pubs.acs.org/Langmuir © 2010 American Chemical Society
Modeling Pure Gas Permeation in Nanoporous Materials and Membranes Suresh K. Bhatia* School of Chemical Engineering, The University of Queensland, Brisbane QLD 4072, Australia Received December 20, 2009. Revised Manuscript Received February 5, 2010 The low-pressure transport of simple fluids in nanopores and in disordered nanoporous networks is analyzed, using a recent oscillator model theory from the author’s laboratory, considering the trajectories of molecules moving in the potential energy field of the fluid-pore wall interaction. The scaling behavior of the single-pore theory is discussed, and it is shown that the Knudsen model provides an upper bound to the diffusivity scaled with the pore radius. The singlepore theory is shown to apply well to ordered materials and successfully interprets recent literature data on the variation of permeability with diffusant molecular size for a DDR zeolite membrane. A peak in permeability is seen at a pore-sizedependent molecular size because of the opposing effects of equilibrium and transport. Application to disordered pore networks is also presented on the basis of a hybrid correlated random walk effective medium theory imbedding the oscillator model at the single-pore level, and a rigorous expression for the tortuosity is derived from the theory. A rich variety of behavior is predicted for the tortuosity, which can increase or decrease with increasing extent of pore size nonuniformity as well as with changes in temperature because the diffusing species preferentially flows through more conducting pores. Weakly adsorbing gases such as helium are seen to have a higher tortuosity than more strongly adsorbing ones. The predicted values of tortuosity are shown to be in line with those obtained from the interpretation of recent experimental mesoporous membrane transport data and are in the range of 5-10 whereas those extracted using the Knudsen model are unrealistically high, in the range of 10-20.
Introduction The subject of the transport of fluids in narrow pores has long been of interest to engineers and scientists because of its importance in catalytic and noncatalytic fluid-solid reactions as well as in adsorptive separations and electrochemical processes. In the last few decades, this interest has been greatly enhanced by the increasing importance of adsorptive molecular sieving and membrane-based processes in industrial separation as well as by the emergence of a host of new applications in adsorptive gas storage, nanofluidics, and lab-on-a-chip technology.1 Complementing these advances has been the explosive growth of a vast array of new ordered nanoporous materials, such as carbon nanotubes,2 MCM-41 silicas and their anologues,3,4 and metalorganic frameworks such as aluminosilicates and aluminophosphates,5,6 which are all considered to hold promise for the above applications. There is also considerable ongoing effort in the synthesis of membranes based on zeolites7-9 and such ordered as well as disordered materials.10,11 The infiltration of fluids into the *E-mail:
[email protected]. (1) Sparreboom, W.; van den Berg, A; Eijkel, J. C. T. Nat. Nanotechnol. 2009, 4, 713–720. (2) Iijima, S. Nature 1991, 354, 56–58. (3) Kresge, C. T.; Leonowicz, M. E.; Roth, W. J.; Vartuli, J. C.; Beck, J. S. Nature 1992, 359, 710–712. (4) Selvam, P.; Bhatia, S. K.; Sonwane, C. G. Ind. Eng. Chem. Res. 2001, 40, 3237–3261. (5) Davis, M. E.; Saldarriaga, C.; Montes, C.; Garces, J.; Crowder, C. Nature 1988, 331, 698–699. (6) Davis, M. E. Nature 2002, 417, 813–821. (7) McLeary, E. E.; Jansen, J. C.; Kapteijn, F. Microporous Mesoporous Mater. 2006, 90, 198–220. (8) Tomita, T.; Nakayama, K.; Sakai, H. Microporous Mesoporous Mater. 2004, 68, 71–75. (9) Zhu, W.; Kapteijn, F.; Moulijn, J. A.; Jansen, J. C. Phys. Chem. Chem. Phys. 2000, 2, 1773–1779. (10) Hinds, B. J.; Chopra, N.; Rantell, T.; Andrews, R.; Gavalas, V.; Bachas, L. G. Science 2004, 303, 62–65. (11) Kim, Y.C.; Jeong, J. Y.; Hwang, J. Y.; Kim, S. D.; Yi, S. C.; Kim, W. J. J. Membr. Sci. 2008, 325, 252–261.
Langmuir 2010, 26(11), 8373–8385
narrow pore spaces of such materials is common to all applications, and this has catalyzed much activity on the understanding of fluid behavior in confined spaces, particularly with regard to structural characterization and adsorption equilibrium.4,12-15 In addition to the equilibrium, the understanding of the diffusion and transport of confined fluids in nanoporous materials is crucial to the modeling and process design of such applications. Nevertheless, little progress has been made since the early work of Knudsen almost a century ago, and the modeling approach is still largely empirical, being based on the dusty gas model (DGM)16,17 that arbitrarily considers the superposition of diffusive and viscous resistance, with the latter being based on the classical Poiseuille flow model. The diffusive part itself superposes Knudsen and activated surface diffusion resistance, along with the molecular diffusion resistance.18 Whereas this approach has been used predominantly for mesopores and macropores, for micropores the surface diffusion component is assumed to dominate the Knudsen part19 and the viscous resistance is considered to be negligible. However, the approach lacks a firm molecular basis and has been largely unsuccessful, unable to explain even a simple experiment such as that of the Stefan tube satisfactorily. Although it has been given some statistical mechanical justification by Mason and Viehland20 in their extension of the model to dense phases, fundamental concerns on key features have been (12) Sonwane, C. G.; Bhatia, S. K. Langmuir 1999, 15, 2809–2816. (13) Sonwane, C. G.; Bhatia, S. K.; Calos, N. Langmuir 1999, 15, 4603-4612. (14) Qiao, S.; Bhatia, S. K.; Nicholson, D. Langmuir 2004, 20, 389–395. (15) Qiao, S.; Bhatia, S. K.; Zhao, X. S. Microporous Mesoporous Mater. 2003, 65, 287–298. (16) Evans, R. B.; Watson, G.; Mason, E. A. J. Chem. Phys. 1962, 36, 1894– 1902. (17) Mason, E. A.; Malinauskas, A. P.; Evans, R. B. J. Chem. Phys. 1967, 46, 3199–3216. (18) Feng, C.; Stewart, W. E. Ind. Eng. Chem. 1973, 12, 143–147. (19) K€arger, J.; Rutheven, D. M. Diffusion in Zeolites and Other Microporous Solids; Wiley: New York, 1992. (20) Mason, E. A.; Viehland, A. J. Chem. Phys. 1978, 68, 3562–3573.
