A Comparison of Atomistic Simulations and Experimental

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A Comparison of Atomistic Simulations and Experimental Measurements of Light Gas Permeation through Zeolite Membranes Travis C. Bowen,† John L. Falconer,† Richard D. Noble,† Anastasios I. Skoulidas,‡ and David S. Sholl*,‡ Department of Chemical Engineering, University of Colorado, Boulder, Colorado 80309-0424, and Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213

We present experimental and theoretical results for single-component permeance of CH4 and CF4 through a supported silicalite membrane at a range of temperatures and pressures. Our theoretical model uses a continuum description of molecular transport through zeolite crystals that is directly parametrized from atomically detailed simulations of molecular adsorption and transport. This approach does not require any assumptions regarding the loading dependence of the adsorbed species’ transport or Maxwell-Stefan diffusivities. Our results are the first direct comparison between a fully atomistic description of intracrystalline transport and permeance measurements for a macroscopic zeolite membrane. These results help to isolate the contributions to the overall flux through polycrystalline zeolite membranes that arise from molecular transport through nonzeolitic pores. We also discuss avenues for future extensions and improvements of our atomistic approach to modeling practical zeolite membranes. Introduction Membranes made from thin films of zeolites have been extensively studied as attractive devices for gasand liquid-phase separations. For reviews of this field, see refs 1-3. Because they are comprised of crystalline inorganic materials, zeolite membranes can be used in a broad range of temperatures, pressures, and chemical environments. In addition to their chemical and thermal stability, zeolites are attractive as membrane materials because of their atomically ordered nanometer-scale pore structures. Because of the strong confinement of molecules inside zeolite pores, the adsorption and transport properties of molecules adsorbed in zeolites can vary enormously as functions of pore size, adsorbate size and shape, temperature, etc.4,5 These variations are one of the key factors in making zeolites useful as membrane materials. At the same time, these variations pose severe challenges to developing predictive models for the performance of zeolite membranes. One strategy for modeling zeolite membranes is to adopt a phenomenological macroscopic approach. Most commonly, diffusive transport through zeolite pores is represented using a Maxwell-Stefan description of nonequilibrium transport.6-10 These models contain a number of parameters such as diffusion coefficients that are in many cases fitted to experimental data for singlecomponent permeation through a membrane. Although these models have been presented in quite general terms, in practical applications a number of simplifying assumptions are typically made. For single-component permeation, for example, these models typically assume that the adsorption follows a Langmuir isotherm and that the generalized Maxwell-Stefan diffusivity of the adsorbed species is independent of loading.6-10 These * Corresponding author. Fax: 412-268-7139. Phone: 412268-4207. E-mail: [email protected]. † University of Colorado. ‡ Carnegie Mellon University.

models have been shown to usefully describe a number of complex situations, including transient effects in multicomponent permeation.6 A disadvantage of these macroscopic methods is that they make no fundamental connection between the performance and the atomicscale structure of the membrane. Models of this type cannot be used, for example, to compare the separation capabilities of two membranes with similar but different pore structures. A second strategy for modeling zeolite membranes is to directly model membranes on atomic scales. Atomistic modeling has been widely used to describe adsorption and self-diffusion of molecules in zeolite pores.6,11 Building upon this work, several groups have developed methods that use atomistic simulations to examine molecular transport through zeolite membranes.12-16 In these models, every atom in the membrane is explicitly represented and the dynamics of gas-phase and adsorbed molecules evolve based on specified interatomic potentials. The transmembrane flux in simulations of this type is determined by counting individual molecules as they move through the membrane. This approach clearly provides the direct connection between membrane performance and microscopic structure that is missing in the macroscopic models described above. Unfortunately, these atomistic methods require enormous computational resources and are therefore limited to extremely thin membranes. The recent work of Martin et al., which appears to be the largest scale simulation of this type to date, used massively parallel computing techniques to simulate CH4 diffusion through silicalite membranes that were 0.016 µm thick.16 For comparison, typical experimental zeolites are 1-100 µm thick.1-3 This limitation is problematic because transport through the ultrathin membranes used in these simulations appears to be dominated by barriers to mass transport at the zeolite-gas interface, not by intracrystalline mass transport.16,17 The importance of these surface effects greatly diminishes as membrane thicknesses increase.17,18 As a result, it is difficult to scale

10.1021/ie010303h CCC: $22.00 © 2002 American Chemical Society Published on Web 02/21/2002

