Self-Diffusion of Methane in Single-Walled Carbon Nanotubes at Sub- and Supercritical Conditions

Transcription

1 Langmuir 2004, 20, Self-Diffusion of Methane in Single-Walled Carbon Nanotubes at Sub- and Supercritical Conditions Dapeng Cao and Jianzhong Wu* Department of Chemical and Environmental Engineering, University of California, Riverside, California Received December 16, In Final Form: February 18, 2004 The diffusivities of methane in single-walled carbon nanotubes (SWNTs) are investigated at various temperatures and pressures using classical molecular dynamics (MD) simulations complemented with grand canonical Monte Carlo (GCMC) simulations. The carbon atoms at the nanotubes are structured according to the (m, m) armchair arrangement and the interactions between each methane molecule and all atoms of the confining surface are explicitly considered. It is found that the parallel self-diffusion coefficient of methane in an infinitely long, defect-free SWNT decreases dramatically as the temperature falls, especially at subcritical temperatures and high loading of gas molecules when the adsorbed gas forms a solidlike structure. With the increase in pressure, the diffusion coefficient first declines rapidly and then exhibits a nonmonotonic behavior due to the layering transitions of the adsorbed gas molecules as seen in the equilibrium density profiles. At a subcritical temperature, the diffusion of methane in a fully loaded SWNT follows a solidlike behavior, and the value of the diffusion coefficient varies drastically with the nanotube diameter. At a supercritical temperature, however, the diffusion coefficient at high pressure reaches a plateau, with the limiting value essentially independent of the nanotube size. For SWNTs with the radius larger than approximately 2 nm, capillary condensation occurs when the temperature is sufficiently low, following the layer-by-layer adsorption of gas molecules on the nanotube surface. For SWNTs with a diameter less than about 2 nm, no condensation is observed because the system becomes essentially one-dimensional. 1. Introduction Understanding the diffusion of gas molecules in micropores is fundamental to industrial design of catalytic and separation processes that utilize microporous materials including zeolites, activated carbons, and rapidly emerging novel nanostructured materials such as MCM- 41. Despite a long history of research, many aspects of the transport phenomena in porous materials remain poorly understood. 1 The problem is difficult and challenging because conventional transport theories are most adequate for describing macroscopic phenomena while the dynamics of molecules in small pores is often strongly affected by the interactions between gas molecules and the confining surfaces. One simple yet broadly applied model for the transport of gases in a micropore is given by the Knudsen flow where both the interaction between gas molecules and the wall potential are neglected. It was shown very recently that when the attraction from the surface is significant as in a typical nanoporous material, the Knudsen model greatly overpredicts the transport coefficients even in the limit of low density where the free path of each gas molecule is much longer than the characteristic dimension of the micropore. 2 The situation becomes even more complicated when the interactions between gas molecules are also considered. 3,4 Single-walled carbon nanotubes (SWNTs) provide an ideal model system for investigating the microscopic details of gas transport in small pores. Since they were first discovered by Iijima and Ichihashi 5 and independently * To whom all correspondence should be addressed. jwu@engr.ucr.edu. (1) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena, 2nd ed.; J. Wiley: New York, (2) Jepps, O. G.; Bhatia, S. K.; Searles, D. J. Phys. Rev. Lett. 2003, 91. (3) Mon, K. K.; Percus, J. K. J. Chem. Phys. 2002, 117, (4) Davis, H. T. In Fundamentals of Inhomogeneous Fluids; Henderson, D., Ed.; M. Dekker: New York, by Bethune et al. 6 about a decade ago, these novel materials have attracted enormous research interests because of their unique geometric, mechanical, and electromagnetic properties. SWNTs are promising for a wide variety of applications that range from probing tips in scanning force/tunneling microscopy to gas storage media for fuel cells. 