Surface energies in a two-dimensional mass-spring model for crystals
Theil, Florian. (2011) Surface energies in a two-dimensional mass-spring model for crystals. ESAIM : Mathematical Modelling and Numerical Analysis, Vol.45 (No.5). pp. 873-899. ISSN 0764-583XFull text not available from this repository.
Official URL: http://dx.doi.org/10.1051/m2an/2010106
We study an atomistic pair potential-energy E((n))(y) that describes the elastic behavior of two-dimensional crystals with n atoms where y is an element of R(2xn) 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((n)) (y) = n E(bulk) + root n E(surface) + o(root n), n -> infinity.
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.
|Item Type:||Journal Article|
|Divisions:||Faculty of Science > Mathematics|
|Journal or Publication Title:||ESAIM : Mathematical Modelling and Numerical Analysis|
|Publisher:||E D P Sciences|
|Page Range:||pp. 873-899|
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