NANO LETTERS
Interactions in Concentric Carbon Nanotubes: The Radius vs the Chirality Angle Contributions
2006 Vol. 6, No. 9 1950-1954
Luca Bellarosa, Evangelos Bakalis, Manuel Melle-Franco, and Francesco Zerbetto* Dipartimento di Chimica “G. Ciamician”, UniVersita` di Bologna, V. F. Selmi 2, 40126 Bologna, Italy Received May 11, 2006; Revised Manuscript Received July 18, 2006
ABSTRACT In multiwall carbon nanotubes in general, and in double wall carbon nanotubes, DWCNTs, in particular, the guest−host interactions depend primarily on the difference of the nanotubes radii, ∆r. The chirality angle mismatch of the two tubes, ∆θ, also matters since it determines the pattern of π-stacking interactions that ultimately is responsible for the shift of graphite layers into the so-called A−B structure. Here we calculate the minimum energy structures of 198 DWCNTs and construct two functions of ∆r and ∆θ that fit the calculated data. Cross terms exists between ∆r and ∆θ. The shape of the functions is rationalized in simple physical terms and can be used to construct minimum energy multiwall nanotubes.
Carbon nanotubes are usually described in shorthand notation by two indices (n,m) that allow one to estimate both their radius, r ) (1.44x3/2π)(m2 + mn + n2)0.5, and their chiral angle, θ ) tan-1(mx3/(m + 2n)). Solid advance has been reached on the experimental characterization of diameters, chiral angles, and indices of individual single wall carbon nanotubes. The assignment of (n,m) indices to electronic transitions in specific types of tubes supported a quantitative analysis of absorption spectra.1 Spectrofluorimetric data in aqueous suspension were fitted to empirical expressions2 and provided a simple method to give the detailed composition of bulk samples, in both tube diameter and chiral angle.3 The nanotube chiral angle was estimated using electrostatic force microscopy.4 Furthermore, the interband transitions of distinct metallic nanotubes were observed and assigned using the Raman spectrum. The results were extrapolated to all metallic carbon nanotubes and shown to be valid also for the electronic structure of semiconducting nanotubes.5 Atomic force, electron transport, and resonant Raman spectroscopy measurements were also used to investigate 25 individual metallic and semiconducting carbon nanotubes with atomic force microscopy.6 Individual semiconducting carbon nanotubes were incorporated as the channel of field-effect transistors7 and a systematic study of the low-frequency current fluctuations of nanodevices consisting of one single semiconducting nanotube was carried out.8 The characterization of multiwall carbon nanotubes, MWCNTs, is less satisfactory. Wide-angle neutron scattering 10.1021/nl061066g CCC: $33.50 Published on Web 07/29/2006
© 2006 American Chemical Society
derived reduced radial distribution functions of the nanotubes and compared them to those determined for graphite and turbostratic carbon, providing evidence that the stacking pattern of graphene tubules in MWCNTs is intermediate between those of the other two carbon forms.9 High-resolution X-ray diffraction of high-purity MWCNTs confirmed that the MWCNTs are concentric cylinders and that there is a nonuniform distribution of inner tube diameters.10 Lacking to date is a set of rules able to evaluate the stability of several tubes grown concentrically, as in MWCNTs. Between two carbon atoms of different tubes, the van der Waals interactions are quantitatively small, but their sum over all pairs of atoms has far reaching consequences, such as in the case of graphite where the hexagonal layers do not overlap perfectly but are offset to form the so-called AB structure. The weak van der Waals interactions are a function of both ∆r and ∆θ and allow the tubes to slide and rotate with respect to each other.11-13 This is a property exploitable in nanomechanical devices14-17 that has triggered a number of studies of the geometrical and energetic parameters that characterize the relative positions of these guest-host systems.18-22 The same interactions exist in double wall carbon nanotubes, DWCNTs, which are simpler and often more easily or better characterized than MWCNTs. DWCNTs have been prepared with a variety of structures. Reported to date were radial differences, ∆r, that range from 3.223 to 4.2 Å;24 inner radii with a minimal value of 4 Å25 and a maximal value of
