Fluid dynamic modelling of crystal growth from vapour

Abstract

Navier-Stokes equations along with no-slip boundary conditions have been widely used, up till now, in modelling vapour crystal growth fluidynamics. In this paper attention is focused on new types of free convection which occur in a gas or a vapour when viscous stresses, due to velocity gradients, are of the same order of magnitude as stresses due to temperature and/or concentration gradients (thermal and/or solutal-stress convection). In this case linear phenomenological relations, based on the classical principles of non-equilibrium thermodynamics, for the diffusive fluxes of momentum, energy and species concentration (conventional Stokes, Fourier, Fick laws; Navier-Stokes fluidynamics) become inadequate and a more general theory must be formulated to account for thermal stresses convection, included in the second-order approximation for the gas-dynamic equations (Burnett equations), and side-wall gas creep, induced by the slip boundary condition in the Knudsen layer. This work deals with the above mentioned gas-kinetic phenomena, presented in the framework of non-equilibrium thermodynamics. Both phenomenological (macroscopic) and kinetic (microscopic) points of view are considered together with the question of the coexistence of these approaches. The Burnett equations are written in non-dimensional form, and a-priori criteria of the non-dimensional order of magnitude analysis lead to the identification of new characteristic velocities and corresponding non-dimensional numbers; new classes of free convection (thermal stress and thermal creep convection) are discussed. The final part of the work is devoted to the evaluation of the orders of magnitude of these new terms during typical crystal growth experiments on Earth and under microgravity, and to the quantitative analysis of thermal stress convection in a simplified geometrical configuration.