ARTICLE pubs.acs.org/JPCC
Open Carbon Nanocones as Candidates for Gas Storage Olumide O. Adisa,* Barry J. Cox, and James M. Hill Nanomechanics Group, School of Mathematical Sciences, University of Adelaide, Adelaide, SA 5005, Australia ABSTRACT: In this Article, we investigate hydrogen, methane, and neon encapsulation and adsorption in open carbon nanocones. We exploit the 612 Lennard-Jones potential function and the continuous approximation to model the surface binding energies and the molecular forces between these gases and carbon nanocones of varying vertex angle and length. Our results show that for a hydrogen or methane molecule, or neon atom, interacting with a carbon nanocone the binding energies of the respective systems are minimized when the gas is encapsulated inside the cone. However, we find that for the shorter carbon nanocones, there is a higher energy barrier preventing methane encapsulation in the nanocone. The present modeling indicates that for the particular apex angle of 112.9° the optimal minimum energy storage for the gases occurs in a nanocone of radius 7.1052 Å and of length 4.7126 Å. Our results agree with recent results suggesting that gas adsorption in carbon nanocones is more favorable at lower temperatures. Overall, our results are in very good agreement with other theoretical studies and molecular dynamics simulations and show that carbon nanocones might be good candidates for gas storage. However, the major advantage of the approach here is the derivation of explicit analytical formulas from which numerical results for varying physical scenarios may be readily obtained.
I. INTRODUCTION Carbon nanostructures have attracted considerable attention1 due to their potential use in many applications including energy storage, gas sensors, biosensors, nanoelectronic devices, and chemical probes. Carbon allotropes such as carbon nanocones and carbon nanotubes have been proposed as possible molecular gas storage devices.29 More recently, carbon nanocones have gained increased scientific interest due to their unique properties and promising uses in many novel applications such as energy and gas storage.1014 Studies on the structural properties of carbon nanocones include Yudasaka et al.,11 who predict that pentagonal defects in carbon nanocones perturb the low-energy electronic structure both locally and globally, defining both a local region of enhanced reactivity and a global phase relation, which has consequences for electron transport around the apex. Shenderova et al.12 employ analytical potentials to show that carbon nanocones can exhibit conventional cone shapes or can form concentric wave-like metastable structures. Their study also predicts that a pentagon in the center of a cone is the most probable spot for emitting tunnelling electrons in the presence of an external field, implying that nanocone assemblies, if practically accessible, could be used as highly localized electron sources for templating on length scales below traditional lithography techniques. Baowan and Hill15 find that the equilibrium location for two distinct and identical cones moves further away from the vertex as the number of pentagons increases. They also observe that the equilibrium configuration is such that one cone is always inside another, therefore predicting the existence of nested double cones. Yu et al.14 employ thermal desorption and photoemission to study the adsorption capacity of hydrogen on carbon nanocones, and they show that the maximum desorption peak is r 2011 American Chemical Society
located near 250 K. They also predict that the desorption process is influenced by site geometry and adsorbateadsorbate interactions. Fullerene interaction with carbon nanocones is investigated theoretically by Suarez-Martinez et al.,10 who conclude that fullerenes weakly bind to the external nanocone wall with an equilibrium spacing of 2.9 Å and binding energy in the range of 0.5 to 0.9 eV and also conclude that the binding energy increases with the number of pentagons at the tip. Majidi et al.13 investigate neon adsorption on carbon nanocones using molecular dynamics simulation, and they find that saturation coverage and saturation pressure both depend on temperature, and under saturation conditions, the maximum values of the interior and exterior coverages are 0.17 and 0.39 neon per carbon, respectively. They conclude that adsorption coverages on carbon nanocones are greater than those on carbon nanotubes, making carbon nanocones a desirable adsorbent. Hydrogen (H2) and methane (CH4), gases at room temperature, hold much promise as sources of clean and affordable energy because they are more environmentally friendly than most hydrocarbon fuels, and they are also readily available.16 However, because of the current difficulties of storage, the widespread commercial use of H2 and CH4 as vehicle fuels has not been achieved. Neon (Ne), unlike other inert gases, discharges electricity even at normal currents and voltages and has major applications in the advertising industry. Using nanotechnology, potentially H2, CH4, and Ne could be stored more efficiently,6,13 thereby increasing their everyday commercial use. Received: July 19, 2011 Revised: November 3, 2011 Published: November 07, 2011 24528
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Table 1. Numerical Values of Constants Used (* and ** Denote Data From Refs 19 and 20, Respectively) parameters
values
radius of hydrogen sphere
a = 1.1 Å
*attractive constant CC *repulsive constant CC
ACC = 17.4 eV Å6 BCC = 29000 eV Å12
**attractive constant CH
ACH = 17.9 eV Å6
**repulsive constant CH
BCH = 17483.5 eV Å12
**attractive constant CNe
ACNe = 13.2 eV Å6
**repulsive constant CNe
BCNe = 12794.6 eV Å12
mean surface density of hydrogen sphere
ηh = 0.263 Å 2
mean surface density of carbon nanocone
ηc = 0.3812 Å2
Table 2. Proposed Base Radii for Carbon Nanocones with Θ 112.9°
Figure 1. Neon, hydrogen, and methane interacting with a singlewalled carbon nanocone of length L2 L1 and apex angle θ.
