No abstract

We extend the phase-field approach to model the solidification of faceted materials. Our approach consists of using an approximate γ-plot with rounded cusps that can approach arbitrarily closely the true γ-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 δ for a γ-plot of the form γ = γ0[1 + δ(| sin θ| + | cos θ|)]. The phase-field results are consistent with the scaling law Λ ∼ V −1/2 observed experimentally, where Λ is the facet length and V is the growth rate. In addition, the variation of V and Λ with δ is found to be reasonably well predicted by an approximate sharpinterface analytical theory that includes capillary effects and assumes circular and parabolic forms for the front and trailing rough parts of the needle crystal, respectively.