Predicting the Adsorption of Gas Mixtures: Adsorbed Solution Theory

Langmuir 2002, 18, 10443-10454

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Predicting the Adsorption of Gas Mixtures: Adsorbed Solution Theory versus Classical Density Functional Theory M. B. Sweatman and N. Quirke* Department of Chemistry, Imperial College, South Kensington, London, SW7 2AY, United Kingdom Received January 10, 2002. In Final Form: October 9, 2002 Accurate prediction of the adsorption properties of fluid mixtures in equilibrium with surfaces and/or nanoporous structures is of considerable scientific and practical importance. Often, while the pure fluid adsorption isotherms are known for each component, those for the mixture are not. Using data from Monte Carlo simulations of model mixtures (including hydrogen and carbon dioxide) adsorbed in graphitic slit pores, for a range of pressures to 1000 bar, we compare theories for mixed adsorption which require pure fluid isotherm data as input. In particular, we develop and evaluate methods based on adsorbed solution theory (AST) and classical density functional theory (DFT). We find that a novel approximate DFT-based model is generally more accurate than AST methods in predicting the adsorption isotherms of mixtures of simple gases.

Introduction In nature most fluids are mixtures. Although the understanding of pure fluid properties is well advanced, that of fluid mixtures is less so, especially if the situation is complicated by the presence of a surface or a confining geometry such as a nanopore. The need to specify the bulk molar concentrations increases the number of degrees of freedom in the problem, making the mapping of the phase diagram of the adsorbed fluid an onerous task, especially if the number of species is large. Theories of adsorption that use adsorption isotherms for pure fluids (even those which might form nonideal bulk mixtures) to accurately predict the adsorption properties of mixtures of the component fluids would be of use both in probing the physical chemistry of mixed adsorption and in practical applications to, for example, adsorbent design. Adsorbents are widely used in industry for a variety of purposes, including the storage, catalysis, and separation of fluids. Our focus is on the formulation of an equilibrium theory that can predict the adsorption of each component of a fluid mixture from knowledge of isotherms of the pure fluid components. We call this the “mixture adsorption problem”. Because of its importance, the search for solutions to the mixture adsorption problem has a long history. The most significant development in this direction was the invention of “ideal adsorbed solution theory”1 (IAST) by Myers and Prausnitz in 1965. This theory does not require a model of the adsorbent or the adsorbates, yet it is accurate for some systems.1 Because of its simplicity and success, it is probably still the most widely used method to treat mixed adsorption, even though its accuracy deteriorates for increasingly nonideal systems.1-3 Other approaches have since been suggested with varying degrees of success. Some4-14 attempt to encode the (1) Myers, A. L.; Prausnitz, J. M. AIChE J. 1965, 11, 121. (2) Vuong, T.; Monson, P. A. Adsorption 1999, 5, 183. (3) Seidel, A.; Gelbin, D. Chem. Eng. Sci. 1988, 43, 79. (4) Sumanayuen, S.; Danner, R. P. AIChE J. 1980, 26, 76. (5) Cochran, T. W.; Kabel, R. L.; Danner, R. P. AIChE J. 1985, 31, 268. (6) DeGance, A. E. Fluid Phase Equilibria 1992, 78, 99. (7) Cheng, L. S.; Yang, R. T. Adsorption 1995, 1, 187.

behavior of adsorbent and adsorbates in terms of a sufficiently flexible mixture isotherm equation. Others15-17 attempt to separate description of the adsorbent and behavior of adsorbates by modeling the adsorbent in terms of a distribution of a specific measure (or measures). For example, the polydisperse slit-pore model attempts to characterize materials in terms of a distribution of ideal slit pores. The mixture problem is then solved “locally” at each value of the specific measure with a sum over the distribution giving the “global” solution. By consideration of this second route in the context of the ideal slit-pore geometry, a flaw in IAST was discovered and corrected by Cracknell and co-workers18 in 1995. In the present work, we take the view that employing explicit models for both adsorbent and adsorbates is likely to be more successful. We consider the mixture problem in the ideal slit-pore geometry and do not concern ourselves with the validity of this geometry with respect to a particular material or how a material can be characterized in terms of a distribution of slit pores. Nevertheless, these are important issues in this context under active investigation.19,20 Our aim is to provide a reliable method for solution of the mixture adsorption problem in slit pores, and we investigate three candidate methods. (8) Choudhury, N. V.; Jasra, R. V.; Bhat, S. G. T. Sep. Sci. Technol. 1995, 30, 2337. (9) Gusev, V.; O’Brien, J. A.; Jensen, C. R. C.; Seaton, N. A. AIChE J. 1996, 42, 2773. (10) Buss, E.; Heuchel, M. J. Chem. Soc., Faraday Trans. 1997, 93, 1621. (11) Staudt, R.; Dreisback, F.; Keller, J. U. Adsorption 1998, 4, 57. (12) Shapiro, A. A.; Stenby, E. H. J. Colloid Interface Sci. 1998, 201, 146. (13) Webster, C. E.; Drago, R. S. Microporous Mesoporous Mater. 1999, 33, 291. (14) Carsky, M.; Do, D. D. Adsorption 1999, 5, 183. (15) Valenzuela, D. P.; Myers, A. L.; Talu, O.; Zwiebel, I. AIChE J. 1988, 34, 397. (16) Hu, X.; Do, D. D. AIChE J. 1995, 41, 1585. (17) Qiao, S.; Wong, K.; Hu, X. Langmuir 2000, 16, 1292. (18) Cracknell, R. F.; Nicholson, D. Adsorption 1995, 1, 16. (19) Sweatman, M. B.; Quirke, N. J. Phys. Chem. B 2001, 105, 1403. (20) Ravikovitch, P. I.; Vishnyakov, A.; Russo, R.; Neimark, A. V. Langmuir 2000, 16, 2311.