7 While the lengths of SWNTs are about several micrometers or even longer, their diameters vary from approximately 0.5 to 4 nm. The diameter and electronic properties of SWNTs are often specified by the nanotube indices, (n, m), where n and m are integers defined by mapping the surface of a cylindrical nanotube to a planar graphite sheet. 8 Diffusion of gases in SWNTs has been studied before. For instance, based on classical molecular dynamic simulations, Mao and Sinnott reported extensive results on the self-diffusions of pure hydrocarbon gases and their mixtures through various SWNTs ranging in length from 2to8nm They also investigated both dynamic and diffusive flows of gas molecules through short SWNTs and predicted novel spiral diffusion paths for the diffusion of nonspherical molecules. 12 Atomic-scale molecular dynamic simulations have also been applied to investigating the self- and transport diffusion coefficients of inert gases, hydrogen, and methane in infinitely long SWNTs. 13,14 It (5) Iijima, S.; Ichihashi, T. Nature 1993, 363, 603. (6) Bethune, D. S.; Kiang, C. H.; Devries, M. S.; Gorman, G.; Savoy, R.; Vazquez, J.; Beyers, R. Nature 1993, 363, 605. (7) Haddon, R. C. Acc. Chem. Res. 2002, 35, 997. (8) White, C. T.; Robertson, D. H.; Mintmire, J. W. Phys. Rev. B 1993, 47, (9) Mao, Z. G.; Sinnott, S. B. J. Phys. Chem. B 2001, 105, (10) Mao, Z. G.; Sinnott, S. B. J. Phys. Chem. B 2000, 104, (11) Mao, Z. G.; Garg, A.; Sinnott, S. B. Nanotechnology 1999, 10, 273. (12) Mao, Z. G.; Sinnott, S. B. Phys. Rev. Lett. 2001, 89, No (13) Ackerman, D. M.; Skoulidas, A. I.; Sholl, D. S.; Johnson, J. K. Mol. Sim. 2003, 29, 677. (14) Skoulidas, A. I.; Ackerman, D. M.; Johnson, J. K.; Sholl, D. S. Phys. Rev. Lett. 2002, /la036375q CCC: $ American Chemical Society Published on Web 04/02/2004

2 3760 Langmuir, Vol. 20, No. 9, 2004 Cao and Wu where b ) nm is the C-C bond length and m and n are the nanotube indices. Both carbon atoms and methane molecules are treated as spherical particles. The pair interaction between gas molecules is specified by the cut-and-shifted Lennard-Jones (LJ) potential φ ff (r) ) { φlj (r) -φ LJ (r c ) r < r c (2) 0 r g r c Figure 1. Computer-generated images of armchair singlewalled carbon nanotubes (SWNTs). was found that the transport rate of an atomic gas in nanotubes is one order of magnitude higher than that in conventional porous materials such as zeolites. 14 Simulations of the dynamic flow of helium and argon atoms through SWNTs revealed that the flow slows rapidly as the temperature falls especially for gas molecules with large size and high molecular weight. 15 Application of nonequilibrium molecular dynamic simulations to the Poiseuille flow in SWNTs predicted very long hydrodynamic slip lengths even for fluids that are strongly adsorbed in the pores. 16 It has been speculated that both ultrafast diffusion and long hydrodynamic slip lengths of fluids in SWNTs are due to the inherent smoothness of the graphite surface potential. 13,16 The purpose of this work is to investigate the effects of temperature and pressure on the diffusivity of gas molecules in idealized micropores with explicit consideration of the intermolecular interaction as well as the surface potentials. The adsorption isotherms of methane in three armchair SWNTs with the nanotube indices of (15, 15), (25, 25), and (30, 30) have been calculated using the conventional grand Monte Carlo (GCMC) simulations. 17 These SWNTs are significantly larger than those investigated before. Classical molecular dynamic (MD) simulations are then used to predict the self-diffusion coefficients in the direction of nanotube axis at pressures up to 8 MPa and temperatures both above and below the bulk critical value. 17 We discuss the relations between the diffusivity of gases in SWNTs and the equilibrium microscopic structures, in particular in terms of the layering transitions and capillary condensation of gas molecules. 