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From The Cover
CHEMISTRY
Graphene nanostructures as tunable storage media for molecular hydrogen
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Steacie Institute for Molecular Sciences, National Research Council Canada, 100 Sussex Drive, Ottawa, ON, Canada K1A 0R6; and Institut für Physikalische Chemie und Elektrochemie, Technische Universität Dresden, D-01062 Dresden, Germany
Edited by James L. Dye, Michigan State University, East Lansing, MI and approved June 2, 2005 (received for review February 9, 2005)
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Abstract |
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equilibrium constants | hydrogen storage | quantum effects
The interaction of nonpolar H2 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 H2 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 H2 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 H2 gas in the 200- to 300-K 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).
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Methods |
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By a fortuitous coincidence, BSSE-uncorrected MP2/cc-pVTZ results for the H2–benzene model system closely mimic the counterpoise-corrected MP2/aug-cc-pVQZ calculations both at the global minimum and in the asymptotic region. Because MP2/cc-pVTZ calculations are significantly less expensive, this basis was used for the PES exploration in larger models (H2–coronene below) and for test calculations with coupled-cluster treatment of the electron correlation.
In the global PES minimum, H2 is aligned along the C6 symmetry axis of benzene at 3.1 Å between the centres of mass of H2 and C6H6. The energy profile shows a typical London-type r-6 long-range behavior (16) (Fig. 1). More sophisticated correlation treatment [CCSD(T)/cc-pVTZ] reduces the H2–benzene interaction by 0.9 kJ/mol (14) (20%) compared with MP2/cc-pVTZ, whereas the geometry remains essentially unchanged.
Interaction of H2 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 H2/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 London-type r-6 long-range potential tail and only slightly underestimates the extrapolated H2/graphene physisorption energy (16).
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, where ri is the distance between H2 centre of mass and carbon atom i. Parameters A, 
, and C6 are fitted to reproduce the MP2/cc-pVTZ results of the C6v-symmetric H2-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 H2–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 H2–PAH interaction energy to be additive and independent on the number of H2 molecules interacting with the PAH (16). For the evaluation of adsorption potential, we have chosen a two-dimensional, periodic graphite sheet with experimental C–C distance (RCC). Contributions to the potential from 30 nearest primitive unit cells are included along the in-plane a and b directions. For the double-layer structures, the potential of two layers ("above" and "below" the H2) is included. For short inter-layer separations, attraction due to the second-neighbor layer may further increase interaction energies. This effect was not considered in our simulations. The potential of the two-dimensionally 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
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where the canonical partition functions qads and qfree,
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are obtained from the energy levels 
i, determined by solving the one-particle Schrödinger equation for H2 motion in the adsorbing potential (qads) and in free space (qfree). Other thermodynamic functions (
E, S, etc.) are obtained from F, 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 H2/PAH computations are given in Tables 3, 4, and 6–8, which are published as supporting information on the PNAS web site.
The time-independent 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 qads and qfree. The rectangular unit cell is obtained by replicating the primitive unit cell (
, bp = 3RCC, RCC = 1.421 Å) in a and b directions. To avoid the attractive artifacts of the exp-6 potential close to the nuclei and reduce the requirements for high-frequency basis set components, the simulation cell extent in the c direction was truncated by dexcl (1.0 
dexcl 2.0 Å) in the vicinity of the graphene sheet(s). For double-layer structures with interlayer spacing d, the unit cell extent in the c direction is, therefore, c = d - 2dexcl. Single-layer 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 adsorption-induced changes in the internal motion (rotation and vibration) of the H2 molecules. The consequences of this approximation can be understood by using the PES of the H2–benzene complex. Because of the stiffness of the H2 molecule, only the change in the zero-point energy can contribute at temperatures of interest. At the C6v minimum of the H2–benzene PES, we calculate H2 harmonic vibrational frequency of 4,494 cm-1, 30 cm-1 below the free H2 molecule. [Indeed, the softening of the H2 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(-v/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 C6v minimum of the H2–benzene PES, the MP2/cc-pVTZ binding energy is -395 cm-1. Binding energies of -261 cm-1 are found for the corresponding C2v-symmetric structures with H2 perpendicular to any of the benzene 
v symmetry planes. Due to the large rotational level spacing of the H2 molecule (2Be 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 H2 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 H2 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 Keq = qads/qfree is converged with respect to the volume of the unit cell. From test calculations on the single-layer graphene structure, we find that unit cell a = 3ap, b = 2bp leads to Keq values converged to better than 15%. All calculations reported presently use these unit cell dimensions. Because tunneling of H2 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 H2–H2 interaction is attractive at intermolecular distance >3.1 Å (28, 29). At higher hydrogen guest densities, H2 can no longer be treated as an ideal gas. Although it is possible to simulate the high-density H2/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 H2 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 H2 nonideality at higher densities from an experimental equation of state (30) as follows: Given the external H2 pressure pext, and the equilibrium constant Keq obtained from the ideal gas simulation, we calculate the effective "internal" H2 pressure pint as
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From pint and the experimental H2 equation of state (30), we obtain the molar volume vmol. The apparent molar volume of H2, characterizing volumetric efficiency of the storage system is then given by
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For dual-layer system, the mass fraction of H2 in the H2/graphene system is given by
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where vmol is in cm3/mol and d, dexcl are in Å. The structural prefactor 
cm3/mol-Å is calculated from the fixed in-plane carbon–carbon distance RCC, the atomic masses mH and mC of hydrogen and carbon, and the Avogadro constant NA. Calculated internal pressure, volumetric and mass weight densities for representative temperatures, and external H2 pressures are collected in Table 1 and Fig. 2.