Experimental work suggests, at least at lower temperatures, that there is a highly organized packing arrangement of hydrogen inside carbon nanocones13,17 and that CH4 molecules are not freely spinning but are somehow self-organized into a confined structure in which there is a local order.18 However, despite the potential use of carbon nanocones as gas storage devices, there is very little available literature on gas storage in carbon nanocones in comparison with that for carbon nanotubes. In this study, we employ the 612 Lennard-Jones potential together with the continuous approximation to investigate the encapsulation of H2, CH4, and Ne in carbon nanocones. In proposing this study, we have in mind gas storage in carbon nanocones. We derive analytical expressions for the binding energies and molecular forces between a single H2 and CH4 molecule and a single Ne atom for carbon nanocones of varying apex angle and length. Finally, we determine the volume available
K integer number
R (Å) radius of
L1 (Å) corresponding point along
of hexagons
carbon nanocone
height H of nanocone
1
1.7763
1.1782
2
3.5526
2.3563
3
5.3289
3.5345
4
7.1052
4.7126
for H2, CH4, and Ne adsorption in carbon nanocones of varying radii. We comment that in these models we account for the physisorption of H2, CH4, and Ne to carbon nanocones, and we assume no chemisorption of these gases onto carbon nanocones. In addition, we comment that this Article provides a new analytical expression for gas encapsulation and adsorption in open carbon nanocones, from which numerical values can be readily deduced to provide an extensive numerical landscape of the problem, which will facilitate new physical insight into this problem. For example, we may readily demonstrate the particular physical dimensions of open carbon nanocones that will readily encapsulate different gases and which describe an entirely new physical phenomena, but which is inherent in the formulas presented here. The Lennard-Jones potential and the continuous approximation employed in deriving the analytical expressions for the binding energies are described in the following section, a discussion of the derivation of the basic results of this study is given in Section III, and Section IV provides a discussion of the results. A brief conclusion is presented in the final section of the Article.
II. METHOD In this section, we outline the procedure employed in our calculations. We begin by defining a complete nanocone with a vertex present as a closed nanocone, whereas a nanocone with no vertex is termed an open nanocone. For closed nanocones, the apex angle is given by sin(Θ/2) = 1 (Np/6), where Np = 1, 2, 3, 4, 5 is the number of pentagons, giving rise to five possible closed nanocone structures.21 Open nanocones, however, may have more flexible structures with more than five different apex angles,22,23 although Eksioglu et al.24 argue that open nanocones should have exactly the same apex angles as those of closed nanocones. We model an open nanocone with base radius R = (3Kσc sin θ)/2 and height H = (3Kσc cos θ)/2, where σc is the CC bond length, K is the number of hexagons as measured 24529
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Figure 2. Binding energy and molecular force for CH4 molecule, two H2 molecules, and four Ne atoms entering single-walled carbon nanocone.
from vertex to vertex along the slant, and θ = Θ/2 is the half angle of the carbon nanocone apex angle Θ, as shown in Figure 1. Employing the formulas for R and H given above, the proposed base radii for the nanocones examined in this study are given in Table 2 using 1.421 Å28 for the CC bond length σc, and L1 is obtained from the above expression for H and is employed in the analytical expression in Section III. In addition, we model a single CH4 molecule assuming that the four hydrogen atoms are smeared over a spherical surface of radius a = 1.1 Å (CH bond length in CH4) with the carbon atom assumed to be situated at the center of the sphere. A major advantage of this CH4 model is that the particular orientation of the hydrogen atoms becomes immaterial, thus providing a significant simplification to the calculation.3 The total energy calculation for the H2, CH4, and Ne to carbon nanocone system is therefore reduced to (i) calculating the interaction energy between the H2, Ne, and CH4 carbon atoms and nanocone and (ii) calculating the interaction energy between the CH4 hydrogen sphere and nanocone. The Lennard-Jones interaction potential for two atoms on two nonbonded molecules is given by νðFÞ ¼ AF6 þ BF12
Figure 3. Binding energy and molecular force for CH4 molecule, two H2 molecules, and four Ne atoms entering single-walled carbon nanocone.
