Continuum Surface Energy from a Lattice Model

TL;DR

This paper derives an explicit formula for continuum surface energy density in deformable crystals from a lattice model, connecting atomistic and continuum descriptions using number theory and bond counting methods.

Contribution

It introduces a new bond counting approach that links lattice models to the lattice point problem, providing explicit surface energy formulas for various domain shapes.

Findings

Explicit formula for surface energy density as a function of deformation and boundary normal

Applicable to domains with polygonal and piecewise smooth boundaries

Bridges atomistic lattice models with continuum surface energy concepts

Abstract

We investigate connections between the continuum and atomistic descriptions of deformable crystals, using certain interesting results from number theory. The energy of a deformed crystal is calculated in the context of a lattice model with general binary interactions in two dimensions. A new bond counting approach is used, which reduces the problem to the lattice point problem of number theory. The main contribution is an explicit formula for the surface energy density as a function of the deformation gradient and boundary normal. The result is valid for a large class of domains, including faceted (polygonal) shapes and regions with piecewise smooth boundaries.