Finite-amplitude instability in growth step trains with overlapping step supply fields

Abstract

We have expanded our numerical model of coupled bulk transport in solution and interfacial kinetics in crystal growth [Vekilov, Lin, and Rosenberger, Phys. Rev. E 55, 3202 (1997)] by explicitly including adsorption on and desorption from terraces between growth steps, surface diffusion, and incorporation into steps. At the steps, the surface (adsorption layer) concentration ${C}_{s}$ is assumed to be either continuous, i.e., have the same values at the top and bottom of a step, or to be discontinuous, i.e., to take on different, respective terrace-width-dependent values. In order to maximize spatial resolution about individual steps, we use a mesoscale grid at the solution-crystal interface, which moves with the step positions and adjusts to the changing terrace widths during the simulation. This model was evaluated with transport and kinetics parameters characteristic for the growth of lysozyme crystals from aqueous solutions. With continuous ${C}_{s}$ at steps, the simulations reproduced the results of our previous model in which the step supply field overlap was only indirectly accounted for by a step-density-dependent deceleration parameter in the step velocity. When discontinuities in ${C}_{s}$ were allowed, significantly higher bunching instability resulted. More importantly, we found that step bunching may or may not occur, depending on the specific step-density perturbation (magnitude, sign and rate of step-density change). This is why linear stability analyses do not predict the unsteady growth behavior observed in our experiments and obtained in our simulations.