Simulations of Nonlinear Strongly Anisotropic, Misfitting Crystals and Thin Films

Abstract

AbstractWe present a new approach for modeling strongly anisotropic crystal and epitaxial growth using regularized, anisotropic Cahn-Hilliard-type equations as a model for the growth and coarsening of thin films. When the surface anisotropy is sufficiently strong, sharp corners form and unregularized anisotropic Cahn-Hilliard equations become ill-posed. Our models contain a high order Willmore regularization to remove the ill posedness at the corners. A key feature of our approach is the development of a new formulation in which the interface thickness is independent of crystallographic orientation. In our previous work, we have provided matched asymptotic analysis to show the convergence of our diffuse interface model to the analytical sharp interface model. In previous models there was no such convergence to sharp interface model when the Willmore energy was considered. We present 2D numerical results using an adaptive, nonlinear multigrid finite-difference method. In particular, we find excellent agreement between the computed shapes using the Cahn-Hilliard approach, with a finite but small Willmore regularization, with dynamical numerical simulations of a sharp interface model. The equilibrium shapes from our diffuse model are compared with an analytical sharp-interface theory recently developed by Spencer [1] at the corners, and there is excellent match. Away from the corners there is an excellent agreement between the diffuse model and the classical Wulff shape. Finally, in order to model the misfit and displacement strains, we add the elastic energy and corresponding forces to our diffuse model. We analyze numerically the effect of elastic stress on the corner regularization in terms of two parameters: one parameter that describes the relative strength of the elastic energy to surface energy and the second that characteristics the strength of the surface energy anisotropy. Adding elastic energy modifies the equilibrium shape and in particular affects the shape of the corners. We can predict different Qdot shapes, such as pyramids and domes, based on the strength of the elastic interactions.