Adsorption and Diffusion of Molecular Nitrogen in Single Wall Carbon

A “wormlike” phase is found for the adsorbed nitrogen in the (15, 0) carbon nanotube at high .... Molecular Sieving Using Single Wall Carbon Nanot...
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Adsorption and Diffusion of Molecular Nitrogen in Single Wall Carbon Nanotubes Gaurav Arora, Norman J. Wagner, and Stanley I. Sandler* Center for Molecular and Engineering Thermodynamics, Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716 Received December 22, 2003. In Final Form: April 19, 2004 Using molecular simulation, the adsorption and self-diffusion of diatomic nitrogen molecules inside a single wall carbon nanotube have been studied over a range of nanotube diameters (8.61-15.66 Å) and loadings at temperatures of 100 and 298 K. Nitrogen adsorption energy is found to increase as the nanotube diameter is reduced toward the molecular diameter of nitrogen. A discrete organization of the nitrogen into adsorbed layers is observed at high loadings that follows a regular progression determined primarily by geometric considerations. The formation of an adsorbate core at the center of the nanotube is found to increase the self-diffusion of nitrogen. A “wormlike” phase is found for the adsorbed nitrogen in the (15, 0) carbon nanotube at high loadings and at 100 K.

I. Introduction Single wall carbon nanotubes (SWCNs) exhibit some unique properties that have led to applications in various fields, including gas separations, gas storage, nanothermometers, DNA recognition, and others.1-5 Because the interior diameter of the SWCN can be comparable to the adsorbate size, it is expected that the extent of adsorption and the adsorbate diffusion in SWCNs will be strong functions of both the nanotube diameter and the adsorbate-nanotube interactions. Previously, both equilibrium and nonequilibrium molecular dynamic simulations (EMD and NEMD, respectively) have been performed to study adsorbent dynamics in microporous materials.6-9 Recently, Skoulidas et al.10,11 reported, on the basis of simulation, that the self- and transport-diffusion coefficients of light gases in SWCNs can be orders of magnitude higher than those in any known microporous materials, approaching free diffusion in the bulk gas. The mass transport of diatomic molecules, that is, oxygen and nitrogen, has been investigated in graphitic slits with the conclusion that the rate of diffusion is highly dependent on the slit width.12,13 A dependence on pore size of the transport properties of carbon dioxide and methane mixtures has also been reported for graphitic cylindrical pores.14 Mao and Sinnott15,16 also performed * Corresponding author. E-mail: [email protected]. Phone: (302)831-2945. Fax: (302)831-3226. (1) Iijima, S. Physica B 2002, 323, 1-5. (2) Takaba, H.; Katagiri, M.; Kubo, M.; Vetrivel, R.; Miyamoto, A. Microporous Mater. 1995, 3, 449-455. (3) Gao, Y. H.; Bando, Y. Nature 2002, 415, 599. (4) Williams, K. A.; Veenhuizen, P. T. M.; de la Torre, B. G.; Eritja, R.; Dekker, C. Nature 2002, 420, 761. (5) Cheng, H.; Pez, G. P.; Cooper, A. C. J. Am. Chem. Soc. 2001, 123, 5845-5846. (6) Maginn, E. J.; Bell, A. T.; Theodorou, D. N. J. Phys. Chem. 1993, 97, 4173-4181. (7) MacElroy, J. M. D.; Boyle, M. J. Chem. Eng. J. 1999, 74, 85-97. (8) Skoulidas, A. I.; Sholl, D. S. J. Phys. Chem. B 2002, 106, 50585067. (9) Duren, T.; Keil, F. J.; Seaton, N. A. Chem. Eng. Sci. 2002, 57, 1343-1354. (10) Skoulidas, A. I.; Ackerman, D. M.; Johnson, J. K.; Sholl, D. S. Phys. Rev. Lett. 2002, 89, article no. 185901. (11) Ackerman, D. M.; Skoulidas, A. I.; Sholl, D. S.; Johnson, J. K. Mol. Simul. 2003, 29, 677-684. (12) Travis, K. P.; Gubbins, K. E. Mol. Simul. 2001, 27, 405-439. (13) Travis, K. P.; Gubbins, K. E. Langmuir 1999, 15, 6050-6059.