Published on Web 03/16/2010
DOI: 10.1021/la9047962
8373
Article
raised by Kerkhof and Geboers.21 These latter authors question the use of a mixture center-of-mass-based frames of reference, indicating difficulties in the convergence of the underlying Enskog treatment of the Boltzmann equation, especially when different species have widely varying mobilities. Another weakness of the approach is the disregard of inhomogeneity and the nonuniformity of the density profile arising from dispersive fluid-solid interactions, which is shown to be particularly significant for mesopores and micropores.22,23 This weakness is also inherent in the modified approach suggested by Kerkhof and Geboers21 as well as the recent Maxwell-Stefan-model-based approach24 developed for microporous materials such as zeolites. The latter is similar to the DGM in spirit and uses empirical pure component diffusivities in the nanoporous material to predict multicomponent transport coefficients. However, the way in which this is done is somewhat questionable because binary diffusivities are empirically estimated using low-density pure component diffusivities in the material, thereby violating established formulations from irreversible thermodynamics25 in which independent binary coefficients are required to satisfy the available degrees of freedom. It is therefore not surprising that the relationship used to estimate binary diffusivities has varied between systems, with the Vignes relation used in some cases26 and modified relations used in others.27 Molecular dynamics (MD) simulation represents the modern recourse for predicting transport coefficients of pure species and mixtures in nanoporous systems28 but is computationally too intensive for use in process design even for simple fluids. The quest for a tractable theory founded on molecular principles is therefore of much significance and has recently attracted some attention. However, rigorous theories29,30 have been largely unsuccessful in achieving this end because mechanical models rapidly become intractable beyond the low-density region. Several more tractable models have been developed in the last two decades,31-34 and they are based on the use of bulk viscosity models with a nonuniform density profile in the fluid phase arising from the inhomogeneous fluid-solid potential field and can be valid only at large densities where the mean free path is much smaller than the pore size (i.e., the low Knudsen number regime). Some work in the large Knudsen number region has been reported for slit pores35 but considers hard sphere systems and has yet to be extended to include dispersive interactions. Considerable success in the development of a practical theory based on molecular fundamentals has been obtained in the author’s laboratory, aided by the finding from molecular dynamics simulations that equilibrium density profiles persist even (21) Kerkhof, P. J. A. M.; Geboers, M. A. M. AIChE J. 2005, 51, 79–121. (22) Bhatia, S. K.; Nicholson, D. Phys. Rev. Lett. 2008, 100, 236103. (23) Bhatia, S. K.; Nicholson, D. J. Chem. Phys. 2008, 129, 164709. (24) Krishna, R; Wesselingh, J. A. Chem. Eng. Sci. 1997, 52, 861–911. (25) Fitts, D. D. Nonequilibrium Thermodynamics: A Phenomenological Theory of Irreversible Processes in Fluid Systems; McGraw-Hill: New York, 1962. (26) Krishna, R. Chem. Eng. Sci. 1993, 48, 845–861. (27) Krishna, R.; van Baten, J. M. J. Phys. Chem. B 2005, 109, 6386–6396. (28) Dubbeldam, D.; Snurr, R. Q. Mol. Simul. 2007, 33, 305–325. (29) Davis, H. T. Fundamentals of Inhomogenous Fluids; Henderson, D., Ed.; Marcel Dekker: New York, 1992. (30) Pozhar, L. A. Transport Theory of Inhomogeneous Fluids; World Scientific: Singapore, 1994. (31) Bitsanis, I.; Vanderlick, T. K.; Tirrell, M.; Davis, H. T. J. Chem. Phys. 1988, 89, 3152–3162. (32) Vanderlick, T. K.; Davis, H. T. J. Chem. Phys. 1987, 87, 1791–1795. (33) Guo, Z.; Zhao, T. S.; Shi, Y. Phys. Rev. E. 2005, 71, 035301. (34) Guo, Z.; Zhao, T. S.; Xu, C.; Shi, Y. Int. J. Comp. Fluid Dyn. 2006, 20, 361– 367. (35) Arya, G.; Chang, H. C.; Maginn, E. J. Phys. Rev. Lett. 2003, 91, 026102. (36) Bhatia, S. K.; Nicholson, D. Phys. Rev. Lett. 2003, 90, 016105. (37) Bhatia, S. K.; Nicholson, D. J. Chem. Phys. 2003, 119, 1719–1730.
8374 DOI: 10.1021/la9047962
Bhatia
during transport.36,37 The incorporation of the assumption of equilibrium density profiles has permitted considerable simplification in mechanical models of transport, and a tractable theory considering dispersive fluid-solid interactions has been possible under conditions of diffuse wall reflections as considered in the Knudsen model. Central to the theory is the oscillator model for low density,38-40 which is exact in the case of diffuse reflection. The theory considers the trajectories of particles oscillating in the fluid-solid potential field and determines the diffusion coefficient based on the momentum gained between diffuse wall collisions. The theory has been extensively validated against molecular dynamics simulations38-42 and has even been exploited to estimate surface reflection coefficients in carbon nanotubes43 where the smooth potential energy landscape leads to the nearly specular reflection of diffusing molecules. It has also recently been extended22,23 to include fluid-fluid intermolecular interactions at higher densities beyond the Henry’s law region through a novel frictional approach that allows coverage of the entire region from low to high Knudsen numbers. Nevertheless, the theory needs further confirmation based on experimental data, and one of the aims of this article is to meet this need. A complexity attending the application of any transport theory to experimental data or its exploitation in process design is the intrusion of pore network and connectivity effects as well as the presence of nonuniform pore sizes, especially in porous materials that are disordered. Effective medium theory (EMT)44-47 provides the necessary machinery for modeling these phenomena, and in this method, a nonuniform network of conductors is replaced by a uniform one in which each conductance is assigned an effective value. This approach has recently been employed by Cai et al.48 in determining effective diffusivities in carbons, with the pores arranged on a cubic lattice. However, EMT has the drawback that the tortuosity of the effective network is unknown and remains an adjustable fitting parameter for disordered systems where the pores cannot be assumed to be arranged on such a regular or symmetric lattice. This problem is avoided by the correlated random walk theory (CRWT) of the author,49,50 which analyzes the meandering of molecules between pore intersections in a random network, while considering the correlation between successive pores traversed because of the finite probability that a molecule retraces it is path. For purely gas-phase diffusion without adsorption, as is essentially the case for large macropores, it has been shown50 that as a result of this correlation the tortuosity increases and can vary with processing conditions such as temperature and pressure as well as the diffusing species. Although this theory can predict the tortuosity, it is intrinsically less accurate than the effective medium theory near the percolation threshold and a hybrid EMT-CRWT approach47 has been found to yield excellent agreement with simulation. (38) (39) 4485. (40) 5406. (41) (42) (43) (44) (45) 1502. (46) (47) (48) 3327. (49) (50)
Jepps, O. G.; Bhatia, S. K.; Searles, D. J. Phys. Rev. Lett. 2004, 91, 126102. Bhatia, S. K.; Jepps, O. G.; Nicholson, D. J. Chem. Phys. 2004, 120, 4472– Jepps, O. G.; Bhatia, S. K.; Searles, D. J. J. Chem. Phys. 2004, 120, 5396– Bhatia, S. K.; Nicholson, D. AIChE J. 2006, 52, 29–38. Kumar, A. V. A.; Bhatia, S. K. J. Phys. Chem. B 2006, 110, 3109–3113. Bhatia, S. K., Chen, H.; Sholl, D. S. Mol. Sim. 2005, 31, 643-649. Kirkpatrick, S. Rev. Mod. Phys. 1973, 45, 574–588. Sahimi, M.; Gavalas, G. R.; Tsotsis, T. T. Chem. Eng. Sci. 1990, 45, 1443– Burganos, V. N.; Sotirchos, S. V. AIChE J. 1987, 33, 1678–1689. Deepak, P. D.; Bhatia, S. K. Chem. Eng. Sci. 1994, 49, 245–257. Cai, Q; Buts, A; Seaton, N. A.; Biggs, M. J. Chem. Eng. Sci. 2008, 63, 3319– Bhatia, S. K. J. Catal. 1985, 93, 192–196. Bhatia, S. K. Chem. Eng. Sci. 1986, 41, 1311–1324.
Langmuir 2010, 26(11), 8373–8385
Bhatia
Article
Although the above studies with the hybrid EMT-CRWT have been carried out for gas-phase diffusion, the transport of adsorbates in nanopore networks such as in membranes has yet to receive attention. Here we adopt this hybrid procedure in conjunction with the oscillator model theory38,39 to investigate transport in nanopore networks in the presence of adsorption in the Henry’s law region and to study the variation of tortuosity with diffusing gas as well as temperature. A rich variety of behavior is predicted for the tortuosity, which can increase or decrease with increasing extent of pore size nonuniformity as well as with changes in temperature, because the diffusing species preferentially flows through more conducting pores. This behavior is governed by the strength of the adsorption, and the theory is corroborated by experimental permeability data for the transport of several gases in a microporous DDR membrane8 as well as in disordered mesoporous silicate glass membranes of varying pore size.51 It is also shown that unreasonably high values of tortuosity, often as large as 15-20,51 arising from the use of the Knudsen model, are avoided when the oscillator theory is used to interpret transport data in disordered materials.