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the observed permeation results from simulations of ultrathin membranes to predict the behaviors that would be observed with membranes of experimentally relevant thicknesses. Neither of the modeling strategies outlined above is currently able to predict the properties of realistic zeolite membranes in a manner that directly links these properties to the membrane’s atomic-scale structure. An alternative strategy has recently been developed that overcomes this difficulty by using a macroscopic description of molecular transport through a crystalline membrane based entirely on results developed using atomistic simulations of molecular adsorption and diffusion in zeolite pores.19 The central purpose of this paper is to present a direct comparison between predictions of this modeling framework with experimental measurements of single-component light-gas permeation through a polycrystalline zeolite membrane. Specifically, we have modeled and measured the single-component permeance of CH4 and CF4 through a supported silicalite membrane over a wide range of pressures and temperatures. These two species were chosen for study because their adsorption isotherms and self-diffusivities in silicalite have been the subject of many previous atomistic simulations.6,11,20-28 The permeation of hydrocarbons, aromatics, and a variety of other molecular species through silicalite membranes has been studied by a number of experimental groups.8,29-43 The theoretical model we present below assumes that all molecular transport through the membrane occurs through zeolite pores. Because practical zeolite membranes are polycrystalline, it is widely acknowledged that permeation through these membranes occurs by a combination of transport through zeolitic and nonzeolitic pores.3,43-45 To develop separation processes that take advantage of zeolites’ atomically ordered pore structures, it is typically desirable to minimize contributions to membrane permeance through nonzeolitic pores. Despite the importance of this issue, unambiguous experimental methods for separating membrane fluxes into zeolitic and nonzeolitic portions are difficult to develop. One significant feature of the atomistic models presented here is that they provide a detailed description of how the membrane studied experimentally would be expected to behave if it was a perfect zeolite crystal. As a result, once the validity of an atomistic model has been established, our methods can be used to isolate the role of transport through nonzeolitic pores in the evaluation of our experimental data. Experimental Methods Membrane Preparation. A silicalite-1 membrane was prepared by hydrothermal synthesis on the inner surface of a porous, tubular, stainless steel support (Mott Corp.). The porous section of the support was 2.6 cm × 0.63 cm i.d. with a nominal pore size of 500 nm and a porosity of approximately 0.27. Nonporous stainless steel tubes were welded to the ends of the support to provide a surface for an O-ring seal. Prior to membrane synthesis, the support was scrubbed for 15 min under tap water and placed in an ultrasonic bath containing deionized water for 40 min. The support was then placed in boiling deionized water for 1 h and dried under vacuum at 373 K for 30 min. The zeolite synthesis gel was prepared using silica sol (Ludox AS-40) as the silica source. The molar composition of the gel was SiO2:TPAOH:H2O ) 12.6:

1.0:293, where TPAOH is tetrapropylammonium hydroxide. Synthesis gel was poured into the support tube, which was sealed on one end with a Teflon cap, and allowed to penetrate into the pores of the support. Gel was added approximately every 30 min until the gel was no longer absorbed by the support. The top end of the support was then sealed with a Teflon cap, and the support was placed vertically in an autoclave at 458 K. A zeolite layer was grown for 24 h, and then the inside of the support was brushed lightly with a nylon brush to remove any crystals that were weakly attached. The membrane was rinsed in distilled water, dried under vacuum at 373 K for 30 min, and refilled with a synthesis gel. A second crystal layer was grown in a manner identical with that of the first layer except that the support was placed upside down in the autoclave to improve the uniformity of the crystal layer thickness. Tuan et al.46 gives a more detailed description of this synthesis procedure. The membrane, after being rinsed with deionized water and dried under vacuum at 373 K, was impermeable to nitrogen at room temperature with a transmembrane pressure drop of 138 kPa. Next, the template was removed by calcination in air, which was carried out using a computer-controlled muffle furnace. The membrane was heated at 0.01 K/s, held at 753 K for 8 h, and then cooled at 0.02 K/s. Adsorption of impurities was minimized by storing the membrane under vacuum when not in use. A silicalite-1 membrane with permeation properties similar to those of the membrane used in these studies was broken and examined using scanning electron microscopy (SEM) and was found to have a zeolite layer approximately 100 µm thick. Membrane Characterization. The membrane was characterized by measuring the n-C4H10/i-C4H10 separation selectivity using a 50/50 feed mixture at temperatures of 373, 423, and 473 K. The feed pressure and transmembrane pressure drop were 239 and 138 kPa, respectively. Permeate and retentate flow rates were measured with soap-film bubble flowmeters, and a gas chromatograph (HP 5890 series II) was used to determine the concentration of each stream. The n/i-C4H10 separation selectivity was 13 at 373 and 423 K but decreased to approximately 4 at 473 K. These selectivities are typical of silicalite membranes made in our laboratory and are comparable to other n/i-C4H10 separation selectivities reported in the literature, as shown in Table 1. Permeation Measurements. Single gas permeation of CH4 and CF4 was measured using the dead-end method (retentate stream blocked and no sweep gas) with feed and permeate pressures controlled by pressure regulators. Permeances were measured over a feed pressure range of 290-2150 kPa. A Bourdon tube pressure gauge ((1.4 kPa) and a differential pressure transducer ((0.7 kPa) measured the feed pressure and transmembrane pressure drop, respectively. A brass module, which held the membrane using silicone O rings, and a 2.6 m preheating section of tubing were in an oven controlled by an Omega temperature controller. The oven heated the feed to 373 and 473 K. Chromelalumel thermocouples ((3 K) measured the temperature of the feed gas in the module and the permeate at the flowmeter. The permeate was at room temperature in the bubble flowmeter ((0.01 cm3) for all permeation measurements, and the flow rate was measured using a stopwatch.

