Theoretical Study of the Contribution of Physisorption to the Low

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Theoretical Study of the Contribution of Physisorption to the Low-Pressure Adsorption of Hydrogen on Carbon Nanotubes E Ä ric Me´lanç on* and Pierre Beńard Institut de recherche sur l’hydroge` ne, Universite´ du Que´ bec a` Trois-Rivie` res, 3351 Boulevard des Forges, C. P. 500, Trois-Rivie` res, Que´ bec, Canada, G9A 5H7 Received December 23, 2003. In Final Form: May 28, 2004 To investigate the contribution of geometry on the adsorption process, we present a theoretical study of the low-pressure physisorption of hydrogen on isolated nanotubes and nanotube bundles through the second virial coefficient, BAS, computed classically with an uncorrugated adsorption potential. The optimal nanotube bundle geometry at low pressure for a Lennard-Jones adsorption potential is obtained by studying the second virial coefficient, BAS, for variable radius or bundle lattice constant. The most favorable bundle adsorption sites at low pressures and temperatures are identified for typical bundle structures and the relative contribution of interstitial sites relative to other sites is discussed as a function of temperature and pressure. The Boyle temperature behavior for the BAS virial coefficient is also discussed as a function of radius for isolated nanotubes. For a given nanostructure, the maximum pressure of applicability of the BAS approach, below which the adsorption isotherm is linear, is estimated as a criterion which depends on temperature.

1. Introduction The quantity of hydrogen adsorbed on carbon nanotubes has been the subject of intense experimental and theoretical study and the object of significant controversy.1-3 The available experimental data suggest that the storage capacity could vary between 0 and 10 wt%, depending on the pressure and temperature regimes3 and whether the nanotubes are doped with metals or not. The exact mechanisms that could explain the large adsorbed quantities are still not clearly understood, although physisorption is certain to play an important role on the adsorption of hydrogen on pure or undoped carbon singlewalled nanotubes (SWNTs). SWNTs are tubular carbon nanostructures found in soots produced by laser evaporation of a graphite target doped with cobalt and nickel4 or by the evaporation of cobalt-doped graphite electrodes in an electric arc discharge.1 They have a diameter on the order of 1 nm and can reach hundreds of micrometers in length. During their synthesis, they tend to self-organize in bundles of hundreds of units.4 Quantum chemistry calculations5 indicate that the expected well depth of the interaction potential between molecular hydrogen and SWNTs is typical of physisorption problems, even though much stronger bonding to the SWNT, as well as a weakening of the C-C bonds, is expected when hydrogen atoms interact with SWNTs.5 The low-pressure limit of the adsorption isotherm is best studied through the second virial coefficient of * Author to whom correspondence should be addressed. Tel: 1-819-376-5108. Fax: 1-819-376-5164. E-mail: eric_melancon@ uqtr.ca. (1) Dillon, A. C.; Jones, K. M.; et al. Nature 1997, 386, 377. (2) Dagani, R. Chem. Eng. News 2002, 80, 25. http://pubs.acs.org/ cen/coverstory/8002/8002hydrogen.html. (3) Simard, B.; Deńommeé, S.; et al. Recent Advances In Carbon Nanotubes Technologies At The National Research Council. 11th Canadian Hydrogen Conference Proceedings, Victoria, BC, June 2001; McLean, G. F., Ed.; Canadian Hydrogen Association: Victoria, BC. (4) Thess, A.; Lee, R.; Nikolaev, P.; et al. Science 1996, 273, 483. (5) Arellano, J. S.; Molina, L. M.; Rubio, A.; et al. J. Chem. Phys. 2002, 117, 2281.

adsorption, BAS, which describes the behavior of the adsorption isotherm in the limit of Henry’s law. A detailed study of BAS can yield valuable information on the adsorbate-adsorbent interactions and the effect of geometry on the sorption properties, although the BAS does not provide information on the interactions between adsorbate molecules and, as such, does not constitute a measure of the storage capacity of the adsorbent. In this work, the classical expression of BAS was employed, thus neglecting the quantum effects arising at low temperatures for light gases. In a first paper,6 Stan and Cole considered the adsorption of rare gases on SWNTs in the limit of Henry’s law through a calculation of the BAS as a function of the radius of the nanotubes using a continuous cylindrical adsorption potential. They compared the adsorbed density of Ne atoms on a SWNT to that on a planar graphene sheet of equivalent surface area as a function of the radius of the nanotubes at a temperature of 10 K. They found that the adsorption inside the SWNT is much greater than that for the graphene sheet, the maximum occurring at the largest potential well depth (Rmin ) 1.086 σ). They showed that the isosteric heat of adsorption qst corresponds to the adsorption potential well depth, D, inside the tube (and to the opposite of the activation energy EA used in our work) for low temperatures and low pressures:

qst ≈ D + kT/2

(1)

