Emergence of Wulff-Crystals from atomistic systems on the FCC and HCP lattices

TL;DR

This paper analyzes the formation of Wulff-shaped crystals from atomistic models on FCC and HCP lattices, deriving continuum limits and showing FCC crystallization is energetically favored for large systems.

Contribution

It provides a rigorous derivation of perimeter-type continuum energies and explicitly computes Wulff shapes for atomistic lattice systems, highlighting FCC preference.

Findings

Continuum limit energies are of perimeter type.

Explicit Wulff shapes are computed.

FCC is energetically preferred over HCP for large N.

Abstract

We consider a system of $N$ hard spheres sitting on the nodes of either the $\mathrm{FCC}$ or $\mathrm{HCP}$ lattice and interacting via a sticky-disk potential. As $N$ tends to infinity (continuum limit), assuming the interaction energy does not exceed that of the ground-state by more than $N^{2/3}$ (surface scaling), we obtain the variational coarse grained model by $\Gamma$-convergence. More precisely, we prove that the continuum limit energies are of perimeter type and we compute explicitly their Wulff shapes. Our analysis shows that crystallization on $\mathrm{FCC}$ is preferred to that on $\mathrm{HCP}$ for $N$ large enough. The method is based on integral representation and concentration-compactness results that we prove for general periodic lattices in any dimension.