Confinement of thermocapillary floating zone flow by uniform rotation

Abstract

The microgravity environment of an orbiting vehicle permits crystal growth experiments with greatly reduced buoyant convection in the liquid melt. Crystals grown in ground-based laboratories do not achieve their potential properties because of dopant variations caused by flow in the melt. The floating zone crystal growing system is widely used to produce crystals of silicon and other materials. However, in this system the temperature gradient on the free sidewall surface of the melt is the source of a thermocapillary flow which does not disappear in a low gravity environment. Smith and Greenspan examined theoretically the idea of using a uniform rotation of the floating zone system to confine the thermocapillary flow to the melt sidewall leaving the interior of the melt passive. These workers considered a half zone, a cylinder of fluid with a constant axial temperature gradient imposed on the cylindrical sidewall. They examined the linearized, axisymmetric flow in the absence of crystal growth. They found that rotation does confine the linear thermocapillary flow. In this paper the simplified model of Smith and Greenspan is extended to a full zone with a more realistic temperature distribution imposed on the sidewall and both linear and nonlinear thermocapillary flows are studied theoretically. Analytical and numerical methods are used for the linear flows and numerical methods for the nonlinear flows. We found that the linear flows in the full zone have more complicated and thicker boundary layer structures than in the half zone, and that these flows are also confined by the rotation. For the model considered and for realistic values for silicon, however, the thermocapillary flow is not linear. We examined the nonlinear flows by first computing a weakly nonlinear flow and then computing the fully nonlinear flow. The weakly nonlinear flow is steady, has less boundary layer character, and penetrates more deeply into the interior than the linear flow but still shows some rotational confinement. The fully nonlinear flow is strong and unsteady (a weak oscillation is present) and it penetrates the interior. Some non-rotating flow results are also presented. Since silicon has a large value of thermal conductivity, we would expect the temperature fields to be determined by conduction alone. This is true for the linear and weakly nonlinear results, but for the stronger nonlinear flow the results show that temperature advection is also important. Thus, this work reveals that for the nonlinear flow, a radiative sidewall boundary condition would be an improvement over the specified temperature boundary condition used in this paper and previously by others. Such a boundary condition would weaken the sidewall axial temperature gradient and hence the thermocapillary flow, allowing the confining effect of rotation to play a stronger role. Hence, uniform rotation may still be a means of confining the flow, and the results obtained define the procedure to be used to examine this hypothesis.