Facet morphology response to nonuniformities in nutrient and impurity supply .2. Numerical simulations

A model for the evolution of facet morphologies in growth from solutions is presented. The numerical model links, for the first time, bulk transport of solute and impurities in a solution growth cell with microscopic interfacial kinetics processes. The macroscopic transport is dealt with as in the 2D model [H. Lin, F. Rosenberger, J.I.D. Alexander and A, Nadarajah, J. Crystal Growth 151 (1995) 153] of a crystallization cell used for lysozyme in our laboratory. The microscopic kinetics is incorporated through a meso-scale continuum model of growth step motion in response to the interfacial concentration distributions. Local growth step velocities are linearly interpolated from the values obtained at the grid points of the bulk transport simulation. Experimentally determined kinetics and transport coefficients are employed. We find that the facets remain macroscopically nat, in spite of the lower nutrient and impurity concentrations in the facet center regions. This stabilization is achieved through the formation of a microscopic depression in the facet, with nonuniform vicinal slope (step density). If the step density in the facet center exceeds a certain value, no further stabilization results on further steepening, and the facet loses its macroscopic morphological stability. This loss of morphological stability depends sensitively on the value of the steps' kinetic coefficient. For pure lysozyme-precipitant solutions, we obtain microscopic depressions with a higher slope at the facet center than at the edge. However, with an impurity that impedes step kinetics and is preferentially incorporated into the crystal, the simulations produce microscopic facet depressions with higher slope at the edge. Impurity depletion at the interface, due to low initial concentration and/or slow diffusion leads to mixed shapes, and eventually to shapes typical of growth from pure solution. Quantitative agreement with facet morphologies observed on lysozyme crystals [P.G. Vekilov and F. Rosenberger 158 (1996) 540] is obtained, assuming overlap of the steps' diffusion fields.