Gamma-expansion for a 1D Confined Lennard-Jones model with point defect

TL;DR

This paper provides a rigorous asymptotic expansion of the energy associated with a point defect in a 1D atomic chain modeled by a Confined Lennard-Jones potential, revealing how defects influence the chain's energy landscape.

Contribution

It introduces a novel 1D Confined Lennard-Jones model with specific convexity assumptions and derives the { extGamma}-limit and first-order expansion, including explicit formulas and decay properties.

Findings

Derived the { extGamma}-limit for the energy functional as the number of atoms increases.

Explicitly characterized the first-order term in the { extGamma}-expansion using an infinite cell problem.

Proved exponential decay of minimizers, indicating defect effects are localized near the defect.

Abstract

We compute a rigorous asymptotic expansion of the energy of a point defect in a 1D chain of atoms with second neighbour interactions. We propose the Confined Lennard-Jones model for interatomic interactions, where it is assumed that nearest neighbour potentials are globally convex and second neighbour potentials are globally concave. We derive the {\Gamma}-limit for the energy functional as the number of atoms per period tends to infinity and derive an explicit form for the first order term in a {\Gamma}-expansion in terms of an infinite cell problem. We prove exponential decay properties for minimisers of the energy in the infinite cell problem, suggesting that the perturbation to the deformation introduced by the defect is confined to a thin boundary layer.