Numerical simulation of a binary alloy of 2D Cahn-Hilliard model for phase separation

Abstract

The interfacial dynamics of a system are significant in understanding the system's behavior. One of the special models of these systems which naturally arises in material sciences is the phase separation (or spinodal decomposition) of binary alloys. A nonlinear evolution equation known as the Cahn-Hilliard equation is used to establish the mathematical modeling of a binary alloy for phase separation. Apart from trivial solutions, the Cahn-Hilliard equation is a fourth-order nonlinear equation without an analytical solution. This study investigates a second-order splitting finite difference scheme based on the 2D Crank-Nicolson technique to approximate the solution of the 2D Cahn-Hilliard problem with Neumann boundary conditions. Using the inherent error estimation of the 2D Crank-Nicolson method, we have shown that the proposed scheme is second-order accuracy and we have proved that the scheme has a unique solution. In addition, we demonstrated that the suggested technique retains mass conservation while decreasing total energy. Furthermore, we chose two numerical examples, one with a special initial value and the other with a random initial value, to confirm the accuracy of the proposed approach. These numerical experiments display physical features of the Cahn-Hilliard model such as coarsening dynamics and spinodal decomposition obtained by experimental studies of binary alloys.