simulations of the diffusive flow of both pure organic and inorganic molecules and their mixtures in carbon nanotubes, showing the dependence of diffusion on molecular interactions. In general, mass transport in nanoporous structures is dependent on both the penetrant-substrate interactions and the thermodynamic state of the penetrant in the confined media, which is strongly affected by these interactions. Recent Grand Canonical Monte Carlo (GCMC) studies and experiments have provided a microscopic picture of nitrogen adsorption on SWCN bundles and in single wall carbon nanohorns.17,18 The presence of a second adsorbed layer or bilayer of Ne, CH4, and Xe on the ridges or grooves between two nanotubes of close-ended SWCN bundles has been established experimentally.19 Here, we present evidence for bilayer adsorption of nitrogen inside SWCNs, which has an important implication for the diffusion of pure gases and gas mixtures.20 In ref 20, NEMD simulations were carried out to study the separation mechanism of a hydrogen/methane mixture in a microporous carbon membrane. It was concluded that the transport occurs with the more strongly adsorbed species near the wall and with the less strongly adsorbed species near the center of the pore. We expect a similar phenomenon to occur in nanoporous materials, determined by geometric and energetic considerations. The purpose of this research is to explore the thermodynamics and self-diffusion of diatomic gas molecules confined in SWCNs, where nanoscale confinement may be expected to lead to significant differences in adsorption and diffusion compared to nitrogen adsorption on bulk (14) Nicholson, D. Mol. Phys. 2002, 100, 2151-2163. (15) Mao, Z. G.; Sinnott, S. B. J. Phys. Chem. B 2001, 105, 69166924. (16) Mao, Z. G.; Sinnott, S. B. J. Phys. Chem. B 2000, 104, 46184624. (17) Yoo, D. H.; Rue, G. H.; Hwang, Y. H.; Kim, H. K. J. Phys. Chem. B 2002, 106, 3371-3374. (18) Ohba, T.; Murata, K.; Kaneko, K.; Steele, W. A.; Kokai, F.; Takahashi, K.; Kasuya, D.; Yudasaka, M.; Iijima, S. Nano Lett. 2001, 1, 371-373. (19) Talapatra, S.; Krungleviciute, V.; Migone, A. D. Phys. Rev. Lett. 2002, 89, article no. 246106. (20) Vieira-Linhares, A. M.; Seaton, N. A. Chem. Eng. Sci. 2003, 58, 4129-4136. (21) Harris, P. J. F. Carbon nanotubes and related structures: new materials for the twenty first century; Cambridge University Press: Cambridge, U.K., 1999.

10.1021/la036432f CCC: $27.50 © 2004 American Chemical Society Published on Web 06/18/2004

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Table 1. Lennard-Jones Potential Parameters for Nitrogen (N) and Carbon (C) σ (Å)22 /kB (K)22 xe (Å)25 K (N/cm)26

N-N

C-N

3.32 36.4 1.10 22.95

3.36 33.4

graphite. GCMC simulations are performed to study adsorption isotherms and adsorption energies. EMD simulations are performed to calculate the self-diffusivities of nitrogen in SWCNs of varying sizes as a function of loading at 100 and 298 K. In the following section, the model, system parameters, and intermolecular potentials used in this work are presented, followed in section III by a description of the key features of the EMD and GCMC simulations. The results are presented and discussed in section IV, and the conclusions are presented in section V. II. Model The simulations were performed at 100 and 298 K for several single, infinitely long SWCNs. The four archetypical “zigzag” nanotubes studied have Hamada indices (11, 0), (15, 0), (17, 0), and (20, 0) with diameters of 8.61, 11.74, 13.31, and 15.66 Å, respectively. The formula relating a nanotube of type (n, m) to its diameter, dt, in angstroms, is21

dt )

2.46xn2 + nm + m2 π

(1)

In our simulations, the SWCN structures were held rigid, with the nanotube carbon atoms “frozen” at their equilibrium crystallographic positions. In both the GCMC and EMD simulations, the N2-N2 and N2-C interactions were modeled using an atom-site 6-12 Lennard-Jones (LJ) intermolecular potential

[( ) ( ) ]

φLJ(rij) ) 4ij

σij rij

12

-

σij rij

6

(2)

where rij is the separation distance between sites i and j, σij is the site collision diameter, and ij is the well depth. The intermolecular potential parameters in Table 1 were obtained by Bojan and Steele22 by fitting the CN values to the adsorption data of nitrogen onto planar graphite sheets. Therefore, any effect of carbon curvature on the N2-C intermolecular potential was not considered here.23,24 Inclusion of the quadrapole moment of nitrogen was found to have no significant effect on the adsorption results inside graphite slit pores at 303 K.18 To our knowledge, no such work has been reported for curved carbon surfaces, and while inclusion of the quadrapole moment of nitrogen might have an effect, it was not considered in this work. In the GCMC simulations, the diatomic gas molecules were modeled to be rigid, and in the EMD simulations, a harmonic bond potential with force constant K was used

