Growth of a needle crystal from an undercooled alloy melt

Abstract

The growth of a free dendrite from a supercooled alloy melt is studied with the phase-field model in the isothermal approximation; the surface tension is assumed to be asymmetric with fourfold anisotropy. For fixed supercooling and when the anisotropy parameter \ensuremath{\gamma} is not too high, the computed tip radius \ensuremath{\rho} and velocity ${v}_{\mathrm{tip}}$ obey the scaling law ${\ensuremath{\rho}}^{2}{v}_{\mathrm{tip}}\ensuremath{\propto}{\ensuremath{\gamma}}^{\ensuremath{-}7/4},$ predicted by the microscopic solvability theory. The solute diffusion across the solid-liquid interface reflects the nonequilibrium conditions of the growth process: the concentration jump deviates from the static value and decreases with increasing interface velocity. We also explore the effect of different diffusivities in the solid and liquid phases. Previous studies based on the equilibrium formulation of the free boundary equations predict a monotonic increase of ${\ensuremath{\rho}}^{2}{v}_{\mathrm{tip}}$ with the solute diffusivity of the solid phase; due to nonequilibrium effects this result is not recovered in the present investigation.