They considered interactions between adatoms in the context of a 1D approximation (valid for narrow SWNTs). In a second paper,7 they considered the effect of the atomicity for a zigzag SWNT (13, 0) of radius 5.09 Å by plotting the hydrogen adsorption potential along a radial line crossing the center of a hexagon or a carbon atom at the surface of the SWNT. The small difference with the potential derived from a continuous distribution of the (6) Stan, G.; Cole, M. W. Surf. Sci. 1998, 395, 280. (7) Stan, G.; Cole, M. W. J. Low Temp. Phys. 1998, 110, 539.

10.1021/la036446l CCC: $27.50 © 2004 American Chemical Society Published on Web 07/30/2004

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carbon atoms on the surface warranted the use of the cylindrical potential obtained in ref 6. They also found that the quantum effects are negligible for T > 50 K by comparing the cylindrical potential of the (13, 0) SWNT with the Feynman effective potential, used to compute a first quantum correction on BAS. In a subsequent article,8 they studied the adsorption of He and Ne inside the interstitial channels present between the SWNTs of the bundles. They found that these atoms are of ideal size for physisorption in the interstitial channels of a bundle of SWNTs of radius 6.9 Å separated by 3.2 Å, since this nearly corresponds to the SWNT lattice spacing leading to the strongest attractive potential in the interstitial channels. When interactions of atoms between neighboring interstitial channels were considered, they showed the existence of exotic anisotropic condensed phases. In ref 9, they identified three different types of adsorption sites for SWNT bundles: the tube sites, located within the SWNTs, the interstitial sites, located inside the interstitial channels within the bundle, and the peripheral sites, located in the peripheral grooves of the external surface of the bundle. They looked at whether a particular gas adsorbs significantly in the nanotubes and/or in the interstitial channels of the bundle. The bundle geometry (lattice constant 17 Å, tube radius 6.9 Å) was assumed to be unaffected by the adsorption process, and a threshold one-dimensional density, corresponding to a mean spacing of ∼30 Å between adsorbate molecules, was used as a criterion for significant adsorption. They found that small atoms and molecules adsorb significantly in SWNTs and interstitial sites, while larger particles only adsorb within the tubes. They also computed the ratio of the amount adsorbed inside an isolated SWNT to that in an interstitial site at 77 K in the low-coverage regime when the molecular size σgg is varied. The groundstate energies and adsorption-potential minima inside the three different adsorption sites of their particular bundle structure were also presented for several gases. The object of this paper is to calculate, using the Stan and Cole analysis and the uncorrugated Lennard-Jones (LJ) interaction potential, the behavior of the classical second virial coefficient for the adsorption of hydrogen on opened and closed single-walled nanotubes, both isolated and in bundles, as a function of temperature, radius, and lattice constant of the SWNT bundle. For the low temperature of 77 K studied here, we associate the various structures present in the BAS curves with the behavior of the adsorption potential. The most favorable adsorption site among the three aforementioned sites is also identified for four different bundle structures. The behavior of the Boyle temperature of the BAS coefficient is also studied for the adsorption on isolated SWNTs, and an estimation of the maximum pressure of applicability of Henry’s law is obtained and compared with classical grand canonical Monte Carlo (GCMC) simulations.

Na ) N - N0 ) BAS(p/kT) + CAAS(p/kT)2 + ... (2)

2. Virial Expansion of the Excess Adsorption Isotherm The excess adsorption isotherm is given, in the lowpressure limit, by the virial expansion:10-12

where

BAS )

∫v {exp[-V(rb)/kT] - 1}drb

(3)

and12

CAAS )

∫∫vexp[- (V(rb1) + V(rb2))/kT]f12drb1drb2 1 {exp[-V(r b1)/kT] + 1}f12dr b1dr b2 2∫∫v

(4)

CAAS has been corrected for the nonideality of the bulk gas far from the adsorbent surface. Intermolecular interactions are included through the term:

