Enthalpy and Entropy Effects in Hydrogen Adsorption on Carbon

The Journal of Physical Chemistry B 2006 110 (19), 9371-9374 ... A 2014 2 (31), 12123-12135 ... The effect of temperature and topological defects on H...
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Enthalpy and Entropy Effects in Hydrogen Adsorption on Carbon Nanotubes Irena Efremenko* and Moshe Sheintuch Department of Chemical Engineering, Technion, Haifa 32000, Israel Received December 30, 2004. In Final Form: April 28, 2005 Interaction energies and entropies associated with hydrogen adsorption on the inner and outer surfaces of zigzag single-wall carbon nanotubes (SWCNT) of various diameters are analyzed by means of molecular mechanics, density functional theory, and ab initio calculations. For a single molecule the strongest interaction, which is 3.5 greater than that with the planar graphite sheet, is found inside a (8,0) nanotube. Adsorption on the outer surfaces is weaker than that on graphite. Due to the steric considerations, both processes are accompanied by an extremely strong decline in entropy. Absence of specific adsorption sites and weak attractive interaction between hydrogen molecules within carbon nanotubes results in their close packing at low temperatures. Using the calculated geometric and thermodynamic parameters in Langmuir isotherms we predict the adsorption capacity of SWCNTs at room temperature to be smaller than 1 wt % even at 100 bar.

1. Introduction Single-wall carbon nanotubes (SWCNTs)1 show, among other unique properties, a high capacity to bind molecules and ions.2-4 That makes them attractive as a potential hydrogen storage medium.5-10 Great efforts, including experimental studies and theoretical calculations, have been undertaken to investigate hydrogen storage in SWCNTs. For an economically viable vehicular hydrogenstorage medium, the adsorption of hydrogen at ambient temperature should attain at least 6.5 wt % (specified by the U.S. Department of Energy) and should be able to release its load at temperatures below 400 K. That translates to a desired hydrogen binding energy of 15-20 kJ/mol.11,12 To date, however, no conclusive results regarding the viability of SWCNTs as hydrogen storage media have been reached. Some groups reported storage capacities in the range of 5-20 wt %.2,13-16 H2 absorption, in these studies, was shown to be caused by dispersion (physisorption) forces with an adsorption heat of ∼19-20 kJ/mol,2,13 a value that greatly exceeds that on activated * Corresponding author. E-mail: [email protected]. (1) Ijima, S. Nature 1991, 354, 56. (2) Dillon, A. C.; Jones, K. M.; Bekkedahl, T. A.; Kiang, C. H.; Bethune, D. S.; Heben, M. J. Nature 1997, 386, 377. (3) Ajayan, P. M.; Iijima, S. Nature 1993, 361, 333. (4) Fan, X.; Dickey, E. C.; Eklund, P. C.; Williams, K. A.; Grigorian, L.; Buczko, R.; Pantelides, S. T.; Pennycook, S. J. Phys. Rev. Lett. 2000, 84, 4621. (5) Darkrim, F. L.; Malbrunot, P.; Tartaglia, G. P. Int. J. Hydrogen Energy 2001, 27, 193. (6) Cheng, H.-M.; Yang, Q.-H.; Liu, C. Carbon 2001, 39, 1447. (7) Ding, R. G.; Finnerty, J. J.; Zhu, Z. H.; Yan, Z. F.; Lu, G. Q. Encycl. Nanosci. Nanotechnol. 2004, 4, 13. (8) Ding, R. G.; Lu, G. Q.; Yan, Z. F.; Wilson, M. A. J. Nanosci. Nanotechnol. 2001, 1, 7. (9) Atkinson, K.; Roth, S.; Hirscher M.; Gru¨nwald, W. Fuel Cells Bull. 2001, 4, 9. (10) Dillon, A. C.; Heben, M. J. Appl. Phys. A 2001, 72, 133. (11) Li, J.; Furuta, T.; Goto, H.; Ohashi, T.; Fujiwara, Y.; Yip, S. J. Chem. Phys. 2003, 119, 2376. (12) Wang, Q.; Johnson, J. K. J. Phys. Chem. B 1999, 103, 4809. (13) Sudan, P.; Zuttel, A.; Mauron, Ph.; Emmenegger, Ch.; Wenger, P.; Schlapbach, L. Carbon 2003, 41, 2377. (14) Gundiah, G.; Govindaraj, A.; Rajalakshmi, N.; Dhathathreyan, K. S.; Rao, C. N. R. J. Mater. Chem. 2003, 13, 209. (15) Chen, P.; Wu, X.; Lin J.; Tan, K. L. Science 1999, 285, 91. (16) Ye, Y.; Ahn, C. C.; Witham, C.; Fultz, B.; Liu, J.; Rinzler, A. G.; Colbert, D.; Smith, K. A.; Smalley, R. E. Appl. Phys. Lett. 1999, 74, 2307.

