# A Discrete-to-Continuum Model of Weakly Interacting Incommensurate Chains

## TL;DR

This paper develops a continuum variational model from a discrete atomic chain model to describe incommensurate layered materials like graphene on a substrate, capturing key behaviors observed in the discrete system.

## Contribution

It introduces a formal discrete-to-continuum derivation for incommensurate atomic chains, bridging microscopic and macroscopic descriptions.

## Key findings

- Continuum model accurately reproduces discrete chain behavior.
- Large commensurate regions separated by incommensurate domain walls.
- Numerical solutions confirm qualitative and quantitative agreement.

## Abstract

In this paper we use a formal discrete-to-continuum procedure to derive a continuum variational model for two chains of atoms with slightly incommensurate lattices. The chains represent a cross-section of a three-dimensional system consisting of a graphene sheet suspended over a substrate. The continuum model recovers both qualitatively and quantitatively the behavior observed in the corresponding discrete model. The numerical solutions for both models demonstrate the presence of large commensurate regions separated by localized incommensurate domain walls.