a Lennard-Jones potential [96-98]. b Feynman-Hibbs perturbative approach [103]. c Silvera-Goldman potential [153], which includes three-body terms. d Crowell-Brown potential [154]; accounts for polarizability of carbon atoms [100]. e Path-integral method of Feynman [155]. f Diep-Johnson anisotropic force field [156].

Note: Primary sources for interaction potentials: W. A. Steele, "The Interaction of Gases with solid Surface." Pergamon, Oxford, U.K., 1974 (used by Yin et al. [151]); T. Kihara, "Intermolecular Forces." Wiley, New York, 1978 (used by Yin et al. [151]).

a Lennard-Jones potential [96-98]. b Feynman-Hibbs perturbative approach [103]. c Silvera-Goldman potential [153], which includes three-body terms. d Crowell-Brown potential [154]; accounts for polarizability of carbon atoms [100]. e Path-integral method of Feynman [155]. f Diep-Johnson anisotropic force field [156].

Note: Primary sources for interaction potentials: W. A. Steele, "The Interaction of Gases with solid Surface." Pergamon, Oxford, U.K., 1974 (used by Yin et al. [151]); T. Kihara, "Intermolecular Forces." Wiley, New York, 1978 (used by Yin et al. [151]).

is the ONIOM [113] method that allows simple molecular mechanics methods to be combined with more accurate quantum mechanics. These new methods are allowing ever more accurate studies to be performed on carbon nanotube systems.

3.3. Binding Sites for Molecular Hydrogen Physisorption on Graphitic Materials

Molecular hydrogen physisorption can occur at a number of sites on graphitic systems. These include on top of atoms, bonds, and the center of the carbon hexagons. One density functional theory study [114] used a local density approximation (LDA), and they found that the most stable configuration for hydrogen molecules was above the center of a hexagon, although the preference for this position was small, indicating that the barriers for classical diffusion would be small. Dubot and Cenedese [115] used semiempirical AM1 simulations to model Li-doped SWNTs, and found that Li adsorbed inside and outside the tube, with the most stable site also being the center of a hexagon. More recently, both static and dynamic calculations were conducted by Arellano et al. [116] to investigate the interaction of molecular and atomic hydrogen with single-wall carbon nanotubes. An ab initio FGI96MD code [117] was selected, in which the DFT was used to compute the electron density and the total energy of the system, and LDA to account for exchange and correlation effects. Molecular physisorption inside and outside the nanotube walls was predicted to be the more stable state of those systems, with the binding energies for physisorption of the hydrogen molecule outside the nano-tube in the range 4-7 kJ • mol-1. From their GCMC study of nanotube bundles, Williams and Eklund [118] showed that the geometry of the site affected the average adsorption potential experienced by the hydrogen molecule. They found that the three endohedral sites they studied had a larger potential than sites inside the nanotube. The largest potential was for endohedral sites in the internal channels (11 kJ • mol-1), followed by the grooves (9 kJ • mol-1) and surfaces (6 kJ • mol-1) on the outer surface of the bundle, with the sites inside the tube being only (5 kJ • mol-1). This means that physical adsorption and desorption of molecular hydrogen from nanotubes are low-energy processes, as many experimental studies have shown. A recent study of the interaction of molecular hydrogen with planar graphite clusters by Okamoto and Miyamoto [119] confirmed that the LDA interaction energy curve was in very good agreement with the results of ab initio second-order M0ller-Plesset (MP2) calculations.

3.4. Hydrogen Density Profile

Lee et al. [29, 120] used SCC-DFTB calculations, and found that hydrogen molecules existed in a number of concentric layers inside the nanotubes. At room temperature, statistical mechanics studies that reported the density profile about the carbon nanotube exhibited a single dense monolayer of hydrogen molecules close to the nanotube wall [100, 101, 106, 121, 122]. This monolayer was present both inside and outside the tube. After this monolayer the density rapidly reduced to the equivalent of a bulk phase at the same compression, indicating that the thermal vibration energy overcomes the ordering enthalpy for the second and subsequent monolayers. Statistical mechanics studies show that at 77 K, the lower thermal vibration energy of the hydrogen molecules did allow a second [99] dense monolayer shell to form, with a density intermediate between the first mono-layer and the bulk phase.

Adding charge to the carbon nanotube increased the interaction with hydrogen molecules, and also induced a second monolayer shell at room temperature [106]. The authors of this study suggest that such a charged system could be created by either metal intercalation or by electrostatic charging. Dubot and Cenedese [115] used semiempirical

AM1 simulations to study molecular hydrogen adsorption in Li-doped SWNTs. The molecular hydrogen bound to the Li atom with a binding energy of 43 kJ • mol-1, and was repelled from the tube wall if the tube was not doped. Froudakis [123] used the ONIOM method to investigate the nature of the hydrogen adsorption in K-doped SWNTs, and to compare it with the adsorption in pure SWNTs. The calculations showed that the charge transfer from the alkali metal to the tube polarizes the hydrogen molecule, and this charge-induced dipole interaction is responsible for higher hydrogen uptake in the doped tubes. This may explain why the hydrogen uptake is improved by the doping of catalysts used in making the carbon nanotube.

