Charge–Metal Interaction of a Carbon Nanotube

Abstract

While the electronic structure of metallic single-walled carbon nanotubes, SWCNTs, is well-understood, only a few, perhaps too rare, steps have been taken to investigate the metallic contribution to their interactions with charged species. 6] This is not to say that interactions between SWCNTs and molecules have not been studied, but rather that the additional metallic contribution to the interaction with a (partial) charge has often been left unquantified or even neglected. And yet, this is not a trivial issue because prior to many of the practical applications advocated for SWCNTs there is the separation of metallic from semiconducting tubes that could be based on the larger adsorption energy of charged molecules when bound to metallic nanotubes. It was, for instance, noticed that for DNA– nanotube hybrids, the negative charge of DNA induces a positive charge on the tube so that the overall charge is smaller to that of pure DNA and, although not quantified, this effect was used to separate metallic from semiconducting nanotubes. Moreover, if the additional metallic contribution is neglected, results of interactions/simulations with tubes of similar radius, such as (10,10) and (17,0), become equivalent, although the former is metallic and the latter semiconducting. Quantum chemical calculations can differentiate between the interaction of a metallic, or a semiconducting, nanotube with a charged species in a way analogous to that of Lu and co-workers who used first-principles calculations to show that the larger electronic polarizability of metallic SWCNTs makes them interact more strongly with adsorbates via p interactions. Quantum chemical calculations, however, routinely handle only a limited number of atoms and may have difficulties with weakly binding van der Waals interactions. If extensive systems of thousands of atoms must be considered and non-bonding interactions play a major role, classical potentials must be applied and they usually do not include this metallic contribution. Quantum chemical calculations, however, routinely handle only a limited number of atoms and if proper periodic boundary conditions are not implemented, that is, a cluster approach is employed, the metallic nature of the SWCNT may disappear. Here we propose an approach based on classical electrostatics that describes the interaction between a charged species and a metallic SWCNT treated as hollow cylinder with a dielectric constant that tends to infinity. It starts from the consideration that for a charge, q, located at cylindrical coordinates ð10; 0; z0Þ (with 1 as radial coordinate from the tube axis, the azimuthal angle and z the cylinder axis) the potential energy is equal to, Uð10; 0; z0Þ 1⁄4 12 qFð10; 0; z0Þ, where F is the electrostatic potential generated by the charges induced on the conductor. F can be obtained through the Green expansion method (see Appendix), the factor 1/2 eliminates double counting of the potential energy. The results of the mathematical treatment are fit to two simple three-dimensional functions of 1) the radius of the tube and 2) the distance of the charge from either its inner or outer surfaces that are readily amenable to use in the treatment of experimental data or in molecular modeling. Arista et al. used a similar approach to investigate the interactions between a metal cylinder and a moving charge, and their results were used by Granger et al. to support the idea of the existence of image states around nanotubes. For a charge outside/inside a hollow cylinder with internal/external radius a1⁄4r x=2 and b1⁄4rþx=2 where r is the tube radius and x is its thickness, the electrostatic potential induced on the position of the charge, can written in terms of a modified Bessel functions Im(x) and Km(x) [12] , see the Appendix for the derivation of Equations (1a) and (1b):