carbon (10.9 kJ/mol)17 or planar graphite (4 kJ/mol).18 This difference was attributed to the curved carbon environment.19,20 Other laboratories, however, reported21-27 very low hydrogen storage capacity of SWCNTs with a heat of adsorption close to that on flat graphite surfaces. The latter studies have implied the necessity of high pressure and/or cryogenic conditions. TGA adsorption-desorption experiments displayed the hysteresis behavior.15 The structure28 and diameter29 of the nanotube were shown to be critical for optimizing hydrogen storage. To resolve the experimentally observed differences we need to understand the key factors governing adsorption and desorption mechanisms. Numerous theoretical studies were conducted to this end. Molecular dynamics (MD)30-34 and Monte Carlo21,35-37 studies predicted high or low capacity of hydrogen in carbon nanotubes depending on the potentials and sampling methods applied in the (17) Valenzuela, D. P.; Myers, A. L. Adsorption equilibrium data handbook; Prentice Hall: Englewood Cliffs, NJ, 1989. (18) Zhou L.; Zhou, Y. Ind. Eng. Chem. Res. 1996, 35, 4166. (19) Kostov, M. K.; Cheng, H.; Cooper, A. C.; Pez, G. P. Phys. Rev. Lett. 2002, 89, 146105. (20) Sznejer, G. A.; Efremenko, I.; Sheintuch, M. AIChE J. 2004, 50, 596. (21) Smith, M. R., Jr.; Bittner, E. W.; Shi, W.; Johnson, J. K.; Bockrath, B. C. J. Phys. Chem. B 2003, 107, 3752. (22) Liu, C.; Fan, Y. Y.; Liu, M.; Cong, H. T.; Cheng, H. M.; Dresselhaus, M. S. Science 1999, 286, 1127. (23) Tibbetts, G. G.; Meisner, G. P.; Olk, C. H. Carbon 2001, 39, 2291. (24) Schimmel, H. G.; Kearley, G. J.; Nijkamp, M. G.; Visser, C. T.; de Jong, K. P.; Mulder, F. M. Chem.sEur. J. 2003, 9, 4764. (25) Lawrence, J.; Hu, G. Appl. Phys. Lett. 2004, 84, 918. (26) Liu, C.; Cheng, H. J. Mater. Sci. Technol. 2002, 18, 124. (27) Liu, C.; Yang, Q. H.; Tong, Y.; Cong, H. T.; Cheng, H. M. Appl. Phys. Lett. 2002, 80, 2389. (28) Shen, K.; Xu, H.; Jiang, Y.; Pietra, T. Carbon 2004, 42, 2315. (29) Kostov, M. K.; Cheng, H.; Cooper, A. C.; Pez, G. P. Phys. Rev. Lett. 2002, 89, 146105. (30) Canto, G.; Ordejon, P.; Cheng, H.; Cooper, A. C.; Pez, G. P. New J. Phys. 2003, 5, 1. (31) Ma, Y.; Xia, Y.; Zhao, M.; Ying, M. Phys. Rev. B 2002, 65, 155430. (32) Wang, Q.; Johnson, J. K. J. Chem. Phys. 1999, 110, 577. (33) Darkrim, F.; Levesque, D. J. Chem. Phys. 1998, 109, 4981. (34) Williams, K. A.; Eklund, P. C. Chem. Phys. Lett. 2000, 320, 352. (35) Wang, Q.; Johnson, J. K. J. Phys. Chem. B 1999, 103, 48094813. (36) Cheng, J.; Yuan, X.; Zhao, L.; Huang, D.; Zhao, M.; Dai, L.; Ding, R. Carbon 2004, 42, 2019. (37) Levesque, D.; Gicquel, A.; Darkrim, F. L.; Kayiran, S. B. J. Phys.: Condens. Matter 2002, 14, 9285.

10.1021/la046757b CCC: $30.25 © 2005 American Chemical Society Published on Web 06/04/2005

