van der Waals interactions in nanotube bundles

Introduction

In the investigation of alkali metal intercalation, it is essential to study the structure and interactions inside nanotube bundles. The simplest case is to consider a triangular geometry built up by infinitely long nanotubes with the same chirality (preferably armchair for simplifying the calculations).

Girifalco et al. studied the potential energies of interaction between graphitic ma-terials [92]. They used Lennard-Jones carbon-carbon potential (LJ) and assumed con-tinuous distribution of atoms in tube surface. Since this is very much similar in na-notubes and graphene, in the following they performed the calculations considering graphene. They found that by using certain reduced parameters, the potential energies of graphene-graphene, tube-tube, C₆₀-C₆₀ and tube-C₆₀ in various arrangements can be plotted on the same curve.

In the following we restrict our considerations to pairs of nanotubes. In the contin-uum model, theφ(R) potential per unit area of interacting tubes between two identical, parallel and infinitely long tubes is:

φ(R) = n²_σ Z

u(x) dΣ1 dΣ2 (3.1)

where nσ is the mean surface density of carbon atoms, x is the distance between two surface elementsdΣ₁ anddΣ₂on the different tubes andRis the perpendicular distance between tube centers. The integrals are independent of tube radius [92].

Girifalco et al. computed the potentials for tubes with diameters between 0.54 and 3.80 nm (armchair tubes in then range from 4 to 28). As expected, the potential has a longer range, and the minimum occurs at a higher reduced distance (the distance of the tubes referred to their diameters) for smaller diameter tubes.

Sun et al. generalized Girifalco’s equation for two not identical tubes [93].

The simplified formula given by Equation 3.3 was used for calculating φ(R) by introducing the reduced parameterRe from [92]:

Re = R−ρ

R₀−ρ (3.2)

whereR₀ = 2r+3.13 [˚A] is the equilibrium distance of the tubes. In our case, definingρ as the sum of the radii,ρ= 2r[˚A], wherer is the tube radius, is a good approximation For calculating |φ(R₀)|, Equation 3.4 is a good approximation [92]:

φ0(r) = −0.1135√

r+ 9.39×10⁻³ [eV/˚A] (3.4)

Results

The potential energy as a function of tube-tube distance was calculated for the tubes listed in Table 3.1. The calculation was performed for all tube types for the mean diameter and for the lower and upper limits of diameter distribution. The min-imum required tube-tube distance was calculated for hosting one K⁺ or one Rb⁺ in

1 . 0 1 . 2 1 . 4 1 . 6 1 . 8 2 . 0 2 . 2 2 . 4 2 . 6 2 . 8 3 . 0

Figure 3.17: van der Waals potentials between two identical nanotubes with mean diameters from Table 2.1. The minimum tube-tube distance for hosting one K⁺ and one Rb⁺ between three tubes is indicated.

the triangular lattice by using simple geometric considerations. For K⁺ the minimum distance is 1.784 nm, for Rb⁺ 1.965 nm.

Figure 3.17 shows the van der Waals potential curves according to Equation 3.3 together with the use of Equation 3.4 for tubes with mean diameter from Table 2.1.

It can be seen that only mean diameter P2 tubes, placed as in Figure 3.16, require no lattice expansion for hosting one K⁺ between three tubes.

Figures 3.18, 3.19 and 3.20 show the van der Waals potential energies for each nanotube samples for their mean diameters and their upper and lower diameter limits.

Presumed states of alkali metal intercalation is shown in Figure 3.21. In the first step, alkali metal atoms condensate onto the surface of a bundle. Then they reduce the nanotubes on the surface. The so-formed alkali cations migrate into the interior of the bundle at the ends of the tubes into the triangular channels. In this model, the activation energy that must be invested is treble of the lattice expansion energy. Latter is the potential energy difference between the intercalated and initial states.

Figures 3.22 and 3.23 show the required lattice expansion and the lattice expansion energy for accomodation a cation in the triangular channel, respectively.

1 . 4 1 . 6 1 . 8 2 . 0 2 . 2 2 . 4 2 . 6 2 . 8 3 . 0 - 4

- 3 - 2 - 1

012345 K +

Potential energy (eV nm-1 )

T u b e - t u b e d i s t a n c e ( n m )

1 . 7 1 . 6 1 . 2

Figure 3.18: van der Waals potentials between two identical nanotubes for P2 tubes with diameters from Table 2.1. The minimum tube-tube distance for hosting one K⁺ between three tubes is indicated.

1 . 0 1 . 2 1 . 4 1 . 6 1 . 8 2 . 0 2 . 2

- 3 - 2 - 1

01234 K +

Potential energy (eV nm-1 )

T u b e - t u b e d i s t a n c e ( n m )

1 . 3 1 . 0 8 0 . 8

Figure 3.19: van der Waals potentials between two identical nanotubes for HiPco tubes with diameters from Table 2.1. The minimum tube-tube distance for hosting one K⁺ between three tubes is indicated.

0 . 8 1 . 0 1 . 2 1 . 4 1 . 6 1 . 8 2 . 0 2 . 2 2 . 4 - 3

- 2 - 1

01234 R b ⁺

Potential energy (eV nm-1 )

T u b e - t u b e d i s t a n c e ( n m )

1 . 1 7 0 . 9 0 0 . 5 7

Figure 3.20: van der Waals potentials between two identical nanotubes for CoMoCat tubes with diameters from Table 2.1. The minimum tube-tube distance for hosting one Rb⁺ between three tubes is indicated.

Figure 3.21: Presumed initial, intermediate and final states of alkali metal intercalation.

0 . 5 1 . 0 1 . 5 2 . 0 2 . 5

Figure 3.22: Lattice expansion values for all tube diameters from Table 2.1 to accomo-date the respective alkali cation.

Figure 3.23: Lattice expansion energies for all tube diameters from Table 2.1 for acco-modating K⁺ and for CoMoCat/Rb⁺.

From these results we can conclude that the required lattice expansion energy for K⁺ has a maximum at around 0.6 nm tube diameter. This value is a bit lower than Kukovecz et al. suggested (0.9-1.2 nm) [68].

Below a certain limit (which is the lattice parameter required to accomodate one K⁺ or one Rb⁺) the required lattice expansion is zero, above this limit it varies linearly with tube diameter. The lattice expansion energies, however, has a maximum value.

This means that K⁺ intercalation becomes more and more energetically favorable for smaller tube diameters than∼0.6 nm.

This simple model can explain the observed anomaly in diameter selectivity for all tube samples. However, in the case of CoMoCat, this model cannot explain the little degree of hydrogenation compared to HiPco. The stability and the lattice expansion energy considerations would lead to a higher degree of hydrogenation compared to HiPco. Since we observed anomalously little hydrogen content in all three parallel experiments, the explanation must be founded on another factor then energetics.

Possible explanations can be the kinetical hindrance of Rb intercalation caused by its larger size compared to K, thus, electron transfer is also only possible on the surface, which means that only a limited number of tubes are involved in the reaction. Another possible reason can be that considering an average size bundle built up by smaller diameter tubes has smaller surface, which can be coated only smaller number of the larger Rb atoms, or the change in the numerical value of energetics of the processes (heat of evaporation, heat of adsorption, ionization energy etc.). Anyhow, this anomaly requires further investigations.