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Graphene nanostructures as tunable storage media for molecular hydrogen

Edited by James L. Dye, Michigan State University, East Lansing, MI (received for review February 9, 2005)
Abstract
Many methods have been proposed for efficient storage of molecular hydrogen for fuel cell applications. However, despite intense research efforts, the twin U.S. Department of Energy goals of 6.5% mass ratio and 62 kg/m^{3} volume density has not been achieved either experimentally or via theoretical simulations on reversible model systems. Carbonbased materials, such as carbon nanotubes, have always been regarded as the most attractive physisorption substrates for the storage of hydrogen. Theoretical studies on various model graphitic systems, however, failed to reach the elusive goal. Here, we show that insufficiently accurate carbon–H_{2} interaction potentials, together with the neglect and incomplete treatment of the quantum effects in previous theoretical investigations, led to misleading conclusions for the absorption capacity. A proper account of the contribution of quantum effects to the free energy and the equilibrium constant for hydrogen adsorption suggest that the U.S. Department of Energy specification can be approached in a graphitebased physisorption system. The theoretical prediction can be realized by optimizing the structures of nanographite platelets (graphene), which are lightweight, cheap, chemically inert, and environmentally benign.
A recent report on hydrogen clathrate hydrate (1) shows that under high pressure, molecular hydrogen can be trapped in the clathrate cavities reaching a mass ratio close to that defined by the U.S. Department of Energy (DOE) (2). However, the hydrogen clathrate is only stable under high pressure or at very low temperature. Simple sterical considerations suggest that the use of a “help gas” to stabilize the clathrate hydrate under less severe thermodynamic conditions would lead to the deterioration of the hydrogen storage mass ratio and may not be viable for mobile applications. On the other hand, there have also been numerous experimental studies on the binding capacity of molecular hydrogen with graphitic substrates (3, 4). At technologically viable conditions, reliably reproducible results are still far from the DOE goal (3, 4). In the attempt to understand and improve the storage capacity of graphitic materials, calculations have been made on many models. Some of the calculations were based on empirical interaction potentials (5–9), and the others used potentials derived from quantum mechanical calculations (10–16). The role of quantum behavior of molecular hydrogen at low temperatures has also been investigated (6, 8, 17–19). Unfortunately, the binding capacity for hydrogen at nearambient conditions has not been calculated, including the quantum effects and accurate, ab initiobased interaction potentials. To date, there has not been a reliable theoretical study indicating that the DOE goal of 6.5% mass ratio can or cannot be achieved in pure graphitic materials.
The interaction of nonpolar H_{2} molecules with physisorption substrates in graphitic system is mainly the London dispersion. Accurate calculations including treatment of electron correlations on model systems, such as polycyclic aromatic hydrocarbons (PAHs) (e.g., benzene and coronene), indicate that H_{2} molecules have adsorption energies between 4 and 7 kJ/mol (12, 14, 16) at an equilibrium distance of ≈3 Å. These binding energies are small and the entropic, and quantum effects cannot be ignored at practical temperatures (≈200–300 K). An adequate description of the motion of light H_{2} molecule in the soft, anharmonic potential of a graphitic system is a prerequisite for accurate prediction of the equilibrium constant and, therefore, the binding capacity of molecular hydrogen in these systems (20). Although the thermodynamic behavior of the free H_{2} gas in the 200 to 300K temperature range is essentially classical, this is no longer true in the presence of soft external potentials. Quantum behavior of hydrogen adsorbed in narrow pores manifests itself in the quantum sieving effects (21), which persist up to 300 K (22). Inclusion of the quantum effects in the free energy is a nontrivial computational problem. It involves solving the Schrödinger equation for the motion of the hydrogen atoms on a complicated potential energy surface (PES).