Ind. Eng. Chem. Res., Vol. 41, No. 6, 2002 1643 Table 1. Comparison of n/i-C4H10 Separation Selectivities Using Silicalite Membranes ref this work Bakker et al.48 Coronas et al.49

Keizer et al.40 van de Graaf et al.50

T (K)

n/i-C4H10 selectivity

372 425 473 295 403 379 517 408 514 298 473 373 423 473

13 13 4.0 27 23 4.1 1.3 11 5.7 52 11 33 17 4.3

Each time the membrane was exposed to air, the permeation system was heated to 473 K under gas flow for at least 1 h to remove impurities that may have adsorbed from the laboratory atmosphere. Calcination at 673 K for 4 h was required periodically to remove strongly adsorbed impurities. Permeation measurements were made over a period of 3.5 months, and gas permeance was reproducible even after repeated calcinations. Some experiments with supported silicalite-1 membranes have been reported to induce nonnegligible pressure gradients across the support.43 To estimate the pressure drop across our support, we performed a series of experiments using a support with no zeolite layer. The permeance of CH4 across this support was too high to measure except with transmembrane pressure drops of less than 2 kPa. To estimate the pressure drop due to the support at the conditions relevant for our experiments with zeolite membranes, several permeance measurements were made with the blank support for pressure drops of less than 2 kPa. We fitted these data by assuming that the flux through the support occurred by viscous flow:47

Jviscous ) C(Pf2 - Pp2)

conditions 120 kPa n-C4/120 kPa i-C4 feed, 101 kPa permeate stainless steel tubular support pressure drop method (no sweep) 50 kPa n-C4/50 kPa i-C4 feed stainless steel disk support, Wicke-Kallenbach method 120 kPa n-C4/120 kPa i-C4 feed, 101 kPa permeate stainless steel tubular support, pressure drop method 120 kPa n-C4/120 kPa i-C4 feed, 101 kPa permeate R-Al2O3 tubular support, pressure drop method 50 kPa n-C4/50 kPa i-C4 feed R-Al2O3 disk support, Wicke-Kallenbach method 50 kPa n-C4/50 kPa i-C4 feed, stainless steel disk support Wicke-Kallenbach method

(1)

Here, Pf and Pp are the support’s feed and permeate pressure, respectively. Using the value of C determined from this fit, we extrapolated to the conditions relevant for our zeolite membrane experiments by substituting the observed flux across a membrane into the expression above. Under the conditions we have examined, this calculation predicts that less than 0.1% of the total pressure drop across our supported membrane is due to the support. Thus, there appears to be effectively no resistance to mass transport that arises solely from the support in our experiments. Theoretical Methods To predict the single-component, steady-state permeance of light-gas species through silicalite-1 membranes directly from atomistic principles, we adopt a macroscopic description of molecular transport through a zeolite crystal and determine all parameters in this model directly from atomistic simulations.19 We assume that the zeolite membrane is a single crystal of thickness L and that the only resistance to mass transport is due to intracrystalline diffusion. We return to the issue of the orientation of the membrane relative to the zeolite’s crystallographic axes below. The molecular flux

of the adsorbed species inside the zeolite is given by J ) -D(c) dc/dz, where c is the concentration of the adsorbed species, D(c) is the concentration-dependent Fickian diffusion coefficient, and z is a coordinate defining the transmembrane direction. The Fickian diffusion coefficient is also often referred to as the transport diffusion coefficient.6,11 We note that this model is entirely equivalent to a Maxwell-Stefan description.6,51,52 The steady-state flux across a membrane using this model is19

J)

1 L

∫cc D(c) dc 0

L

(2)

