Liquid and crystalline films: wetting and evolution

The project aims to provide a unified approach to the study of stress-driven rearrangement instabilities, supported epitaxially-strained thin films, material voids in elastic solids, droplets and capillary drops. The related variational models will be investigated both in the static and evolutionary case. The static setting includes an evaluation of the regularity of film configurations at equilibria and their contact-angle conditions on flat (Youngs law) and rough (Wenzel-Cassie-Baxter law) substrates. The evolutionary framework is devoted to the study of evaporation-condensation processes by adapting the minimizing movement method to solve associated anisotropic film equations. The methodology combines techniques of Geometric Measure Theory and the Calculus of Variations with the methods in Partial Differential Equations. In particular, it includes results from the theory of sets of finite perimeter, (anisotropic) minimal surfaces, gradient flows, homogenization theory, relaxation theory as well as the regularity theory for PDEs, which have already been tested in the establishment of preliminary results of the project. The scientific impact of the project resides on allowing the analysis of more physical situations than those presently available in the literature. For example, the project intends to drop out artificial assumptions on thin film configurations such as subgraph-property, star-shapedness and two-dimensionality, which have been previously adopted in the literature for technical reasons. In addition, previous results will be extended also to rough inhomogeneous substrates that present wettability, expressed by a function admitting both positive and negative values. Finally, from the mathematical point of view, it seems that the results for the evolution of films with contact-angle condition will be the first in the literature where minimizing movement method is implemented with the presence of forcing and boundary terms.

The project aims to study a variational model displaying both elastic and surface energies that simultaneously takes into account the various possible stress-driven rearrangement instabilities. SDRI includes all those material morphologies such as boundary irregularities, cracks, filaments, wetting and dewetting, delaminations, brittle fractures and other surface patterns, which a crystalline material may exhibit in the presence of external forces, such as chemical bonding with adjacent materials. The model provides a unified mathematical treatment of epitaxially-strained thin films, crystal cavities, capillary droplets, as well as Griffith and some failure models, which were previously treated separately in the literature. Furthermore, the possibility of delamination and debonding, i.e., crack-like modes of interface failure at the interface with the substrate is treated in accordance with the analogous models in the literature that were introduced by revisiting in the variational perspective of fracture mechanics. As a consequence the surface energy depends on the admissible deformations and cannot be decoupled from the elastic energy. As a byproduct of our analysis, we extend previous results for the existence of minimal configurations to anisotropic surface and elastic energies, and we relax constraints previously assumed on admissible configurations in the thin-film and crystal-cavity settings. For our SDRI model, graph-like constraints previously considered for the thin-film or crystal-cavity settings are not anymore needed. In addition, previous results in the literature are extended to rough inhomogeneous substrates that present wettability, expressed by a function admitting both positive and negative values. To prove the main results we adapt and/or generalize several tools from Geometric Measure Theory and Calculus of Variations. Also, the proofs cannot be concluded directly from the existing results in the literature for other models for SDRI (such as in the Dirichlet settings or graph-type settings).