On the structural rules of helically coiled carbon nanotubes

We discuss various possibilities for the geometries of helically coiled carbon nanotubes (HCCNTs), especially those derived from toroidal carbon nanotubes (TCNTs) containing nonhexagonal defects. Two schemes for the derivation of HCCNTs from a particular TCNT are studied, namely horizontal and vertical shifting. For both cases there is a parameter characterizing the skewness of the corresponding HCCNTs. In particular, we found that when the horizontal shift parameter (HSP) is larger than a certain value, depending on the parent TCNT chosen, the resulting HCCNT possesses a peculiar geometry that has not been reported, in which the molecular shape cannot be described by a tube rolling on a cylinder. The pitch angle as a function of HSP can be nicely approximated by simple trigonometry functions of HCCNT parameters. In the case of vertical shifting, the dependence of pitch angle on the shift parameter is relatively simpler and more intuitive.