An asymptotic analysis for directional solidification of a diffusion-dominated binary system

Abstract

A model is presented that describes the steady-state transport of heat and solute in a diffusion-dominated, directional solidification system characterized by weak melt flows (e.g. under microgravity conditions). The heat transfer between the ampoule and melt can be asymmetric hence the two-dimensional geometry of the model is restricted to a Cartesian coordinate system. The solution procedure involves a coupled asymptotic/numerical approach. The asymptotic expansions are based upon the assumptions that the ampoule aspect ratio, the heat exchange between the ampoule and sample, and the slope of the liquidus line are small. These scalings lead to geometric boundary layer solutions around the solidifying front. The solidifying interfacial shape, thermal, flow, and solutal profiles can be analytically evaluated as functions of the heater temperature profile, heater translation rate, and material properties of the system. In the regime of interest, the flow is decoupled from the temperature and the concentration fields in the geometric boundary layer. Lateral corrections to the shape of the interface are determined by axial and lateral diffusive transport of heat, and diffusive transport of solute. The latter modify the melting point of the interface and the temperature gradients local to the interface. Lateral solute segregation is predicted to increase with solidification rate.