Evolution of crystalline thin films by evaporation and condensation in three dimensions

TL;DR

This paper models the evolution of crystalline thin film morphologies on substrates through a fourth-order anisotropic curvature flow, incorporating elastic effects and regularization, and proves the existence of solutions over finite times.

Contribution

It introduces a novel mathematical framework combining elasticity and curvature-driven flow for crystalline thin films, with a proof of solution existence.

Findings

Existence of regular solutions for the evolution equation

Inclusion of elastic mismatch effects in the model

Application of minimizing movements for analysis

Abstract

The morphology of crystalline thin films evolving on flat rigid substrates by condensation of extra film atoms or by evaporation of their own atoms in the surrounding vapor is studied in the framework of the theory of Stress Driven Rearrangement Instabilities (SDRI). By following the literature both the elastic contributions due to the mismatch between the film and the substrate lattices at their theoretical (free-standing) elastic equilibrium, and a curvature perturbative regularization preventing the problem to be ill-posed due to the otherwise exhibited backward parabolicity, are added in the evolution equation. The resulting Cauchy problem under investigation consists in an anisotropic mean-curvature type flow of the fourth order on the film profiles, which are assumed to be parametrizable as graphs of functions measuring the film thicknesses, coupled with a quasistatic elastic problem in the film bulks. The existence of a regular solution for a finite period of time is established under periodic boundary conditions by means of employing minimizing movements to exploit the gradient-flow structure of the evolution equation.