10.1021/la0200358 CCC: $22.00 © 2002 American Chemical Society Published on Web 11/27/2002

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We extend previous work by providing a framework, nonideal adsorbed solution theory (NIAST), for solution of the mixture problem for nonideal fluids. We compare this approach with classical density functional theory21 (DFT). Both of these routes naturally require approximation. We propose a novel, simplified DFT approach for slit pores that consists of fitting bulk and adsorbed pure fluid isotherms generated by the model DFT to pure fluid reference data by adjusting model parameters that enter the DFT. The DFT is then solved, using the same model parameters, for the adsorbed mixture. This is essentially the same as using a multicomponent adsorption isotherm equation to solve the mixture problem in slit pores, except that we gain all the advantages of the DFT formalism. Compared to NIAST, an advantage of the DFT approach is its flexibility since the DFT can be solved, with the same model parameters, at any temperature or pressure. This contrasts with NIAST, which can be limited to application at the same temperature as the reference data over a limited pressure range. However, a possible advantage of the NIAST approach is that it does not require knowledge of the geometry of the adsorbent. We test the methods by comparing the predicted adsorption isotherms of three binary mixtures of simple gases in slit pores of various widths against simulated isotherms. We find that the DFT-based method is generally more accurate than approximations based on adsorbed solution theory and can be used even with quite nonideal mixtures. Theory In this section, we give an outline of DFT and show how NIAST is related to it. A series of approximations to these methods is then developed. Density Functional Theory. Density functional theory has played a major role in improving our understanding of fluid phenomena, including adsorption.22 The theory provides a route to the density profile of molecular centers, Fi(r b), from a given intrinsic Helmholtz free energy functional, F[{Fi(r b)}], for an arbitrary external potential, Vxi (r b), and chemical potential, µbi, for each component i. F is dependent only on fluid-fluid interactions, and in principle the formalism is exact. Although there do not yet exist any exact expressions for F for nontrivial threedimensional continuum fluids, DFT can be quite accurate23 for some simple pair-potential fluid interactions. The fundamental quantity in DFT is the grand potential functional,

Ω)F-

∑i ∫drbFi(rb)(µbi - Vxi (rb))

(1)

b), Upon functional minimization with respect to each Fi(r (1) yields the equilibrium Ω, F, and {Fi(r b)}. By splitting F into ideal and excess contributions, F ) Fid + Fex, where Fid is given by the exact relation

Fid ) kBT

∑i ∫drbFi(rb)(ln(Λi3Fi(rb)) - 1)

(2)

we obtain the Euler-Lagrange (EL) equations,

( (

ex b) ) Fbi exp -β Vxi (r b) - µbi + Fai(r

))

δFex δFai(r b)

(3)

(21) Evans, R. In Fundamentals of Inhomogeneous Fluids; Henderson, D., Ed.; Marcel Dekker: New York, 1992.

Here, β ) kBT, kB is Boltzmann’s constant, Λi is the thermal de Broglie wavelength of component i, and we use subscripts a and b to distinguish adsorbed and bulk phases. With an expression for Fex, eq 3 can be solved selfconsistently to give {Fai(r b)}. If Fex is dependent on {Fai(r b)} (and fixed adsorbate model parameters) only then it will be consistent with the Gibbs adsorption equation,24 which b)} is for constant T and {Vxi (r

dΩ + V dP )

∑i ∫dµi ∫Vdrb(Fbi - Fai(rb))

(4)

where P is the pressure and V is the volume of the system. In this work, we are interested in volume-averaged bFai(r b). So we rewrite (3) as quantities, Fai ) V-1 ∫V dr ex ex - µai )) Fai ) Fbi exp(β(µbi

(5)

where

(

ex µai ) ln V-1

(

))

ex

∫V drb exp δFδF(rb) + Vxi (rb) ai

(6)

With the exact relation for the excess chemical potential

βµi ) ln(FiΛi3) + βµex i

(7)

we see that the Euler-Lagrange equation is simply the inhomogeneous description of the fundamental equality of chemical potentials for phases in equilibrium. NIAST. From (7), the chemical potential of the ith component of a mixture, µi, is related to that of the pure fluid, µ0i , at the same T and {Vxi (r b)} by the exact relation

µi(Ω1,{x}) ) µ0i (Ω2i) + kBT ln(ximi(Ω1,Ω2i, {x})) (8) where

mi(Ω1,Ω2i,{x}) )

∑jNj(Ω1,{x}) exp(β(µex(Ω ,{x}) - µ0ex(Ω N0i (Ω2i)

i

1

i

2i)))

(9)

is the mixing coefficient of component i, F ) F∑ixi ) ∑iFi ) ∑iNi/V, xi and Ni are the concentration and number of particles of component i, and a superscript 0 indicates the pure fluid. We use this name and notation for the mixing coefficient rather than “activity” and γi25 to avoid confusion with the activity (which is related to chemical potential) and the surface tension, γ, used in other work.22 Ω1 is the grand potential of the fluid mixture, and Ω2i is the grand potential of the reference pure fluid state. Note that Ω2i can be different for each pure component. We choose the correct reference pure fluid state by drawing on the arguments presented by Rudisill and Levan.25 That is, we define an ideal mixture to exist in the low-density limit and to be defined by m ) 1. We then determine the (22) In Fluid Interfacial Phenomena; Croxton, C. A., Ed.; Wiley: Chichester, 1986. (23) Sweatman, M. B. Phys. Rev. E 2001, 63, 031102. (24) Sweatman, M. B. Mol. Phys. 2000, 98, 573. (25) Rudisill, E. N.; Levan, D. Chem. Eng. Sci. 1992, 47, 1239.

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Langmuir, Vol. 18, No. 26, 2002 10445

correct reference pure fluid state by demanding thermodynamic consistency. We show in the Appendix that this requires Ω2i ) Ω1. Substitution of (8) into (5) with Ω2i ) Ω1 gives the NIAST relations,

z0i (T,Ω,{Vxi (r b)})xaimai ) z0i (T,P)xbimbi

(10)

where the activity is

z0i ) Λi-3 exp(βµ0i )

(11)

and it is understood that Ω is the grand potential of the mixture and P is the pressure of the bulk mixture (for a bulk fluid Ω ) -PV). Because we have yet to make any approximations, the NIAST relations are equivalent to the Euler-Lagrange eqs 3. Thus, explicit information concerning the external potential, density profiles, thermodynamic properties, and model parameters is encapsulated within the equations of state (the activities) and the mixing coefficients. The NIAST equations relate adsorbed concentrations to bulk concentrations in terms of pure fluid equations of state and mixing coefficients. If for the moment we assume that all the mixing coefficients are known, then for fixed b)}, and {xb}, if we can determine all the pure T, Ω, {Vxi (r fluid adsorbed equations of state z0i (T, Ω, H) and bulk equations of state z0i (T, P), then the only remaining unknown quantities are {xa} and P. Since we also know that ∑ixai ) 1, we actually have n + 1 equations with n + 1 unknown quantities. So this set of equations can be solved by matrix inversion. However, the mixing coefficients are generally not known exactly and approximations for them must be sought. The adoption of a density functional prescription for the mixing coefficients will result in the equivalent density functional theory. This will require models for the adsorbent, the adsorbates, and the description of the fluid system in terms of density profiles. Approximate density functional mixing coefficients are not automatically guaranteed to be consistent with the Gibbs adsorption eq 4, re-expressed as

Ni-1 )

∑i

xi(N0i )-1 +

∑i

( )

kBTxi ∂mi mi

(12)

