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Langmuir 1997, 13, 1150-1161
Enthalpy and Entropy Effects of Liquid Mixture Adsorption in Porous Materials: A Study of a One-Dimensional Model† M. Heuchel Institute of Physical and Theoretical Chemistry, University of Leipzig, 04103 Leipzig, Germany Received February 12, 1996. In Final Form: May 21, 1996X Thermodynamic functions characterizing adsorption of binary mixtures in microporous materials are studied by way of the analytical results for inhomogeneous square-well mixtures in one-dimension. The calculations were done for two bulk mixtures: an ideal one and an athermal one. The results show the influence of different adsorbate-wall interactions, pore size, and dispersity in pore sizes on adsorption excess isotherms and thermodynamic functions (Gibbs energy, enthalpy, and entropy of immersion), especially for liquid mixture adsorption. The porous materials are characterized by pore size distributions that range from monodisperse to broadly polydisperse. Additionally, density profiles and Henry’s law constants are presented to illustrate the various effects that occur.
1. Introduction The equilibrium of a fluid mixture inside a porous material with an external bulk phase is important in industrial adsorption applications. The porous materials often have pore dimensions on the same order as the adsorbate molecular size. Mixtures confined to such small regions exhibit microscopic structure and thermodynamics that are sensitive functions of component size and of all the different interactions between fluid particles as well as between fluid particles and pore walls. Furthermore, real adsorbents are not homogeneous in pore size. This “polydispersity” in pore size may produce a distribution of adsorbate-pore interaction potential energies and is also responsible for a distribution of confinement effects in real adsorbents. In the literature1,2 a dispersity of pore sizes is referred to as structural (geometrical) heterogeneity, and nonuniformity of pore potential strength is referred to as energetic heterogeneity. To understand better the thermodynamics of adsorption in such complex systems and to interprete experimental results, it is highly desirable to study simple but qualitatively realistic models of fluids confined to pores. Such theoretical investigations were carried out via computer simulation. Monte Carlo and molecular dynamics techniques3,4 have been widely used for studying adsorption in many types of geometrical pores, such as slitlike, cylindrical, spherical, and zeolite structures. Gas mixtures in such pores have been investigated, e.g., by Karavias and Myers,5 Razmus and Hall,6 Cracknall et al.,7 Piotrovskaya and Brodskaya,8 and van Tassel et al.9 † Presented at the Second International Symposium on Effects of Surface Heterogeneity in Adsorption and Catalysis on Solids, held in Poland/Slovakia, September 4-10, 1995. X Abstract published in Advance ACS Abstracts, September 15, 1996.
(1) Jaroniec, M.; Mady, R. Physical Adsorption on Heterogeneous Solids; Elsevier: Amsterdam, 1988. (2) Rudzinski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: London, 1992. (3) Allen, M. P.; Tildesley, D. J. Computersimulation of Liquids; Clarendon Press: Oxford, 1987. (4) Nicholson, D.; Parsonage, N. G. Computer Simulation and Statistical Mechanics of Adsorption; Academic Press: New York, 1982. (5) Karavias, F.; Myers, A. L. Mol. Simul. 1991, 8, 51. (6) Razmus, D. M.; Hall, C. K. AIChE J. 1991, 27, 769. (7) Cracknell, R. F.; Nicholson, D.; Quirke, N. Mol. Phys. 1993, 80, 885; Mol. Simul. 1994, 13, 161. (8) Piotrovskaya, E. M.; Brodskaya, E. N. Langmuir 1993, 9, 3548. (9) Van Tassel, P. R.; Davis, H. T.; McCormick, A. V. Langmuir 1994, 10, 1257.
S0743-7463(96)00125-4 CCC: $14.00
Special emphasis on adsorption of liquid mixtures is given in a recent Monte Carlo simulation by Dunne et al.,10 who studied N2 and O2 in a model faujasite cavity at 77.5 K. Today, the most advanced theoretical models are density functional theories (DFT) for inhomogeneous fluid mixtures, as they have been developed, e.g., by Sokolowski and Fischer,11 Kierlik and Rosinberg,12 and Tan and Gubbins.13 These theories have the advantage of being computationally faster than full molecular simulations. Both, simulation and DFT, give very detailed molecular level information. But, systematical variations of all the parameters, which are important for the adsorption equilibrium of a mixture, are computationally expensive. Therefore, simulation and DFT are primarily useful for simple fluids and pore geometries; they are much more difficult to apply to more complex systems, e.g., solids with pore size distribution, or zeolites. Accordingly, experimental data have been frequently analyzed on the basis of simple theoretical concepts, where the solid surface or the pore is often modeled as a lattice with distinct adsorption sites. This has been shown, e.g., in a recent review paper by Rudzinski et al.14 about the fundamentals of mixed-gas adsorption on heterogeneous surfaces. Also, the statistical thermodynamics of monolayer adsorption from liquid mixtures on homogeneous surfaces was developed on this basis by Elton,15 Delmas and Patterson,16 Everett,17 and Sircar and Myers.18 In 1987, a review paper was published by Dabrowski et al.19 about multilayer and monolayer adsorption from liquid mixtures on solid surfaces. It was shown that the canonical partition function for a monolayer containing two components only has the same mathematical form as the canonical partition function for binary gas mixtures.20 But, in the case of liquid adsorption one usually assumes that the total number of molecules in the monolayer is (10) Dunne, J. A.; Myers, A. L.; Kofke, D. A. Adsorption 1996, 2, 41. (11) Sokolowski, S.; Fischer, J. Mol. Phys. 1990, 2, 393. (12) Kierlik, E.; Rosinberg, M. L. Phys. Rev. A 1991, 44, 5025. (13) Tan, Z.; Gubbins, K. E. J. Phys. Chem. 1992, 96, 845. (14) Rudzinski, W.; Nieszporek, K.; Moon, H.; Rhee, H.-K. Heterog. Chem. Rev. 1994, 1, 275. (15) Elton, G. A. H. J. Chem. Soc. 1954, 54, 3813. (16) Delmas, G.; Patterson, D. J. Phys. Chem. 1960, 64, 1827. (17) Everett, D. H. Trans. Faraday Soc. 1964, 60, 1803. (18) Sircar, S.; Myers, A. L. J. Phys. Chem. 1970, 74, 2828. (19) Dabrowski, A.; Jaroniec, M.; Oscik, J. Surface and Colloid Science; Plenum Press: New York, 1987; Vol. 14, pp 83-213. (20) Jaroniec, M. In Fundamentals of Adsorption; Myers, A. L., Belford, G., Eds.; American Institute of Chemical Engineers: New York, 1984.
