Mass Transport of O2 and N2 in

Mass Transport of O2 and N2 in Nanoporous Carbon (C168 Schwarzite). Using a Quantum ... Jiang et al.2 and Jiang and Sandler3 studied the adsorption of...
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Langmuir 2006, 22, 4620-4628

Mass Transport of O2 and N2 in Nanoporous Carbon (C168 Schwarzite) Using a Quantum Mechanical Force Field and Molecular Dynamics Simulations Gaurav Arora and Stanley I. Sandler* Center for Molecular and Engineering Thermodynamics, Department of Chemical Engineering, UniVersity of Delaware, Newark, Delaware 19716 ReceiVed NoVember 13, 2005. In Final Form: March 2, 2006 A hierarchical approach is used to calculate the single-component fluxes of N2 and O2 in nanoporous carbon molecular sieves (represented by C168 schwarzite) over a wide range of pressures and pressure drops. The self- and corrected diffusivities are calculated using equilibrium molecular dynamics simulations with force fields for the gas-carbon interactions obtained from quantum mechanical calculations. These results are combined with previously reported adsorption isotherms of N2 and O2 in C168 to obtain transport diffusivities and, by use of the Fick’s equation of mass transport, to obtain single-component fluxes across the membrane. The diffusion coefficients and fluxes are also calculated using an empirical potential, which has been obtained by fitting low coverage adsorption data of N2 and O2 on a planar graphite sheet. By analyzing the diffusivities calculated with the ab initio potential in the limit of infinite dilution over the temperature range from 80 to 450 K, it is observed that the N2/O2 separation is energetically driven and a high selectivity of O2 over N2 can be obtained at low temperatures. However, with the empirical potential both the energetic and entropic contributions to selectivity were found to be close to unity. Similarly, by calculating single-component fluxes and ideal selectivities at 300 K and finite pressures it is found that the ab initio potential better explains the large O2/N2 selectivities of similarly sized molecules that have been observed experimentally. An interesting reversal in ideal selectivity is observed by adjusting the pressure at the two ends of the membrane. As a consequence, we predict that a highly selective kinetic separation in favor of either nitrogen or oxygen could be obtained with the same membrane depending on the operating conditions.

I. Introduction Nanoporous carbon molecular sieves (NPCs) are known to be very efficient for the separation of similarly sized gas molecules and have been of considerable interest during the last two decades as an alternative to existing energy-intensive separations. Foley and co-workers prepared NPCs by ultrasonic deposition and sintering that showed high ideal selectivities for O2, He, and H2 compared to that for N2.1 At room temperature, the pure gas O2/N2 permeability ratio was shown to vary from 2 to 30 depending on the NPC synthesis conditions. NPCs have both negatively and positively curved surfaces containing five-, six-, seven-, and eight-membered carbons rings and a narrow pore size distribution ranging from 0.3 to 0.7 nm. An effective gas separation process can be carried out on the basis of favorable differences in the adsorption equilibrium and/ or the rate of transport of adsorbents through the membrane. Jiang et al.2 and Jiang and Sandler3 studied the adsorption of pure nitrogen, oxygen, and their mixtures in NPCs represented by C168 schwarzite using an empirical potential and grand canonical Monte Carlo simulations. For an equimolar mixture at 300 K, nitrogen was preferentially adsorbed at low pressures; however, a reversal in selectivity was observed as the pressure was increased, resulting in oxygen being preferentially adsorbed at high loadings. Recently, Jiang et al.4 performed a comparative study of the adsorption and thermodynamic properties of pure O2 and N2 and * Corresponding author. E-mail: [email protected]. Tel: (302)-8312945. Fax: (302)-831-3226. (1) Shiflett, M. B.; Foley, H. C. Science 1999, 285, 1902-1905. (2) Jiang, J. W.; Klauda, J. B.; Sandler, S. I. Langmuir 2003, 19, 3512-3518. (3) Jiang, J. W.; Sandler, S. I. Langmuir 2003, 19, 5936-5941. (4) Jiang, J.; Klauda, J. B.; Sandler, S. I. J. Phys. Chem. B 2004, 108, 98529860.

their mixtures in C168 schwarzite using two different potentials: the empirical potential5,6 developed by fitting low-coverage adsorption data of O2 and N2 on planar graphite sheets and an ab initio-based potential7 that incorporates the effects of both carbon curvature and the presence of nonhexagonal carbon rings on the electronic configuration. They demonstrated the importance of the interaction potential in determining the adsorption behavior, with the Steele potential showing very similar adsorption property behavior of N2 and O2 for a 79:21 N2/O2 bulk mixture whereas nitrogen adsorbed much more than oxygen with the potential obtained from quantum mechanical calculations. Similarly, differences in diffusion rates and hence in the rate of macroscopic transport across the membrane can be expected. In this work, we study the importance of the intermolecular interaction potential on the transport of N2 and O2 in NPCs. A hierarchical approach is followed by using a potential derived from quantum mechanical calculations in molecular dynamics simulations to obtain self-, corrected-, and transport diffusivities of N2 and O2 in C168 Schwarzite at both infinite dilution and finite loadings. These results along with previously reported adsorption isotherms4 are then combined in a continuum description of mass transfer to obtain single-component fluxes and ideal selectivities across the membrane. Entropic and energetic contributions to the selectivity have been studied in experiments and by molecular modeling and transition-state theory to understand the key phenomenon behind the large selectivities of the similarly sized O2 and N2.8-11 (5) Bojan, M. J.; Steele, W. A. Langmuir 1987, 3, 116-120. (6) Bojan, M. J.; Steele, W. A. Langmuir 1987, 3, 1123-1127. (7) Klauda, J. B.; Jiang, J.; Sandler, S. I. J. Phys. Chem. B 2004, 108, 98429851. (8) Rallabandi, P. S.; Thompson, A. P.; Ford, D. M. Macromolecules 2000, 33, 3142-3152. (9) Rallabandi, P. S.; Ford, D. M. AICHE J. 2000, 46, 99-109.

