Diffusion during phase separation

Nearest neighbor interactions have been incorporated into novel equations that describe diffusion processes during phase separation and ordering. None of the elements of the model are new, but the gestalt is overdue. Nearest neighbor interactions are included in the free energy of a binary alloy, then the chemical potentials of the two species are derived from this free energy. The diffusion fluxes are driven by the chemical activities, which are given by the chemical potentials. This results in fourth-order difference equations that describe the motion of atoms during phase separation or ordering. Calculations using these equations describe the expected evolution of microstructures, which have often been depicted only with cartoons. In the unstable spinodal region, perturbations grow exponentially with time until the compositions reach the phase boundary compositions, and then the structures coarsen slowly. In the meta-stable region between the spinodal and the phase boundary, composition fluctuations that are large enough generate phase separation by a nucleation process. In the single-phase region outside the two-phase field, there is critical slowing down. Far away from the critical region, the equations become standard diffusion equations, where the diffusion process is driven only by entropy. Numerical solutions describe the temporal evolution of microstructures during spinodal decomposition, precipitation, and ordering. The propagation of instability fronts in the spinodal region have been observed, and their velocities have been determined. More complex interactions between atomic sites are expected to change only the details but not the essential character of the behavior observed in this model. Based on universality, it is expected that method can be applied to many situations where there is an interaction between the volume elements.