Phonons in chiral nanorods and nanotubes: A Cosserat-rod-based continuum approach

Abstract

A Cosserat-rod-based continuum approach is presented to obtain phonon dispersion curves of flexural, torsional, longitudinal, shearing, and radial breathing modes in chiral nanorods and nanotubes. Upon substituting the continuum wave form in the linearized dynamic equations of stretched and twisted Cosserat rods, we obtain an analytical expression of a coefficient matrix (in terms of the rod’s stiffnesses, induced axial force, and twisting moment) whose eigenvalues and eigenvectors give us frequencies and mode shapes, respectively, for each of the above phonon modes. We show that, unlike the case of achiral tubes, these phonon modes are intricately coupled in chiral tubes owing to extension–torsion–inflation and bending–shear couplings inherent in them. This coupling renders the conventional approach of obtaining stiffnesses from the long wavelength limit slope of dispersion curves redundant. However, upon substituting the frequencies and mode shapes (obtained independently from phonon dispersion molecular data) in the eigenvalue–eigenvector equation of the above-mentioned coefficient matrix, we are able to obtain all the stiffnesses (bending, twisting, stretching, shearing, and all coupling stiffnesses corresponding to extension–torsion, extension–inflation, torsion–inflation, and bending–shear couplings) of chiral nanotubes. Finally, we show unusual effects of the single-walled carbon nanotube’s chirality as well as stretching and twisting of the nanotube on its phonon dispersion curves obtained from the molecular approach. These unusual effects are accurately reproduced in our continuum formulation.