© 1997 American Chemical Society
Liquid Mixture Adsorption in Porous Materials
constant and that all adsorption sites are occupied. The extension to energetically heterogeneous surfaces was discussed in detail in a review of Jaroniec et al.,21 dealing with the statistical thermodynamics of monolayer adsorption from gas and liquid mixtures on heterogeneous surfaces. Despite the extension to multilayers, these lattice models have limitations. The adsorbed phase usually contains species of different molecular size and shape. The surface area occupied by the adsorbed molecules depends on the orientation with respect to the surface and the nature of the molecular interactions in this phase. These quantities vary with composition of the adsorbed and bulk phase; but in the theories it is common practice to assume their constancy. So, there is a requirement for theories which describe the packing of mixtures with particle-size difference without the limitations of a rigid lattice concept. Recently, Monson22 has derived the analytical results for inhomogeneous square-well mixtures in one-dimension (1-D). The system Monson studied “is perhaps the simplest exactly solvable model for selective adsorption of a fluid mixture in a porous material, in which attractive interactions as well as molecular-size differences between the adsorbed molecules are taken into account”.22 It considers several key features of real fluid-pore systems: confinement, adsorbate interparticle attractive forces, adsorbate particle excluded volume, adsorbate-adsorbent attractive forces, and spatial inhomogeneity. General advantages and limitations of 1-D systems and the position of Monson’s model in the history of statistical mechanical modeling of adsorption in micropores can be found in the introductory sections of refs 22 and 23. Monson used the Laplace-transform technique for the evaluation of partition functions of 1-D systems with nearest-neighbor interactions to determine the canonical, isobaric-isothermal, and grand partition functions. His paper contains molecular density distribution functions for binary mixtures determined from the grand partition function. The shape of these distribution functions depends significantly on size differences of components and strength of attractive nearest-neighbor and particlewall interactions. With respect to the equilibrium between a confined 1-D binary square-well mixture and such a mixture in the thermodynamic limit (the bulk phase) Monson found two general sources of nonideality in the adsorbed-solution behavior: “First, molecular-size differences act to promote the adsorption of the component with the smaller size. If that component also has weaker wall-particle interactions then this may give rise to azeotropy. Secondly, for cases where the two components have similar wall-particle interactions selective adsorption of the component with the stronger nearest-neighbor interactions will be promoted. If that component also has larger molecular size then azeotropy may also occur. Of the two, the first source of non-ideality is probably the more significant”.22 The analytical solution of the model and hence the fast computation times has made it possible to study additionally the influence of a polydispersity in pore size. For adsorbents represented by various pore size distributions Kaminsky and Monson23 showed how the selectivity in mixture adsorption depends on the ratio of the component Henry’s constants and possible excluded volume effects. In connection with development of theories for mixture adsorption in microporous materials, these authors24 used the 1-D model to clarify the theoretical (21) Jaroniec, M.; Patrykiejew, A.; Borowko, M. In Progress in Surface and Membrane Science; Academic Press: New York, 1981; Vol. 14. (22) Monson, P. A. Mol. Phys. 1990, 70, 401; 1995, 86, 1545. (23) Kaminsky, R. D.; Monson, P. A. Langmuir 1993, 9, 561.
Langmuir, Vol. 13, No. 5, 1997 1151
status of the statistical model adsorption isotherm developed by Ruthven and co-workers,25,26 which is frequently used in the prediction and correlation of mixture adsorption in microporous materials, especially zeolites. In continuing the application of the 1-D model, this paper is devoted to an investigation of thermodynamic functions associated with adsorption of mixtures in small pores. Despite the 1-D model cannot show a condensation transition, the densities in the system could reach typical values for liquid mixtures. (Also, the traditional theories for gas mixture adsorption describe in the limit of saturation pressure the behavior of liquid mixture adsorption.) Therefore, the 1-D model seems interesting and helpful, to study basic effects in the adsorption for liquid mixtures. Using the analytical solution Monson derived, changes in Gibbs energy, enthalpy, and entropy during adsorption processes can be calculated. The goal is a better understanding of experimentally measured thermodynamic functions, like enthalpies of immersion on microporous solids. It would be interesting to know, e.g., how do these functions reflect adsorbate-solid and adsorbate-adsorbate interactions and how do they depend on pore size and a possible polydispersity in pore sizes? To illustrate and to understand the various effects which occur, it will be necessary to include in the discussion adsorption excess isotherms and density profiles. The paper is organized as follows. Section 2 contains a short description of the 1-D model. In section 3 necessary quantities of the phenomenological thermodynamics of mixture adsorption are introduced with respect to the 1-D model. Further, this section contains the path to calculate these thermodynamic quantities with statistical thermodynamics using Monson’s results. Section 4 discusses calculated adsorption excess isotherms and thermodynamic wetting functions for two bulk mixtures. Results are shown for (i) increasing particle-wall interaction of one component at fixed pore size, (ii) increasing pore size at fixed particle-wall potentials, and (iii) different pore size distributions. Section 5 gives a summary and discussion of the results. 2. The Model The system consists of a binary 1-D fluid mixture of components A and B (the bulk phase) in equilibrium with a corresponding 1-D fluid mixture confined to a 1-D pore (i.e., a line segment) of length L (the adsorbed phase). The mixture consists of particles interacting through squarewell, nearest-neighbor potentials. Also, the walls at the end of the pores interact with the fluid through squarewell, nearest-neighbor potentials. The two wall potentials (i.e., adsorbate-adsorbent potentials) are symmetric. The square-well potential is characterized by three parameters: the particle size σ, the potential well depth , and the potential range λ. The binary bulk mixture is characterized by nine parameters: σAA and σBB are the sizes of the A- and B-particles; AA, BB and λAA, λBB are the potential well depths and potential ranges of AA- and BB-pairs, respectively; σAB, AB, and λAB are the corresponding square-well parameters for interactions between AB-pairs. The square-well interaction of A and B with every wall of the pore is represented by six parameters σWi ) σii/2, Wi, and λWi with i ) A, B. All distances are (24) Kaminsky, R. D.; Monson, P. A. AIChE J. 1992, 38, 1979. (25) Ruthven, D. M. Nature Phys. Sci. 1971 232, 70; AIChE J. 1976, 22, 753. Ruthven, D. M.; Loughlin, K. F. J. Chem. Soc., Faraday Trans. 1 1972, 68, 696. Ruthven, D. M.; Loughlin, K. F.; Holborow, K. A. Chem. Eng. Sci. 1973, 28, 701. (26) Ruthven, D. M. Principles of Adsorption and Adsorption Processes; Wiley: New York, 1984.
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measured from the surface for the walls and from the center of the particles. 3. Theory In the 1-D model the pore volume of a solid adsorbent is represented by a line segment of length L with walls on both ends. The thermodynamic and geometric properties of the adsorbent are assumed to be independent of temperature, of pressure of the surrounding fluid, and of the concentration of adsorbed particles. The adsorbent is considered to be inert. The pore is equilibrated with a homogeneous reservoir (bulk phase) of known temperature T, pressure P, and composition xA, and hence of known chemical potentials µA and µB. Under these conditions the particles of the mixture inside the pore form an “adsorbed phase” (index s). Its density and composition depend on a kind of “force field” created by the adsorbent and the chemical potentials of the mixture components in the bulk phase. If the pore contains Nsi molecules of component i (i ) A, B), the fundamental thermodynamic equation for the change of internal energy in the adsorbed phase dUs is
dUs ) T dSs + dWs + µA dNsA + µB dNsB
(1)
where dSs is the entropy change, and µA ) µsA and µB ) µsB are the chemical potentials of A and B in bulk and adsorbed phase. The macroscopic work term dWs for the mixture in the 1-D pore can be written as
dWs ) -PsL dL
(2)
It is the work done by the mixture in the pore, when it expands by an amount dL against the 1-D pressure PsL inside the pore. The pressure PsL depends on the strength of the adsorbate-adsorbent interaction and therefore on pore size L. For L f ∞, the influence of the wall-particle interaction becomes very small and the pressure inside the pore, PsL, is equal to the bulk pressure P. In a threedimensional slitlike pore PsL has the meaning of the normal component of the pressure tensor Pzz (for a detailed discussion see ref 27). Experiments and computer simulations28 have shown that for small pore sizes the pressure PsL (the force per unit length) has an oscillatory behavior and the oscillation period is approximately equal to the molecular diameter of the fluid. Therefore, the pressure PsL may be decomposed into a bulk term P and a pore size dependent disjoining pressure ΠL(L)
PsL(L) ) P + ΠL(L)
(3)
Because usual adsorbent properties (surface area A or adsorbent mass m) are not explicitly defined in the 1-D model, the work term in (2) contains implicitly the total influence of the solid adsorbent on Us. Therefore, the usual contribution in three-dimensional systems -πdA, the work done against spreading pressure π in increasing the surface area by an amount dA, does not appear explicitly. In the 1-D model, it is the contribution -ΠL(L) dL which corresponds to -πdA, if surface A is the extensive variable for the adsorbent. If m is the extensive variable for the adsorbent, than the 1-D term -ΠL(L) dL corresponds to φ dm, where φ is the difference of the specific chemical potentials of the wetted and nonwetted adsorbent at constant P and T. (27) Cracknell, R. F.; Nicholson, D. Adsorption 1995, 1, 7. (28) Christenson, H. K. J. Chem. Phys. 1983, 78, 6906; Chem. Phys. Lett. 1985, 118, 455.
