Interface shapes and thermal fields during the gradient solidification method growth of sapphire single crystals

We present a finite-element model describing the melt-growth of cylindrical sapphire single crystals via the gradient solidification method. The advantage of this model lies in its ability to accurately capture complex physical phenomena associated with heat transfer through the system, while remaining modest in its computational requirements. Internal radiative heat transport through the transparent crystalline phase is accounted for in our formulation, as are details of flow fields evolving in the melt during growth. Both buoyancy and surface-tension-gradient (Marangoni) driven convection effects are considered. Results show a strong dependence of the thermal field in the charge and of melt/crystal interface shapes on operating parameters such as crystal growth rate and furnace temperature gradient. Specifically, the large latent heal value associated with this system, coupled with enhanced radiative cooling through the crystalline phase, causes a dramatic reduction in interface curvature and position for relatively high growth rates and shallow furnace gradients. In addition, effects of fluid flow on the thermal field are shown to be unimportant in this system, even when considering growth in relatively large-diameter crucibles. Trends reported here are in general agreement with experimental observations.