A new phase-field model for anisotropic surface diffusion: Anisotropic Cahn-Hilliard equation with improved conservation

As popular approximations to sharp-interface models, the Cahn-Hilliard type phase-field models are usually used to simulate interface dynamics with volume conservation. However, the convergence rate of the volume enclosed by the interface to its sharp-interface limit is usually at first order of the interface thickness in the classical Cahn-Hilliard model with constant or degenerate mobilities. In this work, we propose a variational framework for developing new Cahn-Hilliard dynamics with enhanced volume conservation by introducing a more general conserved quantity. In particular, based on Onsager's variational principle (OVP) and a modified conservation law, we develop an anisotropic Cahn-Hilliard (ACH) equation with improved conservation (ACH-IC) for approximating anisotropic surface diffusion. The ACH-IC model employs a new conserved quantity that approximates step functions more effectively, and yields second-order volume conservation while preserving energy dissipation for the classical anisotropic surface energy. The second-order volume conservation as well as the convergence to the sharp-interface surface diffusion dynamics is derived through comprehensive asymptotic analysis. Numerical evidence not only reveals the underlying physics of the proposed model in comparison with the classical one, but also demonstrates its exceptional performance in simulating anisotropic surface diffusion dynamics.