1 φv(x) ) K(x - xe)2 2

(3)

where x is the distance between the nitrogen atoms in the (22) Bojan, M. J.; Steele, W. A. Langmuir 1987, 3, 1123-1127. (23) Kostov, M. K.; Cheng, H.; Cooper, A. C.; Pez, G. P. Phys. Rev. Lett. 2002, 89, article no. 146105. (24) Klauda, J. B.; Jiang, J.; Sandler, S. I. To be submitted for publication, 2003.

molecule and xe is the equilibrium bond distance for nitrogen, as given in Table 1.25 The value of K derived from experiment26 is also given in Table 1. Due to the stiffness of the bond, a small time step of 0.5 fs was used in the EMD simulations. Good agreement was observed between the energies and adsorbate structures obtained from independent GCMC and EMD simulations. This in accord with the previous work that shows the bond flexibility of simple homonuclear diatomic molecules does not have a significant effect on the bulk fluid properties.27,28 III. Simulation Methodology A. GCMC Simulations. GCMC29,30 (µVT) simulations were performed to calculate adsorption isotherms and the structure of the adsorbate within the SWCN. The chemical potential was related to the pressure in the bulk phase using the Peng-Robinson31 equation of state with published parameters for molecular nitrogen.32 In the simulations of adsorption, the system was equilibrated for 107 trial moves of molecule displacement, rotation, creation, and deletion. After equilibration, the adsorption density was sampled for another 107 trial moves. For these simulations, a SWCN with a length of 85.2 Å (40 unit cells) was used and periodic boundary conditions were applied only in the axial direction of the nanotube. The adsorbate-adsorbate potential was truncated at 2.5σN-N (8.3 Å), and a cylindrical cutoff length of 4 unit cells (8.52 Å) was used to truncate the adsorbate-adsorbent interactions. The truncation accounts for only 1-3% of the total energy, depending on the conditions. The block transformation technique29 was used to determine the statistical fluctuation in the number of molecules per unit cell, σ〈N〉, which was 1.0 G) X* - 1

( )

(10)

(11)

where FR ) F/FC is the reduced density, TR ) T/TC is the reduced temperature, and, for nitrogen, δ ) 0.012 24 and X* ) 3.261. In developing these relations, Lee et al.35 used experimental self-diffusivities for 27 fluids including nitrogen. They used 519 data points for eq 10 and 526 data points (9 for nitrogen) for eq 11. IV. Results and Discussion A. Adsorption. The results for the adsorption isotherms at 100 K for all the SWCNs studied are shown in Figure 1. There is a sharp initial increase in loading with pressure at very low pressures for the nanotubes with small diameters; the location of this sharp increase increases with the tube diameter. The location is a result of the stronger interactions of the adsorbate with the SWCN with increasing tube curvature (as a result of the decreasing nanotube diameter). However, the small (11, 0) SWCN shows quantitatively different adsorption behavior from that of the other SWCNs, as saturation is achieved at a much lower pressure at 100 K. Figure 2 also shows that the adsorption isotherms at 298 K are similar for the (15, 0), (17, 0), and (20, 0) SWCNs up to ∼5 bar (with a loading of ∼0.75 molecules/unit cell), suggesting a weak effect of curvature of the SWCNs on the adsorption of nitrogen at high temperatures and low loadings. As expected, at both temperatures, the saturation adsorption capacity increases with increasing nanotube diameter and hence increasing pore volume. As mentioned previously, all simulations were performed on a single SWCN with periodic boundary conditions along the axial direction, mimicking infinite length of the nanotubes. Therefore, end-effects and interstitial adsorption were not considered. An adsorbate molecule at the end of a finite length nanotube would experience (35) Lee, H.; Thodos, G. Ind. Eng. Chem. Fundam. 1983, 22, 17-26.

Adsorption and Diffusion of Molecular Nitrogen

Figure 1. Adsorption isotherms of N2 in SWCNs at 100 K.