All of these monolayers appear to have an average thickness of -3.5 A. This agrees nicely with the SCC-DFTB calculations of Tada et al. [124], who found that there was an energy threshold to entering the nanotube that was dependent on the tube diameter, with the barrier reducing to ambient energy levels only when the diameter was above 6 A. The density of the external monolayer is higher than for the same monolayer on an ideal graphite sheet, and the density of the internal monolayer is higher again [121].

3.4.1. Implications of the Density Profile

The increased adsorption of ideal carbon nanotubes appears to be based on the presence of the dense monolayer. These monolayers are dependent on the surface area, which scales with the number of carbons. Alternately, the bulk phase is dependent on the interstitial spaces and nanotube internal volume. These areas are dependent on the square of the number of atoms. This means that a graph of either nano-tube size or nanotube packing separation against system density over bulk density will have a maximum. Therefore, when compared with a simple nonadsorbent system, the carbon nanotubes will have configurations of maximum benefit. For systems for which a single monolayer is predicted, this maximum occurs when one monolayer is adsorbed on the nanotube surfaces (radius -1.2 nm, tube separation -0.7 nm) [101]. No one has quantified the maximum for charged or low-temperature systems where two monolay-ers may exist, but intuitively, it would be expected to be at an additional multiple of 3.5 A, a radius of -1.55 or -1.9 nm and a separation of -1.05 or -1.4 nm. Countering any "ideal" configuration is the random distribution of diameter and nanotube arrangement in experimental systems. A recent theoretical study [125] on randomly generated bundles predicted adsorption isotherms that appeared insensitive to the details of the geometry.

3.5. Isotope Separation

Using Carbon Nanotubes

The separation of atomic isotopes is an energy-intensive and time-consuming process. A recent paper by Challa et al. [126] modeled that separation of the isotopes of hydrogen using a GCMC method including quantum treatment via a path-integral method. They calculated selectivities of 10,000 for T2/H2 and 1000 for D2/H2 at low temperature and pressure (20 K:10-4 Pa), which reduced to, respectively, 6 and 5 at 77 K and 1300 Pa. These results indicate that carbon nanotubes have great potential for use in isotope separation.

3.6. Nanotube Properties

The chemical reactivity of nanotubes has been analyzed by comparing unstrained and strained systems. One group [127] used first principle calculations to look at atomic hydrogen adsorption on a tube that had been elastically squashed to form an elliptic tube. They found a large (-100 kJ • mol-1) difference in the binding energy between the regions of high and low curvature in the ellipse. Another group [128] analyzed, using an empirical method, the reactivity at areas of high curvature caused by twisting or bending the nan-otubes. They also found that sites of higher curvature have higher binding energies (-155 kJ • mol-1) for atomic hydrogen. They also presented some preliminary experimental evidence that nanotubes can preferentially react at strained sites. The finding that the reactivity of carbon nan-otubes is modifiable by physical distortion of the tube structure makes carbon nanotubes quite different from planar graphitic materials where increased reactivity occurs only at defect sites.

3.7. Chemical Adsorption on Carbon Nanotubes

No chemical adsorption was observed in studies using statistical mechanic methods on systems with molecular hydrogen at ambient energy levels. Only the studies with atomic hydrogen discussed below had reactions leading to chemisorption of the hydrogen. None of those systems modeled carbon nanotubes at conditions proposed for hydrogen storage nor proposed a mechanism for release of the chemisorbed hydrogen. However, it is well known that certain metal catalysts promote the formation of atomic hydrogen, and some of these metals are being combined with carbon nanotubes.

Using density functional theory, Jeloaica and Sidis [129] calculated the interaction between atomic hydrogen and a graphite surface. A coronene-like model of the (0001) graphite surface is considered. Two adsorption regions separated by a barrier were found: a physisorption region around 3 A from the surface, and a chemisorption region around 1.5 A. The former is site independent, and compatible with a high mobility of the hydrogen atoms parallel to the surface. The latter is located exclusively on top of a carbon atom, which was consistent again with earlier studies using the semiempirical method [130-133], and more details can be found from our recent review [134]. Yang and Yang [135] focused on the adsorption of H atoms on three faces of graphite: (0001) basal plane, (1010) zigzag edge, and (1121) armchair edge. The relative energies of adsorption (or CH bond energies) follow the order zigzag edge > armchair edge > basal-plane edge. They also found that adsorption on the basal plane sites is exothermic and stable. On the edge sites, the C-H bond energy decreases by nearly 125 kJ • mol-1 when two H atoms are adsorbed on the same site. On the basal plane, the C-H bond energy decreases from 190 kJ • mol-1 when two H are adsorbed on alternating sites to 110 kJ • mol-1 when they are adsorbed on two adjacent sites.