Hydrogen Adsorption on CNTs Enthalpy and Entropy

simulations. Quantum mechanical density functional theory (DFT)21,38-44 and ab initio45 calculations showed that dissociative adsorption is much stronger than a molecular one; however, hydrogen dissociation in the absence of catalytic particles is prohibited by an extremely high activation barrier.38,43 Moreover, it was found that adsorbed hydrogen atoms are subjected to easy recombination within the carbon nanotubes. Special treatment leading to the formation of defects of carbon nanotubes was reported to enhance atomic hydrogen adsorption of up to 1 wt % at high temperatures and in the presence of Pd catalyst.46 Most of such atomically adsorbed hydrogen desorbs only at temperatures as high as 600-1000 K. The dissociated state of adsorbed hydrogen is, therefore, not important from a technological point of view.47 The reported heats of the molecular hydrogen adsorption strongly depend on the applied computational approach: generalized gradient approximation and mixed DFT-HF methods suggest a repulsive interaction between hydrogen and nanotubes while local density approximation and second-order Møller-Plesset (MP2) methods suggest high interaction energies. The differences between the binding energies corresponding to different positions and the diffusion barriers were shown to be small indicating an easy diffusion at low temperature.48 Theoretical predictions of adsorption capacity at room temperature in (10,10) SWCNT (∼1.3 nm in diameter) varied between 1438 and 1.5 wt %.11 The isosteric heat of hydrogen adsorption on the external surface of the nanotubes arranged in the optimized arrays was predicted to be ∼7.2 kJ/mol, and that leads to a gravimetric density of ∼0.8 wt % at 298 K and 50 atm.35 The aim of the present work is to account for the controversy in the experimental and computational results and to estimate the adsorption and desorption kinetics and storage capacity based on the thermodynamics of interaction between a single or multiple hydrogen molecules and SWCNTs. Most notably we account for entropy effects, which were ignored in most previous studies. The importance of entropy effects in the hydrogen-nanotube system is evident since the driving force for adsorption is the difference in chemical potentials between the adsorbed and gas phases, which in turn is determined by the difference between the opposing effects of energetic and entropic terms. Ignoring entropy will introduce only a relatively small error in the description of adsorption on surfaces; however, in the case of a molecule on or inside a narrow nanotube the steric factors could significantly restrict many degrees of freedom.49 We show that the entropy change upon hydrogen adsorption on the inner and outer surfaces of carbon nanotubes inhibits the process. On the other hand, the relatively high hydrogennanotube interaction energy in conjunction with onedimensional molecular transport behavior within nano(38) Lee, S. M.; Lee, Y. H. Appl. Phys. Lett. 2000, 76, 2877. (39) Xia, Y.; Zhao, M.; Ma, Y.; Liu, X.; Ying, M.; Mei, L. Phys. Rev. B 2003, 67, 115117. (40) Tada, K.; Furuya S.; Watanabe, K. Phys. Rev. B 2001, 63, 155405. (41) Froudakis, G. E. J. Phys.: Condens. Matter 2002, 14, R453. (42) Froudakis, G. E. Rev. Adv. Mater. Sci. 2003, 5, 259. (43) Han, S. S.; Lee, H. M. Carbon 2004, 42, 2169. (44) Barone, V.; Heyd, J.; Scuseria, G. E. J. Chem. Phys. 2004, 120, 7169. (45) Okamoto, Y.; Miyamoto, Y. J. Phys. Chem. B 2001, 105, 3470. (46) Yoo, E.; Gao, L.; Komatsu, T.; Yagai, N.; Arai, K.; Yamazaki, T.; Matsuishi, K.; Matsumoto, T.; Nakamura, J. J. Phys. Chem. B 2004, 108, 18903. (47) Haluska, M.; Hirscher, M.; Becher, M.; Dettlaff-Weglikowska, U.; Chen, X.; Roth, S. AIP Conf. Proc. 2002, 633, 601. (48) Arellano, J. S.; Molina, L. M.; Rubio, A.; Alonso, J. A. J. Chem. Phys. 2000, 112, 8114. (49) Myers, A. L. Colloids Surf., A 2004, 241, 9.