∂Ω

b)}. Adoption of an alternative preat fixed T and {Vxi (r scription for the mixing coefficients26-29 that does not require models for the adsorbent, the adsorbates, and the description of the fluid system in terms of density profiles is unlikely to be as accurate as a density functional prescription. Also, dropping the dependence of the mixture coefficients on density means that the Gibbs adsorption equation, (12), will be required to calculate adsorptions. Since this is a differential equation, solution of the NIAST equations along an isotherm is required, unless the righthand-most term of (12) is zero. Furthermore, it is possible that a non-density functional approach will restrict application of the NIAST equations to a pressure range and temperature dependent on the input equations of state. Finally, a non-density functional prescription for the mixing coefficients will probably require the use of (26) Buss, E. Chem. Tech. (Leipzig) 1996, 48, 189. (27) Van der Vaart, R.; Huiskes, C.; Bosch, H.; Reith, T. Adsorption 2000, 6, 311. (28) Sakuth, M.; Meyer, J.; Gmehling, J. Chem. Eng. Process. 1998, 37, 267. (29) Kabir, K.; Grevillot, G.; Tondeur, D. Chem. Eng. Sci. 1998, 53, 1639.

the Gibbs adsorption eq 4 to determine the pure fluid bulk and adsorbed equations of state. Nevertheless, the NIAST route does have two clear advantages over DFT: 1. In principle, if the bulk and adsorbed equations of state for the pure fluids and the mixing coefficients are already known, then models of the adsorbent, the adsorbates or description in terms of densities is not required, that is, these terms do not appear in eqs 4, 10, or 12. This makes the NIAST route attractive for application to real experiments on geometrically complex materials. However, adsorption experiments30 usually measure excess adsorption and so in practice the adsorbent geometry and adsorbent-adsorbate interactions will be required unless Fai . Fbi. 2. There are some simple approximations that allow easy solution of NIAST. These approximations, including ideal mixture adsorbed solution theory (IMAST) and IAST, are detailed below. Approximations This section details some approximation schemes for the NIAST and DFT approach that are computationally efficient. The reliability of these solutions is assessed in the results section. NIAST. Equation 10 is an exact relationship between the concentration of fluid components in the adsorbed phase and those in the bulk phase. Its solution requires knowledge of bulk and adsorbed equations of state for each pure component and all mixing coefficients. We assume that the pure fluid equations of state are known. So approximations for the mixing coefficients are required. The most obvious approximation is to set mai ) mbi, which gives the “equal mixing adsorbed solution theory” (EMAST),

z0i (T,Ω,H)xai ) z0i (T,P)xbi

(13)

This approximation sets interactions to be identical in the bulk and adsorbed phases. Clearly, this approximation will be less useful if adsorbent-adsorbate interactions significantly affect mixing in the adsorbed phase. So we can expect this approximation to be less accurate with increasingly different adsorbent-adsorbate interactions for each component. We note that it is not necessary to set mai ) mbi ) 1; that is, the fluid mixture is not approximated to be ideal (defined by m ) 1). EMAST simply requires mixing to be the same, ideal or not, in the bulk and adsorbed phases. Adsorption can be calculated from the Gibbs adsorption eq 12 provided the function mbi(Ω) is known. IMAST simply sets mai ) mbi ) 1. The advantage of this approximation is that bulk mixing coefficients are not required. It follows that IMAST can be solved at individual values of pressure; that is, integration along an isotherm is no longer required since the right-hand-most term in (12) is zero. IMAST will generate identical results for adsorbed phase concentrations and selectivities to EMAST, but its predictions for adsorption will generally be different. IAST can be obtained from IMAST by using the ideal gas approximation for z0i to give

P0i (T,Ω,H)xai ) Pxbi

(14)

(30) Ruthven, D. M. Principles of Adsorption and Adsorption Processes; Wiley: Chichester, 1984.

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The advantage of this approximation is that bulk isotherms are no longer required. Clearly, this approximation will be less accurate as pressure increases. We can expect EMAST (and IMAST and IAST at sufficiently low pressure) to be increasingly accurate for adsorption in increasingly large pores. Considering that the mixture coefficients encapsulate some information concerning models for the adsorbent, the adsorbates, and density profiles, it might seem surprising that these theories can be useful in narrow pores. However, previous work31,32 concerning the properties of hard-particle fluids in low-dimensional geometries sheds some light on this matter. In particular, it has been shown32 that a mixture of hard rods in one dimension is always an ideal mixture for all uniform external potentials. Interesting work by Talbot31 suggests that this ideality reduces as the hardparticle fluid’s dimensionality increases. So, if a bulk fluid is close to an ideal mixture, then the mixture might also be close to ideality when adsorbed into a narrow slit. In this instance, we can expect bulk and adsorbed mixture coefficients to be nearly equal, and hence EMAST, IMAST, and IAST could be accurate. However, if the bulk mixture is not close to an ideal mixture but the adsorbed mixture is (due to dimensionality constraints), then we cannot expect EMAST, IMAST, or IAST to be accurate in narrow pores. So we expect EMAST (and IMAST and IAST at sufficiently low pressure) to be accurate in narrow pores for significantly supercritical fluids only. Note that EMAST, IMAST, and IAST can be solved very efficiently by matrix inversion. DFT. A DFT method requires models for the adsorbent, adsorbates, and a description of the fluid in terms of density profiles. Over the past three decades or so, increasingly accurate (and complex) density functionals have been proposed for a wide range of fluid systems.23,33-35 One of the most popular prescriptions known to be reasonably accurate for simple fluids (e.g., the LennardJones fluid) involves splitting the intrinsic Helmholtz free energy functional into repulsive and attractive contributions according to the Weeks-Chandler-Andersen (WCA)36 convention. The repulsive contribution is then described by a hard-sphere functional and the attractive contribution by a mean-field term (DFMFT). For a Lennard-Jones (LJ) fluid,

(( ) ( ) )

φLJ ij (r) ) 4ij

12

σij r

-

σij r

6

(15)

where ij and σij are the length and energy scale associated with the interaction of particles of type i and j. The WCA convention splits the pair-potential at its minimum into corresponding repulsive and attractive contributions: LJ φrep ij (r) ) φij (r) + ij

φrep ij (r) ) 0

r < 21/6σij

r > 21/6σij

LJ rep φatt ij (r) ) φij (r) - φij (r)

(16)

(31) Talbot, J. J. Chem. Phys. 1997, 106, 4696. (32) Bakaev, V. A.; Steele, W. Langmuir 1997, 13, 1054. (33) Rosenfeld, Y.; Schmidt, M.; Lowen, H.; Tarazona, P. J. Phys.: Condens. Matter 1996, 8, L577. (34) Rosenfeld, Y.; Schmidt, M.; Lowen, H.; Tarazona, P. Phys. Rev. E 1997, 55, 4245. (35) Tarazona, P. Phys. Rev. Lett. 2000, 84, 694. (36) Weeks, J. D.; Chandler, D.; Andersen, H. C. J. Chem. Phys. 1971, 54, 5237, 5422.

