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Research Papers

# Wall Thickness and Radial Breathing Modes of Single-Walled Carbon Nanotubes

[+] Author and Article Information
R. C. Batra

Department of Engineering Science and Mechanics, M/C 0219 Virginia Polytechnic Institute and State University, Blacksburg, VA 24061rbatra@vt.edu

S. S. Gupta

Department of Engineering Science and Mechanics, M/C 0219 Virginia Polytechnic Institute and State University, Blacksburg, VA 24061ssgupta@vt.edu

J. Appl. Mech 75(6), 061010 (Aug 20, 2008) (6 pages) doi:10.1115/1.2965370 History: Received November 15, 2007; Revised March 11, 2008; Published August 20, 2008

## Abstract

We postulate that an equivalent continuum structure (ECS) of a single-walled carbon nanotube (SWCNT) is a hollow cylinder with mean radius and length equal to that of the SWCNT, and find the thickness of the ECS so that its mechanical response in free vibrations is the same as that of the SWCNT. That is, for mechanical deformations, the ECS is energetically equivalent to the SWCNT. We use MM3 potential to study axial, torsional, radial breathing and bending vibrations of several traction free–traction free SWCNTs of different helicities and diameters and compare them with the corresponding vibrational modes and frequencies of traction free–traction free ECSs obtained by using the three-dimensional linear elasticity theory and the finite element analysis (3D-FEA). The consideration of free ends eliminates the effects of boundary conditions and avoids resolving equivalence between boundary conditions in the analyses of SWCNTs and their ECSs. It is found that the wall thickness of the ECS (and hence of a SWCNT) is $∼1 Å$ and Young’s modulus of the material of the ECS (and hence of the SWCNT) is $∼3.3 TPa$. Both quantities are independent of the helicity and the diameter of the SWCNT. We also study radial breathing mode (RBM) vibrations with the molecular dynamics and the 3D-FEA simulations, and compare them with experimental findings. Accuracy in the assignment of spectral lines for RBMs in the Raman spectroscopy is discussed.

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## Figures

Figure 1

Cylindrical tube equivalent in mechanical response to a SWCNT

Figure 2

Depictions of variables r, θ, and ϕ used in the MM3 potential

Figure 3

Vibrational modes of free-free (5,5) SWCNT: (a) relaxed tube, (b) first bending mode, (c) second bending mode, and (d) third bending mode with zoomed ends indicating the presence of out-of-plane deformations

Figure 4

Radial breathing modes of free-free (8,8) SWCNT corresponding to (a) j=1 and (b) j=2

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