Simple Theoretical Approach

The simplistic idea of obtaining a nanometric grain by ‘cutting’ a nanometric chunk out of an infinite crystal leads one to distinguish the crystalline organizational states in the core of the grain and in those layers influenced by the proximity of the surface (see Fig. 1). Following the argument discussed above with regard to surface relaxation, it is easy to imagine that the lattice parameters may vary continuously as one moves away from the surface, inside the crystal.

Fig. 1. Schematic representation of the distortion of the crystal lattices in layers close to the surface of a nanometric grain. (a) Infinite undistorted 2D lattice. (b) A chunk is cut from the undistorted 2D lattice. (c) Isolated chunk, held in the shape of the undistorted lattice by virtue of fictitious stresses equal to those that were exerted by the rest of the lattice on the same chunk before it was cut away. (d) Isolated chunk after removal of stresses exerted by the rest of the lattice: the case of an expansion after freeing from stresses. (e) Isolated chunk after removal of stresses exerted by the rest of the lattice: the case of a contraction after freeing from stresses