different wall interactions than those experienced by molecules in the central region of the nanotube. However, the region where the end-effect is important only extends to a few molecular diameters.36 The length of the nanotubes made in the laboratory ranges from a few microns to centimeters,37 which is 3-7 orders of magnitude larger than the length of the region where end-effects are important. Therefore, end-effects should have a negligible effect on the adsorption isotherms reported in this work. There has been ongoing work to determine the extent of adsorption in the interstitial channels. Experimental adsorption studies by Talapatra et al. concluded that Xe and Ne do not adsorb on the interstitial channels.38 An experimental study of nitrogen and oxygen adsorption on carbon nanotube bundles showed that a location inside the nanotubes is more favorable for adsorption than the interstitial channels.39 Shi et al. performed simulations reproducing experimental isosteric heats, showing that the gases do adsorb in the interstitial channels of heterogeneous nanotube bundles.40 In a similar study of hydrogen adsorption in nanotube arrays by varying the distance between adjacent nanotubes and for widely spaced nanotubes, Yin and co-workers41 found that (36) Maddox, M.; Ulberg, D.; Gubbins, K. E. Fluid Phase Equilib. 1995, 104, 145-158. (37) Zhu, H. W.; Xu, C. L.; Wu, D. H.; Wei, B. Q.; Vajtai, R.; Ajayan, P. M. Science 2002, 296, 884-886. (38) Talapatra, S.; Zambano, A. Z.; Weber, S. E.; Migone, A. D. Phys. Rev. Lett. 2000, 85, 138-141. (39) Fujiwara, A.; Ishii, K.; Suematsu, H.; Kataura, H.; Maniwa, Y.; Suzuki, S.; Achiba, Y. Chem. Phys. Lett. 2001, 336, 205-211. (40) Shi, W.; Johnson, J. K. Phys. Rev. Lett. 2003, 91, Art. No. 015504. (41) Yin, Y. F.; Mays, T.; McEnaney, B. Langmuir 2000, 16, 1052110527.

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Figure 2. Adsorption isotherms of N2 in SWCNs at 298 K.

Figure 3. Adsorption energies of nitrogen in SWCNs of varying diameters: (a) (20, 0); (b) (17, 0); (c) (15, 0); (d) (11, 0).

adsorption occurs entirely in the interstitial channels. These studies show that interstitial adsorption depends on the homogeneity of the bundle, the adsorbent size, and the spacing between the nanotubes along with the adsorption conditions. Consequently, interstitial adsorption should be taken into account when making comparisons between experiment and the results reported in this work. As shown in Figure 3, the adsorption energies in SWCNs increase with increasing curvature (i.e., in the order (20, 0), (15, 0), (17, 0), and (11, 0)) at both 100 and 298 K. For reference, the adsorption energies for molecular nitrogen on graphitized carbon black at 77 K is 11 kJ/mol.42

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All of the isotherms are non-Langmuirian, which is suggestive of either “multilayer” adsorption or a restructuring of the adsorbed layer with increased loading. This motivated an examination of the adsorbate structure inside the SWCNs. Equilibrium snapshots of SWCNs near saturation loading and at 50% of saturation loading at 100 K (the points marked in Figure 1 by circles and squares, respectively) are shown in Figure 4. Figure 4a and b shows equilibrium snapshots of the locations of the centers-of-mass of the nitrogen molecules in the (20, 0) SWCN; in Figure 4b, two layers, an annular layer and a core, are clearly visible. (Note that the outer circles

correspond to the carbons of the SWCNs.) Similar structures are observed in the (17, 0) SWCN (Figure 4c and d), which shows bilayer formation at the higher pressure. The (15, 0) SWCN does not exhibit bilayer formation, and consequently, the adsorption isotherm is different from the other nanotubes at 100 K near saturation loading. Figure 4f is a snapshot of the locations of the nitrogen molecules in a cross section of this nanotube at 100 K and 10 bar. It is seen that the annular region is not filled homogeneously; instead, the molecules are grouped into four distinct, cylindrically symmetric regions. In contrast, the equilibrium snapshot shown in Figure 4e shows a more homogeneous distribution of molecules, as is the case for the other nanotubes and loadings studied here. This is similar to the inhomogeneous density distribution observed by Jiang et al.43 in Monte Carlo simulations of nitrogen in the C168 Schwarzite structure at high loadings, where there was a discontinuous distribution of the centers-of-mass, first in the small pores and then at higher loadings in the large pores. This “grouping” of molecules is discussed in more detail in the next section. For the small (11, 0) SWCNs, only a cylindrical core of adsorbed molecules is present up to saturation loading, as shown in Figure 4g and h. Insight into the structure of the adsorbed phase can be obtained from an examination of the potential energy surface of the nitrogen inside the nanotube. Figure 5 shows the net interaction energy for a single nitrogen molecule as a function of the radial distance from the nanotube centerline. The solid curve is the interaction energy between the SWCN and a single nitrogen test molecule with its axis parallel to the nanotube axis. The regions marked as R and β are the energy minima of the nitrogen molecules, and the locus of such points gives rise to an annulus where the molecules preferentially locate. The radius of the annular layer decreases with decreasing nanotube diameter. For the SWCN with the smallest diameter, there is only one potential minimum located at the center. Enhancement of the adsorption energy with increasing carbon curvature is evident, as the depths of the potential minima, R and β, decrease with decreasing nanotube diameter. Also shown in Figure 5 as the dashed curve is the interaction energy between the nitrogen molecules in the annular layer and a single nitrogen test molecule as a function of the radial distance, r, from the center of the nanotube. The configuration of the nitrogen molecules in the annular layer was obtained from an equilibrium molecular configuration consisting of 120 nitrogen molecules for both the (20, 0) and (17, 0) SWCNs. The minimum labeled as γ is the location of the additional layer of nitrogen molecules at higher loadings. Above a threshold value, which depends predominantly on the nanotube diameter and temperature, it is energetically favorable for molecules to occupy the central region of the nanotube as a result of steric hindrance in the outer annular region. Large repulsive interaction energies for the test molecule between the tube centerline and the annular layer for the smaller (15, 0) SWCN result in no adsorbed central layer in this nanotube at high loadings. For the (11, 0) SWCN, the potential minima, R and β, merge, and this results in only a cylindrical core of the adsorbate at all loadings. The parameters obtained from fits of the double Langmuir model (eq 4) to the adsorption isotherms are given in Table 2. Note that there is significant adsorption

