Warping caused by surface elasticity in a nanowire under torsion

Abstract

By combining molecular statics simulations and continuum mechanics-based modeling, we show in this paper that the torsion of a 001 single-crystal copper nanowire with a circular cross section gives rise to a warp, i.e., to a displacement field along the wire axis that renders the cross section nonplanar. This behavior, which is in apparent contradiction with what is predicted by continuum mechanics for an isotropic cylinder, can be well explained if we take into account the elastic response of the wire lateral surface. The latter is characterized by the anisotropy of the surface elastic constants and, more specifically in the case of torsion, by the surface shear constant C S 44 whose strength as a function of the local orientation of the lateral surface is estimated independently from atomistic calculations on slabs presenting different vicinal surfaces. The solution of the torsion problem is then obtained by adopting a semi-inverse method in the framework of the finite strain theory in linear elasticity with Gurtin-Murdoch boundary conditions linking surface stress and bulk stress. It is shown that such an approach is well suited to explain quantitatively the warp obtained in our atomistic simulations and to prove the preponderant role played by the surface elastic constant C S 44 .

Understanding the warp in free 〈111〉 metal nanowires by modeling surface elasticity

“…In this Letter we want to shed light on the interpretation of this wrinkling by identifying the physical quantities that are responsible for this phenomenon. Our analysis is based on a recent work [2] we conducted to explain the formation of a warp in twisted 001 copper nanowire, induced by the coupling between the torsion and the surface shear elastic constant (which we will denote C S 44 ). Since wrinkling is finally nothing else than a form of warp existing in the absence of torsion of the nanowire, we wish to test the ability of our continuum mechanics approach based on the Gurtin-Murdoch (GM) theory of surface elasticity to also predict the wrinkling in untwisted 111 wires.…”

“…Since it has been detailed recently in Ref. [2], we will focus only on the developments specific to the case that interests us here, i.e., a free, very long solid circular cylinder composed of a fcc metal and having its main axis oriented along a 111 direction. The surface parameters involved in the model are calculated for two metals, gold and copper, from molecular statics (MS) calculations on slabs with surfaces which are close structurally to those encountered around these wires.…”

“…To account for the difference in elastic behavior on the surface and inside the bulk, one has to determine the surface deformation gradient F S and the surface strain E S . In the simple case of a circular cross section of radius R 0 , F S is directly deduced [2] Assuming that the bulk strain energy ψ (E) per unit undeformed volume and the surface strain energy ψ S (E S ) per unit undeformed area are known, the second Piola-Kirchhoff stress tensors are defined as S = dψ/dE in the bulk and S S = dψ S /dE S at the surface.…”

“…To calculate S S,0 i and C S i j as a function of for an 111oriented wire, we adopt the method described in Ref. [2] where we construct and then deform slabs whose surface initially exhibits a structure similar to that encountered locally on the lateral surface of a 111 wire with circular cross section. Using MS calculations with the tight-binding second moment approximation (SMA) potential (parameters are taken from Refs.…”