Methods
A computationally tractable model of graphene is provided by quantummechanical description of H_{2}–PAH interactions with postHartree–Fock treatment of electron correlation (14, 16). We calculate the H_{2}–benzene PES using secondorder Møller–Plessett (MP2) perturbation theory (16). Interaction energies and the PES shape in the H_{2}–benzene system are sensitive to the choice of the polarization functions and basis set superposition error (BSSE) (16). The BSSE in the interaction energy gradually decreases for larger basis sets, reaching just 0.6 kJ/mol for the augccpVQZ basis at the global minimum of the H_{2}/C_{6}H_{6} PES (see Fig. 5, which is published as supporting information on the PNAS web site). The global minimum is found 4.95 kJ/mol below the separated H_{2} and benzene. The counterpoisecorrected and uncorrected MP2/augccpVQZ equilibrium distances differ by only 0.04 Å.
By a fortuitous coincidence, BSSEuncorrected MP2/ccpVTZ results for the H_{2}–benzene model system closely mimic the counterpoisecorrected MP2/augccpVQZ calculations both at the global minimum and in the asymptotic region. Because MP2/ccpVTZ calculations are significantly less expensive, this basis was used for the PES exploration in larger models (H_{2}–coronene below) and for test calculations with coupledcluster treatment of the electron correlation.
In the global PES minimum, H_{2} is aligned along the C_{6} symmetry axis of benzene at 3.1 Å between the centres of mass of H_{2} and C_{6}H_{6}. The energy profile shows a typical Londontype ≈r ^{6} longrange behavior (16) (Fig. 1). More sophisticated correlation treatment [CCSD(T)/ccpVTZ] reduces the H_{2}–benzene interaction by 0.9 kJ/mol (14) (≈20%) compared with MP2/ccpVTZ, whereas the geometry remains essentially unchanged.
Interaction of H_{2} with larger PAHs of higher polarizabilities and smaller ionization potentials leads to stronger binding. For infinitely large graphene platelets, the interaction has been extrapolated to >7 kJ/mol at the MP2 level (16), whereas the intermolecular distance is unchanged with increase of the PAH size (12, 14, 16). The model system H_{2}/coronene considered here (Fig. 6 and Tables 3 and 4, which are published as supporting information on the PNAS web site) correctly reproduces the expected Londontype r ^{6} longrange potential tail and only slightly underestimates the extrapolated H_{2}/graphene physisorption energy (16).
Direct ab initio evaluation of the sufficiently detailed PES of graphene would have been prohibitively expensive. Instead, we describe the H_{2}/graphene PES by the exp6 form of the LJ pair potential , where r_{i} is the distance between H_{2} centre of mass and carbon atom i. Parameters A, α, and C _{6} are fitted to reproduce the MP2/ccpVTZ results of the C_{6v}symmetric H_{2}coronene model system (Table 5, which is published as supporting information on the PNAS web site). The fitted LJ potential shows good agreement with ab initio results for the H_{2}–benzene system (Fig. 1; and see Figs. 6 and 7, which are published as supporting information on the PNAS web site). The fitted potential slightly underestimates (by 0.4 kJ/mol) the translational barriers between two adjacent minima (Fig. 6). This deviation is small compared with RT and is therefore not expected to affect thermodynamic properties. It is a good approximation to assume the H_{2}–PAH interaction energy to be additive and independent on the number of H_{2} molecules interacting with the PAH (16).
For the evaluation of adsorption potential, we have chosen a twodimensional, periodic graphite sheet with experimental C–C distance (R_{CC} ). Contributions to the potential from 30 nearest primitive unit cells are included along the inplane a and b directions. For the doublelayer structures, the potential of two layers (“above” and “below” the H_{2}) is included. For short interlayer separations, attraction due to the secondneighbor layer may further increase interaction energies. This effect was not considered in our simulations. The potential of the twodimensionally periodic graphite layer is illustrated in Fig. 8, which is published as supporting information on the PNAS web site.
Free energies of adsorption were calculated in the ideal gas approximation where the canonical partition functions q _{ads} and q _{free}, are obtained from the energy levels ε_{i}, determined by solving the oneparticle Schrödinger equation for H_{2} motion in the adsorbing potential (q _{ads}) and in free space (q _{free}). Other thermodynamic functions (ΔE, ΔS, etc.) are obtained from ΔF, using standard expressions (23, 24).