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tubes limits the desorption ability of absorbed molecules. Moreover, theoretical studies mainly concentrate on the hydrogen adsorption within carbon nanotubes while experimentalists usually deal with closed-end nanotube bundles. In the present work we compare the thermodynamics of hydrogen adsorption on and within SWCNTs paying attention mainly to the energetically more effective inner-surface adsorption. The approach we apply consists of calculating the adsorption energies and entropies and then using them for equilibrium capacity calculations based on a meanfield approach. This approach is more effective than MD as Lennard-Jones potentials commonly used in MD simulations do not account for the spatial molecular shape and dimensions, which turn out to be significant if the molecular and nanotube sizes are comparable. The rest of the article is organized as follows. In the next section the computational methods and models are described. Thermodynamics of single and multiple hydrogen molecules adsorbed within and on the outer surfaces of SWCNTs are discussed in sections 3 and 4. In section 5 the kinetics of hydrogen adsorption, desorption, and equilibrium adsorption capacity are estimated based on the computational results. Section 6 concludes. 2. Computational Methods and Models The molecular mechanics (MM) approach is employed as the main tool to model the hydrogen adsorption in SWCNTs since it allows for treatment of reasonably large systems; its application is justified by absence of chemical interaction. The computational method applied is the universal force field50,51 as it is implemented in the Gaussian 98 package.52 In contrast to our previous studies, in the MM simulations each structure was properly relaxed. The most salient features of the nanotubemolecule interactions (nonactivated diffusion, high physisorption energy, and strong change in entropy) are checked by ab initio and DFT calculations. Hydrogen physisorption is governed by van der Waals (VDW) interaction, which is usually evaluated by using secondorder perturbation theory.53 Therefore, MP2 is expected to give a qualitatively correct description for VDW interaction. We also did calculations based on the DFT; however, we found that the energetics of DFT calculations significantly depend on the employed basis set as described below. To retrieve the thermochemistry data for a wide range of temperatures from MM and DFT frequency calculations we applied the THERMO.PL program.54 Details of the calculations are described in ref 55. The (50) Rappe, A. K.; Goddard, W. A. J. Phys. Chem. 1991, 95, 3358. (51) Rappe, A. K.; Casewit, C. J.; Colwell, K. S.; Goddard, W. A.; Skiff, W. M. J. Am. Chem. Soc. 1992, 114, 10024. (52) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Zakrzewski, V. G.; Montgomery, J. A., Jr.; Stratmann, R. E.; Burant, J. C.; Dapprich, S.; Millam, J. M.; Daniels, A. D.; Kudin, K. N.; Strain, M. C.; Farkas, O.; Tomasi, J.; Barone, V.; Cossi, M.; Cammi, R.; Mennucci, B.; Pomelli, C.; Adamo, C.; Clifford, S.; Ochterski, J.; Petersson, G. A.; Ayala, P. Y.; Cui, Q.; Morokuma, K.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Cioslowski, J.; Ortiz, J. V.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Gomperts, R.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.; Peng, C. Y.; Nanayakkara, A.; Gonzalez, C.; Challacombe, M.; Gill, P. M. W.; Johnson, B. G.; Chen, W.; Wong, M. W.; Andres, J. L.; Head-Gordon, M.; Replogle, E. S.; Pople, J. A. Gaussian 98; Gaussian, Inc.: Pittsburgh, PA, 1998. (53) Zaremba, E.; Kohn, W. Phys. Rev. B 1976, 13, 2270. (54) Irikura, K. K. THERMO.PL; National Institute of Standards and Technology: Gaithersburg, MD, 2002 (http://www.nist.gov/ compchem/irikura/prog/thermo.html). (55) Irikura, K. K. In Computational Thermochemistry: Prediction and Estimation of Molecular Thermodynamics; Irikura, K. K., Frurip, D. J., Eds.; ACS Symposium Series 677; American Chemical Society: Washington, DC, 1998.

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Table 1. Calculated and Experimental Standard Thermochemistry Data of a Hydrogen Molecule method

S298, J/(mol‚K)

H298 - H0, kJ/mol

Cp 298, J/(mol‚K)

MM B3LYP/STO-3G MP2/aug-cc-pVDZ experiment56

129.46 129.93 130.53 130.680 ( 0.003

8.677 8.677 8.677 8.468 ( 0.001

29.1 29.1 29.1 28.8

benchmark tests for the thermodynamics of the H2 molecule (Table 1) show very good agreement of MM data with the corresponding ab initio, DFT, and experimental values. Nanopores of various diameters were modeled by zigzag carbon nanotubes varying in size from (6,0) to (24,0) that corresponds to tube diameters (D, defined as a distance between the centers of the carbon atoms) from 4.3 to 18.8 Å. The length of nanotubes was usually set to 7.1 Å (eight carbon layers) with outer C atoms being saturated with hydrogen. Removal of the saturating H atoms causes a negligible (∼0.5%) decrease in the interaction energy. To check the influence of the nanotube length on the energetics of hydrogen-nanotube interactions, several calculations were performed using longer nanotubes. Movement of H2 species through the nanotubes was modeled by the relaxed potential energy surface (PES) scan. The enthalpy and entropy effects for the hydrogen introduction into a nanotube were calculated as a difference between the corresponding values of the system with optimal position of a molecule inside a nanotube and those of the corresponding separated components. 3. Adsorption Thermodynamics of a Single H2 Molecule Within the smallest considered nanotube (6,0) the repulsion of a H2 molecule with the carbon walls prevails and the interaction is energetically strongly unfavorable. With increasing nanotube diameter, the molecule “condenses” on the wall with orientation in the axial direction to maximize the attractive VDW interaction. The interaction energy is quite high as long as a molecule inside the nanotube interacts with the surrounding carbon walls from every quarter; for the optimal nanotube size [nanotube (8,0)] it reaches 23.5 kJ/mol at 0 K and 25.6 kJ/mol at room temperature. These values are even higher than the largest energies obtained in the thermodesorption experiments (∼19.6 kJ/mol).2,13 With increasing nanotube diameter the adsorption energy declines and approaches that on the graphite plane, which may be considered as the limiting zero curvature case (6.8 kJ/mol in the applied MM approach and 8.3 kJ/mol obtained in the DFT calculations48 in comparison with 4 kJ/mol experimentally).18 Adsorption on the outer nanotube surface is even weaker due to the smaller interaction area and increases from 4.4 kJ/mol on a (6,0) nanotube to 5.8 kJ/mol on a (24,0) nanotube. This is in agreement with the ab initio results for H2 interaction with curved graphene clusters45 and isosteric heats of hydrogen adsorption in interstitial cavities of nanotube arrays.35 To check the performance of the applied MM method we calculated the hydrogen adsorption energy in SWCNT (8,0) using the second-order Mo¨ller-Plesset perturbation (MP2) and hybrid density functional (B3LYP) theories with different basis sets (Table 2). For MP2 calculations we used the smallest nanotube (8,0) model cluster C32H16 (one aromatic ring in length) while for DFT and MM calculations a C64H16 cluster (three aromatic rings in length) was used. Heat effects obtained by the DFT method strongly depend on the basis set used: small basis sets

