| The phase field model in two dimensions is used to calculate the operating states (tip velocity, v, and tip radius, {dollar}rho{dollar}) of dendrites grown from pure melts. In this model, the sharp interface that separates solid from liquid is replaced by a diffuse interface described in terms of an auxiliary variable, the phase field. Explicit interface tracking is replaced by the solution of coupled partial differential equations, one for the phase field and another for the temperature. In the asymptotic limit of zero interface thickness, these equations incorporate the Gibbs-Thomson effect as well as linear interface kinetics, consistent with a classical model. Fourfold anisotropies of both surface tension and interface kinetics are treated. The equations are solved by finite difference techniques using vectorized algorithms on a Cray C90 supercomputer. For affordable computation times, results independent of computational parameters can be obtained only at large supercoolings, which results in a dendrite tip radius that is large compared to the thickness of the diffuse interface but small compared to a domain size. Further, the 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 then both are anisotropic. The growth velocity is found to increase as a power law with increasing supercooling, decreasing effect of interface kinetics and increasing anisotropies of surface tension and interface kinetics. Interface kinetics are shown to have a strong effect in that they lead to a smaller velocity and a larger tip radius at a given supercooling. The Peclet number {dollar}P = {lcub}{lcub}vrho{rcub}over2kappa{rcub}{dollar}, where {dollar}kappa{dollar} is the thermal diffusivity, is found to increase with supercooling while the opposite is true for the selection parameter {dollar}sigmasp* = {lcub}2kappa dsb{lcub}o{rcub}over vrhosp2{rcub}{dollar}, where {dollar}dsb{lcub}o{rcub}{dollar} is the capillary length. Their dependence on interface kinetics is found to be strongly influenced by anisotropies. |
