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

Abstract

In the present contribution we review basic mathematical results for three physical systemsinvolving self-organizing solid or liquid films at solid surfaces. The films may undergo astructuring process by dewetting, evaporation/condensation or epitaxial growth,respectively. We highlight similarities and differences of the three systems based on theobservation that in certain limits all of them may be described using models of similarform, i.e. time evolution equations for the film thickness profile. Those equations representgradient dynamics characterized by mobility functions and an underlying energy functional.Two basic steps of mathematical analysis are used to compare the different systems. First,we discuss the linear stability of homogeneous steady states, i.e. flat films, and second thesystematics of non-trivial steady states, i.e. drop/hole states for dewetting films andquantum-dot states in epitaxial growth, respectively. Our aim is to illustrate that theunderlying solution structure might be very complex as in the case of epitaxial growth butcan be better understood when comparing the much simpler results for the dewetting liquidfilm. We furthermore show that the numerical continuation techniques employed can shedsome 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 relateseemingly unrelated physical systems mathematically, but does allow as well for discussingmodel extensions in a more unified way.