabrera-Frank model

Abstract

This paper presents an extension of a classical model of the crystal growth, the Burton–Cabrera–Frank (BCF) model, to a system that includes impurity molecules and discusses the formulation of the amount of the impurity uptake into the crystal as a partition coefficient. The BCF model considers the following surface processes of host molecules: adsorption from a solution to a terrace, desorption from the terrace to the solution, surface diffusion on the terrace, incorporation at a step edge, and release from the step edge to the terrace. The above-mentioned processes have been formulated not only for the host molecules but also for the impurity molecules (Miura, 2020). In this paper, the author analytically solves the previous model and derives an analytic formula for the partition coefficient as a ratio of the uptake of the impurity molecule to that of the host molecule into the crystal. The dependence of the partition coefficient on supersaturation is systematically investigated. The analytic formula is applied to the data obtained in the experiments of protein crystal growth to discuss the behavior of impurity molecules on the growing crystal surface. This model provides a theoretical framework for eliciting quantitative data from the crystal growth experiments on impurity partitioning.