21 Å;26 outer radii with a minimal value of 7.5 Å27 and a maximal value of 27 Å.27 Between these pairs of values others were also found.28-33 While comparatively less attention has been devoted to the variation of chiral angle in DWCNTs, experimentally the following are known: (i) It is possible to determine the chiral indexes of each constituent carbon nanotube independently.34 (ii) Right-handed and left-handed carbon nanotubes are equally distributed for both the inner and outer nanotubes and a preferable handedness relationship between the adjacent layers in DWCNTs was not ruled out.35 (iii) It is possible to prepare DWCNTs with a single type of chirality for the inner tube.36 The chirality of DWCNTs has been linked to their charge transport properties.37 In view of their use in field effect transistors,38-40 or in nanomechanical systems,41,42 the potential energy surface was assessed as a function of the most important parameter, namely, ∆r.20 The stable structures of DWCNTs were calculated with a model similar to the present one for various pairs and the analysis found that the barrier for displacement depends on the chirality of the pair, but surprisingly, the total stability is not a function of ∆θ.20 Moreover, the barriers to relative rotation, sliding, and screwlike motions were classified in terms of interwall distance, nanotube cell length, and chirality angle difference.22 The results allowed the selection of pairs that can be used as nanobearings or bolt-and-nut pairs. Rules for constructing stable multiwall systems as a function of their ∆r and ∆θ have not appeared. Here we examine computationally 198 DWCNTs with radial differences that range from 3.26 to 3.60 Å. The unit cell of each DWCNT was built from those of the two constituent tubes, which were transformed into supercells where each tube has the same length. The tolerance threshold for the difference between the supercells of the inner and outer tube was set to 0.005 times the translational vector of the smaller cell. To describe the carbon structure, we employed the “Brenner” potential.43 The carbon-carbon van der Waals interactions were described by the potential that was calibrated to reproduce the interactions in graphite and in the benzene dimers44 and has proved useful in several application of the stability and dynamics of carbon nanotubes.45-48 This potential energy function reads
[
( ( ))
r V(r) ) v 184000 exp -12 rv
( )]
rv - 2.25 r
equation 2 parameter
equation 3
std error
1644.90 (37.67 -92.15 (2.33 2.49 (0.12 -560.68 (86.02 1917.07 (296.87 (1.66 × 10-3
a b c d f std error of estimate
parameter
std error
1665.51 (39.78 -92.89 (2.47 2.50 (0.13 17.24 (3.98 -58.56 (13.55 (1.76 × 10-3
number of parameters, i.e., 5, and gave the best fits read
(
)
(
)
V(r,θ1,θ2) )
a b c f d + 3+ + 5 sin2 θ1 sin2 θ2 + 6 4 ∆r ∆r ∆r ∆r ∆r (2)
V(r,θ1,θ2) )
d a b c f + + 3+ + 5 sin(θouter 6 4 ∆r ∆r ∆r ∆r ∆r θinner) cos(θouter - θinner) (3)
The five fitted parameters are shown in Table 1. The functions contain a binding contribution proportional to ∆r-3. This is softer than the standard Lennard-Jones binding potential energy function and comes from the ∆r-6 terms in eq 1 summed over all the pairs of atoms. Decrease in the order of the exponent upon summing over a large number of interactions has long been recognized as a general behavior.52 To first order, integration of the attractive component of function 1 over the infinitely long surfaces, s1 and s2, of the two concentric carbon nanotubes gives a quantity whose leading term scales as the ∆r-3 -2.25
∫
()
rv 6 r r dθ dθ dz dz ) r 12 1 2 1 2 6 27π r1rv I(r′,θ1,θ2)(1 + r′) (4) 32 (∆r)3
where I(r′,θ1,θ2) )
6
(1)
where r is the interatomic distance, and rv(C) ) 1.960 Å, v(C) ) 0.056 kcal mol-1. All the calculations were performed with a modified version of the TINKER molecular mechanics/dynamics software package.49-51 The two tubes were maintained concentric during the search of the minimum energy conformation. The interaction energies between the tubes were fitted to several test functions of ∆r and the chiral angles of the two tubes, θ1 and θ2. The two functions that contained the least Nano Lett., Vol. 6, No. 9, 2006
Table 1. Parameters of Equations 2 and 3 Where V Is in kcal mol-1 per Atom and ∆r Is the Difference in Radii in Å
∫02π ∫02π ((r′(cos θ1 - cos θ2) + cos θ2)2 + (r′(sin θ2 - sin θ1) + sin θ2)2)
-
5 2
dθ1 dθ2 (5)
with r′ )
r1 ∆r
Two terms in the equations of the fittings, ∆r-1 and ∆r-6, give repulsive contributions. They are also softer than the original exponential repulsion of eq 1. The longer-range term scales in the same way as the Coulomb potential; however, 1951
Figure 1. (left) Two views of the surface described by eq 2. (right) Two views of the surface described by eq 3. The blue dots are the calculated energies.