To determine thermodynamic functions from statistical thermodynamics, it is convenient to describe the thermodynamics in terms of the grand potential Ωs, defined by
Ωs ) Us - TSs - NsA µA - NsB µB
(4)
The differential form of this equation gives together with eqs 1-3
dΩs ) -Ss dT - P dL - ΠL dL - NsA dµA - NsB dµB (5) Integration over the extensive variables relates the grand potential Ωs to the pressure PsL inside the pore.
Ωs ) -(P + ΠL)L ) -PsLL
(6)
A determination of Ωs (and therefore PsL) is often done in experimental investigations by integration of eq 5 at constant temperature T and pore size L. Necessary information is the number of particles inside the pore, NsA and NsB, and expressions for the chemical potentials, µA and µB. At equilibrium the chemical potential of a component i in the pore is equal to its chemical potential in the coexisting bulk mixture
µsi ) µi(T, P, xi) ) µi*(T, P) + RT ln xi + RT ln γi(T, P, xi) (7) where µi* is the chemical potential of the pure ith component in the bulk phase and γi is the appropriate activity coefficient. Corresponding to the experimental situation in liquid mixture adsorption, where the bulk phase pressure P and temperature T are kept fixed and properties are calculated as functions of mole fraction in the bulk phase xA, the change of the chemical potential of component i is
(dµi)T,P )
( ) ∂µi ∂xi
dxi )
T,P
[ ( )]
∂ ln γi RT 1 + xi xi ∂xi
dxi (8)
T,P
Insertion of eq 8 into eq 5 and use of the Gibbs-DuhemMargules relation for the bulk mixture gives
ΓeA (dµA)T,P (dΩ )T,P,L ) 1 - xA s
(9)
where ΓeA is the adsorption excess of component A, which is
ΓeA ) NsAxB - NsBxA ) (NsA + NsB)(xsA - xA)
(10)
In the last equation the mole fraction of component A in the pore, xsA ) NsA/(NsA + NsB) was introduced. Integration of eq 9 gives
Ωs(xA) - ΩsB* )
∫0
xA
-RT
ΓeA xA(1 - xA)
[ ( 1 + xA
)]
∂ ln γA ∂xA
dxA (11)
T,P
This is the integrated Gibbs adsorption isotherm for mixture adsorption. From an excess adsorption isotherm, ΓeA(xA), the primary experimental quantity in liquid mixture adsorption, the grand potential Ωs can be determined up to a constant representing the adsorption of pure component B.
Liquid Mixture Adsorption in Porous Materials
Langmuir, Vol. 13, No. 5, 1997 1153
To derive associated enthalpy and entropy differences at equilibrium, at constant bulk pressure P and temperature T, the Gibbs energy for the adsorption phase Gs should also be defined, in analogy to the bulk phase quantity G(P,T,xA), with respect to the bulk pressure P, because this quantity (not PsL) is independent of L and kept constant in the usual experiments. Therefore, it is s
s
s
G ≡ U - TS + PL
Gs ) (P - PsL)L + µANsA + µBNsB
( ) ∂Ωs ∂T
P,xA,L
+ NsAsA + NsBsB
(15)
where sA, sB are the partial molar entropies of components A and B in the bulk at composition xA. From the GibbsHelmholtz relation the enthalpy of the adsorbed phase Hs follows
Hs )
[
]
∂(Ωs/T) ∂(1/T)
P,xA,L
+ PL + NsAhA + NsBhB
(16)
where hA, hB are the partial molar enthalpies of A and B in the bulk phase at mole fraction xA. The last equations, eqs 13-16, allow the determination of thermodynamic functions characterizing the process of filling an empty pore by contacting it with an amount of bulk mixture sufficiently large that the change of mole fraction in the bulk phase is very small during the adsorption experiment. At constant T and P, the change ∆wY for every extensive thermodynamic quantity Y during this process of immersion or wetting (index w for wetting) is given by the following relation29
∆wY ) Ys - NsA yA(xA) - NsB yB(xA)
[
]
∂(Ωs/T) ∂(1/T)
+ PL
(18)
P,xA,L
( )
∆wS ) -
∂Ωs ∂T
P,xA,L
∆wG ) Ωs + PL ) (P - PsL)L ) -ΠLL
∑ ∑
Ns
A)0
(19) (20)
A numerical calculation of the grand potential Ωs with (29) Everett, D. H. J. Phys. Chem. 1981, 85, 3263. (30) Johnson, L.; Denoyel, R.; Rouquerol, J.; Everett, D. H. Colloids Surf. 1990, 49, 133.
s
A N B λN A λB
s s B)0 NA!NB!
Z(NsA, NsB, L, T)
Ns
βµA
(22)
βµB
λA ) (σAA/ΛA)e and λB ) (σAA/ΛB)e and are reduced activities of species A and B. ΛA and ΛB are the thermal de Broglie wavelength for the two species. The upper limits in the summation are determined by the maximum number of hard rods of that species that can be accommodated on a line of length L. In general, the configurational integral Z of a binary mixture A and B in a volume V is defined as
Z(NA,NB,V,T) ) 1 exp(-βU(r1,...,rNA+NB)) dr1, ..., NA!NB! V drNA+NB (23)
∫
where U is the total potential energy and β ) 1/kT, where k is the Boltzmann constant. The integral in eq 23 is over the coordinates of all NA + NB particles in the volume V. In the 1-D system the volume is represented by the pore length L. The restriction to nearest-neighbor interactions in 1-D allows one to write the configurational integral Z(NsA,NsB,L,T) in the following convolution
Z(NsA,NsB,L,T) ) *
∑∫0 e_βφ (x ) dyN ∫0 e-βu ∫0y e-βu (x -x ) dy2 ∫0y e-βu l
3
N
3,2
yN
N
3
2
N,N-1(xN-xN-1)
2
dyN-1 ...
