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

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

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topTheil, Florian. "Surface energies in a two-dimensional mass-spring model for crystals." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 45.5 (2011): 873-899. <http://eudml.org/doc/273185>.

@article{Theil2011,

abstract = {We study an atomistic pair potential-energy E(n)(y) that describes the elastic behavior of two-dimensional crystals with natoms where $y \in \{\mathbb \{R\}\}^\{2\times n\}$ characterizes the particle positions. The main focus is the asymptotic analysis of the ground state energy asn 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: $\{\rm min\}_y E^\{(n)\}(y) = n \, E_\{\mathrm \{bulk\}\}+ \sqrt\{n\} \, E_\mathrm \{surface\} +o(\sqrt\{n\}), \qquad n \rightarrow \infty .$ The bulk energy densityEbulk 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 - Modélisation Mathématique et Analyse Numérique},

keywords = {continuum mechanics; difference equations; two-dimensional crystals},

language = {eng},

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/273185},

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 - Modélisation Mathématique et Analyse Numérique

PY - 2011

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 natoms where $y \in {\mathbb {R}}^{2\times n}$ characterizes the particle positions. The main focus is the asymptotic analysis of the ground state energy asn 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: ${\rm min}_y E^{(n)}(y) = n \, E_{\mathrm {bulk}}+ \sqrt{n} \, E_\mathrm {surface} +o(\sqrt{n}), \qquad n \rightarrow \infty .$ The bulk energy densityEbulk 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; two-dimensional crystals

UR - http://eudml.org/doc/273185

ER -

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