Hyper-pre-stress vs. strain-gradient for surface relaxation in diamond-like structures

In this paper, based on the discrete valence force field model for diamond-like crystals, we construct and compare two anisotropic models with cubic symmetry containing an intrinsic internal length. The first model is constructed starting from the discrete valence force field model via lattice dynamics and the classical polynomial approximation of the acoustic branches near the Γ point. We show that the strain-gradient continuum approximation obtained in this way enforces a result obtained previously in Maranganti and Sharma (2007) and leads to internal lengths which are unphysically small. The second model follows a line studied in the one-dimensional context in Charlotte and Truskinovsky (2002). It accounts for incompatible reference lengths in the discrete setting and is able to model both cohesion, surface energy and defects. We study the ground state of an infinite lattice and we conclude that, as expected, for finite structures a boundary-layer effect occurs. We explore numerically using the model with hyper-pre-stress: (a) the average lattice parameter of bulk-like, plate-like and beam-like nano-structures, and (b) the lattice parameter variations as a function of porosity in porous silicon. We conclude that the model with hyper-pre-stress provides a realistic model for the mechanics of nano-structures including size and shape effects.