(154d) Modeling Transport and Kinetics of Crystal Growth from Solution

The rate of growth of a crystal grown from a supersaturated liquid is determined by how fast new layers can be added to its surface. Under most conditions, a screw dislocation acts as a step source, also known as a growth spiral, on the crystal surface. Subsequently, these steps move across the surface, their motion being fed by solute molecules transported through the surrounding liquid phase and, via several possible surface mechanisms, incorporating into the steps. In addition, crystal quality is often compromised during solution growth by morphological instabilities, such as macrosteps, step bunches, and inclusions, which arise from the coupled effects of fluid dynamics, mass transport, and growth. Physically faithful, yet mathematically tractable, models are needed to describe the motion of these steps and thus the growth rate of the entire surface.

In this presentation, we present our work on modeling step motion during liquid-phase solution crystal growth. Our goal is to understand the evolution of nano-scale and meso-scale features of crystal surfaces, namely the behavior of growth spirals and steps, and their interaction with solute transport effects. These models are powerful tools of inquiry, since many of the phenomena important to solution-phase crystal growth, such as surface diffusion of solute molecules along a terrace, are inaccessible via direct experiment.

We present a multi-scale model of the growth of a vicinal crystal surface from a supersaturated liquid solution that couples bulk fluid dynamics with the kinetics of surface step growth. We consider relatively simple flows within boundary layers adjacent to the macroscopic, vicinal surface of a crystal growing from a liquid solution. There is a depletion of solute due to the crystallization at the surface leading to a concentration boundary layer, which is embedded within the momentum boundary layer. At a still smaller scale, we consider a moving group of steps on the crystal surface and construct a growth model involving a series of elementary transport and kinetic processes, starting from material transport in the bulk phase, adsorption and desorption of growth units to and from terraces between steps, followed by surface diffusion and incorporation at discrete step ledges. This phenomenological approach to describe step motion was first proposed in the classical model by Burton, Cabrera and Frank for solution growth. The governing equations are solved numerically by an efficient, moving-boundary, finite element method.