Characterization of a Sample of Single-Walled Carbon Nanotube

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Characterization of a Sample of Single-Walled Carbon Nanotube Array by Nitrogen Adsorption Isotherm and Density Functional Theory Xianren Zhang,† Wenchuan Wang,*,† Jianfeng Chen,‡ and Zhigang Shen‡ Lab of Molecular and Materials Simulation, College of Chemical Engineering, and Research Center of the Ministry of Education for High Gravity Engineering and Technology, Beijing University of Chemical Technology, Beijing 100029, China Received November 28, 2002. In Final Form: May 21, 2003 In this paper the experimental adsorption isotherm and density functional theory (DFT) are used to characterize a sample of single-walled carbon nanotube (SWNT) arrays. Because the adsorption in SWNT arrays is rather complicated, and will occur both inside and outside isolated nanotubes as well as in adsorption sites due to aggregations of nanotubes, it is necessary to introduce adsorption models with different mechanisms. Four adsorption models have been introduced and compared in this work. The pore size distributions, which correspond to the models adopted, are determined by minimizing the deviations between the experimental isotherm and the DFT calculation. Our results indicate that by introducing enough flexibility in model IV proposed, in which adsorption takes place in both the outer surface and inside of the tubes separately or simultaneously, can give a good fit to the experimental isotherm.

1. Introduction Since the discovery of carbon nanotubes, there have been a number of works conducted from both a fundamental and an applied viewpoint. Generally, two different types of carbon nanotubes exist, depending on whether the tube walls are made of one layer (single-walled carbon nanotube, SWNT) or multilayer (multiwalled carbon nanotube, MWNT). Recently, a large scale synthesis of SWNTs has made great progress, since SWNTs can now be obtained in quantity by laser ablation,1 electrical arc,2 and most recently the hydrocarbon catalytic methods.3 However, whatever method has been used for the synthesis of SWNTs, the samples obtained usually include impurities such as catalytic particles, amorphous carbon, and possibly MWNTs, carbon nanoparticles, C60, and other fullerenes. There has been a good deal of interest in the possibility of storage of hydrogen on SWNTs in the past several years. The hope is that these novel carbon materials have such highly uniform pore sizes, high surface areas, and attractive surface potentials, that hydrogen can be adsorbed at high enough densities to reach the US Department of Energy (DOE) targets for vehicular fuel cells. The DOE targets are reported to be about 6.5 wt % and 62 kg/m3 for gravimetric and volumetric capabilities, respectively. There are a number of published experimental studies of * To whom correspondence should be addressed. Fax: +86-1064427626. E-mail: [email protected]. † Lab of Molecular and Materials Simulation, College of Chemical Engineering. ‡ Research Center of the Ministry of Education for High Gravity Engineering and Technology. (1) Thess, A.; Lee, R.; Nikolaev, P.; Dai, H.; Petit, P.; Robert, J.; Xu, C.; Lee, Y. H.; Kim, S. G.; Rinzler, A. G.; Colbert, D. T.; Scuseria, G.; Toma´nek, D.; Fischer, J. E.; Smalley, R. E. Crystalline Ropes of Metallic Carbon Nanotubes. Science 1996, 273 (5274), 483-487. (2) Journet, C.; Maser, W.; Bernier, P.; Loiseau, A.; Lamy de la Chapelle, M.; Lefrant, S.; Deniard, P.; Lee, R.; Fischer, J. E. Large Scale Production of Single-wall Carbon Nanotubes by the Electric Arc Technique. Nature 1997, 388 (6644), 756-758. (3) Cheng, H. M.; Li, F.; Su, G.; Pan, H. Y.; He, L. L.; Sun, X.; Dresselhaus, M. S. Large-scale and Low-cost Synthesis of Single-walled Carbon Nanotubes by the Catalytic Pyrolysis of Hydrocarbons. Appl. Phys. Lett. 1998, 72 (11), 3282-3284.

