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

Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK. f.theil@warwick.ac.uk

Received: 29 January 2010
Revised: 28 October 2010

Abstract

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 E_bulk is given by an explicit expression involving the interaction potentials. The surface energy E_surface 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.