Adsorption Compression - American Chemical Society

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Adsorption Compression: An Important New Aspect of Adsorption Behavior and Capillarity G. L. Aranovich and M. D. Donohue* Department of Chemical Engineering, The Johns Hopkins University, Charles and 34th Streets, Baltimore, Maryland 21218 Received August 22, 2002. In Final Form: December 23, 2002 A previously unstudied aspect of adsorption, compression in confined phases, is discussed. For the equilibrium of a gas on a solid surface, the strong attraction to the surface causes adsorbate molecules to attain much higher densities than those of equilibrium condensed phases. Under these conditions, adsorbate molecules repel each other and heats of adsorption vanish. Adsorption compression is a nanoscale analogue of macroscale compression of air in the gravitational field of the Earth. The difference is that nanoscale compression occurs at immeasurably higher gradients of molecular forces and compression can be extremely large. This compression causes phase transitions and allows a supercompressed state of matter that is nearly impossible to obtain at normal conditions. A statistical mechanical theory of adsorption compression is developed in the framework of the grand canonical ensemble. Simulations and experimental evidence of compressed phases on solid surfaces are discussed. The significance of this phenomenon is not limited to fundamental aspects of adsorption and capillarity; it also plays a crucial role in various applications such as heterogeneous catalysis, membrane separations, and self-assembly on surfaces.

Introduction The adsorption behavior of molecules on solid or liquid surfaces depends on temperature, pressure (or density), and the strength of the adsorbate-adsorbent and adsorbate-adsorbate interactions. The best known and most widely used adsorption isotherm is that of Langmuir,1 which takes into account adsorbate-adsorbent interactions but ignores adsorbate-adsorbate interactions. The Langmuir isotherm is illustrated in Figure 1 (line 1). It predicts that the adsorption rises linearly at low p/ps (i.e., Henry’s law behavior) and then levels off as the surface becomes covered with adsorbed molecules. Since Langmuir’s paper, numerous authors have tried to improve his isotherm by relaxing one or more of his assumptions. Frumkin2 and Fowler and Guggenheim3 allowed lateral interactions between molecules in the adsorbed monolayer and developed an isotherm that could predict a two-dimensional phase transition. This phase transition gives a step in the adsorption isotherm as shown by line 2 of Figure 1. However, this theory2,3 neglected the interactions perpendicular to the surface causing multilayer adsorption and condensation. Brunauer, Emmett, and Teller4 derived an equation for multilayer adsorption by considering adsorbate-adsorbate interactions in the direction perpendicular to the adsorbent surface, but not parallel to the surface. This assumption turns the threedimensional problem into a one-dimensional problem. Therefore, the Brunauer-Emmett-Teller (BET) theory is able to predict multilayer behavior and condensation (see line 3 in Figure 1) but does not predict two-dimensional phase transitions in the adsorbed layers. Other classical contributions in understanding adsorption include the theory of long-range interactions for * To whom correspondence should be addressed. (1) Langmuir, I. J. Am. Chem. Soc. 1916, 38, 2221. (2) Frumkin, A. Z. Phys. Chem. (Leipzig) 1925, 116, 466. (3) Fowler, R. H.; Guggenheim, E. A. Statistical Thermodynamics; Cambrige University Press: London, 1949. (4) Brunauer, S.; Emmett, P. H.; Teller, E. J. Am. Chem. Soc. 1938, 60, 309.

Figure 1. Adsorption isotherms: Langmuir (1), Frumkin/ Fowler-Guggenheim (2), and BET (3). Typical variables are adsorbed amount, a (in number of monolayers), and relative pressure p/ps where p is the absolute pressure and ps is the saturation vapor pressure.

molecules on surfaces,5 the potential theory of adsorption,6,7 application of Laplace’s,8 Kelvin’s,9 and Gibbs’10 concepts of capillarity for understanding adsorption hysteresis,11 and equations of state for adsorbed layers.12 Recent work has built on these classical concepts with molecular simulations and statistical mechanical calculations.13 Steele14 developed potential functions for molecules on surfaces taking into account realistic structures of (5) Lennard-Jones, J. E. In The Adsorption of Gases by Solids; Faraday Society: London, 1932; p 333. London, F. Z. Phys. Chem. 1930, 11, 222. (6) Polanyi, M. Verh. Dtsch. Phys. Ges. 1916, 15, 55. (7) Hill, T. L. Theory of Physical Adsorption. Adv. Catal. 19481952, 4, 211-255. (8) Laplace, P. S. Traite de Mecanique Celeste: Supplement au dixieme livre, Sur l’Action Capillaire; Courcier: Paris, 1806. (9) Tomson, W. Lecture on Capillary Attraction, 1886. In Popular Lectures and Addresses; McMillan: London, 1989; Vol. 1. (10) Gibbs, J. W. On the Equilibrium of Heterogeneous Substances, Trans. Connecticut Acad. In The Scientific Papers of J. Willard Gibbs; Dover: New York, 1961. (11) Defay, R.; Prigogine, I.; Bellemans, A.; Everett, D. H. Surface Tension and Adsorption; Longmans: London, 1966. (12) Toth, J. Acta Chim. Acad. Sci. Hung. 1971, 69, 311.

10.1021/la020739h CCC: $25.00 © 2003 American Chemical Society Published on Web 02/22/2003

