Role of characteristic modes in the dynamics of wave pattern in the Czochralski oxide crystal growth

Abstract

Wave pattern is a flow pattern observed on the free surface of oxide melt in the Czockralski crucible. Wave pattern exhibits regular folds in the azimuthal direction. In this paper three-dimensional flows of the LiNbO 3 melt in a Czockralski crucible of radius r c =100 mm and crystal radius r s =50 mm were investigated. The crucible is still and the crystal is rotating at a rate Ω s . The flow is assumed to be laminar. The governing equations of the melt flows were solved by a block-structured boundary-fitted-coordinate method. Massive numerical simulations were first conducted to obtain several stable n-folded wave patterns by changing aspect ratio A= h/ r c and Ω s , where h is the depth of the oxide melt. The proper orthogonal decomposition was then applied to extract the characteristic modes of the n-folded original wave patterns. The spatial feature of the characteristic modes and the role of the characteristic modes in the dynamic process of the original wave patterns were finally analyzed. Some interesting conclusions are obtained. The characteristic modes of the wave patterns show strong spatial feature. The characteristic modes have more folds in the azimuthal direction than the original wave patterns; the characteristic modes appear in group; with the increase of group order, the number of folds in the azimuthal direction is increased regularly by n for the n-folded original wave pattern. The study of the dynamic process shows that the original wave pattern is a traveling wave, which is formed by the dominant characteristic modes.