From atomistics to continuum: Effects of a free surface and determination of surface elasticity properties

•We establish an atomistic framework for computing surface elasticity properties.•We obtain new restrictions on atomistic models for physical reasonable behaviors.•We achieve an efficient algorithm for surface elasticity properties.

We present an analysis of surface elasticity from the Born–Oppenheimer approximation for monatomic crystals. The analysis shows that the relaxations of crystal planes parallel to a free surface can be sufficiently determined by a low-rank algebraic Riccati equation instead of a full-scale molecular dynamic (MD) simulation, and gives new restrictions on physically reasonable atomistic models and simple criteria for surface reconstructions. In the case of surface relaxations, we calculate surface elasticity properties, i.e., surface tension, residual surface stress and surface elastic stiffness tensor, from atomistic models which are compared with experimental data and prior simulation results. The formulation also proves that surface relaxations always lower surface tension and surface elastic stiffness tensor. Together with the proposed algorithm, the formulation may be useful for investigating a variety of size-dependent phenomena of nano-structures.