its origin is different and similar to that mentioned above for the attractive part of the potential. Derjaguin was probably the first to show that the distance dependence of the energy of interaction between two curved surfaces can be quite different from that of flat surfaces,53 even when the same forces are operating. Decrease of the exponents of the Lennard-Jones, two-atom interactions when integrated over the surfaces of parallel, but nonconcentric, tubes was also found by the analytical model developed by Girifalco and co-workers.54 Interactions that are longer-ranged, that is, have lower negative exponents, than ∆r-6 have also been found for concentric tubes by a density functional theory approach.55 The first three terms of eqs 2 and 3 form a minimum at 3.42 Å, where the terms proportional to ∆r-1 and ∆r-6 tend to balance each other. The fourth term of eqs 2 and 3 couples the chiral angles of the carbon nanotubes and the radial distance between them. Equation 2 is simpler to understand since the angular part is always positive. At 3.42 Å it turns from repulsive to attracting. Because of the positive contribution of the angular part, the mismatch of the chiral angles of the inner and outer tubes does not appear directly in eq 2, where the convolution of all the intertube van der Waals interactions leads only to an additional distance dependence as the two systems tend toward the largest zigzag structures. Although it is borne out of the calculations, we felt that this function is not satisfactory because it simply locates the radial distance 1952
where the mismatch of the hexagonal patterns of the two tubes no longer bear onto the van der Waals stabilization energy. The fourth term of eq 3, instead, changes sign in the radial component at 3.42 Å from binding to antibinding. The behavior agrees with the intuitive notion that we have of graphite, where the AB structure holds together with a shift of the planes. The set of Figure 1 shows two views of function 2 and function 3 together with the interaction energies calculated for the 198 DWCNTs. As the difference in radii increases, the interaction energy goes to zero, and for very small, physically not achievable ∆r’s can even become positive. The functions do not allow the simulation of systems with radial differences so large that the inner tube stays on one side of the inner wall of the outer tube. As far as we know, this case has not been encountered experimentally. Equations 2 and 3 can be used in a different way: (1) Select an (n,m) tube. (2) Calculate its ideal radius and chiral angle. (3) Recast the functions to obtain the potential energy of all possible DWCNTs that contain (n,m) either as an inner or as an outer tube. This is illustrated in the set of Figure 2 where as practical examples we take three of the most studied carbon nanotubes, namely, (10,0), (10,10), and (17,0). If the intertube interactions are described by the potential energy function (3), the global minima for the three carbon nanotubes, when they play hosts, are found at Nano Lett., Vol. 6, No. 9, 2006
Figure 2. Potential energy distribution according to eq 2 for (n,m) carbon nanotubes embedded (left) or embedding (right): (a) (10,0), (b) (10,10), and (c) (17,0).
(10,6)@(10,10) and (7,2)@(17,0). (Notice that (10,0) would have a global minimum at the physically impossible (1,0).) When they are the guests, the global minima are (10,0)@(17,3), (10,10)@(26,0), and (17,0)@(22,6). The present equations are easily applicable to MWCNT and can assist in their quantitative characterization. Since barriers to relative rotation, sliding, and screwlike motions in DWCNTs are a function of the interwall distance and chirality angle differences,20-22 other dynamical properties such as the enhanced Raman active in-phase breathing mode of MWCNTs56 or their bending modulus57 can have a similar dependence. Indeed, using atomically resolved scanning tunneling microscopy and spectroscopy, the nature of interwall interactions within multiwall carbon nanotubes was probed.58 It was concluded that modulations in the tunnel current are introduced by the interwall interactions and can provide information about the stacking nature. The possibility Nano Lett., Vol. 6, No. 9, 2006
of determining the most stable structure of DWCNTs can also assist in the design of new carbon nanostructures such as those recently obtained from DWNTs to form novel carbon bicables.59 Acknowledgment. Partial support from EU projects and the Italian FIRB project on “Nano and microstructured carbon” are gratefully acknowledged. Supporting Information Available: A list of DWCNTs used in this work. This material is available free of charge via the Internet at http://pubs.acs.org. References (1) Hagen, A,; Hertel, T. Nano Lett. 2003, 3, 383-388. (2) Weisman, R. B.; Bachilo, S. M. Nano Lett. 2003, 3, 1235-1238. (3) Bachilo, S. M.; Strano, M. S.; Kittrell, C.; Hauge, R. H.; Smalley, R. E.; Weisman, R. B. Science 2002, 298, 2361-2366. 1953
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NL061066G
Nano Lett., Vol. 6, No. 9, 2006