2. Computational Details 2.1. Potential Models. We assume that SWNTs are infinitely long and all carbon atoms of the nanotubes are fixed in their ideal lattice positions. As suggested by a previous inverstigation, 18 the local dynamics of carbon atoms has little influence on the diffusion of gas molecules within SWNTs. Figure 1 illustrates the computer-generated images of three types of SWNTs considered in this work, with the nanotube indices given by (15, 15), (25, 25), and (30, 30). The diameters of these nanotubes are, respectively, 2.034, 3.39, and nm, calculated from 8 σ c ) b π 3(m2 + mn + n 2 ) (1) (15) Tuzun, R. E.; Noid, D. W.; Sumpter, B. G.; Merkle, R. C. Nanotechnology 1996, 7, 241. (16) Sokhan, V. P.; Nicholson, D.; Quirke, N. J. Chem. Phys. 2002, 117, (17) Frenkel, D.; Smit, B. Understanding Molecular Simulation: From Algorithms to Applications, 2nd ed.; Academic Press: San Diego, (18) Koga, K.; Gao, G. T.; Tanaka, H.; Zeng, X. C. Nature 2001, 412, 802. where r is the center-to-center distance and r c ) 5σ ff is the cutoff radius; i.e., beyond this separation the interaction between gas molecules is assumed to be vanished. The cut-and-shifted potential has been extensively used in the literature for molecular simulations. 19,20 In eq 2, the subscript ff stands for interactions between two fluid molecules, and φ LJ stands for the full LJ potential given by φ LJ (r) ) 4ɛ ff[( σ ff r ) 12 - ( σ ff r ) 6] (3) where ɛ ff and σ ff are, respectively, the energy and size parameters of the LJ potential. For methane, ɛ ff/k B ) K, where k B is the Boltzmann constant and σ ff ) nm. 21,22 The total potential for the interactions between methane molecules and the confining SWNT is calculated using the sitesite interaction method 23 N f N φ fw ) 4ɛ fw i)1 j)1 carbon[( σ 12 fw - r ij ) ( σ fw (4) r ij ) 6] where N f is the total number of methane molecules, N carbon is the total number of the carbon atoms at the wall, and r ij is the centerto-center distance between a methane molecule and a carbon atom from the SWNT. In eq 4, the subscript fw stands for interactions between a fluid molecule and the carbon wall. The cross energy and size parameters, ɛ fw and σ fw, are obtained from the Lorentz-Berthelot combining rules ɛ fw ) ɛ ff ɛ ww (5) σ fw ) (σ ff + σ ww )/2 (6) The energy and size parameters for carbon atom are ɛ ww/k B ) 28.0 K and σ ww ) 0.34 nm, respectively. 23 In eqs 5 and 6, the subscript ww stands for the interactions between two carbon atoms from the tubular wall. As for the interaction between gas molecules, the site-site potential in eq 4 is cut-and-shifted with the cutoff radius equal to 5σ fw Grand Canonical Monte Carlo (GCMC) Simulations. The grand canonical Monte Carlo simulations are used to calculate the adsorption isotherms of methane in SWNTs at various temperatures and pressures. At a given temperature and gas chemical potential, the corresponding bulk pressure of methane in equilibrium with the nanotube is solved from the modified Benedict-Webb-Rubin (MBWR) equation by Johnson et al. 24 In GCMC simulations, the temperature, the chemical potential and the pore volume are specified in advance, and the periodic boundary condition is imposed in the direction of nanotube axis (z direction). All space and energy variables in the simulation are reduced by the LJ parameters for methane. In particular, the reduced temperature is T* ) k BT/ɛ, the reduced axial length of the simulation box is L* ) L/σ, and the reduced cylindrical radius is R* ) R/σ. The volume of the simulation box is πr* 2 L* in dimensionless units. Typically, the simulation box (19) Jiang, S. Y.; Rhykerd, C. L.; Gubbins, K. E. Mol. Phys. 1993, 79, 373. (20) Cao, D. P.; Wang, W. C. Phys. Chem. Chem. Phys. 2001, 3, (21) Cao, D. P.; Wang, W. C.; Shen, Z. G.; Chen, J. F. Carbon 2002, 40, (22) Cao, D. P.; Wang, W. C.; Duan, X. J. Colloid Surf. Sci. 2002, 254, 1. (23) Cao, D. P.; Zhang, X. R.; Chen, J. F.; Wang, W. C.; Yun, J. J. Phys. Chem. B 2003, 107, (24) Johnson, J. K.; Zollweg, J. A.; Gubbins, K. E. Mol. Phys. 1993, 78, 591.