(42) Kruk, M.; Li, Z. J.; Jaroniec, M.; Betz, W. R. Langmuir 1999, 15, 1435-1441.

(43) Jiang, J. W.; Klauda, J. B.; Sandler, S. I. Langmuir 2003, 19, 3512-3518.

Figure 4. Equilibrium snapshots of N2 in SWCNs at 100 K. The state points are marked by circles or squares in Figure 1. The nanotube diameters are to scale, and for clarity, the sizes of the nitrogen and carbon atoms shown are 20% of σN-N and σC-C, respectively.

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Figure 5. Interaction potential curves of N2 in SWCNs: (a) (20, 0); (b) (17, 0); (c) (15, 0); (d) (11, 0). The solid and dashed curves represent the N2-SWCN and N2-N2 interaction potentials, respectively. σC-N is shown by double-headed arrows at the potential minima, R and β. Table 2. Parameters in the Double Langmuir Model at 100 and 298 K for N2 Adsorption in SWCNs T (K) SWCN type 100

298

(20, 0) (17, 0) (15, 0) (11, 0) (20, 0) (17, 0) (15, 0) (11, 0)

Nmax 1

k1 (bar-1)

Nmax 2

k2 (bar-1)

3.35 2.39 1.65 4.85 × 10-1 2.95 1.83 1.28 4.57 × 10-1

1.83 × 9.21 × 103 7.03 × 104 2.27 × 109 0.05 0.11 0.21 9.65

1.10 1.01 4.10 × 10-1 8.53 × 10-2 1.91 2.18 7.58 × 10-1 5.75 × 10-2

4.61 × 101 1.18 7.94 × 101 7.66 × 106 6.65 × 10-4 3.34 × 10-4 1.43 × 10-3 0.12

103

at low pressures, as indicated by the relatively high values of k1, which increases with decreasing tube radius. At is of the same order both 100 and 298 K, the value of Nmax 1 for the (17, 0) and (20, 0) SWCNs, of magnitude as Nmax 2 suggesting that both “layers” make comparable contributions to the maximum loading. However, for the (15, 0) and Nmax differ by 1 order of and (11, 0) SWCNs, Nmax 1 2 magnitude, suggesting that the dual Langmuir isotherm reflects reorganization in the adsorbed layer rather than the existence of two distinguishable adsorption layers. This topic is explored further in the study of the mobility of the adsorbed molecules, which follows. B. Self-Diffusivities. Figure 6 shows averaged mean square displacements (MSDs) for selected EMD runs and illustrates the range of behavior observed with variations in nanotube geometry, temperature, and loading. There are three regimes in the mean square displacement of the diffusant: a ballistic regime for short time scales, followed by a subdiffusive regime, and finally a Fickian diffusive regime in which the MSDs are proportional to time (shown in the inset of Figure 6). Similar distinct regions have been observed for diffusion in polymers and colloids.33,44 The turnover point from the subdiffusive regime to the Fickian regime is taken to occur when the slope

d log〈(z(t) - z(0))2〉 ) 1 ( 0.15 d log t

(12)

is within 15% of the expected linear behavior; this is

Figure 6. Mean square displacements for nitrogen in SWCNs on a log-log plot. The solid lines have a slope of 1, and the inset shows the plots on a linear scale.

indicated by the arrows in Figure 6. Equation 6 is then used to obtain self-diffusion coefficients, which are calculated from the MSDs for 500 ps beyond this turnover point. As expected, the self-diffusion coefficients (shown in Figure 7) initially decrease with increasing loading, as occurs in a bulk gas or fluid with increasing density. However, this behavior differs markedly depending on the temperature and nanotube diameter. For example, the self-diffusivity at 100 K in the (20, 0) SWCN, the nanotube with the largest diameter (15.66 Å) studied, can be divided into two regions, as shown in Figure 7a. The decrease in self-diffusivity with increased loading shows a variation in slope between adjacent loadings upon formation of a central core of nitrogen molecules, whereupon the nitrogen self-diffusivity becomes less sensitive to increases in loading. By following individual molecules (44) Indrani, A. V.; Ramaswamy, S. Phys. Rev. Lett. 1994, 73, 360363.