Adsorption Compression

adsorbents. Nicholson15 modeled properties of molecules in micropores. Gubbins et al.16 developed methods of molecular simulations for phase transitions in confined spaces. Tarazona, Marini Bettolo Marconi, and Evans17 applied density functional theory for interfaces and studied the microscopic structure of the adsorbed phase. Fisher and de Gennes18 analyzed the adsorption behavior near critical points. Swift, Cheng, Cole, and Banavar19 studied critical behavior of adsorbed phases in porous media. Benard and Chahine20 modeled adsorption on microporous adsorbents at supercritical conditions, and de Keizer, Michalski, and Findenegg21 analyzed shifting critical points for fluids in micropores. Recently, the authors have developed lattice density functional theory (LDFT) to study fluids in a confined environment.22-25 This work is based on ideas originally proposed by Ono and Kondo26,27 for density gradients at vapor-liquid interfaces. LDFT is able to predict a wide variety of behavior including multiplayer adsorption,28 hysteresis in micropores,29 adsorption on surfaces with molecular-scale heterogeneities,30 and adsorption in supercritical systems.31-33 In all of the classical and recent models mentioned above, it is generally assumed or derived that adsorbed molecules interact with each other through attractive forces. It is also a typical assumption that molecules in the adsorbed layer(s) are at an intermolecular distance equivalent to that in their liquid state. However, there are theoretical predictions,34,35 molecular simulations,36,37 and experi(13) Nicholson, D.; Parsonage, N. G. Computer Simulation and the Statistical Mechanics of Adsorption; Academic Press: New York, 1982. (14) Steele, W. A. Interaction of Gases with Solid Surfaces; Pergamon Press: Oxford, 1974. (15) Nicholson, D. J. Chem. Soc., Faraday Trans. 1 1996, 92, 1-9. (16) Gubbins, K. E.; Sliwinska-Bartkowiak, M.; Suh, S.-H. Mol. Simul. 1996, 17, 333. Radhakrishnan, R.; Gubbins, K. E. Phys. Rev. Lett. 1997, 79, 2847. (17) Tarazona, P.; Marini Bettolo Marconi, U.; Evans, R. Mol. Phys. 1987, 60, 573. (18) Fisher, M. E.; de Gennes, P.-G. C. R. Acad. Sci. (Paris) 1978, 287, 207. (19) Swift, M. R.; Cheng, E.; Cole, M. W.; Banavar, J. R. Phys. Rev. B 1993, 48, 3124. (20) Benard, P.; Chahine, R. Langmuir 1997, 13, 808. (21) de Keizer, A.; Michalski, T.; Findenegg, G. H. Pure Appl. Chem. 1991, 63, 1495. (22) Aranovich, G. L.; Donohue, M. D. Physica A 1997, 242, 409. (23) Aranovich, G. L.; Donohue, M. D. J. Colloid Interface Sci. 1998, 200, 273. (24) Aranovich, G. L.; Donohue, M. D. Phys. Rev. E 1999, 60, 5552. (25) Aranovich, G. L.; Donohue, M. D. J. Chem. Phys. 2000, 112, 2361. (26) Ono, S.; Kondo, S. Molecular Theory of Surface Tension in Liquids. In Encyclopedia of Physics; Flu¨gge, S., Ed.; Springer: Berlin, 1960; Vol. 10, p 134. (27) Rowlinson, J. S.; Widom, B. Molecular Theory of Capillarity; Clarendon Press: Oxford, 1982. (28) Aranovich, G. L.; Donohue, M. D. J. Colloid Interface Sci. 1997, 189, 101. (29) Donohue, M. D.; Aranovich, G. L. J. Colloid Interface Sci. 1998, 205, 121. (30) Aranovich, G. L.; Donohue, M. D. J. Chem. Phys. 1996, 104, 3851. (31) Aranovich, G. L.; Donohue, M. D. J. Colloid Interface Sci. 1996, 180, 537. (32) Donohue, M. D.; Aranovich, G. L. Adv. Colloid Interface Sci. 1998, 76-77, 137. (33) Donohue, M. D.; Aranovich G. L. Fluid Phase Equilib. 1999, 158, 557. (34) Aranovich, G. L.; Donohue, M. D. Colloids Surf., A 2001, 187188, 95. (35) Rice, S. A.; Gryko, J; Mohanty, U. In Fluid Interfacial Phenomena; Croxton, C. A., Ed.; John Wiley: New York, 1986; Section 6. (36) Rittner, F.; Boddenberg, B.; Bojan, M. J.; Steele, W. A. Langmuir 1999, 15, 1456. (37) D’Evelyn, M. P.; Rice, S. A. Phys. Rev. Lett. 1982, 47, 1844; Discuss. Faraday Soc. 1982, 16, 71; J. Chem. Phys. 1983, 78, 5081; J. Chem. Phys. 1983, 78, 5225.

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ments38-41 indicating a phenomenon that has not been formulated and discussed earlier, that is, compression of molecules in adsorbed phases. In particular, measurements performed by the surface forces apparatus indicate an adsorbed phase of cyclohexane on mica with a density “∼9% above that of the bulk liquid”38 and “the existence of unexpectedly large density variations”.39 An anomalous decrease of the distance (by 0.36 Å) between magnesium chloride molecules on the (111) face of palladium was observed by measurements of low-energy electron diffraction.40 Decrease of the area per molecule with increasing energy of molecule-surface interactions is known for argon, krypton, pentane, and nitrogen on various solids.41,42 These and other results indicate a general phenomenon: the strong field of the adsorbent pulls adsorbate molecules into the surface phase at a density higher than in a normal liquid. At this state, adsorbate molecules repel each other43 until the loss of free energy due to a strong attraction to the surface34 is compensated by the gain in free energy due to this repulsion. Here, we present a simple theory of compression in adsorbed phases for different dimensions: (a) zero-dimensional case where two active sites on a surface are in equilibrium with a gas; if the distance between sites is too large or too small, there is essentially no difference from Langmuir’s model; however, at intermediate distances, both sites can be occupied simultaneously to the expense of compression of molecules sitting on these sites; (b) one-dimensional case where adsorbate molecules fill a trough on a surface, and strong attraction to the surface induces compression of adsorbate molecules; (c) two-dimensional case, which is the classical Langmuir monolayer with one exception: molecules are “soft” and this adds a significant element, surface compression, to the standard picture of the monolayer adsorption; (d) three-dimensional case where molecules fill a nanopore or a nanocapillary. The very first model (a) allows the exact solution for the grand canonical ensemble, and the phenomenon of adsorption compression can be described rigorously. The second case (b) allows the exact solution for the lowtemperature limit. The third case (c) is monolayer adsorption of Lennard-Jones molecules, and we consider an approximation which includes the essential physics but simplifies the mathematics and predicts surface compression. In the fourth case (d), we consider compression in nanopores and nanocapillaries; since a rigorous solution for three dimensions is not possible, we develop a macroscopic approach which is empirical, but it indicates key macroscopic parameters determining compression in adsorbed phases. Confined Fluids Confined fluids have been studied extensively in connection with colloid systems and adsorption phenomena, including capillary condensation in nanoscale pores and surface nanophase transitions.24,44-50 The classical stages of adsorption include the Henry’s law range, twodimensional condensation at low temperatures, capillary (38) Heuberger, M.; Zach, M.; Spenser, N. D. Science 2001, 292, 905. (39) Israelachvili, J.; Gourdon, D. Science 2001, 292, 867. (40) Fairbrother, D. H.; Roberts, J. G.; Rizzi, S.; Somorjai, G. A. Langmuir 1997, 13, 2090. (41) Karnaukhov, A. P. J. Colloid Interface Sci. 1985, 103, 311. (42) Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity; Academic Press: London, 1982. (43) Aranovich, G. L.; Sangwichien, C.; Donohue, M. D. J. Colloid Interface Sci. 2000, 227, 553.

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condensation, and saturation.3,7,42,51-55 All these stages have been studied and understood in detail. In this paper, we consider a new aspect of adsorption behavior: adsorption compression. When there is strong affinity to the adsorbent, the distance between adsorbate molecules becomes less than in a normal liquid and nearest neighbors repel each other. This is possible because the decrease of free energy due to attraction to the adsorbent is greater than the increase of free energy due to repulsions between adsorbate molecules. In other words, a strong attraction to the adsorbent causes compression of the adsorbate. Competition between attraction to the adsorbent and repulsions of neighbors in the adsorbate can result in compression of the adsorbate (adsorption compression) in different situations: on open surfaces (surface compression), in fine pores of a solid (nanopore compression), and in fine capillaries (nanocapillary compression). In the case of nanoscale pores or slits, compression can be larger than on an open surface because the attraction to the walls of a nanopore is greater than on an open surface due to the overlapping of potentials from the walls. Adding molecules to the confined space (surface) is thermodynamically favorable as long as the decrease of the free energy due to the strong attraction is more than its increase due to repulsions between neighbors. The stronger the attraction to walls, the greater the compression. In this paper, we analyze adsorption compression phenomena for Lennard-Jones molecules where the energy of interaction is described by the potential function