2,1(x2-x1) -βφ1(x1)
e
dy1 (24)
where xi is the distance of particle i from the left wall and the variables l and yi are defined by
l ) L - NsAσAA - NsBσBB
(17)
where again the yi ) (∂Y/∂Ni)T,P,Nj*i, i ) A, B represent the partial molar property Y of component i in a bulk mixture with mole fraction xA. In particular, calorimetrically measured enthalpies of immersion, ∆wH, are often used to characterize the adsorption of binary mixtures.30 With eqs 13-17 the changes of entropy, enthalpy and Gibbs energy during an immersion experiment are given by the following relations
∆wH )
s
(13) (14)
(21)
The grand partition function Ξ is defined as
Ξ(µA, µB, L, T) )
The second equation shows the difference between a mixture in the pore and in the bulk phase at fixed experimental bulk pressure P. If pore size L goes to infinity, the pressure PsL goes to the bulk value P, and the Gibbs energy in eq 14 has the usual form for an infinite bulk phase. On differentiating eqs 5 and 7 with respect to T, the entropy of the pore system, Ss, is obtained
Ss ) -
Ωs ) -kT ln Ξ(µA, µB, L, T)
(12)
With eqs 4 and 6 these important relations follow:
Gs ) Ωs + PL + µANsA + µBNsB
statistical thermodynamics is possible using the relation to the grand partition function, Ξ, for a mixture in a pore of length L at temperature T and chemical potentials µA and µB.
(25)
i-1
yi ) xi - σi/2 -
σj ∑ j)1
(26)
Here σj denotes the length of molecule j. The φi(x) in eq 24 denotes the external field acting on molecule i, which will depend on which species molecule i belongs to. Similarly, ui,j(x) denotes the interaction potential between molecules i and j, which will depend on the species types of i and j. The asterisk attached to the summation sign denotes a summation over all possible sequences of the molecules on the line. The exact solution of eq 24 was obtained by Monson22 using the Laplace transformation of eq 24, which can be inverted after performing a binomial expansion. The formula for the the configuration integral Z(NsA,NsB,L,T) (see eq 3.8 in ref 22) is a lengthy analytical expression, but it allows a fast numerical calculation of the grand partition function Ξ. It is this analytical result that opens a path to the practical calculation of all the thermodynamic functions, introduced above, which describe adsorption of mixtures in small pores. It can be used in numerical calculations for pore sizes holding fewer than about 50 particles. Otherwise, rounding errors become too large.
1154 Langmuir, Vol. 13, No. 5, 1997
Heuchel
The average number of particles A in the pore is, e.g. s
〈NsA〉
)Ξ
-1
∑ ∑
Ns
A)1
s
A N B λN A λB
s B)0 (NA
Ns
-
1)!NsB!
Z(NsA,NsB,L,T) (27)
A similar relation is valid for component B. From both particle numbers 〈NsA〉 and 〈NsB〉 follow the mole fraction of component A in the pore, xsA ) 〈NsA〉/(〈NsA〉 + 〈NsB〉) used in eq 10. With Monson’s analytical result for the configurational integral Z(NsA,NsB,T,L) it is possible to derive an analytical expression for (∂Z/∂T)L,NAs,NBs which allows the analytical calculation of the temperature dependence of the grand potential Ξ for the 1-D square-well system. The entropy Ss, Gibbs energy Gs, and enthalpy Hs of the adsorbed phase are calculated using lengthy but analytical expressions:
ln Ξ (∂ ∂T )
Ss ) k ln Ξ + kT
(28)
L,µA,µB
Gs ) NsAµA + NsBµB + PL - kT ln Ξ Hs ) Gs + TSs ) NsAµA + NsBµB + PL + ∂ ln Ξ kT2 ∂T
(
)
L,µA,µB
(29)
(30)
The particle numbers NsA and NsB in eqs 29 and 30 have to be determined from eq 27. The thermodynamic wetting functions in eqs 18-20 are given by the following expressions
∆wH ) -T(NsAsA(xA) + NsBsB(xA)) + ln Ξ (∂ ∂T )
PL + kT2
∆wG ) PL - kT ln Ξ ln Ξ (∂ ∂T )
T∆wS ) kT ln Ξ + kT2
L,µA,µB
(31) (32)
L,µA,µB
T(NsAsA(xA) + NsBsB(xA)) (33) The last relations show that a statistical calculation of the thermodynamical functions requires expressions for the chemical potentials of the bulk 1-D square-well mixture. Applying the isobaric-isothermal ensemble Monson also calculated in his paper22 properties of a 1-D square-well mixture in the thermodynamic limit, i.e., where the boundary effects can be neglected. He got analytical expressions for the chemical potentials, µi(T, P, xA), i ) A, B for a bulk mixture of composition xA at bulk pressure P and temperature T (see eqs 4.6 and 4.7 in ref 22). Further Monson gave an analytical equation of state (see eq 4.10 and erratum in ref 22) which allows one to calculate the number of particles per unit length. The necessary partial molar entropies, sA and sB, and partial molar enthalpies, hA and hB, can be obtained from the analytically-available temperature derivatives of the bulk phase chemical potentials µA and µB. Details are given in an Appendix of this paper. 4. Results and Discussion There are a tremendous variety of systems with different potential parameters that could be studied. To show the kinds of results that can be obtained, two bulk mixtures are used as examples.
Figure 1. Adsorption excess isotherms ΓeA of component A versus bulk mole fraction xA for two binary 1-D square-well mixtures in a pore of length L ) 2.0σAA at constant bulk pressure PσAA/kT ) 2.0: left panel, bulk mixture I, i.e., an ideal mixture with σAA ) σBB ) 1.0 and AA ) BB ) BW ) 1.0; right panel, bulk mixture II, i.e., an athermal mixture with σAA ) 1.0, σBB ) 0.6, and AA ) BB ) BW ) 1.0. The numbers on the curves give the strength of the A-wall interaction parameter AW.
Model mixture I: All square-well interaction parameters of components A and B are equal. σAA ) σBB ) 1.0 and AA ) BB ) 1.0. Model mixture I represents an ideal mixture. Model mixture II: Particles B are smaller than A, with σAA ) 1.0 and σBB ) 0.6. The -values are equal for both components, AA ) BB ) 1.0. Model II represents an athermal mixture. For these two mixtures the adsorption excess isotherms and the thermodynamic wetting functions are calculated. In all calculations the square-well potential range λ is fixed to λij ) σij/2. The bulk phase pressure was always PσAA/kT ) 2.0. All lengths and distances are scaled by σAA. All -values are given in units of kT. To compare results for different pore size L, the adsorption excess ΓeA is reported as particles per unit length of the pore, [N/σAA], and the thermodynamic wetting quantities are given in [kT/σAA]. 4.1. Adsorbate-Wall Interaction. The relative strength of the adsorbate-wall interaction determines significantly the selective adsorption of a mixture component in a pore. As a measure for this effect the adsorption excess isotherm ΓeA(xA), well-known from liquid mixture adsorption, is used. The two panels in Figure 1 show calculated excess isotherms for model mixtures I and II in a small pore of length L ) 2.0. The interactions of the components with the walls of the pore are assumed in the following way: whereas a molecule B interacts with a wall as with an other molecule B, i.e., BW ) BB, the wall interaction parameter for A, AW, is varied between 1.0 and 4.0. The left panel of Figure 1 shows the adsorption excess for model mixture I. For AW ) 1.0, ΓeA ) 0 for all bulk mole fractions xA; i.e., there is no preferred adsorption of one component in the pore, because particle-wall and particle-particle interactions are equal. With stronger A-wall interaction the excess adsorption of A increases. The resulting adsorption excess isotherms are of types I and II in the empirical classification of Nagy and Schay.31 For model mixture II, where the components have different size, a different kind of behavior results. In the case of equal particle-wall interaction, i.e., AW ) BW ) 1.0, the smaller particles B are generally preferred; i.e., the adsorption excess of A is mostly negative, ΓeA < 0. Only for bulk mole fractions xA e 0.2 is there a very small positive excess of A. To understand this behavior, it is important to note that a pore with L ) 2.0 can accom(31) Nagy, L. G.; Schay, G. Magy. Kem. Foly. 1960, 66, 31.
