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

Theil, Florian

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 5, pp. 873-899.

We study an atomistic pair potential-energy E⁽ⁿ⁾(y) that describes the elastic behavior of two-dimensional crystals with n atoms where $y\in {ℝ}^{2\times n}$ 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: ${\min }_y{E}^{\left(n\right)}\left(y\right)=n{E}_{\mathrm{bulk}}+\sqrt{n}{E}_{\mathrm{surface}}+o\left(\sqrt{n}\right),n\to \infty .$ 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.

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