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Figure 7. Self-diffusivity of N2 in (a) a (20, 0) SWCN with a diameter of 15.66 Å, (b) a (17, 0) SWCN with a diameter of 13.31 Å, (c) a (15, 0) SWCN with a diameter of 11.74 Å, and (d) a (11, 0) SWCN with a diameter of 8.61 Å.

in the simulation at a loading of 3.25 molecules/unit cell, it was found that the molecules in the central region diffuse much more rapidly than those in the annular layer. However, separate diffusion coefficients for each region could not be obtained, as there was significant molecular exchange between the two regions. The main conclusion is that the molecular motion in the axial direction is enhanced when a core region is formed. With further increases in loading, the number of molecules in the central core increases and only a moderate increase in the selfdiffusivity is observed. The self-diffusivity at even higher loadings will be determined by a competing effect of the faster motion of molecules at the center and the increasing hindrance to motion within the central core. Qualitatively similar behavior for the self-diffusion coefficient at 100 K is seen in the (17, 0) SWCN (diameter of 13.31 Å), as shown in Figure 7b. Again, formation of the central layer results in a changing slope of the self-diffusion coefficient with increased loading. The self-diffusivity at a loading of 3.0 molecules/unit cell is not reported, as the MSD did not reach the Fickian diffusion limit during the sampling period. The (15, 0) and (11, 0) nanotubes are too narrow to allow bilayer formation. Consequently, in these nanotubes, the self-diffusivity (Figure 7c and d) decreases with increased loading, as is typical for bulk fluids. The self-diffusivity of nitrogen inside SWCNs with varying pressure at fixed temperature is shown in Figure 8. The value of the self-diffusivity inside nanotubes at infinite dilution is ∼3 orders of magnitude faster than that in silicalite (Ds (298 K, c ) 0, silicalite) ) 1.6 × 10-4 cm2/s),45 while at finite loading the difference is only ∼1 order of magnitude (Ds (300 K, c, silicalite) ∼ 10-4 cm2/ s).45 The dramatic drop of the self-diffusivity with increased loading and similar differences in the order of magnitude at both infinite dilution and finite loading were observed previously for H2 and CH410 and Ne and Ar.11 At a fixed adsorbate density, the nitrogen self-diffusivity is similar (45) Makrodimitris, K.; Papadopoulos, G. K.; Theodorou, D. N. J. Phys. Chem. B 2001, 105, 777-788.

Figure 8. Self-diffusivity of N2 in SWCNs of varying diameters at (a) 100 K and (b) 298 K.

in the (15, 0), (17, 0), and (20, 0) nanotubes but markedly larger in the (11, 0) nanotube (Figure 9). As shown in Figure 4g and h, nitrogen in the (11, 0) SWCN forms a structure similar to the interior “core” layer observed in the larger tubes at high pressures rather than the annular

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Figure 9. Self-diffusivity of N2 in SWCNs of varying diameters at constant density obtained by interpolating the simulation results. Table 3. Comparison of Diffusion Coefficients of Nitrogen in Constrained SWCNs and in the Bulk Phase density Dnanotubeb T SWCN loadinga (K) type (per unit cell) (g/cm3) (cm2/s) 100 (20, 0) (17, 0) (15, 0) (11, 0) 298 (20, 0) (17, 0) (15, 0) (11, 0)

c

2.0 3.25 2.0 2.75 1.5 2.0 0.25 0.625 2.0 3.25 2.0 3.0 1.5 2.0 0.25 0.625

0.3674 0.5970 0.5614 0.7719 0.5936 0.7914 0.2521 0.6301 0.3674 0.5970 0.5614 0.7719 0.5936 0.7914 0.2521 0.6301

5.3 × 10-4 2.4 × 10-4 5.1 × 10-4 1.5 × 10-4 2.6 × 10-4 1.4 × 10-5 2.2 × 10-3 1.7 × 10-3 1.4 × 10-3 6.9 × 10-4 1.2 × 10-3 3.8 × 10-4 9.9 × 10-4 7.5 × 10-4 1.6 × 10-2 2.5 × 10-3

Dbulkb (cm2/s)

Dbulk (cm2/s)