φ(r) ) 4[(σ/r)6 - (σ/r)12]

(1)

where σ and are the size and energy parameters. For instance, for argon σ ) 3.499 Å and /k ) -118.13 K.56 The minimum in this function is at r ) r0 ) 21/6σ. Adsorption on Active Sites To illustrate the simplest case of adsorption compression which allows rigorous theoretical treatment, consider two active sites on the surface that can be occupied by molecules as in Figure 2. When the distance between the active sites is large, adsorbate molecules can sit on the sites independently (Figure 2a). When the active sites are close and their attraction of adsorbate molecules is strong, both sites can be occupied, but the adsorbed molecules will repel each other (Figure 2b). In this case, attraction to the active sites must be stronger than repulsion between neighbors. Therefore, adsorption of both molecules simultaneously is thermodynamically favorable; however, (44) Deriagin, B. V. Acta Physicochim. URSS 1940, 12, 181. (45) Saam, W. F.; Cole, M. W. Phys. Rev. B 1975, 11, 1086. (46) Evans, R.; Marconi, U. M. B.; Tarazona, P. J. Chem. Phys. 1986, 84, 2376. (47) Dietrich, S. In Phase Transitions and Critical Phenomena; Domb, C., Lebowitz, J. L., Eds.; Academic Press: London, 1988; Vol. 12, p 1. (48) Cracknell, R. F.; Nicholson, D.; Quirke, N. J. Chem. Soc., Faraday Trans. 1994, 90, 1487. (49) Hess, G. B.; Sabatini, M. J.; Chan, M. H. W. Phys. Rev. Lett. 1997, 78, 1739. (50) Bojan, M. J.; Steele, W. A. Carbon 1998, 36, 1417. (51) Hill, T. L. Proc. Natl. Acad. Sci. U.S.A. 1996, 93, 14328. (52) Valenzuela, D. P.; Myers, A. L. Adsorption Equilibrium Data Handbook; Prentice Hall: Englewood Cliffs, NJ, 1989. (53) Adamson, A. W. Physical Chemistry of Surfaces; John Wiley and Sons: New York, 1976. (54) Brunauer, S. The Adsorption of Gases and Vapors; Princeton University Press: Princeton, 1945. (55) Rouquerol, F.; Rouquerol, J.; Sing, K. S. W. Adsorption by Powders and Porous Solids; Academic Press, London, 1999. (56) Prausnitz, J. M. Molecular Thermodynamics of Fluid-Phase Equilibria; Prentice-Hall: Englewood Cliffs, NJ, 1969.

Figure 2. Molecules on two active sites.

the distance between such molecules is smaller than in a normal liquid and adsorbate molecules repel each other. This is the simplest case of the adsorption compression, and it can occur whenever the lattice spacing for molecules in the adsorbent is smaller than the minimum in the potential function for adsorbate-adsorbate interactions. An example of such a system is the adsorption of xenon (with σ ) 4.1 Å56 and r0 ≈ 4.6 Å) on sodium chloride where the lattice spacing (distance between sodium atoms) is less than 4.0 Å.57 At very small distances between active sites (Figure 2c), this effect disappears: the A molecule blocks the neighboring active site because repulsion between neighbors exceeds attraction to the active sites at this distance. For the grand canonical ensemble, the variables are chemical potential, µ, number of molecules on active sites, N, and absolute temperature, T. To calculate the grand canonical partition function, Ξ, we assume that a gas phase is in equilibrium with two active sites with ψ0 being the energy of molecule-active site interactions and d being the distance between sites. For this model, there are four different states: one state with both active sites empty; the configurational energy of this state is zero; two states where only one of the sites is occupied; the configurational energy of each of these states is ψ0; one state where both sites are occupied and the interaction energy between molecules sitting on these sites is φ(d), 6

12

[(dσ) - (dσ) ]

φ(d) ) 4

(2)

the configurational energy of this state is 2ψ0 + φ(d). The grand canonical partition function for this system can be written as

( )

Ξ ) exp -

(

)

(

)

E′1 E′′1 E0 µ µ + exp + exp + kT kT kT kT kT 2µ E2 exp (3) kT kT

(

)

(57) Vu, N.-T.; Jakalian, A.; Jack, D. B. J. Chem. Phys. 1997, 106, 2551.


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Figure 3. 〈〈N〉〉 as a function of µ/kT at σ ) 3.5 Å, /kT ) -0.5, ψ0/kT ) -7, and d ) 2 Å (a) and d ) 4 Å (b).

where k is Boltzmann’s constant and

E0 ) 0

(4)

E′1 ) E′′1 ) ψ0

(5)

E2 ) 2ψ0 + φ(d)

(6)

The average number of molecules, 〈〈N〉〉, sitting on the two active sites is

〈〈N〉〉 )

[ (

)

(

)

E′1 E′′1 µ µ 1 exp + exp + Ξ kT kT kT kT 2µ E2 2 exp kT kT

(

)]

(7)

Plugging eqs 3-6 into eq 7 gives

(

)

[

]

ψ0 µ 2µ 2ψ0 φ(d) + 2 exp kT kT kT kT kT 〈〈N〉〉 ) ψ0 µ 2µ 2ψ0 φ(d) 1 + 2 exp + exp kT kT kT kT kT (8) 2 exp

(

)

[

]

The average energy of interaction between molecules on the two active sites, 〈〈U〉〉, can be calculated as

〈〈U〉〉 )

[

]

2µ 2ψ0 φ(d) kT kT kT φ(d) (9) ψ µ 0 2µ 2ψ0 φ(d) 1 + 2 exp + exp kT kT kT kT kT exp

(

)

[

]

Figure 3 shows 〈〈N〉〉 as a function of µ/kT predicted by eq 8 at σ ) 3.5 Å, /kT ) -0.5, and ψ0/kT ) -7. In Figure