where c0 and cL are the adsorbate concentrations in the zeolite at the boundaries of the membrane, z ) 0 and L, respectively. Because we are assuming that there are no barriers to mass transport at the boundaries of the membrane, c0 and cL are related to the feed and permeate pressures by the zeolite’s adsorption isotherm. One simple limit of the model described above occurs when both the feed and permeate pressure lie within the Henry’s law regime. In this case if one assumes that the Fickian diffusivity is approximately independent of concentration, the transmembrane flux is found to be proportional to the transmembrane pressure drop. This behavior has been seen experimentally by Vroon et al.31 and Kapteijn et al.8 for permeation of CH4 through silicalite membranes at room temperature. In general, however, to apply eq 2, it is necessary to know both the concentration dependence of D(c) and the adsorption isotherm. A crucial feature of our approach is that all of the parameters in our model can be derived from atomistic simulations of adsorption and diffusion under equilibrium conditions. We have modeled CH4 and CF4 adsorbed in silicalite using atomistic models studied previously by other groups.11,22,23,25,26,28 Silicalite is modeled in its orthorhombic form and is assumed to be rigid. Both molecular species are modeled as single-site Lennard-Jones spheres. The CH4-CH4, CH4-O, CF4CF4, and CF4-O interaction parameters were the same as those in refs 11, 22, 23, 25, 26, and 28. Adsorption isotherms were computed using grand canonical Monte Carlo (GCMC) simulations. Fickian diffusivities were computed using EMD simuations28 following the techniques described by Theodorou et al.11 Further details of our simulations are described in ref 28. The properties of the zeolite membranes in our model are entirely determined by the atomistic model for molecular adsorption and diffusion. It is therefore useful

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Figure 2. Isotherms computed using GCMC for CH4 (filled symbols) and CF4 (open symbols) adsorbed as single components in silicalite at T ) 298 K (circles), 373 K (triangles), and 473 K (diamonds).

Figure 1. Comparisons of isotherm data between experiment and GCMC simulations for (a) CH4 adsorbed in silicalite at 273 K,60 278 K,62 298 K,60 305 K,61 308 K,62 323 K,60 343 K,61 and 352 K62 and (b) CF4 adsorbed in silicalite at 273, 298, and 323 K.60 In each case, the straight line with slope 1 indicates perfect agreement between simulations and experiments.

to examine the level of agreement between our atomistic model and experimental measurements of CH4 and CF4 adsorbed in silicalite. We begin by examining the singlecomponent adsorption isotherms for these species. Figure 1 compares the adsorption isotherms predicted by our GCMC simulations with experimental data over a range of temperatures. The atomistic model for CH4 is in excellent agreement with the experimental data. This model also gives an accurate value for the isosteric heat of adsorption.26 The agreement between the simulated and experimental isotherms for CF4 is less satisfactory, although still reasonably accurate. The atomistic model systematically underestimates the loading at moderate and high pressures for all temperatures where experimental data are available.26 The single-component adsorption isotherms predicted by these atomistic models for CH4 and CF4 at the temperatures of our membrane experiments are shown in Figure 2. Before we discuss the Fickian diffusivities of CH4 and CF4 in silicalite, it is helpful to compare self-diffusivities, Ds, predicted by our atomistic models with experimental data. Self-diffusivities measure the mobility of individual adsorbed molecules and can be measured experimentally using pulsed field gradient (PFG) NMR.6,11 Almost all atomistic simulations of molecular diffusion in zeolites have focused on self-diffusivities.6,11,28 The atomistic model we have used for CH4 predicts selfdiffusivities in excellent agreement with PFG NMR data at 298 K for loadings below 8 molecules/unit cell.22 For higher loadings the model slightly underestimates the

experimentally observed value of Ds.22 The model we have used for CF4 accurately predicts the experimental values of Ds at loadings of 4 and 8 molecules/unit cell at 200 and 300 K.25 At loadings of 12 molecules/unit cell, however, this model underestimates Ds by approximately a factor of 4 at both 200 and 300 K. Note that the adsorption isotherms predicted by this model for CF4 also deviate from experimental observations in this high loading regime. To apply eq 2, we must know the Fickian diffusivity of the adsorbed species as a function of concentration. We have computed Fickian diffusivities for CH4 and CF4 in silicalite at the temperatures of our membrane experiments using equilibrium molecular dynamics (EMD).28 This technique was first described by Theodorou et al.11 and has recently been applied by Sanborn and Snurr to the diffusion of CH4/CF4 mixtures in faujasite.53,54 The details of our simulations are described in ref 28. The pore structure of silicalite is anisotropic, and as a result, molecular diffusion in these pores is anisotropic.11,22,23,25,26,28 To model our random polycrystalline membrane, we assume that the membrane is made up of a single crystal of silicalite of thickness L whose transport properties are orientationally averaged over the three crystallographic directions. The orientationally averaged Fickian diffusivity is defined by D(c) ) [Dx(c) + Dy(c) + Dz(c)]/3, where Di (i ) x, y, and z) is the Fickian diffusivity for transport through an oriented crystal. The orientationally averaged Fickian diffusivities for CH4 and CF4 in silicalite as predicted by our atomistic calculations are shown in Figure 3. Our results for CH4 at 298 K are in excellent agreement with calculations of the same quantity using non-equilibrium MD.23,28 For both species, diffusion in silicalite is an activated process, so the diffusivities increase as the temperature is increased. It can be seen from Figure 3 that the Fickian diffusivities are not independent of the adsorbed concentration; for each species the diffusivity is an increasing function of loading. The Fickian diffusivity, D, is often written in terms of a related quantity known as the corrected diffusivity, Dc:6,11