[

f12 ) exp -

]

b2 - b r 1|) ugg(|r -1 kT

(5)

v is the accessible volume to the adsorbate molecules in the adsorption container, V(r b) is the adsorption potential and ugg(r12) is the adsorbate-adsorbate interaction potential, assumed as a LJ interaction in this work. BAS and CAAS are called the second and third virial coefficient for adsorption, respectively. N is the total number of adsorbate molecules in v with the adsorbent present, while N0 is the number of molecules which would be present in the same volume, v, if no adsorption occurred. BAS depends only on the interaction between a single adsorbate molecule and the surface of the adsorbent. In this work, we study the low-pressure adsorption through BAS, neglecting any interactions between adsorbate molecules. Thus, we are not accounting for the adsorbate-adsorbate interactions ugg(r12) which are considered in the third virial coefficient CAAS. To obtain an estimate of the maximum pressure pmax where Henry’s law is valid so that the isotherm can be described by the virial coefficient BAS, we derived an approximation for pmax by making a rough estimate of the pressure where the quadratic term in (p/kT) of eq 2, arising from the interaction between two adsorbate molecules, becomes comparable to the linear term in (p/kT). The maximum pressure of validity of Henry’s law should then be a fraction of:

pmax (T) ) kT|BAS(T)/CAAS(T)|

(6)

To obtain a simpler expression, the adsorption potential was assumed to be smooth enough to be considered constant on the range of the adsorbate-adsorbate interb2 - b r1|) * 0. CAAS can then be written: action where f12(|r

CAAS ≈

∫vdrb1 exp[-V(rb1)/kT]∫v drb12 exp[-V(rb1)/kT]f12(r12) 1 dr b {exp[-V(r b1)/kT] + 1}∫vdr b12 f12(r12) (7) 2∫v 1

Using the bulk gas second virial coefficient:11 (8) Stan, G.; Crespi, V. H.; Cole, M. W.; Boninsegni, M. J. Low Temp. Phys. 1998, 113, 447. (9) Stan, G.; Bojan, M. J.; Curtarolo, S.; et al. Phys. Rev. B 2000, 62, 2173. (10) Steele, W. A. In The Solid-Gas Interface; Flood, E. A., Ed.; Marcel Dekker: New York, 1967; Vol. 1. (11) Hill, T. L. An Introduction to Statistical Thermodynamics; Dover: New York, 1986. (12) Hill, T. L. Statistical Mechanics: Principles and Selected Applications; Dover: New York, 1987.

B2(T) ) -

∫

1 dr b f (r ) ) -2π 2 v 12 12 12

∫0∞drr2 f12(r)

(8)

which vanishes at kT/gg ) 3.418 (113.8 K for H2) in the case of a LJ 12-6 intermolecular interaction, the maximum pressure, pmax, of applicability of Henry’s law becomes:

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Langmuir, Vol. 20, No. 18, 2004

pmax (T) ≈

|

|

BAS(T) kT 1 2 B2(T) BAS(T/2)

(9)

since BAS is much smaller than BAS(T/2) in the denominator. The behavior of pmax with respect to temperature for a typical SWNT bundle will be discussed in section 5.

where r* ) r/σ, R* ) R/σ, and V* (r*,R*) ) V(r*σ,R*σ)/′s are the reduced distance from the axis, the reduced nanotube radius, and the reduced potential, respectively. For hydrogen adsorption on carbon nanotubes ′s ) πθσ2 ) 371 K ) 3.08 kJ/mol, while for helium adsorption, its value is 160 K. The Mn(x) functions are defined by6

3. Adsorption Potential of SWNTs and SWNT Bundles In this study, as in most adsorption studies on graphite or carbon nanotubes, the adsorption potential of an adsorbate molecule is taken as a sum of effective pairwise isotropic LJ 12-6 interactions between the adsorbate molecule and each carbon atom forming the adsorbent surface:9

V(r b) )

∑j

u(|r b-R B j|)

(10)

Mn(x) )

∫0π(1 + x2 - dφ 2 x cos φ)n/2

which, for the sake of reducing numerical computation time, are represented by the following truncated series, which all give an error less than 8.60 × 10-6%, based on the work of Murata et al.:16

Mn(x) )

π (1 -

7

∑ anix2i (n ) 5, 0 e x e 0.77;

x2)n-1i ) 0

n ) 11, 0 e x e 0.9999) (15)

where

u(x) ) 4{(σ/x)12 - (σ/x)6}

(11)

Mn(x) )