3, plate a is for d ) 2 Å and plate b is for d ) 4 Å. For d ) 2 Å, a molecule sitting on one of the two sites blocks both sites; therefore, in Figure 3a, 〈〈N〉〉 approaches 1 at high chemical potentials. For d ) 4 Å, the two sites act independently, and 〈〈N〉〉 approaches 2 at high chemical potentials (Figure 3b). Figure 3 shows cases when the distance between active sites is small (a molecule sitting on one of the sites blocks both sites) and large (the sites are occupied independently). However, there are intermediate cases. Figure 4 shows 〈〈N〉〉 as a function of µ/kT predicted by eq 8 at σ ) 3.5 Å, /kT ) -0.5, d ) 3 Å, and various values of ψ0/kT. As shown in Figure 4a, at ψ0/kT ) -5, increasing chemical potential results in 〈〈N〉〉 approaching unity and even crossing the line 〈〈N〉〉 ) 1. At higher ψ0/kT, this tendency is more pronounced: the curve crosses line 〈〈N〉〉 ) 1 and there is a significant probability for both sites being occupied, even though d < σ. Therefore, in this situation the molecules are compressed. Figure 5 shows 〈〈U〉〉 as a function of µ/kT predicted by eq 8 for the cases given in Figure 4. As seen from Figure 5, at low chemical potentials, 〈〈U〉〉 is about zero because only one site is occupied; at high chemical potentials, 〈〈U〉〉 becomes positive which means repulsion between molecules sitting on the two sites. Figure 6 illustrates the results shown in Figures 4 and 5: at high chemical potential, the distance between adsorbate molecules is less than in a normal liquid. Figure 4 indicates that compression of molecules on active sites (as shown in Figure 2b) can cause steps in the isotherm. Depending on the ratio d/r0, these steps can be more or less pronounced, but they occur in the range where there is no phase transition (such as wetting or capillary condensation). These steps have been observed in highresolution measurements for various systems. Figure 7 gives an example of such a behavior for nitrogen on pyrocarbon at T ) 77 K (data from ref 58). In Figure 7, the derivative of the adsorption isotherm, D ) dVads/d ln(p/ps), is plotted versus ln(p/ps) where Vads is the adsorbed amount (in cm3/g), p is the pressure, and ps is the saturation vapor pressure. Oscillations in Figure 7 at very low pressures indicate steps in the isotherm similar to those shown in Figure 4. Note that ln(p/ps) gives changes of the relative chemical potential, µ/kT, for an ideal gas. There are other systems with such steps in the adsorption isotherms. These include argon and nitrogen on calcite and apatite at 77 K;59 argon, oxygen, and nitrogen on sodium chloride in the range of temperatures from 70 to 80 K;60 nitrogen on pillared clays at 77 K;61 and acetylene and water on activated carbons at various temperatures.62 The model defined by eqs 2-9 and illustrated in Figure 2 is the simplest one. It includes some of the essential physics but neglects other factors, such as the following: The distance between adsorbate molecules and the surface (i.e., the length of the bond and its angle to the surface) can vary; this would allow relaxing adsorbate-adsorbate bonds in the compressed state. In many real adsorbents, there is a distribution of distances between sites; so, only (58) Villieras, F.; Leboda, R.; Charmas, B.; Bardot, F.; Gerard, G.; Rudzinski, W. Carbon 1998, 36, 1501. (59) Villieras, F.; Michot, L. J.; Bernardy, E.; Chamerois, M.; Legens, C.; Gerard, G.; Cases, J. M. Colloids Surf., A 1999, 146, 163. (60) Orr, W. J. C. Proc. R. Soc. London, Ser. A 1939, 173, 349. (61) Gil, A.; Guiu, G.; Grange, P.; Montes, M. J. Phys. Chem. 1995, 99, 301. (62) Aharoni, Ch.; Romm, F. Langmuir 1995, 11, 1744.

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Figure 4. 〈〈N〉〉 as a function of µ/kT at σ ) 3.5 Å, d ) 3 Å, /kT ) -0.5, and various ψ0/kT values: -5 (a), -7 (b), -10 (c), and -15 (d).

Figure 5. 〈〈U〉〉 as a function of µ/kT at σ ) 3.5 Å, d ) 3 Å, /kT ) -0.5, and various ψ0/kT values: -5 (a), -7 (b), -10 (c), and -15 (d).

some fraction of nearest neighbors can be in the compressed state. There are other factors including directional interactions, complex geometry of molecules, change of electron configurations near the surface, and so forth. Some of these (and other factors) can increase compression, but some can reduce this effect. In such systems, the waves shown in Figure 4 can be smoothed or overlapped. However, the surface compression phenomenon needs be taken into consideration in analysis of the adsorption behavior.

Molecules in a One-Dimensional Box Exact Solution for the Low-Temperature Limit. Consider Lennard-Jones molecules in a one-dimensional box with square-well walls. Though highly idealized, this analysis is not unrealistic because one-dimensional molecular systems can exist, in particular, in nanoscale pores of adsorbents or between chains of polymers. Further, having Lennard-Jones molecules in a one-dimensional square-well box is similar to the assumption used in BET theory of a one-dimensional model for multilayer adsorp-


Langmuir, Vol. 19, No. 7, 2003 2727

Figure 6. Density distribution and Lennard-Jones potential.

Figure 8. The molecule-wall potential, ψ, as a function of the coordinate, x, between walls in a one-dimensional box. The symbol of four ellipses in a rectangle represents the molecules in the one-dimensional box.

Figure 7. Derivative of adsorption isotherm, D ) dVads/d ln(p/ps), for nitrogen on pyrocarbon at T ) 77 K. Here Vads is the adsorbed amount in cm3/g.

tion.51

Considering Lennard-Jones molecules in a onedimensional box with square-well walls enables one to include the essential physics but simplifies the mathematics and allows an analytical solution in the lowtemperature limit. Figure 8 illustrates the molecule-wall potential, ψ, as a function of the coordinate, x, between walls (ends) of the box. Here, ψ0 is the square-well depth and d is the distance between walls (ends). A trough or ditch one molecule deep and one molecule wide on a solid surface is an example of a real system corresponding to the one-dimensional box, as shown in Figure 9. For the grand canonical ensemble of molecules in a onedimensional box that are in equilibrium with a gas phase, the variables are chemical potential, µ, number of molecules in the box, N, and absolute temperature, T. To calculate the grand canonical partition function, Ξ, we assume the following: (a) Interactions occur only between nearest neighbors. (b) There is mechanical equilibrium between molecules in the box; this assumption implies the following two conditions: If d/(N - 1) g r0, then the most probable distance between nearest neighbors is r0 (as in a normal liquid); this corresponds to the minimum in the intermolecular potential energy function and the maximum in the one-dimensional analogue to the radial distribution function; at this distance, the net force on each molecule is zero. If d/(N - 1) < r0, then the average distance between nearest neighbors is d/(N - 1); in this case, the net force on each molecule also is zero because forces from both

Figure 9. Molecules in a trough on a surface: (a) low density; (b) high density.

neighbors are equalized and forces applied to end molecules are equalized by interactions with the walls. (c) The temperature is low enough; so the potential energy, EN, for N molecules in the box is

EN )

{

if N e N0 (N - 1) + Nψ0 d (N - 1)φ + Nψ0 if N > N0 N-1

(

)

}

(10)

where

N0 )

[] [ ] d d ) r0 σ21/6

(11)

In eq 11, the symbol [ ] indicates that the number of molecules must be an integer and the values must be truncated. The grand canonical partition function for this system can be written as ∞

Ξ)

∑

N)0

(

exp

Nµ kT

-

)

EN kT

(12)

The average number of molecules, 〈〈N〉〉, in the box and the average energy, 〈〈φ〉〉, of the nearest neighbor inter-

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Figure 10. Dependence of 〈〈N〉〉 on µ/kT for /kT ) -0.5, ψ0/kT ) -10, σ ) 3.5 Å, and d ) 20 Å.

Figure 12. Dependence of 〈〈N〉〉 on µ/kT for /kT ) -0.5, ψ0/kT ) -20, σ ) 3.5 Å, and d ) 20 Å.