All MP2 and CCSD(T) computations were performed with the nwchem (25) program package. Optimised B3LYP/6–31G^{*} geometries were taken from previous work (16). Coordinates, corresponding energies, and further details on H_{2}/PAH computations are given in Tables 3, 4, and 6–8, which are published as supporting information on the PNAS web site.
The timeindependent Schrödinger equation is solved for a single featureless particle with the mass of a hydrogen molecule. Calculations employ periodic boundary conditions and a plane wave basis set. To accelerate convergence of the free energies with the plane wave cutoff, identical unit cell dimensions and plane wave cutoffs are used for calculation of q _{ads} and q _{free}. The rectangular unit cell is obtained by replicating the primitive unit cell (, b_{p} = 3R_{CC} , R_{CC} = 1.421 Å) in a and b directions. To avoid the attractive artifacts of the exp6 potential close to the nuclei and reduce the requirements for highfrequency basis set components, the simulation cell extent in the c direction was truncated by d _{excl} (1.0 ≤ d _{excl} ≤ 2.0 Å) in the vicinity of the graphene sheet(s). For doublelayer structures with interlayer spacing d, the unit cell extent in the c direction is, therefore, c = d  2d _{excl}. Singlelayer simulations were performed by using the same periodic boundary conditions, but the interlayer spacing d was chosen to be large. The unit cell parameters, plane wave cutoffs, and calculated partition functions are given in Tables 9 and 10, which are published as supporting information on the PNAS web site.
We neglect contributions due to the adsorptioninduced changes in the internal motion (rotation and vibration) of the H_{2} molecules. The consequences of this approximation can be understood by using the PES of the H_{2}–benzene complex. Because of the stiffness of the H_{2} molecule, only the change in the zeropoint energy can contribute at temperatures of interest. At the C_{6v} minimum of the H_{2}–benzene PES, we calculate H_{2} harmonic vibrational frequency of 4,494 cm^{1}, 30 cm^{1} below the free H_{2} molecule. [Indeed, the softening of the H_{2} vibrational mode appears to be a general trend upon physisorption of this molecule (26).] At 200 K, the (neglected) increase in the adsorption constant is exp(Δv/2kT)  1 ≈ 11%, leading to a slight underestimation of the storage capacity.
The assessment of the rotational contribution to the partition function requires knowledge of the anisotropy of the adsorption potential. For the C_{6v} minimum of the H_{2}–benzene PES, the MP2/ccpVTZ binding energy is 395 cm^{1}. Binding energies of 261 cm^{1} are found for the corresponding C_{2v}symmetric structures with H_{2} perpendicular to any of the benzene σ_{v} symmetry planes. Due to the large rotational level spacing of the H_{2} molecule (2B_{e} ≈ 122 cm^{1}) and the effects of the nuclear spin statistics, the explicit treatment of the restricted rotation in this potential leads to a small (≈3%; see Estimation of the Restricted Rotor Contribution to the H_{2} Adsorption Free Energy in Supporting Text, which is published as supporting information on the PNAS web site) increase in the adsorption constants, neglected in our calculations.
Dispersive interactions between the graphene layers were not explicitly considered in our simulations. This approach is equivalent to assuming that the layered host structure remains unchanged upon H_{2} uptake. Because of the significant free energy cost associated with the graphene layer separation process (27), the “empty” layered structure must be stabilized by other means, such as spacers (see below).
Meaningful calculations of the free energy changes require that K _{eq} = q _{ads}/q _{free} is converged with respect to the volume of the unit cell. From test calculations on the singlelayer graphene structure, we find that unit cell a = 3a_{p} , b = 2b_{p} leads to K _{eq} values converged to better than 15%. All calculations reported presently use these unit cell dimensions. Because tunneling of H_{2} through the graphene layer is associated with a very high barrier, we do not need to consider unit cells replication in the c direction.
