Modeling Fluctuations in the Growth Rate of a Single Crystal

Abstract

The growth rate of a single crystal depends on various mesoscopic variables, e.g., irregularities such as defects and steps in the surface. It is highly plausible that such irregularities render the growth rates of crystals of an identical size to vary randomly. This work aims at modeling stochastically the fluctuations in the growth rate of a single crystal in a crystallizer. The crystal size in any of the equally-divided domains is considered as the random variable defining the state of the system. The transition of the crystal size from one state to another is characterized by a set of transition-intensity functions which, as for the case of the deterministic growth rate, exhibit a power-law dependence on the size. The master equation for the system is formulated through probabilistic balance around a particular state by taking into account all mutually exclusive events. The resultant nonlinear master equation has been expanded in power series of a small parameter, i.e., the reciprocal of the maximum crystal size obtainable, by means of the system-size expansion. This has yielded expressions for the means and variances of the size of a single particle. Comparison of the simulated crystal size with available experimental data indicates that the present stochastic model adequately portrays the dynamic behavior of a single crystal.