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Numerical analysis of sapphire crystal growth by the Kyropoulos technique
Presented below are the results of numerical simulation of single leucosapphire crystal growth by the Kyropoulos method using CGSim software. It is a unique feature of the suggested approach that the computations in the crystallization zone involve the turbulent flow of the sapphire melt, laminar gas flow, and radiative heat exchange in the semi-transparent crystal including specular reflectivity at the boundaries.
Improving the hot zone of the Kyropoulos furnace to decrease thermal gradients at the crystallization front results in higher yield ratio and better crystal quality. Numerical simulation comes very helpful in the analysis and optimization of growth systems as it gives us an insight into the processes that are otherwise extremely difficult to observe or measure, such as, the temperature distributions inside the melt and growing crystal. Attempts to apply a simplified model, ignoring semitransparency of the crystal and the melt, gave physically unrealistic results for the crystalization front geometry and temperature gradients at the metl-crystal interface. In the approach developed specifically for the task, the heat fluxes obtained from the heat transfer pre-computation in the whole system were used as the thermal boundary conditions for more detailed computations in the crystallization zone including the crystal, the melt, the crucible, and the gas region around the crystal. To account for sapphire semitransparency, we used an approach developed in [2] in terms of the discrete transfer method for solving problems of the radiation transfer in axisymmetric areas of complex geometry with specular Fresnel's boundaries.
* Account for semitransparency and specular reflection
* 30% quality increase in an industrial setup
Optical properties of sapphire:
* transparent band is 0.5-4.5 μm with the absorption coefficient 19.26 m-1
* refractive index is 1.78
* scattering coefficient is 0
Fig. 1. Sapphire boule with a diameter up to 300 mm and weight of 65 kg grown by Monocrystal Inc. (Stavropol; Russia) using the Kyropoulos technique (left) and the sapphire crystal photo at an intermediate growth stage (right)
Fig. 2. Distributions of the temperature gradient in the crystal, the temperature in the melt and the crucible, and the flow pattern in the melt for Modification 1 (a) and Modification 2 (b)
Using the updated CGSim package, several configurations of the industrial furnace have been considered [1]. In the initial configuration (Modification 1), the melt flow had a two-vortex structure with a larger vortex occupying the melt core and a vortex of lower intensity located near the melt free surface during the lateral crystal growth and disappearing at the cylindrical growth stage, Figure 2. Such flow pattern provided direct delivery of the hot melt to the crystallization front, resulting in high temperature gradients along the melt/crystal interface. After considering several hot zone modifications, we found a furnace configuration providing one-vortex flow structure in the melt (Modification 2). Such flow pattern results in gradual cool-down of the melt as it approaches the growing crystal, thus, decreasing the temperature gradients in the crystal by up to 30%, Figure 3.
Fig. 3. Temperature gradient distribution along the crystallization front from the axis of symmetry to the periphery for Modification 1 (a) and Modification 2 (b)
The model was verified using available experimental data. A good agreement between the computational and the experimental crystallization shapes indicates that the model provides an adequate prediction of the temperature fields in the reactor.
1. "Numerical analysis of sapphire crystal growth by the Kyropoulos technique", S.E. Demina, E.N. Bystrova, M.A. Lukanina, V.V. Kalaev, V.M. Mamedov, V.S. Yuferev, E.V. Eskov, M.V. Nikolenko, V.S. Postolov, to be published in Journal of Optical Materials in 2006.
2. "Numerical solution of problems with radiation transfer in axisymmetric areas of a complex shape with specular Fresnel's", V.M.Mamedov, S.? Rukolaine, Math. Modeling, vol. 16, 10 (2004) pp.15-28 |
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