Computation of the dendritic operating state at large supercoolings by the phase field model.

Abstract

The phase field model in two dimensions is used to calculate numerically the operating states (tip velocity $v$ and tip radius $\ensuremath{\rho}$) of dendrites grown from pure melts. At large supercoolings, a dendrite has a nearly hyperbolic envelope close to its tip, as opposed to being nearly parabolic, as at small supercoolings. The corresponding tip radius increases with supercooling for anisotropy only in surface tension, decreases for anisotropy only in interface kinetics, and displays a mixture of these behaviors when both are anisotropic. The growth velocity is found to increase as a power law with increasing supercooling, to decrease with increased effect of interface kinetics, and to increase with increased anisotropies of surface tension and interface kinetics. Interface kinetics are shown to have a strong effect in that a smaller kinetic coefficient leads to a smaller velocity and a larger tip radius at a given supercooling. The P\'eclet number $P=\frac{v\ensuremath{\rho}}{(2\ensuremath{\kappa})}$, where $\ensuremath{\kappa}$ is the thermal diffusivity, is found to increase with supercooling while the opposite is true for the selection parameter $\ensuremath{\sigma}=2\frac{\ensuremath{\kappa}{d}_{0}}{(v{\ensuremath{\rho}}^{2})}$, where ${d}_{0}$ is the capillary length. The dependencies of $P$ and $\ensuremath{\sigma}$ on interface kinetics are found to be influenced strongly by anisotropies.