On anisotropy function in crystal growth simulations using Lattice Boltzmann equation

TL;DR

This paper demonstrates how the Lattice Boltzmann method can effectively simulate crystal growth with various shapes by accurately modeling anisotropy functions and interface normals, ensuring isotropy and realistic growth patterns.

Contribution

It introduces a lattice-specific directional derivatives method for computing interface normals, improving isotropy in crystal growth simulations using LB methods.

Findings

Directional derivatives method ensures isotropic solutions.

Simultaneous growth of multiple crystals modeled successfully.

Various 3D crystal shapes simulated with different anisotropy functions.

Abstract

In this paper, we present the ability of the Lattice Boltzmann (LB) equation, usually applied to simulate fluid flows, to simulate various shapes of crystals. Crystal growth is modeled with a phase-field model for a pure substance, numerically solved with a LB method in 2D and 3D. This study focuses on the anisotropy function that is responsible for the anisotropic surface tension between the solid phase and the liquid phase. The anisotropy function involves the unit normal vectors of the interface, defined by gradients of phase-field. Those gradients have to be consistent with the underlying lattice of the LB method in order to avoid unwanted effects of numerical anisotropy. Isotropy of the solution is obtained when the directional derivatives method, specific for each lattice, is applied for computing the gradient terms. With the central finite differences method, the phase-field does not match with its rotation and the solution is not any more isotropic. Next, the method is applied to simulate simultaneous growth of several crystals, each of them being defined by its own anisotropy function. Finally, various shapes of 3D crystals are simulated with standard and non standard anisotropy functions which favor growth in <100>-, <110>- and <111>-directions.