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It is important to understand the approximations made in the nonideal gas estimation of the storage capacities vre and wre. 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 pint. At the same time, using the free gas equation of state to calculate the molar volume neglects changes in H2–H2 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 vid and wid 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 H2/graphene interaction potential and the corresponding LJ fit. Based on the basis set and method convergence, we estimate this error at ±25% in vre and wre. 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.
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Results and Discussion |
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S300). The physisorption free energy corresponds to an equilibrium constant of Keq = exp(-F/RT) 1.6. In other words, at room temperature a single graphite layer increases the H2 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 H2 storage.
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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 H2/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 fine-tuning of Keq 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 H2 pressure inside the layered structure (see Table 1). As a result, the target of 62 kg/m3 (31 cm3/mol) volumetric storage density, set by the DOE, can be approached at a moderate external H2 pressure (10 MPa) even at room temperature.
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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 H2–H2 interactions becomes increasingly uncertain at lower temperatures (see Methods), the simulations indicate that gravimetric storage capacities of 5.0–6.5 wt % of H2 should be achievable at technologically acceptable conditions, e.g., at T = 250 K, Pext = 10 MPa or T = 200 K, Pext = 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.
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9 Å). At low pressures, van der Waals close-packing arguments suggest that at most two H2 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 H2 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 close-packing limit (30). At the same time, our approach to H2–H2 interactions is crude, so that the pronounced maximum in H2 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 H2 storage capacity of mono- and bilayer graphite is easily understood by comparing the density-of-states (DOS) (Fig. 9, which is published as supporting information on the PNAS web site) for the two models. The single-layer graphite potential creates relatively few H2 bound states, which are localized at the graphite surface. For the majority of states, the DOS is very similar to the DOS of free H2, reflecting the short-range nature of the binding potential. For the double-layer structure, DOS has many binding states, with the bulk of the available states shifted to low energies. In other words, whereas free H2 can move away from a single layer and hence on average experiences little attraction, H2 inside a double-layer structure is always in an attractive potential. Fig. 3 shows probability densities of H2 in between two graphite layers, for a few lowest states. Even low-energy modes are delocalized on the surface. The abundance of low-energy surface modes on graphite and in sandwiched structures suggests the presence of a two-dimensional H2 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 H2 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 H2 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 H2 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 H2 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 N2, CO, and CO2, which typically have a higher graphene binding energy than H2 and may reduce storage capacity (21). For example, we compute the binding energy of the N2/benzene model system to be two times larger than the comparable H2/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 H2 storage capacity requires a stage-1 intercalation compound, with guest molecules appearing after each graphene sheet. Stable stage-1 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 cross-linked to the graphene host, followed by elimination of the weakly bound smaller guest. Chemical cross-linking may partially disrupt the graphene 
-system, 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.
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Acknowledgements |
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Footnotes |
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Abbreviations: DOE, U.S. Department of Energy; DOS, density-of-states; LJ, Lennard–Jones; MP2, second-order Møller–Plessett; PAH, polycyclic aromatic hydrocarbon; PES, potential energy surface.
To whom correspondence should be addressed. E-mail: tse{at}ned.sims.nrc.ca.
© 2005 by The National Academy of Sciences of the USA
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