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Figure 2. Thermodynamic functions of immersion or wetting yw ≡ (∆wY(xA) - ∆wYB*)/L versus bulk mole fraction xA for two bulk mixtures with respect to pure component B. (For ∆wY see eq 17). Mixture I (left panel), mixture II (right panel). Pore length is L ) 2.0σAA. The bulk pressure is PσAA/kT ) 2.0. The A-wall interaction parameter AW is changed from 1.0 to 4.0 and BW ) 1.0 is constant. Symbols are y ≡ g (line), y ≡ h (circles), and y ≡ -Ts (triangles). (For the square-well parameters of both mixtures see Figure 1.)
modate easily three particles of B, but only one particle of A can move freely in the pore (the single state with two A’s in the pore does not contribute significantly to the distribution function, because it has a very low probability). In a pore with L ) 2.0 a single A-particle can interact only with one of the walls, so the situation for A-particles in the pore is disadvantageous in comparison to the bulk mixture II, where every A has two neighbors with an interaction equal in strength to the A-wall interaction. With increasing AW-value a more positive excess of A results at bulk concentrations where A is dissolved in B. The resulting excess isotherms belong to types IV and V of the Schay-Nagy classification. How do the thermodynamic wetting functions in eqs 31-33 reflect different particle-wall interactions? In accordance with the experimental situation in liquid mixture adsorption (see eq 11) the wetting functions are calculated here with respect to pure component B. As a shorthand notation, the wetting function difference per unit length of the pore yw is introduced for the difference between wetting values for the mixture with bulk composition xA, ∆wY(xA), and the pure component B, ∆wYB*
yw ≡ (∆wY(xA) - ∆wYB*)/L
(34)
If both components of mixture I, A and B, interact equally strong with the pore walls, all yw-values are zero (see upper left of Figure 2). With increasing preferred adsorption of A (higher AW-values) the plots for the difference of Gibbs energy gw, and enthalpy hw are stronger decreasing functions. The drop of the curves at low bulk mole fractions becomes steeper with increasing AW-value, and at AW ) 4.0 the enthalpy difference of wetting hw becomes nearly constant for xA > 0.25. Positive values for the wetting entropy differences -Tsw mean a stronger ordered structure in the pore phase with respect to the bulk phase. The maximum value of the -Tsw curve is shifted to smaller bulk mole fractions xA with increasing preferred adsorption of A. Now it is interesting to see how a reduction in size of one component, e.g., B, effects the wetting functions yw.
For the athermal mixture II the gw(xA)-curves (see right panel of Figure 2) have a minimum at a bulk mole fraction xA where the azeotrope appears in the excess isotherm. With increasing particle-wall interaction of A, the difference of wetting enthalpies hw(xA) shows minima at smaller mole fractions xA. For AW ) 1.0, i.e., equal interaction of A and B with the walls, the preferred adsorption of the smaller component B in the pore is responsible for the positive gw-value at xA ) 1.0. 4.2. Pore Size. Now that it has been shown how thermodynamic wetting functions depend on different particle-wall interactions, in this section the particlewall interaction strengths, i.e., AW and BW, are kept constant and only the pore size L is changed. Again, the discussion is restricted to the two example mixtures I and II. For the ideal bulk mixture I it is assumed that A interacts twice as strong as B with the walls, i.e., AW ) 2BW, and BW ) AA ) BB ) 1.0. To measure the strength of the interaction of single particles with the pore, the Henry’s law constant KH can be used. It is defined as the value of the one-particle configurational integral Z1 ≡ Z(N)1,T,L) (see eq 24) divided by the pore length L. For the 1-D square-well system KH is (see also ref 23):
KH ) 0
0eLeσ
)
L - σ 2βW e L
)
2(L - σ - λW) βW 2λW - L + σ 2βW e + e L L
σ e L e σ + λW
σ + λW e L e σ + 2λW
)
L - σ - 2λW 2λW βW σ + 2λW e L + e L L
(35)
The parameters for single components A and B are w )
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Figure 3. Henry’s law constants KH as a function of pore size L for components A (b) and B (O) of bulk mixture I (left panel, AW ) 2.0, BW ) 1.0) and of bulk mixture II (right panel, AW ) 1.0, BW ) 1.0). (For the square-well parameters of both mixtures, see Figure 1.) Figure 5. Component-density distributions for mixture I (see Figure 1) between square-well walls with AW ) 2.0 and BW ) 1.0 for three pore lengths L/σAA ) 1.8, 2.0, and 2.2. The full line is the density of component A and the dotted line that of component B. The corresponding bulk properties are xA ) 0.5 and PσAA/kT ) 2.0.
Figure 4. Adsorption excess of component A, ΓeA, for bulk mixture I (see Figure 1) as a function of pore size L and bulk mole fraction xA at constant bulk pressure PσAA/kT ) 2.0. The particle-wall interactions are AW ) 2.0 and BW ) 1.0.
iW, λW ) λiW, and σ ) σiW with i ) A, B. The left panel of Figure 3 shows KH as function of pore size L for both components of mixture I and illustrates the four cases of eq 35. For pore sizes smaller than σ the pore is so small that no particle can enter it. The second case with increasing KH values corresponds to pores of a size where the two wall potentials completely overlap. The maximum KH is reached at a pore size L ) σ + λW. The following drop of KH belongs to the third case where the wall potentials start to overlap only partially, and the fourth case corresponds to where the wall potentials do not overlap at all. The picture shows clearly that KH becomes more strongly dependent on L for L < σ + 2λW and the effect is stronger for large values of W. Figure 3 shows also that the 1-D model produces a simple distinction between very small pores (complete overlap of wall potentials) and small pores (only partial overlap), and therefore it supports the distinction of micropores and sub-micropores.33,34 A comparison of the KH values for both components of mixture I leads one to expect a preferential adsorption of component A for all pore sizes. The positive adsorption excess ΓeA in Figure 4, plotted as function of pore size L and bulk mole fraction xA, confirms this. For larger pores the excess adsorption becomes very (32) Vanderlick, T. K.; Davis, H. T.; Percus, J. K. J. Chem. Phys. 1989, 91, 7136. (33) Lastoskie, C.; Gubbins, K. E.; Quirke, N. J. Phys. Chem. 1993, 97, 4786. (34) Sing, K. S. W.; Everett, D. H.; Haul, R. A. W.; Moscou, L.; Pierotti, R. A.; Rouquerol, J.; Siemieniewska, T. Pure Appl. Chem. 1985, 57, 603.