Table 2. Binding Energy (BE ) ENT + EH2 - EH2 in NT, kJ/mol) for H2 Interaction with the (8,0) Nanotube Calculated at Different Levels of Theory B3LYP/ STO-3G

B3LYP/ 3-21G

B3LYP/ cc-pVDZ

MP2/ aug-cc-pVDZ

MM

-8.32

-4.96

2.61

24.02

23.50

Table 3. Entropy Change (∆S, J mol-1 K-1) for H2 Insertion into the (8,0) Nanotube at 298.15 K and Its Components Calculated at Different Levels of Theory total translational rotational vibrational

B3LYP/STO-3G

B3LYP/3-21G

MM

-101.3 -117.4 -12.4 28.6

-105.2 -117.4 -12.8 25.1

-87.1 -117.4 -12.0 42.3

bring about repulsive H2-nanotube interaction; improvement of the basis set leads to an increase of the physisorption energy by more than 10 kJ/mol. The MP2 approach leads to even higher interaction energy, in good agreement with the MM result. While energetic effects suggest strong hydrogen adsorption within SWCNT, the sterically restricted situation of a molecule inside a nanotube results in an extremely strong decline in entropy, when compared to the separated H2 and SWCNT. The main contribution to the decrease of total entropy change is due to the translational component (Table 3 and Figure 1a); the rotational component is also slightly negative. The vibrational component partially compensates the total decline of the entropy while a sharp decrease of ∆S in small tubes results mainly from restricted vibrations. The electronic partition function is not affected by the interaction due to the absence of chemical interaction. A similar sharp decline in entropy (∆S0 ) -90 ( 5 J mol-1 K-1) was measured for hydrogen adsorption on the zeolite Li-ZSM-5.57 Hydrogen adsorption on the outer nanotube surface is also accompanied by a strong entropy decline originated mainly from the translational partition function (Figure 1b). It can be understood by taking into account that translation of the H2 molecule in any direction other than along the nanotube wall leads to its separation from the surface. With increasing nanotube diameter the entropy change diminishes and approaches that on a flat surface. Such a strong change in the entropy results in an extremely small probability of hydrogen insertion into a nanotube or its adsorption on the external surface at relatively high temperatures. The calculated change in the Gibbs free energy is negative for all the nanotube sizes, as well as for a flat graphite surface, only at 77 K (Figure 2a). With increasing temperatures the range of the tube diameters characterized by ∆G < 0 diminishes (it is between 5.6 and 13.5 Å at 150 K). The ∆G values are higher for adsorption on the external surfaces due to the smaller adsorption heats and are positive even at the temperature as low as 77 K. We conclude, therefore, that at temperatures higher than 77 K hydrogen absorption into nanotubes with diameter D g 14 Å and adsorption on nanotubes with D e 20 Å are hampered by the entropy effects. This is in agreement with the experimentally observed low adsorption capacity of SWCNTs at these temperatures. In short nanotubes the most stable adsorption site is located near the center of the nanotube length. Figure 3a demonstrates the energy change when moving a hydrogen (56) CRC Handbook of Chemistry and Physics, 85th ed.; CRC Press: Boca Raton, FL, 2004-2005. (57) Otero, A. C.; Manoilova, O. V.; Bonelli, B.; Rodriguez, D. M.; Turnes, P. G.; Garrone, E. Chem. Phys. Lett. 2003, 370, 631.

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Figure 1. Size dependence of the total entropy change and its translational, rotational, and vibrational components for hydrogen absorption within (a) and on the outer surfaces (b) of SWCNTs at 298.15 K calculated by the MM method. Points on the right border present the calculated asymptotic values as the nanotube diameter D tends to infinity (graphite sheet).

Figure 2. Influence of the SWCNT diameter on the Gibbs free energy change upon hydrogen adsorption on the inner (a) and outer (b) surfaces at various temperatures. Points on the right borders present the calculated asymptotic values as the nanotube diameter D tends to infinity (graphite sheet).