hydrogen adsorption in SWNTs,4-6 and high hydrogen storage capacities have been reported. On the other hand, theoretical studies of hydrogen adsorption in SWNTs have been carried out to verify the hydrogen storage capacity of SWNTs by molecular simulation,7-14 the density functional theory (DFT),15,16 and other calculations.17 Interestingly, many gravimetric storage capacities reported from experiments are diversified and some of them greatly exceed the values predicted by simulations and (4) Dillon, A. C.; Jones, K. M.; Bekkedahl, T. A.; Kiang, C. H.; Bethune, D. S.; Heben, M. J. Storage of hydrogen in single-walled carbon nanotubes. Nature 1997, 386, 377-379. (5) Ye, Y.; Ahn, C. C.; Witham, C.; Fultz, B.; Liu, J.; Rinzler, A. G.; Colbert, D.; Smith, K. A.; Smalley, R. E. Hydrogen adsorption and cohesive energy of single-walled carbon nanotubes. Appl. Phys. Lett. 1999, 74 (16), 2307-2309. (6) Liu, C.; Fan, Y. Y.; Liu, M.; Cong, H. T.; Cheng, H. M.; Dresselhaus, M. S. Hydrogen storage in single-walled carbon nanotubes at room temperature. Science 1999, 286, 1127-1129. (7) Wang, Q. Y.; Johnson, J. K. Optimization of carbon nanotube arrays for hydrogen adsorption. J. Phys. Chem. B 1999, 103, 4809-4813. (8) Wang, Q. Y.; Johnson, J. K. Molecular simulation of hydrogen adsorption in single-walled carbon naotubes and idealized carbon slit pores. J. Chem. Phys. 1999, 110 (1), 577-586. (9) Rzepka, M.; Zlamp, P.; de la Casa-lillo, M. A. Physisorption of hydrogen on microporous carbon and carbon nanotubes. J. Phys. Chem. B 1998, 102, 10894-10898. (10) Darkrim, F.; Levesque, D. Monto Carlo simulation of hydrogen adsorption in single-walled carbon nanotubes. J. Chem. Phys. 1998, 109, 4981-4984. (11) Darkrim, F.; Levesque, D. Monto Carlo simulation of hydrogen adsorption in single-walled carbon nanotubes. J. Phys. Chem. B 2000, 104, 6773-6776. (12) Yin, Y. F.; Mays, T.; McEnaney, M. Molecular simulations of hydrogen storage in carbon nanotube arrays. Langmuir 2000, 16, 10521-10527. (13) Gu, C.; Gao, G. H.; Yu, Y. X.; Mao, Z. Q. Simulation study of hydrogen storage in single walled carbon nanotubes. Int. J. Hydrogen Energy 2001, 26, 691-696. (14) Williams, K. A.; Eklund, P. C. Monte Carlo simulation of H2 physisorption in finite-diameter carbon nanotube ropes. Chem. Phys. Lett. 2000, 320, 352-358. (15) Gordon, P. A.; Saeger, R. Molecular modeling of adsorptive energy storage: Hydrogen storage in single-walled carbon nanotubes. Ind. Eng. Chem. Res. 1999, 38, 4647-4655. (16) Zhang, X. R.; Wang, W. C. A density functional study of hydrogen adsorption in single walled carbon nanotube arrays. Acta Chim. Sin. 2002, 60, 1395-1404. (17) Stan, G.; Cole, M. Hydrogen adsorption in nanotubes. J. Low Temp. Phys. 1998, 110, 539-544.

10.1021/la026924c CCC: $25.00 © 2003 American Chemical Society Published on Web 07/15/2003

Single-Walled Carbon Nanotubes

DFT calculations. At this point it is still not clear whether carbon nanotubes will have real technological applications in hydrogen storage applications. Therefore, the storage capacities reported in the literature need to be verified based on well-characterized materials and a detailed understanding of the storage mechanism. Moreover, most reported simulations and theoretic calculations have been in idealized systems, in which the SWNT lattice is infinite in three dimensions and no heterogeneity has been introduced. Some authors, for example, have investigated the possibility that infinite lattices with novel symmetries or large intertube spacing might enhance the storage capacity. To our knowledge, Williams and Eklund14 considered the finite-diameter “ropes (bundles)” of parallel SWNTs, which contain a few to tens of parallel nanotubes. They found that the gravimetric adsorption of hydrogen was strongly dependent on the diameter of SWNT ropes and suggested that delamination of SWNT ropes should increase the gravimetric storage capacity. Their work shows the importance of characterization of SWNT arrays to narrow the discrepancy between observed hydrogen storage capacities and the results of physisorption simulations and DFT calculations in the literature. Understanding of the structure and heterogeneity of solids is extremely important for practical applications, such as adsorption and catalytic reaction processes. As a result, characterizing the microporous structure and heterogeneity of a microporous material is a very challenging task. Compared with the X-ray diffraction and transmission electron microscopy data, physical adsorption isotherms have the advantage of producing a macroscopic average measurement, from which such characteristics as the specific surface, the pore volume, and the pore size distribution (PSD) are obtained. Until now, PSD still can be accepted as a general means to characterize adsorption materials of geometrical and energetic heterogeneities. In particular, amorphous materials, even some more uniform mesoporous, e.g., MCM-41,18 are frequently characterized using nitrogen adsorption isotherms at 77 K to obtain the pore size distribution. Although nanotubes in SWNT arrays always show some ordered structures, i.e., the well-aligned tubes in bundles and narrow distributions for inner diameters of tubes. The structures of SWNT arrays are far from uniform. In addition to the energy heterogeneity and impurities, SWNT arrays display some characteristics of heterogeneous structures. For instance, the ends of tubes are open or closed with a ratio, and tubes are isolated or assembled into bundles with varying numbers and sizes. All these factors, of course, affect the capacity of gas storage in SWNT arrays significantly. From the work of Cheng et al.,19 it is reasonable to speculate that adsorption in SWNT arrays may occur in the inner cavities of isolated nanotubes (endohedral adsorption), in the range of micropores, or on the outer surfaces of isolated nanotubes (nonpore adsorption, namely, exohedral adsorption). In addition, adsorption would happen in other pores or surfaces due to aggregation of nanotubes, such as in the interstices of SWNT bundles (micropores), on the outer surfaces of a SWNT bundles (nonpore), and even pores formed by neighboring bundles. Because of its rather complicated adsorption mechanism and its heterogeneity, it is difficult to compare adsorption isotherms by theoretic work with those from experiments. Even for a commonly studied system, hydrogen adsorbed in SWNT arrays, there is a qualitative discrepancy20 between the shape of the experimental isotherm and those from molecular simulation and the DFT method. For example, the simulations7,8,9,14 and DFT method16 exhibit

