General rolled-up and polyhedral models for carbon nanotubes

Abstract

In many computational studies of carbon nanotubes, the minimum energy configuration frequently settles on a structure for which the bond lengths are distinct. Here, we extend both the rolled-up and the polyhedral models for SWCNTs to produce general models incorporating either distinct bond lengths and the same bond angle, or distinct bond lengths and distinct bond angles. The CNTs considered here are assumed to be formed by sp2 hybridization but with different bond lengths so that the nanotube structure is assumed to comprise irregular hexagonal patterns. The polyhedral model with distinct bond lengths and distinct bond angles is based on the two postulates that all bonds lying on the same helix are equal in length and that all atoms are equidistant from a common axis of symmetry. The polyhedral model with distinct bond lengths and the same bond angle has the additional postulate that all the adjacent bond angles are equal. We derive exact formulae for the geometric parameters and we present asymptotic expansions for the polyhedral model with distinct bond lengths and distinct bond angles to the first two orders of magnitude. Good agreement is demonstrated for the predictions of the polyhedral model compared with the results obtained from other computational studies.