D(c) ) Dc(c)

∂ ln f ∂ ln c

(3)

Here, f is the fugacity of the adsorbed species with

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Figure 3. Orientationally averaged transport diffusivities computed as described in the text for CH4 (filled symbols) and CF4 (open symbols) adsorbed in single components in silicalite. Symbols are the results of EMD simulations at 298 K (circles), 373 K (triangles), and 473 K (diamonds). The solid (dashed) curve shows the result predicted by the Darken approximation for CH4 (CF4) at T ) 298 K.

adsorbed concentration c. For single-component systems, the corrected diffusivity is identical with the Maxwell-Stefan diffusivity. Equation 3 is exact provided that Dc is allowed to be a function of concentration.6,11 It is well-known that the Fickian diffusivity, D, the self-diffusivity, Ds, and the corrected diffusivity, Dc, are only equal in the limit of dilute adsorbed concentrations.6,11 In models of zeolite membranes based on the Maxwell-Stefan approach,6-10 eq 3 is usually simplified by assuming that Dc is independent of concentration:

D(c) = Dc(0)

∂ ln f ∂ ln c

(4)

We will refer to this approximate expression as the Darken approximation.28 The accuracy of the Darken approximation for CH4 and CF4 in silicalite at 298 K is shown in Figure 3 by comparing the predictions of eq 4 with the true Fickian diffusivities. We emphasize that all of the data in Figure 3 was calculated from the same atomistic models, so this is an entirely self-consistent test of eq 4. Figure 3 shows that eq 4 accurately describes the diffusion of CH4 in silicalite but substantially overestimates the Fickian diffusivity of CF4 for loadings greater than a few molecules per unit cell. This is a consequence of the fact that the corrected diffusivity of CF4 in silicalite is not independent of the concentration.28 These qualitative conclusions are also valid at the other temperatures examined in Figure 3.28 We note that Makrodimitris et al.63 have recently used atomistic simulations of self-diffusivities in zeolites to relate atomistic models of CO2 and N2 in silicalite to macroscopic membrane data by invoking eq 4. The fact that eq 4 is qualitatively inaccurate for CF4 in silicalite, as described above, may be important in understanding the sizable discrepancies that were found between the models of Makrodimitris et al. and experimental data. Before we compare our theoretical predictions with experimental data from polycrystalline membranes, it is useful to compare our results with those measured using oriented single-crystal membranes. By using a silicalite single-crystal membrane 100 µm in length oriented along its crystallographic z axis, Talu et al. measured the corrected diffusivity of CH4 along that axis, Dc,z, to be 3.19 × 10-5 cm2/s at low loadings ( 200 K. The theoretical predictions in Figure 5 are cubic-


Figure 5. Experimental and theoretical data for temperature dependence of CH4 permeance with a transmembrane pressure drop of 138 kPa and feed pressures of 292, 499, 706, and 912 kPa.

Figure 7. Experimental and theoretical data for feed pressure dependence of CF4 permeance with a transmembrane pressure drop of 138 kPa at T ) 298, 373, and 473 K.

write the total flux through the membrane as a combination of intracrystalline, Knudsen, and viscous contributions:

Jtot(Pf,∆P) ) Jzeolite + A∆P + B∆P(2Pf - ∆P) (9)

Figure 6. Experimental and theoretical dependence of CH4 permeance on a transmembrane pressure drop at 298 K.

spline fits to computed permeances at 200, 298, 373, and 473 K. At 298 K, our model overpredicts the variation in the permeance as the feed pressure is varied (cf. Figure 4). At 473 K, our model underpredicts the total permeance due to the contributions of nonzeolitic pores discussed above. As the temperature is reduced, the dependence of the permeance on the feed pressure increases in both the experimental and theoretical data. This can be understood by realizing that the four feed pressures chosen sample a wider range of the adsorption isotherm as T is reduced (cf. Figure 2). Another test of our model’s predictive ability is to examine the CH4 permeance when the temperature is fixed and the transmembrane pressure drop, ∆P, is varied. The results of performing this comparison at 298 K are shown in Figure 6. For feed pressures above 500 kPa, there is no observable variation in the experimental permeance with the three pressure drops shown in Figure 6. Our theoretical model predicts that the permeance varies slightly with ∆P in this regime, but these variations are only slightly larger than the scatter in our experimental data. At feed pressures lower than 500 kPa, we can see that the experimental permeance decreases noticeably as ∆P is increased. This is the opposite of what is predicted by our theoretical model. Because eqs 7 and 8 suggest that contributions to the total membrane flux due to viscous flow may increase as T is lowered, it is interesting to examine if the behavior in Figure 6 can be explained by including viscous flow contributions. To do this, we empirically

