Continuum Surface Energy from a Lattice Model

Details Continuum Surface Energy from a Lattice Model Continuum Surface Energy from a Lattice Model

2012-01-03

arxiv.org/abs/1201.0712v1

Physics

Mathematical Physics

10 pages, no figures

Scientific paper

The energy of a homogeneously deformed, faceted crystal is calculated in the context of a central force lattice model in two dimensions. It is shown that the energy equals the bulk elastic energy, plus the integral over the boundary of a surface energy density, plus the sum over the vertices of a corner energy function. This is an exact result when the interatomic potential has finite range; for an infinite-range potential it is asymptotically valid as the lattice parameter tends to zero. The surface energy density is obtained explicitly as a function of the deformation gradient and boundary normal. The corner energy is found as an explicit function of the deformation gradient and the normals of the two facets meeting at the corner. A new bond counting approach is used, which allows the problem to be reduced to the well known lattice point problem of number theory.

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