πxn-2 (1 -

7

∑ anix-2i (n ) 5, x g 1.30;

x2)n-1i ) 0

n ) 11, x g 1.0001) (16)

for the adsorbate-carbon interaction while

ugg(r12) ) 4gg{(σgg/r12)12 - (σgg/r12)6}

(12)

for the adsorbate-adsorbate interaction (neglected in our r and R B j are the position of the adsorbate BAS study). b molecule and of the carbon atom, respectively. The LJ parameters of the hydrogen-hydrogen interaction are taken from ref 13 as gg/k ) 33.3 K and σgg ) 2.97 Å, and the parameters for the C-H2 interaction are obtained from the same reference by using the Lorentz-Berthelot combining rules: /k ) 30.5 K and σ ) 3.19 Å. While for helium, the parameters recommended by Steele14 were used for the C-He interaction: /k ) 15 K and σ ) 2.98 Å. We computed numerically the minimum of the adsorption potential V(r), given by eq 10, for a hydrogen molecule inside a discrete nanotube (12, 7) of radius 6.515 Å and of length 588.9 Å to check when we can use a continuous cylinder having a uniform surface density, θ ) 0.382 Å-2, of carbon atoms to represent the nanotubes. The nanotube’s carbon atoms were generated by using the program given in ref 15, and the value of the minimum was computed as we move along the nanotube axis. We found that the minimum varies between -692.2 and -679.4 K for this discrete nanotube, while its value is -681.6 K for the corresponding continuous nanotube (from eq 13). Thus, for temperatures of practical interest (>77.4 K), the kinetic energy of the adsorbate molecules will be much greater than the energy barrier between neighbor adsorption sites on the tube surface (12.8 K), and the adsorption will be delocalized, thus justifying the use of continuous nanotubes in following computations. For continuous nanotubes, the adsorption potential (10) becomes:6

V*(r*,R*) ) 21 1 3 32 R*

[ ( )

10

M11(r*/R*) -

(14)

(R*1 ) M (r*/R*)] (13) 4

5

(13) Rzepka, M.; Lamp, P.; de la Casa-Lillo, M. A. J. Phys. Chem. B 1998, 102, 10894. (14) Neimark, A. V.; Ravikovitch, P. I. Langmuir 1997, 13, 5148. (15) Mathematica notebook used for plotting nanotubes: Brandbyge, M. Atomic and Electronic Structure of Carbon Nanotubes; Mikroelectronik Centret (MIC), Danish Technical University. http:// library.wolfram.com/infocenter/MathSource/384/.

The coefficients, ani, of this series are given in Table 1. The reduced adsorption potential of a bundle of Ntubes nanotubes of reduced radius R* centered on an hexagonal lattice visible in Figure 1 is obtained by summing up the contributions (13) from each nanotube: Ntubes / (r b*, d*, R*, Ntubes) ) Vbundle

∑ V * (|rb* - RB /pq|, R*)

nodes (p,q)

(17)

B /pq are the positions of the centers of the SWNTs where R on the bundle’s hexagonal lattice of reduced lattice constant d*. Figure 1 shows the bundle’s adsorption sites (a) and the reduced adsorption potential surface (b) for hydrogen adsorbed on a bundle of 19 SWNTs. The SWNTs have a radius of 6.5 Å, and the spacing between their walls is 3.7 Å. The reduced lengths R* ) 2.038 and d* ) 5.235 were used for plotting, and the reduced equipotentials values of Figure 1(a) are (-2.25, -1.75, -1.25, -0.75, -0.25, 0.25, 10.00). The adsorption potential well depth inside the different adsorption sites will be equal to the opposite of the activation energy, EA, of the adsorption process on bundles. We thus give the minima of potential Vmin for hydrogen inside the different adsorption sites shown in Figure 1(a) for bundles made of seven SWNTs in Table 2b,c. The values found were nearly independent of Ntubes. Our values of the minimum inside an isolated SWNT of radius 6.5 Å (see Table 2a) and inside the peripheral sites of the corresponding two bundles of Table 2b are in good agreement with the corresponding reduced values (-1.727 and -2.022) deduced from ref 9 for a bundle having SWNTs of reduced radius 2.14 and lattice constant 5.27. However, our values for the interstitial sites do not agree with their value (-2.072) and show a great sensitivity to the size of the bundle. We also see that even the most attractive adsorption potential, -13.78 kJ/mol, obtained for tubes of optimal radius 3.46 Å separated by 3.15 Å (see Table 2c) is inadequate to explain the large heat of adsorption (16) Murata, K.; Kaneko, K.; Steele, W. A.; et al. J. Phys. Chem. B. 2001, 105, 10210.


