Modelling of directional solidification: from Scheil to detailed numerical simulation

Abstract Solute segregation in crystals grown by variations on the directional solidification method has long been correlated by the simple idea of a well mixed melt and a uniform stagnant film adjacent to the interface. All the details of convection in the melt are hidden in the single parameter, the diffusion layer thickness δ. Although extremely useful as a qualitative measure of convection, this description potentially oversimplifies the complex interactions of the furnace geometry, heat transfer and buoyancy driven convection in setting the flow pattern and solute segregation. Today, detailed numerical simulations of directional solidification are feasible that include all of the complexity introduced by the presence of the melt/crystal interface, convection in the melt and heat transfer throughout the system. This paper reports results of simulations of directional solidification of dilute alloys in a prototypical vertical Bridgman system. The predictions of these calculations are compared directly with the stagnant film model of segregation. It is demonstrated that although the diffusion layer thickness can be used to correlate the transition between intense convection in the melt and diffusion-controlled growth, it does not necessarily correspond to a physical picture of the solute transport in the bulk and does not predict the dependence of the radial uniformity of the composition on the flow. Direct comparison between the calculations and growth experiments for gallium-doped germanium demonstrate the accuracy of the numerical simulations for predicting the behavior of real systems.