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

An elastic string-elastic shell model is developed to study the coupled vibration of a carbon nanowire 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 with vibration of the CNT only for vibration modes with circumferential wavenumber n = 1. In other cases, such as axisymmetric modes (n = 0) or higher-order vibration modes with n >= 2, total resultant van der Waals (vdW) force acting on the C-chain due to the innermost tube always vanishes, and therefore vibration of the CNT does not cause vibration of the inserted C-chain, although the existence of the C- chain does have an effect on the vibration of the CNT through the chain-CNT vdW forces acting on the innermost tube. The present model predicts that non-coaxial vibration between the C-chain and the innermost tube does not occur due to negligible bending rigidity of the C-chain. In addition, it is found that the C-chain has most significant effect on the lowest frequency 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 known experimental and modeling results available in the literature. The present work offers systematic modeling results on the effects of an inserted C-chain on the vibration of CNTs.