Three-dimensional instabilities in Czochralski process of crystal growth from silicon melt

摘要

This paper deals with axisymmetry breaking instabilities in Czochralski process of crystal growth from silicon melt (Prandtl number Pr = 0.01). Numerical 3D linear stability analysis was carried out on the axisymmetric bulk flow model. Stability diagram of critical Grashof numbers Gr_c dependent on aspect ratio α( = height/radius) in the range 0.4 ≤ α ≤ 1.0 was computed. Computations were carried out using the spectral element method in the meridional plane with Fourier decomposition in the azimuthal direction. It was found that convective instability sets in through an Hopf bifurcation displaying oscillations in time. Sensitivity of mode transitions was observed at parameter range of α > 0.65 and in some regions modes were observed approaching each other closely. Dangerous modes in the 0.65 < α ≤ 0.8 region were 2, 3, 4. For 0.4 ≤ α ≤ 0.85 dispersion relation analysis reveals convective instability effects while for larger α rotational effects appear. Analysis for the case of no heat convection (Pr = 0) was carried out showing different behavior for 0.4 ≤ α ≤ 0.85, thus, validating the analysis of Pr = 0.01.