Figure 1. Schematic diagram of the simple density functional model.

We obtain the excess intrinsic Helmholtz free energy functional

Fex ) Fex b)}] + HS[{Fi(r 1 2

b1 - b r 2|) ∑ij ∫∫drb1 drb2Fi(rb1)Fj(rb2)φatt ij (|r

(17)

With a suitable choice for Fex HS, such as the fundamental measure functional for hard spheres,37-39 and for the set b), µbi, ij, σij, dij}, where dij is an appropriate choice {T, Vxi (r for the effective hard-sphere diameters, this density functional theory can be solved to yield Fi(r b) and hence Fia. This type of theory is popular in the literature and has been used to study adsorption and selectivity of binary LJ mixtures.40,41 However, for an isotherm solution of this DFT is a lengthy calculation relative to both IMAST and IAST. In this work, we seek to simplify the DFT model so that reasonably accurate results can be obtained quickly, as for IMAST and IAST. In this work, we focus on solution of the mixture problem in slit pores. We consider slit pores formed from two identical parallel surfaces of area A separated by a distance Hp. This simplifies the external potential model so that it can be described in terms of Hp and parameters describing the range and strength of each adsorbentadsorbate interaction. We simplify the DFT by imposing constraints on the density profile and external potential for each component. We choose a model that has sufficient flexibility that it can accurately fit a wide range of adsorption isotherms. For this, the model must be able to exhibit a range of confined fluid phenomena such as wetting, capillary condensation, and so forth.21,22 Yet it must also have as few variable parameters as possible so that solution of the model is sufficiently fast. Our model is presented schematically in Figure 1 with the density profile represented by the vertically shaded region. It symmetrically parametrizes density profiles in terms of the set {Hp, δH, F1i, F2i, F3i, σbi, δσbi, z*}, where each element is non-negative. The external potential is symmetrically parametrized by Vxi which describes the strength of each gas-surface interaction. Parameters with a subscript i can be different for each fluid component; otherwise they are the same for all components. We choose (37) Rosenfeld, Y. Phys. Rev. Lett. 1989, 63, 980. (38) Kierlik, E.; Rosinberg, M. L. Phys. Rev. A 1990, 42, 3382. (39) Phan, S.; Kierlik, E.; Rosinberg, M. L.; Bildstein, B.; Kahl, G. Phys. Rev. E 1993, 48, 618. (40) Kierlik, E.; Rosinberg, M. L. Phys. Rev. A 1991, 44, 5025. (41) Tan, Z.; Gubbins, K. E. J. Phys. Chem. 1992, 96, 845.


Langmuir, Vol. 18, No. 26, 2002 10447

this model for the following reasons. The slit width, Hp, describes the physical width of the slit; that is, Hp is the distance normal to the walls between wall atom centers. The region where the density is zero, described by δH ) Hp - Hc, prohibits fluid particles from overlapping wall particles. The three regions or “slabs” of density represent (1) a monolayer of fluid strongly adsorbed at the wall, (2) a thick layer of adsorbed fluid, and (3) the fluid in the center of the slit. Slab 1 has a thickness equal to

z1i ) min(σbi + δσbi,Hc/2)

(18)

which represents the thickness of a bulk particle, σbi, plus the effective width of the external potential well experienced by particles of type i, δσbi. Slab 3 has a thickness

z* ) max(Hc/2 - z2,0)

(19)

while slab 2 fills in the remainder of the slit and represents a thick adsorbed or wetting layer. Since z1i can be different for each component, slab 2 also allows for the effective adsorption of smaller particles onto the surface of larger particles that are strongly adsorbed at the wall surface. The hard-sphere excess Helmholtz free energy functional can be represented by the local density approximation21 (LDA). It is the use of the LDA for the hard-sphere contribution that requires slab 1 to have width given by (18). We employ the Percus-Yevick compressibility solution for a bulk 3-D hard-sphere fluid mixture (written in terms of fundamental measures)

b)}] ≈ A βFex HS[{Fi(r A

∫0

Hp

(

∫0H

p

dzβfex PYHS({Fi(z)}) )

dz -n0 ln(1 - n3) +

n1n2 (1 - n3)

+

n23

)

24π(1 - n3)2 (20)

with

nR )

∑i Fi(z)R(R) i

R ) 0, 1, 2, 3

(21)

and

R(0) i ) 1 R(1) i ) di/2 R(2) i

) πdi

where di is the effective hard-sphere diameter of fluid component i. We set di ) σai, where σai is the effective size of an adsorbed particle of type i. We also define ai, the effective interaction energy of an adsorbed particle of type i. We allow bulk and adsorbed particles of the same type to have different LJ parameters for bulk and adsorbed phases so that the adsorbed equation of state can mimic that of LJ disks for very narrow slits. We expect σai f σbi and ai f bi as Hp f ∞.

(23)

we obtain

Fex

∫0H

)

A

p

dzβfex PYHS({Fi(z)}) +

∑ij {Fatt(F1i,F1j,0,Hc) -

2Fatt(F1i,F1j - F2j,z1j,Hc - z1j) 2Fatt(F1i,F2j - F3j,z2,Hc - z2) + Fatt(F1i - F2i,F1j - F2j,|z1i - z1j|,Hc - 2 min(z1i,z1j)) + 2Fatt(F1i - F2i,F2j - F3j,z2 - z1i,Hc - 2z1i) + Fatt(F2i - F3i,F2j - F3j,0,H - 2z2)} (24) where

Fatt(Fni,Fmj,a,b) )

(

3 9x2σij2 2 8x2σij3 (c1 - c2) + (c3 - c42) 9 20 σij6 -2 σij12 -8 1 4 (c3 - c44) + (c1 - c2-2) (c - c2-8) 24 6 180 1

-2πijFniFmj

)

and

c1 ) max(b,21/6σij)

c2 ) max(a,21/6σij)

c3 ) min(b,21/6σij)

c4 ) min(a,21/6σij)

Finally, to completely specify the grand potential functional we need the ideal gas and external potential contribution to the Helmholtz free energy and the term dependent on the bulk chemical potential (see (1) and (2)). In our model, the external potential acts solely on the strongly adsorbed slab 1. If we write the external potential contribution as A∑iF1iz1iVxi , that is, if we “smear” the contribution of the external potential across the whole width of slab 1, then we will generally severely underestimate the strength of the fluid-surface interaction. This is because the width of slab 1 represents the width of the fluid-surface interaction plus the width of a particle. It will lead to poor performance for prediction of calorimetric properties and mixture properties. Instead, for both the external potential and ideal gas contributions of slab 1 to the grand potential we transform slab 1 so that it has width δσbi while conserving the total number of particles, / ) F1iz1i/δσbi. This is represented that is, it has density F1i by the horizontally shaded region in Figure 1. Putting all contributions together gives the grand potential functional

