Density-functional study of curvature energy of individual carbon nanotube: inverse near-quartic radius dependence

While carbon nanotubes (CNTs) have a well defined radius that is dictated by the chiral index (n,m), CNT structure defined this way will have deviations in its C-C bond length as a sheet of graphene is mapped onto a ideal cylinder. Assuming the deviations in C-C bond length to effect curvature energy of CNTs, we have calculated curvature energies of CNTs with given chiral index while varying its radius using density functional theory (DFT). Calculating energies of CNT with various radii and corresponding strained graphene we analyzed the curvature energy of armchair, zigzag and chiral CNTs. We observed that while with varying chiral index the curvature energy is proportional to the inverse of radius squared, when chiral index is given and radius is changed, curvature energy shows an approximate quartic inverse dependence on the radius.