We study an atomistic pair potential-energy E(n)(y) that describes the elastic behavior of two-dimensional crystals with n atoms where 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: 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.
@article{M2AN_2011__45_5_873_0,
author = {Theil, Florian},
title = {Surface energies in a two-dimensional mass-spring model for crystals},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {45},
year = {2011},
pages = {873-899},
doi = {10.1051/m2an/2010106},
mrnumber = {2817548},
zbl = {1269.82065},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2011__45_5_873_0}
}
Theil, Florian. Surface energies in a two-dimensional mass-spring model for crystals. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 45 (2011) pp. 873-899. doi : 10.1051/m2an/2010106. http://gdmltest.u-ga.fr/item/M2AN_2011__45_5_873_0/
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