Surface energies in a two-dimensional mass-spring model for crystals

Florian Theil

ESAIM: Mathematical Modelling and Numerical Analysis (2011)

  • Volume: 45, Issue: 5, page 873-899
  • ISSN: 0764-583X

Abstract

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We study an atomistic pair potential-energy E(n)(y) that describes the elastic behavior of two-dimensional crystals with n atoms where y 2 × n characterizes the particle positions. The main focus is the asymptotic analysis of the ground state energy as n tends to infinity. We show in a suitable scaling regime where the energy is essentially quadratic that the energy minimum of E(n) admits an asymptotic expansion involving fractional powers of n: min y E ( n ) ( y ) = n E bulk + n E surface + o ( n ) , n . The bulk energy density Ebulk is given by an explicit expression involving the interaction potentials. The surface energy Esurface can be expressed as a surface integral where the integrand depends only on the surface normal and the interaction potentials. The evaluation of the integrand involves solving a discrete algebraic Riccati equation. Numerical simulations suggest that the integrand is a continuous, but nowhere differentiable function of the surface normal.

How to cite

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Theil, Florian. "Surface energies in a two-dimensional mass-spring model for crystals." ESAIM: Mathematical Modelling and Numerical Analysis 45.5 (2011): 873-899. <http://eudml.org/doc/197544>.

@article{Theil2011,
abstract = { We study an atomistic pair potential-energy E(n)(y) that describes the elastic behavior of two-dimensional crystals with n atoms where $y \in \{\mathbb R\}^\{2\times n\}$ characterizes the particle positions. The main focus is the asymptotic analysis of the ground state energy as n tends to infinity. We show in a suitable scaling regime where the energy is essentially quadratic that the energy minimum of E(n) admits an asymptotic expansion involving fractional powers of n: $\{\rm min\}_y E^\{(n)\}(y) = n \, E_\{\mathrm\{bulk\}\}+ \sqrt\{n\} \, E_\mathrm\{surface\} +o(\sqrt\{n\}), \qquad n \to \infty.$ The bulk energy density Ebulk is given by an explicit expression involving the interaction potentials. The surface energy Esurface can be expressed as a surface integral where the integrand depends only on the surface normal and the interaction potentials. The evaluation of the integrand involves solving a discrete algebraic Riccati equation. Numerical simulations suggest that the integrand is a continuous, but nowhere differentiable function of the surface normal. },
author = {Theil, Florian},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Continuum mechanics; difference equations; continuum mechanics; two-dimensional crystals},
language = {eng},
month = {2},
number = {5},
pages = {873-899},
publisher = {EDP Sciences},
title = {Surface energies in a two-dimensional mass-spring model for crystals},
url = {http://eudml.org/doc/197544},
volume = {45},
year = {2011},
}

TY - JOUR
AU - Theil, Florian
TI - Surface energies in a two-dimensional mass-spring model for crystals
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2011/2//
PB - EDP Sciences
VL - 45
IS - 5
SP - 873
EP - 899
AB - We study an atomistic pair potential-energy E(n)(y) that describes the elastic behavior of two-dimensional crystals with n atoms where $y \in {\mathbb R}^{2\times n}$ characterizes the particle positions. The main focus is the asymptotic analysis of the ground state energy as n tends to infinity. We show in a suitable scaling regime where the energy is essentially quadratic that the energy minimum of E(n) admits an asymptotic expansion involving fractional powers of n: ${\rm min}_y E^{(n)}(y) = n \, E_{\mathrm{bulk}}+ \sqrt{n} \, E_\mathrm{surface} +o(\sqrt{n}), \qquad n \to \infty.$ The bulk energy density Ebulk is given by an explicit expression involving the interaction potentials. The surface energy Esurface can be expressed as a surface integral where the integrand depends only on the surface normal and the interaction potentials. The evaluation of the integrand involves solving a discrete algebraic Riccati equation. Numerical simulations suggest that the integrand is a continuous, but nowhere differentiable function of the surface normal.
LA - eng
KW - Continuum mechanics; difference equations; continuum mechanics; two-dimensional crystals
UR - http://eudml.org/doc/197544
ER -

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