10.1021/la053062h CCC: $33.50 © 2006 American Chemical Society Published on Web 04/05/2006

Mass Transport of O2 and N2 in C168 Schwarzite

Energetic selectivity is generally known to favor oxygen in zeolite 4A and NPCs, whereas the entropic selectivity for oxygen can be either greater or less than 1, depending on the membrane sample. Rallabandi and Ford9 have compiled the range of reported energetic and entropic selectivities in CMS used in different studies and zeolite 4A. The two selectivities were found to be strongly inversely correlated, with a large value of one corresponding to a smaller value of the other. Theodorou et al.12 demonstrated that the transport diffusivities in microporous materials could be calculated from EMD simulations, and this approach has been used to calculate transport properties of pure components and mixtures in various microand nanoporous materials.13-20 Such simulations combined with a continuous description of mass transport were first used by Sholl to calculate single-component permeances across a zeolite membrane.21 In this approach, atomistic simulations are first used to obtain transport diffusivities and adsorption isotherms. These results can be then combined with a suitable theoretical method, for example, Fick’s equation of mass transport, to calculate fluxes across the membrane for any given membrane thickness and pressure drop. Sholl and co-workers also used this framework to calculate transport diffusivities and permeances of pure CH4 and CF4 and their mixtures through silicalite and found the results to be in good agreement with experiment.22-25 We have recently calculated macroscopic mass transport properties of N2 and O2 across the carbon nanotube-based membranes26,27 and now apply the same formalism to N2 and O2 transport across C168 schwarzite membranes in order to better understand the key mechanism for the effective separation of N2 and O2. This article is organized as follows. The force fields and the simulation model are discussed in the following section. Section III explains the simulation methodology, parameters, and definitions of the diffusion coefficients along with the procedure to calculate macroscopic membrane properties. The findings of this work are reported and discussed in section IV, followed by a summary in section V.

II. Simulation Model and Interaction Potential A crystalline periodic structure of C168 schwarzite28 was used to represent amorphous NPCs. For simulation purposes, the known atomic coordinates of C168 schwarzite in a periodic cell are used to model NPCs whose exact morphology is difficult to obtain because of their amorphous nature. Similar to NPCs, C168 schwarzite has both negative and positive curvature, and most of the carbon atoms are believed to be sp2 hybridized. A cubic (10) Singh, A.; Koros, W. J. Ind. Eng. Chem. Res. 1996, 35, 1231-1234. (11) Acharya, M.; Foley, H. C. AICHE J. 2000, 46, 911-922. (12) Theodorou, D. N.; Snurr, R. Q.; Bell, A. T. Compr. Supramol. Chem. 1996, 7, 507-548. (13) Arora, G.; Wagner, N. J.; Sandler, S. I. Langmuir 2004, 20, 6268-6277. (14) Sanborn, M. J.; Snurr, R. Q. Sep. Purif. Technol. 2000, 20, 1-13. (15) Skoulidas, A. I.; Sholl, D. S. J. Phys. Chem. B 2001, 105, 3151-3154. (16) Skoulidas, A. I.; Sholl, D. S. J. Phys. Chem. A 2003, 107, 10132-10141. (17) Skoulidas, A. I.; Ackerman, D. M.; Johnson, J. K.; Sholl, D. S. Phys. ReV. Lett. 2002, 89, 185901. (18) Chen, H. B.; Sholl, D. S. J. Am. Chem. Soc. 2004, 126, 7778-7779. (19) Skoulidas, A. I.; Sholl, D. S. J. Phys. Chem. B 2005, 109, 15760-15768. (20) Krishna, R.; van Baten, J. M. J. Phys. Chem. B 2005, 109, 6386-6396. (21) Sholl, D. S. Ind. Eng. Chem. Res. 2000, 39, 3737-3746. (22) Skoulidas, A. I.; Bowen, T. C.; Doelling, C. M.; Falconer, J. L.; Noble, R. D.; Sholl, D. S. J. Membr. Sci. 2003, 227, 123-136. (23) Jobic, H.; Skoulidas, A. I.; Sholl, D. S. J. Phys. Chem. B 2004, 108, 10613-10616. (24) Bowen, T. C.; Falconer, J. L.; Noble, R. D.; Skoulidas, A. I.; Sholl, D. S. Ind. Eng. Chem. Res. 2002, 41, 1641-1650. (25) Chong, S. S.; Jobic, H.; Plazanet, M.; Sholl, D. S. Chem. Phys. Lett. 2005, 408, 157-161. (26) Arora, G.; Sandler, S. I. J. Chem. Phys. 2005, 123, 044705. (27) Arora, G.; Sandler, S. I. J. Chem. Phys. 2006, 124, 084702. (28) Vanderbilt, D.; Tersoff, J. Phys. ReV. Lett. 1992, 68, 511-513.

