Kinetics of phase segregation in a quenched alloy

Abstract

We model the time evolution of a lattice gas or binary alloy quenched from infinite temperature to a temperature T less than Tc, the critical temperature. The alloy is represented on a simple cubic lattice of N sites by the Ising Model with Kawasaki dynamics assuming nearest neighbour attraction. The Becker-Doring equations are used to model the rate of change of the distribution of cluster sizes in the quenched lattice gas. The coefficients of these equations are calculated from first principles by solving a diffusion problem for the concentration of particles near a given cluster. These coefficients are found for cluster sizes up to 6, and extrapolated to larger cluster sizes. The resulting version of the Becker-Doring equations are then solved numerically for T = 0.59Tc, for three separate densities, p, of the lattice gas: p = 0.05, 0.075 and 0.10. Simulations of the lattice gas with these parameters were carried out by Marro and others. AT each density, the differential equations give good predictions of the cluster size distribution in the corresponding simulation, when the critical cluster size, usually denoted by l*, is the same in both. The cluster size distribution in terms of l* predicted by the Becker-Doring equations also compares very well with that observed in real alloys (Ni-Al), and is an improvement on the theory of Lifshitz and Slyozov, which consistently underestimates the number of large clusters. The critical cluster size, l*(t), is compared between simulations and the differential equations at the same value of the time t. For the lower density, p = 0.05, l*(t) is very well predicted by the Becker-Doring equations. For the higher densities, p = 0.075 and 0.10, l* (t) was found to be approximately linear over the whole time range. However, the Becker-Doring equations underestimate the rate of growth of l*(t) by a factor of 0.3.