The weak H_{2}–H_{2} interaction is attractive at intermolecular distance >3.1 Å (28, 29). At higher hydrogen guest densities, H_{2} can no longer be treated as an ideal gas. Although it is possible to simulate the highdensity H_{2}/graphene system directly, using the grand canonical path integral Monte Carlo approach, such calculations are rather laborious and may be hard to interpret. At the same time, for slowly varying potentials adsorption free energy of an ideal gas is a good approximation to the adsorption free energy of the real gas at the same number density (see Estimation of the Adsorption Free Energy of Nonideal H_{2} Gas from the Results of an Ideal Gas Simulation and the Experimental Equation of State in Supporting Text and refs. 23 and 24). We estimate the corrections due to H_{2} nonideality at higher densities from an experimental equation of state (30) as follows: Given the external H_{2} pressure p _{ext}, and the equilibrium constant K _{eq} obtained from the ideal gas simulation, we calculate the effective “internal” H_{2} pressure p _{int} as
From p _{int} and the experimental H_{2} equation of state (30), we obtain the molar volume v _{mol}. The apparent molar volume of H_{2}, characterizing volumetric efficiency of the storage system is then given by
For duallayer system, the mass fraction of H_{2} in the H_{2}/graphene system is given by where v _{mol} is in cm^{3}/mol and d, d _{excl} are in Å. The structural prefactor cm^{3}/molÅ is calculated from the fixed inplane carbon–carbon distance R_{CC} , the atomic masses m_{H} and m_{C} of hydrogen and carbon, and the Avogadro constant N_{A} . Calculated internal pressure, volumetric and mass weight densities for representative temperatures, and external H_{2} pressures are collected in Table 1 and Fig. 2.
The procedure outlined above is equivalent to treating graphene structure as a “nanopump,” increasing the hydrogen pressure inside the structure, but otherwise leaving H_{2} guest gas unaffected. Although approximate, this treatment allows us to completely sidestep the (still controversial) question of the choices H_{2}–H_{2} interaction potentials, treatment of the quantum effects for H_{2}–H_{2} interactions, and the associated convergence issues.
It is important to understand the approximations made in the nonideal gas estimation of the storage capacities v _{re} and w _{re}. Eq. 3 requires the use of fugacities f, rather than gas pressures p. As the f/p ratio decreases with pressure (at temperatures and pressures considered presently), Eq. 3 is expected to underestimate the internal pressure p _{int}. At the same time, using the free gas equation of state to calculate the molar volume neglects changes in H_{2}–H_{2} radial distribution function due to the adsorption potential and, thus, overestimates the compressibility of the guest. Although these two defects may be expected to partially offset each other, the overall nonideality correction is at best semiquantitative. To illustrate the effect of the nonideality corrections, Table 1 also includes v _{id} and w _{id} values calculated from the ideal gas equation of state. When the real and ideal gas values are close, the residual error in the storage capacity is determined by the remaining uncertainties in the ab initio H_{2}/graphene interaction potential and the corresponding LJ fit. Based on the basis set and method convergence, we estimate this error at ±25% in v _{re} and w _{re}. When the real and ideal gas values deviate significantly from each other, the real gas result is clearly more reliable but should still be treated only as a semiquantitative estimate.
Results and Discussion
The spatial distribution of molecular hydrogen adsorbed on graphene is very delocalized (Fig. 3). This observation is in agreement with previous pathintegral and Monte Carlo simulations (5, 15, 18) and indicates an essentially free lateral motion of H_{2}. Our calculations indicate a slightly attractive (1.2 kJ/mol) H_{2}graphene free reaction energy at 300 K (see Table 2). The entropic contribution to the free energy is significant (ΔS _{300}). The physisorption free energy corresponds to an equilibrium constant of K _{eq} = exp(ΔF/RT) ≈ 1.6. In other words, at room temperature a single graphite layer increases the H_{2} abundance by only ≈60%. The enhancement factor does not significantly change at lower temperatures (Fig. 4) or higher pressures. Considering the volume taken up by graphene itself, graphite surfaces are unsuitable for practical H_{2} storage.