Langmuir, Vol. 22, No. 10, 2006 4621 Table 1. Atomic Site Lennard-Jones Potential Parameters for Oxygen (O), Nitrogen (N), and Carbon (C)

σ(Å) /kB (K) l (Å) K (N/cm)

Steele potential

ab initio potential

O-O

N-N

C-O

C-N

C-O

C-N

2.99 52.0 1.208 11.77

3.32 36.4 1.10 22.95

3.19 37.6

3.36 33.4

3.269 35.128

3.542 36.902

unit cell of C168 schwarzite has 672 carbon atoms and is 21.8 Å in length. The presence of six- and seven-membered carbon rings in C168 schwarzite is believed to result in an electronic configuration similar to that in real NPCs. Although the two different pore sizes of 7 and 9 Å of C168 are larger than the mean pore size of 5 Å of NPCs, the pore cavities and intersectional channels of C168 provide a very similar environment to the tortuous pathways in the NPCs. The two differently sized pores of C168 are separated from each other by carbon walls but are connected to pores in the adjacent layers through intersection channels at an angle of approximately 45°. Two different force fields were used to describe N-C and O-C interaction potentials: the empirical force field developed by Bojan and Steele6 and an ab initio potential obtained by Klauda et al.7 Bojan and Steele obtained their force field by fitting lowcoverage adsorption data of N2 and O2 on a planar graphite sheet, and this force field is referred as the Steele potential here. Although the Steele potential has been used in the past to model N2 and O2 interactions with curved carbon surfaces, it has been shown using an ab initio-based potential in adsorption studies of N2 and O2 in C168 schwarzite4 and N2 adsorption on C60 and C70 fullerene molecules29,30 that the Steele potential might not accurately represent the change in the electronic configuration and therefore the force field of the carbon adsorbents as a result of the curved surfaces and the presence of nonhexagonal carbon rings. Klauda et al. have used quantum mechanical calculations that incorporate these factors to obtain a force field describing the interactions of N2 and O2 with C168.7 Here we perform molecular dynamics simulations using this potential and compare the calculated diffusivities and macroscopic mass transport with those obtained from the Steele potential. The adsorbate-adsorbate and adsorbate-adsorbent interactions in the Steele and ab initio potentials were described using a Lennard-Jones 6-12-type potential, which is satisfactory because of the small range of interatomic distances accessible in an NPC. Also, because all interactions have been included in the quantum mechanical calculations, the effects of quadrupole and all other dispersion interactions are implicitly included in the values of the Lennard-Jones parameters. Nitrogen and oxygen were treated as flexible molecules, and a harmonic potential was used to model the intramolecular interactions. The values of Lennard-Jones constants, σij and ij, along with the intramolecular harmonic spring constant, K, are given in Table 1. A detailed comparison of the interaction energies of O2 and N2 as a function of the distance from the center of the channel intersection of C168 schwarzite is given elsewhere.4 It was observed that there is a small difference between the N2 and O2 interaction energies using the Steele potential, whereas the differences are larger with the ab initio potential. This suggests that the ab initio potential might be able to describe the very different permeation rates of the similarly sized N2 and O2 molecules in NPCs observed by Foley et al.,1 which cannot be predicted using the Steele potential. (29) Jiang, J.; Klauda, J. B.; Sandler, S. I. J. Phys. Chem. B 2005, 109, 47314737. (30) Arora, G.; Klauda, J. B.; Sandler, S. I. J. Phys. Chem. B 2005, 109, 17267-17273.

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Arora and Sandler

Also given in Table 1 are the gas-gas interaction parameters for nitrogen31 and oxygen32 that have been obtained by fitting experimental bulk properties.

III. Simulation Model and Theory Equilibrium molecular dynamics (EMD) simulations were carried out in a canonical ensemble (NVT) using a Nose-Hoover thermostat33 to obtain the diffusion coefficients of pure N2 and O2 in C168 at 300 K. The Verlet neighbor list method was used to speed up the simulations.34 A spherical cutoff distance of 13 Å was used to truncate adsorbate-adsorbate and adsorbateadsorbent interactions with periodic boundary conditions applied in all three dimensions. All simulations were performed on a large periodic structure of C168 containing 5376 carbon atoms whose crystallographic positions were fixed during the simulation. First, grand canonical Monte Carlo simulations were performed to obtain the initial configuration of the adsorbate molecules at a desired loading. Then, EMD simulations were performed for an equilibration period of 0.5 ns followed by a sampling period of 5-10 ns, which took from 72 to 240 h on a single AMD opetron 2 GHz processor depending on the adsorbate loading. Infinite dilution results at various temperatures were obtained by performing simulations with 100 adsorbate molecules in C168 schwarzite and turning off the adsorbate-adsorbate interactions. Results were averaged over 10 independent simulations at infinite dilution and also at each loading to obtain the desired level of statistical accuracy. The self-diffusion coefficient, which describes the mobility of individual tagged particles, can be calculated from EMD simulations using the Einstein relation

Ds(c) )