Here, Jzeolite is the intracrystalline flux, A and B are constants (at constant T), and the remaining two terms have been written in terms of Pf and ∆P to facilitate comparison with Figure 6. Equation 9 shows that the Knudsen contribution to the permeance is independent of ∆P. At constant Pf, the viscous permeance in eq 9 decreases as ∆P is increased by an amount that is independent of Pf. Thus, our experimental observation that the permeance is almost independent of ∆P for high feed pressures in Figure 6 suggests that if the permeance through our membrane includes contributions from viscous flow at 298 K, then these contributions are very small. Single-Component Permeation of CF4 The single-component, steady-state permeance of CF4 through our silicalite membrane when the transmembrane pressure drop was held constant at 138 kPa is shown in Figure 7 at 298, 373, and 473 K. As for CH4 under these conditions, the permeance decreases with increasing feed pressure at 298 and 373 K but increases slightly with feed pressure at 473 K (cf. Figure 4). The observed permeances are almost independent of temperature when the feed pressure is approximately 750 kPa. Figure 7 also shows the permeance predicted by our atomistic model of CF4 in silicalite using eq 2 and the membrane thickness determined above. Although these theoretical predictions capture some of the qualitative features of the experimental data, they are not in quantitative agreement with the experiments. At 298 K, the theoretical prediction for the permeance decreases too rapidly as the feed pressure is increased and at high feed pressures substantially underestimates the true permeance. At higher temperatures, the dependence of the predicted permeance on the feed pressure is less severe. An alternative way to test our theoretical predictions is to examine the ideal selectivity of CH4 relative to CF4. The ideal selectivity is defined to be the ratio of the single-component permeances. We consider here only

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Figure 8. Experimental and theoretical data for CH4/CF4 ideal selectivity as a function of feed pressure with a transmembrane pressure drop of 138 kPa at 298 K. The solid curve is the result of the full theory, the dashed curve is the theoretical prediction when the Darken approximation is used, and the symbols are the experimental data.

the ideal selectivity at 298 K because we can carefully compare our atomistic model to single-crystal adsorption and diffusion studies at this temperature. The experimentally observed ideal selectivity at 298 K is shown in Figure 8 as a function of feed pressure with a constant transmembrane pressure drop of 138 kPa. The ideal selectivity gradually increases with increasing feed pressure, reaching approximately 5.5 at the highest feed pressures we examined. The ideal selectivity predicted by our fully atomistic model is shown in Figure 8 as a solid curve. We note that the ideal selectivity predicted by our model is independent of the membrane thickness, L (see eq 2), and therefore contains no parameters apart from the interatomic potentials used to specify our atomistic models. As can also be seen from comparison of Figures 4 and 6, our model overestimates the ideal CH4/CF4 selectivity. We believe that part of the discrepancy between our theoretical and experimental results can be traced to known inaccuracies in the atomistic model we have used for CF4. We showed above that our atomistic model for CH4 in silicalite gives adsorption isotherms and self-diffusivities in good agreement with available experimental data at 298 K. We also noted that the self-diffusivities predicted by the atomistic model we adopted for CF4 in silicalite underestimate the experimental values by up to a factor of 4 at loadings above 8 molecules/unit cell.25 It can be seen from the adsorption isotherms for this atomistic model (Figure 2) that this loading is relevant for all pressures higher than roughly 300 kPa. That is, this atomistic model is known to underestimate the true self-diffusivity of CF4 at all of the feed pressures shown in Figure 8. We have emphasized above that, for nondilute pore loadings, the self-diffusivity and Fickian diffusivity of an adsorbed species are not identical. Nevertheless, if we make the reasonable assumption that the atomistic model of CF4 also underestimates the Fickian diffusivity at high pore loadings, we see from eq 2 that our theoretical model will underestimate the flux of CF4 across a silicalite membrane. An underestimation of the CF4 flux is equivalent to an overestimation of the CH4/CF4 ideal selectivity. Thus, the known inaccuracy of the atomistic model we have used to describe CF4 adsorbed in silicalite may account for much of the discrepancy between our experimental and theoretical results in Figure 8.

