▪ 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.

In this paper we present a phase-field model to describe isothermal phase transitions between ideal binary-alloy liquid and solid phases. Governing equations are developed for the temporal and spatial variation of the phase field, which identifies the local state or phase, and for the composition. An asymptotic analysis as the gradient energy coefficient of the phase field becomes small shows that our model recovers classical sharp interface models of alloy solidification when the interfacial layers are thin, and we relate the parameters appearing in the phase-field model to material and growth parameters in real systems. We identify three stages of temporal evolution for the governing equations: the first corresponds to interfacial genesis, which occurs very rapidly; the second to interfacial motion controlled by diffusion and the local energy difference across the interface; the last takes place on a long time scale in which curvature effects are important, and corresponds to Ostwald ripening. We also present results of numerical calculations. PACS number(s): 81.30.Bx, 82.65. Dp, 68.10.Gw, 64.7Q. Dv and Hilliard [4 -6] have used this approach to model interfacial energies, nucleation, and spinodal decomposition in a binary alloy. Also Langer and Sekerka [7] have modeled the motion of a planar interface using this approach. More generally, various models that employ these ideas are reviewed by Halperin, Hohenburg, and Ma [8], particularly in regard to the study of critical phenomena. The model C given by Halperin e$ a/. has been adapted by Langer [9], and most prolifically by Caginalp [10, 11],to derive the so-called "phase-field model" of solidification which describes the phase change of a pure material.Caginalp has studied this model, and its variations [12,13], extensively. In this model the phase field is required to evolve according to 45 7424Work of the U. S. Government Not subject to U. S. copyright

No abstract

No abstract