Ring formation of single-walled carbon nanotubes:  Competition between conformation energy and entropy

We investigate ring formation of single-walled carbon nanotubes theoretically in this paper. The conformation energy of the nanotubes is identified as the curvature elastic energy of a curved graphite and the intertube van der Waals interaction. The entropy of ring nanotubes depends on the total number of carbon atoms. Assuming continuity, we find that the ring single-walled carbon nanotube is stable only when the radius of the tube is valued 6.958 {}. The ring diameter ranges from 200 to 700 nm.