Predicting Single-Component Permeance through Macroscopic

Ind. Eng. Chem. Res. 2000, 39, 3737-3746

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Predicting Single-Component Permeance through Macroscopic Zeolite Membranes from Atomistic Simulations David S. Sholl* Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213

The atomically ordered crystalline pores of zeolites endow membranes made from zeolites with many desirable properties. We present a theoretical framework that allows predictions of singlecomponent permeance through zeolite membranes directly from atomistic models of the adsorbate and zeolite. In contrast to previous atomistic approaches to zeolite membranes, this framework does not attempt to include an entire membrane in a single nonequilibrium atomistic simulation. Rather, equilibrium molecular simulations are used to derive those data necessary for solving a general macroscopic description of molecular transport through a membrane. This approach allows the behavior of membranes of realistic thicknesses to be assessed over a broad range of temperatures and pressures. To illustrate this new framework, the steady-state permeances of Xe through an AlPO4-31 membrane and CF4 through an AlPO4-5 membrane are predicted directly from atomistic models of these systems. 1. Introduction Recent developments in the reliable synthesis of membranes made from zeolites have led to a rapid growth of interest in these membranes. For recent reviews of this area, see refs 1 and 2. Zeolites exhibit a number of macroscopic properties that make them attractive as membrane materials. They are stable in high-pressure and -temperature environments, and they also exhibit excellent chemical stability. These properties allow zeolite membranes to operate in conditions that would be too harsh for most polymeric membranes. Zeolites also offer an attractive feature by virtue of their atomically ordered pores. The crystalline nature of zeolites means that their pore-size distribution is monodisperse. In contrast, membranes made from disordered porous materials such as porous carbon or polymers exhibit polydisperse pore-size distributions.3,4 The highly ordered pores in zeolite membranes allow the possibility of highly selective separations of species with only slightly different adsorption and transport properties. Furthermore, as hundreds of zeolite structures are known,5 there are, in principle, a large variety of possible membrane materials that can be considered to achieve a desired separation. The development of zeolite membranes for new applications faces at least two significant challenges. First, devising synthesis strategies for a new membrane material remains difficult. This fact greatly limits the screening of possible membrane materials to find membranes that are ideal for a desired separation. Second, accurate interpretation of behaviors observed with existing membranes is often difficult. This state of affairs greatly hampers performance optimization. Progress toward overcoming these difficulties would be greatly aided if accurate theoretical models were available for predicting the performance of realistic zeolite membranes. The aim of this paper is to demonstrate a theoretical approach that takes some initial steps toward this goal. * Author to whom correspondence should be addressed. E-mail: [email protected]. Fax: 412-268-7139.

Before describing the new results of this work, it is useful to briefly review previous theoretical approaches to modeling transport through zeolite membranes. The most successful models to date have used a macroscopic approach based on a Maxwell-Stefan formulation of the transport problem.6-10 Practical applications of this model have typically assumed that the single- and multicomponent adsorption isotherms have the Langmuir form, that the single-component Maxwell-Stefan diffusivities are constant, and that the cross-species Maxwell-Stefan diffusivities are zero.6-10 With these approximations, the unknown parameters in the model can be fit using single-component permeation experiments. This technique has been successfully applied, for example, to the transport of mixtures of ethane and ethene through silicalite membranes.8 Although this theoretical approach has great value for aiding the interpretation of existing experimental results, it cannot be used in a truly predictive fashion. That is, these models cannot be used to predict the performance of a membrane that cannot currently be synthesized, and therefore, they cannot be used for computational screening of possible membrane materials. An alternative approach to modeling zeolite membranes is to use molecular simulation methods to represent the membrane atomistically and to directly simulate adsorbate transport across the membrane.11-13 The obvious advantage of this approach over the macroscopic methods described above is that it explicitly incorporates the atomic-scale structure of the zeolite’s ordered pores. As a result, direct atomistic modeling can, in principle, be used to model zeolite membranes in a completely predictive manner and to screen a variety of possible membranes for effective separations capabilities. Unfortunately, fully atomistic simulations of flow through a membrane are very computationally demanding. As a result, applications of this approach have been limited to extremely thin membranes with extremely large pressure drops across the membrane. For example, in the simulations of Takaba et al.,12 the membrane thickness was 2.3 nm, and the effective pressure drop across the membrane was approximately 100 atm. In

10.1021/ie000301h CCC: $19.00 © 2000 American Chemical Society Published on Web 08/24/2000

