▪ 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

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Sn-rich alloys in the Sn-Ag-Cu system are being studied for their potential as Pb-free solders. Thus, the location of the ternary eutectic involving L, (Sn), Ag 3 Sn and Cu 6 Sn 5 phases is of critical interest. Phase diagram data in the Sn-rich corner of the Sn-Ag-Cu system are measured. The ternary eutectic is confirmed to be at a composition of 3.5 wt % Ag, 0.9 wt % Cu at a temperature of 217.2 ± 0.2 °C (2σ). A thermodynamic calculation of the Sn-rich part of the diagram from the three constituent binary systems and the available ternary data using the CALPHAD method is conducted. The best fit to the experimental data is 3.66 wt % Ag and 0.91 wt % Cu at a temperature of 216.3 °C. Using the thermodynamic description to obtain the enthalpy-temperature relation, the DTA signal is simulated and used to explain the difficulty of liquidus measurements in these alloys.

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Hot tearing in castings is closely related to the difficulty of bridging or coalescence of dendrite arms during the last stage of solidification. The details of the process determine the temperature at which a coherent solid forms; i.e., a solid that can sustain tensile stresses. Based on the disjoining-pressure concept used in fluid dynamics, a theoretical framework is established for the coalescence of primaryphase dendritic arms within a single grain or at grain boundaries. For pure substances, approaching planar liquid/solid interfaces coalesce to a grain boundary at an undercooling (⌬T b ), given bywhere ␦ is the thickness of an isolated solid-liquid interface, and ⌬⌫ b is the difference between the grain-boundary energy, ␥ gb , and twice the solid/liquid interfacial energy, 2␥ sl , divided by the entropy of fusion. If ␥ gb Ͻ 2␥ sl , then ⌬T b Ͻ 0 and the liquid film is unstable. Coalescence occurs as soon as the two interfaces get close enough (at a distance on the order of ␦ ). This situation, typical of dendrite arms belonging to the same grain (i.e., ␥ gb ϭ 0), is referred to as "attractive". The situation where ␥ gb ϭ 2␥ sl is referred to as "neutral"; i.e., coalescence occurs at zero undercooling. If ␥ gb Ͼ 2␥ sl , the two liquid/solid interfaces are "repulsive" and ⌬T b Ͼ 0. In this case, a stable liquid film between adjacent dendrite arms located across such grain boundaries can remain until the undercooling exceeds ⌬T b . For alloys, coalescence is also influenced by the concentration of the liquid film. The temperature and concentration of the liquid film must reach a coalescence line parallel to, but ⌬T b below, the liquidus line before coalescence can occur. Using one-dimensional (1-D) interface tracking calculations, diffusion in the solid phase perpendicular to the interface (backdiffusion) is shown to aid the coalescence process. To study the interaction of interface curvature and diffusion in the liquid film parallel to the interface, a multiphase-field approach has been used. After validating the method with the 1-D interface tracking results for pure substances and alloys, it is then applied to twodimensional (2-D) situations for binary alloys. The coalescence process is shown to originate in small necks and involve rapidly changing liquid/solid interface curvatures.