Breakdown of structural models for vibrations of single-wall zigzag carbon nanotubes

Free vibrations of zigzag single-wall carbon nanotubes SWCNTs of aspect ratio length/diameter 6 and with ends traction-free have been studied using molecular mechanics MM simulations with the MM3 potential. It is found that the frequencies of inextensional the Love and the Rayleigh modes of an n, 0 SWCNT saturate at the circumferential wave number of either n−1 /2 or n /2 where n is odd or even. This is explained in terms of its molecular structure. Since the frequencies of the inextensional modes of vibration of a thin cylinder made of an isotropic linear elastic material do not saturate with an increase in the circumferential wave number, a continuum structure cannot represent all modes of vibration of a zigzag SWCNT. This result is independent of the value assigned to the wall thickness of the SWCNT. We have also found values of material and geometric parameters of a shell and a hollow cylinder by equating their frequencies of the inextensional, the radial breathing, the axial and the torsional modes of vibrations to the corresponding ones of a zigzag SWCNT, and by taking their mean diameter and length equal to those of the SWCNT. The frequencies of the extensional modes of oscillations of the two continuum structures for various axial half wave numbers and circumferential wave numbers are found to match well with those of the SWCNT obtained from the MM simulations. However, the frequencies of the inextensional modes of the continuum structures deviate noticeably from those of the SWCNT, and this deviation increases with an increase in the circumferential wave number. © 2009 American Institute of Physics. doi:10.1063/1.3232206