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contrast, realistic membranes are typically 10-100 µm thick,1,2 and most experimental investigations use pressure drops on the order of 1 atm. The enormous disparity between these thicknesses and pressure drops raises questions about how accurately the simulation results can be related to practical situations. A more subtle difficulty with these models is that the performance of a simulation at one set of physical conditions does not give any immediate insight into how the membrane would behave at other conditions. In contrast, once a macroscopic model such as the MaxwellStefan model mentioned above has been parametrized, it can immediately be used to predict the membrane’s performance over a wide range of operating conditions. It is important to note that although direct atomistic modeling of zeolite membranes is problematic for the reasons given above, this type of modeling has proven enormously successful in providing a detailed understanding of other processes involving zeolites. In particular, atomistic simulations have been widely used to study equilibrium adsorption isotherms and intracrystalline diffusion, especially tracer diffusion.14 In this paper, an alternative approach to the theoretical modeling of molecular transport through zeolite membranes is demonstrated. This approach is intended to incorporate the advantages of both of the approaches outlined above while avoiding many of their disadvantages. The aim of this approach is to use equilibrium atomistic simulations to directly parametrize a macroscopic description of membrane transport. The only input into this approach is the atomistic model of the zeolite and adsorbates, as in previous atomistic approaches to these materials. As a result, the final predictions of this method are directly determined by the atomic structure of the membrane, as required for any approach that aims to predictively describe a range of zeolites. The central difference between this work and previous atomistic modeling of zeolite membranes is that no attempt is made here to directly model the entire membrane atomistically. Instead, small-scale atomistic simulations are used to derive the information needed to fully define a general macroscopic description of molecular transport across the membrane. Once this description is complete, predictions can easily be made of the membrane’s performance for any combination of membrane thickness, feed, and permeate pressure. This paper illustrates the approach just described for two systems, CF4 transport through an AlPO4-5 membrane and Xe transport through an AlPO4-31 membrane. These two systems were chosen because many details of the necessary atomistic simulations have been developed previously. In section 2, the general formalism for macroscopic transport of a single component across a zeolite membrane is presented. In section 3, the atomistic models that have been used for CF4/ AlPO4-5 and Xe/AlPO4-31 are defined. Sections 4 and 5 describe the adsorption isotherms and Fickian diffusivities, respectively, for the systems of interest, as computed from atomistic simulations. The combination of the macroscopic transport theory and the atomistic simulations to give predictions for single-component membrane permeances is described in section 6. The paper concludes in section 7 with a summary of our results and discussion of future applications. 2. General Theory In this paper, we restrict our attention to isothermal membranes comprising a single zeolite crystal of thick-

ness L oriented along the z direction in contact with bulk single-component gas phases at upstream (downstream) pressure P0 (PL). The zeolite unit cell’s cross-sectional area perpendicular to z will be denoted a. There are a number of equivalent approaches to modeling macroscopic transport in microporous materials.15,16 We adopt the Fickian approach, so the flux of the adsorbed species within the membrane is given by

J ) -D(c)

dc dz

(1)

where c(z) is the concentration of the adsorbed species and D(c) is the concentration-dependent Fickian diffusion coefficient. If c is given in units of number of molecules per unit cell, eq 1 gives the flux in terms of the number of molecules passing through a membrane with cross-sectional area a per second. This flux will be denoted Juc. This molecular flux is related to the more usual macroscopic molar flux, denoted simply J, by J ) (nuc/NA)Juc, where nuc ) 1/a is the number of unit cells per square meter of the membrane-gas interface and NA is Avogadro’s constant. Under steady-state conditions, eq 1 can be integrated to give the steady-state flux

J)

1 L

∫cc D(c) dc 0

L

(2)

where c0 (cL) is the adsorbate concentration in the zeolite at z ) 0 (z ) L). Below, we will frequently refer to the adsorbate concentrations in the pores as the loading. Thus, to determine the steady-state flux, it is only necessary to know the loading-dependent Fickian diffusion coefficient, D(c), and the boundary loadings. If there are no barriers to mass transfer at the interfaces between the zeolite crystal and the gas phases, the boundary loadings in eq 2 are given by the zeolite’s equilibrium adsorption isotherm. This assumption of local thermodynamic equilibrium at the membrane boundaries has been made in several previous models of zeolite membranes6-10 and will also be made throughout this paper. We note, however, that this assumption cannot be precisely correct.17 Barrer showed, using a phenomenological model for zeolite membranes, that the presence of surface barriers to mass transport reduces the net flux through a membrane.17 Results developed under the assumption of local equilibrium at membrane boundaries are, therefore, most accurately viewed as upper bounds on the membrane’s flux. Experimental study of surface barriers, while possible, has been confined to a small number of systems.18-21 One direction for future theoretical studies of zeolite membranes will be the development of atomic-scale models that can be used to critically assess current assumptions dealing with membrane interfaces. As noted above, previous approaches to describing zeolite membranes using atomic-scale models have relied on simulations in which the entire membrane is represented atomistically.11-13 Equations 1 and 2 present an important alternative to this approach. If atomically detailed simulation techniques can be used to compute an adsorbate’s equilibrium adsorption isotherm and the loading-dependent Fickian diffusion coefficient, then the flux through a membrane of arbitrary thickness with arbitrary upstream and downstream pressures can be predicted.

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It is useful to briefly review two standard approximations to the loading-dependent Fickian diffusivity and their implications for membrane transport. It can be shown analytically that the Fickian diffusivity of a noninteracting lattice gas is independent of loading and equal to the zero-loading tracer diffusivity, D0.22-24 The adsorption isotherm for this model is the Langmuir isotherm25

c(P) KP ) csat 1 + KP

(3)

Here, csat is the pore’s saturation loading and K ) KH/ csat, where KH is the system’s Henry’s law constant measured in molecules per unit cell per unit pressure. Combining these results with eq 2 shows that the steady-state flux through a single pore for this model is

(P0 - P1) JL ) csatD0K (1 + KPL)(1 + KP0)

(4)

An alternative approach is to assume that adsorption is governed by the Langmuir isotherm given above and that the Maxwell-Stefan diffusivity is constant.6-10 A constant Maxwell-Stefan diffusivity gives

D(c) )

D0

(5)

(1 - c/csat)

The steady-state flux through a single pore for this model is

JL ) csatD0 ln

(

)

1 + KP0 1 + KPL

(6)

