The effects of an inserted linear carbon chain on the vibration of a carbon nanotube

Abstract

An elastic string–elastic shell model is developed to study the coupled vibration of a carbonnanowire made of a linear carbon chain (C-chain) inserted inside a carbon nanotube(CNT). It is shown that the vibration of the inserted C-chain is coupled withvibration of the CNT only for vibration modes with circumferential wavenumbern = 1. In other cases, such as axisymmetric modes(n = 0) or higher-ordervibration modes with n≥2, total resultant van der Waals (vdW) force acting on the C-chain due to the innermosttube always vanishes, and therefore vibration of the CNT does not cause vibration of theinserted C-chain, although the existence of the C-chain does have an effect on the vibrationof the CNT through the chain–CNT vdW forces acting on the innermost tube. Thepresent model predicts that non-coaxial vibration between the C-chain and theinnermost tube does not occur due to negligible bending rigidity of the C-chain. Inaddition, it is found that the C-chain has most significant effect on the lowestfrequency associated with the radial vibration mode for circumferential wavenumber 2(n = 2). In particular, the effect of the C-chain on the axisymmetric radial breathing frequencies(n = 0) predicted by the present model is found to be in reasonable agreement with knownexperimental and modeling results available in the literature. The present work offerssystematic modeling results on the effects of an inserted C-chain on the vibration of CNTs.