Single-Component Permeation Maximum with Respect to

Ind. Eng. Chem. Res. 2006, 45, 5501-5511

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Single-Component Permeation Maximum with Respect to Temperature: A Lattice Density Functional Theory Study Daniel Matuszak, Gregory L. Aranovich, and Marc D. Donohue* Department of Chemical and Biomolecular Engineering, The Johns Hopkins UniVersity, Baltimore, Maryland 21218

Membrane permeability and flux of pure gases can exhibit maxima with respect to temperature. For zeolites, this has been explained as a competition between surface and nonsurface diffusion within pores and as a process that depends on the diffusive activation energy and the heat of adsorption. This behavior is reproduced for nanoscale pores by using the lattice density functional theory approach for modeling diffusion. The approach can give expressions for the permeability of noncondensable fluids through nanoscale pores in terms of bulk densities and intermolecular interactions; it also gives the following molecular explanation for the permeation maximum: with decreasing temperature, (i) attractions at the pore’s entrance increase permeation because molecules in the pore experience difficulty in back-diffusing to the feed, and (ii) at even lower temperatures, attractions at the pore’s exit reduce permeation because molecules in the pore experience difficulty in escaping from the walls at the end of the pore. The approach shows that permeation maxima can occur without competition between surface and nonsurface diffusion. However, when this competition occurs, the maximum in the surface flux leads to the overall permeation maximum with respect to temperature. 1. Introduction The temperature dependence of membrane permeability can be considered to be a diagnostic tool, at least for zeolites and foam films.1,2 The permeability can increase monotonically with temperature, or it can exhibit a maximum, minimum, or both.3-5 The general trend is illustrated in Figure 1, at least for zeolites membranes. For zeolite membranes, single-component permeation trends have been explained in terms of (i) a competition between gas and surface diffusion,6 (ii) a competition between the equilibrium amount adsorbed and the diffusivity,7 and (iii) the ratio of the diffusive activation energy to the heat of adsorption.1,8 For foam films, the permeability of gases is explained in terms of a competition of resistances between a surfactant monolayer and a central aqueous core,2 and of factors that influence the surfactant adsorption density and the film thickness.9 The molecular mechanisms that cause these permeation trends, however, are not always obvious, even for weakly adsorbing gases in zeolites;10 for Newton black foam films stabilized by dodecyl maltoside, the temperature dependence of permeability still is considered “irregular”.9 Therefore, modeling can make a contribution by filling the gap between the macroscopic performance and molecular characteristics of the membranes.11 Besides simulations based on molecular dynamics and Monte Carlo, analytical approaches have been used to study the selfdiffusion behavior within zeolites; these include the MaxwellStefan formulation12,13 and the mean-field theory by Coppens et al.14 Due to the range of industrial applications of zeolites, much focus has been placed on the sorbate-loading dependence of the diffusivity.15 As for foam films, in addition to experiments, a mean-field theory analogous to that of van der Waals has been developed to describe (phase) changes of film thickness.16,17 However, few theoretical studies have focused directly on permeation trends with respect to temperature. The most relevant theoretical study that we have found addresses * To whom correspondence should be addressed. E-mail: mdd@ jhu.edu. Tel.: (410) 516-7761. Fax: (410) 516-5510.

Figure 1. Illustration of the typical permeation trend with respect to temperature for pure gases through zeolite membranes.6 The diffusive flux Jˆ (diffusive flow per area) is the ordinate.

zeolites and uses a Langmuirian kinetic model, explaining the permeation maximum with respect to temperature as a competition between the diffusivity and the amount adsorbed.7 We have not found analogous theoretical studies for foam films. Furthermore, there are other examples of multicomponent permeability (and selectivity) behavior whose temperature dependence may be clarified by theoretical modeling.18,19 The primary objectives of this paper are (i) to provide a simple theoretical modeling approach based on lattice density functional theory (LDFT), (ii) to demonstrate this approach by reproducing and analyzing permeation trends with respect to temperature, and (iii) to propose a molecular explanation for permeation maxima that are observed with respect to temperature at least in zeolites. After presentation of the approach, this paper is limited to examining simple nanoporous membranes, which represent zeolites. These membranes exhibit permeation maxima and minima with respect to temperature, as do foam films. We do not claim that the molecular explanations are the same for foam films, but for modeling foam films and other systems, the simple theoretical approach is the same. Density functional theory (DFT) approaches have become increasingly important for describing dynamic systems, generally by combining a transport equation with equilibrium DFT.20-23 As an extension of equilibrium LDFT,24-35 we have developed36 an LDFT approach for modeling diffusion that gives correct

10.1021/ie051039l CCC: $33.50 © 2006 American Chemical Society Published on Web 02/18/2006

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Ind. Eng. Chem. Res., Vol. 45, No. 16, 2006

equilibrium limits, Fickian behavior with correct departures for interacting systems, and qualitatively correct, nonequilibrium phase behavior.36 It can be shown37 that the LDFT approach is consistent with the Giacomin-Lebowitz-Marra theory38 of nonequilibrium phase transitions. Furthermore, the LDFT approach gives the correct formulation of the Maxwell-Stefan equations for systems with more than two components (ref 39 shows a detailed example for a quaternary system). The LDFT approach can give quantitative agreement with molecular dynamics simulations, at least for interdiffusing WeeksChandler-Anderson fluids in an external field.40 Last, the LDFT approach correctly gives a vanishing diffusive flux at the critical point and agreement with the theory of nonequilibrium thermodynamics.41 The subsequent sections (1) provide the LDFT equations for diffusion, (2) reproduce the mentioned permeation trends using simple models of nanopore permeation, (3) show how to obtain analytical expressions for the flux and permeability of noncondensable fluids in terms of the bulk densities and the intermolecular interactions, (4) explain the permeation trends on a molecular level, (5) show the influence of adsorption and diffusion along the outer surfaces of the membrane, and (6) demonstrate the competition between gas and surface diffusion for pores that are wider than one molecular diameter.