3. Atomistic Models for CF4/AlPO4-5 and Xe/ AlPO4-31 In the remainder of this paper, the prediction of membrane fluxes using our combined atomic-scale/ macroscopic approach is illustrated for two example systems: CF4 diffusion through AlPO4-5 and Xe diffusion through AlPO4-31. Although oriented thin films and membranes have been created using aluminophosphate materials,26-31 no experimental permeation data for these two systems are currently available. We emphasize that the aim of this paper is to demonstrate that it is possible to predict macroscopic membrane fluxes from atomically detailed models, not to provide a detailed comparison between the results of this approach and experimental measurements. As such, the two examples treated here were chosen because detailed atomic-scale models have been developed in previous studies of other aspects of intracrystalline diffusion.32-35 The application of these methods to physical systems for which direct comparisons to experimental data are possible is underway, and the results of these comparisons will be reported elsewhere.36 In this section, the models used to represent these two systems atomistically are summarized, together with some of each system’s key physical features. AlPO4-5 and AlPO4-31 both have unidimensional pores with nominal diameters of 7.3 and 5.4 Å, respectively.5 Detailed images of the unit cells and pore structures of these materials can be found at http:// www.iza-structure.org. The membranes that are mod-

Figure 1. Equilibrium adsorption isotherms for Xe in AlPO4-31 at the temperatures indicated. Symbols are results from GCMC simulations, and solid curves are Henry’s law results.

eled below are assumed to be oriented along the direction of these pores. Both of the adsorbates examined here perform single-file diffusion, that is, two molecules cannot pass one another inside a pore.32,37,38 One implication of this observation is that the tracer diffusion coefficient of these adsorbates is undefined for nonzero pore concentrations, as the mean square displacement of a single molecule is not linearly proportional to time at long times.32,37,38 It is important to note, however, that the single-component Fickian diffusivity is well-defined in single-file systems.23,24 To perform atomistic simulations of CF4/AlPO4-5 and Xe/AlPO4-31, the pore structure is defined using the experimentally determined crystal structures.5 The pore structure is assumed to be rigid in all of the simulations discussed below. The adsorbates are modeled as spheres, and adsorbate-adsorbate and adsorbate-pore interactions are respresented by Lennard-Jones potentials. The parameters for these potentials were taken from previous work.32-35 This type of model has been successfully applied in a wide range of atomistic simulations of nonpolar molecules adsorbed in noncationic zeolites.14 4. Adsorption Isotherms As discussed above, one necessary quantity for predicting the flux across a membrane is the equilibrium adsorption isotherm. The computation of adsorption isotherms with atomistic models can now be routinely accomplished using the grand canonical Monte Carlo (GCMC) technique and variants of this method. Standard GCMC simulations can be applied to spherical adsorbates and mixtures of spherical adsorbates.39-42 The development of configurational bias techniques has made the computational treatment of linear and branched chain molecules feasible.43,44 Techniques for accurately including quantum effects on the adsorption of light molecules are also available.45 Simulations using these methods have shown that atomistic models accurately reproduce experimental isotherms for a variety of adsorbate/zeolite pairs.41,44 The adsorption isotherms for Xe in AlPO4-31 and CF4 in AlPO4-5 at a range of temperatures are shown in Figures 1 and 2, respectively. Each data point in these figures was computed using standard GCMC methods. The uncertainties in the measured pore loadings are similar to the symbol sizes. The pressure of the bulk Lennard-Jones phase was calculated using a second virial expansion with the exact second virial coefficient

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To predict membrane permeance, it is convenient to have a functional form for the adsorption isotherm that can be evaluated at any pressure. Initially, we attempted to fit our computed isotherms using the Langmuir and Langmuir uniform distribution isotherms.49 Neither of these isotherms gave accurate fits over the entire pressure range of our calculations. As a result, in the calculations reported below involving adsorption isotherms, we defined the isotherms by linear interpolation between GCMC data points in the regions where these data are shown in Figures 1 and 2. The Henry’s law results shown in Figures 1 and 2 were used for pressures lower than the lowest-pressure GCMC point on each isotherm. Figure 2. Equilibrium adsorption isotherms for CF4 in AlPO4-5 at the temperatures indicated. Symbols are results from GCMC simulations, and solid curves are Henry’s law results.

for Lennard-Jones fluids.46 We note that bulk-phase pressures for more complex atomistic models can readily be computed using bulk GCMC. We also evaluated the Henry’s law coefficient for each system using direct Monte Carlo evaluation of a single adsorbed molecule’s configurational integral.47 The Henry’s law predictions using these results are shown as solid curves in Figures 1 and 2. Agreement between the Henry’s law prediction and the low-pressure GCMC results is excellent, as should be expected. No hysteresis was observed in any of the simulated isotherms, as is generally expected for these quasi-one-dimensional systems.48 One useful quantity for describing the pore loading of a unidimensional pore is the ideal one-dimensional loading. This is the loading corresponding to a purely one-dimensional array of adsorbates arranged so that the adsorbate-adsorbate potential energy between adjacent molecules is minimized. If Xe atoms are arranged along the center of an AlPO4-31 pore in this fashion, they are spaced 4.552 Å apart, leading to a pore loading of 1.107 molecules/unit cell. It can be seen from Figure 1 that this loading accurately characterizes the highest loading observed when T ) 100 K. For high gasphase pressures when T ) 200 and 300 K, an adsorbed phase with more than 1.107 molecules/unit cell is observed. Analysis of this phase shows that it is formed by maintaining spacings between adjacent particles at approximately the optimum distance for the LennardJones potential but displacing molecules away from the pore center. A similar situation occurs for CF4 in AlPO4-5 (Figure 2), whose ideal one-dimensional loading is 1.607 molecules/unit cell. For this loading, the optimum distance between CF4 molecules is 5.28 Å. At the saturation loadings observed in Figure 2, the average interadsorbate spacing is only 5.00 Å, and the average spacing along the pore axis is 4.24 Å. In this configuration, molecules are displaced appreciably from the pore center. The spacing of molecules along the pore in this saturation state is not surprising, as the potential energy profile for a single CF4 molecules in AlPO4-5 pore has minima spaced every 4.24 Å along the pore,32 displaced slightly from the pore center. At lower loadings, the intermolecular spacing in this system is dominated by the intermolecular potential rather than the adsorbate-lattice potential.32 Analysis of configurations from our GCMC simulations showed that the CF4-CF4 optimum distance accurately represented the intermolecular spacing at low loadings and that the particles remained close to the pore center.

