Carbon nanotubes, buckyballs, ropes, and a universal graphitic potential

The potential energies of interaction between two parallel, infinitely long carbon nanotubes of the same diameter, and between ${\mathrm{C}}_{60}$ and a nanotube in various arrangements, were computed by assuming a continuous distribution of atoms on the tube and ball surfaces and using a Lennard-Jones (LJ) carbon-carbon potential. The constants in the LJ potential are different for graphene-graphene and ${\mathrm{C}}_{60}\ensuremath{-}{\mathrm{C}}_{60}$ interactions. From these, the constants for tube-${\mathrm{C}}_{60}$ interactions were estimated using averaging rules from the theory of dispersion forces. For tubes in ropes, the cohesive energy per unit length, the compressibility, and the equilibrium separation distance were computed as a function of tube radius. For a ${\mathrm{C}}_{60}$ molecule interacting with tubes, the binding energy inside a tube was much higher than on a tube or at the tube mouth. Within a tube, the binding energy was highest at a spherically capped end. The potential energies for tubes of all radii, as well as for interactions between ${\mathrm{C}}_{60}$ molecules, for a ${\mathrm{C}}_{60}$ molecule outside of a nanotube, between a ${\mathrm{C}}_{60}$ molecule and a graphene sheet, and between graphene sheets, all fell on the same curve when plotted in terms of certain reduced parameters. Because of this, all the potentials can be represented by a simple analytic form, thereby greatly simplifying all computations of van der Waals interactions in graphitic systems. Binding-energy results were all consistent with the recently proposed mechanism of peapod formation based on transmission electron microscopy experiments.