Langmuir 2004, 20, 3759-3765
3759
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 92521
Received December 16, 2003. 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 MCM41. 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 Ichihashi5 and independently
* To whom all correspondence should be addressed. E-mail:
[email protected].
(1) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena,
2nd ed.; J. Wiley: New York, 2002.
(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, 2289.
(4) Davis, H. T. In Fundamentals of Inhomogeneous Fluids; Henderson, D., Ed.; M. Dekker: New York, 1992.
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
2 to 8 nm.9-11 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, 5485.
(9) Mao, Z. G.; Sinnott, S. B. J. Phys. Chem. B 2001, 105, 6916.
(10) Mao, Z. G.; Sinnott, S. B. J. Phys. Chem. B 2000, 104, 4618.
(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. 278301.
(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, 89.
10.1021/la036375q CCC: $27.50 © 2004 American Chemical Society
Published on Web 04/02/2004
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Langmuir, Vol. 20, No. 9, 2004
Cao and Wu
where b ) 0.142 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
{
φLJ(r) - φLJ(rc) r < rc
r g rc
0
φff(r) )
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 4.068 nm, calculated from8
σc )
b
x3(m2 + mn + n2)
π
(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, 8531.
(17) Frenkel, D.; Smit, B. Understanding Molecular Simulation: From
Algorithms to Applications, 2nd ed.; Academic Press: San Diego, 2002.
(18) Koga, K.; Gao, G. T.; Tanaka, H.; Zeng, X. C. Nature 2001, 412,
802.
(2)
where r is the center-to-center distance and rc ) 5σff 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 φLJ stands for the full LJ
potential given by
[( ) ( ) ]
φLJ(r) ) 4ff
σff
r
12
-
σff
r
6
(3)
where ff and σff are, respectively, the energy and size parameters
of the LJ potential. For methane, ff/kB ) 148.1 K, where kB is
the Boltzmann constant and σff ) 0.381 nm.21,22
The total potential for the interactions between methane
molecules and the confining SWNT is calculated using the sitesite interaction method23
Nf Ncarbon
∑∑
φfw ) 4fw
i)1 j)1
[( ) ( ) ]
σfw
12
-
rij
σfw
rij
6
(4)
where Nf is the total number of methane molecules, Ncarbon is the
total number of the carbon atoms at the wall, and rij 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, fw and σfw, are obtained from
the Lorentz-Berthelot combining rules
fw ) xff ‚ ww
(5)
σfw ) (σff + σww)/2
(6)
The energy and size parameters for carbon atom are ww/kB )
28.0 K and σww ) 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 5σfw.
2.2. 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* ) kBT/, 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*2L* 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, 3150.
(21) Cao, D. P.; Wang, W. C.; Shen, Z. G.; Chen, J. F. Carbon 2002,
40, 2359.
(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, 13286.
(24) Johnson, J. K.; Zollweg, J. A.; Gubbins, K. E. Mol. Phys. 1993,
78, 591.
Self-Diffusion of Methane in Carbon Nanotubes
Langmuir, Vol. 20, No. 9, 2004 3761
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, 2 × 107 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.
2.3. 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
algorithm25 with the time step set to 0.005 in dimensionless units,
defined as t* ) t ‚ x/mσ2.17 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 30 000 time steps and
another 20 000 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 geometry17
〈∆z2(t)〉 ) 2Dzt
(7)
where ∆z2(t) represents the square of displacement in the
z-direction as a function of time (t), Dz 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, 1997.
(26) Muris, M.; Dufau, N.; Bienfait, M.; Dupont-Pavlovsky, N.; Grillet,
Y.; Palmari, J. P. Langmuir 2000, 16, 7019.
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 7.4 × 10-9, 3.1 × 10-9,
and 3.8 × 10-9 m2/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 (Tc ) 190.4 K, Pc ) 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 (Tc ) 190.4 K). The amount of adsorption is
defined by
Γ ) mCH4/mSWNT
(8)
where mCH4 is the (average) total mass of adsorbed
methane and mSWNT 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, 0.038 and 0.061 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, 8720.
(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.
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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 CH4/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 reports32 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 adsorption34
Figure 5. Effect of temperature on the pressure at the maximum excess adsorption. The solid lines are for the guidance of
the eye.
where m°CH4 is the amount of methane if the confining
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
analysis17 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.
(31) Lozano-Castello, D.; Alcaniz-Monge, J.; de la Casa-Lillo, M. A.;
Cazorla-Amoros, D.; Linares-Solano, A. Fuel 2002, 81, 1777.
(32) Gelb, L. D.; Gubbins, K. E.; Radhakrishnan, R.; SliwinskaBartkowiak, M. Rep. Progr. Phys. 1999, 62, 1573.
(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.
Γexcess ) (mCH4 - m°CH4)/mSWNT
(9)
Self-Diffusion of Methane in Carbon Nanotubes
Langmuir, Vol. 20, No. 9, 2004 3763
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* ) Fσ3 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 20-30 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 ) 0.005 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.
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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.
3.2. 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 (∼1.0 × 10-10 m2/s). In particular,
for the (15, 15) SWNT, the diffusion coefficient reduces to
the magnitude of 1.0 × 10-11 m2/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 4.0 × 10-9 to 1.8 ×
10-8m2/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.
Self-Diffusion of Methane in Carbon Nanotubes
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.
Langmuir, Vol. 20, No. 9, 2004 3765
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. 69757), by Lawrence
Livermore National Laboratory (Grant No. MSI-04-005)
and by the National Science Foundation (Grant No.
CTS-0340948).
LA036375Q