On anisotropy function in crystal growth simulations using Lattice Boltzmann equation

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.

Modeling ice crystal growth using the lattice Boltzmann method

“…The kinetics of phase-field are described by a parameter φ designating solid and fluid phases respectively when reaching +1 (ice) and -1 (vapor) and the reduced supersaturation of water vapor defined by U = (c − c S sat )/c S sat , where c S sat (T ) is the saturation number density of vapor of ice at temperature T . The space/time evolution equations are written as [27,41,42,43]:…”

“…To validate the model and subsequent implementation, we use the generic system with four-fold symmetry studied in [41,53]. To provide both qualitative and quantitative proof, we compare the shape of the dendrites and the evolution of the tip velocity.…”

“…In that same spirit a number of work have proposed LBMbased formulations to solve the phase-field evolution equation. Younsi et al [41] modified the standard lattice Boltzmann (LB) equations and equilibrium distribution function to introduce anisotropic surface tension and growth rate effects. The proposed model was successfully used to simulate anisotropic and dendritic growth.…”

“…Figure3: Iso-contours of φ = 0 at time t = 1 × 10 5 δt. Black circles are reference results from[41].…”

Multiple-Relaxation-Time Lattice Boltzmann scheme for fractional advection–diffusion equation

“…( 1) in several particular cases (see e.g. [31] and [26] for the ADE and [32,33] for the phase-field models of crystallization) provided the equilibrium function | f f f eq is appropriately chosen.…”