Theory Oscillator Model for Transport at Low Density in a Single Pore. Fundamental to our analysis of adsorbate transport in pore networks, the main thrust of the present work is the oscillator model,38,39 which provides the diffusion coefficient at the singlepore level. In the present work, the individual pores are considered to be cylindrical and infinitely long, with an arbitrary radius, rp, measured between the centers of diametrically opposite solid surface atoms. The model considers the motion of fluid particles subjected to an axial chemical potential gradient while under the influence of a 1D potential field, φ(r), governed by the fluid-wall interaction. Cross-sectional equilibrium is assumed on the basis of molecular dynamics evidence36,37 that equilibrium density profiles prevail even during transport. Under conditions of diffuse wall reflection, the product of the axial driving force, -rμ, and the mean time of the trajectory, Æτæ, will be balanced by the momentum gained between collisions, leading to Æuz æ ¼
Do ð -rμÞ Æτæ ð -rμÞ ¼ m kB T
ð1Þ
where m is the molecular mass and Do is the corrected transport (i.e., collective) diffusivity based on a chemical potential driving force. Equation 1 provides the transport coefficient Do ¼
kB T Æτæ m
ð2Þ
In the conservative force field of the 1D fluid-wall potential, φ(r), the angular momentum as well as the energy Eðr, pr , pθ Þ ¼ φðrÞ þ
pr 2 pθ 2 þ 2m 2mr2
ð3Þ
are constant, leading to the result for the time duration of a trajectory between wall collisions:38,39 Z τðr, pr , pθ Þ ¼ 2m
rc1 ðr, pr , pθ Þ
rco ðr, pr , pθ Þ
dr0 pr ðr0 , r, pr , pθ Þ
ð4Þ
(51) Markovic, A.; Enke, D.; Schl€under, E.-U.; Stoltenberg, D.; SeidelMorgenstern, A. J. Membr. Sci. 2009, 336, 17–31.
Langmuir 2010, 26(11), 8373–8385
Here pr(r0 , r, pr, pθ) is the radial momentum at position r0 for a particle having radial momentum pr at position r and is given by 8 !91=2 < 2 = 2 p r θ pr ðr0 , r, pr , pθ Þ ¼ 2m½φðrÞ - φðr0 Þ þ pr 2 ðrÞ þ 2 1 - 02 : r r ; ð5Þ In equation 4, rc1(r, pr, pθ) and rco(r, pr, pθ) are the radial bounds of the trajectory of the particle, with the former corresponding to the location of the diffuse wall reflection and the latter corresponding to its radius of closest approach to the center. These are given by the values of r0 corresponding to the solution of pr ðr0 , r, pr , pθ Þ ¼ 0
ð6Þ
The transport coefficient is now obtained by averaging the trajectory time in eq 4 with a canonical distribution of energies and substituting this result for Æτæ in eq 2. This yields Z ¥ Z ¥ 2 2 -βjðrÞ e dr e -βpr =2m dpr Do ¼ πmQ 0 0 Z ¥ Z rc1 ðr, pr , pθ Þ dr0 2 2 ð7Þ e -βpθ =2mr dpθ 0 0 rco ðr, pr , pθ Þ pr ðr , r, pr , pθ Þ R -βφ(r) where Q = ¥ dr and pr(r0 , r, pr, pθ) follows eq 5. Whereas 0 re eq 7 is general and can be used with any arbitrary field φ(r), in the present work we employ the Tjatjopoulos et al.52 potential Z φðrÞ ¼ 16εfs Fs rp
Z
¥
dz 0
0
π
"
σfs r0
12 6 # σfs dR - 0 r
ð8Þ
which considers the pore surface to comprise randomly distributed sites interacting with fluid particles by a Lennard-Jones 6-12 potential. Here r0 = [z2 þ r2 þ rp2 - 2rrp cos(R)]1/2, εfs and σfs are fluid-solid Lennard-Jones parameters, and Fs is the areal density of pore surface sites. Although the result for φ(r) can be expressed52 in terms of hypergeometric functions, here we have used eq 8 with numerical integration of the double integral to obtain φ(r). Transport in Pore Networks. An important aim of this article is to investigate the transport of adsorbates in nanoporous materials that have disordered networks, with the above oscillator model employed at the individual pore level. To this end, we consider a random network of interconnected pores having a coordination number N. This represents the number of pores meeting at an intersection and is assumed to be uniform throughout the network. Effective medium theory44,45 provides a convenient tool for describing transport in the network and replaces the network by an effective one having pores of uniform conductance. Following the hybrid EMT-CRWT procedure,47 the use of the CRWT on this effective network then provides its tortuosity and enables the calculation of its transport properties. Previous work along these lines47 has been concerned with gaseous flows without adsorption and is further developed here for adsorbed phases. For a pore of arbitrary radius rp, the molecular current is given as i ¼
πrp 2 Fa Do ðrp Þ ð -rμÞ Rg T
ð9Þ
(52) Tjatjopoulos, G. J.; Feke, D. L.; Mann, J. A. J. Phys. Chem. 1988, 92, 4006– 4007.
DOI: 10.1021/la9047962
8375
Article
Bhatia
which may be expressed in terms of a pseudobulk pressure drop to provide πrp 2 Kðrp ÞDo ðrp Þ ð -ΔPÞ ð10Þ i ¼ Rg Tl where l is the pore length and P is the pseudobulk pressure at which the bulk fluid will be in equilibrium with the adsorbate of density Fa. Here Do(rp) is the corrected pore diffusivity obtained from the oscillator model described above. Although the present development is more generally applicable, here we consider only the low-density case for which the bulk fluid is ideal and the equilibrium constant K(rp) is given by Z 2 rp expð -φðrÞ=kB TÞr dr ð11Þ Kðrp Þ ¼ 2 rp 0 Equation 10 provides the conductance for the pore of radius rp as λ ¼
πrp Kðrp Þ Dðrp Þ l 2
where Æ•æ represents a number average over the pores. For the effective medium, the pore flux may be expressed as λe l 2 ð -rPÞ πrp 2 Rg T
ð14Þ
and upon assuming the pseudobulk pressure to be locally uniform in the network (i.e., the pores are locally at equilibrium) we obtain the net flux as46 j ¼
ελe Æl 2 æ ð -rPÞ γπRg TÆrp 2 l æ
ð15Þ
where ε is the network porosity in the solid and γ is a tortuosity. For a disordered network of pores with identical conductances, the tortuosity is given as49,50 γ ¼
3ðN þ 1Þ ðN -1Þ
ð16Þ
in which the value of 3 for random orientation is modified by the factor (N þ 1)/(N - 1) to account for correlation between successive pores traversed by a diffusing molecule as it meanders between pore intersections. Equations 15 and 16 yield j ¼
ελe ðN -1ÞÆl 2 æ ð -rPÞ 3πðN þ 1ÞRg TÆrp 2 l æ
ð17Þ
In practice, when interpreting experimental data it is common to evaluate the diffusivity and flux considering only a single pore size, rp, such as a suitable mean or modal pore size, so that eq 15 yields j ¼
εKðrp Þ Do ðrp Þ ð -rPÞ γapp Rg T
ð18Þ
where γapp is an apparent tortuosity that is generally obtained by fitting the measured flux to eq 18. Comparing eqs 17 and 8376 DOI: 10.1021/la9047962
18, we obtain the theoretical value of this apparent tortuosity to be
ð12Þ
It is assumed that on the nanoscale local equilibrium prevails so that the pseudobulk pressure drop (-ΔP) is considered to be pore-size-independent. The effective medium conductance, λe, is now given44-47 by the solution to ðλ -λe Þ ¼0 ð13Þ ½λ þ ðN=ð2 -1ÞÞλe
jp ¼
Figure 1. Variation of the scaled diffusivity with the dimensionless pore radius for various gases at different values of the scaled energy parameter.
γth app ¼
3πðN þ 1ÞÆrp 2 l æKðrp ÞDo ðrp Þ ðN -1ÞÆl 2 æλe
ð19Þ
Here we explore the variation of this theoretical apparent tortuosity with pore size distribution parameters as well as temperature and gaseous species. It is shown that the values and trends compare well with those obtained from actual experimental data.