5. Fickian Diffusivities Atomistic simulations have been widely used to study diffusive molecular transport in zeolite pores.14 Almost without exception, these studies have focused on tracer diffusion coefficients, and numerous examples of agreement between simulations and experimental measurements of this quantity are available.14,50 However, as shown above, to describe molecular transport across a zeolite membrane, we need to know the Fickian diffusivity. To reiterate a well-known fact, the Fickian and tracer diffusion coefficients are only equal in the limit of zero loading.14 Determining Fickian diffusivities from simulations is considerably more challenging than determining the tracer diffusivity, as the former involves collective relaxations of many molecules. We know of only two studies that have used atomistic simulations to derive Fickian diffusivities for adsorbed species in zeolites. Maginn et al. used several nonequilibrium molecular dynamics (NEMD) techniques to measure the Fickian diffusivity of methane in silicalite at T ) 300 K.47 In these calculations, an artificial driving force for diffusion is introduced across the simulation cell, and the resulting particle transport is measured. Maginn et al. demonstrated that Fickian diffusivities can be successfully measured using NEMD, but their work indicated that care must be taken with these methods to ensure that the simulated system lies within the linear response regime.47 More recently, Sanborn and Snurr have used equilibrium MD to examine binary transport diffusivities in an equimolar mixture of CF4 and CH4 in faujasite at T ) 300 K.51 In addition to these atomistic studies, Fickian diffusivities have been studied in a variety of idealized models. In particular, Coppens et al. have recently performed an extensive study of the effects of pore loading and network topology on Fickian diffusivities using an idealized lattice model.52 We have computed Fickian diffusivities for our atomistic models of Xe in AlPO4-31 and CF4 in AlPO4-5 using an approach different from those of refs 47 and 51. The first step in our calculations was to derive an atomically accurate coarse-grained model of the adsorbed molecule’s dynamics. The derivation of these models has been described in detail elsewhere,35 so we only summarize the principal details here. The most crucial step in deriving an accurate coarse-grained model is identification of the microscopic processes that control molecular mobility in the fully atomistic model. For the systems we consider here, it has been shown, using a combination of molecular mechanics and molecular dynamics, that molecular transport in these pores is dominated by the concerted motion of weakly


bound one-dimensional clusters of molecules.33,34 It is important to note that traditional lattice-gas models that define molecular transport in terms of single molecules hopping between fixed adsorption sites23,25,52 do not include this type of concerted cluster diffusion. Once cluster diffusion was identified as the relevant diffusion mechanism, fully atomistic MD simulations were used to directly compute the microscopic mobilities of these molecular clusters and the dissociation rates of all possible cluster dissociation pathways.34,35 This information was used to define coarse-grained models in which the mobile species are molecular clusters of variable size and the rates of all processes available to these clusters were determined from atomistic MD.35 These MD simulations used standard methods for modeling nonpolar adsorbates in noncationic zeolites. The zeolites were assumed to be rigid with the experimentally determined crystal structures. The adsorbates were represented by single-site spheres with LennardJones potentials used for both adsorbate-adsorbate and adsorbate-lattice interactions. The dynamics of the resulting coarse-grained models are simulated using kinetic Monte Carlo. Because atomistic details are not explicitly represented in these coarse-grained models, simulations can be performed on time and length scales orders of magnitude longer than atomistic MD simulations. We have previously used our coarse-grained models to examine the loading dependence of singlefile mobilities (the equivalent of tracer diffusivities in these one-dimensional systems) using simulations that tracked the dynamics of thousands of distinct molecules on microsecond time scales.35 One restriction on these coarse-grained models is that they were derived assuming a purely one-dimensional packing of the adsorbed molecules. As was discussed above, this assumption is not valid for the highest pore loadings observed in the equilibrium adsorption isotherms. To emphasize this restriction, we report the results from our coarsegrained models in terms of fractional loading, θ, relative to the ideal one-dimensional pore packing, θ ) 1. When θ ) 1, adsorbates are uniformly spaced along the pore with spacing 21/6σAA, where σAA is the Lennard-Jones adsorbate-adsorbate length scale. This spacing optimizes the adsorbate-adsorbate interactions. For Xe in AlPO4-31, θ ) 1 corresponds to 1.107 molecules/unit cell, and for CF4 in AlPO4-5, θ ) 1 corresponds to 1.607 molecules per unit cell. We have computed Fickian diffusion coefficients from the coarse-grained models outlined above using an elegant method due to Mak et al.53 To use this method, we define the cosine transform of the N adsorbed particle’s positions, zi(t), by N

ck(t) )

cos[kzi(t)] ∑ i)1

(7)

Fluctuations of this transform from its mean value are denoted Ck(t) ) ck(t) - 〈ck〉. The quantity that is actually analyzed is the autocorrelation function of these fluctuations

〈〈Ck〉〉 )

〈Ck(t)Ck(0)〉 〈Ck(0)Ck(0)〉

(8)