Figure 13. Dependence of 〈〈φ/kT〉〉 on µ/kT for /kT ) -0.5, ψ0/kT ) -20, σ ) 3.5 Å, and d ) 20 Å. Figure 11. Dependence of 〈〈φ/kT〉〉 on µ/kT for /kT ) -0.5, ψ0/kT ) -10, σ ) 3.5 Å, and d ) 20 Å.

actions are

〈〈N〉〉 )

1

N0

[ [

Nµ

∑ N exp kT ΞN)0

(N - 1)

-

]

Nψ0

+ kT kT Nψ0 Nµ N - 1 d 1 ∞ φ N exp (13) ΞN)N0+1 kT kT N-1 kT

∑

〈〈φ〉〉 )

1

N0

[

Nµ

∑ exp kT ΞN)0

(N - 1)

( ) -

]

]

Nψ0

+ kT kT Nψ0 1 ∞ d Nµ N - 1 d φ φ exp ΞN)N0+1 N - 1 kT kT N-1 kT (14)

∑

( ) [

( )

]

Figure 10 shows the dependence of 〈〈N〉〉 on µ/kT predicted by eq 13 for /kT ) -0.5, ψ0/kT ) -10, σ ) 3.5 Å, and d ) 20 Å. These parameter values are typical of argon adsorbed on activated carbon at room temperature. As seen in Figure 10, at high negative chemical potentials (low gas-phase pressure), the number of particles in the box is very small. As gas-phase pressure and µ/kT go up, 〈〈N〉〉 goes up. Then, there is a range of saturation at µ/kT between -8 and -6. However, at µ/kT > -6, 〈〈N〉〉 goes up again. In this range, 21/6σ(N - 1) becomes greater than d, and particles cannot be in the box without a significant compression. Figure 11 gives the dependence of 〈〈φ〉〉 on µ/kT predicted by eq 14 for /kT ) -0.5, ψ0/kT ) -10, σ ) 3.5 Å, and d

) 20 Å. As seen from Figure 11, at µ/kT > -6 (where ∂φ/∂r > 0), the interactions between molecules in the box become repulsive. Hence, Figures 10 and 11 indicate that at high chemical potential, the box becomes overcrowded and molecules repel each other. Note that at /kT ) -0.5, Lennard-Jones fluid crystallizes as µ/kT > -3; however, as seen from Figures 10 and 11, there is a range of µ/kT between -6 and -3 where there is a significant compression of the adsorbed phase but the bulk phase is still a fluid. As seen from Figures 10 and 11, at µ/kT > -3, compression in the adsorbed phase becomes very large (〈〈φ〉〉/kT even changes its sign from negative to positive at µ/kT ≈ -2.6); this is in the range of solid nanophases (nanoclusters). In Figures 10 and 11, we used ψ0/kT ) -10. However, for molecules in nanopores, the attraction to the walls can be much stronger due to overlapping of adsorption potentials and multiple energetic bonds with the surrounding walls. Experimentally measured heats of physical adsorption of various gases on activated carbons can be as strong as ∼60 kJ/mol.52 At room temperature, this gives ψ0/kT of about -25. At liquid nitrogen temperature, this value is about -100. Adamson (see Table XIV-1 in ref 53) gives a typical range of adsorbate-adsorbent energies for physical adsorption down to about ∼38 kJ/mol. At liquid nitrogen temperature, this gives ψ0/kT as low as -57. For hydrogen bonding and chemisorption, ψ0/kT can be even stronger.53 Figures 12 and 13 give 〈〈N〉〉 and 〈〈φ〉〉 as functions of µ/kT for /kT ) -0.5, ψ0/kT ) - 20, σ ) 3.5 Å, and d ) 20 Å. As seen from Figures 12 and 13, the stronger attraction to the wall (ψ0/kT ) -20 instead of -10 in Figures 10 and 11) shifts the effect of compression to a much lower range of the chemical potentials.


Langmuir, Vol. 19, No. 7, 2003 2729

Figure 14. Dependence of 〈〈N〉〉 on µ/kT for σ ) 3.5 Å, d ) 20 Å, and various /kT and ψ0/kT values: (a) -0.5 and -20; (b) -1.0 and -40; (c) -2.0 and -80; (d) - 4.0 and -160; (e) -8.0 and -320; (f) -16.0 and -640. At these /kT and ψ0/kT, each temperature is twice as low as the previous one.

As mentioned previously, eqs 13 and 14 become rigorous in the low-temperature limit. Figures 14 and 15 show the behavior of 〈〈N〉〉 and 〈〈φ〉〉 as functions of µ/kT as /kT and ψ0/kT go from -0.5 and -20 down to -16 and -640, respectively, which is equivalent to decreasing temperature by a factor of 32. In Figures 14b and 15b, T is 2 times lower than in Figures 14a and 15a; in Figures 14c and 15c, T is 2 times lower than in Figures 14b and 15b, and so forth. In Figures 14f and 15f, T is 32 times lower than in Figures 14a and 15a. As seen from Figures 14 and 15, at low temperatures, there are two steps in the dependence of 〈〈N〉〉 on µ/kT. The first step reflects saturation of the adsorption capacity, and the second step indicates the phenomenon of the compression similar to that shown in Figure 4. Gradually increasing temperature will make eqs 13 and 14 less rigorous because they are based on a low-temperature limit. However, there is no reason to believe that the phenomenon of the nanocapillary compression shown in Figures 14 and 15 will disappear. Monolayer Adsorption Figure 2 illustrates the simplest case of adsorbate compression, that is, the compression of two molecules on two active adsorption sites. This case is important because

the A and B molecules shown in Figure 2b may react in a way or at a rate different from uncompressed molecules, and this is a key factor in heterogeneous catalysis. However, there are more sophisticated cases of surface compression, including compression of molecules in the pores of an adsorbent and compression of molecules in a monolayer. In each of these cases, the attractive field of the adsorbent causes compression of the adsorbed phase. Figure 16 illustrates this point for a monolayer. It is favorable for molecules to move into the monolayer until repulsions from neighbors balance the attraction to the surface. Hence, in a full monolayer, adsorbate molecules repel each other. This is possible only if the distance between neighbors is smaller than in a normal liquid, so the adsorbed phase is compressed. We think that the phenomenon of surface compression is of a fundamental significance because it changes the classical picture of adsorption. In particular, the adsorption capacity, am, is the amount adsorbed at the point where adsorbate-adsorbent attractions are balanced by adsorbate-adsorbate repulsions. Classical models3,7,42,53-55 treat am as a constant which does not depend on energies of adsorbate-adsorbate and adsorbate-adsorbent interactions. Because of surface compression, the differential

2730



Figure 15. Dependence of 〈〈φ〉〉 on µ/kT for σ ) 3.5 Å, d ) 20 Å, and various /kT and ψ0/kT values: (a) -0.5 and -20; (b) -1.0 and -40; (c) -2.0 and -80; (d) -4.0 and -160; (e) -8.0 and -320; (f) -16.0 and -640. At these /kT and ψ0/kT values, each temperature is twice as low as the previous one.