Figure 2. Illustration of a pore and spherically symmetric noncondensable penetrant molecules. Pores are perfectly straight with a diameter of approximately one penetrant diameter and with a length of N penetrant diameters. Penetrant molecules having a diameter larger than the pore diameter experience repulsion within the pore relative to the bulk phase; otherwise, they experience attraction within the pore.

2. LDFT Equations for Diffusion

Equation 3 is analogous to the Metropolis Monte Carlo algorithm, in which one molecule at site r is moved to a vacant rs site s with a probability equal to eEˆ n . However, this work is not based on Monte Carlo dynamics: molecular motion occurs rs throughout the system, and the terms Fn, eEˆ n , and τnin eq 2 are averaged quantitiessreaders interested in describing Fn with ensemble averages can do so for equilibrium as in the literature,42,43 but in this paper, the variables are considered to be time averages. The subsequent section applies eq 1-5 in a model of a nanoporous membrane.

The LDFT equations for diffusion can be summarized in the following molecular balance at the general lattice site r,

3. Simple Nanoporous Membranes

∂Fn ∂t

|

z

)

r

∑s (jsrn - jrsn )

(1)

where the number density is limited to 0 e Fn(r) e 1. Onedirectional molecular flows (in units of molecules per time) occur between sites r and s in the lattice reference frame,

j rs n )

1

Fn(r)[1 -

zτn(r)

Fm(s)]eEˆ ∑ m

rs

n

(2)

which is related to the flux (defined as the flow per crosssectional area) of species n between sites r and s; the characteristic time τn is related to the mean lifetime of n-type molecules at site r; and the functional Eˆ rs n defines the influence of the reduced potential-energy field on the migration of species n from site r to site s,

Eˆ rs n )

{

Eˆ n(r) > Eˆ n(s) Eˆ n(r) e Eˆ n(s)

0 -|Eˆ n(r) - Eˆ n(s)|

(3)

where

Eˆ n(i) )

n(i) + φn(i) ; i ) r,s kT(i)

(4)

The term φn(r) is a particular long-range interaction or an external potential energy applied to molecules at site r, and the intermolecular interaction energy n(r) is a sum of nearestneighbor contributions. For a system with only first nearestneighbor interactions, the interaction energy can be written z

n(r) )

nm∑Fm(s) ∑ m s

(5)

In many instances, membranes have pore diameters that are on the order of one penetrant molecule. In MFI-type zeolite membranes, for example, a p-xylene isomer enters the pore freely, whereas o-xylene feels a slight repulsion at the pore entrance due to its slightly larger diameter.44 This sieving ability, combined with molecule-surface attraction, affects the temperature dependence of the permeability. The typical permeability trend with respect to temperature for pure gases through zeolite membranes6 is illustrated in Figure 1. Beginning at a low temperature, the permeability increases with respect to temperature; it passes through a permeation maximum, after which it can pass through a minimum. This paper focuses on the permeation maximum. It has been proposed1,8 that this permeation maximum occurs when 0 < |Ed| < |Qa| where Ed is the activation energy for diffusion and Qa is the heat of adsorption. To complement and add resolution to the available macroscopic explanations, the following sections reproduce and explain single-component permeation maxima with respect to temperature by using the LDFT approach. Section 3.1 describes an analytical model of a straight nanopore that has a diameter of approximately one penetrant molecule. Section 3.2 describes a model of straight nanopores with larger diameters. The models were chosen for their simplicity and ability to capture the essential physics, allowing reproduction and insight of the vapor permeation maxima with respect to temperature. 3.1. Analytical Model of a Simple Nanopore. Figure 2 illustrates the modeled system. The model is constrained to puregas permeation, steady-state, and isothermal conditions. Equations 1-5 are used to describe the diffusion; there is no applied potential energy, φ ) 0; and penetrant’s characteristic time τ is a function of temperature. At this point, it is not necessary to specify the dimensionality. The pore is straight with a length of N penetrant diameters and with a diameter of approximately one penetrant molecule. Penetrant molecules having a diameter larger than the pore

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diameter (as illustrated) experience repulsion within the pores there is an increase in the energy when a molecule enters the pore from the site immediately outside the pore, i.e., ∆ > 0, where (r) is the potential energy at a general site r and has the same meaning as the potential energy defined by eq 5. The entry mechanism for repulsive pores is the same as that in the surface barrier effect: a fraction of molecules striking the open pore mouth have enough energy to overcome the barrier imposed by the membrane.45 On the other hand, penetrant molecules can experience attraction within the pore if the penetrant diameter is smaller than the pore diameter; in this case, ∆ < 0. The penetrant can adsorb and diffuse along the surfaces outside the pore. Except for size-exclusion, penetrant molecules do not interact with each other; the only contributions to ∆ come from surfaces. Except at the entrance and exit, the potential energy gradient ∇ ) 0 inside this pore. In this zone, the one-directional diffusive flows defined by eq 2 are jrs ) (1/zτ)F(r)[1 - F(s)] and jsr ) (1/zτ)F(s)[1 - F(r)], producing the net flow (jrs - jsr) of Jrs ) (1/zτ)[F(r) - F(s)]. For a constant net flow in a pore of N sites, one can combine N - 1 such equations to write

J)

[

]

1 F(1) - F(N) zτ N-1

(6)

where J is the net (diffusive) flow through the pore and F(1) and F(N), respectively, are the number densities at the first and last sites within the pore. Equation 6 is a version of Fick’s first lawsfor no gradients in the potential energy, the diffusivity is constant. At the pore boundaries, however, the potential energy can change (|∆| g 0) as stated earlier. The pore is considered to be repulsive when ∆ > 0 and to be attractive when ∆ < 0, where ∆ ) [(1) - (0)] ) [(N) - (N + 1)]; the sites 0 and N + 1 are located immediately outside the pore. Equation 2 can be used to write the net flow at both boundaries of a repulsive pore as

J) J)

