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. E $53,$ 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 $\gamma ={\gamma }_0[1+\delta (|\sin \theta |+|\cos \theta |\left)\right].$ The phase-field results are consistent with the scaling law $\Lambda \sim V^{-1/2}$ 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.

• Received 17 February 2003

DOI:https://doi.org/10.1103/PhysRevE.68.041604