1 N lim 〈 [ri(t) - ri(0)]2〉 6N tf∞ t i)1 1

∑

(1)

[ ] -Ea RT

(2)

where T is the temperature and R is the gas constant. The frequency factor can be expressed in terms of entropic barrier Sa using the relation35

Df ) eδ2

()

kBT Sa exp h R

Ds,N2

) exp

[

] [

(Sa,O2 - Sa,N2) R

exp

(3)

where e is the base of natural logarithm and δ is the average diffusive jump length. Because the kinetic diameters of N2 and O2 are very similar, δ is approximately equal for the two. From eqs 2 and 3, the N2-O2 diffusive selectivity can be calculated using10 (31) Murthy, C. S.; Singer, K.; Klein, M. L.; McDonald, I. R. Mol. Phys. 1980, 41, 1387-1399. (32) Klein, M. L.; Levesque, D.; Weis, J. J. Phys. ReV. B 1980, 21, 57855792. (33) Frenkel, D.; Smit, B. Understanding Molecular Simulation; Academic Press: London, 1996. (34) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Oxford University Press: Oxford, U.K., 1987. (35) Glasstone, S.; Laidler, K. J.; Eyring, H. The Theory of Rate Processes, 1st ed.; McGraw-Hill: New York, 1941.

]

-(Ea,O2 - Ea,N2) RT

(4)

where the first exponential term describes the entropic selectivity and is simply the ratio of the frequency factors of N2 and O2 and the second exponential term describes the energetic selectivity. Unlike entropic selectivity, the effect of the energetic selectivity is temperature-dependent. From eq 4, the entropic and energetic contributions to the selectivity can be calculated separately by regression analysis of the self-diffusion coefficients at various temperatures. Another quantity used to describe the transport of adsorbate molecules in porous media is the corrected diffusivity, Do(c), which can be obtained by calculating the mean square displacement of the center of mass of the adsorbed molecules as a function of time, which is linear in the limit of long time.12,15

Do(c) )

1 N lim 〈( [ri(t) - ri(0)])2〉 6N tf∞ t i)1 1

∑

(5)

Therefore, unlike eq 1, which is obtained by performing an ensemble average over all of the adsorbed molecules, eq 5 results in poorer statistics and requires averaging over a large number of independent EMD simulations. An average over 10 to 20 independent EMD trajectories has been found to provide statistically reliable results,14,17,27,36 as was also observed in this work. The single-component flux J of a species across a membrane with a concentrations gradient can be described in terms of transport (Fickian) diffusivity using Fick’s equation of mass transport

dc dz

J ) -Dt(c)

where N is the number of particles in the system and ri(t) is the position vector of particle i at time t. The diffusion in C168 schwarzite can be described as an activated process with the temperature dependence of the self-diffusion coefficient described by an Arrhenius-type relation in terms of frequency factor Df and activation energy Ea

Ds ) Df exp

Ds,O2

(6)

where the transport diffusivity, Dt(c), is related to corrected diffusivity by the following equation:12,37

Dt(c) ) Do(c)

(∂∂ lnln cf )

T

(7)

Note that the self-, corrected, and transport diffusivities are all different functions of concentration and are equal only in the limit of infinite dilution when the thermodynamic nonidealities are absent and there is no positional correlation between the adsorbate molecules.12,15 The self- and corrected diffusivities (eqs 1 and 5) can be calculated directly from EMD simulations; however, the calculation of transport diffusivity using eq 7 requires information on the concentration dependence of the fugacity. In this work, the thermodynamic factors (∂ ln f /∂ ln c)T were obtained by differentiation of the adsorption isotherms of pure N2 and O2 in C168 at 300 K recently reported by Jiang et al.4 Once the concentration dependence of the transport diffusivity and the adsorption isotherms are known, eq 6 can be integrated across a membrane of length L to obtain the steady-state flux21,24,27

J)

1 L

∫cc

in

out

Dt(c) dc

(8)

where cin and cout are the concentrations at the upstream and (36) Ackerman, D. M.; Skoulidas, A. I.; Sholl, D. S.; Johnson, J. K. Mol. Simul. 2003, 29, 677-684. (37) Karger, J.; Ruthven, D. M. Diffusion in Zeolites and Other Microporous Solids; John Wiley: New York, 1992.

Mass Transport of O2 and N2 in C168 Schwarzite

Langmuir, Vol. 22, No. 10, 2006 4623

downstream ends of the membrane, respectively. These concentrations can be obtained from the adsorption isotherm assuming that the mass transfer resistance is across the membrane and negligible at the membrane boundaries. There have been various studies to estimate the entrance and exit resistances of the zeolite membrane and also their dependence on the membrane orientation, temperature, permeate conformation and size, adsorbate density, and membrane thickness.38-41 For silicalite membranes, these effects were appreciable compared to the intracrystalline resistance only for membranes less that 1 µm thick and therefore, as here, can be neglected for synthesized membranes that are considerably thicker than 1 µm. The effect of membrane support and sweep gas, if any, was also neglected in this work.42 A measure of membrane performance is the ideal selectivity, which is defined as the ratio of the fluxes of the pure components across a membrane under identical conditions

Ri/j )

Ji Jj

(9)