Results and Discussion Scaling of Single-Pore Diffusivity. The oscillator model in eqs 1-7 provides the transport coefficients at the single-pore level and has been analyzed in some depth as well as validated against molecular dynamics simulation in earlier articles.39-42 Equations 5-7 provide the result for the low-density diffusivity with diffuse wall reflection in a pore of diameter rp in the presence of the fluid solid potential field and replaces the established Knudsen result that does not consider dispersive interactions. As an illustration of its generality, at high temperature, where the kinetic energy dominates the interaction, the result in eq 7 has been shown39 to lead to the Knudsen formula. An interesting feature noted here is that for the potential energy profile considered here eqs 5-8 provide the scaling ! rffiffiffiffiffiffiffiffiffi Do m rp Fs εfs σ fs 2 , ¼f rp kB T σfs kB T
ð20Þ
For the high-temperature Knudsen limit, the function on the right-hand side of the above equation is constant and has a value of 1.064. To investigate the form of this function in the presence of dispersive interactions, calculations were carried out here for various gases at 298 K and therefore various values of (Fsεfsσfs2/kBT) in silica pores of various sizes, and the results are illustrated in Figure 1. Table 1 lists the Lennard-Jones parameters for the different gases considered here, whereas the parameters for the surface sites were obtained from the values Fsεfs = 2253 K/nm2 and σfs = 0.317 nm for N2 on an all-silica MCM-41 obtained by Ravikovitch et al.53 The Lorentz-Berthelot rule was employed to relate all binary interaction parameters to those of the individual components. The solid lines in Figure 1 represent the results for the various gases at 298 K, with the corresponding different values of (Fsεfsσfs2/kBT) indicated in parentheses. As a (53) Neimark, A. V.; Ravikovitch, P. I.; Grun, M.; Schuth, F; Unger, K. K. J. Colloid Interface Sci. 1998, 207, 159–169.
Langmuir 2010, 26(11), 8373–8385
Bhatia
Article
Table 1. Fluid-Fluid Lennard-Jones Parameters Used in Calculations parameter
He
H2
Ar
O2
N2
CH4
Xe
CF4
σff (nm) 0.2551 0.2915 0.341 0.3467 0.3572 0.381 0.4047 0.4662 120.0 106.7 93.98 148.2 231.0 134.0 ∈ff/kB (K) 10.22 38.0
confirmation of the scaling represented in eq 20, results were also obtained for H2 at 656 K, for (Fsεfsσfs2/kBT ) = 0.176 corresponding to He at 298 K and H2 at 84 K, and for (Fsεfsσfs2/kBT ) = 1.376 corresponding to Xe at 298 K. These are given by the dashed and dotted lines, which match the corresponding curves for He and Xe at 298 K, respectively. While demonstrating the scaling behavior of the diffusivity and the approach to the Knudsen limit at high temperature and large pore size, Figure 1 also shows a peak in the value of the scaled diffusivity on the right-hand side of eq 20 in the dimensionless pore radius range of about 1.2-1.4. This is close to the size at which the phenomenon of levitation is manifested,42,54,55 for which the potential energy profile undergoes a transition from a single to a double potential well and the diffusing molecule becomes confined to a narrow region near the wall as the pore size increases. This leads to a decrease in the time of an oscillation in eq 4 and hence to a decrease in diffusivity with an increase in pore radius. However, as the pore radius is increased, the width of the potential wells near the surface increases and the diffusivity goes through a minimum before again increasing with pore size. This behavior does not occur at high temperature,42 where the molecules have sufficient kinetic energy to oscillate over the whole pore width and are not confined to the potential well near the wall. Thus, the minimum is absent for the curve for He at 298 K or for H2 at 656 K, for which Fsεfsσfs2/kBT = 0.176. Apparent Tortuosity of Pore Networks. Equation 19 provides the apparent low-pressure tortuosity of a porous material composed of randomly interconnected pores of arbitrary size distribution having a uniform coordination number N at each node. Here we investigate the effect of the pore size distribution (PSD) and temperature on this tortuosity for various gases in microporous as well a mesoporous networks, with the single pore diffusivities given by the oscillator model and the equilibrium constant following eq 11. To this end, we consider the Rayleigh distribution of pore sizes f ðrp Þ ¼
ðrp - ro Þ 2
ðrm - ro Þ
2
2
e -½ðrp - ro Þ =ðrm - ro Þ =2 , ro e rp < ¥
ð21Þ
where f(rp) is the number density of pores of radius rp, ro is the minimum pore radius, and rm is the modal pore radius. The standard deviation of this distribution is readily seen to be s ¼ 0:7024ðrm - ro Þ
ð22Þ
Figure 2 depicts this distribution for three different modal pore sizes, with ro chosen such that the relative standard deviation, s/rm, has a value of 0.5. The distribution is skewed, and with increaseing modal pore radius, it is seen to be wider. The maximum allowable relative standard deviation is 0.7024, for which ro = 0. In practice, however, pores smaller than about 0.55 nm in diameter are essentially inaccessible to most gases and have virtually zero conductance. Hence, we have considered distributions with ro >0.275 nm, for which s/rm < 0.7024(1 - (0.275/rm)). (54) Derouane, E. G.; J-M. Andre, J.-M.; A.A. Lucas, A. A. J. Catal. 1988, 110, 59–73. (55) Yashonath, S.; Santikary, P. J. Phys. Chem. 1994, 98, 6368–6376.
Langmuir 2010, 26(11), 8373–8385
Figure 2. Illustration of the shape of the Raleigh pore size distribution for different values of modal and minimum pore radii.
Figure 3. Effect of coordination number and model pore radius on the variation of apparent tortuosity with the relative standard deviation of the pore size distribution.
Furthermore, for purposes of evaluating the apparent tortuosity we take the mean pore size, rp, to be that based on the pore volume to surface area ratio given by R¥ rp ¼ Rro¥
rp 2 l f ðrp Þ drp
ro rp l
f ðrp Þ drp
ð23Þ
and assume that the pore length, l , is uniform and independent of the radius. For most porous materials it may be expected that the coordination number, N, is about 3 or 4, which is consistent with percolation theory characterisations of slit pore carbons.56-58 For uniform networks, one expects a tortuosity of 6 when N = 3 and 5 when N = 4, following eq 16. However, this will be modified by the presence of a nonuniform PSD because of the short-circuiting effect of pores with high conductance. Figure 3 depicts the variation of apparent tortuosity, following eq 19, with the relative standard deviation for Ar at 298 K in disordered silica pore networks having rm = 0.5 nm (microporous) and 1.3 nm (mesoporous) and a coordination number of 3 or 4. In both cases, (56) Ismadji, S.; Bhatia, S. K. Langmuir 2000, 16, 9303–9313. (57) Ismadji, S.; Bhatia, S. K. J. Colloid Interface Sci. 2001, 244, 319–335. (58) Lopez-Ramon, M. V.; Jagiello, J.; Bandosz, T. J.; Seaton, N. A. Langmuir 1997, 13, 4435–4445.
DOI: 10.1021/la9047962
8377
Article
Bhatia
Figure 4. Variation of pore conductance with pore radius: (a) over the pore radius range of 0-12 nm and (b) an expanded view over the pore radius range of 0.2-1.6 nm showing a peak in conductivity.
increases in the coordination number reduce the tortuosity, as is to be expected, because with increasing N there are more alternate paths available and the chance of a molecule retracing its path decreases. For the larger modal pore radius of 1.3 nm, it is seen that the tortuosity increases monotonically with increasing standard deviation for both values of N and has a minimum value for the uniform network (s = 0) following eq 16. This increase is due to the increasing availability of large pores with high conductance, which serve as short circuit paths, as the width of the distribution is increased. Remarkably, however, for the microporus network with rm=0.5 nm the tortuosity has a small initial increase and goes through a maximum before decreasing as the value of s is increased and the distribution widens. The above unexpected behavior is best understood through a plot of the variation of conductance with pore size, as depicted in Figure 4 with the pore length assumed to be 10 nm. It is noted that the conductance corresponds to that for a pore of given radius, rp, and is not that for a network with a pore size distribution. Furthermore, the choice of a 10 nm pore length is arbitrary, purely to illustrate the behavior of the conductance, and is not necessarily that corresponding to a real system. In fact, for nearly specular diffusion a very long pore length, compared to the pore diameter, may be needed for the trajectories to decorrelate and a pore-length-independent diffusivity (corresponding to an infinitely long pore) to be attained. When the actual pore length is too small for the trajectories to decorrelate, then it would be necessary to perform molecular dynamics simulations for the short pores to obtain the pore diffusivity before network effects can be considered. However, this is not important for our purpose, which is purely to interpret the results in Figure 3 for a network of long pores of uniform length, and as is readily determined from the derivation in eqs 12-19, the tortuosity is independent of the pore length when this is uniform. Hence, the arbitrary choice of a 10 nm pore length has no influence on the 8378 DOI: 10.1021/la9047962
Figure 5. Variation of apparent tortuosity with the relative standard deviation in the pore size distribution for various gases at 298 K: (a) at a modal pore radius of 0.5 nm and (b) at a modal pore radius of 1.3 nm.