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Figure 1. Nitrogen adsorption isotherm of the sample SWNT array at 77 K.

a maximum or a plateau at 77 K, while the experimental data5 show a monotonic increase in adsorption. In this paper, we first measure an experimental isotherm of nitrogen at 77 K for a sample of SWNT arrays provided by Cheng’s group.3 Then, four models of different mechanisms of adsorption are used to characterize the pore structure of the sample. The PSD corresponding to the model is solved by minimizing the deviations between the experimental isotherm and the DFT calculations. Finally, we recommend a flexible and effective model for the description of adsorption in the sample of SWNT arrays. 2. Experimental Results of Nitrogen Adsorption at 77 K A sample of SWNT array, which was synthesized by Cheng et al.3 from catalytic decomposition of hydrocarbons, was used for experimental measurements of the adsorption isotherm of nitrogen at 77 K. The measurements were carried out in the apparatus of ASAP 2010 (Micrometritics Instrument). Nitrogen dose amount for every measured point was set to 3 monolayers STP (standard conditions of temperature and pressure, 273 K and 1.013 × 105 Pa). We set the adsorption pressure and maintained it for hours until pressure fluctuation was within 1.3 × 10-4 Pa. Then, the data point was recorded. The range of the relative pressure, P/P0, where P0 is the saturated vapor pressure, was from 10-6 to 1, and 54 data points in this range were measured. The experimental results are shown in Figure 1. According to the IUPAC classification, the isotherm is of a mixed shape of type I (at low pressures) and type II (at medium pressures), indicating the presence of a certain amount of micropores as well as the development of mesopores. The BET specific surface area is 179 cm2/g. This value is smaller than that reported in the work of Ye et al. (285 cm2/g)5 and Alain et al. (302 cm2/g).21 The (18) Ravikovitch, P. I.; Wei, D.; Chueh, W. T.; Haller, G. L.; Neimark, A. V. Evaluation of pore size structure parameters of MCM-41 catalyst supports and catalysts by means of nitrogen and argon adsorption. J. Phys. Chem. B 1997, 101, 3671-3679. (19) Cheng, H. M.; Li, F.; Sun, X.; Brown, S. D. M.; Pimenta, M. A.; Marucci, A.; Dresselhaus, G.; Dresselhaus, M. S. Bulk morphology and diameter distribution of single-walled carbon nanotubes synthesizes by catalytic decomposition of hydrocarbons. Chem. Phys. Lett. 1998, 289, 602-610. (20) Simonyan, V. V.; Johnson, J. K. Hydrogen storage in carbon nanotubes and graphitic nanofibers. J. Alloys Compd. 2002, 330-332, 659-665. (21) Alain, E.; Yin, Y. F.; Mays, T.; McEnaney, M. Molecular simulations and measurement of adsorption in porous carbon nanotubes. Stud. Surf. Catal. 2000, 128, 313-322.

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Table 1. Parameters for Potential Models for Nitrogen and SWNTa N2

SWNTs

σff/nm

ff/k/K

σss/nm

ss/k/K

Fsurf/nm-2

0.375

95.2

0.34

28.0

38.2

ss, σss, and Fsurf are taken from ref 15. ff and σff are taken from ref 25. a

different structure characteristics of SWNT arrays synthesized by the catalytic decomposition of hydrocarbons may be one of the reasons for the lower BET surface area, compared with those by both the laser vaporization and electric arc methods. According to the work of Cheng et al.,19 there are large quantities of ropes and ribbons with diameters of about 100 µm, consisting of many roughly aligned small SWNT bundles. This structure reduces the surface area for adsorption and even may make some volume in the array inaccessible to nitrogen molecules. 3. Theory 3.1. Potential Model. The Lennard-Jones (LJ) potential is adopted here to represent the interaction between a pair of nitrogen molecules

ULJ(r) ) 4

12

6

[(σr) - (σr) ]

(1)

where r is the interparticle distance, is the well depth, and σ is the distance between a pair of molecules when the potential is zero. The interaction between the wall and a nitrogen molecule inside a tube of SWNTs, v(r,R), can be represented by the smoothed potential functions proposed by Tjatjopoulos et al.22 and Fischer et al.23 Here, the potential function of Tjatjopoulos et al. is adopted and is expressed as

[ [ ( )] [ ( ) ] [ ( )] [ ( ) ]]

63 r r -10 2F -9/2, -9/2;1; 32 σsf R r -4 r 2 r -3 2F -3/2,-3/2;1; 1 σsf R R

v(r,R) ) π2sfFsurfσsf2 1-

r R

2

(2a) where F[R,β;γ;χ] is the hypergeometric series, R is the radius of a tube, r is the distance from the wall, and Fsurf is the surface density of carbon atoms on the wall. The parameters sf and σsf for solid-fluid interactions are simply obtained by the Lorentz-Berthelot (LB) combining rules suggested by Steele.24 Because the interaction between carbon and nitrogen molecules is nonpolar in nature, this approximation is acceptable. The parameters used to represent the N2-N2 and N2-adsorbent interactions are listed in Table 1. Similar to eq 2a, when adsorption takes place outside a single and isolated nanotube, the adsorbate-adsorbent interaction potential vout is obtained by integration of the LJ potential over the outer surface of a nanotube and is expressed as15

vout(r,R) )

