Journal of Chemical Physics, Vol.109, No.6, 2503-2508, 1998
Free convection and surface kinetics in crystal growth from solution
As a crystal grows from solution, there is ordinarily a boundary layer depleted in solute, which forms at the crystal-solution interface. When the normal to the growing crystal surface is oriented in any direction other than parallel to gravity, the boundary layer is set into motion by the force of buoyancy. Using a similarity transformation and a boundary layer approximation, we have solved the Navier-Stokes equation and the equation for convective diffusion for a crystal in the form of a flat plate growing with normal perpendicular to gravity. Parameters in the theory include solute concentration, c(0), and diffusion coefficient, D; solution shear viscosity, mu, mass density, rho, and logarithmic density derivative with respect to concentration, alpha; crystal solubility, c(s), height, h, and linear growth rate, k(G); the specific rate, k (sticking coefficient), of the reaction which transfers molecules from the solution to the crystal and the kinetic order, n, of this reaction; and the acceleration due to gravity, g. We find these parameters to be related by the equation log[1-Sh/a (Sc) (1/4)(Gr) (1/4)phi(s)(1/4)]= (1/n) log[a(5/4) (n)(D/hkc (n-1)(0))(Sc) (1/4)(Gr) (1/4)] + [(5/4-n)/n]log phi(s), where a=0.9, Sh=k(G)h/D, Sc=mu/rho D, Gr=g alpha h(3)rho(2)/4 mu(2), and phi(s)=(c(0)-c(s))/c(0). Given a knowledge of the solution physical properties, if Sh is measured as a function of phi(s) and the results plotted in accord with the above equation, both n and k can be determined.