small, because the importance of particle-wall interactions decreases and the compositions in pore and bulk become very similar. The striking feature of the twodimensional plot in Figure 4 is the oscillations in ΓeA(L) for fixed bulk composition xA. Corresponding xsA(L) plots at fixed bulk concentrations have been presented for hard rod mixtures by Vanderlick et al.32 and for square-well mixtures by Monson.22 The oscillations are related to the fact that only a whole number of particles may be in a given pore. Also, the different interactions of the configurations of these particle-arrangements are important. The strongest effect appears at L ) 2.0. To get more insight into what is responsible for the minimum values of ΓeA at this special pore size, the density profiles of the single components have been calculated. Recently, Monson22 has shown how molecular distribution functions for square-well systems may be obtained in the grand ensemble. In the case of a square-well mixture between square-well walls at distance L, he obtained the following expression for the density of component A at distance x from the left pore wall
ρA(x) ) λAe-βφA(x)
ΞWA(x - 1/2σA)ΞAW(L - x - 1/2σA) ΞWW(L)
(36)
and similarly for component B
ρB(x) ) λBe-βφB(x)
ΞWB(x - 1/2σB)ΞBW(L - x - 1/2σB) ΞWW(L)
(37)
The index WW denotes the walls at the end of the pore, WA denotes that the right wall interacts with the particles in the system as if it were a particle of sort A, etc. Figure 5 shows the density profiles for L ) 1.8, 2.0, and 2.2 at bulk mole fraction xA ) 0.5. A pore with L ) 1.8 can hold only one particle of mixture I. The highest local density F is in a range of 0.2σAA with overlapping wall-potentials in the middle of the pore. The double strength of particlewall interaction of A with respect to B (AW ) 2BW) leads to a six-times higher density of A in the pore. Also, for L ) 2.0 there is essentially only one particle in the pore.
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Figure 6. Thermodynamic immersion or wetting functions yw ≡ (∆wY(xA) - ∆wYB*)/L (see Figure 2) versus pore size L for the binary square-well mixture I (see Figure 1) between squarewell walls with AW ) 2.0 and BW ) 1.0 at constant bulk pressure PσAA/kT ) 2.0 and bulk mole fraction xA ) 0.5: y ≡ g (line), y ≡ h (circles), y ≡ -Ts (triangles).
But at this pore size there is no range of overlapping particle-wall potentials. Therefore, the double strength of the A-wall interaction leads only to a double local density of A with respect to B and the resulting adsorption excess ΓeA is smaller than for the smaller pore with L ) 1.8. For a pore size of L ) 2.2 two particles can enter the pore. On both walls A-particles are preferred so that a density proportion of 4:1 in favor of A results, and the excess of A becomes again higher with respect to the smaller pore with L ) 2.0. In accordance with the adsorption excess isotherms ΓeA, the thermodynamic wetting function differences yw also show oscillations with respect to pore size L. Figure 6 contains the difference in Gibbs energy gw, enthalpy hw, and entropy -Tsw for bulk mole fraction xA ) 0.5. The largest “amplitudes” belong to the smallest pores a particle can enter. To interpret the behavior, including the cusps at L ) 2.0, it is useful to remember that yw values represent differences of wetting function for a bulk mixture and for a pure component B (see eq 34). In Figure 7 both single contributions to these differences, ∆wY(xA ) 0.5) and ∆wYB*, are shown together with the value for pure component A, ∆wYA*. All these functions oscillate with pore size L. The upper panel of Figure 7 shows the Gibbs energy of wetting (or immersion) which is, according to eq 20, equal to the difference of bulk pressure P and local pressure in the pore PsL, i.e., ∆wG/L ) P - PsL. Pores with L < 1.0 cannot be entered by mixture particles. Therefore, the local pressure PsL is zero, PsL ) 0, and ∆wG/L ) P is positive and has the value for the bulk pressure. Immersion or wetting of larger pores with pure A shows a negative Gibbs energy of immersion, ∆wG < 0, i.e., the adsorption of pure A is spontaneous. This is not true for pure component B. Especially in the pore range between L ) 1.7 and 2.5 there is no spontaneous immersion with pure B. This is understandable, because with the selected model parameters the particle-wall interaction is equal to the particleparticle interaction. So, entering pores of this size range means for bulk B particles losing one interaction. In the bulk phase the density is higher, and therefore the net interaction with neighboring particles is greater. This view is supported by the ∆wH curve for pure B in Figure 7, which goes back to zero at L ) 1.7. Likewise, the ordering state in the pore becomes lower. If the pore size increases, beginning with L > 4.0, the part of pore space without particle-wall interaction becomes larger and therefore the thermodynamic states of particles in the pore and in the bulk become more and more similar. The effect of wetting or immersion is decreasing, as can be seen on all curves of wetting functions in Figures 6 and
Figure 7. Wetting functions ∆wG/L, ∆wH/L, and -T∆wS/L versus pore size L for bulk mixture I (see Figure 1) for pure A (b), pure B (O), and a bulk mixture with xA ) 0.5 (s) in pores with square-well walls (AW ) 2.0, BW ) 1.0) at constant bulk pressure PσAA/kT ) 2.0.
7. The ∆wY-curves for the equal molar bulk mixture xA ) 0.5 in Figure 7 develop similarly to the respective curves for pure component A. Therefore, the yw curves in Figure 6 belonging to the same bulk mole fraction can be more or less understood as differences of the immersion functions ∆wY of both pure components. Further, a comparison of enthalpy and entropy curves in Figure 7 shows that the local minima of the wetting enthalpy are related to maxima in -T∆wS, i.e., stronger ordering. Therefore, the preferred adsorption of A in the pore produces more ordered configurations with respect to the pure bulk mixture. A further feature, which all ∆wY(L)-plots show, is the cusps at L ) 2.0. They can be explained with respect to the minimum of local densities for this pore size discussed before. As a second example for pore size effects on thermodynamics of mixture adsorption, the athermal bulk mixture (model mixture II) was selected. To see the role of nonequal particle size without the complication of different particle-wall interactions, all -values have been taken to be equal, i.e., ij ) 1.0, i, j ) A, B, W. The only remaining distinction between particles A and B is their different size. It follows that the Henry’s law constants for A and B (right panel of Figure 3) have the same maximal height, but the positions of the maxima are shifted on the L-axis. From the KH values one expects a selective adsorption of B in pores with L < 1.0, because these pores cannot be entered by A-particles. For pore sizes between 1.3 and 1.9 the KH value, a measure for interaction strength (affinity) of the pore to a single particle, is greater for A than for B. For larger pores the KH values are very similar. Figures 8-10 show the pore size dependence of adsorption excess and of different thermodynamic wetting functions. A comparison of Henry’s constants in Figure 3 with the adsorption excess of A, ΓeA, in Figure 8 shows that selective adsorption can be explained only partly by different KH values of A and B. The excess adsorption ΓeA(L, xA), shown in Figure 8, is a two-dimensional curve rich in valleys and peaks. The first valley is for L < 1.0, because only B particles can
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Figure 8. Adsorption excess of component A, ΓeA, for the athermal bulk mixture II (see Figure 1) as a function of pore size L and bulk mole fraction xA at constant bulk pressure PσAA/ kT ) 2.0. The particle-wall interactions are AW ) 1.0 and BW ) 1.0.
Figure 10. Wetting functions ∆wG/L, ∆wH/L, and -T∆wS/L versus pore size L for bulk mixture II (see Figure 1) for pure A (b), pure B (O), and a bulk mixture with xA ) 0.5 (s) in pores with square-well walls (AW ) 1.0, BW ) 1.0) at constant bulk pressure PσAA/kT ) 2.0.