2.7 × 10-4 1.2 × 10-4 1.3 × 10-4 5.6 × 10-5 1.2 × 10-4 5.2 × 10-5 4.2 × 10-4 1.1 × 10-4 5.9 × 10-4 2.8 × 10-4 3.2 × 10-4 1.6 × 10-4 2.9 × 10-4 1.5 × 10-4 9.3 × 10-4 2.5 × 10-4

2.4 × 10-4 c 1.0 × 10-4 c 1.2 × 10-4 c 4.3 × 10-5 c 1.1 × 10-4 c 3.8 × 10-5 c 3.5 × 10-4 d 9.0 × 10-5 c 7.9 × 10-4 c 3.9 × 10-4 c 4.5 × 10-4 c 2.0 × 10-4 c 4.0 × 10-4 c 1.8 × 10-4 c 1.1 × 10-3 d 3.5 × 10-4 c

a Range of loading in MD simulations. b From MD simulations. From eq 10. d From eq 11.

structure that appears in the larger SWCNs at lower pressures. The formation of such a core region, whether it is a consequence of direct confinement as in this narrow SWCN or of bilayer formation as in the (17, 0) and (20, 0) SWCNs, results in substantially greater adsorbate mobility than that in the loaded annular adsorbate layers. This observation is in qualitative agreement with the findings of Skoulidas et al.10 Increasing the temperature at fixed loading increases the self-diffusivities. The diffusivities and their dependence on loading is qualitatively similar at 298 K to those at 100 K for the (20, 0) and (17, 0) nanotubes in which bilayers form (Figure 7a and b). Also, the diffusivities in the (15, 0) and (11, 0) SWCNs are similar at these two temperatures (Figure 7c and d). Skoulidas et al.10 observed rapid transport of H2 and CH4 inside carbon nanotubes, which they attributed to the relative smoothness of the nanotube potential energy surface (PES), resulting in low activation energy for diffusion. We find that the self-diffusivity of N2 inside carbon nanotubes is generally higher than that in the unconfined bulk fluid at comparable densities and temperatures, as shown in Table 3. An exception to this observation was found for the (15, 0) carbon nanotube for nitrogen loadings close to saturation at 100 K, which is to be discussed in detail next. As the nanotube diameter decreases and the wall curvature increases, the PES becomes smoother, as shown in Figure 10. The increased smoothness of the PES and the confinement of nitrogen

Figure 10. Minimum energy, Eact, required to move a nitrogen atom along the nanotube axis, calculated at a distance of 21/6σC-N (3.77 Å) from the nanotube wall for a SWCN (a) with a diameter of 8.61 Å (11, 0) and (b) with a diameter of 13.31 Å (17, 0).

molecules in the core region results in self-diffusivities up to 1 order of magnitude higher for the (11, 0) SWCNs compared to the bulk at comparable densities. For reference, the activation energies of CH4 and H2 in a (10, 10) SWCN with a diameter of 13.6 Å are 0.054 and 0.066 kJ/mol, respectively. (Separately, there is good agreement between the bulk phase self-diffusivities obtained from simulation and the correlations of eqs 10 and 11 for the self-diffusivity in the bulk phase (Table 3).) In the (15, 0) SWCN at 100 K, the self-diffusivity decreases to a value of 1.4((0.1) × 10-5 cm2/s at a loading of up to 2 molecules/unit cell (Figure 7c), which is considerably lower than the self-diffusivity in the bulk, 5.2((0.1) × 10-5 cm2/s (Table 3). This is a consequence of the structural rearrangement in which the adsorbed nitrogen molecules were found to be packed into four distinct, cylindrically symmetric groups in the annular region, as was also observed in the GCMC results presented in Figure 4f. This semiordered structure of nitrogen in the SWCNs is described by the radial density distribution, F(r), and the functions g(|φ|) and g(|z|) that were defined earlier. There is little difference in the values of F(r) for the two loadings of 1.75 and 2 molecules/unit cell shown in Figure 11a. The location of the peak corresponds to the radius of the annular layer, which is nearly identical for both of the loadings. The spread of the distribution corresponds to the width of the annular region, which is also very similar at both loadings, as the molecules have the same thermal energy. The peak heights are directly proportional to the loading. However, there is a difference in the functions g(|φ|) and g(|z|) at the two densities plotted in Figure 11b and c. Figure 11b shows

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Figure 12. Semiordered structure of nitrogen in a SWCN of 11.74 Å at 100 K and a loading of 2 molecules/unit cell. Nitrogen is shown correct to its molecular size, and a color scheme is used for better visualization.