Here, the double angled brackets denote an average over many independent trajectories with the same pore loading. In the limit of large system sizes

〈〈Ck〉〉 ) exp(-Dk2t)

(9)

where D is the Fickian diffusion coefficient at the loading of the simulations.53 One advantage of this approach is that it examines the decay of spontaneous density fluctuations in an equilibrium system, so no assumptions about the applicability of linear response theory need to be made.47 A second advantage is that it is straightforward to monitor density fluctuations with a range of wavelengths simultaneously. If a simulation has fully converged, the diffusivities derived from fluctuations with different wavelengths should be the same. Some sample results of applying the formalism of Mak et al.53 to our coarse-grained models of molecular transport are shown in Figure 3. These data come from simulations of CF4 in AlPO4-5 at T ) 200 K at a fractional pore loading of θ ) 0.3. In these simulations we simulated a pore containing 1000 CF4 molecules, giving a pore length of L ) 1758.5 nm. Periodic boundary conditions were applied at the ends of the simulated pore. Cosine transforms were computed for

kn )

2πn L/17

(10)

with n ) 1, 2, 3, and 4. The transform with n ) 1 examined density fluctuations with a wavelength of approximately 100 nm. The data shown in Figure 3 were obtained from 10 independent trajectories. For each trajectory, the adsorbed molecules were initially arranged to fill the pore with uniform spacings between them. The particle’s positions were then equilibrated by following their dynamics for 0.1 µs. Following this equilibration procedure, the system’s dynamics were simulated for an additional 0.5 µs while data were collected.35 Note that both the time and length scales of these simulations are orders of magnitude larger than the scales accessible with fully atomistic MD simulations.35 In Figure 3, each point at time t was computed by splitting the total trajectory of each simulated system into independent subtrajectories of length t and averaging over all possible subtrajectories. It is clear that the data for all of the wavelengths shown in Figure 3 conform to the prediction of eq 9, so these data can be used to compute the system’s Fickian diffusivity. By repeating the procedure described above for multiple pore loadings, the loading-dependent Fickian diffusivity of an adsorbed species can be computed. For the results discussed below, data were collected from 10 independent runs including 1000 molecules each at each loading examined. The equilibration procedures and run lengths were identical to those described in ref 35. As above, cosine transforms with four wavelengths were examined, with the longest wavelength chosen to be approximately 100 nm. The precise wavelengths examined varied with loading because the total length of the simulated pore must be an integer multiple of the longest wavelength examined.53 Fickian diffusivities were computed by fitting the data from each wavelength with ln(〈〈Ck〉〉) g -1 independently and averaging these results. The error bars shown on our computed diffusivities are the sample standard deviation from our four computed values. Our computed Fickian diffusivities for Xe in AlPO431 at a range of temperatures are shown in Figure 4. We have verified that the measured values of D ap-

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Figure 3. Cosine autocorrelation functions for CF4 in AlPO4-5 at T ) 200 K and fractional loading θ ) 0.3. Density fluctuations with four different wavelengths were monitored. See text for details.

Figure 4. Computed Fickian diffusivities for Xe in AlPO4-31 as a function of fractional loading, θ. The data sets are for T ) 300, 200, and 100 K from top to bottom. The data points are results from coarse-grained dynamics, and the solid curves are empirical curve fits described in the text. Note that a logarithmic scale is used for the diffusivity.

proach the zero-loading tracer diffusivity, D0, measured in independent MD simulations35 as θ f 0. The most dramatic feature of these results is the rapid decrease in D with increasing θ at low pore loadings. At higher loadings, the observed diffusivities reach a minimum and finally begin to increase again. At all of the temperatures examined in Figure 4, the Fickian diffusivity is less than D0 for all nonzero pore loadings. Because Xe diffusion in AlPO4-31 is a weakly activated process,35 it is not surprising that D increases with increasing temperature at any fixed loading. As with the adsorption isotherms, it is convenient to have analytic expressions for D at arbitrary loadings. We tested the suitability of many functional forms for this purpose, but none were able to effectively fit our data over its entire range. We chose instead to adopt the following piecewise function

{

D(θ) ) D0 + c1θ if θ e 0.01 c c2θ 3 if 0.01 e θ e 0.1 c4(θ - 0.1)2 + c5(θ - 0.1) + c6 if θ g 0.1 We fit our data to this function with the constraint that

Figure 5. Comparison of computed and approximate Fickian diffusivities for Xe in AlPO4-31 at T ) 300 K. The solid symbols are the results from the full coarse-grained dynamics. The open symbols are results for the restricted dynamics. The dashed line is the prediction of a simple lattice gas model, and the dotted curve is the result for a system with constant Maxwell-Stefan diffusivity.

