Dewetting of thin solid films

This dissertation is devoted to the mathematical study of solid state dewetting and deals with various mathematical topics such as phase field modeling, the derivation of corresponding sharp interface limits, existence of solutions, numerical simulations and linear stability analysis of the dewetting front. We start with the formulation of a two-dimensional anisotropic phase field model for solid state dewetting on a solid substrate. The evolution is described by a Cahn-Hilliard type equation with a bi-quadratic degenerate mobility and a polynomial homogeneous free energy. We propose an anisotropic free boundary condition at the film/substrate contact line which correspond to the natural boundary condition from the variational derivation. We show via matched asymptotic analysis that the resulting sharp interface model is consistent with the pure surface diffusion model. In addition, we show that the corresponding natural boundary conditions at the substrate imply a contact angle condition which is known as Young-Herring condition. We provide an existence result for the present degenerate partial differential equation on a simplified domain with homogeneous Neumann boundary conditions. Under the assumption that the strength of the anisotropy is sufficiently small, we establish certain convexity properties and higher order bounds of the strongly non-linear anisotropic operator. This enables to prove existence of weak solutions. Furthermore, we show that solutions are bounded by one without having a maximum principle. Completing the part which is concerned with the phase field representation, we consider the numerical simulation of the present model, where we apply a diffuse boundary approximation to handle the boundary conditions at the substrate. The reformulated equation can be solved by a standard finite element method. A matched asymptotic analysis shows that solutions of the reformulated equations formally converge to those of the original equations. We provide numerical simulations which confirm this analysis. In addition, we address the previously discussed question of how the mobility influences the evolution and simulate dewetting scenarios for different mobilities and anisotropies. In the last main chapter we consider a generalized class of thin film equations, including the case which corresponds to the small slope approximation of the sharp interface model for isotropic solid state dewetting. We present an improved method for the linear stability analysis of unsteady, non-uniform base states in thin film equations which exploits that the initial fronts evolve on a slower time-scale than the typical perturbations. The result is a unique value for the dominant wavelength which is different from the one obtained by the frequently applied linear stability analysis with "frozen modes". Furthermore we show that for the present class of stability problems the dispersion relation is linear in the long wave limit, which is in contrast to many other instability problems in thin film flows.

Author dc:creator

• Dziwnik, Marion

Advisors dc:contributor.advisor

• Wagner, Barbara
• Münch, Andreas

Licence dc:rights.uri

• http://rightsstatements.org/vocab/InC/1.0/

Language dc:language.iso

en

Identifier URI

http://dx.doi.org/10.14279/depositonce-5187

OAI identifier oai:identifier

oai:depositonce.tu-berlin.de:11303/5558