Lattice Growth Models

Abstract

Since the 1960’s, Monte Carlo simulations with lattice gas models have been used to study a variety of surface phenomena including roughening transitions [1–3] and the properties of adsorbed monolayers [4]. These lattice gas models are based on the Ising model [4,5]. Lattice sites are assigned occupation variables, c i , with values 1 and 0, if the site is occupied or empty, respectively. The Hamiltonian for the lattice generally depends on the interactions between occupied nearest- (n) and next-nearest neighbor (nn) sites. This approach has also been used to model equilibrium and growth morphologies of crystals [6–9] and growth rate dependence on temperature and supersaturation [10,11]. These studies focused on interface kinetics and in some cases included surface diffusion. Recently [12–16], the interplay between bulk transport and interface kinetics has been included in lattice growth models. Transport of the growth species in the nutrient phase is accounted for by considering the random walk of a potential growth unit. At the interface, attachment, surface diffusion and detachment kinetics are considered. Transition probabilities for surface processes are calculated from pair interaction energies, growth temperature and the nutrient chemical potential at the surface. In this sense, the model may be considered as a combination of the Gilmer-Bennema model [10] and the diffusion limited aggregation (DLA) model of Witten and Sander [17]. Examples of 2D and 3D simulations are presented which demonstrate the ability of this approach to incorporate both microscopic and macroscopic contributions to the evolution of growth morphologies.