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On the Hierarchy of Scales in Modeling of Weakly Interacting Chains of Atoms

Dmitry Golovaty, J. Patrick Wilber

TL;DR

The paper develops a two-stage variational convergence framework for a Frenkel-Kontorova-type one-dimensional chain with weak substrate coupling. First, as $\delta\to 0$, a mesoscale continuum energy $F$ with a single diffuse domain wall emerges, with $F[\xi] = \int \xi'^2 dx + \int w(\xi) dx$ and $w$ periodic, capturing a smooth transition between registries. Then, by introducing a second scale $\varepsilon$ and letting $\varepsilon\to 0$, a macroscale Modica–Mortola-type BV-energy is obtained, where discontinuities correspond to sharp domain walls and multiple walls are possible under general boundary data. This hierarchical upscaling clarifies when domain walls are diffuse or sharp and provides a rigorous link from microscopic atomistic models to mesoscopic and macroscopic descriptions relevant to moiré patterns and registry effects in layered materials.

Abstract

In the first part of this paper, we apply a well known discrete-to-continuum approach to a Frenkel-Kontorova-type model of an infinitely long one-dimensional chain of atoms weakly interacting with a line of fixed atoms. The rescaled model contains a small parameter $δ$ that is the ratio of the strengths of the weak interaction and the elastic interaction. After replacing discrete displacements with piecewise affine functions to define continuum versions of the discrete energies, we prove that these energies $Γ$-converge to a continuum energy as $δ\rightarrow 0$. This limiting process represents a transition from the microscale, at which individual atoms are resolved, to a mesoscale with a single diffuse domain wall. In the second part of this paper, we introduce an additional rescaling $\varepsilon$, and an associated limiting process that converts our problem to the macroscale. The $\varepsilon$-limiting energy is finite for piecewise constant functions of bounded variation. In the context of our problem, each point of discontinuity of a minimizer of the limiting energy corresponds to a sharp domain wall.

On the Hierarchy of Scales in Modeling of Weakly Interacting Chains of Atoms

TL;DR

The paper develops a two-stage variational convergence framework for a Frenkel-Kontorova-type one-dimensional chain with weak substrate coupling. First, as , a mesoscale continuum energy with a single diffuse domain wall emerges, with and periodic, capturing a smooth transition between registries. Then, by introducing a second scale and letting , a macroscale Modica–Mortola-type BV-energy is obtained, where discontinuities correspond to sharp domain walls and multiple walls are possible under general boundary data. This hierarchical upscaling clarifies when domain walls are diffuse or sharp and provides a rigorous link from microscopic atomistic models to mesoscopic and macroscopic descriptions relevant to moiré patterns and registry effects in layered materials.

Abstract

In the first part of this paper, we apply a well known discrete-to-continuum approach to a Frenkel-Kontorova-type model of an infinitely long one-dimensional chain of atoms weakly interacting with a line of fixed atoms. The rescaled model contains a small parameter that is the ratio of the strengths of the weak interaction and the elastic interaction. After replacing discrete displacements with piecewise affine functions to define continuum versions of the discrete energies, we prove that these energies -converge to a continuum energy as . This limiting process represents a transition from the microscale, at which individual atoms are resolved, to a mesoscale with a single diffuse domain wall. In the second part of this paper, we introduce an additional rescaling , and an associated limiting process that converts our problem to the macroscale. The -limiting energy is finite for piecewise constant functions of bounded variation. In the context of our problem, each point of discontinuity of a minimizer of the limiting energy corresponds to a sharp domain wall.
Paper Structure (5 sections, 4 theorems, 89 equations, 2 figures)

This paper contains 5 sections, 4 theorems, 89 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Description of Discrete Problem
  3. Convergence to a Mesoscale Model
  4. Convergence to a Macroscale Model
  5. Conclusion

Key Result

Lemma 3.1

Let $C$ and $\delta$ be positive constants. Let $\{\xi_{i}\}$ be a sequence and let $\xi^\delta$ be the associated piecewise affine function on $P_{\delta}$. Suppose that either $E^{\delta}[\{\xi_{i}\}]<C$ or that $E_{\delta}\left[\xi^\delta\right]<C$. Then there is a constant $\hat{C}$ that depends

Figures (2)

  • Figure 2.1: Reference Configuration of Discrete System.
  • Figure 4.1: Construction of $\hat{\xi}_{n}$.

Theorems & Definitions (5)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Theorem 3.3
  • Theorem 4.1