1 {F(0)[1 - F(1)]β - F(1)[1 - F(0)]} zτ

(7)

where a Boltzmann factor is defined as β ) e-∆/kT for convenience and simplicity of the final result. Note that 0 < β < 1 in this case, acting as a resistance to molecules entering the pore from sites 0 and N + 1; in general, β can have any positive value. On the other hand, for an attractive pore with the same magnitude of ∆,

J)

{

{

1 1 F(0)[1 - F(1)] - F(1)[1 - F(0)] zτ β

}

1 1 F(N)[1 - F(N + 1)] - F(N + 1)[1 - F(N)] zτ β

( )[

(9)

}

(10)

where 0 < (1/β) < 1 for an attractive pore. The term (1/β) acts as a resistance to molecules leaving the pore from sites 1 and N. The net diffusive flow through the pore may be written in terms of the outer densities F(0) and F(N + 1), by using either eqs 7 and 8 or eqs 9 and 10 to replace the variables F(1) and F(N) in eq 6; one equation suits both attractive and repulsive pores:

]

F(0)β F(N + 1)β 1 1 zτ N - 1 1 - (1 - β)F(0) 1 - (1 - β)F(N + 1) 10 1 1 1 1+ e-Eˆ + N - 1 1 - (1 - β)F(0) 1 - (1 - β)F(N + 1)

{ ( )[

] }

(11)

where Eˆ 10 has the form of eq 3. It should be noted that the outer densities F(0) and F(N + 1) do not necessarily equal the bulk densities due to surface diffusion and adsorption that can occur outside the pore. This can be modeled with the LDFT approach, as shown in a subsequent section. Instead, we limit this section to a well-stirred bulk phase and a membrane with a low porosity, such that the outer densities approximately are equal to the bulk densities. This is the simplest model that demonstrates the permeation maximum with respect to temperaturesit can include but does not require adsorption and diffusion along the outer surfaces of the pore. Most experimental measurements of the permeation maximum, however, are performed at a constant pressure drop across the membrane. The pressure for a noninteracting lattice fluid can be written40,46 P ) -(kT/b) ln(1 - F), where b is the excluded volume and is equal to the lattice size λ3. Therefore, the pressure difference across the membrane is

P(0) - P(L) )

[

]

kT 1 - F(N + 1) ln b 1 - F(0)

(12)

For noninteracting molecules in a well-stirred bulk phase (or in the one-dimensional treatment of this problem), it is a fair approximation that the measurable pressure drop across the membrane ∆P = P(0) - P(L). Thus, combining eqs 11 and 12 allows one to formulate the net flux Jˆ (the net diffusive flow per area) in the form

J Pe = ∆P λ2 λN

Jˆ )

(13)

where P e is the effective permeability coefficient;47 in this case Pe )

1 {F(N)[1 - F(N + 1)] - F(N + 1)[1 - F(N)]β} (8) zτ

J)

J)

( )[ ]{ (

λ2

[

N N-1

1 - F(N + 1) zτkT ln 1 - F(0)

]

F(0)β F(N + 1)β 1 - (1 - β)F(0) 1 - (1 - β)F(N + 1)

)[

1 1+ N-1

] }

10 1 1 + e-Eˆ 1 - (1 - β)F(0) 1 - (1 - β)F(N + 1)

(14)

Thus, one can obtain analytical expressions for the flux and permeability in terms of the bulk densities and intermolecular interactions. 3.1.1. Results of Analytical Model. Previous research has discussed that the permeation minima occurring with increasing temperature are due to an increasing flux in the Knudsen limit.8,10 The flux increases as a result of an increasing molecular velocity λ/τ, where λ is the characteristic length and the characteristic time τ is a function of temperature, e.g., for gases at low density τ ) λxπµ/8RT where µ is the molecular mass. Instead, this paper is concerned with the permeation maxima with respect to temperature; for this reason, a model is not proposed for τ(T) other than that it is a monotonic function of temperature. The permeation maxima are examined by studying diffusive flow in the dimensionless form Jh ) zτJ. Figures 3 and 4 show the reduced diffusive flow Jh with respect to temperature for a constant pressure drop of 100 kPa across a pore that is 5000 penetrant diameters in length; this length is equivalent to a membrane thickness of 1.9 µm for

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Figure 3. Temperature dependence of the reduced diffusive flow for a simple nanopore in which molecules feel an attractive potential energy (∆/k < 0 K) relative to the bulk phase. The pore length was arbitrarily set to N ) 5000, which is equivalent to a length of 1.9 µm for λ ) 3.8 Å. A constant pressure difference of 100 kPa is imposed across the pore. The subscript L represents the stage N + 1.

Figure 4. Temperature dependence of the reduced diffusive flow for a simple nanopore in which molecules feel a repulsive potential energy (∆/k > 0 K) relative to the bulk phase. As in Figure 3, the pore length is N ) 5000, the pressure difference is constant at 100 kPa, and the subscript L represents the stage N + 1.

λ ) 3.8 Å. Under these conditions, the diffusive flow and the permeability exhibit identical trends at a constant pressure difference, as shown in eq 13. The negative ∆/k in Figure 3 indicates an attractive pore relative to the bulk phases; the positive ∆/k in Figure 4 indicates a repulsive pore relative to the bulk phases, where k is Boltzmann’s constant. Figure 3 shows qualitatively correct permeation maxima for attractive pores.1,6 Figure 4 also shows permeation maxima, but for repulsive pores. The data in Figures 3 and 4 can be shown on one figure by plotting the reduced diffusive flow versus the dimensionless group ∆/kT, as shown for different pore lengths in Figure 5. Figure 5 shows permeation maxima for both attractive and repulsive pores. However, these maxima can be misleading because much of this behavior is caused by the variability in bulk-phase densitysalthough the pressure drop ∆P is held constant, the density difference ∆F is a function of temperature as can be seen in eq 12. For example, as T f ∞ at a constant ∆P, it follows that ∆F f 0 and, therefore, the reduced diffusive flow approaches zero. For the same reason, Figures 3-5 additionally are misleading because they show a vanishing reduced diffusive flow in the high-temperature limit. This can be avoided by showing data at a constant density difference rather than a constant pressure difference. Figure 6 shows the reduced diffusive flow as a function of ∆/kT for a constant density difference ∆F, not a constant pressure difference. Permeation maxima occur only for attractive pores that are longer than one molecular diameter (N > 1). Permeation maxima do not occur for repulsive pores at a

Figure 5. Reduced diffusive flow at a constant pressure difference (100 kPa) versus the dimensionless group ∆/kT, for various values of pore length (N). The term ∆ < 0 for pores that are attractive with respect to the bulk phase, and ∆ > 0 for pores that are repulsive.