Figure 16. Thermodynamic reasons for surface compression. The free energy of the system, F, goes up due to repulsions between neighbors; F goes down due to attraction to the surface. Having an extra molecule on the surface is thermodynamically favorable as long as attraction to the surface is larger than repulsions between neighbors.

heat of adsorption must vanish as the adsorbed amount approaches am, and this has been observed in highresolution microcalorimetric experiments.63 Classical models3,7,42,53-55 predict finite values of differential heat of adsorption at this limit. Generally, surface compression is a new aspect of the molecular behavior influencing the (63) Phillips, J.; Kelly, D.; Radovic, L. R.; Xie, F. Microcalorimetric Study of the Influence of Surface Chemistry on the Adsorption of Water by High Surface Area Carbons. Second TRI/Princeton International Workshop “Characterization of Porous Materials”, June 19-21, 2000.

adsorption isotherm, the heat of adsorption, the intermolecular interactions, and the structure of the adsorbed layer. It also has important practical implications. One is in the measurement of surface areas for dispersed and porous materials by adsorption; another is in heterogeneous catalysis, where A and B molecules shown in Figure 2b may react in ways that are unachievable for noncompressed molecules. Model. Consider adsorption of a gas on a solid with the following assumptions: the surface is homogeneous; adsorbate molecules are in a monolayer on the surface of the adsorbent; there is some average energy of interaction, φs, between each adsorbate molecule and the surface; the energy of interaction between adsorbate molecules can be described by the Lennard-Jones potential function (eq 1). As shown previously,34 the monolayer adsorption isotherm for soft molecules is

(a/am)(1 - xb) kT ln + φs + Ua - Ug ) 0 (1 - a/am)xb

(15)

where xb is the reduced density of adsorbate molecules in


Langmuir, Vol. 19, No. 7, 2003 2731

the bulk (far from the surface), a is the density of the adsorbed layer (number of molecules adsorbed per square meter), Ua is the energy of interaction between a central molecule and surrounding molecules in the two-dimensional adsorbed layer, and Ug is the energy of interaction between a central molecule and surrounding molecules in the three-dimensional gas. After algebraic manipulations, eq 15 can be written in the following form:

xb a ) (16) am xb + (1 - xb) exp[(φs + Ua - Ug)/kT] The energy of interaction, Ua, between a central molecule and surrounding molecules in a two-dimensional fluid is

Ua ) a

∫0∞ φ(r)g(r)2πr dr

(17)

where g(r) is the two-dimensional radial distribution function. For adsorption from a gas, Ug is small compared to Ua and can be ignored. Under this condition, eqs 16 and 17 give

xb a ) am x + (1 - x ) exp{[φ + a b b s

∫0∞ φ(r)g(r)2πr dr]/kT}

(18)

Neglecting the adsorbate-adsorbate interactions in eq 16 gives the Langmuir isotherm1 as a special case of eq 16:

xb a ) am xb + (1 - xb) exp(φs/kT)

(19)

Another important special case is when Ua is a linear function of the monolayer density, a. This happens when C ) ∫∞0 g(r)φ(r)2πr dr is constant and

Ua ) Ca

(20)

This gives the Frumkin/Fowler-Guggenheim isotherm:3,64

xb

a ) am xb + (1 - xb) exp[(φs + Ca)/kT]

(21)

as another special case of eq 16. The isotherm given in eq 21, with the linear approximation for Ua (a), has been studied in detail.3,64 However, deviations from the linearity in eq 20 have not. Since eq 18 is written in terms of a general expression for Ua, it can be used to study nonlinear effects. It will be shown that these effects include a change in the average energy of adsorbate-adsorbate interactions from attractive to repulsive, compression of the monolayer (compared to a normal fluid) as the monolayer becomes nearly full, and a dependence of the monolayer capacity on the adsorbate-adsorbent interaction energy. The latter has been observed previously42,65 but has not been studied in any detail. (64) Frumkin, A. Z. Phys. Chem. (Leipzig) 1925, 116, 466. (65) Karnaukhov, A. P. J. Colloid Interface Sci. 1985, 103, 311.

In the nearest neighbor approximation,34

g(r) ) g0δ[(r - r*)/r*]

(22)

where r* is the average distance between the central molecule and its nearest neighbors, δ is the delta function, and g0 is a coefficient not depending on r. Plugging g(r) from eq 22 into eq 17, we obtain

Ua ) 2πr*2g0aφ(r*)

(23)

This result is consistent with the fact that Ua is nearly linear with density, a, at low and moderate densities;66,67 however, it allows for deviations from this linear behavior at high densities. Using eqs 1 and 22, we obtain from eq 18

a ) am xb xb + (1 - xb) exp[φs/kT - 8πg0a3(σ12K5a3 - σ6K2)/kT] (24) where K is the packing factor.34 Equation 24 determines a as a function of xb and, hence, is an isotherm for adsorption of Lennard-Jones molecules. Monolayer Capacity. am is considered to be a constant in lattice theories3,7,23,26,68 and in techniques for determining surface areas from adsorption isotherms.4,69 However, real molecules and Lennard-Jones molecules are soft; hence, am is not constant. To determine am, consider very low temperatures where the entropic term of eq 15 is not significant. At these temperatures, the monolayer capacity depends only on the adsorbate-adsorbent and adsorbateadsorbate interactions. If φs + (2πg0/K)φ(r0) < 0, adding molecules in the monolayer is thermodynamically favorable because each molecule added to the adsorbed layer decreases the free energy of the system. When φs + (2πg0/ K)φ(r0) > 0, adding molecules to the monolayer is thermodynamically unfavorable because each additional molecule decreases the free energy of the system. So, the monolayer capacity can be approximated from the condition

φs + (2πg0/K)φ(r0) ) 0

(25)

which gives for a ) am

φs - 8πg0(σ12K5am6 - σ6K2am3) ) 0

(26)

There are multiple solutions to eq 26, but the only physically meaningful solution is

am ) K-1σ-22-1/3[1 + (1 + Kφs/2πg0)1/2]1/3 (27) Since φs < 0 and < 0, eq 27 always gives a positive value of am. The monolayer capacity for hard molecules, amh, does not depend on the adsorption affinity. Defining amh as am at φs ) 0, we obtain

amh ) K-1σ-2

(28)

(66) Donohue, M. D.; Bokis, C. P. J. Phys. Chem. 1995, 99, 12655. (67) Lee, L. L. Molecular Thermodynamics of Nonideal Fluids; Butterworths: Boston, 1988; p 254. (68) Lane, J. E. Aust. J. Chem. 1968, 21, 827. A. R. Altenberger, A. R.; Stecki, J. Chem. Phys. Lett. 1970, 5, 29. (69) Aranovich, G. L. Langmuir 1992, 8, 736.

2732



Figure 18. Adsorption isotherm for soft molecules.

Figure 17. s0 versus the BET constant, C.

