We present a phase-field model (PFM) for solidification in binary alloys, which is found from the phase-field model for a pure material by direct comparison of the variables for a pure material solidification and alloy solidification. The model appears to be equivalent with the Wheeler-Boettinger-McFadden (WBM) model [A.A. Wheeler, W. J. Boettinger, and G. B. McFadden, Phys. Rev. A 45, 7424 (1992)], but has a different definition of the free energy density for interfacial region. An extra potential originated from the free energy density definition in the WBM model disappears in this model. At a dilute solution limit, the model is reduced to the Tiaden et al. model [Physica D 115, 73 (1998)] for a binary alloy. A relationship between the phase-field mobility and the interface kinetics coefficient is derived at a thin-interface limit condition under an assumption of negligible diffusivity in the solid phase. For a dilute alloy, a steady-state solution of the concentration profile across the diffuse interface is obtained as a function of the interface velocity and the resultant partition coefficient is compared with the previous solute trapping model. For one dimensional steady-state solidification, where the classical sharp-interface model is exactly soluble, we perform numerical simulations of the phase-field model: At low interface velocity, the simulated results from the thin-interface PFM are in excellent agreement with the exact solutions. As the partition coefficient becomes close to unit at high interface velocities, whereas, the sharp-interface PFM yields the correct answer.

We developed an efficient computation scheme for the phase-field simulation of grain growth, which allows unlimited number of the orientation variables and high computational efficiency independent of them. Large-scale phase-field simulations of the ideal grain growth in two-dimensions (2D) and three-dimensions (3D) were carried out with holding the coalescence-free condition, where a few tens of thousands grains evolved into a few thousand grains. By checking the validity of the von Neumann-Mullins law for individual grains, it could be shown that the present simulations were correctly carried out under the conditions of the ideal grain growth. The steady-state grain size distribution in 2D appeared as a symmetrical shape with a plateau slightly inclined to the small grain side, which was quite different from the Hillert 2D distribution. The existence of the plateau stems from the wide separation of the peaks in the size distributions of the grains with five, six, and seven sides. The steady-state grain size distribution in 3D simulation of the ideal grain growth appeared to be very close to the Hillert 3D distribution, independent of the initial average grain size and size distribution. The mean-field assumption, the Lifshitz-Slyozov stability condition, and all resulting predictions in the Hillert 3D theory were in excellent agreement with the present 3D simulation. Thus the Hillert theory can be regarded as an accurate description for the 3D ideal grain growth. The dependence of the growth rate in 3D simulations on the grain topology were discussed. The large-scale phase-field simulation confirms the 3D growth law obtained from the Surface Evolver simulations in smaller scales.

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We show that a finite interface thickness in an isothermal phase-field model for alloy solidification introduces a chemical potential gradient in the interfacial region and the solute trapping effect, and the relationships between material properties and parameters in the phase-field equation can be determined at a low interface velocity limit condition. Also we show that there are upper bounds on the interface thickness and the interface kinetics coefficient, which vary with thermophysical properties of alloys. The predicted compositions of solid and liquid at the interface with this model are in good agreement with those obtained by numerical simulation of one-dimensional isothermal solidification of an Al-2-mole-%-Si alloy at 870 K. ͓S1063-651X͑98͒01309-9͔

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