3 Self-Diffusion of Methane in Carbon Nanotubes Langmuir, Vol. 20, No. 9, contains about 600 gas molecules at equilibrium. The initial configuration is randomly generated with the number of gas molecules in the simulation box estimated from the bulk density. The new configurations are subsequently created according to the Metropolis algorithm by changing the position of a randomly selected gas molecule and by insertion or removal of molecules from the system. For each combination of temperature and chemical potential, configurations are generated, half of which are for the system to reach the equilibrium whereas the other configurations are divided into 10 blocks to average the pertinent structural and thermodynamic properties. 17 The uncertainties on the ensemble averages of the number of adsorbate molecules in the box and the total potential energy are estimated to be less than 2% according to the block error analysis. 17 For methane adsorption in (15, 15) and (30, 30) SWNTs at T ) 140 and 267 K, the uncertainties are also calculated by repeated simulations Molecular Dynamics (MD) Simulations. The average number densities of methane molecules obtained from GCMC simulations are used as the input for the canonical (NVT) MD simulations to calculate the parallel self-diffusion coefficient of methane inside SWNTs. In the MD simulations, the equations of motion are solved using the fifth-order Gear predictor-corrector algorithm 25 with the time step set to in dimensionless units, defined as t* ) t ɛ/mσ As in GCMC, the periodic boundary condition is applied only to the axial direction of the carbon nanotube (i.e., z direction). The temperature scaling method is used to maintain the fixed temperature. 17 The relaxation of initial configuration to equilibrium evolves time steps and another steps are used to sample the diffusion coefficient. The fixed carbon atoms are not included in the calculation of temperature for methane molecules. Because the motion of methane molecules is limited in both x and y directions, we consider only the diffusion in the axial direction of the nanotube, i.e, the z direction. For all nanotubes investigated in this work, we found that the motion of methane molecules follows the normal-mode diffusion; i.e., the selfdiffusion coefficients can be calculated from the Einstein relation for one-dimensional geometry 17 z 2 (t) ) 2D z t (7) where z 2 (t) represents the square of displacement in the z-direction as a function of time (t), D z is the z direction selfdiffusion coefficient and the brackets stand for the ensemble average. Figure 2 presents some representative results on the dependence of the mean-square displacement (MSD) as defined by eq 7 for the motion of methane molecules in the (15, 15) nanotube at 207 K. The uncertainties of diffusion coefficients were obtained by repeated calculations at the same simulation conditions. 3. Results and Discussion 3.1. Adsorption of Methane from GCMC Simulations. There have been a few reports on the adsorption of methane in the pores of SWNTs or on the out surfaces of SWNT bundles and interstices by means of both experiments and molecular simulations. Using adsorption volumetry and microcalorimetry, Muris et al. investigated the adsorption isotherms and phase transitions of methane and krypton in the interstitial channels and on the surfaces of close-end SWNT bundles at T ) 77 and 110 K. 26 Adsorption of methane on close-end SWNTs was also investigated by Talapatra et al. 27,28 Wang and co-workers performed molecular simulations and density-functional theory calculations on methane adsorption in organized arrays of SWNTs. 23,29 An optimized arrangement of SWNTs for methane storage at room temperature was also reported. 23 Tanaka et al. studied the adsorption of (25) Gear, C. W. Numerical Integration of Ordinary Differential Equations; Prentice Hall: New Jersey, (26) Muris, M.; Dufau, N.; Bienfait, M.; Dupont-Pavlovsky, N.; Grillet, Y.; Palmari, J. P. Langmuir 2000, 16, Figure 2. Mean-square parallel displacement of methane in (15, 15) SWNT at T ) 207 K. The curves are from molecular dynamic simulations and the straight lines are fitted using the Einstein equation (eq 7). The slopes 0.74, 0.31, and 0.38 correspond to the diffusion coefficients , , and m 2 /s, respectively. methane inside and outside of isolated SWNTs using a nonlocal density functional theory over a range of pore diameters and pressures. 