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Table 1. Coefficients of the Series Used to Compute the Functions Mn(x) n

an0

an1

an2

an3

an4

an5

an6

an7

5 11

1 1

9/4 81/4

9/64 3969/64

1/256 11025/256

9/16384 99225/16384

9/65536 3969/65536

49/1048576 441/1048576

81/4194304 81/4194304

Table 2. Values of the Minima of the Adsorption Potential Vmin (in kJ/mol) for Hydrogen Inside the Different Adsorption Sites of an Isolated SWNT (a) and for a Bundle of Seven SWNTs of Radius 6.501 Å (2.038) (b), or of Radius Rmin (c)a (a) Isolated SWNT Vmin(kJ/mol) (Vmin*) R(Å) (R*)

SWNT interior

SWNT exterior

6.501 (2.038) 3.464 (1.086)

-5.450 (-1.769) -12.52 (-4.065)

-3.080 (-1.000) -2.793 (-0.9067)

(b) Bundle of 7 SWNTs (radius ) 6.501 Å) Vmin(kJ/mol) (Vmin*) d(Å) (d*)

tube sites

interstitial sites peripheral sites

16.70 (5.235) -5.843 (-1.897) -9.249 (-3.002) -6.180 (-2.006) 16.15 (5.063) -5.980 (-1.941) -4.452 (-1.445) -6.181 (-2.006) (c) Bundle of 7 SWNTs (radius ) Rmin) Vmin(kJ/mol) (Vmin*) d(Å) (d*)

tube sites

interstitial sites peripheral sites

10.63 (3.332) -13.42 (-4.355) 3.810 (1.237) -5.650 (-1.834) 10.08 (3.159) -13.78 (-4.473) 70.58 (22.91) -5.657 (-1.836) a The reduced values (by σ or ′ ) are given in parentheses and s Rmin ) 3.464 Å (1.086) is the radius of the isolated SWNT having the most attractive interior.

Figure 1. Adsorption sites (a) and adsorption potential surface (b) of a bundle with 19 SWNTs (R* ) 2.038; d* ) 5.235).

of 19.6 kJ/mol found by Dillon et al. for nonoptimal tubes of radius ∼6 Å.1 4. Adsorption on Isolated Single-Walled Nanotubes The reduced second virial coefficient B/AS, independent of the specific nature of adsorbent and adsorbate (independent of , σ, and θ), for an isolated SWNT of length L and of reduced radius R* is given by:

B/AS (T*, R*) ) BAS ∞ 1 ) 2π a* exp - V*(r*, R*) - 1 r* dr* (18) 2 T* Lσ

∫{

(

) }

where a* ) 0 for an opened SWNT and a* ) R* for a closed SWNT. T* ) kT/′s is the reduced temperature of the adsorbate. Figure 2 shows the behavior of B/AS of nanotubes as a function of the reduced radius R*. When the radius is too narrow (R* < 0.93), only the external surface contribute to adsorption. At low temperature (T* ) 0.2089; H2, 77.4

Figure 2. B/AS behavior of an isolated SWNT with respect to radius. The SWNT reduced radii corresponding to the C20 and C60 fullerene cages are indicated by the vertical lines in the case of hydrogen adsorption.

K) B/AS of opened SWNT reaches its maximum value near the maximum potential well depth inside a SWNT (R*min ) 1.086). B/AS is then 3 orders of magnitude greater than the B/AS for the corresponding planar graphene sheet. At larger radii, curves for opened and closed SWNTs get closer, indicating that SWNTs behave like planar graphene sheets. It is worthwhile to note here that the theoretically smallest possible nanotube radius in Figures 2, 3, and 7 (computed for continuous cylinders) is 2.0 Å. This corre-

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Figure 3. Reduced Boyle temperature of an isolated SWNT with respect to radius. The coordinates of the extrema are also given in the form (R*, T*B). The SWNT reduced radii corresponding to the C20 and C60 fullerene cages are indicated by the vertical lines for the case of helium.

sponds to a zigzag (5, 0) nanotube capped with half cages of the smallest possible fullerene C20.17 However, until now, these small tubes have only been observed as the innermost shell of multiwalled nanotubes.17,18 The smallest SWNT observed for now have a radius of 3.5 Å, the same as that of a C60 buckyball structure.19 Thus, the nanotube’s reduced radii corresponding to C20 and C60 are respectively indicated in these figures by the full and dashed vertical lines, for the cases of hydrogen or helium adsorption. As discussed in detail in ref 14, the experimental excess adsorption isotherms are given by the difference between the experimentally determined total number of adsorbate molecules in the adsorption container and the number of molecules which would occupy the available gas volume, v, with the uniform density Fg of the bulk gas phase. The available volume is usually determined by means of the helium calibration procedure. It is the volume that the total number of helium atoms NHe, measured in the adsorption container, would occupy at the density of the bulk helium gas, Fg,He, that is, vHe ) NHe/Fg,He. In general vHe * v since the density of helium is inhomogeneous due to its adsorption by the adsorbing surface. Thus, helium calibration must be performed under conditions such that there is no excess adsorption in order to give an adequate estimate of the available gas volume, v. From eq 2, the calibration should therefore be performed at the Boyle temperature TB, for which BAS ) 0. Figure 3 shows the dependence of the reduced Boyle temperature T*B, with respect to SWNT radius. For a given radius, B/AS > 0 for T* < T*B and B/AS < 0 for T* > T*B. At temperatures below T*B, B/AS is positive, owing to the predominant effect of the attractive part of V(r b) on the integral of B/AS given by eq 3. On the other hand, for high temperatures, the attractive part of V(r b) becomes greatly attenuated while the hard core repulsive part continues to contribute significantly to the B/AS integral, thus making B/AS < 0. This phenomenon is shown in Figure 4 for the case of adsorption inside a slit pore of reduced width H* ) 4.0, formed by two planar graphene sheets. Thus, the Boyle temperature, in the context of the virial expansion of the excess adsorption isotherm, indicates the maximum temperature for which there is gain (17) Peng, H. Y.; Wang, N.; Zheng, Y. F.; et al. Appl. Phys. Lett. 2000, 77, 2831. (18) Sun, L. F.; Xie, S. S.; Liu, W.; et al. Nature 2000, 403, 384. (19) Ajayan, P. M.; Iijima, S. Nature 1992, 358, 23.

