Harmonic oscillators of carbon nanotube arrays

The harmonic properties of single-walled carbon nanotube (10, 10) arrays in cantilever geometry of lengths, L = 1,000 to 10,000 nm and diameters, D = 15 to 70 nm have been measured recently, and a linear relationship between the first natural frequency and the ratio of array diameter and the square of the span length, D/L2 was postulated. In the present work the authors show that this relationship is highly nonlinear, especially for large values of the ratio, D/L2. In addition, for a given array length, L = 1000 nm, the first natural frequency of the cantilever is shown to vary little with diameters more than 30 nm and to become asymptotic to a value of 22 MHz as it is further increased. The present study is based on earlier work of the authors wherein the flexural stiffness of the single-walled carbon nanotube (CNT) array of hexagonal symmetry and of non-covalent bonding, due to van der Waals interactions, was predicted in terms of the chirality of the nanotubes and the shearing transfer efficiency between nanotubes when subjected to flexural deformation. In addition, predictions are shown to be in agreement with the experimental evidence wherein the flexural modulus of the CNT array decreases with an increase in array diameter.