Toroidal molecules formed from three distinct carbon nanotubes

In order to design nanotori for nanomechanical systems, perhaps involving oscillating components, precise physical parameters for the nanotori are necessary. Toroidal shaped molecules of carbon have been investigated previously by the present authors as constructed by connecting elbow sections formed from joining armchair and zigzag nanotubes through a pentagonal–heptagonal pair defect. In this paper, we extend this design by constructing the elbow structures from three distinct carbon nanotubes. Since for a toroidal molecule, there is a constraint on the bend angles in the elbow sections to add up to 360°, particular elbow types which can accommodate this requirement are (5,0)–(4,4)–(7,0) and (3,3)–(6,0)–(4,4). We adopt a least squares approach for the bond length to minimise the variation from the ideal carbon–carbon bond length, which is taken to be σ = 1.42 Å. Moreover, formulae for the mean generating radius of the nanotori and the mean radius of the nanotubes are obtained from certain integral expressions. This purely geometrical approach can be formally directly related to certain numerical energy minimisation methods used by a number of authors.