Center for Nonlinear Analysis
CNA Home
People
Seminars
Publications
Workshops and Conferences
CNA Working Groups
CNA Comments Form
Summer Schools
Summer Undergraduate Institute
PIRE
Cooperation
Graduate Topics Courses
SIAM Chapter Seminar
Positions
Contact |
Seminar Abstracts
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
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 |