Asymptotic solutions for an axisymmetric, stagnant film model of directional solidification

Abstract

A model is presented to describe the time-dependent transport of heat and solute in a directional solidification system for a binary alloy. The stagnant film concept is used to include the effects of melt flow and the transport of solute by convection. The geometry of the model is described by an axisymmetric coordinate system. The solution procedure involves a coupled asymptotic/numerical approach. The asymptotic expansions combine quasi-steady limits with a small ampoule aspect ratio. The boundary layer solutions around the solidifying front are obtained under additional assumptions that the heat exchange between the ampoule and sample is small, the latent heat is large, the solute diffusivity in the solid phase is small, and the thickness of the stagnant film is small. The core melt and solid temperature profiles and the mean position of the solidifying front are calculated numerically as functions of the latent heat, heater profile, and solute profile in the liquid phase. Due to the inclusion of curvature effects on the melting point, the non-planar shape of the interface is described by a superposition of an oscillatory profile onto a parabolic profile. The parabolic profile is set via the balance of axial and radial diffusive transports of heat. The oscillatory profile develops in the case of undercooling as in the classical morphological instability of a planar interface during directional solidification. The above results are obtained for varying stagnant film thickness to study melt regimes ranging from diffusion—to convection—controlled growth. The dependence on the film thickness of axial and radial segregation and the morphological stability of the interface are considered.