Surface stress effects on the bending properties of fcc metal nanowires

Abstract

The major purpose of this work is to investigate surface stress effects on the bending behavior and properties of $⟨100⟩/{100}$ gold nanowires with both fixed/fixed and fixed/free boundary conditions. The results are obtained through utilization of the recently developed surface Cauchy-Born model, which captures surface stress effects on the elastic properties of nanostructures through a three-dimensional, nonlinear finite element formulation. There are several interesting findings in the present work. First, we quantify the stress and displacement fields that result in the nanowires due to bending deformation. In doing so, we find that regardless of boundary condition, the stresses that are present in the nanowires due to deformation induced by surface stresses prior to any applied bending deformation dominate any stresses that are generated by the bending deformation unless very large $(\ensuremath{\approx}5%)$ bending strains are applied. In contrast, when the stresses and displacements induced by surface stresses prior to bending are subtracted from the stress and displacement fields of the bent nanowires, we find that the bending stresses and displacements do match the solutions expected from bulk continuum beam theory, but only within the nanowire bulk, and not at the nanowire surfaces. Second, we find that the deformation induced by surface stresses also has a significant impact on the nanowire Young's modulus that is extracted from the bending simulations, where a strong boundary-condition dependence is also found. By comparing all results to those that would be obtained using various linear surface-elastic theories, we demonstrate that a nonlinear, finite deformation formulation that captures changes in both bulk- and surface-elastic properties resulting from surface stress-induced deformation is critical to reproducing the experimentally observed boundary-condition dependence in Young's modulus of metal nanowires. Furthermore, we demonstrate that linear surface-elastic theories based solely on the surface energy erroneously predict an increase in Young's modulus with decreasing nanowire size regardless of boundary condition. In contrast, while the linear surface-elastic theories based upon the Gurtin and Murdoch formalism can theoretically predict elastic softening with decreasing size, we demonstrate that, regardless of boundary condition, the stiffening due to the surface stress dominates the softening due to the surface stiffness for the range of nanowire geometries considered in the present work. Finally, we determine that the nanowire Young's modulus is essentially identical when calculated via either bending or resonance for both boundary conditions, indicating that surface effects have a similar impact on the elastic properties of nanowires for both loading conditions.