Continuum Surface Energy from a Lattice Model

We investigate some connections between the continuum and atomistic descriptions of de- formable crystals, using some interesting results from number theory. The energy of a deformed crystal is calculated in the context of a lattice model with binary interactions in two dimensions. A new bond counting approach is used, which reduces the problem to the lattice point problem of number theory. When the crystal shape is a lattice polygon, we show that the energy equals the bulk elastic energy, plus the boundary integral 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 infinite-range potentials it is asymptotically valid as the lattice parameter 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. For more general convex domains with possibly curved boundary, the surface energy density depends on the unit normal in a striking way. It is continuous at irrational directions, discontinuous at rational ones and nowhere differ- entiable. This pathology is alarming since it renders the surface energy minimization problem (under domain variations) ill-posed. An alternative approach of defining the continuum region is introduced, that restores continuity of the surface energy density function