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 (T = ∞) to T < T c, the critical temperature. The alloy is represented on a simple cubic lattice of N sites by the Ising Model with Kawasaki dynamics assuming a nearest neighbour attraction. The basic kinetic process is the interchange of two unlike particles on adjacent sites, and is Markovian. The unit of time t is taken as one attempted interchange per lattice site. The differential equation used is that of Becker-Doring which assumes the droplets of the new phase to grow or shrink by absorbing one particle at a time. For each size of droplet the equation contains two kinetic coefficients a& and b& which are related to the probability that an &-droplet absorbs or emits one particle. The two coefficients are related by a detailed balance condition. We first find the coefficient a& in the limit of zero density a&(0), assuming steady state diffusion as in the Lifshitz Slyozov theory. We express a&(0) as the solution of a lattice diffusion problem describing the motion of the other particles near a given &-cluster with suitable boundary conditions at infinity and at the surface of the cluster. The differential equation is solved numerically for densities p = 0.05, 0.075, 0.10. The solution compares well with histograms from real alloys (Ni.- Al) and computer simulation of alloys, at the same value of &*, which characterizes the supersaturation. To determine &* (t), we need to know the variation of a& with &*. We find a& is negative for small &* (or high supersaturation) indicating the presence of spinodal decomposition initially. For &* > 26, however, a& is approximately constant (2.0) for 0 t > 5000, however, the differential equation underestimates &* because of coalescence.