present results. As seen in Figure 4a,b, in the mesopore range the conductance increases monotonically with pore size; however, it has a sharp maximum for micropores of about 0.375-0.4 nm radius for all gases except He and H2 (Figure 4b). This is due to the strong adsorption with a sharp maximum in the equilibrium constant near this pore radius for most of the gases except helium. At smaller pore size, repulsive effects reduce adsorption, and at larger pore size, the dispersive interaction weakens. This size is also close to that at which a weak maximum in the diffusivity occurs, discussed above as the levitation effect, and both of these factors combine to provide the strong maximum in the conductance near this size. Furthermore, there is a weak minimum in conductance near a pore radius of 0.5 nm, with only a small gradual increase in conductance with increasing pore size beyond this minimum. Thus, in the microporous network with rm = 0.5 nm, as the width (i.e., standard deviation) of the PSD increases, both smaller and larger pores offer short circuit paths of higher conductance compared to that at pore size rm. This leads to the weak maximum followed by a decrease in apparent tortuosity for Ar, with increasing width of the PSD. Nevertheless, one may expect an increase in tortuosity at sufficiently large values of s/rm for the other gases because of the steeper decrease in conductance for pore sizes smaller than that at which the maximum in conductance occurs, as seen in Figure 3. This is indeed found to be the case, as is evident from Figure 5a, which depicts the variation in apparent tortuosity with the relative standard deviation following eqs 12, 13, 19, and 21, for a modal pore radius, rm, of 0.5 nm and N = 3 for various gases at 298 K. Although Langmuir 2010, 26(11), 8373–8385
Bhatia
Figure 6. Variation of pore conductance with temperature for various gases: (a) at a modal pore radius of 0.5 nm and (b) at a modal pore radius of 1.3 nm.
weakly interacting gases He and H2 show monotonic increases in tortuosity, all other gases besides Ar show a minimum in tortuosity at a value of s/rm of between 0.2 and 0.3. Figure 5b depicts the variation of tortuosity with s/rm for the mesoporous network having rm= 1.3 nm and N = 3, showing a monotonic increase in tortuosity with increasing s/rm for all gases except the most strongly adsorbing ones (CF4 and Xe). In the latter case, the tortuosity decreases mildly from the uniform network value of 6 because of the relatively modest increase in conductance with increasing s/rm, as seen in Figure 4a,b. This leads to only a low level of short circuit paths being created, and the effect of increased conductivity as s/rm increases leads to a reduction in apparent tortuosity. In Figure 5a,b, we see that the apparent tortuosity is strongly dependent on the adsorptive species and is highest for the more weakly adsorbed ones for which the conductance increases more strongly with increasing pore size and the effect of short circuiting dominates. Figure 6a,b depicts the variation of pore conductance with temperature for the various gaseous adsorptives in silica, for pore radii of 0.5 nm (micropore) and 1.3 nm (mesopore) respectively, with pore length assumed to be uniform at 10 nm. As in Figure 4, the conductance corresponds to the single capillary of given radius, rp, and not for a network. For the smaller pore radius of 0.5 nm in Figure 6a, it is seen that the conductance of He increases mildly with temperature whereas that of all other gases decreases with temperature. This decline in conductance with temperature reflects the importance of adsorption at low temperature for these gases because with increasing temperature the equilibrium Langmuir 2010, 26(11), 8373–8385
Article
Figure 7. Variation of apparent tortuosity with temperature for various gases for (a) a modal pore radius of 0.5 nm and a relative standard deviation of 0.29 and (b) a modal pore radius of 1.3 nm and a relative standard deviation of 0.4.
constant K decreases and the diffusion coefficient Do increases. For the most strongly adsorbed species, Xe and CF4, this decline in conductance is quite steep below about 300 K and milder at higher temperatures because of the reduced importance of adsorption equilibrium. This suggests that as the temperature is increased, at some point when the adsorption is sufficiently weak the conductance must increase. This is indeed borne out for the higher pore size of Figure 6b, where the conductance of H2 increases with temperature over the entire range whereas that of Ar and N2 increases beyond about 300 K. The most interesting observation from Figure 6a,b is that at sufficiently low temperatures strongly adsorbed heavier gases can have a higher conductance than more weakly adsorbed lighter gases because of the importance of the equilibrium constant in the product KDo on which the conductance is based. Thus, for the smaller pore radius of 0.5 nm all gases have higher conductances than even helium below 500 K, whereas for a pore radius of 1.3 nm CH4, CF4, and Xe have higher conductances than helium below about 300 K, despite being heavier and much less mobile. However, the lighter H2 is both more strongly adsorbing and more mobile than helium and has a higher conductance at all temperatures. Similarly, for rm = 0.5 nm, both Ar and N2 have a higher conductance than H2 below about 360 K and a lower conductance above this temperature, reflecting the coupling of adsorption equilibrium and transport. Figure 7a depicts the variation of apparent tortuosity, following eqs 12, 13, 19, and 21, for the different gases with temperature DOI: 10.1021/la9047962
8379
Article
for a microporous network with N = 3, rm = 0.5 nm, and s/rm = 0.29, and Figure 7b depicts the variation of apparent tortuosity for a mesoporous network with N = 3, rm = 1.3 nm, and s/rm = 0.4. As before, the tortuosity is largest for the weakest adsorbing gas, He, and decreases with increasing strength of the adsorption. In general, the tortuosity increases with increasing temperature. This is because of the larger loss in conductance of the small pores compared to that of the large pores as temperature increases in a network with a pore size distribution (i.e., s > 0), as evident from Figure 6a,b. This leads to an increased propensity for transport through larger pores and thereby an increase in tortuosity. An interesting observation from Figure 7a,b is that strongly adsorbing gases can have an apparent tortuosity that is less than that given by eq 16 because of the presence of a maximum in conductance at small pore radius as seen in Figure 4a,b. As a result, in the presence of a PSD both small and large pores offer short circuit paths of larger conductance than the pores of the uniform network and this leads to a reduction in apparent tortuosity. Indeed, these complexities lead to a minimum in the apparent tortuosity near 300 K for the most strongly adsorbed gases, CF4 and Xe, for rm = 0.5 nm. It is evident from the above discussion that the competing effects of adsorption equilibrium and transport on the pore conductance lead to a rich variety of behavior in pore networks that is not envisaged in existing treatments considering the tortuosity as a constant and independent of adsorptive as well as processing conditions. In what follows, we examine recent literature data on transport in ordered microporous DDR zeolites and disordered mesoporous silicate glass membranes with a view to validating the oscillator model and also to demonstrating the improved results based on the use of this approach compared to the established Knudsen formula for the low-pressure diffusivity. Application to Literature Data. The oscillator model provides a theoretical advance over the existing Knudsen theory by considering dispersive interactions between the fluid molecules and the pore surface and is central to our investigation of network effects. Direct validation of this model is, however, somewhat problematic because of the absence of suitable model materials with uniformly sized and ideally shaped nonintersecting pores. At the micropore level, zeolites are perhaps the most attractive for validation, although their structures are complex and their transport is governed by windows or short channels whose cross section is often slightly elliptical rather than perfectly cylindrical. For mesopores, whereas ordered mesoporous materials such as MCM-41 silicas do exist they have complex multiscalar structures spanning a wide range of length scales,12 and transport coefficients in the narrow nanoscale pores cannot easily be directly determined independent of resistance on other scales. However, conventional disordered silicas present issues of network connectivity, but these can be addressed using the hybrid EMTCRWT formulation discussed here. Hence, here we consider the application of the oscillator model to the interpretation of the experimental data on the transport of various gases in ordered microporous DDR zeolite8 and disordered mesoporous silicate glass51 membranes. Ordered Microporous DDR Zeolite Membrane. This zeolite has recently attracted much attention8,9,59,60 because of its narrow pore spaces offering the potential for molecular sieving. Its accessible pore space comprises 19-hedra cages connected by narrow channels of a nearly circular cross-section composed of (59) van den Bergh, J.; Ban, S. A.; Vlugt, T. J. H.; Kapteijn, F. J. Phys. Chem. C 2009, 113, 17840–17850. (60) Jee, S. E.; Sholl, D. S. J. Am. Chem. Soc. 2009, 131, 7896–7904.
