Numerical Simulation of Heat And Fluid Flows for Sapphire Single Crystal Growth by The Kyropoulos Method

Numerical computation has been performed to investigate temperature and velocity distributions for different stages of the Kyropoulos sapphire single crystal-growth process. The finite-element method is employed to solve the governing equations with proper boundary conditions. In the power history considered here, a vortex appears in the melt during growth, and its strength decreases as the input power is reduced. Isotherms in the melt are distorted by flow motion. The crystal–melt interface is always convex towards the melt and in early stages the convexity increases as the input power decreases. When the crystal–melt interface is close to the bottom of the crucible, this interface is flat near the apex because of reduction in growth rate near the upper region caused by input heat from the bottom of the crucible. Therefore, convexity of the crystal–melt interface decreases the input power decreases. The crystal shape predicted by the present simulation is similar to that of crystals grown in the industry.

In the present work, 3D features of melt convection during sapphire growth of 100 mm diameter Cz and of 200 mm diameter Ky crystals are studied. The approach accounting for radiative heat exchange with absorption and a specular reflection in the crystal, which we applied in 2D modeling 1, 2 and 3, has been extended to 3D computational domains and coupled to 3D heat transfer in the melt, crystal, and crucible. 3D melt unsteady convection together with crystallization front formation are taken into account within the Direct Numerical Simulation (DNS) approach. Results of 3D modeling are discussed in detail and quantitatively compared to the previously reported data of 2D modeling and experiments 2 and 3. It has been found that the features of unsteady melt convection during the “before seeding”, “seeding”, and “shouldering” growth stages are quite different from each other, which necessitates a flexible control of the radial and vertical temperature gradients in the crucible to provide optimal conditions for stable growth of high quality sapphire crystals.


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