Phase-Field Approach for Faceted Solidification

TL;DR

This paper extends the phase-field method to model faceted solidification, demonstrating convergence for equilibrium shapes and analyzing needle crystal growth, with results aligning with experimental scaling laws and analytical predictions.

Contribution

The authors develop an approximate gamma-plot approach with rounded cusps for faceted solidification within the phase-field framework, enabling accurate modeling of faceted growth.

Findings

Phase-field approach converges for equilibrium shapes.

Facet length scales inversely with square root of growth rate.

Analytical theory reasonably predicts growth dynamics.

Abstract

We extend the phase-field approach to model the solidification of faceted materials. Our approach consists of using an approximate gamma-plot with rounded cusps that can approach arbitrarily closely the true gamma-plot with sharp cusps that correspond to faceted orientations. The phase-field equations are solved in the thin-interface limit with local equilibrium at the solid-liquid interface [A. Karma and W.-J. Rappel, Phys. Rev. E53, R3017 (1996)]. The convergence of our approach is first demonstrated for equilibrium shapes. The growth of faceted needle crystals in an undercooled melt is then studied as a function of undercooling and the cusp amplitude delta for a gamma-plot of the form 1+delta(|sin(theta)|+|cos(theta)|). The phase-field results are consistent with the scaling law "Lambda inversely proportional to the square root of V" observed experimentally, where Lambda is the facet length and V is the growth rate. In addition, the variation of V and Lambda with delta is found to be reasonably well predicted by an approximate sharp-interface analytical theory that includes capillary effects and assumes circular and parabolic forms for the front and trailing rough parts of the needle crystal, respectively.