Microscopical Justification of the Winterbottom problem for well-separated Lattices

We consider the discrete atomistic setting introduced in \cite{PiVe1} to microscopically justify the continuum model related to the \emph{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 \cite{PiVe1} to characterize the \emph{wetting} and \emph{dewetting regimes} and to perform the \emph{discrete to continuum passage}. In particular, all results of \cite{PiVe1} 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.