density, the total potential energy is given by E ¼ η1 η2
Z S1
S2
νðFÞ dS1 dS2
ð2Þ
where η1 and η2 represent the average atomic surface densities (namely, number of atoms/surface area) of a carbon nanocone and a CH4 molecule, respectively, and ν(F) is the Lennard-Jones potential function for two nonbonded atoms with typical surface elements dS1 and dS2 at a distance F apart. The original sources and the numerical values of the constants used in this study are as given in Table 1. We comment that the numerical data taken are from two different sources19,20 because there is no single source currently available. We further comment that the present modeling does not take into account quantum effects for hydrogen binding into carbon nanocones, which is an issue that is not fully understood in the literature.2527 Following a similar calculation to that given by Thornton et al.,6 the volume available for H2, CH4, or Ne adsorption in the free cavity of a single-walled carbon nanocone is assumed to be given by
ð1Þ
where A and B denote the attractive and repulsive constants, respectively, and F is the distance between the two atoms. Using the continuous approximation that assumes that the carbon atoms on the carbon surface can be replaced by smearing the atoms over the entire surface and using a uniform atomic surface
Z
Vad ¼
Z L2 L1
πr 2 ð1 ejEmin j=kT Þ dz
where Emin is the minimum binding energy of H2, CH4, or Ne to a single-walled carbon nanocone, H is the height of the cone, r = z tan θ is the variable cone radius, k is the Boltzmann constant, and T is the temperature in Kelvin. We note that the upper and lower limits 24530
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We comment that for the open nanocone, the positions of the two edges are z = L1 for the smaller open-end and z = L2 for the larger open-end as shown in Figure 1. The numerical values for L1 employed in this study are as given in Table 2 and L2 is assumed to be infinite in length. Substituting 3 into 1 and 2, we obtain the total binding energy for a single atom with a carbon nanocone as given by Etot atom ¼ ηc tan θ sec θ
Z L 2 L1
zð AF6 þ BF12 Þ dz
ð4Þ
where ηc is the atomic surface density of the carbon nanocone. Following a similar calculation by Lee et al.,29 the total binding energy Etot atom can be evaluated analytically and is given by Etot atom ¼ Aηc J3 þ Bηc J6
ð5Þ
where Jm is defined as
" # z ¼ L2 ðz2 2zZ cos2 θ þ Z2 cos2 θÞ1 m Jm ¼ sin θ 2ðsec2 θÞm 1 ð1 mÞ z¼L Z sin θ cos θ þ 2m 2 2m 1 Z ðsin θÞ
1
z ¼ L2 z ¼ L1
ðcos2 ωÞ
m1
dω
ð6Þ
where L1 and L2 are the extremities of the carbon nanocone previously defined and ω = arctan(z Z cos2 θ/Z sin θ cos θ). As previously described, we now locate a hydrogen sphere at (0,0,Z). The interaction energy for an atom interacting with the hydrogen sphere is derived in Cox et al.28 and is given by " ! πaηh A 1 1 GðFÞ ¼ 2 ðF aÞ4 ðF þ aÞ4 F !# B 1 1 þ ð7Þ 5 ðF aÞ10 ðF þ aÞ10 Figure 4. Binding energy and molecular force for CH4 molecule, two H2 molecules, and four Ne atoms entering single-walled carbon nanocone.
of the above integral L2* and L1* correspond to the maximum and minimum points along the single-walled nanocone axis for which the binding energy remains negative.