[[ [ ]

π2sfFsurfRσsf2r-1 2

) ] - 3 r σ- rR

r R

2

sf

]

63 r2 - R2 32 σsfr 2 -4

[

-10

[

(

F -9/2,-9/2;1; 1 -

(

F -3/2,-3/2;1; 1 -

) ]] (2b)

r R

2

3.2. Density Functional Theory. For a system in an external field v(r), where r is the local position vector

within the pore space, at a fixed temperature T and the chemical potential µ, the grand potential functional Ω[F(r)] is written as

Ω[F(r)] ) F[F(r)] +

∫ dr [v(r) - µ]F(r)

(3)

where F(r) is the local density and F[F(r)] is the intrinsic Helmholtz free energy functional. The criterion for equilibrium is that the grand potential is a minimum, namely, the density profile F(r) of the adsorbate within the pore satisfies the condition

dΩ[F(r)]/dF(r) ) 0

(4)

The smoothed density for the description of inhomogeneous fluids was introduced by Johnson and Nordholm,26 Fischer and Methfessel,27 and Tarazona et al.28,29 In this work we use the version of DFT proposed by Tarazona, and the attractive forces are included via the attractive mean force approximation.30 The density functional theory applied here is identical to that described in our previous work.16 The free energy of a fluid in eq 3 is split into attractive and repulsive contributions. The attractive part of the fluid-fluid potential Φattr(|r - r′|) is given by the Weeks-Chandler-Anderson perturbation scheme31 for a cut and shifted LJ potential

Φattr(|r - r′|) ) -ff - ULJ(rc) ΦLJ(|r - r′|) - ULJ(rc) 0

{

|r - r′| < rm rm < |r - r′| < rc (5) |r - r′| > rc

where rm ) 21/6σff, rc ) 5σff, and r and r′ are the position vectors of particles. The free energy per particle of hard spheres is calculated from the Carnahan-Starling equation of state for a system of hard spheres32 by using a smoothed density Fj

fex(Fj(r)) ) kTη

(4 - 3η) (1 - η)2

(6)

where η ) (π/6)Fjd3, and the effective diameter d of a hard sphere is calculated as a function of temperature (22) Tjatjopoulos, G. J.; Feke, D. L.; Mann, J. A. Molecule-micropore interaction potentials. J. Phys. Chem. 1988, 92, 4006-4007. (23) Fischer, J.; Bohn, M.; Ko¨rner, B.; Findenegg, G. H. Supercritical gas adsorption in porous materials II. Prediction of adsorption isotherms. Ger. Chem. Eng. 1983, 6, 84-91. (24) Steele, W. A. The physical interaction of gases with crystalline solids I. Gas-solid energies and properties of isolated adsorbed atoms. Surf. Sci. 1973, 36, 317-352. (25) Maddox, M. W.; Olivier, J. P.; Gubbins, K. E. Characterization of MCM-41 using molecular simulation: Heterogeneity effects. Langmuir 1997, 13, 1737-1745. (26) Johnson, M.; Nordholm, S. Generalized van der Waals theory. VI. Application to adsorption. J. Chem. Phys. 1981, 75, 1953-1957. (27) Fischer, J.; Methfessel, M. Born-Green-Yvon approach to the local densities of a fluid at interfaces. Phys. Rev. A 1980, 22, 28362843. (28) Tarazona, T. Free-energy density functional for hard spheres. Phys. Rev. A 1985, 31, 2672-2679; Phys. Rev. A 1985, 32, 3148. (29) Tarazona, P.; Marini Bettolo Marconi, U.; Evans, R. Phase equilibria of fluid interfaces and confined fluids: Nonlocal versus local density functional. Mol. Phys. 1987, 60, 573. (30) Fischer, J.; Heinbuch, U.; Wendland, M. On the hard sphere plus attractive mean field approximation for inhomogeneous fluids. Mol. Phys. 1987, 61, 953-961. (31) Weeks, J. D.; Chandler, D.; Anderson, H. C. Role of repulsive forces in determining the equilibria structure of simple liquids. J. Chem. Phys. 1971, 54, 5237. (32) Carnahan, N. F.; Starling, K. E. Equation of state for nonattracting rigid spheres. J. Chem. Phys. 1969, 51, 635.