Figure 6. Reduced diffusive flow at a constant density difference versus the dimensionless group ∆/kT, for various values of pore length (N). For all values of ∆/kT, the feed density is constant at F0 ) 0.00131 and the permeate density is constant at FL ) 0. Permeation maxima occur only for attractive pores.

constant density difference. The transition between Figures 5 and 6 suggests that variations in the bulk phase are responsible for the maxima in Figure 4 and, correspondingly, in the repulsive range of Figure 5. This may be an additional explanation for experimentally observed permeation maxima at a constant pressure difference: a variable bulk density suppresses permeation in the high-temperature limit, and strong forces (attractive or repulsive) of the membrane suppress permeation in the lowtemperature limit; a permeation maximum occurs at intermediate temperatures. Figure 6 shows a discontinuity in the slope at ∆/kT ) 0. This is due to the mechanism change between attractive and repulsive regimes; i.e., at a finite temperature, a small reduction of pore diameter to a value less than one molecular diameter leads to repulsion, which causes a sharp decrease in the diffusive flow. A small increase of pore diameter to value slightly larger than one molecular diameter leads to attraction, which can cause an increase in the diffusive flow. For attractive pores that are one molecular diameter in length (N ) 1), a permeation maximum does not occur; the molecular explanation in the next section clarifies the reason. For attractive pores that are longer than one molecular diameter, a slight attraction enhances the flux and very strong attraction suppresses the flux; a permeation maximum occurs betweensthis is the permeation maximum that this paper attempts to explain on a molecular level; it is caused by the physics in the pore, not the variation of the bulk-phase densities. 3.1.2. Molecular Explanation of Maxima for Attractive Pores. To obtain a molecular explanation of the permeation maxima shown in Figures 3 and 6 for attractive pores, this


section reconsiders the previous derivation, conveniently using F(N + 1) ) 0 and labeling the Boltzmann factors as β1 and βN to indicate the first and last sites of the pore, respectively. For example, eqs 9 and 10 are J ) (1/6τ){F(0)[1 - F(1)] - F(1)[1 - F(0)](1/β1)} and J ) (1/6τ)F(N)(1/βN). The factors 1/β1 and 1/βN are resistances to molecules leaving the pore from sites 1 and N, respectively, and they have the same value that is equal to or greater than unity; however, they are kept distinguishable. With this as the context, the net diffusive flow for attractive pores is

[

]

F(0) 1 1 zτ N - 1 1 1 + 1F(0) β1 β1 J) 1 1 + βN 1+ 1 N-1 1 + 1F(0) β1 β1

(

{

(

)

)

[

(

)

(

)

]}

(15)

which is a particular case of eq 11. For very strong attraction within the pore relative to the bulk phase (β1 ) βN f ∞), the diffusive flow J f 1/(zτβN). Resistance at the last stage (1/βN) impedes molecules from exiting the pore, suppressing permeation. Molecules attempting to escape are compelled to return to the pore’s interior. Terms containing β1 do not contribute in this limit. On the other hand, for very weak attraction within pores (β1 ) βN ≈ 1), eq 15 can be expanded as a combination of Taylor series centered at β1 ) 1 and βN ) 1,

1 J) zτ

{

(

F(0) + F(0)[1 - F(0)] 1 N-1

)

}

1 + ... β1

(16)

where the ellipsis includes terms that are negligible when β1 ) βN ≈ 1 and N . 1. Resistance at the first stage (1/β1) impedes molecules from returning to the feed, enhancing permeation. Molecules attempting to escape are inclined to return to the pore’s interior. Terms containing βN do not contribute in this limit. This analysis indicates that the molecular explanation for the permeation maximum in attractive pores is the following: (i) An enhanced permeation with decreasing temperature can be attributed to the energetic barrier in going from the pore entrance back to the upstream bulk phase; i.e., attractions at the pore’s entrance (relative to the bulk phase) increase permeation because molecules in the pore experience difficulty in back-diffusing to the feed phase. (ii) At lower temperatures, a decreased permeation with decreasing temperature can be attributed to the energetic barrier between the pore exit and the downstream bulk phase; i.e., attractions at the pore’s exit (relative to the bulk phase) reduce permeation because molecules in the pore experience difficulty in escaping from the last stage. Between (i) and (ii), the permeation maximum occurs. The influence of the energetic barrier in item (ii) is the same as the pore exit effects described by Arya et al.48 Although the energetic resistances |∆/kT| to exiting the pore from the first and last stages are equal, the relative importance of these resistances can vary. For example, the upstream resistance is more important relative to the downstream resistance in part (i) of this explanation. The physical reason for this is that the density at the first stage is greater than the density at the last stage, within the temperature range of part (i), placing more importance on the physics associated with back-diffusion to the feed phase. Thus, the upstream resistance to back-diffusion enhances permeation (while the downstream resistance attempts

Figure 7. Reduced diffusive flow at a constant density difference versus the dimensionless group ∆/kT, for various values of pore length (N). For all values of ∆/kT, the feed density is constant at F0 ) 0.73 and the permeate density is constant at FL ) 0.72. A density of 0.73 corresponds to 100 MPa for λ ) 3.8 Å.