A (22)

ij ) xij

σij ) (σi + σj)/2

Ω

2

3 R(3) i ) πdi /6

Using the Lorentz-Berthelot (LB) combining rules

)

Fex A

+

∑i {2δσbi F1i/ (Vxi + ln(Λ3F1i/ ) - µbi - 1) +

2(z2 - z1i)F2i(ln(Λ3F2i) - µbi - 1) + 2z* F3i(ln(Λ3F3i) - µbi - 1)} (25) Minimization of (25) with respect to the density with all other parameters held fixed, that is, variation of {F1i, F2i, F3i, z*} at fixed {T, Hp, δH, σbi, δσbi, bi, σai, ai, Vxi , µbi}, gives the equilibrium state. The bulk LJ parameter set, {σbi, bi}, can be determined by fitting to pure bulk fluid reference pressure-density isotherms for a given temperature, T, with Hp f ∞. The adsorbed parameter set is determined as follows. First,

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we assume that Hp is known and we estimate δH. Next, the fluid-surface interaction width, δσbi, is estimated at the given T and the fluid-solid interaction strength, Vxi , is determined from the low-density limit of a reference pure fluid adsorption isotherm at the same temperature. Alternatively, δσbi and Vxi can be fixed by requiring that the DFT model reproduces the reference isosteric heat of adsorption and adsorption in the low-density limit. Clearly, we need to demonstrate that the DFT model is insensitive to small variations in the estimated parameters, δH and δσbi. This DFT model cannot be applied to very narrow slits where Hc < δσbi. We assume that the average pore density is zero for each fluid component where this condition holds. Finally, the adsorbed LJ parameters, {σai, ai} are determined by fitting to reference pure fluid adsorption isotherms where µbi is determined from the bulk LJ parameters and the bulk density. For prediction of mixture adsorption, the same DFT model is solved to determine {F1i, F2i, F3i, z*} with all the other fixed parameters and the bulk fluid composition, xbi, known. For fixed T, Fbi, and xbi, µbi is given by

( ) ∂fex b ∂Fbi

µbi ) β-1 ln(Λi3Fbi) +

(26)

T,V

where ex fex b ) fPYHS({Fb}) -

16πx2 9

∑ij ijσij3FbiFbj

(27)

The bulk pressure is then

P)

∑i Fbi(µbi - β-1 ln(Λi3Fbi) + β-1) - fexb

(28)

Numerical Details. The grand potential functional is minimized with a Newton-Raphson method42 for each free parameter {F1i, F2i, F3i, z*}. Solution of a binary mixture adsorption isotherm consisting of 50 values for pressure is accomplished in less than a second on a standard desktop PC. When fitting LJ parameters to reference data, we minimize the rms (root-mean-square) deviation of the fitted and reference isotherms. To fit the bulk LJ parameters {σbi, βbi}, we use a grid-search method with a final resolution of 2.0 × 10-3 where σbi is measured in nanometers and βbi is dimensionless. Following this, we use a downhill simplex method, initialized at the best point from the grid-search, to find the local minimum. The local minimum is defined to be found when the average distance between successive vertexes of the “amoeba”43 has contracted to less than 1.0 × 10-5. We use the same downhill simplex algorithm, initialized near to {σbi, βbi}, to find the local minimum of the fit for the adsorbed LJ parameters {σai, βai} to reference data for each slit width. The amoeba is initially very small to ensure that only the local minimum is found. Our algorithm allows the amoeba to grow and contract. Alternatively, a gradient method could be used. We have checked our fitting search method against a grid-search method with a resolution of 1.0 × 10-5 for each application in the results section. We find that when (42) Frink, L. J. D.; Salinger, A. G. J. Comput. Phys. 2000, 159, 425. (43) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical recipes in Fortran 77: The Art of Scientific Computing; Cambridge University Press: Cambridge, 1992.

fitting the bulk LJ parameters our downhill simplex method always finds the global best fit. For the adsorbed LJ parameters, it always finds the local best fit in the neighborhood of the bulk parameters. We also find that there is always only one local minimum, at the resolution of the grid-search, in this neighborhood; that is, all other local minima must be reached by initially going uphill from the initial point {σbi, βbi}. This means that any other fitting method that can robustly and accurately find the global best fit for the bulk LJ parameters and the local best fit (when started from the bulk LJ parameters) for the adsorbed LJ parameters will locate the same LJ parameters as our method. We choose a resolution of 1.0 × 10-5 because the local minima are quite “flat” on this scale, that is, the DFT model is quite insensitive to variations of 1.0 × 10-5 in the LJ parameters. A typical downhill simplex search will consist of 50200 iterations. Fitting to pure bulk data and pure adsorption data in narrow pores is typically completed in a fraction of a second with a standard desktop PC. Fitting to pure adsorption data in wide pores can take as long as 10 s. The IAST and IMAST calculations both require a grand potential isotherm as well as a method for interpolating between reference data points. Grand potential isotherms are constructed from the reference data by integrating the Gibbs adsorption eq 4. When the reference data are of high resolution, the precise method by which integration and interpolation are performed is not significant. But in this paper, our reference isotherms often consist of about 10-20 data points over several decades of pressure. This means that significant differences can be observed dependent on the integration and interpolation method used. This is a general problem with the AST method. In this paper, we choose to use our DFT model to fit the data. We generate isotherms consisting of 100 data points from the DFT model and use the trapezium rule and linear interpolation for the IAST and IMAST calculations. This means that our results for IAST and IMAST are dependent on the DFT model. However, in each of the applications below the DFT model always fits the reference data so well that the influence of the DFT model should not be significant. Results We compare results for IAST, IMAST, and the DFT model for three different simulated fluid mixtures. We simulate three equimolar binary fluid mixtures and their pure fluid components using grand canonical ensemble (GCEMC) simulation. The first two mixtures are LennardJones binaries, one with a length-scale ratio of 21/3 and a second with an energy scale ratio of 2. A third binary mixture is composed of nonspherical molecules constructed from Lennard-Jones and partial charge sites. The molecular model parameters for each fluid are detailed in Table 1. For the first two mixtures, we choose a supercritical temperature, T ) 295 K, while for the third we choose T ) 298 K. Thus the first mixture corresponds to a model of methane (CH4) at T* ) kBT/ ) 2.0, mixed with a component twice as large (B), and the second mixture corresponds to a model of methane at T* ) 2.0 mixed with a component (C) at T* ) 4.0. The third mixture corresponds to a model of an equimolar mixture of hydrogen (H2) and carbon dioxide (CO2) at a temperature that is marginally subcritical for pure CO2 but very supercritical for pure H2. For the first two mixtures, we cut and shift all pairinteractions at a cutoff radius, rc, of 1.492 nm, while for


