Exact and approximate geometric parameters for carbon nanotubes incorporating curvature
Introduction
The scientific literature presently includes tens of thousands of articles dealing with various aspects of carbon nanotubes. In all but a few of these articles, such as Machón et al. [1], Popov [2], Samsonidze et al. [3] and Xiao et al. [4], curvature effects are ignored and such nanostructures are assumed to be constructed by rolling up a plane sheet of graphene, which in its plane state is assumed to comprise a network of perfect hexagons, in the sense that all bond lengths and all bond angles are assumed to be identical. In the cylindrical “rolled-up” state, it is generally recognised that the bond lengths and bond angles are no longer equal, but it is generally believed that such discrepancies are small, and may be ignored in regard to the calculation for the radius and length of the carbon nanotube. Here, we construct a new model for carbon nanotubes, which adopts as its basic hypothesis that the hexagons in the cylindrical “rolled-up” state should be perfect in the sense that all bond lengths and all bond angles are identical and furthermore that all atoms are equidistant from a common axis of symmetry. These three assumptions give rise to a new geometric structure for which all covalent bonds play a truly equal role, unlike conventional theory for which, strictly speaking, certain bonds play a privileged role. We show that the new exact model gives rise to all the conventional formulae as the leading terms, but in addition we may include correction terms for which simple analytical expressions are given. In general, the smaller the tube radius, the larger these correction terms become. As a byproduct of the new geometric structure a simple expression may be determined for the nanotube thickness, for which in many mechanical models becomes paramount to determine accurately. Admittedly, the new correction terms are very small, but nevertheless, given the widespread interest and applications of carbon nanotubes, knowledge of the exact dimensions could be vital in many instances. In terms of understanding carbon nanotube behaviour, conceptualising carbon nanotubes correctly as facetted polyhedral surfaces and knowledge of the detailed and exact geometric structure could be important and may explain many subtle phenomena.
In the following section we summarise the major results for the conventional “rolled-up” model. The section thereafter has two parts. The first part deals with the geometric parameters for the new “polyhedral” model and the second part gives the asymptotic expansions of these parameters, the full details of which are given in Appendix A. Section 4 contains the derivation of the new model and Section 5 summarises the numerical results of the new model and compares them with the conventional model and with a number of ab initio studies. Concluding remarks are made in the final section of the paper.
Section snippets
Conventional “rolled-up” model
Carbon nanotubes are molecules which, according to the conventional model [5], [6], [7], can be considered as a sheet of graphene which is rolled into a cylinder. The orientation of the hexagons which make up the nanotube is characterised by two integers (n, m) which prescribe the chirality of the nanotube. The direction of the chiral vector is termed the chiral angle θ0, which is defined as the angle from the base vector for the graphene plane and is given by the expression
Exact geometric parameters
In order to describe the new model, we need to introduce a further two angles which we term the subtend semi-angle ψ, and the incline angle ω, and a further three parameters λ, μ and ν, which are defined in terms of the various angles by Eqs. (26), (27), (28). The first step in determining the exact geometric structure is to determine the subtend angle 2ψ, subtended at the axis of the cylinder by the base of a single equilateral triangle. This angle is determined as a root of the following
Carbon nanotubes as facetted polyhedra
With all atoms of the carbon nanotube lying on the surface of a right circular cylinder the hexagons themselves cannot be faces of a polyhedron because necessarily, all vertices of each hexagon cannot be coplanar. To describe the nanotube as a polyhedron we must begin with the usual tessellation of regular hexagons where the vertices of the tessellation represent the point masses of the atoms and lines of the hexagons represent covalent bonds. We overlay on this a second tessellation of
Results
The magnitude of the discrepancies between the conventional “rolled-up” model and the new exact model is a function of the curvature of the nanotube. When the effect of curvature is examined, such as Sánchez-Portal et al. [10], Machón et al. [1], Cabria et al. [11] and Popov [2], nanotubes with radii in the range from 2 to 5 Å are considered and that general idea is followed here. If the radius of a nanotube is much greater than 5 Å then the effects of curvature are quite small and generally in
Conclusion
Assuming purely symmetric geometrical constraints we derive an analytical expression for the true radius of atoms (modelled as point masses) in a carbon nanotube which takes into account the curvature of the tube and maintains the position of all atoms as equidistant from a common axis, all covalent bond lengths are kept to a fixed constant and the angle between all bonds is equal throughout the molecule. The analysis shows that significant differences may exist in the various dimensions of the
Acknowledgements
The support of the Australian Research Council, both through the Discovery Project Scheme and for providing an Australian Professorial Fellowship for J.M.H. is gratefully acknowledged.
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