the resulting function be continuous. The values of D0 were taken from our previous MD simulations.32 The results of these data fittings are shown in Figure 4 as solid curves. We emphasize that these fitted curves are only empirical fits to our computed data. It is interesting to compare the results presented above to the predictions of common approximations for D(θ). The results for a noninteracting lattice gas [D(θ) ) D0]22-24 and for a system with a Langmuir adsorption isotherm and constant Maxwell-Stefan diffusivity [D(θ) ) D0/(1 - θ)]7-10 are compared to our computed results for T ) 300 K in Figure 5. The latter theory is included because Langmuir isotherms have frequently been assumed in previous empirical models of zeolite membranes.7-10 For both approximate theories we have used the zero-loading tracer diffusivity derived from singleparticle MD simulations. It is clear from Figure 5 that neither of these approximate theories gives an accurate representation of the actual loading dependence of D. This conclusion is quite different from the results of Maginn et al., who found that their detailed MD simulations of methane in silicalite were well approximated by a constant Maxwell-Stefan diffusivity.47 The observed deviations from the simple lattice-gas predictions are somewhat less surprising, as deviations from this result are well-known from simulations of twodimensional surface diffusion in models with energetic interactions between particles.54,55 As mentioned above, Xe atoms move through the pores of AlPO4-31 by concerted diffusion mechanisms involving weakly bound clusters of adsorbed atoms. To assess the impact of these cluster diffusion mechanisms on Xe diffusion, we have performed a series of simulations of a restricted version of our coarse-grained dynamics model in which only events involving single Xe atoms were allowed to take place.35 The diffusivities computed with this model are shown in Figure 5 as open symbols for θ g 0.1. These results show that the motion of clusters containing multiple atoms is an important contribution to the overall diffusivity. For θ > 0.5, more than half of the observed Fickian diffusivity can be attributed to these cluster motions. This observation is consistent with our previous study of single-file mobilities in this system, where we found that including all relevant cluster motions in our coarse-grained model


Figure 6. Computed Fickian diffusivities for CF4 in AlPO4-5 as a function of fractional loading, θ. The data sets are for T ) 200, 100, and 50 K from top to bottom. The data points are results from coarse-grained dynamics, and the solid curves are empirical curve fits described in the text. The dotted curve is the result of an approximate Maxwell-Stefan theory at T ) 200 K. The open symbols (connected by dashed lines to guide the eye) are results from the coarse-grained model with restricted dynamics for T ) 100 K.

substantially increased single-file mobilities relative to the restricted model.35 We have also used our coarse-grained dynamics models to compute the Fickian diffusivity of CF4 in AlPO4-5 at a range of temperatures. Our results are summarized in Figure 6. As for Xe/AlPO4-31, these results approach the zero-loading tracer diffusivity, D0, as θ f 0. The activation energy for diffusion of an isolated CF4 molecule in AlPO4-5 is larger than the activation energy for Xe/AlPO4-31,35 so the variation in D0 with temperature is more apparent in Figure 6 than in Figure 4. Similarly to Xe/AlPO4-31, the CF4 diffusivity decreases as a function of loading for low pore loadings then increases at higher loadings. In contrast to Xe/AlPO4-31, the CF4 diffusivity is considerably larger than D0 for high loadings at all of the temperatures we examined. The solid curves in Figure 6 are fits to the data using the empirical expression D(θ) ) D0 exp(aθ + bθ2), which fits the data quite effectively over the entire range of the data. It is apparent from Figure 6 that the noninteracting lattice gas approximation, D(θ) ) D0, is a poor approximation to the true loadingdependent diffusivity, as it was for Xe/AlPO4-31. The Maxwell-Stefan approximation, D(θ) ) D0/(1 - θ), gives a somewhat better approximation to the correct result, although it does not predict the decrease in D(θ) with increasing θ for low loadings. The Maxwell-Stefan approximation for CF4/AlPO4-5 for T ) 200 K (in conjunction with the value of D0 evaluated using MD) is shown in Figure 6 as a dotted curve. As with Xe/AlPO4-31, the motion of CF4 molecules due to concerted motions of weakly bound clusters of adsorbates is a significant contribution to the observed diffusivity. The diffusivity for a restricted version of our coarse-grained dynamics in which only processes involving single molecules are allowed is shown as a function of loading at T ) 100 K in Figure 6 using open symbols. It is clear that the loading dependence of the Fickian diffusivity is qualitatively different for the full and restricted coarse-grained dynamics. The diffusivity of the restricted model is substantially less than that for the full dynamics, particularly at moderate and high loadings. Concerted cluster motions have a similar impact on single-file mobilities in this system.35

Figure 7. Predicted steady-state membrane permeance for Xe transport through a 10-µm AlPO4-31 single-crystal membrane with zero downstream pressure. The pressure indicated on the figure is the upstream gas pressure. Dotted lines indicate regions where θ > 1.

One interesting feature of both Figures 4 and 6 is that, in each case, D(θ) is a nonmonotonic function, initially decreasing and then increasing as θ increases. This feature cannot be captured by the two models discussed in Figure 5 or by the more flexible model of Chen and Yang,56 all of which predict that D(θ) varies monotonically with θ. It is not clear precisely what leads to the nonmonotonicity seen in Figures 4 and 6. Figure 6 suggests that the diffusion of molecular clusters (as opposed to monomer diffusion) plays an important role in this behavior, although the restricted model in Figure 5 also gives nonmonotonic results. We note that nonmonotonic changes in D with surface coverage have been observed in simulations of two-dimensional surfaces with nontrivial energetic interactions between particles.55 6. Predicted Membrane Permeances The results of the preceding sections provide all of the information necessary for predicting single-component permeances through single-crystal membranes directly from our atomistic models. To accomplish this task, we specify the membrane’s upstream and downstream pressures and compute the steady-state membrane flux using eq 2. In this equation, the boundary pore concentrations and loading-dependent Fickian diffusivities are defined using the data from our atomistic simulations described above. The steady-state permeance is defined by the steady-state flux divided by the pressure drop across the membrane. For all of the results presented in this section, we fix the membrane thickness at 10 µm, noting that we can rescale our results to membranes of other thicknesses using eq 2. We reiterate that our model assumes that the membrane is made from an oriented single-crystal in which all pores are accessible for molecular transport across the membrane. The predicted steady-state permeances of Xe across a 10-µm-thick AlPO4-31 membrane are shown for a broad range of pressures and temperatures in Figure 7. For these results, the downstream pressure is assumed to be zero, so the pressure drop across the membrane is equal to the upstream pressure, P. The low-pressure results can be understood by noting that, when the highest pore loading is very small, the boundary loading is given by Henry’s law, c0 ) KHP,