The spatial configuration of the nitrogen molecules in the dense, solidlike phase can be seen in more detail in Figure 12. The structure is “wormlike” with four intertwined rows of nitrogen molecules. A similar disorderto-order transition has been reported previously for the water molecules inside carbon nanotubes, where alignment of the water molecules into spiral columns was observed by varying the nanotube diameter.46 Each quadrant in the cross-sectional view in Figure 4f corresponds to a nitrogen column in the wormlike structure. The density of nitrogen inside the nanotube is 0.79 g/cm3. Nitrogen is a liquid in the bulk at the same density and temperature. This density is much less than that of crystalline nitrogen in the β phase (∼1.4 g/cm3 at 300 K and 2.8 GPa),47 which has a hexagonal close packing and exists from 33.6 K up to the melting curve.48 V. Conclusions

Figure 11. Structural properties of nitrogen in a SWCN of 11.74 Å at 100 K: (a) F(r); (b) g(φ); (c) g(z).

g(|φ|) for a (15, 0) SWCN at 100 K for loadings of 1.75 and 2 molecules/unit cell. At a loading of 2 molecules/unit cell, g(|φ|) indicates orientational correlations between molecules at 45° increments, consistent with Figure 4e, while, at a loading of 1.75 molecules/unit cell, g(|φ|) indicates negligible orientational correlation. Figure 11c shows g(|z|) plotted for half of the nanotube length. At 1.75 molecules/ unit cell, the adsorbent phase is fluidlike, but at a loading of 2 molecules/unit cell, a periodic structure is observed. The period of oscillation of g(|z|) at 2 molecules/unit cell is approximately half that at 1.75 molecules/unit cell, suggesting that nitrogen molecules align preferentially perpendicular to the nanotube axis in the former and parallel to the axis in the latter.

On the basis of our simulations, molecular nitrogen is more strongly adsorbed in SWCNs than on graphite. Due to the nanoscale confinement, an interesting range of nanoscale structuring occurs depending on the diameter of the SWCN and temperature. For tubes with moderate diameters, nitrogen was found to occur in an annulus formed near the carbon walls, rather than being homogeneously distributed throughout the nanotube diameter. A bilayer structure of nitrogen molecules is formed in the (20, 0) and (17, 0) SWCNs at high loadings due to the steric hindrance in the annular layer. This results in a central core of nitrogen molecules with higher mobility than that of the molecules in the annular layer. The formation of a semiordered, wormlike structure at loadings close to saturation was observed in the (15, 0) SWCN at 100 K, which greatly suppresses the molecular diffusion of nitrogen compared to its bulk diffusion under similar conditions. The highest diffusivities at loadings near saturation were found in the nanotubes with the smallest diameters, as the nitrogen molecules are confined to a corelike arrangement, similar to the central core in the (20, 0) and (17, 0) SWCNs. For nanotubes with smaller (46) Noon, W. H.; Ausman, K. D.; Smalley, R. E.; Ma, J. P. Chem. Phys. Lett. 2002, 355, 445-448. (47) Meijer, E. J.; Frenkel, D.; Lesar, R. A.; Ladd, A. J. C. J. Chem. Phys. 1990, 92, 7570-7575. (48) Mills, R. L.; Schuch, A. F. Phys. Rev. Lett. 1969, 23, 1154.

Adsorption and Diffusion of Molecular Nitrogen

diameters, we would expect the qualitative behavior of nitrogen to be similar to that found in the (11, 0) SWCN until the nanotube diameter is sufficiently small such that nitrogen adsorption is energetically unfavorable. Our results suggest that the localization of nitrogen into the core region, either by direct confinement in narrow tubes or by bilayer formation at higher loadings in the larger nanotubes, can significantly enhance molecular diffusion in SWCNs at higher loadings. The formation of this core region is a consequence of steric confinement of the adsorbate by the carbon nanotube. We note, however, that the quadrapole moment of nitrogen was not taken into account. It would be interesting to see if its inclusion would affect the results, possibly at low temperatures and high loadings where the existence of the wormlike structure was observed. Moreover, these are the results of simulation using atomic potential parameters optimized

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for graphite. Recently, Kostov et al.23 developed SWCN curvature dependent potential parameters for hydrogen, and ab initio quantum mechanical calculations have found a stronger interaction between nitrogen and carbon in curved carbon structures than in graphite.24 This suggests that nanotube curvature may directly affect the SWCNadsorbate atomistic potentials, which is not considered herein but will be the subject of future research based on the intermolecular potentials obtained from quantum chemical calculations. Acknowledgment. This research was supported by Grant No. EEC-0085461 from the National Science Foundation. The authors are grateful to Jeffery Klauda and Jaeeon Chang for useful discussions. LA036432F