Curvature energy and circumferential deformation energy of carbon nanotubes from first principles

Carbon nanotubes (CNTs), while resembling rolled-up sheets of graphene, show subtle structural differences due to the curvature effect. When a CNT is made by rolling up a sheet of graphene and then its atomic structure is optimized by minimizing its total energy in the density functional theory (DFT), its curvature energy is decreased by the increase of its cylindrical radius while its circumferential deformation energy is induced by the deviation of its atomic structure on the cylindrical surface from that of the rolled-up graphene. We analyzed the curvature energy and the circumferential deformation energy of armchair, zigzag, and chiral CNTs of various radii in order to obtain their dependences on the cylindrical radius. We also compared the circumferential deformation energy of CNTs with the strain energy of graphene-like structures made by unrolling of fully relaxed CNTs.