Despite the discrepancies between our atomistic predictions and experimental results in Figure 8, this figure illustrates an important reason for using atomistic models to describe molecular adsorption and diffusion in zeolites. As mentioned above, many empirical descriptions of single-component permeation through zeolites describe intracrystalline diffusion by assuming that the generalized Maxwell-Stefan diffusivity of the adsorbed species is independent of loading.6-10 This assumption is equivalent to assuming that the Darken approximation (eq 4) is accurate.28 In Figure 8 we show the ideal selectivity that is predicted by using the Darken approximation to approximate the Fickian diffusivity of CH4 and CF4 with a dotted curve. As shown in Figure 3, the Darken approximation is quite accurate for CH4 at 298 K but is quite inaccurate for CF4 under the same conditions. As a result, the ideal selectivity predicted using the Darken approximation in Figure 8 is significantly different from the correct result for this atomistic model. The difference between a selectivity of 2 and a selectivity of 10 or more can obviously be important in practical applications. This example, therefore, suggests that using models based on the Darken approximation may lead to qualitatively incorrect estimates of the ability of zeolite membranes to separate light gases. It is tempting to conclude from Figure 8 that the Darken approximation leads to results that are in better accord with our experimental findings than the full theoretical treatment. This is not a physically significant result. Because the Darken approximation is not accurate for our atomistic model of CF4 in silicalite, the results based on this approximation in Figure 8 should be thought of as applying a correction known to be inaccurate to an atomistic model that is known to not perfectly describe the true physical situation. In this case the Darken approximation for the atomistic model fortuitously cancels some of the differences between the underlying atomistic model and the real material. Conclusion We have presented experimental and theoretical results for the single-component permeance of CH4 and CF4 through a silicalite membrane over a range of pressures and temperatures. Our theoretical predictions are based directly on atomistic models for the adsorbed molecules by directly computing the Fickian diffusivities and adsorption isotherms for these models. As a result, the only parametric input into our theory is the interatomic potentials describing the adsorbed molecules, the crystal structure of the zeolite, and the thickness of the membrane.19 Although numerous previous studies have modeled gas permeance through zeolite membranes, we believe that our results are the first that directly compare the predictions of a fully atomistic approach with macroscopic experimental results. The theoretical model we have presented here treats the zeolite membrane as a single crystal with a random crystallographic orientation. Practical polycrystalline membranes such as the one we have used experimentally are clearly more complex than this. Our results underscore the conclusion from many previous studies that the total flux through a polycrystalline zeolite membrane includes contributions from flow through both zeolitic and nonzeolitic pores. The modeling methods we have used here can be viewed within the context of polycrystalline membranes in two complementary


ways. First, our methods can be used to provide detailed descriptions of the flux through perfect zeolite crystals as a function of temperature, feed pressure, and transmembrane pressure drop. These results alone can help to refine our understanding of how the zeolitic and nonzeolitic pores in a practical membrane combine to define a membrane’s overall performance. Alternatively, our methods can be seen as one step toward detailed modeling of polycrystalline membranes. If the microstructure of a membrane is known from, for example, confocal microscopy,59 our atomistically based transport model could be applied individually to each crystal in the membrane. If this approach is combined with an accurate description of gas permeation through the nonzeolitic pores that separate zeolite crystals, the overall permeance through a polycrystalline membrane could be investigated. We note that current models of permeation through nonzeolitic pores such as those we have discussed above are only strictly applicable to pores that are many times wider than the adsorbed species molecular diameter, so there is considerable work to be done before accurate models of gas permeation through small nonzeolitic pores are available. An important feature of our atomistic model is that it points out possible weaknesses in empirical models for gas permeance through zeolite membranes. Many empirical models of gas permeation through zeolite membranes assume that the single-component MaxwellStefan diffusivities of the adsorbed species are independent of the pore loading.6-10 We have shown that this is a poor assumption for the atomistic model we have used for CF4 adsorbed in silicalite. If this assumption is used to predict the permeance through silicalite membranes based on known diffusivities at low loadings, qualitatively incorrect predictions for the CH4/CF4 ideal selectivity at 298 K result. As we have discussed elsewhere,28 CF4 adsorbed in silicalite does not appear to be an anomalous example; it has a Langmuir-like isotherm, and its self-diffusivity decreases with increasing pore loading. As a result, it seems likely that there are other chemical species whose Maxwell-Stefan diffusivities in zeolite vary significantly with pore loading. Our results suggest several fruitful avenues for future work. We attribute much of the discrepancy between our experimental and theoretical results for CF4 permeance to known inaccuracies in the atomistic model for adsorbed CF4. It would, therefore, be interesting to derive an atomistic model for adsorbed CF4 that is in better agreement with the known adsorption isotherms and self-diffusivities than the model we have used here to see whether such a model would also improve the predictions of our membrane results. Preliminary efforts to achieve this by varying the Lennard-Jones potential parameters in the atomistic model have so far not lead to a more satisfactory atomistic model. We are currently exploring other possible forms for the interatomic potential of a CF4 molecule interacting with the atoms making up a silicalite crystal. It would also be interesting to extend our theoretical and experimental results to binary gas mixtures. The possibility of extending the atomistic approach we have examined here to the intracrystalline diffusion of binary mixtures has recently been demonstrated by Sanborn and Snurr.53,54 Acknowledgment Work at the University of Colorado was supported by the NSF I/U CR Center For Membrane Applied Science and Technology (MAST) and the Department of Educa-

