Two least squares analyses of bond lengths and bond angles for the joining of carbon nanotubes to graphenes

In order to transmit signals from future nanoelectromechanical graphene sheets to other materials, connections with carbon nanotubes need to be effected. Here, we examine three particular perpendicular connections of carbon nanotubes employing two simple distinct least squares approaches and using Euler’s theorem. Firstly, for (8, 0) and (4, 4) carbon nanotubes, we apply a least squares approach to the bond lengths. Sixteen distinct defects and two possible orientations for the armchair tube (4, 4) are identified. Assuming that only pentagons, hexagons, heptagons and occasionally octagons are accepted, the number of possibilities are greatly reduced. By excluding octagonal rings, the number of possible configurations may be further reduced to only one and two most likely configurations for the zigzag (8, 0) and the armchair (4, 4) tubes, respectively. Secondly, for (6, 0) and (8, 0) carbon nanotubes, we apply a least squares approach to bond angles, and for one particular (8, 0) junction, we show that the two least squares approaches produce similar structures in terms of atom locations.These purely geometric approaches can be formally related directly to certain numerical energy minimisation methods used by a number of authors.