to suppress permeation) within the temperature range in part (i) of this explanation. If the density at the last stage was increased (e.g., by reducing the pore length), the physics associated with back-diffusion to the feed phase would become relatively less important (until the downstream resistance becomes dominant). The limiting case is that of N ) 1. For N ) 1, the pore has one stage; i.e., the densities at the “first” and “last” stage are equal since they are at the same location. There is less importance on the upstream resistance (which attempts to enhance permeation with decreasing temperature) than there is on the downstream resistance. Thus, permeation decreases with decreasing temperature for N ) 1, as shown in Figure 6. Another case in which the upstream resistance equals but is less important than the downstream resistance occurs at lower temperatures in the second part (ii) of the explanation. In this case, penetrant molecules can enter the pore from the feed phase much more rapidly than they can escape from either side. As such, the density gradient is small within the pore. A reduction in temperature suppresses permeation (as described for N ) 1 in the previous paragraph) due to the downstream resistance. 3.1.3. Possible Maxima for Repulsive Pores. The previous results show that repulsive pores do not exhibit permeation maxima when there is a constant and small density difference ∆F at low densities. However, the molecular explanation for attractive pores, as well as eq 11, suggests that such permeation maxima are possible for repulsive pores at higher densities. One example is shown in Figure 7. The molecular explanation for permeation maxima in repulsive pores is very similar to that of attractive pores: beginning at the high-temperature limit and decreasing in temperature, (i) permeation increases because molecules on the permeate side of the membrane experience resistance in re-entering the pore; at low enough temperature, (ii) permeation decreases with decreasing temperature because molecules in the feed experience resistance in entering the first stage of the pore. Between (i) and (ii), the permeation maximum occurs. The following analysis confirms the explanation. When tracking Boltzmann factors, as done earlier for the attractive pore, the net flux for repulsive pores can be written

J) F(0)β1 F(N + 1)βN 1 1 zτ N - 1 1 - (1 - β1)F(0) 1 - (1 - βN)F(N + 1) 1 1 1 1+ + N - 1 1 - (1 - β1)F(0) 1 - (1 - βN)F(N + 1) (17)

( )[

{ ( )[

]

]}

5506


which is a particular case of eq 11. The factors β1 and βN are equal but are kept distinguishable. For mild repulsion in the pore relative to the bulk phases (β1 ) βN ≈ 1), eq 17 can be expanded as a combination of Taylor series around the points β1 ) 1 and βN ) 1,

J)

( )( {

1 1 [F(0) - F(N + 1)] + zτ N - 1 F(N + 1)[1 - F(N + 1)](1 - βN) + ... (18) -F(0)[1 - F(0)](1 - β1)

} )

Equation 18 shows that the term which increases diffusive flow contains βN; i.e., the resistance at the last stage prevents the permeate from re-entering the pore. On the other hand, for strong repulsion relative to the bulk phases, β1 ) βN , 1. In this case, eq 17 can be expanded as a combination of Taylor series around the points β1 ) 0 and βN ) 0,

J)

( ){

}

F(0)β1 - F(N + 1)βN 1 1 + ... (19) zτ N - 1 [1 - F(0)][1 - F(N + 1)]

Equation 19 shows that the flux is suppressed by diminishing β1 for strong repulsions; i.e., increasing repulsion at the feed side of the pore suppresses the flux. 3.1.4. Heat of Adsorption versus the Diffusive Activation Energy. For solids with slitlike nanopores, adsorption increases with increasing porosity especially when intermolecular interactions create phase transitions in the adsorbed layer.49 If experimental adsorption behavior in porous solids is modeled in the framework of Langmuir’s isotherm, the energy of adsorption (obtainable from the Henry’s law range) can appear much greater than the adsorbate-surface energy and the activation energy for diffusion; i.e., it appears that 0 < |Ed| < |Qa|, which has been proposed as a criterion for the permeation maximum. This section demonstrates and explains such behavior within the context of this analytical model for nanopore permeation. Equilibrium LDFT can be used to describe the adsorption50 onto the outer (O) and inner (I) surfaces of the pore in Figure (2) as

F(i) )

KiF∞ (1 - F∞) + KiF∞

; i ) O,I

(20)

where F∞ is the density of the bulk phase that is in contact with the system. Equation 20 can describe systems with adsorbateadsorbate interactions, but for this analytical model KO ) exp(-AS/kT) and KI ) exp(-∆/kT). For attractive pores, both KO and KI are greater than unity. In the low bulk-density limit, eq 20 is Langmuirian. The energy of adsorption usually is determined in this limit. Empirically, the effective energy of adsorption can be determined by applying

Feff )

KeffF∞ 1 + KeffF∞

(21)

where Keff ) exp(Qa/kT) and Qa is the effective energy of adsorption.1 In the framework of eq 21, a full monolayer occurs on the surfaces when Feff ) 1. For a membrane of attractive pores with a pore-to-pore separation of W g 3, a pore width of approximately one penetrant diameter, and a pore length of N, the effective energy of adsorption can be determined by

combining eqs 20 and 21 with the material balance

Feff )

2(W - 1)F(O) + NF(I) 2W

(22)

where the denominator 2W represents the combined areas of full monolayers on the outer surfacessmonolayer formation on a flat surface is implied in the Langmuir adsorption isotherm characterized by eq 21. It follows that

lim Keff ) KO + F∞f0

N K 2W I

(23)

when W ≈ W - 1. In this limit, Keff ) KO for a nonporous membrane and Keff . KI for a highly porous membrane. Additionally, eq 23 shows that Keff usually is greater than both KO and KI for porous solids that attract molecules into their pores; i.e., usually, |Qa| > |AS| and |Qa| > |∆| for attractive pores. On the other hand, for a membrane of moderately or strongly repulsive pores (e.g., as in a dense solid with very low porosity), Keff ≈ KO and KI ≈ 0; adsorption occurs only on the outer surfaces giving |Qa| ≈ |AS| for such pores. The activation energy for diffusion is obtained by measuring the temperature dependence of the diffusivity and using D ) D0 exp(-Ed/kT).1 Within this analytical model of nanopore permeation, the activation energy for diffusion appears within the term β in eq 11; i.e., |Ed| ) |∆|. This can be seen from the low-density and high-N limit of eq 11. Thus, the analysis in this section shows that in the analytical model (section 3.1)

