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

ESAIM: Mathematical Modelling and Numerical Analysis (2011)

- 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 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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