Microscopical justification of the Winterbottom problem for well-separated lattices

We consider the discrete atomistic setting introduced in Piovano and Velcic (2022) to microscopically justify the continuum model related to the Winterbottom problem, i.e., the problem of determining the equilibrium shape of crystalline film drops resting on a substrate, and relax the rigidity assumption considered in Piovano and Velcic (2022) to characterize the wetting and dewetting regimes and to perform the discrete to continuum passage. In particular, all results of Piovano and Velcic (2022) are extended to the setting where the distance between the reference lattices for the film and the substrate is not smaller than the optimal bond length between a film and a substrate atom. Such optimal film-substrate bonding distance is prescribed together with the optimal film-film distance by means of two-body atomistic interaction potentials of Heitmann-Radin type, which are both taken into account in the discrete energy, and in terms of which the wetting-regime threshold and the effective expression for the wetting parameter in the continuum energy are determined.