Carbon-nanotube geometries: Analytical and numerical results

Abstract

We investigate carbon-nanotubes under the perspective of geometry optimization. Nanotube geometries are assumed to correspond to atomic configurations which locally minimize Tersoff-type interaction energies. In the specific cases of so-called zigzag and armchair topologies, candidate optimal configurations are analytically identified and their local minimality is numerically checked. In particular, these optimal configurations do not correspond neither to the classical Rolled-up model [5] nor to the more recent polyhedral model [3]. Eventually, the elastic response of the structure under uniaxial testing is numerically investigated and the validity of the Cauchy-Born rule is confirmed.

Graphene ground states

“…By reducing to nearest-neighbor interactions, it can nonetheless be checked to be a local minimizer, under specific assumptions on the interaction potentials [23]. This stability analysis allows to tackle other carbon nanostructures as well, including nanotubes [11,17,18], fullerenes [12,23], diamond [23], carbyne stratified configurations [16].…”

Characterization of Optimal Carbon Nanotubes Under Stretching and Validation of the Cauchy–Born Rule

“…Nanotubes with µ = µ us will be referred to as unstretched nanotubes. The aim of [55,56] was to prove that G α us is a local minimizer. This has been illustrated numerically in [55] and checked analytically in [56], for a restricted class of perturbations.…”

“…The aim of [55,56] was to prove that G α us is a local minimizer. This has been illustrated numerically in [55] and checked analytically in [56], for a restricted class of perturbations. Our stability result Theorem 3.3 below delivers an analytical proof of stability with respect to all small perturbations.…”

“…The proof for the case λ 1 = λ 2 = 1 was detailed in [55]. The extension to our setting is a straightforward adaption and is therefore omitted.…”

“…This work is the culmination of a series on the geometry and mechanics of nanotubes [55,56]. The theoretical outcomes of this paper have been predicted computationally in [55], where stability of periodic configurations have been investigated with Monte Carlo techniques, both for zigzag and armchair topologies under moderate displacements. A first step toward a rigorous analytical result has been obtained in [56] for both zigzag and armchair topologies under no stretching.…”