To improve the binding capacity, it is possible to sandwich H_{2} between graphite layers. Binding energies of up to 30 kJ/mol have been reported for H_{2} inside carbon nanotube tips (11). We have calculated quantummechanical physisorption free energies for interlayer distances d, between two graphite layers ranging from 4 to 14 Å. As can be seen from the results in Table 2, at separations above 7 Å the zerotemperature enthalpy of the H_{2}–graphite interaction is very similar for bi and monolayer structures (12). Even at these separations, the free energy is strongly affected by the presence and position of the second layer (Table 2 and Fig. 4). We predict an increase in H_{2} binding free energy with decreasing layer separation up to a maximum of ≈10 kJ/mol for an interlayer distance of ≈6 Å. For smaller separations, exchange repulsion reduces the free energy considerably (Fig. 4), which becomes positive for interlayer separations <5 Å. The thermodynamics of the H_{2}/graphite system at shorter separations is purely repulsive (31). The calculated equilibrium constant shows a sharp “peak” at 6–7 Å (Fig. 2 Upper). Consequently, experimental finetuning of K _{eq} should be taken with care; too small a separation could readily lead to a collapse in the adsorption free energy. At ambient conditions (T = 298 K, P = 0.1 MPa), we estimate maximum equilibrium constant of ≈30, for a graphite–graphite interlayer distance of 7 Å. The very favorable adsorption free energies for d = 6–7 Å effectively creates a “nanopump,” increasing the internal H_{2} pressure inside the layered structure (see Table 1). As a result, the target of 62 kg/m^{3} (≈31 cm^{3}/mol) volumetric storage density, set by the DOE, can be approached at a moderate external H_{2} pressure (≈10 MPa) even at room temperature.
At the same time, our simulations indicate that a pure carbonbased storage system cannot achieve the DOE gravimetric storage target (6.5 wt % H_{2}) at room temperature and moderate pressures. Due to the increasingly nonideal behavior of H_{2} at high internal pressures, roomtemperature storage capacity is limited to 2–3 wt % at 5 MPa, and 3–4 wt % at 10 MPa. Although not yet achieving the DOE target, this storage capacity is already competitive with best known physisorption substrates. We calculated similar roomtemperature gravimetric storage capacities for a wide range of interlayer spacing (d = 7–12 Å; see Fig. 2), which should simplify practical design of the storage material.
Lower but still technologically acceptable temperatures can increase the equilibrium constant by at least one order of magnitude. Fig. 4 illustrates the temperature dependence of the equilibrium constants for a bilayer with d = 8 Å. Although our treatment of H_{2}–H_{2} interactions becomes increasingly uncertain at lower temperatures (see Methods), the simulations indicate that gravimetric storage capacities of 5.0–6.5 wt % of H_{2} should be achievable at technologically acceptable conditions, e.g., at T = 250 K, P _{ext} = 10 MPa or T = 200 K, P _{ext} = 5 MPa (Table 1 and Fig. 2). Because these storage capacities are only weakly sensitive to the graphite interlayer separation, the design of the necessary layered graphite structure does not appear to be an insurmountable challenge.
Caution should be applied in interpreting the results of our simulations for small interlayer distances (d ≤ 9 Å). At low pressures, van der Waals closepacking arguments suggest that at most two H_{2} monolayers (6.6 wt %) can fit between the graphene layers separated by d = 9 Å. For d = 6 Å, just one monolayer can fit in the structure, indicating a maximum H_{2} storage capacity of 3.3 wt % (3, 4). Although this argument holds at low pressures, the large adsorption free energy calculated for small interlayer spacing can drive the internal pressure beyond 2 kbar (1 bar = 100 kPa), even at moderate external pressures (Table 1). In the free hydrogen gas, the density at such pressures exceeds the closepacking limit (30). At the same time, our approach to H_{2}–H_{2} interactions is crude, so that the pronounced maximum in H_{2} storage capacity we find at interlayer spacing 6 Å ≤ d ≤ 7.5 Å should be considered only as a tantalizing possibility. Confirming or disproving its existence would require further, more elaborate simulations (or experiment).