30 The GCMC simulations presented in this work are focused on methane adsorption inside of infinitely long SWNTs at temperatures both above and below the critical point (T c ) K, P c ) 4.6 MPa). We select these simulation conditions in considering that the supercritical conditions are related to the potential application of SWNTs for methane storage while the confined methane at sub-critical temperatures may exhibit rich phase behavior that is greatly different from that in the bulk phase. Figure 3a,b presents the adsorption isotherms of methane in four SWNTs at T ) 118 and 140 K. The error bars are comparable to the symbol size according to the block-error analysis or repeated simulations. In both cases, the temperature is below the bulk critical temperature for methane (T c ) K). The amount of adsorption is defined by Γ ) m CH4 /m SWNT (8) where m CH4 is the (average) total mass of adsorbed methane and m SWNT is the mass of the SWNT. Two-step isotherms, corresponding to the capillary condensation of methane, are observed in (30, 30), (25, 25), and (20, 20) nanotubes. The jumps become less distinctive at 140 K for all cases, probably due to approaching the critical points of the capillary condensations. At 118 K, the approximate condensation pressures are 0.015, and MPa for the (20, 20), (25, 25), and (30, 30) SWNTs, respectively. Figure 3 indicates that the condensation pressure in- (27) Talapatra, S.; Migone, A. D. Phys. Rev. Lett. 2001, (28) Talapatra, S.; Migone, A. D. Phys. Rev. B 2002, 65. (29) Zhang, X. R.; Wang, W. C. Fluid Phase Equilib. 2002, 194, 289. (30) Tanaka, H.; El-Merraoui, M.; Steele, W. A.; Kaneko, K. Chem. Phys. Lett. 2002, 352, 334.

4 3762 Langmuir, Vol. 20, No. 9, 2004 Cao and Wu Figure 4. Excess adsorption isotherms of methane in various SWNTs at supercritical temperatures T ) 207, 237, and 267 K. The solid lines are for the guidance of the eye. Figure 3. Adsorption isotherms of methane in SWNTs of different sizes at (a) T ) 118 K and (b) T ) 140 K. The solid lines are for the guidance of the eye. creases with temperature and with the diameter of the confining nanotubes. At T ) 140 K, capillary condensation occurs at about 0.10, 0.22 and 0.27 MPa for (20, 20), (25, 25), and (30, 30) SWNTs, respectively. In terms of application of SWNTs for methane storage, it might be worth mentioning that at 118 K, the adsorbed amount in the (30, 30) nanotube reaches 430 g CH 4 /kg C at 0.08 MPa, nearly 3-folds of the storage capacity targeted by the U.S. Department of Energy. 31 Only Langmuir-type isotherms are observed for methane adsorption in the (15, 15) nanotube, implying capillary condensation is prohibited in small SWNTs. This is consistent with previous reports 32 that capillary condensation occurs only when the pore is lager enough to hold more than 3-4 layers of molecules. That is to say, in a pore with less than 2 layers of adsorption, capillary condensation is prohibited. Similar observations have also been reported by Maddox et al. on the adsorption of argon and nitrogen in a SWNT of 1.02 nm in diameter at 77K. 33 The adsorption isotherms of methane at supercritical temperatures are often represented by the excess amount of adsorption 34 Γ excess ) (m CH4 - m CH4 )/m SWNT (9) where m CH4 is the amount of methane if the confining (31) Lozano-Castello, D.; Alcaniz-Monge, J.; de la Casa-Lillo, M. A.; Cazorla-Amoros, D.; Linares-Solano, A. Fuel 2002, 81, (32) Gelb, L. D.; Gubbins, K. E.; Radhakrishnan, R.; Sliwinska- Bartkowiak, M. Rep. Progr. Phys. 1999, 62, Figure 5. Effect of temperature on the pressure at the maximum excess adsorption. The solid lines are for the guidance of the eye. space is filled with the coexisting bulk phase. Figure 4 shows the adsorption isotherms at temperatures T ) 207, 237, and 267 K, all above the bulk critical temperature for methane. The