## Stephen J. Watson, Engineering Sciences and Applied Math, McCormick School of Engineering and Applied Science, Northwestern University.

"Coarsening dynamics of the convective Cahn-Hilliard equation and faceted crytal growth"

#### Abstract

The coarsening dynamics of a faceted vicinal crystalline surface growing into its melt by attachment kinetics is considered. The convective Cahn-Hilliard equation ( ) is derived as a small amplitude expansion of such surface evolutions restricted to 1-D morphologies. It takes the form

 ( )

where the local surface slope serves as the order parameter, subscripts denote partial derivative with respect to time and space respectively, and denotes the -derivative. The effective free energy takes the form of a symmetric double well with minima at , thereby capturing the anisotropy of the crystal surface energy. Also, the dimensionless small parameter multiplying the convective term is a dimensionless measure of the growth strength.

A sharp interface theory for is derived through a matched asymptotic analysis. The theory predicts a nearest neighbor interaction between two non-symetrically related phase boundaries (kink and anti-kink), whose characteristic separation grows as coalescing kink/anti-kinks annihilate one another. Theoretical predictions on the resulting (skew-symetric) coarsening dynamical system include

• The characteristic length , provided is appropriately small with respect to the Peclet length scale .
• Binary coalescence of phase boundaries is impossible
• Ternary coalescence may only occur through the kink-ternary interaction; two kinks meet an anti-kink resulting in a kink.
Direct numerical simulations performed on both and confirm each of these predictions.

Last, a linear stability analysis of identifies a pinching mechanism as the dominant instability, which in turn leads to kink-ternaries. We propose a self-similar period-doubling pinch ansatz as a model for the coarsening process, from which an analytical coarsening law for the characteristic length scale emerges. It predicts both the scaling constant of the regime, i.e., , as well as the crossover to logarithmically slow coarsening as crosses . Our analytical coarsening law stands in good qualitative agreement with large scale numerical simulations that have been performed on .

In part, joint work with Felix Otto and Stephen H. Davis.

April 15, 2003
Time: 4:30 P.M.
Location: PPB 300