Lee et al. [29, 120] used an SCC-DFTB method, and identified two chemisorption sites for atomic hydrogen on SWNTs: similarly to graphite, the carbon atom top sites at the exterior and the interior of the tube wall. Gulseren and co-workers [127, 136] and Tada et al. [124] found that the binding energy depends sensitively on the curvature of the nanotubes, either zigzag or armchair, and was proportional to the inverse of the radius. They found that the sp3 rehy-bridization of a single carbon atom from sp2 was always exothermic, provided the new bonding orbital was exohe-dral, and predicted that no chemisorption could occur inside the nanotube. However, Tada et al. reported repulsive interactions between hydrogen and a planar graphite layer that are not in agreement with the results of Arellano et al. [114]; the discrepancy between the results of the two calculations may be due to the different treatments (GGA by Tada et al. and LDA by Arellano et al.), with GGA known to overestimate repulsive interactions. Gulseren and co-workers [127, 136] also found that the CnHn nanotubes formed were direct band insulators with a gap of 1.5-2.0 eV at the G point of the electronic band structure, in contrast to pure nanotubes which are metallic or semiconductors.

Bauschlicher [137] used a (10,0) carbon nanotube for studying the hydrogen and fluorine binding using the ONIOM method (B3LYP/4-31G:UFF). The addition of two or four hydrogen atoms (from molecular hydrogen) to a (10,0) tube is computed to be endothermic. As a comparison, fluorine atoms appear to favor bonding next to existing fluorine atoms, and the adsorption is quite exothermic. The reason may be that the H2 bonding energy (460 kJ • mol-1 at B3LYP level) is larger than F2 bonding energy (140 kJ • mol-1). In another study, Bauschlicher [138] examined the maximum coverage of the tube wall, finding that the average C-H bond energy for the more stable 50% coverage is 240 kJ • mol-1, which is higher than that of 100% coverage (160 kJ • mol-1). This indicates that it is very difficult to achieve 100% coverage on a (10,0) tube. The favorable 50% coverage corresponds to about 4% by weight storage of hydrogen.

Froudakis [139] applied the ONIOM approach to a 200-atom (4,4) SWNT, treating up to 64 carbons and 32 hydrogens with the higher level of theory. The small diameter of the tube together with the large number of atoms considered allow the higher level model to include a cylindrical part of the tube. The calculations showed that hydrogen atoms bond to the tube walls, and do not enter the tube interior. This binding takes places in zigzag rings around the tube walls, and not in lines toward the tube axis, changing the tube shape and causing an enlargement of the tube volume by 15%. After the tube walls are half filled with hydrogens, the energetically more favorable procedure of hydrogen insertion in the tube is obtained.

These results for hydrogen chemisorption on the exterior wall of SWNT are in fair agreement with that on the basal plane of graphite. The values of 110 kJ • mol-1 from Yang and Yang [135] and 160-240 kJ • mol-1 from Bauschlicher [138] agree well with 96 kJ • mol-1 from experimental desorption of hydrogen from MWNT. However, more studies, both experimental and theoretical, need to be made to eliminate the remaining differences. Furthermore, these large energy barriers somewhat preclude hydrogen chemisorption as a useful storage method at ambient temperatures.

3.8. Hydrogen Storage Inside Closed SWNT

In another series of studies [140, 141], hydrogen atoms were used to bombard the wall of a model nanotube in a molecular dynamics study. They found that atoms with kinetic energy in the range 90-360 kJ • mol-1 were chemically adsorbed on the surface. Atoms with energy higher than this but lower then 1.3 MJ • mol-1 bounced off the tube. Atoms with energy higher than 1.3 MJ • mol-1 began to pass through the tube wall; some did this without damaging the wall, some would remain in the tube while others passed out the other side, and some entered and damaged the wall. The study also showed that any damage was "healed" within a short time (1-3 ps). The studies proposed that carbon nanotubes could be used as molecular-sized containers for hydrogen, or deuterium and tritium, particularly where very high densities are required.

Lee et al. [142] addressed the issue of a possible mechanism for the insertion of hydrogen molecules into the nanotubes using an SCC-DFTB method. According to their hypothesis, a hydrogen atom bonded to a carbon atom in an arch-type geometry could push the carbon atoms down, and then flip into the C-C midbond. The C-C bond recovers after the H atom flips into the internal space. Once the first atoms has flipped in, the nearest neighbor top site atoms can flip more easily because of the lower activation barrier, leading to a continuous flip-in process. Then the zigzag flip-in process, that is, the continuous flip-in process in the second nearest neighbor top site, ultimately results in the formation of zigzag geometry. A kick-in mechanism was proposed to explain hydrogen insertion in the nanotubes. Repeating the kick-in process eventually leads to the formation of molecular hydrogen inside the tubes. The storage mechanism is completed by a similar hydrogen extraction mechanism with a low energy barrier. Similarly to Gulseren and co-workers [127, 136] mentioned earlier, the calculations revealed that C-H bond formation changes the electronic structures of metallic carbon nanotubes to semiconductors during the electronic chemical storage process. This suggests that doping or adding conductive materials to the nano-tube electrode was required in the electrochemical storage experiments.

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