Figure 3. PES cross sections for the movement of the H2 molecule from the most stable position within SWCNT (8,0) toward the tube end calculated at the MM and B3LYP/STO-3G levels of theory for frozen and optimized geometries of the nanotube of 7.1 Å in length (a) and MM PES cross sections for the movement of the hydrogen molecule from the center of SWCNTs (10,0) of various lengths with the frozen nanotube geometry.

molecule from its most stable position along the axis of the (8,0) SWCNT of 7.1 Å in length. One can see that MM and DFT provide almost the same energy changes and

that optimization of the nanotube geometry results in a very small difference in the relative energies. In such a narrow tube the hydrogen molecule following the lowest-

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Figure 4. Connolly surfaces (surfaces available for solvent; H2 molecular size is set as a size of solvent molecules) calculated for optimized geometries of hydrogen molecules within the (16,0) nanotube (D ) 12.53 Å) upon consequent filling (a); top and side views of the optimal arrangement of 14 hydrogen molecules within the (16,0) nanotube (carbon atoms are not shown; b).

energy diffusion pathway is oriented along the tube axis. In wider tubes the hydrogen molecule moves along the walls and changes its orientation so as to increase the interaction with carbon atoms. In several nanotubes it follows almost a helical trajectory similar to that predicted for the diffusion of small organic molecules.58 In nanotubes longer than 7.1 Å in the absence of steric restrictions [(8,0) and wider] the adsorption energy remains almost constant along the nanotube length indicating nonactivated diffusion (Figure 3b). Near the tube ends the energy of the system increases. For a single hydrogen molecule the adsorption energy in the most stable position shows negligible increase with increasing the nanotube length. These features are tied to the nature of the physisorption forces, which involve the attraction between temporarily induced dipoles in nonpolar molecules. As the polarizability increases for larger, more dispersed electron clouds, these forces become stronger as the nanotube becomes longer. It was shown that the static polarizabilities of carbon nanotubes reach their saturation levels at lengths of ∼100 unit cells.59 Molecular hydrogen diffusions on the flat graphite surface and on the outer surface of SWCNTs are characterized by activation barriers smaller than 1 kJ/mol. 4. Thermodynamics of Multiple H2 Molecules in SWCNTs We address now the energy change when several hydrogen molecules adsorb in the same axial position or in the different axial positions on the nanotube walls and in the interstitial area. As it was shown earlier60 the interaction of adsorbed molecules in quasi-one-dimensional phases confined within a bundle of nanotubes differs significantly from that in the gas phase. Hydrogen is a nonspherical, nonpolar molecule that may exhibit different orientations in multilayer adsorption. At low coverage the energetically preferential position of absorbed hydrogen molecules is near the longitudinal center of the nanotube with axial orientation at a distance of 3.0 Å from the tube wall. Figure 4 shows the optimal arrangements of H2 molecules within the (16,0) SWCNT upon consequent filling. In the absence of steric restrictions, the interaction between hydrogen molecules inside a (58) Mao, Z.; Sinnott, S. B. Phys. Rev. Lett. 2002, 89, 278301. (59) Jensen, L.; Astrand, P.-O.; Mikkelsen, K. V. J. Phys. Chem. A 2004, 108, 8795. (60) Simonyan, V. V.; Diep, P.; Johnson, J. K. Chem. Phys. Lett. 2000, 332, 26.

Figure 5. Adsorption energy of the incident H2 molecule for the consequent filling of carbon nanotubes of various diameters.

Figure 6. VDW electron density surfaces for the SWCNT (10,0) (a) and for the optimized arrangement of seven hydrogen molecules inside the SWCNT (16,0) shown in Figure 4b (b).

nanotube is attractive and that leads to their condensation with an H2-H2 nearest-neighboring distance of 2.7-2.9 Å in comparison with 3.51 Å obtained from neutrondiffraction studies of H2 adsorbed on graphite with a coverage of ∼one monolayer.61 The pair interaction energies of ∼0.02-0.06 and ∼0.6-1 kJ/mol in the nearsurface and interstitial areas, respectively (Figure 5), are in line with the reduction of the intermolecular potential in the confined phase;60 the former value is even smaller than that predicted for outer nanotube surfaces. After filling all the surface sites we find that hydrogen molecules occupy the inner nanotube space with an optimal distance from the hydrogen molecules adsorbed on the nanotube walls of ∼3.5 Å. These adsorption positions are energetically less favorable, which is consistent with the generally observed rule that the energy of VDW interaction between the molecules of the adsorbate and those of the adsorbent is of the same order of magnitude as, but is usually greater than, the energy of condensation of the adsorbate. The increase in the entropy for such inner positions is much smaller than that for the nanotube surface sites and even smaller than for adsorption on the planar graphite surface. These dissimilarities result from the different electron density distributions within the cylinder formed by adsorbed hydrogen molecules in comparison with the carbon nanotube of similar inner diameter (Figure 6) resulting in much smaller polarizability. For the same reason (lower electron density) the molecular hydrogen adsorption on the outer surfaces (61) Nielsen, M.; McTague, J. P.; Ellenson, W. J. Phys. 1977, 38, C4.

