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 0.50.3em0exto0.3em0ex20.3em0exnm (curvatures from 40.3em0exto0.3em0ex10.3em0exnm^-1) validate the present analysis. When the curvature is less than 40.3em0exnm^-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_ωa₀²∕(8d²) where C_ω, a₀, 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∕d², 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.