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Surface energies in a two-dimensional mass-spring model for crystals
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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. doi:10.1051/m2an/2010106 ISSN 0764-583X.
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Official URL: http://dx.doi.org/10.1051/m2an/2010106
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 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 | ||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Journal or Publication Title: | ESAIM : Mathematical Modelling and Numerical Analysis | ||||
Publisher: | E D P Sciences | ||||
ISSN: | 0764-583X | ||||
Official Date: | 2011 | ||||
Dates: |
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Volume: | Vol.45 | ||||
Number: | No.5 | ||||
Page Range: | pp. 873-899 | ||||
DOI: | 10.1051/m2an/2010106 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published |
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