error bars estimated from block-error analysis 17 are smaller than the symbol size as shown in the case for T ) 267 K. In all these temperatures, the excess amount of adsorption increases with the nanotube diameter, and the disparity is more significant at high pressures. At supercritical conditions, no capillary condensation is observed even when the SWNTs are essentially fully loaded. Interestingly, the excess amount of adsorption exhibits a maximum at each temperature, suggesting an optimum pressure for methane storage. Figure 5 presents the approximate optimal pressures at three supercritical temperatures considered in this work. (33) Maddox, M. W.; Ulberg, D.; Gubbins, K. E. Fluid Phase Equilib. 1995, 104, 145. (34) Zhang, X. R.; Cao, D. P.; Chen, J. F. J. Phys. Chem. B 2003, 107, 4942.

5 Self-Diffusion of Methane in Carbon Nanotubes Langmuir, Vol. 20, No. 9, Figure 7. Snapshots of methane molecules inside the (30, 30) SWNT at T ) 118 K and different pressures. Figure 6. Pressure dependence of the radial density profiles of methane in the (30, 30) SWNT at (a) T ) 118 K and (b) T ) 207 K. Here the reduced density is defined as F* )Fσ 3 and the reduced distance from the tubular axis is r* ) r/σ. It is shown that the optimal pressure for methane storage increases monotonically with temperature. Extrapolation of the data points in Figure 5 to ambient conditions indicates that the optimum pressure for methane storage is around 4 MPa in (15, 15) and 8 MPa in (25, 25) and (30, 30) SWNTs, these values are well below that involved in conventional natural gas storage using high strength vessels (about MPa). To provide further insights on the adsorption behavior, we also calculated the microstructures of methane inside carbon nanotubes. Panels a and b of Figure 6 present, respectively, the local density profiles of methane in the (30, 30) nanotubes at T ) 118 and 207 K, below and above the critical temperature, respectively. At P ) MPa, methane molecules adsorb only at the inner surface of the nanotube; i.e., a monolayer absorption is observed. Adsorption at the second, third, and higher layers evolves as the pressure is increased. At T ) 118 K, the first peak is essentially independent of pressure and the layer-by-layer adsorptions can be clearly distinguished. Near capillary condensation (P 0.06 MPa, see Figure 3a), the highorder layers appear almost simultaneously, signaling a huge jump in the adsorption isotherm. At T ) 207 K, however, only the first two layers are clearly defined and the adsorptions on the remaining layers are more or less continuous. In this case, no capillary condensation was observed (see Figure 4). Figure 7 shows the corresponding snapshots of methane molecules in the (30, 30) nanotube at 118 K. At low pressure, all molecules are located at the inner surface of the nanotubes, corresponding to a monolayer adsorption. The second layer is formed by the increase in pressure only after the first layer is completely filled. The second layer peak is always smaller than the first layer peak because the potential energy at the second layer is always smaller than that at the first layer. As discussed before, 34 the difference in the magnitude of a potential energy determines the difference of the molecular numbers at the first two layers. The observation of local density profiles is consistent with the results from other simulations. 35 At the condition of capillary condensation, large amount of gas molecules fill the nanotube all together, signaling a big jump in the adsorption isotherm. 36 After capillary condensation, the number density becomes insensitive as the pressure is further increased. Both the local density profiles and the snapshots clearly demonstrate the layer-by-layer adsorptions prior to the capillary condensation. Why the excess amount of adsorption as a function of pressure exhibits a maximum? As shown in Figure 6, the amount of adsorption increases with pressure when methane molecules are adsorbed on the first and second (35) Wang, Q. Y.; Johnson, J. K. J. Chem. Phys. 1999, 110, 577. (36) Cao, D. P.; Wang, W. C. Phys. Chem. Chem. Phys. 2002, 4, 3720.