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η1kT/ff + η2 d ) σff η3kT/ff + η4

(7)

where the parameters ηi can be referred to the literature.33 The smoothed density is given by the Tarazona recipe.28,29 3.3. Calculation of Pore Size Distributions from the DFT Method. The DFT method predicts the adsorption isotherms in individual pores as well as the pressure values of P/P0 (note that in our calculations we set P0 to 0.1013 MPa), corresponding to the equilibrium thermodynamic transitions. In a cylindrical pore, the endohedral excess isotherm per unit of mass of adsorbent, NDFT, is calculated here from the dimensionless density profile by the DFT method as

a standard least-squares approach with non-negativity constraints to minimize the objective function fobj

fobj[φ] )

∑P [N(P/P0) - ∫D

Dmax min

dD φ(D)NDFT(D,P/P0)]2 (12)

By representing the pore size distribution, φ(D), in terms of a discrete function, eq 12 can be rewritten as

fobj[φ] )

∑ P i

Dmax

[N(Pi/P0) -

∆Djφ(Dj)NDFT(Dj,Pi/P0)]2 ∑ D min

(13)

The solution for the eq 13 is

2 NDFT(P/P0) ) DC-CFsurf

∫0

Dint/2

r dr (F(r) - Fb)

(8a)

For exohedral

NDFT(P/P0) )

2 DC-CFsurf

∫DD /2/2 + 19σ ext

ff

ext

r dr (F(r) - Fb) (8b)

where Fb is the equilibrium bulk density, which is the homogeneous gas density in the bulk phase at the same temperature and chemical potential with the fluid in the pore, and Fsurf is the surface particle density of the pore. Dint is the internal pore diameter for endohedral adsorption

Dint ) DC-C - ∆

(9a)

and Dext is the external pore diameter for exohedral adsorption

Dext ) DC-C + ∆

(9b)

In eqs 8 and 9, DC-C is the diameter measured between the centers of the first layer of solid atoms and ∆ is the effective diameter of solid atoms, treated as hard spheres. The effective diameter of solid atoms is defined from the combining rule

∆ ) 2(σsf - 0.5σff)

(10)

Note that 19σff is chosen here somewhat artificially to calculate excess exohedral adsorption of an isolated carbon nanotube. However, it must be large enough so that the effects of walls of SWNT at this distance can be neglected. In our calculations, we computed the bulk density of nitrogen at 77 K first, while assuming that the density of nitrogen be of the bulk value at the distances from the wall larger than 19σff. Having a set of isotherms in individual model pores, the experimental isotherm is then expressed by a superposition of the calculated isotherms and the pore size distribution

N(P/P0) )

∫DD

max

min

φ(D)NDFT(D,P/P0) dD

(11)

where N(P/P0) is the experimental isotherm, NDFT(D,P/ P0) is the theoretical isotherm in the pore of diameter D, ranging from the minimum pore size Dmin to the maximum pore size Dmax, and φ(D) is the pore size distribution. Equation 11 represents a Fredholm integral, and its inversion is well-known to present an ill-posed problem. The most straightforward method to solve eq 11 is to use

x ) (ATA) - 1ATy

(14)

where yi ) N(Pi/P0), xj ) φ(Dj), and Aij ) NDFT(Dj,Pi/P0)∆Dj, where ∆Dj is set to 0.2 nm. Equation 14 is simply the least-squares solution to a set of linear equations

y ) Ax

(15)

To obtain the information for internal structure of SWNTs, we set the minimum pore size, Dmin, to 0.7 nm in which adsorption takes place, and the maximum pore size, Dmax, to 6.1 nm according to the literature.34 If we consider a set of D of m members and a vector N(P/P0) of members n, n g m must hold. In this work we use an overdetermined matrix, in which n > 2m. More data points can be obtained by interpolation between the available experimental data. An interpolating cubic spline method was used to obtain 200 data points from the 54 experimental data points. It is noted that interpolation was performed only at pressures higher than the filling pressure of the micropore, because the cubic spline method would not applied to the locations for phase transitions. The interpolated values are also presented along with the experimental data in Figure 1. 4. Characterization of SWNT Arrays Using Different Adsorption Models As is well-known, an SWNT array consists of many tubes assembled. To characterize a real sample SWNT array, it is important to understand the mechanism governing the adsorption in the array. Then, the heterogeneity described by PSD should be taken into account. Consequently, we propose four adsorption models, as shown in Figure 2, in a SWNT array as follows. Model I. Each tube in the array behaves as an isolated tube. Only adsorption inside the tube (endohedral) occurs. The uptake is the sum of the endohedral adsorption of all the isolated tubes. In addition, the tubes are of a PSD, which can be determined by fitting to an experimental isotherm, as is mentioned above. Model II. Similar to model I, but fluid molecules are adsorbed on the outer surface of isolated tubes alone (exohedral). The model is based on the fact addressed by Williams and Eklund14 that the existence of nonpore is of importance for hydrogen adsorption, and some of the tubes are of closed ends. Therefore, the uptake is assumed be (33) Lastoskie, C.; Gubbins, K. E.; Quirke, N. Pore size distribution analysis of microporous carbons: A density functional theory approach. J. Chem. Phys. 1993, 97, 4786-96. (34) Lebedkin, S.; Schweiss, P.; Renker, B.; Malik, S.; Hennrich, F.; Neumaier, M.; Stoermer, C.; Kappes, M. M. Single-walled carbon nanotubes with diameters approaching 6 nm obtained by laser vaporization. Carbon 2002, 40, 417-23.

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Figure 3. Pore size distribution obtained by considering endohedral adsorption for each tube in the array, model I.

