Microscopic validation of a variational model of epitaxially strained crystalline film

TL;DR

This paper rigorously derives a continuum model for epitaxially strained crystalline films from atomistic interactions using $ ext{Gamma}$-convergence, validating macroscopic models through microscopic analysis.

Contribution

It provides a discrete-to-continuum analysis of crystalline films, justifying continuum models from atomistic interactions with rigorous mathematical proofs.

Findings

Established a discrete rigidity estimate for small lattice mismatches.

Proved the limiting deformation is a single rigid motion.

Derived continuum energy models consistent with existing literature.

Abstract

A discrete-to-continuum analysis for free-boundary problems related to crystalline films deposited on substrates is performed by $\Gamma$-convergence. The discrete model here introduced is characterized by an energy with two contributions, the surface and the elastic-bulk energy, and it is formally justified starting from atomistic interactions. The surface energy counts missing bonds at the film and substrate boundaries, while the elastic energy models the fact that for film atoms there is a preferred interatomic distance different from the preferred interatomic distance for substrate atoms. In the regime of small mismatches between the film and the substrate optimal lattices, a discrete rigidity estimate is established by regrouping the elastic energy in triangular-cell energies and by locally applying rigidity estimates from the literature. This is crucial to establish pre-compactness for sequences with equibounded energy and to prove that the limiting deformation is one single rigid motion. By properly matching the convergence scaling of the different terms in the discrete energy, both surface and elastic contributions appear also in the resulting continuum limit in agreement (and in a form consistent) with literature models. Thus, the analysis performed here is a microscopical justifications of such models.