Two-dimensional model for thermal and solutal convection in multizone physical vapor transport

Abstract

Analytical and numerical solutions are proposed for the prediction of the thermal and solutal convection in a long horizontal ampoule in the presence of local temperature gradients at the walls as involved in the multizone furnace applications. A finite difference method is used for a Boussinesq fluid in a two-dimensional cavity (aspect ratio l = H/L ) with wall temperature profiles, locally applied and characterized by the reduced gradient width, γ = L ΔT /L at a given ΔT . Analytical laws for the velocity are extracted from the asymptotical theory in the middle of these gradient zones, and compared to the numerical solutions. The results are carried out without interfacial mass flux but for conditions relevant to crystal growth by vapor transport in closed ampoules: Gr T / l 3 ≲l.4×10 7 ,∣Gr M ∣/ l 3 ≲ 2×l0 4 , Pr = 0.73, Sc = 1.0 with 0.10 ≤ l ≤ 0.5 and 0.05 ≤ γ ≤ 1. Emphasis is put on typical aspects of conduction regimes corresponding to low Grashof numbers. The effect of an irregularity in the wall temperature profile is treated. Special kinds of behaviour are shown to result from the competition between solutal and thermal convection. Also, interfacial mass fluxes have been considered to analyze their effect on the buoyancy driven flow.