A model for diffusion - controlled solidification of ternary alloys in mushy layers †

We describe a model for non-convecting diffusion-controlled solidification of a ternary (three-component) alloy cooled from below at a planar boundary. The modelling extends previous theory for binary alloy solidification by including a conservation equation for the additional solute component and coupling the conservation equations for heat and species to equilibrium relations from the ternary phase diagram. We focus on growth conditions under which the solidification path (liquid line of descent) through the ternary phase diagram gives rise to two distinct mushy layers. A primary mushy layer, which corresponds to solidification along a liquidus surface in the ternary phase diagram, forms above a secondary (or cotectic) mushy layer, which corresponds to solidification along a cotectic line in the ternary phase diagram. These two mushy layers are bounded above by a liquid layer and below by a eutectic solid layer. We obtain a one-dimensional similarity solution and investigate numerically the role of the control parameters in the growth characteristics. In the special case of zero solute diffusion and zero latent heat an analytical solution can be obtained. We compare our predictions with previous experimental results and with theoretical results from a related model based on global conservation laws described in the Appendix. Finally, we discuss the potentially rich convective behaviour anticipated for other growth conditions.