Figure 2. Models representing different adsorption mechanisms in an SWNT array: model I, endohedral adsorption only; model II, exohedral adsorption only; model III, endohedral and exohedral adsorption simultaneously; model IV, the combination of model I, model II, and model III.

the sum of the exohedral adsorption of all the isolated tubes in the model. A PSD can also be determined by fitting to the experimental isotherm. Model III. Adsorption takes place both inside tubes and on the outer surfaces of the isolated tubes simultaneously. The uptake is the sum of the endohedral and exohedral adsorption of all the isolated tubes. Of course, a different PSD can be obtained. Model IV. In fact, a real SWNT array material is of a more complicated picture. The total uptake of a sample SWNT array can break down into (1) endohedral and exohedral adsorption for isolated tubes separately or simultaneously; (2) adsorption in the bundles that are assembled by tubes, the uptake for a bundle of tubes consists of the adsorption inside the tubes and in the interstices formed by the tubes in the bundle, which can take place simultaneously or separately; (3) adsorption on the outer surfaces of SWNT bundles; (4) adsorption may also occur in the pores formed by neighboring bundles, although no one has discussed this possibility. If all the phenomena above are considered during characterization of an SWNT array, it would be impractical and even impossible. Therefore, in model IV, it is assumed that tubes in an array are grouped into three catalogs: (1) the tubes of endohedral adsorption only, (2) the tubes of exohedral adsorption only, and (3) the tubes of endohedral and exohedral adsorption simultaneously. In fact, model IV is a combination of models I, II, and III, as is shown in Figure 2. We expect that by introducing enough flexibility, the model can provide an approximate picture of the adsorption of SWNT arrays in an effective way. Similarly, the PSD for model IV can also be derived by fitting calculated results from the DFT method to the experimental adsorption isotherm. 5. Results and Discussion 5.1. Characterization of the Sample SWNT Array Using Model I. Only the endohedral adsorption is considered in model I. Figure 3 shows the PSD by solving eq

Figure 4. Comparison of adsorption isotherm of nitrogen calculated from model I and experimental results, T ) 77 K.

15. It shows that three peaks are located in the regions of micropores, 0.7-0.9 nm and 1.9-2.1 nm, and mesopores, 5.7-5.9 nm, respectively. A comparison of the fitted isotherm from the PSD and the DFT method with the experimental data is shown in Figure 4. As is expected, a large difference between the adsorption isotherm calculated and the experimental data is found, especially at high pressures. Obviously, it is attributed to the oversimplified mechanism in this model. Several authors have speculated the existence of nonpore gas adsorption in SWNT arrays. Using molecular simulation, Alain et al.21 studied exohedral adsorption and simultaneous endohedral and exohedral adsorption of nitrogen in carbon nanotubes with different diameters. They found the exohedral adsorption might be more important than endohedral adsorption in the application to, e.g., gas storage. 5.2. Characterization of the Sample SWNT Array Using Model II. As is discussed above, exohedral adsorption plays an important role for gas storage and a large deviation would occur, if we calculate the total adsorption of gases in SWNT arrays without considering exohedral adsorption, especially at high pressures. In model II only the exohedral adsorption is taken into account to get the PSD, shown in Figure 5. There is only one peak of the PSD ranging from 5.9 to 6.1 nm. A comparison of the fitted isotherm and the experimental data is shown in Figure 6. As is expected, even more significant discrepancies between the adsorption isotherm fitted by the PSD and experimental data are observed at


Langmuir, Vol. 19, No. 15, 2003 6093

Figure 5. Pore size distribution obtained by considering exohedral adsorption for each tube in the array, model II.

Figure 8. Calculated endohedral and exohedral adsorption isotherms of nitrogen from the DFT method for isolated carbon nanotubes at 77 K, diameters of the tubes are (a) 4.0 nm, (b) 2.0 nm, and (c) 1.0 nm.

Figure 6. Comparison of adsorption isotherm of nitrogen calculated from model II and experimental results, T ) 77 K.

Figure 7. Fluid-wall interaction for a nitrogen molecule inside and outside different size carbon nanotubes, d ) 1.0, 2.0, and 4.0 nm. Note that we take the centers of nanotubes as origin and define the reduced potential energy being u* ) u/ff and the reduced distance being r* ) r/σff, respectively.

both low and high pressures, compared with model I. This indicates that the adsorption on the outer surfaces, i.e., for nonpore, is not the governing mechanism for SWNT arrays. It is interesting to investigate the variation of endohedral and exohedral adsorption with the diameters of a carbon tube. Figure 7 presents the interaction potentials between a nitrogen molecule and the walls of the tubes of 1.0, 2.0, and 4.0 nm, respectively. It is found in Figure

