Carbon nanotori and nanotubes encapsulating carbon atomic-chains

In this paper we investigate toroidal carbon nanotubes (carbon nanotori) encapsulating a single symmetrically located carbon atomic-chain. The interaction energy of the carbon chain is found from the Lennard-Jones potential using the continuous approach which assumes that atoms are uniformly distributed over the surface of the torus and the line of the circular chain with constant atomic surface and line densities, respectively. We assume that the chain is centrally located and that the carbon nanotorus is synthesized from a perfect carbon nanotube. We predict that the carbon chain can be encapsulated inside the carbon nanotorus when the cross-sectional radius \(r\) of the nanotorus is larger than 3.17 Å. At the minimum energy, a value of the toroidal radius \(R\) lies between 3.6 and 3.7 Å corresponding to each value of \(r\) . We also investigate the energy of carbon chains inside carbon nanotubes, which are (4,4), (5,5) and (10,0) tubes. We find that they are energetically favourable in (5,5) and (10,0) tubes, but not in a (4,4) tube, because it is geometrically too small, and these results are in agreement with existing studies. The same results for these three carbon nanotubes can also be obtained from the corresponding nanotori when \(R\) goes to infinity.