| A phase-field model is developed for simulating microstructural pattern formation in solidification of dilute binary alloys with coupled heat and solute diffusion. The model reduces to the sharp-interface equations in a thin-interface limit where (i) the width of the diffuse interface is smaller than the radius of curvature of the interface but much larger than the real width of a solid-liquid interface, and (ii) kinetic effects are negligible. A recently derived anti-trapping current [Karma, A. 2001. Phase-field formulation for quantitative modeling of alloy solidification. Physical Review Letters 87, (September): 115701-1--115701-4] is used in the species equation to recover local equilibrium at the interface and to eliminate spurious effects that arise when the solutal diffusivities are unequal in the solid and liquid. Results are compared to analytical solutions for one-dimensional steady state solidification. Two-dimensional dendritic growth simulations with vanishing solutal diffusivity in the solid show that the solute profile in the solid is accurately modeled by the present approach. Results are presented that illustrate the capacity of the model for simulating dendritic solidification for the large ratios of the liquid thermal to solutal diffusivities typical of alloys. Two-dimensional simulations are used to test the theories of Lipton, Glicksman and Kurz (LGK) and Lipton, Kurz and Trivedi (LKT) for free dendritic growth of binary alloys. Good agreement between numerically predicted and theoretical values of the growth Peclet numbers is obtained. The numerical results show, however, a strong variation of the selection parameter sigma* with the initial concentration in the liquid. This finding stands in contrast with the theories, in which sigma* is supposed to remain constant. Nonetheless, it was found that for pure substances and alloys that solidify isothermally, the selection parameter is approximately the same. Additionally, it was found that for sufficiently high values of the Lewis number there is indeed a tip velocity maximum for varying concentrations, as predicted by the theoretical models. Strong dependencies of sigma* on the Lewis number and the melt undercooling are also found. |