7 that the potential inside the tubes is increased pronouncedly with decreasing tube diameter. This observation complies with the results of Fischer et al.23 In contrast, the depth of the potential outside the tubes increases slightly with the tube diameter. This observation shows that endohedral adsorption and exohedral adsorption would vary with the diameter in different ways. Figure 8 shows the calculated endohedral and exohedral isotherms for nitrogen in the isolated tubes of 1.0, 2.0, and 4.0 nm from the DFT method, respectively. The endohedral adsorption in the smallest pore of 1.0 nm shows an adsorption feature of micropores and no observable increment in the range of pressures we investigated. For the tubes of 2.0 and 4.0 nm, there are stepwise changes of the adsorption, corresponding to monolayer completion and capillary condensation, respectively. Saturated endohedral adsorption can be obtained at lower pressures for smaller tubes, for the stronger potential energy experienced by the adsorbate molecules inside the tube, as is show in Figure 7. However, the larger tube gives higher maximum uptake, because it provides larger volume to host more molecules. For exohedral adsorption, the situation is different. It shows that the type II isotherm rather than type I and type IV is found for the endohedral adsorption. Some stepwise changes of the adsorption that is associated with monolayer completion in exohedral adsorption emerge. It is found that at low pressures, endohedral adsorption has higher nitrogen uptake against the exohedral, while at high pressures, exohedral adsorption exceeds the endohedral. This indicates that in applications of SWNTs for gas storage at high pressures, exohedral adsorption or adsorption on outer surfaces of SWNT bundles might be significant and comparable with that inside the tubes, as is pointed out by Alain et al.21 and Williams and Eklund.14

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Figure 9. Pore size distribution obtained by considering endohedral adsorption and exohedral adsorption taking place simultaneously for each tube in the array, model III.

Zhang et al.

Figure 11. Pore size distribution obtained by model IV. In model IV, exohedral adsorption and endohedral adsorption for each tube in the array take place separately or simultaneously.

5.4. Characterization of the Sample SWNT Array Using Model IV. In this model, endohedral and exohedral adsorption can take place in the tubes of a SWNT array either separately or simultaneously, as is shown in Figure 2d. In fact, model IV is a combination of models I, II, and III, covering all the cases in these models. The problem now becomes to find the solution vector of the pore size distribution x′ in the linear equation, which is similar to eq 15

y ) A′x′

(16)

where we define

x′ ) Figure 10. Comparison of adsorption isotherm of nitrogen calculated from model III and experimental results, T ) 77 K.

The analyses above suggest that either model I or model II is an oversimplified picture of the sample SWNT array. As a result, it is reasonable to take into account both the endohedral and exohedral adsorption for a tube, which is addressed in model III as below. 5.3. Characterization of the Sample SWNT Array Using Model III. In model III, endohedral and exohedral adsorption takes place simultaneously for each tube, which corresponds to the open isolated carbon nanotubes (see Figure 2c). Equation 15 is again used to solve the PSD, shown in Figure 9. A comparison of the calculated isotherm from the DFT method and the PSD with the experimental data is shown in Figure 10. It is found that from the PSD obtained not only micropores but also mesopores exist in the SWNT array. The agreement between the two isotherms is much improved compared with model I and model II. This suggests that introduction of exohedral adsorption would improve adsorption at high pressures against model I, while introduction of endohedral adsorption would improve adsorption at low pressures against model II. There are two separated peaks of pore size distribution at 1.7-1.9 nm and 5.3-5.5 nm. The first peak agrees well with the main diameters (1.69 ( 0.3 nm) of the nanotubes from their high-resolution transmission electron microscopy (HRTEM) measurement.19 Because only one SWNT of diameter larger than 3.2 nm (4.3 nm) was observed during 74 isolated SWNTs by Cheng et al.,19 it suggests that the peak at 5.3-5.5 nm does not reflect the real diameters of nanotubes. However, its appearance can give a good fit to adsorption isotherm at high pressure.

() x X

and A′ ) (AB). Here x(Dj) and X(Dj) are the solution vectors for each pore size Dj for endohedral and exohedral adsorption, respectively, while A(Dj,Pi/P0) and B(Dj,Pi/P0) are matrix elements of values for a pore size Dj at pressure Pi/P0 for endohedral and exohedral adsorption, respectively. The least-squares method is also used to solve eq 16; thus the PSD is obtained. Note that in this model the structure characteristics of the SWNT array include not only the adsorption in the pores but also the outer surface nonpore adsorption. Then, the PSD obtained by eq 16 is no longer the PSD that is commonly used to characterize the structure heterogeneity of the pores. The calculated PSD and a comparison of the adsorption isotherms fitted by the DFT method and the PSD with the experimental data are shown in Figure 11 and Figure 12, respectively. Again, the PSD shows that micropore, mesopore, and nonpore all appear in the SWNT array. Impressively, the calculated and experimental isotherms coincide very well. Compared with model I, model II, and model III, a pronounced improvement is attained. The peaks of the PSD ranging from 1.5 to 1.7 nm both for exohedral and endohedral adsorption appear again, which complies with the HRTEM direct observation (1.69 ( 0.3 nm) by Cheng et al.19 However, other peaks for endohedral and exohedral adsorption emerge. For example, a highest peak in the range of 0.7-0.9 nm suggests that there are a large number of micropores smaller than 1.5 nm existing in the SWNT array. According to the work by Cheng et al.,19 however, most of carbon nanotubes there have diameters ranging from 1.2 to 2.0 nm. Because carbon nanotubes in this work are assumed be isolated from each other, not arranged in bundles, in which many nanotubes are well-aligned, it is


Figure 12. Comparison of adsorption isotherm of nitrogen calculated from model IV and experimental results, T ) 77 K.