Combining eqs 27 and 28 gives

am/amh ) 2-1/3[1 + (1 + Kφs/2πg0)1/2]1/3

(29)

Equation 29 shows the difference in the monolayer capacity between hard molecules and soft (Lennard-Jones) molecules. It follows from eq 29 that the monolayer capacity for Lennard-Jones molecules is always greater than that of hard molecules. This is because soft molecules can pack more closely than hard molecules. From eq 27, it follows that the area, s0, occupied by one molecule of adsorbate in the full monolayer is

s0 )

1 21/3Kσ2 ) am [1 + (1 + Kφ /2πg )1/2]1/3 s 0

(30)

The important result of eq 30 is that the area per molecule depends on the energy of adsorbate-adsorbent interaction. Though there have been experiments which suggest that s0 depends on adsorbate-adsorbent interaction energy,42,65 this is the first theoretically based model for the area per molecule, s0, that shows it is a function of φs. Figure 17 illustrates experimental data for s0 as a function of the BET constant for argon, nitrogen, and krypton on various adsorbents.65 These experimental results show that s0 goes down as the energy of the adsorbate-adsorbent interactions goes up, in qualitative agreement with eq 30. Adsorption Behavior of Soft Molecules. Figure 18 illustrates the adsorption isotherm predicted by eq 24 for σ ) 3.5 Å, /kT ) -0.5, K ) 1, 2πg0 ) 6, and φs/kT ) -6. Also shown is the energy of adsorbate-adsorbate interactions as a function of adsorption. As seen from Figure 18, at small and moderate adsorption, nearest molecules attract each other in the monolayer; however, at high adsorption, molecules of the adsorbate repel each other. Figure 19 illustrates the distribution of adsorbateadsorbate interaction energies determined from Monte Carlo simulation for nitrogen on the (110) face of rutile at T ) 77 K and a pressure of 1 Torr.36 At these conditions, the molecules of nitrogen are in a full monolayer.36 As seen from Figure 19, the energies of adsorbate-adsorbate interactions are predominantly positive (repulsive). This is in agreement with the behavior shown in Figure 18.

Figure 19. Distribution of the adsorbate-adsorbate interaction energies obtained from Monte Carlo simulations for nitrogen on the (110) face of rutile at T ) 77 K and a pressure of 1 Torr (data from ref 36).

Equation 23 can be written as

Ua ) λaφ(r*)

(31)

where λ ) 2πr*2g0. For Ug, this equation can be written in a similar form:

Ug ) λ′xbφ(r*′)

(32)

where λ′ and r*′ are three-dimensional analogues of λ and r* for the gas phase. Plugging eqs 31 and 32 into eq 15 gives

(a/am)(1 - xb) φs λaφ(r*) λ′xbφ(r*′) + )0 + ln kT kT kT (1 - a/am)xb (33) In eq 33, the first two terms dominate and can be combined to

(a/am)(1 - xb) Y ) ln H(1 - a/am)xb

(34)

H ) exp(-φs/kT)

(35)

where


Langmuir, Vol. 19, No. 7, 2003 2733

Figure 20. Adsorption isotherms for different systems in coordinates of eq 36: (a) benzene on silica gel at T ) 303 K, data from ref 70; (b) butane on a carbon molecular sieve at T ) 324 K, data from ref 71; (c) CO2 on h-mordenite at T ) 303 K, data from ref 72; (d) CO2 on a zeolite molecular sieve at T ) 298 K, data from ref 73; (e) CO on zeolite at T ) 228 K, data from ref 74; (f) ethylene on activated carbon at T ) 311 K, data from ref 75; (g) ethylene on a carbon molecular sieve at T ) 279 K, data from ref 71; (h) acetylene on activated carbon at T ) 293 K, data from ref 76; (i) methane on activated carbon at T ) 213 K, data from ref 52.

and eq 33 can be represented in the following form:

λφ(r*) a λ′φ(r*′) Y )+ xb kT xb kT

(36)

Plotting Y/xb as a function of a/xb, we can get the value of λφ(r*) from the slope of the graph. The sign of λφ(r*) gives information about the sign of adsorbate-adsorbate interactions, that is, whether they are attractive or repulsive. Note that plotting Y/xb as a function of a/xb also gives the intercept, λ′φ(r*′), which can be used to verify whether the assumption that the term Ug in eq 15 can be neglected and approximation 18 are reasonable. Figure 20 shows experimental isotherms for adsorption of various vapors on different adsorbents in the coordinates of eq 36. As shown by Figure 20, in these coordinates, experimental data are straight lines near monolayer coverage (at small a/xb). An interesting thing is that the slope of the straight lines in Figure 20 is negative which indicates positive φ(r*). So, in all these cases, there are repulsions between molecules in the adsorbed layer near monolayer coverage. As seen from Figure 18, adsorption saturation (at small a/xb, where a becomes almost constant) occurs with an energy of adsorbate-adsorbate interactions which is repulsive if the affinity to the adsorbent is high. Note that the intercepts in Figure 20 are near zero; this

validates the assumption of neglecting Ug in eq 15 and using approximation 18. The quantity Y/xb should go to zero at high values of a/xb. This can be seen from eq 34. In the limit of xb f 0, this equation is Y ) ln(a/am)/Hxb. Since H ) limxbf0(a/ am)/xb, then Y ) 0 at this limit. This deviation from linear behavior at high a/xb is seen in frames a-d of Figure 20. A more careful analysis of the behavior at high a/xb (i.e., low xb) will require a large number of accurate experimental data points at low levels of adsorption. The data in Figures 17-20 demonstrate the macroscopic consequences of surface compression. As illustrated in Figure 16, having an extra molecule on the surface is thermodynamically favorable as long as attraction to the surface is larger than repulsions between neighbors. Hence, the differential heat of adsorption, Ha, must vanish as the adsorbed amount approaches the adsorption capacity. This phenomenon has been observed experimentally by Phillips, Kelly, Radovic, and Xie.63 Figure 21 shows the heat of adsorption as a function of adsorbed amount, a, for water on carbon Norit-C treated by nitrogen at 950 °C. As seen from Figure 21, Ha goes to zero as a reaches the adsorption capacity (∼3.7 mmol/g). Compression in Nanopores and Nanocapillaries As shown in Figures 17 and 18, eqs 8, 9, 13, 14, and 27 predict the compression of a fluid in a strong external

2734



Figure 23. Dependence of ∆p on β for φ0 ) 5 × 108 J/m3. Figure 21. Differential heat of adsorption as a function of adsorption for water on carbon Norit-C treated in nitrogen at 950 °C. The data are from ref 63.

Since ∆E ) -∆U, then ∆V2/(2βSd0) ) φ0Sd0 which gives

∆V ) x2βφ0 V0

(41)

From eqs 38 and 41, we obtain Figure 22. Compressed fluid between two walls.

∆p )

field. Though these predictions are for idealized systems and for particular cases, this phenomenon will occur for any compressible fluid in a strong external field that is in equilibrium with a large mass of bulk fluid. In particular, this situation will occur whenever a fluid is in contact with a nanoporous solid. Because of the strong field of the adsorbent in nanopores, the fluid is pulled into the pores until attraction to the adsorbent balances the repulsions between the molecules of fluid. Consider a fluid in a slitlike pore, between two solid walls as shown in Figure 22. Let φ0 be the average energy density for the fluid-wall interaction. Then, the total energy of the fluid-wall interaction, ∆E, is

∆E ) φ0Sd0

(37)

where d0 is the distance between walls and S is the area of the wall. As discussed earlier, at complete filling of the pore, the attraction of adsorbate molecules to walls is balanced by their compression. The compressibility coefficient for the fluid, β, is

β)

1 dV V0 dp

(38)

where V0 ) Sd0. The compressed fluid can be considered as a spring in the range of the linear dependence between force, ∆F, and change of the length, ∆x. Then, eq 38 can be written in terms of mechanical parameters of this spring:

∆F ) -K*∆x

(39)

where ∆F ) S∆p, ∆x ) ∆V/S, and K* ) -S/(βd0). The potential energy, ∆U, of the compressed spring is