3744


and D = D0 throughout the membrane. Combining these results with eq 2 shows that the low-pressure limit of the membrane permeance is constant and equal to D0KH/L. As T is increased, the Henry’s law constant decreases much more rapidly than D0 increases in this system, so the low-pressure permeance is a rapidly decreasing function of temperature. At high pressures, the variation in the permeance with temperature is dominated by the diffusion coefficient, so the permeance decreases with increasing temperature at high pressures. The range of pressures that is accessible with our methods is limited by the fact that the saturation loading possible in our coarse-grained model is slightly less than that observed in the adsorption isotherms, as discussed above. The results in Figure 7 for which some portion of the membrane includes pore loadings with θ > 1 are indicated with dashed lines. The Fickian diffusivity was assumed to be equal to D(θ ) 1) when θ > 1. The predictions of Figure 7 cannot be directly compared to experimental results, as no measurements of permeance through AlPO4-31 membranes are currently available. It is still instructive to compare our results with experimentally measured permeances of other small molecules through zeolite membranes. Dong and Lin measured room-temperature permeances of 1-5 × 10-7 mol m-2 s-1 Pa-1 for Ar, H2, and CH4 permeance through a P-type zeolite membrane of nominal thickness 10 µm.57 Rescaling reported experimental data from several experimental groups for CH4 permeance through silicalite membranes at room temperature to a 10-µmthick membrane using the membrane thicknesses reported by each group yields permeances ranging from 5 × 10-8 to 1.4 × 10-6 mol m-2 s-1 Pa-1, with considerable variation between membranes used in different studies.8,58,59 Permeances in this range have also been reported for N2 and SF6 transport through silicalite membranes at room temperature.60 The predicted permeances in the high-pressure region of Figure 7 are consistently higher than these experimental values, but are not radically different in magnitude. A detailed comparison between theoretical and experimental results will be more informative once the methods outlined here have been extended to a system for which experimental data are available. Work is underway to make such a comparison feasible.36 It was shown above that, if the Henry’s law constant and the zero-loading tracer diffusivity are known, the steady-state flux across a membrane for any pressure can be predicted within a simplified Maxwell-Stefan approximation by eq 6. Figure 8 compares the predictions of this approximate model with the results of our full calculations for Xe passing through a 10-µm-thick AlPO4-31 membrane at T ) 300 K. The approximate theory overestimates the steady-state permeance by as much as a factor of 25 at high pressures because of the strong deviations between the true and approximated loading-dependent diffusivity (cf. Figure 5). It is important to note that, in this application of the approximate Maxwell-Stefan theory, both the Henry’s law constant and the zero-loading tracer diffusivity have been obtained directly from atomistic simulations, in contrast to previous applications of this theory in which these quantities were fit to experimental data.6-10 The predicted steady-state permeance of CF4 across a 10-µm-thick AlPO4-5 membrane is shown in Figure 9 as a function of temperature and pressure. As in Figure

Figure 8. Predicted steady-state membrane permeance for Xe transport through a 10-µm AlPO4-31 single-crystal membrane with zero downstream pressure at T ) 300 K. The solid curve is the result of our full model, and the dotted line is the result of the Maxwell-Stefan approximation.

Figure 9. Predicted steady-state membrane permeance for CF4 transport through a 10-µm AlPO4-5 single-crystal membrane with zero downstream pressure. The pressure indicated on the figure is the upstream gas pressure. Dotted lines indicate regions where θ > 1.

7, regions in which some portion of the membrane has pore loadings with θ > 1 are indicated with dotted lines. The shoulder in the predicted permeance at T ) 100 K is due to the nontrivial shape of the measured adsorption isotherm. The general features of the permeance as functions of temperature and pressure are similar to those for Xe/AlPO4-31. In particular, the permeance is a decreasing function of temperature at low pressure, but this trend is reversed at high pressure. 7. Conclusion This paper has demonstrated a qualitatively new theoretical framework for modeling molecular transport through zeolite membranes. Previous approaches to this topic have relied on either empirically parametrized macroscopic descriptions6-10 or large-scale nonequilibrium MD simulations.11-13 Our new approach blends the advantages of these previous methods. Specifically, atomistic detail is retained by using moderately sized equilibrium molecular simulations to compute adsorption isotherms and loading-dependent diffusion coefficients. This information is then combined with a macroscopic description of the nonequilibrium transport problem. This combination of methods allows, for the first time, the direct prediction from atomistic models


