Modelling anisotropic Cahn-Hilliard equation with the lattice Boltzmann method

TL;DR

This paper introduces a novel multiple-relaxation-time lattice Boltzmann method for solving the anisotropic Cahn-Hilliard equation, effectively modeling nanoscale crystal surface faceting with validated numerical simulations.

Contribution

It develops a new lattice Boltzmann scheme tailored for the anisotropic Cahn-Hilliard equation, addressing nonlinear challenges and ensuring energy dissipation.

Findings

Successfully models droplet deformation and spinodal decomposition

Recovers macroscopic governing equations via Chapman-Enskog analysis

Demonstrates adherence to energy dissipation law

Abstract

The anisotropic Cahn-Hilliard equation is often used to model the formation of faceted pyramids on nanoscale crystal surfaces. In comparison to the isotropic Cahn-Hilliard model, the nonlinear terms associated with strong anisotropic coefficients present challenges for developing an effective numerical scheme. In this work, we propose a multiple-relaxation-time lattice Boltzmann method to solve the anisotropic Cahn-Hilliard equation. To this end, we reformulate the original equation into a nonlinear convection-diffusion equation with source terms. Then the modified equilibrium distribution function and source terms are incorporated into the computations. Through Chapman-Enskog analysis, it successfully recovers the macroscopic governing equation. To validate the proposed approach, we perform numerical simulations, including cases like droplet deformation and spinodal decomposition. These results consistent with available works, confirming the effectiveness of the proposed approach. Furthermore, the simulations demonstrate that the model adheres to the energy dissipation law, further highlighting the effectiveness of the developed lattice Boltzmann method.