6 3764 Langmuir, Vol. 20, No. 9, 2004 Cao and Wu Figure 9. Diffusion coefficients of methane in various SWNTs at T ) (a) 207, (b) 237, and (c) 267 K. Figure 8. Diffusion coefficients of methane in various SWNTs at (a) T ) 118 K and (b) T ) 140 K. layers. However, the local density profiles become insensitive to the pressure once the SWNT is filled while the density of the bulk phase increases remarkably as the pressure is further increased. As a result, the excess amount of adsorption falls with pressure, and the maximum excess adsorption arises. However, no maximum appears in the adsorption isotherms in terms of the total amount of adsorption because the total adsorption amount always increases as the pressure is increased Self-Diffusion Coefficients from Molecular Dynamic Simulations. Panels a and b of Figure 8 show the self-diffusion coefficients of methane at T ) 118 and 140 K, respectively, both below the bulk critical temperature. The error bars shown in Figure 8b are estimated by repeated simulations. At low pressure, the diffusion coefficient increases with the diameters of nanotubes, in qualitative agreement with the prediction of the Knudsen flow. However, the trend becomes more obscure as the pressure is further increased, particularly at high temperatures (as shown later). In the range of pressures corresponding to a monolayer adsorption (see Figure 6), the diffusion coefficient falls rapidly as the pressure rises, probably due to the increased collisions of methane molecules in the axial direction of the nanotube. With further increase in pressure, the diffusion coefficient exhibits an oscillatory behavior, corresponding to the layer-by-layer adsorption of methane molecules as discussed earlier. Interestingly, the self-diffusion coefficient for gases in a confined geometry may increase with pressure, in contrast to the intuitive expectation based on the bulk diffusion behavior. As reported earlier, the local diffusion coefficient of methane near the surface of SWNTs is smaller than that in the center of the pore. 37 As a result, the overall parallel diffusion coefficient may increase upon the formation of the second or third layers. At T ) 118 K, the diffusion coefficients in all SWNTs are close to that corresponding to a solid ( m 2 /s). In particular, for the (15, 15) SWNT, the diffusion coefficient reduces to the magnitude of m 2 /s. Both panels a and b of Figure 8 indicate that even though the amount of adsorption does not vary significantly with pressure, the diffusion coefficients may be drastically decreased as the pressure increases. Figure 9 shows the diffusion coefficients of methane in SWNTs at T ) 207, 237, and 267 K, all above the bulk critical temperature. As in the low temperature case, the diffusion coefficient falls drastically as the pressure increases from 0.1 to about 2.0 MPa. With further increase in pressure, the diffusion coefficient reaches a plateau with some small fluctuations. This can be explained by the saturation of gas adsorption in the SWNTs as exhibited in the density profiles (Figure 6). Different from the lowtemperature cases, the limiting diffusion coefficients are insensitive to the pore diameters. This directly contradicts to the prediction of the Knudsen flow (which, of course, is not applicable at high pressures). However, as predicted by the Knudsen flow, the high-pressure limits of the diffusion coefficients vary with temperature. As the temperature is increased from 207 to 267 K, the limiting self-diffusion coefficient arises from to m 2 /s. Apparently temperature plays an important role in the motion of gas molecules even at supercritical conditions. A comparison of Figures 8 and 9 shows that while the self-diffusion coefficients decrease smoothly with pressure at temperatures above the bulk critical point, the trend is more subtle at low temperatures due to the layering transition and capillary condensations inside the SWNTs. (37) Cao, D. P.; Zhang, X. R.; Chen, J. F.; Yun, J. Carbon 2003, 41, 2686.

