On the importance of surface elastic contributions to the flexural rigidity of nanowires

We present a theoretical model to calculate the flexural rigidity of nanowires from three-dimensional elasticity theory that incorporates the effects of surface stress and surface elasticity. The unique features of the model are that it incorporates, through the second moment, the heterogeneous nature of elasticity across the nanowire cross section, and that it accounts for transverse surface-stress-induced relaxation strains. The model is validated by comparison to benchmark atomistic calculations, existing one-dimensional surface elasticity theories based on the Young–Laplace equation, and also three-dimensional surface elasticity theories that assume homogeneous elastic properties across the nanowire cross section via three examples: surface-stress-induced axial relaxation, resonant properties of unstrained, strained and top-down nanowires, and buckling of nanowires. It is clearly demonstrated that the one-dimensional Young–Laplace models lead to errors of varying degrees for all of the boundary value problems considered because they do not account for transverse surface stress effects, and it is also shown that the Young–Laplace model results from a specific approximation of the proposed formulation. The three-dimensional surface elasticity model of Dingreville et al. (2005) is found to be more accurate than the Young–Laplace model, though both lose accuracy for ultrasmall (<5nm diameter) nanowires where the heterogeneous nature of the cross section elasticity becomes important. Overall, the present work demonstrates that continuum mechanics can be utilized to study the elastic and mechanical behavior and properties of ultrasmall nanowires if surface elastic contributions to the heterogeneous flexural rigidity are accounted for.