Hydrogen Adsorption on CNTs Enthalpy and Entropy

of SWCNTs is slightly weaker than that on the flat graphite plane. Increasing coverage along nanotube walls leads to some strengthening (∼2 kJ/mol) of the inner adsorption positions; however, neither at low nor at high hydrogen loadings we get adsorption energies as large as those measured by Dillon et al.2 in a nanotube of 18.5 Å in diameter that approximately corresponds to the nanotube (24,0). The optimal position for the hydrogen adsorption on the outer nanotube surface is at the distance of 2.5 Å from the surface in the center of aromatic ring with orientation of the molecular axis along the nanotube length. Consequent filling of these positions in the longitudinal section causes a decrease of the intermolecular distances from 4.5 to 3.1 Å when moving from the (6,0) to the (24,0) nanotube with corresponding H2-H2 attraction energies from 0.08 to 0.38 kJ/mol. 5. Adsorption Capacity In this section we apply the thermodynamic parameters calculated for the hydrogen adsorption in one longitudinal ring to estimate the nanotube storage capacity at various temperatures. Let us consider first the geometry of maximal adsorption. We assume that the capacity in such a ring is independent of its position or the nanotube length. We also assume that all hydrogen molecules are oriented along the nanotube axis and neglect small intermolecular interactions. The maximal storage capacity is calculated from geometric considerations. The actual temperaturedependent capacity is calculated by applying the Langmuir isotherm for the surface layer and in larger nanotubes also by adding the capacity of the interstitial volume of a second and possibly a third layer. The following geometric considerations were applied: Only one hydrogen molecule can be packed at every longitudinal section in the nanotube (10,0) and in smaller ones; with increasing coverage the absorbed molecules change their orientation so as to allow for maximal adsorption (Figure 5). From 3 to 14 molecules could be adsorbed in a longitudinal section on the walls of nanotubes of increasing size from (12,0) to (24,0). Additional space is available for hydrogen molecules in the interstitial cavity of nanotubes (16,0) and larger. Using the geometrical model with close packing of hydrogen molecules (Figure 4) within nanotubes of various sizes leads to a maximal storage capacity of 0.5% for adsorption in the nanotube of 7 Å in diameter, and it increases to 4.2 wt % in the nanotube of 18.8 Å in diameter where additional filling of the interstitial cavity occurs. Similar geometrical estimates62 based on close packing of hydrogen molecules of kinetic diameter 2.9 Å on the inner walls and in the volume of a (10,10) tube (∼1.3 nm in diameter) led to 3.3 wt % hydrogen adsorption within the tube and 0.7 wt % adsorption within the interstitial space (total 4.0 wt %). Our estimation for the nanotube of the same size is notably smaller (total of ∼2.5 wt %) since we consider the hydrogen molecules as prolate spheroids with the axis lengths of 2.93 × 3.67 Å in the near-wall area and 3.5 × 3.67 Å in the interstitial space. Adsorption on the external nanotube surface is site-specific and leads to an H2/C ratio of 1:2 (8.3 wt %), independent of the nanotube size. To estimate the storage capacity at specific conditions we need to determine the dynamics of the hydrogen charging and release. The results presented above demonstrate that both adsorption on and desorption from the inner and outer surfaces of SWCNTs are thermodynamically limited processes with the former one being restricted (62) Dresselhaus, K. A. W.; Eklund, P. C. MRS Bull. 1999, 24, 45.

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by entropy effects and the latter one having an activation barrier for desorption of Ed ) -∆Hads. Adsorption within the nanotube is controlled by the probability to enter its mouth. If Fads )Pref/(2πNa2mkT)1/2 is the Henz-Knudsen flux of incident hydrogen molecules at Pref ) 1 bar, then for a single nanotube the adsorption frequency is ka ) χFadsNaSn at that pressure, where χ ) exp(∆S/R) is the “sticking coefficient”, which characterizes the probability of a molecule to enter the tube of radius rn (Sn ) πrn2 is the cross section). The adsorption frequency on a single adsorption center of the planar graphite and of the external nanotube surface could be calculated by the same way taking Sn ) 5.24 × 10-20 m2 (the surface of one aromatic ring). Desorption is controlled by the kinetic energy needed to overcome the dispersive interaction; its frequency is assumed to follow kd ) (kT/h) exp(-Ed/RT)i, where h is Planck’s constant, m is the mass of a hydrogen molecule, k is Boltzmann’s constant, and i is the number of hydrogen molecules per cross section of nanotube. Comparison of the frequencies characterizing the two steps at a pressure p ) P/Pref, kap/kd ) pPref(Sn/i)h/[kT(2πmkt)1/2]e-∆G/(RT) (Figure 7), shows that at 100 bar occupancy is high (kap/kd . 1) for all the nanotube diameters only at 77 K and for a narrow range of nanotubes at 150 K; with increasing temperature ka/kd quickly drops. For hydrogen interaction with the external nanotube surface the desorption is much faster than adsorption for all temperatures and all the nanotube diameters. To translate this information to storage capacity we apply the Langmuir isotherm θ ) (kap/kd)/(1+ kap/kd) for the inner surface. In the limit of low coverage we find θ ) nλ(Sn/m)e∆G/RT, where n is the gas-phase density and λ ) h/(2πmkt)1/2 is the thermal de Broglie wavelength. This should be compared to the number density of hydrogen in the nanotube, derived from thermodynamic equilibrium of chemical potential to be Ftherm ) nλ2e∆E/RT (refs 63 and 64), where entropy effects are neglected (i.e., ∆G ) ∆E). Assuming maximal packing of x(Sn/i) in distance, the ratio of our Fkinetic ) θ/x(Sn/i) to Ftherm approach is