7 Self-Diffusion of Methane in Carbon Nanotubes Langmuir, Vol. 20, No. 9, Figure 10. Diffusion coefficient of methane in (15, 15) SWNT changing with the inverse of the temperature at P ) 1.0 M Pa. At a supercritical temperature, the motion of methane molecules is predominately determined by the kinetic energy and intermolecular interactions. At low temperature, however, the motion of gas molecules is strongly affected by the wall potential. As a result, the high pressure limits of the self-diffusion coefficients are sensitive to the diameters of SWNTs at low temperatures while they are essentially constant at supercritical conditions. Figure 10 shows the relationship of diffusion coefficient changing with the inverse of the temperature at P ) 1.0 MPa. Different from an earlier report by by Keffer, 38 it appears that the diffusivity versus 1/T does not follow a linear relation. 4. Conclusions We have investigated the adsorption and diffusion of methane in various SWNTs by a combination of GCMC and classical MD simulations. The adsorption isotherms from GCMC provide insights for the potential application of SWNTs as gas storage media and, more importantly, the starting point for the investigation of the self-diffusion coefficients of methane in SWNTs at different temperatures and pressures. We found that both the adsorption behaviors and diffusion coefficients of methane in SWNTs are considerably different at supercritical and subcritical temperatures. At a low temperature, capillary condensation occurs in SWNTs with indices (20, 20), (25, 25) and (30, 30), following a layer-by-layer adsorption of methane molecules (38) Keffer, D. Chem. Eng. J. 1999, 74, 33. from the surface to the center of nanotube. In particular, it was observed that capillary condensation leads to the simultaneous formation of multilayers. As reported by others, capillary condensation is inhibited in small nanotubes when the confined methane becomes essentially a one-dimensional system. At a supercritical temperature, the amount of adsorption smoothly increases with pressure. In this case, distinctive layering is observable only very close to the inner wall of SWNTs as seen in the local density profiles. When the nanotube pores are fully loaded with methane, the density profiles of methane molecules inside of the SWNTs are insensitive to the pressure changes while the bulk density still varies with pressure. Consequently, the excess amount of gas adsorption varies nonmonotonically with pressure, exhibiting a maximum amount of methane storage. This optimum pressure increases monotonically with temperature and depends on the diameters of SWNTs. At room temperature, the estimated optimum pressure is 4 MPa for (15, 15) and 8 MPa for (25, 25) and (30, 30) SWNTs, both are significantly lower than the pressure applied in compressed nature gas storage. In the pressure range corresponding to a monolayer adsorption, the diffusion coefficients fall drastically as the pressure increases, probably due to the collision of gas molecules. At a subcritical temperature, the diffusion coefficient is strongly affected by the interactions between gas molecules and the confining wall. The local diffusion coefficient near the inner surface of a SWNT is considerable smaller than that in the middle. As a result, the overall self-diffusion coefficient may increase with the loading of methane molecules. Because of the strong effects of the wall potential, the diffusion coefficients are also sensitive to the size of SWNTs even at high pressure. At a supercritical temperature, the results are drastically different. In this case, the diffusion coefficient falls smoothly as the pressure increases and reaches a plateau that is essentially independent of the diameters of SWNTs. When the SWNTs are fully loaded, the self-diffusion coefficient is insensitive to the wall potential. Acknowledgment. We thank Dr. Anil Kumar for insightful comments on the diffusion coefficients of methane at low temperatures. This work has been sponsored in part by the University of California Research and Development Program (Grant No ), by Lawrence Livermore National Laboratory (Grant No. MSI ) and by the National Science Foundation (Grant No. CTS ). LA036375Q