Thin film evolution equations from (evaporating) dewetting liquid layers to epitaxial growth

TL;DR

This paper reviews mathematical models of thin film evolution in systems like dewetting, evaporation, and epitaxial growth, highlighting their similarities, differences, and complex solution structures through stability analysis and numerical methods.

Contribution

It introduces a unified mathematical framework for different thin film systems, comparing their stability and steady states, and demonstrates the usefulness of numerical continuation techniques.

Findings

Similar models describe different physical systems.

Complex solution structures are revealed through stability analysis.

Numerical continuation aids in understanding steady states.

Abstract

In the present contribution we review basic mathematical results for three physical systems involving self-organising solid or liquid films at solid surfaces. The films may undergo a structuring process by dewetting, evaporation/condensation or epitaxial growth, respectively. We highlight similarities and differences of the three systems based on the observation that in certain limits all of them may be described using models of similar form, i.e., time evolution equations for the film thickness profile. Those equations represent gradient dynamics characterized by mobility functions and an underlying energy functional. Two basic steps of mathematical analysis are used to compare the different system. First, we discuss the linear stability of homogeneous steady states, i.e., flat films; and second the systematics of non-trivial steady states, i.e., drop/hole states for dewetting films and quantum dot states in epitaxial growth, respectively. Our aim is to illustrate that the underlying solution structure might be very complex as in the case of epitaxial growth but can be better understood when comparing to the much simpler results for the dewetting liquid film. We furthermore show that the numerical continuation techniques employed can shed some light on this structure in a more convenient way than time-stepping methods. Finally we discuss that the usage of the employed general formulation does not only relate seemingly not related physical systems mathematically, but does as well allow to discuss model extensions in a more unified way.