Langmuir, Vol. 18, No. 26, 2002 10449

Table 1. Simulation Model Parametersa parameter

CH4

B

C

H2

CO2

σ (nm)

0.373

0.47

0.373

H: 0.27

/kB (K)

147.5

147.5

73.75

H: 7.5

LLJ (nm)

0

0

0

H: (0.0371

Lq (nm)

shifted? rc (nm) ramped? ra, rc (nm) σgs (nm)

yes, 1.492 no 0.3465

yes, 1.492 no 0.405

yes, 1.492 no 0.3465

Hq0: 0 Hq1: (0.0371 Hq0: 0.98 Hq1: -0.49 no yes, 1.35, 1.5 H: 0.305

gs/kB (K)

64.265

64.265

45.442

H: 14.491

Q (e)

C: 0.275 O: 0.3015 C: 28.3 O: 81.0 C: 0 O: (0.1149 Cq: 0 Oq: (0.1149 Cq: 0.6512 Oq: -0.3256 no yes, 1.35, 1.5 C: 0.3075 O: 0.3275 C: 28.150 O: 47.624

a Each molecular model is linear. b L LJ and Lq describe the distance of a Lennard-Jones and charge site, respectively, from the center of the molecule. Q is the charge on a charge site.

Table 2. DFT Model Parameters parameter

CH4

B

C

H2

CO2

σb (nm) b/kB (K) σa (Hp ) 1.0 nm) (nm) a/kB (Hp ) 1.0 nm) (K) Vx1/kB (Hp ) 1.0 nm) (K) σa (Hp ) 5.0 nm) (nm) a/kB (Hp ) 5.0 nm) (K) Vx1/kB (Hp ) 5.0 nm) (K)

0.34828 131.52 0.35606 99.961 -1614.2 0.35497 136.47 -1357.4

0.45273 140.93 0.50889 150.38 -2436.7 0.45668 144.73 -1714.4

0.34410 67.002 0.35027 32.141 -1204.0 0.35294 75.635 -1052.2

0.25753 19.382 0.26986 21.933 -679.35 0.26051 21.544 -666.33

0.37437 203.69 0.36781 235.61 -1806.5 0.35484 222.53 -1553.7

the third we linearly “ramp” each site-site potential to zero between ra ) 1.35 nm and rc ) 1.5 nm. We consider each mixture in graphitic slits of physical width Hp ) 1.0 and 5.0 nm. We employ the Steele potential44 to model the interaction between a Lennard-Jones site and a graphitic surface,

Vsi (z) )

(( ) ( )

2πFc∆gsσgs2

2 σgs 5 z

10

-

σgs z

4

-

σgs4

)

3∆(z + 0.61∆)3

(29)

where Fc ) 114 nm-3, ∆ ) 0.335 nm, and z is the distance of the LJ site from the plane of the carbon atom centers in the first layer of the surface. The gas-surface interaction parameters, gs and σgs, are also given in Table 1. The slit potential is given by

Vxi (z) ) Vsi (z) + Vsi (Hp - z)

(30)

DFT Model Parameters. To solve the DFT, we need to specify values for the estimated parameters δH and δσbi. In this work, we have the privileged position of knowing that the external potential is created from the Steele potential. By analyzing this potential for slits of relevant widths together with LJ test particles corresponding to the bulk LJ parameter set {σbi, bi}, we find that the “half-width” of the density profile of the first adsorbed layer at low pressure can be approximated quite well with δσbi ) σbi/4. The half-width is the difference in the z-components of the points where the density profile at low pressure is half that of the maximum density. These points are located at z ) δH/2 + σbi/2 and z ) δH/2 + σbi/2 + δσbi for each component (there are also two more points (44) Steele, W. A. Surf. Sci. 1973, 36, 317.

found by reflecting these points in Hp/2). By analyzing the Steele potential again, we find that a good approximation for each component is δH ) 0.285 nm. Of course, estimates for these parameters might often not be as straightforward to obtain. We will demonstrate that the DFT model is not unduly sensitive to these estimated parameters. Table 2 presents the DFT parameters obtained from the fitting procedure. Figure 2a-e shows the fit for each pure fluid in the bulk and in each slit. The quality of these fits is generally very good. The fit to the CO2 adsorption isotherm at Hp ) 5.0 nm is significantly worse than the others. This is due to both statistical errors in the simulation data and the increased complexity of the isotherm, which indicates that the system is in close proximity to a capillary condensation transition. The poorer quality of the fit might result from the slab approximation imposed on the DFT model. It might also indicate that it is not possible to accurately fit a relatively simple DFT model to simulation data for a “realistic” model fluid that incorporates multiple Lennard-Jones sites and significant quadrupolar interactions. CH4/B Mixture. Figure 3a-d compares results of the three methods against GCEMC simulation for the CH4/B mixture at 295 K in slits with Hp ) 1.0 and 5.0. For Hp ) 1.0 nm, we see that IAST is in slightly better agreement with simulation up to about 100 bar. But this is reversed above 100 bar with IMAST and the DFT model being more accurate at high pressures. This suggests that both the bulk and adsorbed mixtures are close to ideal. We see that the larger molecules are preferentially adsorbed at pressures up to about 100 bar, but this trend is reversed at higher pressures. This pattern is observed because at low pressure the larger molecules experience a stronger attraction within the slit, but at high pressures it is easier to insert a small molecule, rather than a large one, into the dense adsorbed fluid. This pattern is reflected in

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Figure 2. (a) CH4 isotherms at 295 K from the GCEMC simulation (diamonds) and the fitted DFT model (lines). The solid line is the bulk isotherm; the dashed lines are adsorption isotherms in ideal graphitic slits with Hp ) 1.0 nm (upper dashed line) and Hp ) 5.0 nm (lower dashed line). The bulk density and pressure are on a logarithmic scale. Simulation molecular model parameters are presented in Table 1, while fitted DFT molecular model parameters are presented in Table 2. Statistical errors are less than the size of the symbols. (b) As for (a), except that we have fluid B at 295 K. (c) As for (a), except that we have fluid C at 295 K. (d) As for (a), except that we have fluid CO2 at 298 K. (e) As for (a), except that we have H2 at 298 K.

the selectivity shown in Figure 3b, which for a binary mixture is

Fa1Fb2 S12 ) Fa2Fb1

(31)

