  The continuum model related to the Winterbottom problem, i.e., the problem of
determining the equilibrium shape of crystalline drops resting on a substrate,
is derived in dimension two by means of a rigorous discrete-to-continuum
passage by $\Gamma$-convergence of atomistic models taking into consideration
the atomic interactions of the drop particles both among themselves and with
the fixed substrate atoms. As a byproduct of the analysis effective expressions
for the drop surface anisotropy and the drop/substrate adhesion parameter
appearing in the continuum model are characterized in terms of the atomistic
potentials, which are chosen of Heitmann-Radin sticky-disc type. Furthermore, a
threshold condition only depending on such potentials is determined
distinguishing the wetting regime, where discrete minimizers are explicitly
characterized as configurations contained in a layer with a one-atom thickness,
i.e., the wetting layer, on the substrate, from the dewetting regime. In the
latter regime, also in view of a proven conservation of mass in the limit as
the number of atoms tends to infinity, proper scalings of the minimizers of the
atomistic models converge (up to extracting a subsequence and performing
translations on the substrate surface) to a bounded minimizer of the
Winterbottom continuum model satisfying the volume constraint.