Figure 9. Wetting functions yw ≡ (∆wY(xA) - ∆wYB*)/L versus pore size L for the binary square-well mixture II (see Figure 1) between square-well walls with AW ) 1.0 and BW ) 1.0 at constant bulk pressure PσAA/kT ) 2.0 and mole fraction xA ) 0.5: y ≡ g (line), y ≡ h (circles), y ≡ -Ts (triangles).
enter such small pores, and therefore the adsorption excess of A is strongly negative. For pores with a size between 1.0 < L < 1.5 the KH value for A is higher than for B, and because of its stronger interaction with the walls, component A is preferentially adsorbed. For larger pores the influence of particle-wall interactions decreases, and because of confinement effects, the smaller component B is adsorbed preferentially at most bulk mole fractions. Figure 9 illustrates how confinement determines thermodynamic wetting function differences yw for a bulk mixture with xA ) 0.5. Layering and packing effects related to the finite size of the mixture particles cause the yw values to vary with pore size on a length scale the size of the fluid particles. The functions in Figure 9 are more structured than the corresponding functions for bulk mixture I (see Figure 6). All three yw-plots are rapidly decreasing with pore size. Already for pores larger than 3.5σAA the yw-values are very small, because the immersion effects are so small. For mixture II the gw(L) curve is positive, except in the range 1.2 e L e 1.5; i.e., at equal molar bulk composition component B is adsorbed preferentially at most pore sizes. The hw curve and -Tsw curve show oscillations with a shift in periodicity of about 0.4σAA. The local minima of gw(L) are nearly at the same positions as local minima of hw(L) and the local maxima of -Tsw. Figure 10 shows ∆wY functions for the equal molar bulk mixture and for both pure components. It is striking that all plots for both pure components show
stronger oscillations than the plots belonging to the equalmolar bulk mixture, especially for L > 2.5. A comparison of corresponding ∆wY curves for the equal-molar bulk mixture and pure component B explains the oscillations in the yw(L) curves of Figure 9 as a result of the subtraction of the reference state, pure B. The wetting functions ∆wYB* in Figure 10 have a periodicity of about 0.6 with pore size L and produce therefore the oscillations with a periodicity of about 0.4 ) σAA - σBB in the yw curves of Figure 9. 4.3. Polydisperse Systems. The analytical treatment of the 1-D square-well system also allows model studies of the influence of porosity on the wetting functions. A solid with pores of different size is assumed to be represented through a pore size distribution f(L), where L is the pore length and f(L) dL is the fraction of the system’s total length which is composed of pores of lengths between L and L + dL. In selection of an analytical model pore size distribution, the work by Kaminsky and Monson23 was followed, in which a Pearson type III distribution was chosen
f(L) )
2
( ) ()
2L 2 ν〈L〉 ν〈L〉Γ ν
(2/ν-1)
(
exp -
2L ν〈L〉
)
(38)
Lg0 where Γ(x) is the gamma function, 〈L〉 is the mean of the distribution, and ν is a dimensionless measure of the distribution’s variance. Changing ν in eq 38 allows the representation of a broad range of distributions; e.g., ν ) 0 represents a uniform distribution as in zeolites; small values, such as ν ) 0.25, give a narrow distribution similar to those found in certain real adsorbents such as silica gels, alumina gels, and carbon molecular sieves; and ν ) 1.0 stands for the gamma distribution, which is an approximation used for some activated carbon systems. Since in 1-D there is no pore connectivity, every property y for polydisperse pore size systems can be calculated by
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Figure 11. Model pore size distributions (eq 38) for mean pore sizes 〈L〉 ) 2.0 and 4.0 and for values of ν ) 0.0, 0.25, 1.0.
Figure 12. Adsorption excess ΓeA(xA) for bulk mixture I (see Figure 1) in porous materials characterized by a pore size distribution eq (38) with 〈L〉 ) 2.0 (left panel) and 〈L〉 ) 4.0 (right panel) and ν ) 0.0, 0.25, and 1.0 at constant bulk pressure PσAA/kT ) 2.0. The particle-wall interactions are AW ) 2.0 and BW ) 1.0.
integrating the behavior for isolated pores over the pore size distribution
y(P,T,xA) )
∫0∞yL(P,T,xA,L)f(L) dL
(39)
where yL is the property in a single isolated pore of size L. As examples two values for the mean pore size, 〈L〉 )
2.0, and 〈L〉 ) 4.0, were selected. This choice was made since in three-dimensional pores two molecular diameters across and four molecular diameters across are reasonable values for small and larger micropores, respectively. The distributions with ν ) 0.0, 0.25, and 1.0 are shown in Figure 11. The adsorption excess ΓeA of ideal model mixture I in materials characterized by the six chosen pore size distributions are given in Figure 12. The general shape of the excess isotherms is not changed much by porosity. In the case of 〈L〉 ) 2.0, the highest excess is calculated for ν ) 0.25, followed by ν ) 1.0 and ν ) 0.0. Using Figure 11 for a comparison of these three distributions with the same mean pore size 〈L〉 ) 2.0, one can see that the excess adsorption of A is higher when the portion of small pores with L < 2.0 is greater. For materials with the higher mean pore size, 〈L〉 ) 4.0, the influence of ν and therefore of the porosity is smaller. The highest excess of component A results now for ν ) 1, because again the material with this value of ν has the greatest proportion of small pores. The respective wetting function differences yw (see eq 34) in Figure 13 support this behavior. The biggest difference in gw between pure A and pure B results for 〈L〉 ) 2.0 and ν ) 0.25. For the three distributions with 〈L〉 ) 4.0 the influence of ν is smaller than in the case of the lower mean pore size, 〈L〉 ) 2.0. This supports the results for the excess isotherms discussed before. Finally, Figures 14 and 15 show adsorption excess isotherms ΓeA and wetting function differences yw for the athermal model mixture II. In general, the dependence on the distribution parameters, 〈L〉 and ν, is similar to that for mixture I. The special feature of a partially preferred adsorption of the larger component A, despite equal particle-wall interactions of A and B, was explained for (ν ) 0.0) as a result of the confinement in small pores. This effect is abolished if the adsorbent has a significant dispersion in pore sizes L. The adsorption excess isotherms in Figure 14 for adsorbents with ν ) 0.25 and 1.0 show for all bulk compositions xA a preferred adsorption of the smaller component B. The highest contribution to the preferred adsorption of particles B results from pores
Figure 13. Wetting functions yw ≡ (∆wY(xA) - ∆wYB*)/L for binary mixture I on porous materials. Every plot belongs to one of the adsorption excess curves of Figure 12, which gives also the parameters of the respective pore size distributions: y ≡ g (line), y ≡ h (circles), and y ≡ -Ts (triangles).
