A Numerical Implementation of the Finite-Difference Algorithm for solving Conserved Cahn–Hilliard Equation

Abstract

Phase field modelling technique is critical to contextualizing material microstructures and to represent the composition of microstructural evolution. This work utilizes the periodic boundary condition to numerically solve the Cahn-Hilliard equation. To enhance computation and improve flexibility, Python programming language is introduced to develop and implement the proposed approach. The numerical implementation considered a hypothetical binary system of element A and B using the finite difference method on the conserved order parameter. The work also validates the concentration dependent gradient of the system and the energy coefficient which serves as the first step to show spinodal decomposition in a system. The implementation involves solving the Cahn Hillard equation in multi-dimensions capturing minimal time steps evolution, thus serving as an esplanade, an approach into crystallization. The system shows that the element A and B can be used to describe evolution phases alpha (α) and beta (β) through a persisting thermodynamic variable to form a single phase. The time-dependent phase morphology of the studied system, and the concentration and mobility effects are discussed in this paper.