8380 DOI: 10.1021/la9047962
Bhatia
eight-membered rings of size 0.632 0.705 nm2 (O-O center-tocenter distance) based on XRD structure data.61 The accessible porosity is 0.21,9 predominantly lying in the cages, with only a small portion of this attributable to the interconnecting channels, and transport in this zeolite is controlled by the resistance to flow through the narrow channels interconnecting the cages.60 The latter resistance has recently been modeled by Jee and Sholl60 considering the channels to be windows, the flow through of which could be conveniently analyzed using transition-state theory. Their calculations confirmed that the transition state occurs at the eight-membered ring,62 where the channels have a cross section of size 0.632 0.705 nm2 based on the O-O center-to-center distance, which provides a critical open size of 0.36 nm,63 considering an oxygen atom diameter of 0.27 nm that is typical for zeolites.64 Although these channels are strictly not cylindrical, one may expect the transport through these to be governed by the critical dimension of 0.63 nm (or 0.36 nm open size), and it is of interest to compare the changes in permeability with diffusing gas molecular size predicted by the oscillator model for a cylindrical channel of this diameter with experimental data for this zeolite. Tomita et al.8 have investigated the transport of various gases in a DDR membrane supported on a porous alumina tube at 301 and 373 K and reported the permeances, defined as p* = j/(-ΔP), versus the kinetic diameter. The permeance may be converted to permeability, p, following p = p*l , where l is the thickness of the membrane layer. This membrane layer thickness was given as 5-10 μm whereas for the support tube the internal diameter was 17 mm and the pore size was 0.6 μm. On the basis of these values, it is readily determined that the support will offer negligible resistance compared to the active zeolite layer, even considering a support thickness as large as 1.5 mm. For this, we determine the support permeance (DKε/γLsRgT þ rp2Pε/8LsμγRgT) considering both Knudsen and viscous flow. For example, for N2 at 300 K, assuming a support porosity of 0.25 and tortuosity of 3, this permeance is estimated to be 3.5 10-6 mol 3 s/kg 3 m and is 3 orders of magnitude above the measured permeance of 2.1 10-9 mol 3 s/kg 3 m. Thus, the data can be directly compared with the model predictions. Following eq 10, we may define the pore permeability for diffusion through a cylindrical channel as p ¼
j Do K ¼ ð -rPÞ Rg T
ð24Þ
and to convert the experimental data of permeance into the pore permeability it is necessary to scale it by the factor l γ/εchan, where εchan is the porosity attributable to the cylindrical channels and γ is their tortuosity. The latter can be taken as 3 because the three channels connecting each cage are symmetrically arranged so that there is no directional correlation effect (i.e., the factor (N þ 1)/ (N - 1) in eq 16 is not applicable); however, the value of the porosity contribution of the intercage cylindrical channels, εchan, is unclear. Because the porosity of the DDR crystals is 0.21,9 the minimum value of γ/εchan is 14.3. In practice it will be significantly larger because the channels offer only a small fraction of the total porosity and also because in useful membranes there is a considerable intergrowth of crystals to mimimize intercrystalline macropores. This creates blockages, which reduce the membrane (61) Gies, H. Z. Kristallogr. 1986, 175, 93–104. (62) Jee, S. E.; Sholl, D. S. Personal communication, December 2009. (63) Foster, M. D.; Rivin, I.; Treacy, M. M. J.; Delgado Friedrichs, O. Microporous Mesoporous Mater. 2006, 90, 32–38. (64) Baerlocher, C.; Meier, W. M.; Olson, D. H. Atlas of Zeolite Framework Types, 5th ed.; Elsevier: Amsterdam, 2001.
Langmuir 2010, 26(11), 8373–8385
Bhatia
Figure 8. (a) Comparison of the model prediction with experimental data8 for the variation of permeability with molecular size for a DDR zeolite membrane and (b) the predicted variation of the Henry’s constant and diffusivity with molecular size at 301 K in a 0.63-nm-diameter channel.
microporosity below the true intracrystalline value. Furthermore, surface barriers, not considered here, are often present in zeolite samples and may also play a role in reducing the membrane permeability.65 In interpreting the data of Tomita et al.,8 it was found that a value of γ/εchan = 133 with a membrane layer thickness of l = 7.5 μm was appropriate for matching the oscillator model predictions of pore permeability for cylindrical channels of 0.63 nm diameter. The effective value of the factor γ/εchan = 133 may be slightly more or less depending on the active membrane layer thickness, which varied between 5 and 10 μm. Figure 8a depicts the variation of pore permeability with the Lennard-Jones interaction diameter of the diffusing species, determined as above from the data of Tomita et al.9 and for the oscillator model at two experimental temperatures of 301 and 373 K. In the latter case, the potential parameters in Table 1 were used, with the Tjatjopoulos et al.52 fluid-wall interaction potential in eq 8 and the parameters for silica discussed above on the basis of the values of Ravikovitch et al.53 for MCM-41, except that σO-O was very slightly increased by rounding off to a value of 0.28 nm. The calculations were made for all gases except CO2, which is not well modeled as a single-site molecule, particularly for diffusion in such a narrow, molecularly sized space. Nevertheless, the model curve matches the CO2 data, and it is evident that excellent agreement is obtained over the entire range of molecular size for the pore diameter of 0.63 nm, with γ/εchan taken as 133. With a slight reduction in the pore diameter to 0.62 nm, it is seen that the sieving effect for methane is too strong whereas increasing the pore diameter to 0.64 nm yields too high a permeability for the larger species. These results also confirm that the trajectories of the diffusing molecules moving through the transition state can be effectively considered to be confined to a cylindrical channel of the critical diameter. (65) Henke, L.; Kortunov, P.; Tzoulaki, D.; K€arger, J. Adsorption 2007, 13, 215– 223.
