Analytical validation of the Yound-Dupr\'e law for epitaxially-strained thin films

TL;DR

This paper analytically validates the Young-Dupré law for epitaxially strained thin films, providing insights into the regularity of optimal film profiles and the effects of elastic properties and surface tensions.

Contribution

It offers the first analytical validation of the Young-Dupré law in a continuum mechanics model for thin films, extending regularity analysis methods.

Findings

Regularity results for film profiles in different growth modes.

Validation of Young-Dupré law at the contact point.

Conditions for the optimal wetting angle.

Abstract

We present here an analysis of the regularity of minimizers of a variational model for epitaxially strained thin-films identified by the authors in the companion paper [Davoli E., Piovano P., Derivation of a heteroepitaxial thin-film model. Submitted 2018]. The regularity of energetically-optimal film profiles is studied by extending previous methods and by developing new ideas based on transmission problems. The achieved regularity results relate to both the Stranski-Krastanow and the Volmer-Weber modes, the possibility of different elastic properties between the film and the substrate, and the presence of the surface tensions of all three involved interfaces: film/gas, substrate/gas, and film/substrate. Finally, geometrical conditions are provided for the optimal wetting angle, i.e., the angle formed at the contact point of films with the substrate. In particular, the Young-Dupr\'e law is shown to hold, yielding what appears to be the first analytical validation of such law for a thin-film model in the context of Continuum Mechanics.