Formation mechanism of toroidal carbon nanotubes

In order to study the formation mechanism of toroidal carbon nanotubes (TCNTs), we have examined their elastic energies by utilizing the continuum elastic theory. It is found that the allowed shapes of TCNTs should be determined by the constraint that the shape formation free energy should be equal to zero. We have also found that there exists a threshold condition under which a straight carbon nanotube will be curved and the TCNTs are one of the allowed shapes. The calculated torus radii for a toroidal multi-wall carbon nanotube and single-wall carbon nanotube bundle are in good agreement with the experimentally measured results. This indicates that the continuum elastic theory persists well into the small-radius tubes and the formation of TCNT is a self-organization process controlled by their shape formation free energy.