0 < |Ed| < |Qa| attractive pores 0 < |Qa| < |Ed| moderately/strongly repulsive pores This is in agreement with the currently accepted explanation1,8 because attractive pores exhibit permeation maxima and repulsive pores do not (at least for the low-density limit in which Ed and Qa are measured). 3.2. Numerical Model Containing Outer-Surface Diffusion and Wider Pores. Although many real materials have pores that are larger than one molecular diameter, the previous model for simple nanopores additionally is useful because it shows that a permeation maximum can occur without (i) Gibbs adsorption and diffusion on the outer surfaces of the membrane and without (ii) the competition between surface diffusion and gas diffusion within the pore. However, outer-surface diffusion and the gas-surface diffusion competition within the pore can affect the position of the permeation maximum. This section examines these factors, i and ii, by using the LDFT approach. Figure 8 illustrates the modeled system, which is an N × M pore positioned at the center of an L × W frame with cylindrical boundary conditions; i.e., molecular flux occurs between the top and bottom layers of the system, and the distance between adjacent pores is W. The model is constrained to pure-gas permeation, a steady-state, and isothermal conditions. Equations 1-5 are used to describe the diffusion; they are solved simultaneously for each site in the illustrated system. There is no applied potential energy, φ ) 0; the penetrant’s characteristic time τ is a monotonic function of temperature. The modeled system is two-dimensional. The model considers integer values of N and M, where M ∈ {1,2,3,4,5} allowing comparison of pores that have both surface and nonsurface diffusion and of pores that only have surface diffusion. The previous results for the narrow attractive pore of length N are retrieved exactly, regardless of the parameter W, by


Figure 8. General illustration of an N × M pore that is positioned at the center of an L × W frame with cylindrical boundary conditions. The distance between adjacent pore centers is W. As illustrated, N ) 5, M ) 3, L ) 12, and W ) 8.

solving eqs 1-5 while setting the following parameters:

M)1 L)N+2

F0 ) 0.00131 FL ) 0

Figure 9. Reduced diffusive flow at a constant density difference versus the dimensionless group ∆/kT for the system illustrated in Figure 8 using M ) 1, N ) 10, W ) 9, and different values of L. Gibbs adsorption and diffusion along the outer surfaces can occur for L ) 14, and they are switched on or off in the model. Gibbs adsorption and diffusion on the outer surfaces are not accounted for in the case of L ) 12 and in the analytical model (curve) are represented by the label “eq 11”, because the boundary conditions fix the outer-surface and pore-entrance densities.

(24)

where F0 is the number density set at the leftmost column (0) in Figure 8 and FL is the number density set at the rightmost column (L) in the same figure. 3.2.1. Gibbs Adsorption and Surface Diffusion Outside of the Pore. This section demonstrates the departure from the previously described analytical model upon introducing outersurface diffusion or molecule-surface attraction outside of the pore. This molecule-surface attraction leads to outer-surface densities in excess of those that occur when the penetrantsurface interaction energy is zero (AS ) 0). Since penetrant molecules do not interact with each other (AA ) 0) in this model, this excess adsorption is conveniently identified with the Gibbs adsorption. The modeled system uses N ) 10, M ) 1, W ) 9, and different values of L; these parameters are illustrated in Figure 8. As an initial test of the numerical model, the parameters are set as in eq 24, and the results are compared to the analytical model represented by eq 11. Then, the length of the frame is changed to L ) (N + 4) ) 14 so that Gibbs adsorption and diffusion can be incorporated on both outersurfaces of the membrane. To examine the influence of outersurface diffusion, the diffusive flows among the outer-surface sites were switched either “on” or “off” by manipulating the material balances at these sites, i.e., by adding or removing diffusive flows in eq 1. Gibbs adsorption at the outer surfaces (but not within the pore) also was examined by switching the penetrant-surface interaction energy AS “on” or “off” in eq 5. Figure 9 shows the reduced diffusive flow through the central cross section of the membrane; it is the analogue of Figure 6 for this numerical model. When L ) (N + 2) ) 12, as noted before eq 24, the numerical model retrieves the analytical results represented by eq 11 for the attractive pore. In this case, the boundary conditions constrain the outer-surface densities (and the pore entrance and exit densities) to F0 and FL. On the other hand, for L ) 14, the outer-surface densities can change due to adsorption and diffusion outside of the pore. In transitioning from L ) 12 to any case in which L ) 14, as shown in Figure 9, the reduced diffusive flow maintains the same dependence on ∆/kT as the analytical model. When Gibbs adsorption and outer-surface diffusion are absent (in this case, L ) 14, AS ) 0, surface densities are nonzero, and outer-surface flows are constrained to zero), the reduced diffusive flow has a smaller magnitude than in the analytical and numerical (L ) 12) models, as indicated by the squares in