Unlike eq 4, which is valid only at dilute concentrations (i.e., for ideal solution behavior), eq 9 can also be used for a finite pressure drop across the membrane. Moreover, eq 4 provides a measure of the relative magnitudes of the diffusion coefficients whereas the ideal selectivity of eq 9 also contains the contributions from the adsorptive strength of the membrane for the two single components. To include the contribution of the adsorptive strength of the membrane at infinite dilution, either eq 9 or the following expression that relates eqs 9 and 4 can be used10

RO2/N2 )

Ds,O2 Ds,N2

×

S O2 SN2

(10)

where Si is the solubility of the adsorbent. In the limit of low pressure, eq 10 reduces to

RO2/N2 )

Ds,O2 Ds,N2

×

KH,O2 KH,N2

Figure 1. Dependence of the logarithm of the self-diffusion coefficient on inverse temperature in the infinite dilution limit for N2 and O2 in C168 schwarzite using (a) the Steele potential and (b) the ab initio potential. Self-diffusion coefficients are in units of cm2/s and have a statistical error of ∼10%. Table 2. Diffusion Activation Energies and Preexponential Factors of Nitrogen and Oxygen in C168 Schwarzite Calculated Using Steele and ab Initio Potentials Steele potential

(11) (10-4

where KH,i is Henry’s law constant of component i. A detailed discussion of the calculation of the Henry’s law constant and adsorptive selectivity for N2 and O2 in C168 schwarzite using both the Steele and ab initio potentials can be found elsewhere.4

IV. Results and Discussion Nitrogen was found to adsorb more strongly than oxygen in C168, and as in chromatographic separations, this suggested that O2 will have a higher diffusion coefficient than N2.4 This argument is substantiated by calculating the infinite dilution diffusion coefficients of N2 and O2 at 80, 100, 150, 200, 250, 300, 350, and 450 K. As mentioned previously, in the infinite dilution limit the self-, corrected, and transport diffusivities are all equal and will be referred to here simply as the diffusion coefficients. As shown in Figure 1, the diffusion coefficients of O2 were found to be higher than that of N2 over the entire range of temperature using both Steele and ab initio potentials. (38) Arya, G.; Maginn, E. J.; Chang, H. C. J. Phys. Chem. B 2001, 105, 2725-2735. (39) Ahunbay, M. G.; Elliott, J. R.; Talu, O. J. Phys. Chem. B 2002, 106, 5163-5168. (40) Ahunbay, M. G.; Elliott, J. R.; Talu, O. J. Phys. Chem. B 2004, 108, 7801-7808. (41) Newsome, D. A.; Sholl, D. S. J. Phys. Chem. B 2005, 109, 7237-7244. (42) Skoulidas, A. I.; Sholl, D. S. AICHE J. 2005, 51, 867-877.

cm2/s)

Df Ea (kJ/mol)

ab initio potential

N2

O2

N2

O2

5.00 1.91

5.99 1.73

6.43 3.12

4.86 1.74

In C168 schwarzite, the interaction energies are slightly more attractive for O2 with the ab initio potential than with the Steele potential; the difference was much more significant for N2, with the ab initio potential being substantially more attractive.4 As a result, the diffusivities of O2 calculated using the ab intio potential were found to be marginally lower than those calculated using the Steele potential, and as expected, the difference was greater for N2, particularly at low temperatures. Through regression analysis, a good linear fit was obtained of the logarithm of diffusion coefficients with inverse temperature. The slope and intercept of the fit were then used to obtain the frequency factor and activation energy (eq 2) for N2/O2 diffusion using both the ab initio and Steele potentials. The larger σC-N value of the ab initio potential results in a larger entropic barrier to diffusion as shown in Table 2. Also, because the values of σC-N and C-N are larger in the ab initio potential than in the Steele potential, the minimum of the interaction potential curve between an N2 molecule and the C168 channel intersection decreases significantly from approximately 2.9 to 2.0 Å with much more attraction for r < 2.9 Å, where r is the distance of the molecule from the center of the channel intersection.4 The shift of the minima closer to the C168 wall results in a significant

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Figure 2. Energetic selectivity of O2 compared to that of N2 in the infinite dilution limit in C168 schwarzite as a function of temperature using (a) the Steele potential and (b) the ab initio potential. Also shown is the temperature-independent entropic selectivity for the two potentials.

increase in the activation energy for diffusion with the ab initio potential, and this is reflected in the results in Table 2. The minima in the interaction energy between an O2 molecule and the C168 channel intersection using both potentials are nearly the same at 2.9 Å, with similar attraction for r < 2.9 Å resulting in similar activation energy. However, a greater repulsion for r > 2.9 Å using the ab initio potential4 results in a greater decrease in the entropic barrier than with the Steele potential. The entropic selectivities at infinite dilution for O2 compared to those for N2 (eq 4) calculated using the Steele and ab initio potentials were found to be 1.20 and 0.76, respectively. That is, as a result of the entropic effect, the diffusivity of O2 is higher than for N2 using the Steele potential, whereas it is higher for N2 using the ab initio potential as shown in Figure 2. However, energetic selectivity clearly favors oxygen with either potential and is higher with the ab initio potential at all temperatures. At 300 K, energetic selectivities of 1.07 and 1.74 were found using the Steele and ab initio potentials, and this difference increased with decreasing temperature. These results suggest that the energetic effect might be the dominant factor in O2 and N2 separations, which is in agreement with the previous molecularlevel and transition-state theory findings.9,11 Two independent center-of-mass trajectories of N2 and O2 at infinite dilution and 300 K in C168 schwarzite are shown in Figure 3. For clarity, the equilibrated trajectories are plotted only for a duration of 100 ps. The nitrogen and oxygen molecules are intermittently trapped in the pores; however, once these molecules have sufficient energy to overcome the trapping energetic barrier and have a favorable orientation to move through the pore window,

