Multi-scale model of the stability of a laterally heated liquid bridge under the influence of an axial magnetic field.

摘要

Thermocapillary flows, occurring in floating-zone semiconductor crystal growth, are studied using numerical and analytical linear stability analyses. In this process, a polycrystalline feed rod is passed through an optical heating source, where it melts and then re-solidifies into a single crystal. In microgravity, the system is characterized by a molten region suspended between the feed rod and the crystal, and held in place by surface tension. The temperature gradients on the molten free surface drive thermocapillary flow, which is susceptible to instabilities. These instabilities can lead to defects in the crystal lattice, and nonuniform dopant distribution, effectively ruining the properties of the crystal. This study investigates the use of magnetic fields in suppressing these instabilities. When a crystal is grown in the presence of a magnetic field, electric currents are generated in the conducting melt. These currents interact with the magnetic field, producing Lorentz body forces, which damp the thermocapillary flow, and delay the onset of transition. While this is the accepted explanation of magnetic damping in crystal growth, the process has yet to be successfully modeled. Furthermore, the mechanism of energy transfer, from the axisymmetric base state, to the perturbed flow, has not been described. The primary obstacle to modeling this process is obtaining sufficient resolution of the thin Hartmann boundary layers, which form perpendicular to the magnetic field at the solidification and melting fronts. A secondary challenge is due to the segregation of the flow into a strong convection cell near the free surface, and a nearly stagnant core region. To overcome these modeling obstacles, a multi-scale solution technique has been developed, which takes advantage of the Hartmann layers and flow segregation, by introducing an asymptotic solution in the strong magnetic field limit. This solution offers physical insight into the transition mechanism. At weak field strengths, improved numerical methods are used to extend the analysis by an order of magnitude over that of previous studies.