For Hp ) 5.0 nm, we see that both IMAST and the DFT model are more accurate than IAST over the entire pressure range shown. IAST is in good agreement with simulation up to 200 bar but loses accuracy at higher pressures. The selectivity reveals that the DFT model is

the most accurate theory, although each theory performs well up to 400 bar. CH4/C Mixture. Figure 4a-d compares results of the three methods against GCEMC simulation for the CH4/C mixture at 295 K in slits with Hp ) 1.0 and 5.0 nm. For Hp ) 1.0 nm, we see that each method is in good agreement with simulation up to high pressure. Again, this suggests that both the bulk and adsorbed mixtures are close to ideal. The DFT model is in best agreement with simulation, and its predictions remain accurate at pressures above the maximum pressure to which either IMAST or IAST


Langmuir, Vol. 18, No. 26, 2002 10451

Figure 3. (a) Adsorption isotherms for CH4 and fluid B in an ideal graphitic slit with Hp ) 1.0 nm from an equimolar bulk mixture at 295 K. Circles denote GCEMC simulation data for CH4; triangles are the same for fluid B. Lines are the predictions of theory: DFT (solid line), IMAST (dashed line), and IAST (dotted line). Pressure is on a logarithmic scale. Simulation molecular model parameters are presented in Table 1, while fitted DFT molecular model parameters are presented in Table 2. Where error bars are not drawn, statistical errors are less than the size of the symbols. (b) As for (a), except that selectivity is shown with squares denoting simulation data. (c) As for (a), except that Hp ) 5.0 nm. (d) As for (c), except that selectivity is shown with squares denoting simulation data.

can extend (about 340 and 170 bar, respectively). Figure 4b shows the selectivity, and we see once again that the DFT model is the most accurate in this instance. For Hp ) 5.0 nm, we see good agreement between simulation, IMAST, and the DFT model. IAST is accurate up to about 100 bar but loses accuracy above this pressure. Figure 4d shows that IMAST and the DFT model in particular are accurate for the selectivity up to high pressure. Once again, this suggests that the mixing coefficients in the bulk and adsorbed phases are nearly equal. CO2/H2 Mixture. Figure 5a-d compares results of the three methods against GCEMC simulation results for the CO2/H2 mixture in slits with Hp ) 1.0 and 5.0 nm, respectively. We see that the DFT method is in best agreement with simulation for both slit widths. Both IMAST and IAST perform poorly for H2 above a few bar for Hp ) 1.0 nm and above about 50 bar for Hp ) 5.0 nm, and these theories are limited to pressures well below the maximum pressure of the input CO2 simulation data for Hp ) 1.0 nm. In contrast, the DFT results continue to be accurate up to at least 100 bar. Indeed, for Hp ) 5.0 nm the DFT predicts the sharp rise in CO2 adsorption, indicating the close proximity of a capillary condensation

transition, with good accuracy. This indicates that the DFT method is able to mimic with reasonable success the near-critical fluid behavior of nonspherical molecular models incorporating significant quadrupolar interactions in slit pores. Figure 5b shows that although the DFT model cannot predict very accurately the very high selectivity of CO2 over H2 for Hp ) 1.0 nm, it is much more accurate than either IAST or IMAST. In Figure 5d, we see that the DFT model is accurate for the selectivity for Hp ) 5.0 nm up to high pressure. In contrast, both IMAST and IAST are unable to predict the large increase in selectivity of CO2 seen in both slits. This indicates that this mixture is significantly nonideal. Note that the unusual, almost discontinuous, behavior of IAST and IMAST at high pressure for Hp ) 5.0 results from a combination of the structure of these theories, the proximity of the system to a capillary condensation transition, and the sensitivity of these theories to numerical integration and interpolation schemes. Table 3 presents results for the sensitivity of the DFT model to small variations in the estimated parameters, δH and δσbi. These results were obtained by comparing results from the DFT model with an increase in these

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Figure 4. (a) As for Figure 3a, except that we have a bulk equimolar mixture of CH4 and fluid C (triangles) at 295 K. (b) As for (a), except that selectivity is shown with squares denoting simulation data. (c) As for (a), except that Hp ) 5.0 nm. (d) As for (c), except that selectivity is shown with squares denoting simulation data.

estimated parameters of 10%; that is, two further sets of results were obtained: one with δH ) 0.3135 nm and one with δσbi ) 0.275σbi. The values given in Table 3 are the rms variation in the DFT model’s predictions (compared with the base values used for the results in Figures 3a5d) over the entire pressure range shown in Figures 3a5d. From these results, we conclude that the DFT model is almost completely insensitive to these estimated parameters for all of the applications in this paper except for the CO2/H2 system with Hp ) 5.0 nm. For this system, the sensitivity of H2 adsorption, and hence the selectivity, is appreciable. Figure 6 shows the three predictions for selectivity of the DFT model with different estimated parameters compared against simulation results. Clearly, this system is most sensitive to small changes in the “external potential length” parameter, δσbi. This parameter’s primary effect is to control the strength of the fitted external potential, Vxi . It is important to note from Figure 6 that the statistical errors in the simulation results for this system are also large, indicating considerable density fluctuations in the simulations. So in this case we conclude that the DFT model is actually mimicking the real sensitivity of the CO2/H2 system to the strength of the external potential for Hp ) 5.0 nm. Indeed, there appears to be a degree of agreement between the DFT model and simulation concerning the bulk density-sensitivity relationship. This could be regarded as a useful feature of the DFT model. Also note that it is likely that very large

changes in the estimated parameters are needed to force the DFT model to give worse predictions than either IAST or IMAST. So we conclude that the DFT model is not unduly sensitive to the estimated parameters. Conclusions We present three theories for the prediction of mixture adsorption isotherms from pure component adsorption isotherms. We find that a relatively simple DFT method generally outperforms the IMAST and IAST approaches. For significantly nonideal bulk mixtures, such as the equimolar model CO2/H2 mixture, we find that the DFT model is significantly more accurate than IMAST and IAST. However for nearly ideal bulk mixtures, such as the CH4/B and CH4/C equimolar mixtures, we find that all three theories perform well, with the DFT model being generally more accurate than IMAST and IMAST generally more accurate than IAST. Of course, IAST loses accuracy as the ideal gas approximation becomes invalid at high pressure. The IMAST and IAST methods have the advantage that knowledge of adsorbent geometry and particle interactions is not required. So these methods are attractive for predictions concerning poorly characterized materials, such as active carbons,45 when adsorbed densities are (45) Jankowska, H.; Swiatkowski, A.; Choma, J. Active Carbon; Ellis Horwood: New York, 1991.


