Asymmetry in crystal facet dynamics of homoepitaxy by a continuum model

In the absence of external material deposition, crystal surfaces usually relax to become flat by decreasing their free energy. We study an asymmetry in the relaxation of macroscopic plateaus, facets, of a periodic surface corrugation in 1+1 dimensions via a continuum model below the roughening transition temperature. The model invokes a highly degenerate parabolic partial differential equation (PDE) for surface diffusion, which is related to the weighted-$H^{-1}$ (nonlinear) gradient flow of a convex, singular surface free energy in homoepitaxy. The PDE is motivated both by an atomistic broken-bond model and a mesoscale model for steps. By constructing an explicit solution to the PDE, we demonstrate the lack of symmetry in the evolution of top and bottom facets in periodic surface profiles. Our explicit, analytical solution is compared to numerical simulations of the PDE via a regularized surface free energy.Comment: 23 pages, 5 figures, comments welcome! Text slightly modified, references updated in Version 2. Referee comments addresse

Regularity and monotonicity for solutions to a continuum model of epitaxial growth with nonlocal elastic effects

We study a nonlocal 4th order degenerate equation deriving from the epitaxial growth on crystalline materials. We first prove the global existence of evolution variational inequality solution with a general initial data using the gradient flow structure. Then with a monotone initial data, we prove the subdifferential of the associated convex functional is indeed single-valued, which gives higher regularities of the global solution. Particularly, higher regularites imply that the strict monotonicity maintains for all time, which provides rigorous justification for global-in time monotone solution to epitaxial growth model with nonlocal elastic effects on vicinal surface

Global existence and decay to equilibrium for some crystal surface models

ABSTRACT: In this paper we study the large time behavior of the solutions to the following nonlinear fourth-order equations @tu=Δe-Δu;@tu=-u2Δ2(u3) These two PDE were proposed as models of the evolution of crystal surfaces by J. Krug, H.T. Dobbs, and S. Majaniemi (Z. Phys. B, 97,281-291, 1995) and H. Al Hajj Shehadeh, R. V. Kohn, and J. Weare (Phys. D, 240, 1771-1784, 2011), respectively. In particular, we find explicitly computable conditions on the size of the initial data (measured in terms of the norm in a critical space) guaranteeing the global existence and exponential decay to equilibrium in the Wiener algebra and in Sobolev space

Analysis of a fourth order exponential PDE arising from a crystal surface jump process with Metropolis-type transition rates

We analytically and numerically study a fourth order PDE modeling rough crystal surface diffusion on the macroscopic level. We discuss existence of solutions globally in time and long time dynamics for the PDE model. The PDE, originally derived by the second author, is the continuum limit of a microscopic model of the surface dynamics, given by a Markov jump process with Metropolis type transition rates. We outline the convergence argument, which depends on a simplifying assumption on the local equilibrium measure that is valid in the high temperature regime. We provide numerical evidence for the convergence of the microscopic model to the PDE in this regime.Comment: 14 pages, 4 figures, comments welcome! Revised significantly thanks to very thorough referee reports. Some previous discussions have been removed and will be reported in a separate result by one of the author