Analysis of Carbon Nanotubes and Graphene Nanoribbons With Folded Racket Shapes

Abstract

Folding of carbon nanotubes and graphene nanoribbons into a shape that looks like a tennis racket is considered. An elastic continuum model is utilized in two types of analysis. The first is called an “adhesion model,” in which the adjacent sides of the racket handle are assumed to be straight and bonded together with constant or no separation. The nanotube or nanoribbon is represented as an elastica. This model has been treated in the literature, but new analytical results are derived here, involving the geometry, work of adhesion, and bending and adhesion energies. Expressions are determined for (i) the length for which the total energy is the same as for the straight unstrained equilibrium configuration and (ii) for the minimum length for existence of a stable racket equilibrium shape. The second type of analysis uses the Lennard-Jones potential to model the attractive (van der Waals) and repulsive forces between the two sides of the racket. A nanoribbon is investigated, and the derivative of the interatomic potential is integrated along the length and across the width. Numerical solutions of the integro-differential equations are obtained with a new technique utilizing the finite difference method and minimization of the squares of the resulting algebraic equations. The results are presented for two cases with different flexural rigidities. The separation between the two sides of the handle decreases in the direction of the racket head (loop), and the handle experiences internal compression under the external attractive and repulsive forces. For the adhesion model, the dimensions of the head are proportional to the square root of the flexural rigidity, and this relationship is approximately satisfied in the numerical results based on the Lennard-Jones model.