/ Figure 4. Behavior of the integrand f ) exp(-Vslit /T*) - 1 of / BAS for a slit pore of reduced width H* ) 4.0 with respect to temperature.

over compression (Na > 0) in the low-pressure limit. For opened and closed SWNT, this maximum temperature is 1.214 and 1.240 (He, 194 and 198 K; H2, 450 and 460 K), respectively. For a T*B between 0.9409 and 1.089, there are three radii giving B/AS ) 0 for an opened SWNT. This is also visible in Figure 2 on the T* ) 1.00 (H2, 371 K) curve. The behavior of the Boyle temperature curves is shown in Figure 3 for the case of opened and closed SWNTs, along with the Boyle temperature arising from the potential inside the SWNT. For SWNT of radii smaller than R0* ) 0.9322, the adsorption potential inside the tube is repulsive. The integrand f ) exp(-V* (r b)/T*) - 1 of the expression for B/AS is therefore negative everywhere inside the tube, contributing a negative value of B/AS. The reduced Boyle temperature, T*B, that could be deduced from the potential inside a SWNT would be undefined for radii smaller than R0*. However, for opened and closed SWNTs, T*B remains defined even for very small radii. This is due to the contribution of the external potential of the SWNT, which always has an attractive part. The latter can overcome the repulsive contributions of the overall B/AS integral at sufficiently low temperatures. For very small radii (and unphysical nanotubes), the curves of opened and closed SWNTs merge since, in this case, only the external surface contributes to adsorption. As the SWNT radius becomes larger, the Boyle temperature tends toward its limiting value T*B ) 1.178 (He, 188 K) for a planar graphene sheet. The overall behavior of T*B for the opened SWNT is a composite of the behavior for the interior volume and the exterior surface of the SWNT. For narrow radii, the opened SWNT curve merges with the closed SWNT curve, while for larger radii, it follows the SWNT interior curve. The transition between the two regimes occurs for R* between 0.3 and 1.0. 5. Adsorption on Bundles of Single-Walled Nanotubes The reduced second virial coefficient B/AS for a bundle of Ntubes SWNTs of radius R* having their centers separated by a distance d* is given by

B/AS(T*, d*, R*) )

∫A

/ acc

BAS Lσ2

{exp(- T*1 V

)

/ b*, bundle(r

) }

d*, R*, Ntubes) - 1 dA* (19)


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Table 3. Dominant Hydrogen Adsorption Sites of an Isolated SWNT (a) and for Bundles of 25 SWNTs of Radius 6.501 Å (2.038) (b), or of Radius 3.464 Å (1.086) (c) (a) Isolated SWNT dominant adsorption site R(Å) (R*)

opened SWNT

closed SWNT

6.501 (2.038) 3.464 (1.086)

tube interior tube interior

tube exterior tube exterior

(b) Bundle of 25 SWNTs (radius ) 6.501 Å) dominant adsorption site d(Å) (d*)

bundle opened SWNTs

16.70 (5.235) interstitial sites 16.15 (5.063) tube sites peripheral sites

bundle closed SWNTs interstitial sites peripheral sites

Figure 6. B/AS behavior with respect to lattice constant (for H2, R ) 6.5 Å; T ) 77.4 K).

