Equilibrium structure and strain energy of single-walled carbon nanotubes

The equilibrium structure and strain energy per atom of $(n,n)$ and $(n,0)$ single-walled carbon nanotubes (SWNTs) are analyzed by developing a simple force-field based atomistic model, in which the pyramidalization angle is used to characterize the inversion term energy associated with the curvature at an atom. Their closed-form expressions are obtained. Good agreements with existing numerical results based on ab initio calculations for various tube diameters from $0.5\text{to}2\mathrm{nm}$ (curvatures from $4\text{to}1{\mathrm{nm}}^{-1}$) validate the present analysis. When the curvature is less than $4{\mathrm{nm}}^{-1}$, the present results show that the strain energy is mainly due to the inversion term, the effects of helicity and bond angle variation are smaller than 10%. Also, the first term of the Taylor expansion of the strain energy expressions is dominant in the strain energy, which is $C_{\omega }{a}_0^2∕\left(8d^2\right)$ where $C_{\omega }$, $a_0$, and $d$ are the force constant associated with the pyramidalization angle, bond length in graphene sheet and tube diameter. Based on the proportional relation of the strain energy and $1∕d^2$, the bending stiffness of SWNTs is obtained by comparing with the strain energy of a corresponding cylindrical shell. However, when the curvature becomes larger, the helicity effect and the energy due to the bond angle variation will become significant. As a result, the nonlinear behavior of the tube bending may occur.

Figure 1

Schematic diagrams for the three bond lengths, three bond angles, and the common angle between the POAV ($\pi$ orbital axis vector) and the three $\sigma$ bonds around a representative atom.

Figure 2

Schematic diagrams for the bond lengths and angles and the rolling angle from graphene sheet to tube, (a) for $(n,n)$ tubes and (b) for $(n,0)$ tubes.

Figure 3

The variations of the two inequivalent bond angles within (a) $(n,n)$ and (b) $(n,0)$ tubes with the tube diameter.

Figure 4

The strain energy variations with tube diameter predicted by various methods.