Fkinetic xSn/i (∆G-∆E)/RT e ) Ftherm λ and if the entropy effect is neglected then this value varies from 0.1 to 0.2 with the minimum at D ∼ 7.8 Å. In the larger nanotubes the model should differentiate between the adsorption in the interstitial space and that close to the surface using ka and kd parameters that correspond to each domain. This approach is somewhat more elaborate, but we expect the interstitial positions to be occupied only after the surface positions have been occupied. The adsorption capacities within the nanotubes and on the outer nanotube surface are presented in Figure 8. Almost maximal (geometric) storage capacity could be achieved at P ) 100 bar and 77 K for the inner hydrogen absorption; on the outer surface storage capacity is very low even at this temperature while the maximal adsorption (8.1 wt %) is available on the planar graphite surface. With increasing temperature the Gibbs free energies increase resulting in a sharp decline of hydrogen storage capacity: only 0.5 wt % could be adsorbed at room temperature within the (10,0) nanotube, which is characterized by the lowest ∆G values; hydrogen adsorption (63) Wang, Q.; Challa, S. R.; Sholl, D. S.; Johnson, J. K. Phys. Rev. Lett. 1999, 82, 956. (64) Challa, S. R.; Sholl, D. S.; Johnson, J. K. Phys. Rev. B 2001, 63, 245419.

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Figure 7. Influence of the SWCNT diameter on the adsorption-desorption equilibrium of a single hydrogen molecule adsorbed on the inner (a) and outer (b) nanotube surfaces at various temperatures and at atmospheric pressure. The asymptotic values as D f ∞ calculated by the same method for the planar graphite plane are shown on the right borders of each panel.

Figure 8. Size dependence of the calculated hydrogen storage capacity of SWCNTs at various temperatures and pressure P ) 100 bar.

at this temperature does not exceed 0.1 wt % on the inner and outer surfaces of larger nanotubes. 6. Conclusions Thermodynamic parameters for molecular hydrogen adsorption within SWCNTs and on their outer surfaces are calculated using the MM method. We found three distinguishing features characterizing hydrogen adsorption on the nanotube surfaces: (i) Adsorption energy is high in narrow nanotubes (in nanotube (8,0) it is 3.5 times higher than that on the graphite plane), and it declines with increasing nanotube size. Similar results were obtained in previous studies.65 Adsorption on the external nanotube surfaces is weaker than that on graphite and strengthens with increasing diameter. (65) Rzepka, M.; Lamp, P.; de la Casa-Lillo, M. A. J. Phys. Chem. B 1998, 102, 10894.

(ii) There are no specific adsorption sites within the carbon nanotubes; consequently, the diffusion is nonactivated, and hydrogen molecules can be close packed at low temperatures owing to weak attractive interaction between adsorbed molecules. The volume captured by each molecule is much larger than that implied by the LennardJones potential and depends on the adsorption position (inner or outer surface or interstitial cavity). On the planar graphite and external nanotube surfaces the preferential adsorption positions are in the center of the aromatic ring separated by activation barriers smaller than 1 kJ/mol. (iii) Most importantly, we found that entropy changes are significant for hydrogen adsorption on both inner and outer nanotube surfaces, the effect that was neglected in previous studies. This results in a sharp increase in ∆G with increasing temperature and hampers the hydrogen adsorption. These main features are confirmed by ab initio and DFT calculations for small systems. Application of the obtained geometric and thermodynamic parameters for estimation of the hydrogen storage capacity of SWCNTs leads to the conclusion that up to 4.2 wt % could be stored within SWCNTs and up to 8.3 wt % can be stored on their outer surface and on one side of the graphite sheet. However, these values can be achieved only for adsorption at low temperatures (77 K) and high pressures (e.g., 100 bar). At room temperature and 100 bar the adsorption on all the considered surfaces is smaller than 1 wt %. Acknowledgment. This work was supported by the Water Research Institute. I.E. gratefully acknowledges the partial financial support of the Center for Adsorption in Science, Ministry of Immigrant Absorption, State of Israel. LA046757B