(c) Bundle of 25 SWNTs (radius ) 3.464 Å) dominant adsorption site d(Å) (d*)

bundle opened SWNTs

bundle closed SWNTs

10.63 (3.332) 10.08 (3.159)

tube sites tube sites

peripheral sites peripheral sites

Figure 7. B/AS behavior with respect to SWNTs radius (for H2, d ) 16.7 Å; T ) 77.4 K). The SWNT reduced radii corresponding to the C20 and C60 fullerene cages are indicated by the vertical lines in the case of hydrogen adsorption.

Figure 5. B/AS behavior with respect to temperature’s inverse (R* ) 2.038; d* ) 5.063).

where A/acc represents the projection of the accessible volume to the adsorbate molecules on a plane perpendicular to the nanotubes composing the bundle. The bidimensional numerical integration of eq 19 has been made with two nested recursive trapezoidal rules20 on a circular region including the whole bundle. By comparing the different activation energies, E*A, obtained from the slope of the linear part of the graphs of ln B/AS versus T*-1 with the adsorption sites potential well depths given in Table 2b,c, we can find which sites dominate the physisorption of hydrogen in bundles at low temperatures. The results are shown in Table 3 for isolated SWNTs and different bundle structures. Figure 5 shows the determination of activation energies for SWNTs of reduced radius 2.038 (6.5 Å for H2) both isolated and in bundles having a reduced lattice constant of 5.063 (16.15 Å for H2). The relative discrepancies, ∆, of E*A with respect to the potential minimum inside the corresponding adsorption sites of the isolated nanotube (1 SWNT) or nanotube bundle are also given. (20) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; et al. Numerical Recipes in C, the art of scientific computing, 2nd ed.; Cambridge University Press: Cambridge, 1992.

By comparing the corresponding carbon nanostructures of Tables 2 and 3, we see that our computations predict that the most attractive site available to the hydrogen molecules will also adsorb the most in the limit of low pressure and temperature physisorption. This is in agreement with the theoretical results of Stan et al. obtained in ref 9. The interstitial site for their bundle having a lattice constant of 17 Å and a tube radius of 6.9 Å is the most attractive one (see section 2) and also gives rise to significant hydrogen adsorption. The corresponding one-dimensional mean spacing between hydrogen molecules is then smaller than 32.3 Å inside the interstitial channels and within the tubes for a pressure of 1.81 × 10-4 atm and a temperature of 37 K. Figure 6 shows the behavior of B/AS as a function of the distance between SWNTs forming the bundle. The B/AS of the opened bundle (black curve) exhibits a sharp maximum close to the origin. This structure is due to the outer potential well of the neighboring nanotubes, which deepens the potential inside the nanotubes. The peak at d* ) 5.31 is due to the merging of the potential minima at the center of the interstices between the tubes. As the tubes move away from each other, secondary minima present in the interstices will begin to disappear at d* ) 6.00, thus creating the small peak in the B/AS curve. For still higher d*, the second virial coefficient stabilizes to its limiting value for 25 independent nanotubes, which is higher for opened SWNTs since the tube internal sites are more attractive than their external surface (see Table 2a). Figure 7, compares the B/AS of isolated SWNTs (dashed curves) with the B/AS of the bundles (solid curves). When the SWNTs are too narrow (R* < 0.97), the internal tubes sites are repulsive and only the external surface contributes to adsorption. The peak at R* ) 1.12 for opened SWNTs and bundle is reached when the potential well inside the nanotubes reaches its maximum depth. The second peak, at R* ) 1.97, is due to the merging of the

