Last-stage solidification of alloys: Theoretical model of dendrite-arm and grain coalescence

Abstract

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 primary-phase 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 (ΔT _b ), given by

$$\Delta T_b = \frac{{\Delta \Gamma _b }}{\delta } = \frac{{\gamma _{gb} - 2\gamma _{sl} }}{{\Delta s_f }}\frac{1}{\delta }$$

where δ is the thickness of an isolated solid-liquid interface, and ΔГ _b is the difference between the grain-boundary energy, γ _gb , and twice the solid/liquid interfacial energy, 2γ _sl , divided by the entropy of fusion. If γ _gb <2γ _sl , then ΔT _b <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 δ). This situation, typical of dendrite arms belonging to the same grain (i.e., γ _gb =0), is referred to as “attractive”. The situation where γ _gb =2 γ _sl is referred to as “neutral”; i.e., coalescence occurs at zero undercooling. If γ _gb >2γ _sl , the two liquid/solid interfaces are “repulsive” and ΔT _b >0. In this case, a stable liquid film between adjacent dendrite arms located across such grain boundaries can remain until the undercooling exceeds ΔT _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 ΔT _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 two-dimensional (2-D) situations for binary alloys. The coalescence process is shown to originate in small necks and involve rapidly changing liquid/solid interface curvatures.