▪ 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.