Arora and Sandler

Figure 3. Two independent center-of-mass trajectories at 300 K in the infinite dilution limit in the periodic C168 schwarzite structure (gray) for N2 (dark and light blue) and O2 (red and orange).

they follow a relatively smooth trajectory in the channel intersections joining the pores. Similar analysis of the trajectories at lower temperatures showed the molecules to be trapped in the pores for a much longer time. The diffusion coefficients calculated in the infinite dilution limit do not take into account the decorrelation of the momentum as a result of the presence of other adsorbate molecules. Because flux measurements are performed at finite loadings, we next consider the effect of the adsorbate loading and the intermolecular potential on the diffusivities at finite pressures. EMD simulations of N2 and O2 through C168 schwarzite at 300 K were performed at various loadings using the Steele and ab initio potentials. The ranges of loadings were obtained from the adsorption isotherms and together with the results at the infinite dilution limit span bulk pressures ranging from near 0 to 500 kPa. Similar to the results in the infinite dilution limit, the mobility of the O2 molecules was higher than for N2 over the entire loading range with both potentials (Figure 4). Self-diffusivities of O2 with the ab initio potential were on average 13% lower than with the Steele potential; for N2, the average decrease was 34% as a result of the more attractive ab initio potential. Self-diffusivities in all cases decreased with increased loading (pressure) due to increasing steric hindrance. The corrected diffusivity, which is a measure of the collective mobility of molecules, is shown as a function of loading in Figure 5. The qualitative dependence of the corrected diffusivity on loading calculated using the Steele and ab initio potentials is similar to the results discussed above for self-diffusivities. Also,

Mass Transport of O2 and N2 in C168 Schwarzite

Langmuir, Vol. 22, No. 10, 2006 4625

Figure 4. Self-diffusivity of O2 and N2 as a function of loading at 300 K in C168 schwarzite using the Steele and ab initio potentials.

Figure 6. Thermodynamic factor of O2 and N2 as a function of loading at 300 K in C168 schwarzite using (a) the Steele potential and (b) the ab initio potential.

Figure 5. Corrected diffusivity of O2 and N2 as a function of loading at 300 K in C168 schwarzite using the Steele and ab initio potentials.

oxygen shows a higher collective mobility than nitrogen using both potentials. As discussed previously, the statistical uncertainties in corrected diffusivities shown in Figure 5 are larger than those of the self-diffusivities in Figure 4 over the entire range of loading. The property that is of more interest in describing mass transport across a membrane is the Fickian diffusivity given by eq 7. The thermodynamic factor required to calculate Fickian diffusivity was obtained by the direct differentiation of adsorption isotherms previously reported by Jiang et al.4 and is shown in Figure 6 as a function of adsorbate loading. Because of the very similar shape of the adsorption isotherms of O2 with Steele and ab initio potentials, the dependence of the thermodynamic factor on loading is also very similar for O2 with these two potentials. However, thermodynamic factors of N2 computed using the ab initio potential differ significantly from those resulting from the Steele potential and for both potentials for O2 at high loadings because of the very different adsorption behavior over the entire pressure range. It should be noted that unlike the self- and corrected diffusivities the thermodynamic

Figure 7. Transport diffusivity coefficients of O2 and N2 as a function of loading at 300 K in C168 schwarzite using Steele and ab initio potentials.

factor for N2 is always greater than that for O2 at all loadings for both potentials and affects the transport diffusivities accordingly. Transport diffusivities of N2 and O2 at 300 K in C168 are shown in Figure 7 as a function of adsorbate loading. The values obtained result from the competing effect of decreasing corrected diffusivities and increasing thermodynamic factors. In general, transport diffusion coefficients were found to increase with increased loading, except for N2 using the ab initio potential in the loading range of 30-50 molecules/unit cell. This can be

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Table 3. Parameters in the Polynomial Used to Fit the Transport Diffusivities of N2 and O2 in C168 Schwarzite at 300 K Dt(c) Steele ab initio

N2 O2 N2 O2

a0

a1

a2

2.23 2.97

0.074 0.047

-6.26 × 10-4 -2.97 × 10-3

0.112 0.034

-2.24 × 10 2.55 × 10-4

1.75 2.57

a3 8.34 × 10-5

-3

understood by the relatively low rate of increase of the thermodynamic factor (Figure 6) in this loading range and the substantial decrease in the corrected diffusion coefficients (Figure 5). Single-component fluxes were then calculated in a membrane allowing mass transport only in the z direction. The adsorbate concentrations at upstream and downstream ends of the membrane were obtained from the adsorption isotherms assuming that the adsorbed phase is in thermodynamic equilibrium with the bulk phase and that there is no mass transfer resistance at the membrane surfaces. To calculate steady-state flux across a membrane at any given pressure drop, the transport diffusivity is required as a function of concentration (eq 8). Because of the isotropic structure of the C168 schwarzite, diffusion in C168 is also isotropic (i.e., Dt(c) ) Dt,x(c) ) Dt,y(c) ) Dt,z(c)), and this was also verified by independent test simulations. Shown as solid curves in Figure 7 are the fits of transport diffusivities to a polynomial of degree n