Figure 9. The smaller magnitude of diffusive flow generally occurs for all cases of L ) 14 due to a lowered density at the pore entrance, in comparison with the case of L ) 12. Gibbs adsorption occurs in the absence of outer-surface diffusion when penetrant-surface attractions are switched on at the outer surfaces while constraining the outer-surface flows to zero. This does not cause a significant change in the reduced diffusive flow, as indicated by the triangles in Figure 9. However, when penetrant molecules are allowed to diffuse along the outer surfaces, the reduced diffusive flow is greater than when the outer-surface flows are constrained to zero, as indicated by the diamonds and stars in Figure 9. It appears that Gibbs adsorption does not change the reduced diffusive flow significantly, at least for the conditions studied heresthese systems are much less sensitive to the penetrant-surface interactions occurring outside of the pore than they are to outersurface diffusion. Therefore, for attractive membranes in which the pore diameter is approximately one penetrant diameter, outer-surface diffusion increases permeation without changing the behavior with respect to temperature. At least for the types of systems described in this section, attractive penetrant-surface interactions with the outer surfaces do not lead to a significant change in permeation when there is outer-surface diffusion. 3.2.2. Wider Pores. This section describes a two-dimensional model for porous membranes as illustrated in Figure 8 with the parameters N ) 10, L ) 14, W ) 9, and different values of M. The case of M ) 1 was described in the previous sections. Both adsorption and surface diffusion occur outside and inside the pore. The main objective of this section is to examine the competition between surface diffusion and gas (nonsurface) diffusion within the pore; this competition has been used to explain the permeation maxima with respect to temperature. Figure 10 shows permeation trends for pores with various pore diameters, indicated by the parameter M. All diffusive flows shown and discussed in this section occur through the central cross-sectional area of the pore. For the total flows and surface flows, increasing M leads to increased flows; on the other hand, changes in M do not influence the nonsurface flows significantly. The total flows and surface flows exhibit maxima with respect to the molecule-surface interaction AS/kT, which does not equal the ∆/kT used in the previous figures; e.g., when M ) 1, a two-dimensional pore has ∆ ) 2AS and a three-dimensional pore has ∆ ) 4AS, but ∆ is not meaningful when M > 1. As

5508


Figure 10. Reduced diffusive flows (total, surface, and nonsurface) at a constant density difference versus the dimensionless molecule-surface interaction energy AS/kT for the two-dimensional system illustrated in Figure 8 using N ) 10, L ) 14, W ) 9, and different values of M. Gibbs adsorption and diffusion along the outer surfaces occur in all trials. The nonsurface flows for M ) 3, 4, and 5 have the same shape and are insignificant in the range of the permeation maximum; trials with M ) 1 and 2 do not have nonsurface flows.

shown in Figure 10, in the competition between surface diffusion and gas (nonsurface) diffusion, the surface diffusion dominates in the range of the overall permeation maximum. However, the surface and nonsurface flows reverse their roles at greater (more negative) values of AS/kT that are too strong to be considered relevant to the permeation maximum. At least for the membranes examined in this section, a competition between surface and nonsurface diffusion occurs within the pores but it is not relevant to the overall permeation maximassurface diffusion dominates in the range of the overall permeation maxima. Permeation maxima occur overall and at the surfaces but not in the nonsurface “gas-phase” region. The nonsurface flows exhibit minima with respect to AS/kT. These permeation minima in the nonsurface regions are not the same as the permeation minima that occur at temperatures approaching the Knudsen limit as described earlier and illustrated in Figure 1. The following analysis shows that the minima in the (reduced) nonsurface flows are caused by diffusion occurring perpendicular to the pore length, mostly within the pore in the radial direction. Only the case of M ) 5 shown in Figure 10 is examined here; of the five regions parallel to the pore wall, one region contains the (central) axial flow, two regions contain surface flows, and two regions contain flows that are “medial” (in between the surface and the pore axis). When the radial flows within the pore are switched off by removal from the material balances in the numerical model, while all other flows remain on (including those at the outer surfaces of the membrane), the minima in the nonsurface (axial and medial) flows are much less significant than in the (2D) case in which radial flows occur, as shown in Figure 11. Furthermore, in the one-dimensional (1D) case when all the flows perpendicular to the pore length are switched off, the axial and medial flows are equal and constant with respect to temperature, as shown by the solid line in Figure 11. Therefore, radial diffusion within the pore is the major cause of the (reduced) nonsurface permeation minima with respect to temperature. 4. Discussion One useful explanation of membrane permeation maxima invokes the solution-diffusion mechanism;51 i.e., the competition between diffusivity and solubility may cause maxima, as demonstrated using a Langmuirian model for zeolites.7 However, if a permeation maximum can occur experimentally at high

Figure 11. Reduced nonsurface flows (axial and medial) at a constant density difference versus the dimensionless molecule-surface interaction energy AS/kT for the system illustrated in Figure 8 using N ) 10, L ) 14, W ) 9, and M ) 5. For M ) 5, there is one (central) axial flow and two medial flows (between the center and the surface). When fluxes (j) occur throughout the system in two dimensions (2D), these nonsurface flows have the same values as those shown in Figure 10. When the radial fluxes (jradial) inside the pore are turned off, these nonsurface flows exhibit a weaker permeation minimum than in the 2D case. In the one-dimensional (1D) case, which has no diffusion perpendicular to the pore length, both nonsurface flows are equal and constant with respect to AS/kT.

pressures in repulsive pores, as suggested by the analytical model in section 3.1.3 of this paper, then the concept of solution-diffusion competition would not be as useful because both the solubility and diffusivity are high at high temperatures and low at low temperatures. To test for such maxima, the model suggests using a dense solid membrane and a supercritical fluid at pressures on the order of 100 MPa. Some support of this comes from the simulations and model of Ford and Glandt, who show that steric hindrance at the entrance to small pores (i.e., the “surface barrier effect”) can lead to nonmonotonic behavior in the mass transfer coefficient.45 Ford and Glandt also describe the relevant experimental studies.45,52 Another macroscopic explanation uses the criterion that the heat of adsorption must be greater in magnitude than the activation energy for diffusion, 0 < |Ed| < |Qa|. At least for zeolites membranes operating with noncondensable fluids near atmospheric pressures, this criterion is very useful and (as shown in section 3.1.4 of this paper) it is in agreement with the molecular explanation that this paper proposes for the permeation maximum. The molecular explanation proposed in this work is based on the resistances to forward- and back-diffusion to the bulk phases. For this explanation, the system can include but does not require Gibbs adsorption or diffusion at the outer surfaces of the membrane; also, a competition between surface and gas (nonsurface) diffusion within the pore is not necessary. Gibbs adsorption and surface diffusion at the outer surface of the membrane generally occur in real systems. At least for the nanopores examined in section 3.1 of this paper, outersurface diffusion can enhance permeation significantly, regardless of Gibbs adsorption on these surfacessthese systems are relatively insensitive to the penetrant-surface interaction energy AS. When attractive penetrant-surface interactions are switched on or off on the outer surfaces, Gibbs adsorption changes but there is no significant change in the permeation rate. More importantly, for all the scenarios in which Gibbs adsorption and diffusion on the outer surfaces were turned on or off, the permeation trends with respect to temperature were the same; although the permeation rates are different, the maxima occur at the same temperature (i.e., at the same value of ∆/kT). Again, these permeation maxima occur in the absence of competition between surface and gas (nonsurface) diffusion.