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5. Summary and Conclusions
Figure 14. Adsorption excess ΓeA(xA) for the athermal bulk mixture II (see Figure 1) in the same porous materials considered already in Figure 12, and at the same bulk pressure. The particle-wall interactions are AW ) 1.0 and BW ) 1.0.
with L < 1.0, because these pores cannot be entered by A and additionally the particle-wall interaction for B is high because of the wall-potential overlap. Therefore the highest excess values result for ν ) 1.0, because the respective distributions have the largest portion on those small pores. The wetting functions yw of the athermal mixture II, shown in Figure 15, are smaller than those for mixture I. The disappearance of the adsorption azeotrope for adsorbents with ν ) 0.25 and 1.0 corresponds to the disappearance of the minimum in the gw curve. The positive gw values signal the preferred adsorption of component B. Except for a pore size distribution with 〈L〉 ) 2.0 and ν ) 1.0, the immersion enthalpy difference hw is very small. The reason is the equal particle-wall interaction strength of A and B. The exception for 〈L〉 ) 2.0 and ν ) 1.0 results from the higher part of very small pores with L < 1.0 where B-particles can interact partly with both pore walls. Because of the small enthalpic effects, the selective adsorption of the athermal mixture II in adsorbents with a pore size distribution is governed by an entropic effect, as can be seen from the similar course of the gw and -Tsw curves.
The 1-D square-well mixture in a confined geometry represents an interesting model to study the dependence of thermodynamic functions on size and interaction effects for mixture adsorption in pores. It was shown that similar to the adsorption excess the thermodynamic wetting functions show strong confinement effects in small pores with a size of up to 4-6 times the particle size. Particle-size differences act to promote adsorption of the particle with the smaller size. If the smaller particle interacts more weakly with the wall, azeotropy results. The wetting functions reflect also a dispersity in pore size. The effect is stronger for materials with smaller mean pore size 〈L〉. The investigation of mixture adsorption on solid materials with a pore size distributions is still in its beginning. In this work only the principal approach in using an advanced but simple model for thermodynamic adsorption studies on such materials was shown. Very often, experimental results, e.g., excess isotherms or immersion enthalpies, are discussed with help of general assumptions about size differences of components, pore size, and the different interaction energies occurring in the system. The value of the 1-D model lies in the possibility to test such general molecular ideas in very fast calculations, which can easily cover the whole range of bulk compositions and which also yield results for the relevant experimentally measurable thermodynamic functions. In further studies, the 1-D model should be applied to experimental data for simple nonpolar mixtures. Suitable should be data from liquid mixture or gas mixture adsorption on energetically homogeneous surfaces, e.g., carbon molecular sieves. The pore size distribution of such materials could be determined experimentally from accurate gas adsorption isotherms using an advanced numerical procedure, which inverts the integral equation of adsorption via regularization method and calculates the local adsorption in uniform slitlike pores, e.g., according to the density functional theory.35 Other candi-
Figure 15. Wetting functions yw ≡ (∆wY(xA) - ∆wYB*)/L for binary mixture II on porous materials. Every plot belongs to one of the adsorption excess curves of Figure 14, which gives also the parameters of the respective pore size distributions: y ≡ g (line), y ≡ h (circles), and y ≡ -Ts (triangles).
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dates of porous solids for application of the 1-D model to experiments could be aluminosilicate frameworks with small cylindrical pores. Recently, it was possible to model the adsorption of xenon gas in pores of mordernite with a 1-D hard-rod model.36 Acknowledgment. The author wishes to thank P. Bra¨uer, M. v. Szombathely, and R. Q. Snurr for helpful discussions during this work. Appendix: Temperature Dependence of Bulk Phase Properties The analytical expressions for the chemical potentials of a 1-D square-well bulk mixture of mole fraction xA ) 1 - xB at bulk pressure P and temperature T have been derived in the literature (see eqs 4.6 and 4.7 in ref 22). With respect to the Lewis form (eq 7) the analytical expressions for the chemical potential of pure components A and B are
βµ*A ) ln ΛA + βPσAA - ln ξAA(βP)
(A1)
βµ*B ) ln ΛB + βPσBB - ln ξBB(βP)
(A2)
DA )
ξij(s) )
e
βij
- (e
ln γA ) -2 ln xA + ln(xA - 1/2xAB) - 1/2DA ln ζ 1
1
1
1
DA[ /2 ln(xA - /2xAB) + /2 ln(xB - /2xAB) - ln( /2xAB)] (A4)
∂xAB ∂xB
(A8)
xA,P,T
1
/2 ln(xB - 1/2xAB) - ln(1/2xAB)] (A9)
To represent first derivatives with respect to temperature in a simple way, the quantities y′ are introduced
y′ ≡
(∂β∂y)
(A10)
P,xA
The first derivatives with respect to temperature of eqs A3 and A6-A are
ξ′ij(βP) )
ijeβij + e-βPλij[(Pλij - ij)eβij - Pλij] ξij βP β (A11)
[
ζ′ ) ζ
(A3)
s
xB,P,T
( )
xB ln(xB - 1/2xAB) - xAB[1/2ln ζ + 1/2 ln(xA - 1/2xAB) +
x′AB )
- 1)e
DB )
βgE ) -2(xA ln xA + xB ln xB) + xA ln(xA - 1/2xAB) +
-sλij
where i, j ) A, B, W. The equations for the corresponding bulk phase activity coefficients, γA and γB, are
1
∂xAB ∂xA
The excess Gibbs enthalpy for the bulk mixture is
where ΛA and ΛB are the thermal de Broglie wavelengths for the two components, β is 1/kT, and ξAA and ξBB are the Laplace transforms of the nearest-neighbor square-well interaction, which is in general βij
( )
D′A )
D′B )
]
2ξ′AB ξ′AA ξ′BB ξAB ξAA ξBB ζ′(2xAxB - 1)
(1 - ζ)[ζ2 + 4ζ(1 -ζ)xAxB]1/2
ζ′xAB(2xB - xAB) - ζx′AB[2xB(ζ - 1) - ζ] [xAB(1 - ζ) + ζ]2 ζ′xAB(2xA - xAB) - ζx′AB[2xA(ζ - 1) - ζ] [xAB(1 - ζ) + ζ]2
(A12)
(A13)
(A14)
(A15)
ln γB ) -2 ln xB + ln(xB - 1/2xAB) - 1/2DB ln ζ -
With these relations, analytical expression are available for, e.g., the excess enthalpy of the bulk mixture
DB[1/2 ln(xA - 1/2xAB) + 1/2 ln(xB - 1/2xAB) - ln(1/2xAB)] (A5)
hE ) -
where the quantity xAB ) NAB*/(NA + NB) measures the maximum number of AB-pairs, N* AB. It is
xAB )
-ζ + [ζ2 + 4ζ(1 - ζ)xAxB]1/2 1-ζ ζ)
ξ2AB(βP) ξAA(βP)ξBB(βP)
(A6)
xABζ′ - x′AB[1/2 ln ζ + 1/2 ln(xA - 1/2xAB) + 2ζ 1 /2 ln(xB - 1/2xAB) - ln(1/2xAB)] (A16)
and the partial molar enthalpy of component A in the bulk mixture
ξ′AA + β(ln γA)′ ξAA
βhA ) 1/2 + βPσAA - β
(A7)
The quantities DA and DB in eqs 42-43 are abbreviations for (35) Olivier, J. P.; Conklin, W. B.; v. Szombathely, M. Stud. Surf. Sci. Catal. 1994, 87, 81. (36) Kim, S.-C.; Suh, S.-H. J. Chem. Soc., Faraday Trans. 1995, 91, 2945.
(ln γA)′ ) -
(A17)
x′AB D′A ln z DAz′ 2xA - xAB 2 2z
D′A [1/2 ln(xA - 1/2xAB) + 1/2 ln(xB - 1/2xAB) - ln (1/2xAB)] + DAx′AB(4xAxB - xAB) 4(xA - 1/2xAB)(xB - 1/2xAB)xAB LA960125A
(A18)