The qualitative difference in the H_{2} storage capacity of mono and bilayer graphite is easily understood by comparing the densityofstates (DOS) (Fig. 9, which is published as supporting information on the PNAS web site) for the two models. The singlelayer graphite potential creates relatively few H_{2} bound states, which are localized at the graphite surface. For the majority of states, the DOS is very similar to the DOS of free H_{2}, reflecting the shortrange nature of the binding potential. For the doublelayer structure, DOS has many binding states, with the bulk of the available states shifted to low energies. In other words, whereas free H_{2} can move away from a single layer and hence on average experiences little attraction, H_{2} inside a doublelayer structure is always in an attractive potential. Fig. 3 shows probability densities of H_{2} in between two graphite layers, for a few lowest states. Even lowenergy modes are delocalized on the surface. The abundance of lowenergy surface modes on graphite and in sandwiched structures suggests the presence of a twodimensional H_{2} gas. The lateral modes should facilitate diffusion inside the storage material, thus ensuring easy loading and unloading.
The estimated limit of accuracy of our calculations at d ≥ 8 Å is within ±25% (see Methods). Even considering this large margin of error, we conclude that an H_{2} storage enhancement material approaching, or possibly even exceeding, the DOE specification can be produced by encapsulating molecular hydrogen in graphite layers with an appropriate interlayer spacing. The significant dependence of the equilibrium constant on the graphite interlayer distance indicates that the H_{2} abundance of a nanostructured graphite system is intimately controlled by the nano and mesoscopic structure. Together with the pronounced temperature dependency, the controversial issue of the qualitatively different amounts of H_{2} adsorption in the literature of the last decade (5, 10, 32–34) on graphite systems may well arise from slight microstructural variations both within and between different samples.
It is hence an experimental challenge to provide a synthesis of nanostructured graphite with sufficiently uniform and reproducible interlayer distances suitable for H_{2} storage. One possibility is to introduce well defined spacers. Aside from the tuning possibilities, the spacers may provide additional benefits, such as an increase in the stability of the storage media. Spacers can also act as molecular sieves, preventing penetration of larger gas molecules such as N_{2}, CO, and CO_{2}, which typically have a higher graphene binding energy than H_{2} and may reduce storage capacity (21). For example, we compute the binding energy of the N_{2}/benzene model system to be two times larger than the comparable H_{2}/benzene model.
One possibility for the experimental realization of the layered graphene storage system is provided by graphite intercalation compounds, which exhibit a wide range of interlayer separations (35). Achieving the optimal H_{2} storage capacity requires a stage1 intercalation compound, with guest molecules appearing after each graphene sheet. Stable stage1 species with interlayer separations from 3.7 to 12 Å are known (35), offering almost unlimited possibilities for chemical tuning. One of the experimental challenges in synthesizing our storage system lies in minimizing the interlayer volume, excluded by the spacers. This goal can be achieved by preparing a mixed intercalation compound with two guests of a different size. The larger guest can be chemically or photochemically crosslinked to the graphene host, followed by elimination of the weakly bound smaller guest. Chemical crosslinking may partially disrupt the graphene πsystem, somewhat decreasing the van der Waals interactions and the adsorption free energy. At the same time, sufficiently polar spacers can increase electrostatic interactions enough to more than compensate for this loss (15).
Another prototype for the storage system is carbon foam (36) with the structure based on rigidly interconnected segments of graphite. In principle, the interlayer separation can be adjusted by extending the interlayer carbon skeleton. The resulting “foam” structures cover the structural phase space extending from hexagonal diamond to graphite. A previous theoretical study (36) has shown that this hybrid system possesses unusually high structural stability and low mass density.
Acknowledgments
S.P. thanks Dr. D. D. Klug for helpful discussions. This work was supported by the Deutsche Forschungsgemeinschaft, the Stiftung Energieforschung BadenWürttemberg, and the National Sciences and Engineering Research Council of Canada.
Footnotes

↵ ‡ To whom correspondence should be addressed. Email: tse{at}ned.sims.nrc.ca.

This paper was submitted directly (Track II) to the PNAS office.

Abbreviations: DOE, U.S. Department of Energy; DOS, densityofstates; LJ, Lennard–Jones; MP2, secondorder Møller–Plessett; PAH, polycyclic aromatic hydrocarbon; PES, potential energy surface.
 Copyright © 2005, The National Academy of Sciences
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