Modelling The Rotating Magnetic Field With The Skin Effect And Its Application To Convection Control On Thermocapillary Flow In A Shallow Czochralski Configuration

In crystal growth,the flow stability of melt convection plays a significant role in determining the quality of crystals,and therefore it is crucial to control the melt flow.Thanks to high electrical conductivity of semiconductors,external magnetic fields have been widely utilized in controlling the flow instability.In particular,rotating magnetic field(RMF),due to low energy consumption and satisfactory performance,has drawn great attention.When a RMF is introduced into melt flow,the so-called skin effect,which highly depends on the electrical conductivity,the magnetic permeability and the radius of the melt,as well as the frequency of the RMF,may affect the RMF distribution.To be specific,a dimensionless shielding parameter,K,can be used to determine whether the RMF could penetrate the whole melt without any change(K<<1)or not.The former assumption was applied in most previous studies,and however,turns unreasonable in practice when growing large size crystals.There were few studies attempting to exploit effects of the skin effect on the distribution of a given magnetic field.These studies did not consider effects of melt flow or geometric model on the Lorentz force,nor did they investigate the convection control or flow stability under RMF.In this thesis,the main objective is to address the skin effect encountered in melt convection under RMF.With the skin effect accounted for,the calculation of the Lorentz force was developed according to the melt flow and geometric model,and resulted in a complementary mathematical model for convection control under RMF.Furthermore,the stability of thermo-capillary flow under RMF in a shallow Czochralski(Cz)configuration was investigated and the physical mechanisms by which the convection loses its stability were explored.As a result,the main contributions and findings of the thesis go to the following,which are believed to enrich the scientific gap that was missing in crystal growth under RMF.(1)Based on the equations describing magnetic diffusion in magnetohydrodynamics and the phasor method for time-harmonic electromagnetic fields,the magnetic field distribution in melt flow was derived and was found to follow the power series expansion.For easier calculating the Lorentz force and programing,the power series expansion was cut off at different expansions,which lead to several approximations of the magnetic fields following polynomial functions.The applicability of these approximate magnetic fields was estimated mathematically as well.(2)The RMF finite ?1-?2 model was proposed to take into account the skin effect and to calculate the Lorentz force.The corresponding mathematical model considering the skin effect of RMF was then developed and implemented in an existing CFD package.Using the CFD package,the applicability of each approximate magnetic field was validated aganist the RMF finite ?1-?2 model.Moreover,the condition for the assumption that the skin effect may be negligible was extended from K?1 to K?0.8.(3)The influence of the skin effect on the convective flow induced by RMF was examined quantitatively for a varying value of K.It was found that for a larger value of K the maximal azimuthal velocity increases while the maximal axial and radial velocity decreases.(4)For the RMF induced azimuthal convection in the cylinder cavity,there exists a critical value of K that maximizes the melt velocity.Compared to the critical value K=5.52 reported by Volz,the present study suggests the critical value K=6 when considering the melt flow and geometric model.(5)Effects of RMF strength on the thermo-capillary flow in a three-dimensional shallow Czochralski configuration were investigated.A transition from two-dimensional axisymmetric to three-dimensional steady flow was observed first in the melt.Afterwards the evolution of the thermo-capillary flow was found as the following from 1 to 8.7 mT: initial three-dimensional steady flow ? periodic rotating oscillatory flow(I)? three-dimensional steady flow(II)? periodically rotating oscillatory flow(III)? three-dimensional steady flow(IV)? periodically rotating oscillatory flow(V).For periodical oscillatory flows,because of the changes in their structure,the main frequency rises dramatically at the transitions in(III).With regards to the cause of the flow instability,during phases(I)and(III),the hydrothermal-wave instabilities caused by inconsistent velocity and temperature oscillations can be concluded as the physical mechanism of the flow instability.In contrast,in phase(V),the strong RMF forced convection is the reason for the flow instability.For steady flows,the increasing RMF suppresses the thermo-capillary flow and reduces the vortex intensity beneath the interface of crystal and free surface.Further increasing the RMF results in a reduction in temperature gradient in the vicinity of the interface of crystal and free surface.As the RMF exceeds a certain extent,the temperature gradient rises again at the interface of crystal and free surface.(6)Considering whether or not the skin effect,the thermo-capillary flow under a RMF was compared in a shallow Czochralski configuration.The most significant effect was found to incur in the critical region where the flow turns from stable to instable.In non-critical regions,the skin effect can greatly affect the intensity of melt flow for steady phases or the frequency and magtinude for oscillations.