A cluster growth model for heterogeneous nucleation

In previous works, we showed that the Keller-Rubinow model of Liesegang ring formation could be regularized by hypothesizing a bistable transition associated with the heterogeneous nucleation of the dichromate on impurities within the gel of the experiment. This hypothesis eliminated an ill-posedness associated with this model, wherein the 'rings' which formed were of infinitesimal thickness, owing to a discontinuity in the model formulation. In the present paper, we consider a discrete stochastic model for nucleation, and show that it can provide a basis for our earlier hypothesis. The result relies on the idea that the detachment rate of the dichromate ions from the impurity surface depends on already present clusters, through a thermodynamic unmixing coefficient. We provide a Monte Carlo simulation of the process, and we derive a stochastic model of it as a sequence of differential equations for the saturation probabilities. Solution of the model gives results which are in agreement with the results of the simulations, and also with a much simpler mean field approximation.