Dt(c) ) a0 + a1θ + a2θ2 +...+ anθn

(12)

where ai, i ) 0, 1, 2, or 3, represents the fitting parameters and θ is the concentration of the adsorbate in units of molecules/unit cell. The parameters in the fit are given in Table 3. The single-component fluxes of N2 and O2 using the ab initio and Steele potentials across an NPC membrane of 10 µm length at 300 K are shown in Figure 8a as a function of upstream pressure with the downstream pressure held constant at 1 kPa. At a given pressure drop, the concentration gradients of O2 and N2 across the membrane are very similar using the Steele potential. Moderately higher transport diffusivities of O2 than N2 with the Steele potential over the entire concentration range (Figure 7) result in a slightly higher flux of O2 than N2. A significant increase in the concentration gradient of N2 using the ab initio potential compared to that of the Steele potential dominates the slight decrease in the transport diffusivity of N2 using the ab initio potential and results in a large increase in the N2 flux for a given pressure drop across the membrane. The flux of O2 does not change very much because of the similar adsorption behavior and concentration dependence of the transport diffusivity using either the ab initio or Steele potential. Similar calculations were repeated for different downstream pressures to obtain more information on the dependence of the ideal selectivity on the pressures at the membrane surfaces. The range of upstream pressures for a fixed downstream pressure was selected such that the flux is allowed only from the feed to the permeate side. Again, because of very similar concentration gradients across the membrane for a given pressure drop and slightly larger transport diffusivities of O2 than N2 with the Steele potential, the ideal selectivity of N2 over O2 is slightly less than 1. This is true for varying upstream pressures and downstream pressures held constant at different values as shown in Figure 8b. The ideal selectivity for N2 of 0.88 in the low-pressure regime calculated using eq 9 is in good agreement with the value of 0.94 ( 0.15 obtained from eq 11. The values of Henry’s law constants for N2 and O2 at 300 K were obtained from ref 4.

Figure 8. (a) Fluxes of N2 and O2 across a C168 schwarzite membrane at 300 K using the ab initio and Steele potentials at varying upstream pressures. Downstream pressure was held constant at 1 kPa. (b) Selectivity of N2 over O2 across a C168 schwarzite membrane at 300 K at fixed downstream pressures of 1 kPa (solid), 10 kPa (dashed), and 100 kPa (dotted) using ab initio (blue) and Steele potentials (red). The insets show the selectivity of N2 over O2 with the Steele potential on a larger scale.

However, with the ab initio potential an interesting range of ideal selectivity is observed that favors either N2 or O2 depending on the conditions (Figure 8b). A high ideal selectivity of N2 over O2 can be obtained by operating in the lowpressure regime when the downstream pressure is held constant at 1 kPa and the upstream pressure is varied up to 20 kPa. Under these conditions, the concentration gradient of N2 is significantly greater than that of O2, and this dominates the moderately higher oxygen diffusivities. As a result, an ideal selectivity for N2 of 7.58 is predicted at a low-pressure drop in C168 schwarzite at 300 K, which is again in good agreement with RN2/O2 equal to 8.46 ( 1.33 obtained from eq 11. However, the selectivity drops as upstream pressure is increased because of the decreasing difference between the concentration gradients of N2 and O2. Similar behavior is observed when the downstream pressure is held constant at 10 kPa, but the selectivity of N2 over O2 is less compared to a fixed downstream pressure of 1 kPa because of the decreased difference in the concentration gradients of N2 and O2 under these conditions. However, when the downstream pressure is held constant at 100 kPa and the upstream pressure is changed from 100 to 500 kPa, the concentration gradient of O2 is greater than that of

Mass Transport of O2 and N2 in C168 Schwarzite

Langmuir, Vol. 22, No. 10, 2006 4627

potential better explain the large ideal selectivities observed experimentally for the similarly sized nitrogen and oxygen molecules.1

V. Conclusions

Figure 9. (a) Fluxes of N2 and O2 across a C168 schwarzite membrane at 300 K using the ab initio and Steele potentials at varying downstream pressures. Upstream pressure was held constant at 500 kPa. (b) Selectivity of N2 over O2 across a C168 schwarzite membrane at 300 K at fixed upstream pressures of 10 kPa (solid line), 100 kPa (dashed line), and 500 kPa (dotted line) using ab initio (blue) and Steele potentials (red). The inset shows the selectivity of N2 over O2 with the Steele potential on a larger scale.