Pores that are wider than a few molecular diameters can experience a competition between surface and nonsurface diffusion. If this competition causes the permeation maximum with respect to temperature, one would expect that surface diffusion dominates on one side of the maximum and gas (nonsurface) diffusion dominates on the other side. At least for the nanopores examined in section 3.2 of this paper, surface and nonsurface diffusion reverse their roles at temperatures that are outside of the range of the permeation maximum observed here. In the range of this permeation maximum, surface diffusion dominates the gas-surface competition. Furthermore, both the overall diffusive flow and the surface diffusive flow exhibit maxima in the same range of temperatures; on the other hand, the nonsurface flows exhibit permeation minima in the same temperature range. These are not the same minima that are shown in Figure 1 and that are caused by approaching the Knudsen limit. The permeation minima in the nonsurface regions (occurring in the range of the overall permeation maximum) are caused mainly by diffusion in the radial direction inside the pore. This would not be observed in a conventional model for pore diffusion in which the chemical potentials at the surface and gas (nonsurface) phases are set equal in each cross section.53 The molecular explanations for permeation maxima are not necessarily the same for both zeolites and foam films, but they may be similar. For example, the criterion of a constant pressure drop across a film may lead to a permeation maximum with respect to temperature because of the variability in the bulkphase densities, as demonstrated for nanopores in section 3.1.1 of this paper. Without additional modeling, we cannot be certain that the other molecular explanations are the same for foam films. This paper focuses on permeation maxima with respect to temperature, and it has avoided characterization of the permeation minima (in approaching the Knudsen limit) by examining the reduced diffusive flow, Jh ) zτJ. As mentioned in section 3.1.1 of this paper, the minimum occurs at high temperatures due the temperature dependence of the characteristic time τ(T), which is a monotonic function of temperature. We would like to note that both permeation maxima and minima can be described with the LDFT approach by incorporating a model for τ(T). In fact, Figure 1 was generated by using the analytical model for an attractive pore (section 3.1) using N ) 10, F0 ) 0.00131, FL ) 0, and the additional constraint that τ ) λxπµ/8RT where λ ) 3.8 Å and the mass µ ) 39.9 g/mol. Conclusions This paper describes and demonstrates applications of the LDFT approach for modeling diffusion on a molecular scale. This approach can provide molecular explanations for permeation trends that are not well understood, at least for nanoporous membranes. The analytical model in this work shows that permeation maxima at constant pressure drops can be caused by the variability of the bulk-phase densities with respect to temperature. To check if this prediction is correct, permeation should be measured experimentally as a function of temperature at a constant density difference across the membrane. At least for membranes in which the majority of permeation occurs through nanoscale pores, the pores of the membrane can be considered to be repulsive to the penetrant if maxima are not observed; otherwise, the conclusion depends on the bulk densities as follows. If permeation maxima are observed at relatively low densities (e.g., at densities of noncondensable gases at atmospheric conditions), the pores of the membrane

can be considered to be attractive to the penetrant. For very high pressures, a conclusion cannot be made. Most experimental observations of the permeation maxima of noncondensable gases are made near atmospheric pressures, and therefore, the criterion proposed in the literature (0 < |Ed| < |Qa|) is a useful onesit is in agreement with the proposed explanation that only attractive pores exhibit permeation maxima with respect to temperature at low bulk densities and with a constant density difference. Nanopores with diameters of approximately one penetrant molecule exhibit permeation maxima but do not have a competition between surface diffusion and gas (nonsurface) diffusion. For such nanopores, permeation increases with decreasing temperature because back-diffusion to the feed phase is suppressed; at lower temperatures, permeation decreases with decreasing temperature because diffusion toward the permeate side of the membrane is suppressed. The gas-surface diffusion competition occurs in wider nanopores. However, in the range of the permeation maximum with respect to temperature, the surface diffusive flux (flow per area) is much larger than the nonsurface flux, at least for the systems studied in this paper. Both the surface flux and the overall flux exhibit permeation maxima, but the nonsurface fluxes exhibit permeation minima due to radial diffusion within the pore. Acknowledgment Support by the U.S. Department of Energy, under Contract No. DE-FG02-87ER13777, is gratefully acknowledged. D.M. acknowledges Building Service 32BJ Thomas Shortman Training, Scholarship and Safety Fund for a graduate fellowship. Nomenclature b ) molecular volume Eˆ ) potential-energy functional E ) activation energy (for diffusion) j ) one-directional diffusive flow J ) (net) diffusive flow Jˆ ) (net) diffusive flux (flow/area) Jh ) (net) reduced diffusive flow k ) Boltzmann’s constant L ) frame length in numerical model M ) pore width N ) pore length P ) pressure Pe ) effective permeability Qa ) heat of adsorption r ) general lattice site s ) lattice site adjacent to site r t ) time T ) temperature z ) coordination number Greek Letters β ) energetic barrier at pore entrances ) intermolecular potential on lattice φ ) externally applied potential energy λ ) lattice spacing µ ) molecular mass F ) number density (dimensionless) τ ) characteristic time for diffusive flow Superscripts rs ) direction from site r toward site s sr ) direction from site s toward site r

5510


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ReceiVed for reView September 15, 2005 ReVised manuscript receiVed January 16, 2006 Accepted January 18, 2006 IE051039L

Single-Component Permeation Maximum with Respect to

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