An Application of the Finite Differences Method To a Dynamical Interface Problem

Abstract

A multiple-order-parameter model for Cu-Au system on a face cubic centered lattice was recently developed in the presence of anisotropy. In that model, three order parameters (non-conserved) and one concentration order parameter (conserved), which has been taken as a constant, were considered. Later on, the model has been extended, so that, concentration has been taken as a variable. It has been seen that two models were in a good agreement near critical temperature since the non-conserved order parameter behaves like a constant near critical temperature in both models. Thus, we extended the rst model to a dynamical diffuse interface model near critical temperature. After writing the free energy of the system in terms of the order parameters, minimizing the energy with respect to the order parameters and Langevin equation yield the non-linear system of parabolic equations. The finite differences method was implemented to solve this non-linear system of parabolic equations. The forward difference discretization was applied for the rst derivative of the solution with respect to time and centered difference discretization was applied for the second order derivative of the solution with respect to spatial variable. We obtained stability criteria and nd the error bound. The orientation dependence proles, variation of interfacial energy and the effect of the degree of the anisotropy on the width of the diffuse interface are simulated when the time evolves.