Theory and Simulation of 3-D Crystal Growth

Abstract

 

In this talk, we consider the quasi-steady evolution of growing crystals in 3-d. The case of quasi-steady crystal growth is a fundamental problem both in phase transitions and in diffusion dominated growth. The growth of a spherical germ from a supercooled melt or supersaturated solution (with isotropic surface tension) was first analyzed by Mullins and Sekerka in 1963. It was found that a growing sphere is linearly unstable to large wavelength perturbations. Moreover, as the sphere becomes larger, the unstable wavelengths become smaller and smaller. This provides a heuristic explanation for the dendritic and highly complex shapes typically observed in freezing processes in nature (e.g. snowflakes). A re-examination of this problem, however, reveals that the Mullins-Sekerka instability may be suppressed by appropriately varying the undercooling (far-field temperature) in time. For example, in 3 dimensions, by imposing the far-field temperature flux (rather than a temperature condition), a class of asymptotically self-similar, non-spherical growing crystals can be found. Simulations show that this class of solutions is robust with respect to perturbations and is well-predicted by solutions of the linearized equations. To simulate the problem numerically, we use a boundary element method with a fully adaptive surface triangulation. This enables us to simulate three dimensional crystals stably and accurately well into the nonlinear regime. Simulations of both stable and unstable crystal growth will be presented. This work has important implications for shape control in processing applications; in fact, experiments are currently being designed (by Stefano Guido at the University of Naples) to test this possibility.  This work is joint with Dr. Vittorio Cristini (School of Math, Dept Chem. Eng. and Mat. Sci, U. Minn.)