Modeling flows and solute redistribution resulting from small transverse accelerations in Bridgman growth

Abstract

An approximate analytical model has been developed to describe the two-dimensional flow field and its effect on solute redistribution during horizontal Bridgman growth at very small Grashof numbers. The purpose of the model is to examine the effects of small residual accelerations in attempts to grow crystals under diffusion limited conditions in a microgravity environment. Also, the model provides insight into the interplay of the various parameters involved and allows scaling laws to be formulated. The model is quite accurate provided the buoyancy-induced flows do not significantly alter the density field. For dilute systems this requires Gr Pr < O(100) or for non-dilute systems, Gr Sc < O(10). The amount of segregation along the growth interface is characterized by a segregation parameter ζ defined as the maximum difference in composition divided by the average composition. This parameter is shown to be directly proportional to the product of Grashof and Schmidt numbers (Gr Sc) and to a complicated expression involving the segregation coefficient, the ratio of the ampoule width to the diffusion length, and the ratio of ampoule width to the characteristic length of the density field. Conditions for maximum coupling are identified for various values of the segregation coefficient. Under these worst case conditions, it is shown that to achieve near diffusion-limited growth conditions (ζ < 0.01), the Gr Sc < 2. This places a severe constraint on the tolerable transverse acceleration, especially for systems with high Sc numbers. Steady or quasi-steady components of the residual acceleration considerably less than 0.1 micro- g in the transverse direction can produce significant solute redistribution, even in systems of modest size ( ∼ 1 cm in diameter). Some relief can be obtained by configuring the system to lessen the coupling between the flow and solute fields as discussed in this paper. Also, transverse accelerations can be minimized by orienting the furnace axis along the residual acceleration vector. If properly oriented, the residual acceleration might also help stabilize the system against the smaller transverse accelerations that can arise due to the small variations in the residual acceleration vector.