Langmuir, Vol. 19, No. 15, 2003 6095

for a tube of a proper size. Here, the endohedral adsorption in the tubes of 0.7-0.9 nm is, in fact, an equivalent description of the adsorption in the interstices of the array tubes. Likewise, we can speculate that the peaks of exohedral adsorption in the ranges of 2.8-3.3 and 4.9-5.3 nm do not reflect the real existence of the tubes of these sizes. It just reflects the adsorption characteristics of a certain structure of the array due to the aggregation of nanotubes. Therefore, the combination of endohedral and exohedral adsorption in the ranges of 4.95.3, 2.8-3.3, 1.5-1.7, and 0.7-0.9 nm give an equivalent expression of the array. Consequently, although in our model the adsorption in the interstices and on the outer surfaces of SWNT bundles, which will make the DFT calculations tedious and the computer time too long to bear with, is not taken into account, by introducing enough flexibility as in model IV, the model on the isolated nanotubes can be used to characterize the SWNT array of interest. 6. Conclusions

Figure 13. Comparison of endohedral adsorption of nitrogen for an SWNT of 0.8 nm from the DFT method and adsorption in the interstices for the tubes of 1.6 nm in a triangle lattice of an SWNT array from the GCMC simulation, T ) 77 K. The distance between two nearest tubes in the lattice is 0.334 nm (the van der Waals gap).

suggested that the appearance of the peak may result from or partly from the adsorption in the interstices of the tubes in the SWNT array. To prove this, we performed grand canonical Monte Carlo (GCMC) simulations to get the adsorption isotherm of nitrogen in the interstices of an SWNT array at 77 K. In this array the tubes are of 1.6 nm and arranged in an ideal configuration, namely, in a triangular lattice, and the distance between two nearest nanotubes is 0.334 nm (the van der Waals gap). The potential models for N2-N2 and N2-SWNT as well as the potential parameters used in GCMC are the same as those adopted in our DFT calculations. Details of GCMC can be found elsewhere.35 A comparison of the adsorption isotherms of nitrogen in the interstices of the array by molecular simulation with that for the interior of the tube of 0.8 nm by DFT calculations is shown in Figure 13. Both isotherms are of the characteristics of micropores. Moreover, the uptakes of 2.5 and 2.2 nm-1 (N2 molecules adsorbed per unit length) for the two cases are comparable and present a similar variation with pressure. This observation indicates that one can find an effective way to describe the adsorption on outer surfaces in terms of the endohedral adsorption (35) Zhang, X. R.; Wang, W. C. Adsorption of linear ethane molecules in single walled carbon nanotube arrays by molecular simulation. Phys. Chem. Chem. Phys. 2002, 4, 3048-54.

The density functional theory method for the calculation of the pore size distribution in activated carbon and other amorphous adsorption materials is applied to study the adsorption characteristics of a sample of SWNT arrays. An adsorption isotherm of N2 at 77 K was first measured by experiment. As is well-known, adsorption in an SWNT array is of a rather complicated picture. Adsorption would take place inside and/or outside isolated nanotubes, in the sites due to aggregation of nanotubes, and in the interstices and on the outer surfaces of SWNT bundles with finite diameters. Therefore, it is necessary to introduce adsorption models with different mechanisms in characterization of SWNT arrays. Four models are proposed in this work. In these models we assume that the total adsorption for an array is composed of the contributions from all the isolated tubes, where the endohedral adsorption and exohedral adsorption occur separately or simultaneously. Moreover, a pore size distribution, which corresponds to the model adopted, can be obtained by minimizing the deviations of those calculated by the DFT method and experimental isotherms. Then, based on different models and corresponding PSDs, the calculated isotherms by the DFT method are compared with the experimental ones to test the performances of the models. It is found in Figure 12 that enough flexibility, model IV proposed here, gives an excellent fit to the experimental data for the sample array. As is shown in Figure 11, in model IV the peak for 1.5-1.7 nm in the PSD is in agreement with the HRTEM measurement, 1.69 ( 0.3 nm, by Cheng et al.19 However, the other peaks for endohedral and exohedral adsorption in Figure 11, for example, have highest peaks in the range of 0.7-0.9 nm, suggesting that there are a large number of micropores in the SWNT array. In fact, the endohedral adsorption in the tubes of 0.7-0.9 nm is an equivalent description of the adsorption in the interstices of the array tubes. Likewise, we can speculate that the peak of exohedral adsorption in the range of 4.9-5.3 nm and the peak of endohedral adsorption of 2.8-3.3 nm do not represent the real existence of the tubes of these sizes. It just reflects the adsorption characteristics of the structure of the array due to the aggregation of nanotubes. In summary, the combination of endohedral and exohedral adsorption described in model IV gives an equivalent expression of the performance of the sample array. Consequently, model

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IV proposed in this work can be used for characterization of an SWNT array in an effective way. Acknowledgment. This work was supported by the State Key Fundamental Research Plan (No. G2000048010), NSF China (No. 20236010, 20276004), the Key Research

Zhang et al.

of Science & Technology of the Ministry of Education, China (No. 0202), the Research Center for High Gravity Engineering and Technology of the Ministry of Education, China, and the China Postdoctoral Science Foundation. LA026924C

Characterization of a Sample of Single-Walled Carbon Nanotube

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