of macroscopic properties such as the flux across a membrane over a large range of temperatures and pressures. This framework provides a powerful new method for achieving a detailed understanding of the performance and capabilities of zeolite membranes. We conclude by commenting on restrictions that currently apply to our methods and on directions for future extensions of this work. One serious restriction on the methods developed here is that they apply only to single-crystal membranes. This statement also applies to previous theoretical models of zeolite membranes. In contrast, all practical zeolite membranes to date are polycrystalline. The two model membranes examined here were chosen because their atomistic models had been thoroughly developed in the author’s previous work, and it is not possible to directly compare the results for these membranes to experimental measurements. As such, it seems premature to undertake a detailed discussion of the extension of current theories to polycrystalline membranes. We only note that the interplay between transport through zeolite pores and effects due to membrane polycrystallinity are difficult to unambiguously characterize experimentally. The existence of theoretical methods that can make detailed predictions regarding single-crystal membranes should provide a useful tool in future studies of these issues. Our current work is focusing on extending our theoretical methods to systems for which direct comparisons can be made with experimental data from polycrystalline membranes36 and single-crystal membranes.61 All of the results presented here deal with singlecomponent systems. In practical applications of membranes, it is obviously necessary to deal with multicomponent systems. Although multicomponent systems are necessarily more complex than the single-component case, there is no intrinsic conceptual barrier to extending the methods presented here to multicomponent systems. One useful formulation of the macroscopic multicomponent transport problem uses Onsager transport coefficients.14 In this case, one must determine the multicomponent adsorption isotherms and a symmetric matrix of loading-dependent transport coefficients from atomistic simulations to apply the methods of this paper. GCMC has been used successfully to perform atomistic simulations of adsorbed binary mixtures.41,42,44 Sanborn and Snurr have recently demonstrated that Onsager transport coefficients for binary adsorbed mixtures in zeolite pores can be computed for atomistic models using equilibrium MD simulations.51 As in previous macroscopic models of zeolite membranes, the results presented here have assumed local thermodynamic equilibrium at the zeolite membrane interfaces.6-10 An important advantage of methods based on atomistic models for the adsorbate and zeolite compared to empirical macroscopic methods is that this assumption can, in principle, be tested by performing simulations of the membrane interface. Developing a detailed understanding of barriers to mass transfer at zeolite membrane interfaces using atomistic simulations will be an important area for future work in this area. Finally, we note that only steady-state situations have been examined in this paper. There are obviously situations in which transient behaviors are important.62 It is straightforward to extend the methods developed here to the modeling of transient results.

Acknowledgment This work was partially supported by the donors of the Petroleum Research Fund, administered by the ACS, and by the Carnegie Mellon Faculty Development Fund. The assistance of J. Karl Johnson, Sivakumar Challa, and Qinyu Wang with the GCMC code is gratefully acknowledged, as are helpful discussions with Anastasios Skoulidas. Literature Cited (1) Matsukata, M.; Kikuchi, E. Zeolitic membranes: Synthesis, Properties, and Prospects. Bull. Chem. Soc. Jpn. 1997, 70, 2341. (2) Coronas, J.; Santamaria, J. Separations Using Zeolite Membranes. Sep.Purif. Methods 1999, 28, 127. (3) Shiflett, M. B.; Foley, H. C. Ultrasonic deposition of highselectivity nanoporous carbon membranes. Science 1999, 285, 1902--1905. (4) Kesting, R. E.; Fritzsche, A. K. Polymeric Gas Separation Membranes; John Wiley and Sons: New York, 1993. (5) Meier, W. M.; Olson, D. H. Atlas of Zeolite Structure Types; Butterworths: London, 1987. (6) Krishna, R. Multicomponent Surface Diffusion of Adsorbed Species: A Description Based on the Generalized Maxwell-Stefan Equations. Chem. Eng. Sci. 1990, 45, 1779. (7) Krishna, R. Problems and Pitfalls in the Use of the Fick Formulation for Intraparticle Diffusion. Chem. Eng. Sci. 1993, 48, 845. (8) Kapteijn, F.; Bakker, W. J. W.; Zheng, G.; Poppe, J.; Moulijn, J. A. Permeation and separation of light hydrocarbons through a silicalite-1 membrane: 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) 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. (12) Takaba, H.; Koshita, R.; Mizukami, K.; Oumi, Y.; Ito, N.; Kubo, M.; Fahmi, A.; Miyamoto, A. Molecular dynamics simulation of iso and n-butane permeations through a ZSM-5 type silicalite membrane. J. Membr. Sci. 1997, 134, 127-139. (13) Pohl, P. I.; Heffelfinger, G. S. Massively parallel molecular dynamics simulation of gas permeation across porous silica membranes. J. Membr. Sci. 1999, 155, 1-7. (14) 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. (15) Curtiss, C. F.; Bird, R. B. Multicomponent diffusion. Ind. Eng. Chem. Res. 1999, 38, 2515. (16) Benes, N.; Verweij, H. Comparison of macro- and microscopic theories describing multicomponent mass transport in microporous media. Langmuir 1999, 15, 8292. (17) 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, 1991; p 257. (18) Ka¨rger, J.; Caro, J. Interpretation and Correlation of Zeolitic Diffusivities Obtained from Nuclear Magnetic Resonance and Sorption Experiments. J. Chem. Soc., Faraday Trans. 1977, 73, 1363. (19) Buelow, M.; Struve, P.; Finger, G.; Redszus, C.; Ehrhardt, K.; Schirmer, W.; Ka¨rger, J. Sorption kinetics of n-hexane on MgA zeolites of different crystal sizes. J. Chem. Soc., Faraday Trans. 1980, 76, 597. (20) Ka¨rger, J.; Pfeifer, H. Nuclear magnetic resonance measurement of mass transfer in molecular sieve crystallites. J. Chem. Soc., Faraday Trans. 1991, 87, 1989. (21) Micke, A.; Struve, P.; Bu¨elow, M.; Kocirı´k, M.; Zika´nova´, A. Sorption kinetics of p-ethyltoluene in NaH-ZSM-5 crystalss simultaneous effects of intracrystalline diffusion and a mass transport resistance within the crystal surface. Collect. Czech. Chem. Commun. 1994, 59, 1525.

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Received for review March 8, 2000 Revised manuscript received July 5, 2000 Accepted July 11, 2000 IE000301H

Predicting Single-Component Permeance through Macroscopic

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