III. INTERACTION OF H2, CH4, AND NE WITH A SINGLEWALLED CARBON NANOCONE In this section, we examine the binding energy between H2, CH4, and Ne interacting with a single-walled carbon nanocone for interactions, as shown in Figure 1. We assume that the H2 and CH4 molecule, and Ne atom, are centered along the axis of the single-walled carbon nanocone and with the origin situated at the vertex of the extended carbon nanocone. Therefore we locate a point on the nanocone surface in Cartesian coordinates at (r cos ϕ, r sin ϕ, z), where ϕ is the angle in the xy plane measured anticlockwise from the x axis, and r = z tan θ is the surface radius. Similarly, the coordinates of the H2, CH4 carbon, and Ne atom can be located in Cartesian coordinates at (0,0,Z), where Z is the horizontal location of the H2, CH4 carbon, or Ne atom along the z axis, as shown in Figure 1. The distance F between the center of the H2, CH4 molecule, or Ne atom to any point on the nanocone surface is therefore given by Fatom 2 ¼ z2 tan2 θ þ ðz ZÞ2
ð3Þ
where ηh is the atomic surface density of the hydrogen sphere. Therefore, the total interaction energy can be found by integrating eq 7 over the surface of a carbon nanocone and is given by A B tot Ehsphere ¼ bηh ηc sin θ ðQ4þ Q4 Þ ðQ10þ Q10 Þ 2 5 where Qn((n = 4, 10) are given by " #z ¼ L2 cos2 θ ðF ( aÞð1 nÞ Qn( ¼ ð1 nÞ þ Z cos3 θ
Z z¼L 2
z ¼ L1
dλ n z ¼ L1 ðZ sin θ cosh λ ( aÞ
ð8Þ
where F as a function of z is defined by eq 3.
IV. RESULTS AND DISCUSSION In this section, we present the results showing the binding energy and molecular force for H2, CH4, and Ne entering carbon nanocones of varying apex angle and size. The numerical calculations are performed using the algebraic software package MAPLE, and we give specific results for an apex angle of 112.9°13 and L2 is assumed to be infinite. The numerical values for L1, the corresponding radii R, and the integer number of hexagons K employed in this study are derived from Section II and are as given in Table 2 using 1.421 Å28 for the CC bond length σc. Describing an encapsulation process from the smaller end of the 24531
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Figure 5. Volume for adsorption for a CH4 molecule, two H2 molecules, and four Ne atoms in a single-walled carbon nanocone for two distinct nanocone radii.
nanocone, as shown in Figure 2a, the positive energy profile for the CH4 molecule indicates that the molecule may not pass through the smaller open end of the cone because of the presence of an energy barrier at the cone entrance, whereas the H2 molecules and Ne atoms will be attracted inside the cone because of the negative energy profile, as the energetically most favorable position for the H2 molecules and Ne atoms is inside the cone even though they are slightly repulsed after entering the cone. The molecular force that is calculated by differentiating the total binding energy with respect to the horizontal distance Z is shown in Figure 2b. The molecular force profile shows that the H2 molecules and Ne atoms experience an acceleration on entering the cone resulting in H2 and Ne encapsulation. The gas molecules however, decelerate after entering, as indicated by the negative force, whereas the CH4 molecule experiences a large energy barrier at the entrance of the cone resulting in repulsion. We comment that the initial minimum energy well of 0.0278 eV at the entrance of the cone for a Ne atom during encapsulation in Figure 2a is in agreement with the binding energy value of 0.02073 eV for exohedral adsorption derived from atomistic molecular dynamics simulation by Majidi et al.13 We further comment that although the force plot gives only the slope of the energy plot it does allow for the visualization of phenomenon such as the magnitude of the maximum force, which is more easily obtained than the mere measurement of the gradient of the energy profile. Figures 3 and 4 describe the binding energies and molecular forces as H2, CH4, and Ne are being encapsulated
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inside carbon nanocones of lengths L1 = 3.5345 Å and L1 = 4.7126 Å, respectively. As shown in these Figures, the three gases experience a slight repulsive force just at the entrance of the cones, but they are then attracted inside the cone, as indicated by the positive force gradients in these Figures. After encapsulation, they then decelerate again similar to the motion paths of the H2 molecules and Ne atoms in Figure 2b. The presence of a slight energy barrier coupled to the minimum energy configuration inside the cone suggests that if the attractive force is high enough then the gases would overcome the repulsive force at the cone entrance and would be encapsulated as the energetically most favorable position for the gases is inside the cone, an observation which is consistent with CH4 encapsulation inside carbon nanotubes.3 For a carbon nanocone of length L1 = 3.5345 Å, the average minimum energy is 0.0048 eV for a H2 molecule encapsulated inside the nanocone, 0.0131 eV for a CH4 molecule, and 0.00175 eV for a Ne atom. We comment that encapsulation from the larger open end is in the opposite direction to the encapsulation process from the smaller openend. For example, for a carbon nanocone of length L1 = 3.5345 Å, the gases would accelerate to the entrance of the cone, experience a deceleration by the cone entrance, accelerate again, and then become encapsulated before decelerating. We comment that as the size of the carbon nanocones increases, the minimum binding energy and molecular force decrease, an observation seen in CH4 encapsulation in varying sized carbon nanotubes.3 In Figure 5, we plot the relationship between the volume available for H2, CH4, and Ne adsorption and temperature. It can be seen from the Figure that desorption occurs and increases with an increase in temperature. As shown in Figure 5a, at lower temperatures, the volume available for Ne adsorption decreases quicker than that for H2 and CH4, whose volume available for adsorption starts to decrease at slightly higher temperatures for a cone of radius 5.3289 Å. Similarly, for a cone of radius 7.1052 Å, desorption commences and increases as the temperature increases. This observation is in good agreement with Yu et al.,14 who employ thermal desorption and photoemission to predict gas desorption from carbon nanocones at higher temperatures. We further comment that as might be expected, an increase in the cone radius results in an increase in the volume available for adsorption.