tion Government Assistantships in Areas of National Need (GAANN) Program. Work at CMU was supported by the NSF CAREER program under Grant CTS9983647 and by Air Products and Chemicals Inc. D.S.S. acknowledges support from the Alfred P. Sloan Foundation. A.I.S. acknowledges support from the Alexandros Onasis Foundation. We thank Dr. Vu A. Tuan for synthesizing the membranes used in this study and Megan E. Fortado for experimental assistance. Literature Cited (1) Matsukata, M.; Kikuchi, E. Zeolitic Membranes: Synthesis, Properties and Propects. Bull. Chem. Soc. Jpn. 1997, 70, 2341. (2) Coronas, J.; Santamaria, J. Separations Using Zeolite Membranes. Sep. Purif. Methods 1999, 28, 127. (3) Tsapatsis, M.; Gavalas, G. R. Synthesis of Porous Inorganic Membranes. MRS Bull. 1999, March, 30. (4) Ka¨rger, J.; Ruthven, D. M. Diffusion in Zeolites and Other Microporous Materials; John Wiley: New York, 1992. (5) Chen, N. Y.; Degnan, T. F.; Smith, C. M. Molecular Transport and Reaction in Zeolites; VCH: New York, 1994. (6) Keil, F. J.; Krishna, R.; Coppens, M.-O. Modeling of Diffusion in Zeolites. Rev. Chem. Eng. 2000, 16, 71. (7) Krishna, R. Multicomponent Surface Diffusion of Adsorbed Species: A Description Base on the Generalized Maxwell-Stefan Equations. Chem. Eng. Sci. 1990, 45, 1779. (8) Kapteijn, F.; Bakker, W. J. W.; Zheng, G.; Poppe, J.; Moulijn, J. A. Permeation and separation of light hydrocarbons through a silicalite-1 membranes: Application of the generalized MaxwellStefan equations. Chem. Eng. J. 1995, 57, 145. (9) Krishna, R.; van den Broeke, L. J. P. The Maxwell-Stefan Description of Mass Transport Across Zeolite Membranes. Chem. Eng. J. 1995, 57, 155. (10) van den Broeke, L. J. P. Simulation of diffusion in zeolitic structures. AIChE J. 1995, 41, 2399. (11) Theodorou, D. N.; Snurr, R. Q.; Bell, A. T. Molecular dynamics and diffusion in microporous materials. In Comprehensive Supramolecular Chemistry; Alberti, G., Bein, T., Eds.; Pergamon Press: New York, 1996; Vol. 7, pp 507-548. (12) Pohl, P. I.; Heffelfinger, G. S.; Smith, D. M. Molecular dynamics computer simulation of gas permeation in thin silicalite membranes. Mol. Phys. 1996, 89, 1725. (13) Takaba, H.; Koshita, R.; Mizukami, K.; Oumi, Y.; Ito, N.; Kubo, M.; Fahmi, A.; Miyamoto, A. Molecular dynamics simulation of iso- and n-butane permeation through a ZSM-5 type silicalite membrane. J. Membr. Sci. 1997, 134, 127. (14) Pohl, P. I.; Heffelfinger, G. S. Massively parallel molecular dynamics simulations of gas permeation across porous silica membranes. J. Membr. Sci. 1999, 155, 1. (15) Mizukami, K.; Kobayashi, Y.; Morito, H.; Takami, S.; Kubo, M.; Belosludov, R.; Miyamoto, A. Molecular Dynamics Studies of Surface Difference Effect on Gas Separation by Zeolite Membranes. Jpn. J. Appl. Phys. 2000, 39, 4385. (16) Martin, M. G.; Thompson, A. P.; Nenoff, T. M. Effect of Pressure, Membrane Thickness, and Placement of Control Volumes on the Flux of Methane through Thin Silicalite Membranes: A Dual Control Volume Grand Canonical Molecular Dynamics Study. J. Chem. Phys. 2001, 114, 7174. (17) MacElroy, J. M. D.; Boyle, M. J. Nonequilibrium molecular dynamics simulation of a model carbon membrane separation of CH4/H2 mixtures. Chem. Eng. J. 1999, 74, 85. (18) Barrer, R. M. Zeolites as Membranes: The Role of the GasCrystal Interface. In Catalysis and Adsorption by Zeolites; Ohlmann, G., Pfeifer, H., Fricke, R., Eds.; Elsevier: Amsterdam, The Netherlands, 1991; p 257. (19) Sholl, D. S. Predicting Single-Component Permeance Through Macroscopic Zeolite Membranes from Atomistic Simulations. Ind. Eng. Chem. Res. 2000, 39, 3737. (20) June, R. L.; Bell, A. T.; Theodorou, D. N. Prediction of Low Occupancy Sorption of Alkanes in Silicalite. J. Phys. Chem. 1990, 94, 1508. (21) June, R. L.; Bell, A. T.; Theodorou, D. N. Molecular Dynamics Study of Methane and Xenon in Silicalite. J. Phys. Chem. 1990, 94, 8232.

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Received for review April 5, 2001 Revised manuscript received July 18, 2001 Accepted December 28, 2001 IE010303H

A Comparison of Atomistic Simulations and Experimental

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