Approximate Solutions for a Self-Folding Problem of Carbon Nanotubes

This paper treats approximate solutions for a self-folding problem of carbon nanotubes. It has been observed in the molecular dynamics calculations (Buehler, Kong, Gao, and Huang, 2006, “Self-Folding and Unfolding of Carbon Nanotubes,” ASME J. Eng. Mater. Technol., 128, pp. 3–10) that a carbon nanotube with a large aspect ratio can self-fold due to the van der Waals force between the parts of the same carbon nanotube. The main issue in the self-folding problem is to determine the minimum threshold length of the carbon nanotube at which it becomes possible for the carbon nanotube to self-fold due to the van der Waals force. To the best of the author’s knowledge, no exact solution for this problem has been obtained. In this paper, an approximate mathematical model based on the force method is constructed for the self-folding problem of carbon nanotubes, and is solved exactly as an elastica problem using elliptic functions. Additionally, three other mathematical models are constructed based on the energy method. As a particular example, the lower and upper estimates for the critical threshold (minimum) length are determined based on both methods for the (5,5) armchair carbon nanotube.