Statistical distribution model for crystal growth kinetics under impurity pinning

Impurities play a critical role in crystal growth by adsorbing onto growth steps and reducing step advancement rates. In this study we introduce the Statistical Distribution Model (SDM), which incorporates the random distribution of immobile impurities using Poisson statistics. We define two key parameters: the velocity reduction factor (A), which quantifies how strongly an impurity reduces step velocity once pinned, and the pinning probability scaling constant (B), which reflects how likely it is for impurity pairs to fall within a critical pinning distance. Using these two parameters, we classify impurities into at least five types, ranging from those that weakly reduce velocity and rarely pin to those that strongly inhibit growth through frequent pinning events. Depending on impurity strength and coverage, we show that SDM accurately describes three commonly observed behaviors during the growth of crystals in impure solutions: complete growth arrest, partial inhibition, and linear reduction in velocity. The model shows excellent agreement with experimental data from ADP, KBr, and sucrose systems, offering improved insight into impurity-induced growth kinetics.