N2, resulting in the selectivity of O2 over N2. This reversal in ideal selectivity is enhanced by the decreasing dependence of the transport diffusivity of N2 on concentration in this pressure range. Similar calculations were repeated with the upstream pressure held constant at several different values and varying downstream pressures (Figure 9). Similar to the varying upstream pressure case, the effect of the potential is more prominent on the N2 fluxes than on the O2 fluxes (Figure 9a) and using the Steele potential results in moderately selective O2 to N2 permeation over the entire range of downstream pressure (Figure 9b). Again, with the ab initio potential a high N2 selectivity is observed for conditions when both the upstream and downstream pressures are in the low-pressure range of up to 10 kPa, and a reversal in ideal selectivity favoring O2 is predicted to occur for operations at conditions of 100 to 500 kPa. These results show that Steele potential predicts a small ideal selectivity in favor of oxygen with varying either upstream or downstream pressure and the pressure on the other side of the membrane being held constant. The results using the ab initio

We have used C168 schwarzite to represent nanoporous carbon membranes and have calculated the diffusivities and singlecomponent fluxes of nitrogen and oxygen through a membrane using equilibrium molecular dynamics simulations and a macroscopic description of mass transport. Two types of interaction potentials were used to model adsorbate-adsorbent interactions: an empirical Steele potential based on adsorption on planar graphite sheets and an ab initio-based potential that incorporates the effect of carbon curvature and the presence of nonhexagonal rings in C168 schwarzite. By analyzing the calculated diffusivities at low pressure over the temperature range from 80 to 450 K, the energetic and entropic contributions to the diffusive selectivity were calculated. It was observed that the N2/O2 separation is energetically driven with the ab initio potential resulting in selectivities as high as 8 for O2 over N2 at low temperature, whereas the entropic selectivity was found to be close to 1. However, predictions based on the Steele potential were that both energetic and entropic selectivities are close to unity over the entire range of pressure so that if this potential were to be correct no significant diffusive separation is likely. We next considered finite pressures and studied the effect of the choice of potential and varying adsorbate loading on the self-, corrected, and transport diffusivities of N2 and O2 in C168 schwarzite at 300 K. There was a marginal decrease in the selfand corrected diffusivities over the entire range of loading for O2 with the ab initio potential when compared to the Steele potential. However, there was a larger decrease in these properties for nitrogen because of the much stronger adsorption of N2 with the ab initio potential than with the Steele potential. Similar results were also found for the transport diffusivities. However, unlike the self- and corrected diffusivities, the transport diffusion coefficient of N2 using the ab initio potential was found to decrease at loadings higher than 30 molecules/unit cell because of a dominant decrease in the corrected diffusivities over the small increase in the thermodynamic factor. At all loadings, the mobility of O2 was found to be higher than the N2 using either the ab initio or the Steele potential. Once the concentration dependences of adsorption and transport were known, Fick’s equation of mass transport was used to calculate the fluxes of N2 and O2 and the selectivities across the C168 membrane at varying upstream and downstream pressures. Predictions based on the Steele potential showed a small selectivity in favor of oxygen over the entire range of pressure drop. However, it was found that the ab initio potential better explains the large O2/N2 selectivities of the similarly sized molecules that had been observed experimentally.1 Further, by suitably adjusting the pressures at the two ends of a membranes a high ideal selectivity in favor of either nitrogen or oxygen can be obtained, depending on the conditions. Similar to the conclusions obtained with the ab initio potential for adsorption-based separation of N2/O2 mixtures using a C168 schwarzite membrane,4 this work also predicts that these membranes can be useful for kinetic-based separations. Last, we note that this work focuses only on ideal selectivities, which provide an effective way of quantitatively describing the relative permeation rates of single components. For the purpose of separation of N2/O2 mixtures, say, for example, air into its major components, additional insight is obtained by calculating

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Arora and Sandler

mixture selectivity, which unlike ideal selectivity also includes the effects on the flux of individual components due to the presence of the second component. A detailed description of the EMD approach for calculating transport diffusivities and fluxes for binary mixtures can be found in studies of CF4/n-alkanes mixtures in faujasite by Sanborn and Snurr,14,43 CH4/CF4 mixtures in single-walled carbon nanotubes by Chen and Sholl,18,44 and our work on N2/O2 mixtures in single-walled carbon nanotubes.26,27 Alternatively, the mixture diffusivities can also be predicted using single-component data as shown for CH4/CF4 binary mixtures in the MFI zeolite45 and binary, ternary, and quaternary mixtures of methane, ethane, propane, and n-butane in the FAU zeolite.20 Under identical operating conditions, the ideal selectivity can be very different from the mixture selectivity

because in the latter case the presence of a second component can have a significant effect on the relative diffusion and adsorption strengths of individual components. The cross-term diffusivities of mixtures, if significant, also play an important role in calculating mixture fluxes. Further discussion of the differences between ideal and mixture selectivity can also be found in refs 22, 46, and 47.

(43) Sanborn, M. J.; Snurr, R. Q. AICHE J. 2001, 47, 2032-2041. (44) Chen, H. B.; Sholl, D. S. J. Membr. Sci. 2006, 269, 152-160. (45) Skoulidas, A. I.; Sholl, D. S.; Krishna, R. Langmuir 2003, 19, 79777988.

(46) Bakker, W. J. W.; Kapteijn, F.; Poppe, J.; Moulijn, J. A. J. Membr. Sci. 1996, 117, 57-78. (47) Keizer, K.; Burggraaf, A. J.; Vroon, Z.; Verweij, H. J. Membr. Sci. 1998, 147, 159-172.

Acknowledgment. This research was supported by grant CTS0083709 from the National Science Foundation and contract DE-FG02-85ER13436 from the Basic Energy Sciences Division of the U.S. Department of Energy. G.A. acknowledges the graduate fellow scholarship offered by the University of Delaware. LA053062H