▪ Abstract  An overview of the phase-field method for modeling solidification is presented, together with several example results. Using a phase-field variable and a corresponding governing equation to describe the state (solid or liquid) in a material as a function of position and time, the diffusion equations for heat and solute can be solved without tracking the liquid-solid interface. The interfacial regions between liquid and solid involve smooth but highly localized variations of the phase-field variable. The method has been applied to a wide variety of problems including dendritic growth in pure materials; dendritic, eutectic, and peritectic growth in alloys; and solute trapping during rapid solidification.

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Rapid advances in atomistic and phase-field modeling techniques as well as new experiments have led to major progress in solidification science during the first years of this century. Here we review the most important findings in this technologically important area that impact our quantitative understanding of: (i) key anisotropic properties of the solid-liquid interface that govern solidification pattern evolution, including the solid-liquid interface free energy and the kinetic coefficient; (ii) dendritic solidification at small and large growth rates, with particular emphasis on orientation selection; (iii) regular and irregular eutectic and peritectic microstructures; (iv) effects of convection on microstructure formation; (v) solidification at a high volume fraction of solid and the related formation of pores and hot cracks; and (vi) solid-state transformations as far as they relate to solidification models and techniques. In light of this progress, critical issues that point to directions for future research in both solidification and solid-state transformations are identified.

A phase-field model is developed for simulating quantitatively 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 computationally tractable thin-interface limit where (i). the width of the diffuse interface is about one order of magnitude smaller than the radius of curvature of the interface but much larger than the real microscopic width of a solid-liquid interface, and (ii). kinetic effects are negligible. A recently derived antitrapping current [Phys. Rev. Lett. 87, 115701 (2001)]] is used in the solute conservation equation to recover precisely local equilibrium at the interface and to eliminate interface stretching and surface diffusion effects that arise when the solutal diffusivities are unequal in the solid and liquid. Model results are first compared to analytical solutions for one-dimensional steady-state solidification. Two-dimensional thermosolutal dendritic growth simulations with vanishing solutal diffusivity in the solid show that both the microstructural evolution and the solute profile in the solid are accurately modeled by the present approach. Results are then presented that illustrate the utility of the model for simulating dendritic solidification for the large ratios of the liquid thermal to solutal diffusivities (Lewis numbers) typical of alloys.

A multiphase/multiscale model is used to predict the columnar-to-equiaxed transition (CET) during solidification of binary alloys. The model consists of averaged energy and species conservation equations, coupled with nucleation and growth laws for dendritic structures. A new mechanism for the CET is proposed based on solutal interactions between the equiaxed grains and the advancing columnar frontas opposed to the commonly used mechanical blocking criterion. The resulting differences in the CET prediction are demonstrated for cases where a steady state can be assumed, and a revised isotherm velocity (V T ) vs temperature gradient (G) map for the CET is presented. The model is validated by predicting the CET in previously performed unsteady, unidirectional solidification experiments involving Al-Si alloys of three different compositions. Good agreement is obtained between measured and predicted cooling curves. A parametric study is performed to investigate the dependence of the CET position on the nucleation undercooling and the density of nuclei in the equiaxed zone. Nucleation undercoolings are determined that provide the best agreement between measured and calculated CET positions. It is found that for all three alloy compositions, the nucleation undercoolings are very close to the maximum columnar dendrite tip undercoolings, indicating that the origin of the equiaxed grains may not be heterogeneous nucleation, but rather a breakdown or fragmentation of the columnar dendrites.

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of alloy species due to the relative movement or ow of segregated liquid and solid during solidi cation. Modelling and simulation of macrosegregationThere are numerous causes of uid ow and solid has experienced explosive growth since the pioneering studies of Flemings and co-workers in movement in casting processes: the mid-1960s. This paper presents a review of d ow that feeds the solidi cation shrinkage and the recent macrosegregation models, with particular contractions of the liquid and solid during cooling emphasis on their application to selected d buoyancy induced ows due to thermal and solutal industrially relevant casting processes. The gradients in the liquid; the thermal and solutal successes and shortfalls of the models in buoyancy forces can either aid or oppose each predicting measured macrosegregation patterns other, depending on the direction of the thermal are noted. Although it is still early in their gradient and whether the rejected solutes cause an development, advanced macrosegregation models increase or a decrease in the density of the liquid that include a detailed consideration of the d forced ows due to pouring, motion of gas bubbles, solidification microstructure are also reviewed. Recent studies involving direct numerical applied magnetic elds, stirring, rotation, simulation of macrosegregation related vibration, etc. phenomena on a microscopic scale are d movement of free (equiaxed) grains or solid fraghighlighted. Important issues deserving future ments that have heterogeneously nucleated in the research attention are identified. IMR/378