Vibrational analysis of thermal oscillations of single-walled carbon nanotubes under axial strain

Visualization of electronic density

“…The width range they quote is between 0.69 and 0.0617 nm. In two of our papers [24,28] we found via two distinct selfconsistent procedures with simulations of nanotubes of multiple lengths and radii that 0.067 nm is consistent with all our simulations and with earlier studies [30,31]. However we wish to be able to estimate changes in this value directly when we place adsorbed molecules to be weighed, hence the present investigation.…”

“…For a nanotube of radius r, length L, and volume V the frequency of the nth mode, f n , of its vibrations depend on the nanotube's width, w and radius r via two quantities used in its calculation: the moment of inertia, I = π rw(4r 2 + w 2 ) and mass linear density ρ L = 2π rwρ V (ρ L is the density per unit length and ρ V the density per unit volume) which are related to the tubes' Young's modulus, E. The exact relation depends on the boundary conditions and model; and the main point of [24][25][26][27][28] was to deduce these frequencies without explicit use of any of the competing analytic models. Literature estimates of E range between 1-6 Tpa, and this is known as Yakobson's paradox [30].…”

“…The essential parameter for model analysis is the width of the nanotube wall, and it is the electronic cloud around the atomic nuclei that determines this. In a series of papers and a thesis, Pine and coworkers [23][24][25][26][27][28][29] reviewed the literature and carried out extensive classical molecular dynamics simulations at the atomistic scale to carefully determine the limits of applicability of the analytic theory. While values for the wall width were deduced indirectly by us and many others, direct estimation is of course more desirable.…”

Tight-binding vibrational analysis of single-wall carbon nanotubes

“…[6] time-averaged dynamical matrices for finite CNTs are derived from MD trajectories at ultra low temperatures to calculate the eigenfrequencies. Recently, Pine et al reported comparative MD simulations (using the Brenner potential for hydrocarbons and Fourier analysis for the frequency spectrum) and classical beam calculations for suspended single-walled CNTs under axial strain [8,9,10]. The low resonance frequencies resulted from classical continuum theories are, generally, in fair agreement with their counterparts obtained from the dynamical matrix or MD simulations, but gradually deviate as the effective wavelength becomes of the order of the interatomic distances.…”

“…Previous theoretical work was essentially carried out by: (a) Normal mode analysis based on the dynamical matrix, constructed using force constants [5,6] or tight-binding force fields [7]; (b) Molecular dynamics (MD) simulations using interatomic potentials and vibration spectra derived as Fouriertransformed correlation functions [8,9,10]; (c) Continuum beam models based on the Euler-Bernoulli or Timoshenko equations, which, however, do not address the structural atomistic particularities of the CNTs [8,9,10,11,12,13].…”