V. CONCLUSIONS With a view to modeling gas storage in carbon nanostructures, we investigate the encapsulation of three different gases inside single-walled carbon nanocones of varying vertex angle and size. We model these systems in terms of the parameters such as the nanocone base radius, the nanocone length, the initial distance between the carbon nanocone and the gas molecules, and the nanocone apex angle. The interaction binding energies and molecular forces for the encapsulation of hydrogen, methane, and neon in carbon nanocones are derived from the 612 Lennard-Jones potential and the continuous approximation, resulting in surface integrals and giving rise to analytical expressions for the total binding energy between either hydrogen, methane, or neon interacting with a single-walled carbon nanocone. Our results suggest that a hydrogen molecule or a neon atom will experience a positive force, resulting in the gas being accelerated inside the cone during encapsulation and then it will come to rest inside the cone. For larger cones, a methane molecule will experience a similar motion during encapsulation, whereas for smaller cones, there is a high-energy barrier at the 24532
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’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT We are grateful to the Australian Research Council for support through the Discovery Project Scheme and the University of Adelaide for the provision of an ASI for OOA. ’ REFERENCES (1) Iijima, S. Nature 1991, 354, 56–58. (2) Adisa, O. O.; Cox, B. J.; Hill, J. M. Micro Nano Lett 2010, 5, 291–295. (3) Adisa, O. O.; Cox, B. J.; Hill, J. M. Physica B 2011, 406, 88–93. (4) Zhao, J.; Buldum, A.; Han, J.; Lu, J. P. Nanotechnology 2002, 13, 195–200. (5) Albesa, A. G.; Fertitta, E. A.; Vincente, J. L. Langmuir 2010, 26, 786–795. (6) Thornton, A. W.; Nairn, K. M.; Hill, J. M.; Hill, A. J.; Hill, M. R. J. Am. Chem. Soc. 2009, 131, 101–108. (7) Severin, E. S.; Tildesley, D. J. Mol. Phys. 1980, 41, 1401–1418. (8) Adisa, O. O.; Cox, B. J.; Hill, J. M. Eur. Phys. J. B 2011, 79, 177–184. (9) Adisa, O. O.; Cox, B. J.; Hill, J. M. Carbon 2011, 49, 3212–3218. (10) Suarez-Martinez, I.; Monthioux, M.; Ewels, C. P. J. Nanosci. Nanotechnol. 2009, 9, 1–5. (11) Yudasaka, M.; Iijima, S.; Crespi, V. H. Top. Appl. Phys. 2008, 111, 605–629. (12) Shenderova, O. A.; Lawson, B. L.; Areshkin, D.; Brenner, D. W. Nanotechnology 2001, 12, 191–197. (13) Majidi, R.; Tabrizi, K. G. Physica B 2010, 405, 2144–2148. (14) Yu, X.; Tverdal, M.; Raaen, S.; Helgesen, G.; Knudsen, K. D. Appl. Surf. Sci. 2008, 255, 1906–1910. (15) Baowan, D.; Hill, J. M. J. Math. Chem. 2008, 43, 1489–1504. (16) U.S Energy Policy Act (EPAct); U.S. Department of Energy, 1992. (17) Krungleviciute, V.; Mercedes Calbi, M.; Wagner, J. A.; Migone, A. D. J. Phys. Chem. C 2008, 112, 5742–5746. (18) Murata, K.; Miyawaki, J.; Yudasaka, M.; Iijima, S.; Kaneko, K. Carbon 2005, 43, 2817–2833. 24533
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