Langmuir, Vol. 18, No. 26, 2002 10453

Figure 5. (a) As for Figure 3a, except that we have an equimolar bulk mixture of CO2 (circles) and H2 (triangles) at 298 K. Both pressure and adsorbed density are on logarithmic scales. (b) As for (a), except that selectivity is shown on a linear scale with squares denoting simulation data. (c) As for (a), except that Hp ) 5.0 nm. (d) As for (c), except that selectivity is shown on a linear scale with squares denoting simulation data. Table 3. Root Mean Square (rms) Deviation in the Adsorbed Density (G) and Selectivity (S) When Estimated Parameters in the DFT Model Are Changed by 10%a δH ) 0.3135 nm

system Hp (nm) 1.0 5.0 1.0 5.0 1.0 5.0 a

1 B B CH4 CH4 CO2 CO2

2 CH4 CH4 C C H2 H2

rms F1

(nm-3) 10-3

2.1 × 4.6 × 10-3 2.3 × 10-3 2.0 × 10-3 3.4 × 10-3 1.9 × 10-2

rms F2

(nm-3) 10-3

2.7 × 6.1 × 10-3 1.9 × 10-3 4.5 × 10-3 3.2 × 10-3 8.2 × 10-3

δσbi ) 0.275σbi rms S12 10-2

1.6 × 3.3 × 10-3 9.6 × 10-3 1.7 × 10-3 1.1 × 10-0 3.7 × 10-2

rms F1

(nm-3)

rms F2 (nm-3)

rms S12

10-7

2.1 × 10-6 1.3 × 10-2 1.1 × 10-7 1.9 × 10-3 8.8 × 10-4 4.7 × 10-2

6.1 × 10-6 8.2 × 10-3 4.7 × 10-7 6.8 × 10-3 3.3 × 10-1 1.8 × 10-1

9.1 × 8.5 × 10-3 9.9 × 10-7 1.0 × 10-2 4.4 × 10-4 3.7 × 10-2

The largest relative change occurs for the selectivity of the CO2/H2 system with Hp ) 5.0 nm.

expected to be much greater than bulk densities, that is, at low pressure. However, when the geometry of the adsorbent can be approximated in terms of the polydisperse slit-pore model19 and if the bulk mixture is significantly nonideal or of significant density, then the DFT method is expected to be more accurate. We have shown that the DFT model is also robust for the applications in this paper; that is, the DFT model is not unduly sensitive to small variations in the estimated input data. This method is also more flexible since it is not limited to pressure and temperature ranges defined by the pure component data. It can be used to locate thermodynamic phase transitions and provide information concerning isosteric heat (isosteric heat is thought to be a sensitive probe of pore structure46).

However, the DFT model might encounter problems when its underlying approximations are no longer adequate. For example, the spherical Lennard-Jones model potentials might be inadequate to fit the adsorption isotherms of complex fluids that either are very nonspherical (such as stiff long-chain alkanes or polymers) or have very strong short-ranged or significant long-ranged interactions (such as colloidal particles or salts in solution). Also, the Lorentz-Berthelot mixing rules (eq 23) might be inadequate to describe interactions between highly (46) Nicholson, D.; Quirke, N. In Characterisation of Porous Solids V: Proceedings of the 5th International Symposium on the Characterisation of Porous Solids; Unger, K. K., Kreysa, G., Baselt, J. P., Eds.; Studies in Surface Science and Catalysis Vol. 128; Elsevier: Amsterdam, 2000; p 11.

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with the grand potential per unit area, σ. From the fundamental Legendre transform

Ω)F-

∑i µiNi ) E - TS - ∑i µiNi

(A1)

and the full work term

dW ) T dS +

∑i µi dNi + w dv

(A2)

we obtain a Gibbs-Duhem relation

dΩ ) -S dT -

∑i Ni dµi + w dv

(A3)

This gives Ω ) wv, which when substituted back into (A3) gives Figure 6. Selectivity of CO2/H2 from the three results for the DFT model with different estimated parameters and from Monte Carlo simulation (squares). The solid line is the same as in Figure 5d; the dashed line is for δσbi ) 0.275σbi, while the dotted line (barely visible against the solid line) is for δH ) 0.3135 nm. Pressure is on a logarithmic scale.

nonideal mixtures. For example, the LB rules might be inadequate for modeling the cross interaction between methane and water since methane does not form hydrogen bonds. In this case, bulk fluid mixture reference data (for example, the solubility) can be used to define crossinteraction parameters. Clearly, the LJ model with LB mixing rules is sufficient for describing the simple molecular fluids studied in this paper, including CO2 which in our simulations is modeled to be nonspherical with a significant electrostatic quadrupole moment. But further work is required to establish the limits of the present theory. Despite the undoubted limitations of the present theory, we expect the general principle that a DFT model can outperform IAST to hold provided the DFT model (including the model potentials) is developed with the specific fluids in mind. We have also presented, but not assessed, another approach (EMAST) that could be more accurate for adsorption isotherms than both IAST and IMAST when the bulk mixture is significantly nonideal. Appendix First, we require that the work on infinitesimally increasing the amount (mass) of adsorbent at constant µi and T is given by dW ) w dv, where v and w are a pair of extensive and intensive conjugate thermodynamic quantities. For example, within the ideal slit-pore geometry we can associate v with the surface area, A, and w

v dw ) -

∑i Ni dµi

(A4)

at constant T. Substitution of (A4) into (8) in the ideal mixture (m ) 1), low-density limit gives

v dw(Ω1,{x}) ) -

∑i Ni dµ0i (Ω2i)

(A5)

By varying {x} at constant T, v, and w, we have constant Ω1 and the left-hand side of (A5) is zero. Thermodynamic consistency is guaranteed by the choice Ω2i ) Ω2i(T, Ω1). This proves that the reference state can only depend on the grand potential and temperature of the mixture; that is, it cannot depend on other thermodynamic quantities such as the Helmholtz or Gibbs energies. Note that this conclusion is slightly different to that concluded by Rudisill and Levan,25 that is, that Ω2i ) Ω1. Their conclusion is over-restrictive at this stage, although it is eventually found to be correct. To obtain their result (Ω2i ) Ω1), we must look again at the expression for the mixing coefficients, (9), which shows that in the low-density limit 0ex 0 where µex i (Ω1,{x}) - µi (Ω2i) ) 0, we must have Ni (Ω2i) ) ∑jNj(Ω1,{x}). From the Gibbs adsorption equation, (4), we see that this is only satisfied in the low-density limit if Ω2i ) Ω1. This last part of the proof is similar to the requirement of Rudisill and Levan that the selectivity rule is obeyed in the Henry’s Law (low-density) regime. So we have proved the assertion of Cracknell and Nicholson18 and others2 that the correct potential at which to define the pure reference state is the same grand potential and temperature as the mixture. LA0200358

Predicting the Adsorption of Gas Mixtures: Adsorbed Solution Theory

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