New and Improved Analytical Solutions for the Self-Folding Problem of Carbon Nanotubes

Abstract

It has been observed experimentally and computationally that a carbon nanotube with a large aspect ratio can self-fold because of the van der Waals force between parts of the same carbon nanotube. The primary issue in the self-folding problem is to determine the minimum threshold length of a carbon nanotube at which it becomes possible for a carbon nanotube to self-fold because of the van der Waals force. In this paper, approximate mathematical models based on both energy and force methods are constructed for the self-folding problem of carbon nanotubes and they are solved exactly as an elastica problem using elliptic functions. This paper is a sequel to previous papers by the writer and a more realistic and accurate deformation of a self-folded CNT (carbon nanotube) is used in the models. The primary result of this paper is determination of the critical threshold (minimum) length of a carbon nanotube as a function of geometry, material parameters, and force field parameters for particular atomic potentials used in the models. The secondary but equally important issue is the actual self-folded shape of the CNT. Because the research reported in this paper uses a more accurate mathematical model for a self-folded CNT and obtains an exact solution, the self-folded shape is improved. As a particular example, estimates for the critical threshold (minimum) length are obtained for (5,5), (6,6), (8,8), (10,10), (15,15), and (20,20) armchair carbon nanotubes based on both methods.