Langmuir 2010, 26(11), 8373–8385
Article
It is clear from the above that the oscillator model can represent the sieving effect of micropores quite adequately. A feature of particular interest is the maximum in permeability with molecular size, evident in both the theoretical predictions and the experimental data. As with the maximum in pore conductance with pore size, depicted in Figure 4b, this is due to the combination of adsorption equilibrium and diffusion, which has previously been discussed. Indeed, Figure 4b also clearly indicates this maximum in permeability with molecular size, with the conductance being maximized for Xe at a pore radius of about 0.4 nm. The combination of adsorption and diffusion in the present case can most simply be illustrated by considering the theoretical variation of the adsorption equilibrium constant, K, and the diffusity, Do, with molecular diameter, as depicted in Figure 8b. For molecules smaller than argon, the value of the equilibrium constant increases sharply with molecular size whereas the diffusivity decreases steeply. However, for larger molecules the diffusivity increases weakly with molecular size whereas the equilibrium constant decreases sharply. These opposing effects of molecular size on adsorption and diffusion lead to the maximum in permeability in Figure 8a. The factor of 133 used for the value of γ/εchan, although exceeding the minimum value by a factor of about 9.3, is quite reasonable considering that blockages must exist in the membrane and that the channel volume is small compared to the porosity of 0.21 used in estimating the minimum value. Although we have used the Tjatjopoulos et al.52 potential incorporating a Lennard-Jones 12-6 site-site potential, as in eq 8, to model the fluid-wall interactions, the recent study of Jee and Sholl60 investigating the adsorption and diffusion of CO2 and CH4 in DDR zeolite crystals proposes that the site-site interaction may vary more steeply with separation in the repulsive region. These authors empirically suggest a modified 18-6 potential to obtain a match for both adsorption and diffusion data by simulation. In the present study, attempts to replace the 12-6 potential in the integrand of eq 6 by an 18-6 term similar to that of Jee and Sholl were unsuccessful in providing good correlation of the data over the entire range of molecular size, even when the energy parameter of the interaction was varied, with the permeabilities of Ar and O2 increasing much too dramatically. In this connection, it may be observed that the Tjatjopoulos et al. potential used here, which considers only interactions with surface sites, ignores dispersive interactions with the internal sites of the framework and is therefore in effect more repulsive than one that considers these interactions. Thus, there is no clear disagreement with the proposal of Jee and Sholl that the site-site interaction is inherently more repulsive than the 12-6 LJ potential. Another uncertainty to be considered is the surface momentum accommodation factor because diffuse reflection is assumed in the oscillator model. This factor modifies the predicted low-density diffusivity as follows40 ð2 -RÞ Do ðR ¼ 1Þ Do ðRÞ ¼ ð25Þ R where R is the momentum accommodation coefficient and Do(R = 1) corresponds to the diffusivity value for the diffuse reflection case. Unity for the diffuse reflection, R, is very small for smooth surfaces such as those of carbon nanotubes for which values on the order of 0.001 are estimated.41,43 The present agreement would suggest that it is much closer to unity in the present case and probably for zeolites in general, given their relatively uneven energy landscape due to the wider spacing of the oxygens compared to that of covalently bonded carbons. In this region, the surface accommodation coefficient will also be much less sensitive to molecular size than in the regime of nearly DOI: 10.1021/la9047962
8381
Article
specular reflection as in carbon nanotubes, where it varies by a factor of nearly 5 between H2 and CH4.43 Consequently, the multiplicative factor (2 - R)/R will be essentially unchanged between the gaseous species in Figure 8a and near unity. Under this assumption, the consideration of the Tjatjopoulos et al.52 potential with diffuse reflection as used here would appear to be appropriate. It may be noted that Tomita et al.8 also provided permeance data for larger molecules, n-C4H10, i-C4H10, and SF6, for which the permeance was relatively independent of molecular size and almost 2 orders of magnitude smaller than that of methane. Given the much larger size of these molecules compared to the open critical channel size of 0.36 nm, their permeance is clearly representative of the small leakage flux through the defects and intercrystalline macropores in the membrane, so this part of the data has not been considered. To my knowledge, the oscillator model is the only rigorous approach that predicts the transport in this “configurational” regime, albeit for spherical molecules as in Figure 8a. One of the most well known of the existing approaches is the activated diffusion model of Xiao and Wei,66 but it is somewhat empirical in that it is an arbitrary modification of the classical Knudsen model by an Arrhenius-type factor containing an activation-energy term. The present approach is completely derived from first principles without such arbitrary considerations. Disordered Mesoporous Membranes. As another example of the application of the present approach, we consider the data of Markovic et al.51 for the low-pressure transport of various gases in unsupported mesoporous glass membranes prepared from the phase separation and leaching of heat-treated sodium-rich borosilicate glass. Membranes of three different pore radii, 1.15, 1.55, and 2.1 nm, were prepared, and the diffusion of He, Ar, N2, C3H8, and CO2 was studied dynamically at various temperatures in the range of 290-440 K and at pressures below 1 bar. Of the data, only those for the first three gases were considered for interpretation using the present approach because C3H8 and CO2 are nonspherical molecules and cannot yet be treated by the current analysis that assumes a single interacting site in the diffusing molecule. Furthermore, CO2 has a strong quadrupole moment and was reported by the authors to have significant electrostatic interactions with their glass membranes because of the large surface concentration of hydroxyl groups. Such interactions are also not included in the current single-site interaction model, and hence only the data for He, Ar, and N2 is examined here. This data has been provided in the form of permeability at various temperatures for each gas. Markovic et al.51 correlated their data using a model superimposing Knudsen and Poiseuille model-based viscous flows, with fluid-solid interactions neglected so that K = 1, and obtained strikingly high tortuosity values of 20.1, 19.6, and 15.8 for membranes with pore radii of 1.15, 1.55, and 2.1 nm, respectively. However, it is readily seen that the viscous flow resistance is very large compared to that due to the Knudsen contribution. Following eq 24, for axial flow in a single pore the diffusional resistance per unit length, with K = 1, is given as RgT/ Do whereas that due to Poiseuille flow is 8μRgT/rp2P. The ratio of these resistances is therefore 8μDo/rp2P and may be readily estimated for any species. For example, for Ar at 298 K in the largest pore of 2.1 nm radius for which μ = 2.2 10-5 Pa 3 s and with Do taken as that predicted by the Knudsen model, one obtains this ratio as 222 at a pressure of 1 bar. Similarly, for He, for which μ = 2.0 10-5 Pa 3 s at 298 K, one estimates this ratio to (66) Xiao, J.; Wei, J. Chem. Eng. Sci. 1992, 47, 1123–1141.
8382 DOI: 10.1021/la9047962
Bhatia
be 638. Because viscous and diffusive flows largely occur in parallel, with such large a ratio of their resistances it is clear that diffusive flow will dominate and the impact of viscous flow will be negligible. For the smaller pore sizes, the ratio will be even larger. Nevertheless, even with viscous flow neglected the key issues are those of the assumption of negligible adsorption effects (i.e., K = 1) and the applicability of the Knudsen model. To address the issue of neglecting adsorption, we consider eq 24 with the equilibrium constant expressed in terms of the enthalpy change as K exp(-ΔH/RgT) and the diffusivity expressed in terms of an activation energy as Do exp(-ED/RgT). Equation 24 then yields 1 ln pRg T
! ¼ Aþ
ðED þ ΔH o Þ Rg T
ð26Þ
where A is a constant, including the effects of tortuosity and accessible porosity. With adsorption neglected, ΔHo = 0 and a plot of ln(1/pRgT ) against 1/T should yield an effective activation energy, ED, that is positive and consistent with that for Knudsen diffusion if the expression for the latter holds for the diffusion. However, if adsorption is important, then ΔHo < 0 and the effective activation energy (ED þ ΔHo) will be reduced and can even be negative. Figure 9a-c depicts such plots based on the data of Markovic et al.51 for the three pore sizes used by them. For pore radii of 1.15 and 1.55 nm, both Ar and N2 show negative effective activation energies, and even for the largest diameter of 2.1 nm, the effective activation energies are only slightly positive and much smaller than the expected value of 1.47 kJ/mol for Knudsen diffusion. Even for helium, the effective activation energies are significantly less than the Knudsen value in all cases, demonstrating the importance of considering adsorption even for this gas. Following the above discussion, calculations of the apparent tortuosity were made on the basis of the data of Markovic et al. and the oscillator model predictions for Do, with eq 11 used to estimate K at each pore size. For this, we used the expression of eq 18 to obtain the theoretical permeability p ¼
εKðrp Þ Do ðrp Þ γapp Rg T
ð27Þ
which was compared with the data to yield γapp. Although the pore size distribution was not given by the authors, the mean pore sizes given were those based on the pore volume to surface area ratio, as in the traditional BJH analysis of the PSD67 and as also considered here in eq 23 and in the related analysis of tortuosity in pore networks. Because this represents the accessible pore diameter, the center-to-center diameter needed for the oscillator model calculations was approximated by adding the surface oxygen diameter of 0.27 nm to the nominal BJH diameter specified by Markovic et al.51 Furthermore, porosities of the membranes were provided by these authors and were adjusted for the center-center-based pore diameter required by the model. Figure 10a-c depicts the variation of apparent tortuosity with temperature for the membranes with three different pore sizes. Remarkable similarity of these results is seen with that theoretically determined for the temperature variation of the tortuosity for these gases in a 1.3-nm-radius pore and depicted in Figure 7b. This pore size corresponds to the experimental open pore radius of 1.15 nm after considering the oxygen diameter of 0.27 nm. (67) Gregg, S. G.; Sing, K. S. W. Adsorption, Surface Area and Porosity; Academic Press: New York, 1982.
Langmuir 2010, 26(11), 8373–8385
Bhatia
Article
Figure 10. Variation of apparent tortuosity with temperature 51
Figure 9. Variation of permeability with temperature
for He, N2, and Ar in mesoporous glass membranes with pore diameters of (a) 2.3, (b) 3.1, and (c) 4.2 nm.