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interstitial minima for the particular lattice constant used here (d* ) 5.235). The increase (R* > 2.13) observed for the opened bundle can be attributed the effect of the neighboring tubes, which deepens the well depth inside a given nanotube. The knee observed for the closed bundle at R* ) 1.65 appears when the radius is large enough to permit the formation of secondary potential minima in the interstices. The B/AS of independent SWNTs asymptotically reaches a limiting value as the radius increases to infinity, where the SWNTs behave like planar graphene sheets. For low temperatures, we see from Figures 6 and 7 that near interstitial maxima (d* ) 5.31 and R* ) 1.97, respectively) the attractive interstitial sites dominate the adsorption in the low temperature and pressure limit since curves for bundles of opened and closed SWNTs merge together despite the presence of the internal tube sites and peripheral sites. However, at higher temperatures, interstitial sites are expected to contribute less because the number of molecules adsorbed inside the tubes and inside the peripheral sites will become comparable to the number adsorbed within an interstice on a temperature scale determined by their energy difference: 410 and 370 K for hydrogen within the tubes and peripheral sites, respectively (see Table 2b). Also, for larger pressures, the repulsive interactions between adsorbate molecules, which are not accounted for in the BAS analysis, will favor the larger sites located inside the tubes and in the peripheral groove sites since these interactions introduce a size effect, σgg, which limits the number of molecules which we can store in a narrow volume. The quantum effects, which were not included in our study, are also expected to decrease interstitial adsorption significantly even at temperatures as high as 298 K where the bulk hydrogen is classical.21 In ref 21, Wang and Johnson showed that for a bundle of (9, 9) opened nanotubes (radius 6.1 Å) with an inter-wall spacing of 3.2 Å, the total amount of adsorbed hydrogen within the interstices is 40% higher in the classical simulations than in quantum simulations at 50 atm due to the fact that the very narrow interstices have a high zero-point energy.21 The maximum pressure, pmax, where Henry’s law holds for hydrogen adsorbed on a bundle of opened SWNTs of radius 6.5 Å with a lattice spacing of 16.15 Å has been estimated from eq 9 as a function of temperature. For temperatures lower than the Boyle temperature, TB, of the BAS adsorption virial coefficient, the natural logarithm of BAS is linearly dependent on the inverse temperature (Figure 5), with a slope given by the adsorption well depth, D, of the most attractive adsorption site. From Table 2b, this D is equal to 743.3 K for hydrogen. The BAS ratio of eq 9 then reduces to e-D/kT. Figure 8 shows the dependence of pmax on temperature. As expected, the isotherm is linear for higher pressures as the temperature increases, with a local peak in the pmax curve corresponding to the bulk gas Boyle temperature (B2(T) ) 0). Close to the Boyle gas temperature, the expression for pmax is expected to fail due to the approximation used to calculate CAAS. For a fixed temperature, pmax decreases exponentially as D increases. Figure 9 compares the absolute hydrogen adsorption isotherms computed from BAS with classical GCMC simulations for a bundle of seven opened SWNTs of radius 6.5 Å, having their centers separated by 16.15 Å. The bundle was contained in a rectangular volume, v, whose lateral dimensions are 100 Å and length along the bundle’s tubes is 70.9 Å. The BAS isotherms are in excellent (21) Wang, Q.; Johnson, J. K. J. Chem. Phys. 1999, 110, 577.


Figure 8. Maximum pressure of validity of Henry’s law of adsorption with respect to temperature for hydrogen adsorbed on an opened SWNT bundle (R ) 6.5 Å; d ) 16.15 Å).

Figure 9. Comparison of absolute hydrogen adsorption isotherms for a bundle of seven opened SWNTs, computed with BAS virial coefficient and GCMC simulations at a temperature of 77 K (a) and 300 K (b). (R ) 6.5 Å; d ) 16.15 Å).

agreement with GCMC simulations at low pressures for the two temperatures studied (77 and 300 K). We see that the maximum pressure pmax of validity of Henry’s law computed from eq 9, indicated by the vertical dashed line, gives in these cases a good criteria for estimating the pressure for which the linear isotherm departs from GCMC isotherms, which includes the LJ 12-6 hydrogenhydrogen interactions. 6. Conclusions We have studied the contribution of physisorption of hydrogen on isolated SWNTs and SWNT bundles in the low-pressure limit by using the second virial coefficient, BAS, where any interactions between hydrogen molecules are negligible. The potential well depth inside the three


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different bundle’s adsorption sites was obtained from an uncorrugated LJ interaction. At low temperatures, the various structures present in the curves of BAS as a function of radius or lattice spacing could be associated with specific properties of the potential behavior. It was also possible to identify the most favorable bundle’s adsorption sites by comparing the activation energy with the potential well depth of the different sites. The study of the Boyle temperature for isolated SWNTs gives us the maximum temperature (450 and 460 K for opened and closed SWNT, respectively) where there is an advantage of using isolated SWNTs instead of compressed hydrogen. It is possible to obtain an adequate estimate of the available gas volume in adsorption experiments if the helium calibration procedure is performed at the Boyle temperature (190 K) determined for isolated SWNTs of radius 6.5 Å. The linear

absolute adsorption isotherms deduced from BAS have been shown to be in excellent agreement with classical GCMC simulations at low pressures for a typical SWNT bundle at two temperatures of interest (77 and 300 K). A criterion has also been established to estimate the maximum pressure of validity for the Henry’s law of adsorption. Acknowledgment. The authors thank Philippe Lachance for doing low-pressure GCMC simulations on SWNT bundles. The support of the Fonds que´bećois de la recherche sur la nature et les technologies, of NSERC, of the Ministe`re des ressources naturelles du Que´bec, of NRCAN, and of the Auto 21 Center of Excellence is greatly appreciated. LA036446L

Theoretical Study of the Contribution of Physisorption to the Low

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