Calculus of Variations and Geometric Measure Theory

L. Kreutz - P. Piovano

Microscopic validation of a variational model of epitaxially strained crystalline films

created by kreutz on 21 Jan 2019
modified by piovano on 19 May 2023


Published Paper

Inserted: 21 jan 2019
Last Updated: 19 may 2023

Journal: SIAM J. Math. Anal.
Volume: 53
Number: 1
Pages: 453-490
Year: 2021


A discrete-to-continuum analysis for free-boundary problems related to crystalline films deposited on substrates is performed by Γ-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 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 of 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.