Simple and practical method for the simulations of two-component PFC models for binary colloidal crystals on curved surfaces

Abstract

Herein, we propose a practical numerical method for solving two-component phase-field crystal model with vacancy effect on curved surfaces and perform various numerical simulations. In the present work, the curved surfaces are defined by the zero level-set of signed distance functions. By adopting the pseudo-Neumann boundary condition and closest-point type method, we can extend the computation into a three-dimensional narrow band domain which contains the curved surfaces. The spatial discretization is performed by using the standard finite difference method and the seven-point Laplacian operator can be used to replace the surface Laplace–Beltrami operator. Therefore, the numerical implementation is simple and efficient. The semi-implicit discretization with stabilization term is considered to update the solutions in time. By simulating the pattern formations on curved surfaces with various shapes, we validate the good performance of the proposed method.