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A discrete-to-continuum analysis for free-boundary problems related tocrystalline films deposited on substrates is performed by $\Gamma$-convergence.The discrete model here introduced is characterized by an energy with twocontributions, the surface and the elastic-bulk energy, and it is formallyjustified starting from atomistic interactions. The surface energy countsmissing bonds at the film and substrate boundaries, while the elastic energymodels the fact that for film atoms there is a preferred interatomic distancedifferent from the preferred interatomic distance for substrate atoms. In theregime of small mismatches between the film and the substrate optimal lattices,a discrete rigidity estimate is established by regrouping the elastic energy intriangular-cell energies and by locally applying rigidity estimates from theliterature. This is crucial to establish pre-compactness for sequences withequibounded energy and to prove that the limiting deformation is one singlerigid motion. By properly matching the convergence scaling of the differentterms in the discrete energy, both surface and elastic contributions appearalso in the resulting continuum limit in agreement (and in a form consistent)with literature models. Thus, the analysis performed here is a microscopicaljustifications of such models.

Microscopic validation of a variational model of epitaxially strained crystalline film