Hydrodynamic Instabilities in Czochralski Process of Crystal Growth - Effect of Varying The Seed to Crucible Radii Ratio

Abstract

This paper deals with axisymmmetry breaking instabilities in Czochralski process of crystal growth. Numerical linear stability analysis was carried out using the axisymmetric bulk flow model. Stability diagrams of critical Grashof numbers Gτc and frequencies ωc dependent on aspect ratio α( = height/radius), 0.4 ⩽ α ⩽ 1.0 and Prandtl number Pτ = 0.01 are shown. 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 bifuraction. For Pτ = 0.01 computations were carried out for the first 10 modes, only the first 5 (0,1,2,3,4) were important. Sensitivity of mode transitions was observed at parameter range of α > 0.65 and in some regions modes were observed approaching each other closely. For 0.4 ⩽ α ⩽ 0.85 and Pτ = 0.01 dispersion relation analysis reveals convective instability effects while for larger α rotational effects appear. Comparative study shows that for seed to crucible ratio β = Rx/Rc = 0.4, 0.5 different pathologies are observed. While mode switches at β = 0.4 are many and modes approach each other closely, different behaviour is observed at β = 0.5 which is quite regular.