As predicted in Figure 7b, He has a significantly higher tortuosity than N2 and Ar, and the values for the latter two gases are close to each other, which is experimentally validated. Although the predicted tortuosity for Ar is slightly larger than that of N2 in Figure 7b, the experimental tortuosity of Ar is seen to be slightly lower. This can be attributed to the quadrupole moment of N2 and electrostatic interactions with surface hydroxyl groups in the membrane, which are not considered in the model. As a result of these interactions, N2 will adsorb more strongly than considered in the model, reducing its apparent tortuosity. As indicated by Markovic et al.,51 such electrostatic interactions were indeed inferred for the adsorption of CO2. A further effect observed for all gases in Figure 10a,b and for N2 in Figure 10c is the presence of a minimum in the tortuosity in the range of 320-350 K. This unexpected behavior is due to the low permeabilities reported at the lowest temperature in these cases, which is reflected in the corresponding points in Figure 9 where a conspicuous increase in the value of 1/pRgT is observable compared to the linear fit. Given that it is consistently observed, except for He and Ar at the largest pore size, it is suspected that this reflects a change in the mechanism that is probably related to an Langmuir 2010, 26(11), 8373–8385
based on the application of the oscillator model to the experimental permeability data of Markovic et al.51
entry barrier68 and strong adsorption at the pore mouth at the lowest temperature. This effect may also be related to pore shape nonideality because it is highly unlikely that pores created by leaching out a precipitated phase will be cylindrical as assumed in the theory. In support of these arguments, it may be noted that a decrease in the nominal pore diameter to about 1.4 nm reduced the permeability by 30-200 times, as is evident from data in a companion publication by Markovic et al.69 Such a large decrease in this pore size range could not be explained by a cylindrical pore model and suggests structural irregularities and/or pore mouth effects. More detailed studies and characterizations of the structure are needed to confirm this. For pore radii of 1.15 and 1.55 nm, the magnitudes of the tortuosities are also roughly in the same range as those theoretically predicted for N = 3 in Figure 7b and show a similar temperature increase, although it may be noted that the PSDs and their standard deviations are not known for the experimental membranes. With increasing BJH pore radius from 1.15 to 1.55 nm, the tortuosity increases for each gas, similar to that between parts a (68) Nguyen, T. X.; Bhatia, S. K. Langmuir 2008, 24, 146–154. (69) Markovic, A.; Enke, D.; Schl€under, E.-U.; Stoltenberg, D.; SeidelMorgenstern, A. J. Membr. Sci. 2009, 336, 32–41.
DOI: 10.1021/la9047962
8383
Article
Bhatia
temperatures. If one insists on using the using the Knudsen model while considering adsorption so that K > 1 in eq 27, then the tortuosity estimated from the data will be even higher. Indeed, accounting for adsorption is an acknowledgment of the presence of the fluid-solid force field and the associated density profiles, so the Knudsen model (which overlooks this force field and assumes linear molecular trajectories as well as a uniform density) can no longer hold. Thus, if adsorption is to be considered then a diffusion model that considers the force field must also be included in the theory. Doing so reduces the theoretical diffusivity because the oscillator model values are significantly smaller than those predicted by the Knudsen expression, which reduces the tortuosity estimated from experimental data to more realistic levels. All of these results provide strong support for the oscillator model at the single-pore level, with the hybrid EMT-CRWTbased network model discussed here.
Figure 11. Variation of apparent tortuosity with temperature based on the application of the Knudsen model to the experimental permeability data of Markovic et al.51
and b of Figure 7, albeit for different pore sizes. For the 2.1-nmradius pore, the tortuosity is somewhat lower than that for the smaller pore sizes and is likely related to better connectivity (i.e., a higher value of N) and/or smaller pore entry barriers because of the larger amount of leaching and phase separation achieved in the generation of the larger pore size and possibly a different (lower) standard deviation and shape of the PSD compared to that assumed in the theory following eq 21. In the absence of more detailed characterizations of the structure, more precise comparisons cannot be made. Nevertheless, the general conclusion that the results are consistent with theoretical expectations is quite evident. In support of the above conclusion, it should be emphasized that if the Knudsen model with negligible adsorption effect was indeed applicable then the conductance would be proportional to T1/2 and the temperature term would cancel out of eq 19 because both Do(rp) and λe would then be proportional to T1/2. In such a case, the experimental tortuosity would be temperature-independent, which is contradicted by Figure 10. As further evidence, the apparent tortuosities based on the data and a purely Knudsen diffusion model with K = 1 have also been determined and are depicted in Figure 11a-c. Not only is a strong temperature dependence evident, but also the tortuosity values are unrealistically high with values for helium approaching 18 at the highest 8384 DOI: 10.1021/la9047962
Conclusions We have analyzed here the low-pressure transport of simple gases in nanoporous materials using a statistical mechanical model incorporating dispersive fluid-solid interactions. This represents a significant advance over the conventional Knudsen model, which considers only the hard sphere case, and it is seen that the latter provides an upper limit to the scaled diffusivity expressed as (Do/rp)(m/kBT)1/2, which is approached at large pore size and high temperature. The scaled diffusivity depends on two variables;a dimensionless pore radius and a temperaturedependent scaled well depth parameter;and is seen to exhibit a maximum with respect to the scaled pore size at sufficiently large values of the dimensionless energy parameter, consistent with the levitation effect discussed in the literature. The application of this model to nanoporous materials comprising ordered and disordered pore networks is also discussed. In the former case, the successful interpretation of the data of Tomita et al.8 on the variation of the permeability with the molecular size of the diffusing species for a microporous DDR zeolite membrane is demonstrated. As in the experimental data, the theory shows a peak in permeability with molecular size, and this is found to be due to the competing effects of equilibrium and transport on the permeability. Large molecules adsorb strongly but diffuse slowly whereas the converse is true for small molecules. Such successful theoretical predictions of experimental molecular sieving data in an actual nanoporous material are rare. The theory is also applied to investigate transport in disordered solids based on a hybrid correlated random walk-effective medium theory of transport in pore networks exploiting the oscillator model at the single-pore level. It is shown that the tortuosity depends on the diffusing species, temperature, and pressure and is not a porous medium property alone. A rich variety of behavior is predicted for the tortuosity, which can increase or decrease with increasing extent of pore size nonuniformity as well as with a change in temperature because pores with higher conductance offer short circuit paths for the diffusing molecules. The application of the oscillator model to recent data on the transport of various gases in mesoporous silicate glass membranes having different pore sizes yields tortuosities that depend on the temperature and diffusing species, consistent with the theoretical predictions both in the trend and in range of magnitude. Light gases such as helium have the highest tortuosities, with heavier, more strongly adsorbing species generally having smaller tortuosities because of lower short-circuiting effects, which is indicated by both theory and the experimental data. In general, the tortuosity increases with increasing temperature. However, the use of the Knudsen model in the hybrid Langmuir 2010, 26(11), 8373–8385
Bhatia
CRWT-EMT approach for the transport yields a constant tortuosity that is purely a porous medium property, which is inconsistent with the data, and yields unrealistically high tortuosities. The new model offers promise for application to membrane process design, catalysis, adsorption, and nanofluidics, where the transport of fluids confined in molecularly sized spaces plays a critical role. The oscillator model utilized here is developed for spherical nonpolar molecules, and at present, there is no tractable extension that can treat more complex species or electrostatic interactions with the pore walls. Thus, it is best suited for small molecules such as He, H2, O2, N2, CH4, and Ar and others such as CO270 that can be reasonably approximated by a spherical nonpolar molecule (70) Skoulidas, A. I.; Sholl, D. S.; Johnson, J. K. J. Chem. Phys. 2006, 124, 054708.
Langmuir 2010, 26(11), 8373–8385
Article
approximation. Nevertheless, besides their importance in the characterization of transport properties, such molecules are increasingly the subject of molecular sieving and membranebased separation processes. Hence the model investigated here has considerable potential for application. The current alternative largely relies on the Knudsen model that, as shown here, is significantly less accurate whereas the more accurate MD simulation technique is still computationally too intensive for routine application. Acknowledgment. This research has been supported by a grant from the Australian Research Council under the Discovery Scheme. I am grateful to Professor Andreas Seidel-Morgenstern and Ana Markovic at the Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany, for sharing their extensive data on diffusion in porous glass membranes.
DOI: 10.1021/la9047962
8385