∆U ) K*∆x2/2 ) -∆V2/(2βSd0)

(40)

x

2φ0 β

(42)

In this equation, φ0 is in units of J/m3 and β is in units of Pa-1. Therefore, ∆p is in units of Pa. For adsorption of water on silica gel, φ0 is about 104 J/mol,42 which is equivalent to about 5 × 108 J/m3 (energy of adsorption per cubic meter of liquid water). The compressibility of liquid water, β, is about 50 × 10-11 L/Pa.77 Putting these numbers in eq 42 gives ∆p of about 104 atm. This is the pressure on the walls of nanopores as the pores are saturated with water. Figure 23 shows the dependence of ∆p on β predicted by eq 42 at φ0 ) 5 × 108 J/m3. As seen from Figure 23, the phenomenon of nanocapillary compression can cause enormous internal forces in nanoporous structures. This effect depends on the size of pores because φ0 goes down and vanishes very quickly if the sizes of pores go up. There is a certain analogy between the adsorption compression phenomenon and the variation of pressure with altitude in air and with depth in water.81 The common feature of these two phenomena is that the density of the fluid increases if the fluid is in a strong field. In the first case, it is the field of the adsorbent; in the second case, it is the gravitational field. The difference between these two cases is that compression in gravitational field is a macroscale phenomenon and adsorption compression is a nanoscale phenomenon. The phenomenon of nanocapillary compression is important for understanding nanocapillarity. Figure 24 illustrates the difference in the behavior of a fluid between a nanocapillary and a micron-sized capillary. In large capillaries, capillary forces are inward (plate a in Figure (70) Sircar, S.; Myers, A. L. AIChE J. 1973, 19, 159. (71) Nakahara, T.; Hirata, M.; Omori, T. J. Chem. Eng. Data 1974, 19, 310. (72) Talu, O.; Zwiebel, I. AIChE J. 1986, 32, 1263. (73) Hyun, S. H.; Danner, R. P. J. Chem. Eng. Data 1982, 27, 196. (74) Nolan, J. T.; McKeehan, T. W.; Danner, R. P. J. Chem. Eng. Data 1981, 26, 112. (75) Kaul, B. K. Ind. Eng. Chem. Res. 1987, 26, 928. (76) Szepesy, L.; Ilies, V. Acta Chim. Hung. 1963, 35, 37. (77) Perry, R. H.; Chilton, C. H. Chemical Engineers’ Handbook; McGraw-Hill: New York, 1973.


Figure 24. Fluid in micron-sized (a) and in nanosized (b) capillaries. In microcapillaries, the capillary forces are inward, pulling the walls in; in nanocapillaries, capillary compression causes forces on the walls from inside, i.e., outward. This is just the opposite of the Kelvin’s equation prediction (ref 53).

24); in nanoscale capillaries, capillary forces press on walls from the inside, that is, they are outward (plate b in Figure 24), which is the opposite of the prediction by Kelvin’s equation.53 Conclusion Molecules, such as Lennard-Jones molecules, with a soft intermolecular potential can become compressed in a strong external field. The degree of this compression depends on the energy imposed by the field and on the compressibility of the fluid. This phenomenon has various manifestations, including compression of molecules sitting on neighboring active sites of a solid surface, compression of adsorbed surface layers, and compression in nanopores. A simple model for adsorption on neighboring active sites on a solid surface allows an exact solution in the grand canonical ensemble and predicts surface compression (Figure 2). A one-dimensional model (Figure 8) allows an exact solution in the low-temperature limit and indicates a similar phenomenon: compression in the confined phase. Monolayer adsorption of soft molecules (Figure 16) is another example where surface compression is of a fundamental significance because it contradicts the classical assumption that the adsorbed molecules are attracted to each other. Though compression in confined phases has not been discussed previously in the literature, there is considerable experimental evidence of this phenomenon. In thin films on solid substrates, the adsorbed layer of molecules is found to be significantly compressed if the adsorbatesubstrate interaction is strong. This has been observed using low-energy electron diffraction for films of magnesium chloride on the (111) face of palladium.78 In this case, in the compressed phase the distance between magnesium chloride molecules is about 10% (0.36 Å) smaller than in the ionic crystal. This results in a density that is about 20% higher than the bulk density. There also are experimental data on adsorption of gases on solids indicating the surface compression phenomenon. First, the area per molecule in the full monolayer goes (78) Fairbrother, D. H.; Roberts, J. G.; Rizzi, S.; Somorjai, G. A. Langmuir 1997, 13, 2090.

Langmuir, Vol. 19, No. 7, 2003 2735

down as energy of the molecule-surface interactions goes up. This was observed in BET measurements for pentane, argon, krypton, and nitrogen on various adsorbents.42,65 Second, the differential heat of adsorption vanishes as adsorption approaches the adsorption capacity.63 This is because the attractions balance repulsions when the adsorbed layer is filled. Third, experimental isotherms for many systems (such as nitrogen, carbon dioxide, methane, butane, ethylene, acetylene, benzene on silica gel, activated carbon, and zeolites) show that the adsorbed phase is compressed and nearest neighbors repel each other.34 Fourth, gases (argon, oxygen, nitrogen, helium, carbon monoxide, carbon dioxide, vapors of water, acetaldehyde, acrolein, and isopropyl alcohol) adsorbed on a variety of porous catalysts (magnesium oxide, magnesium hydroxide, cobalt-molybdenum, carbonyl iron) cause significant internal forces that significantly change the mechanical properties of porous materials.79 In some cases, this has been shown to lead to disintegration of catalyst particles (grains of alumina-chromium oxide-potassium oxide) by the adsorption of vapors (butane and butene).80 In addition to these experimental results, there are Monte Carlo simulations that show repulsions between Lennard-Jones molecules in adsorbed layers. In particular, such results were reported by W. Steele and coauthors36 for nitrogen on the (110) face of rutile at T ) 77 K and a pressure of 1 Torr. There are several reasons why this nanocapillary compression phenomenon deserves attention: It is a fundamental aspect in the mechanism of fluid-adsorbed phase equilibria. There is no theory for this phenomenon. Capillary compression can play a significant role in a variety of applications (such as catalysis), and its understanding can lead to technological improvements or even new applications. Note that nanocapillary compression can have both positive and negative effects. In catalysis, its effect will be positive if it increases the rate of the catalytic reaction but its effect will be negative when it results in pressure on the walls of nanoporous materials that is sufficient to destroy the fine structure of the catalyst. Degradation of catalysts is a major problem in petrochemical refining. Acknowledgment. Support by the Division of Chemical Sciences of the Office of Basic Energy Sciences, U.S. Department of Energy, under Contract DE-FG0287ER13777, by the National Science Foundation under Grant BES-9910174, and by E.I. du Pont de Nemours and Company is gratefully acknowledged. LA020739H (79) Shchukin, E. D.; Margolis, L. Ya.; Kontorovich, S. I.; Polukarova, Z. M. Russ. Chem. Rev. 1996, 65, 813. (80) Paranskii, S. A.; Medvedev, V. N.; Veden’eva, A. I.; Bessonov, A. I.; Sterligov, O. D.; Shchukin, E. D. Kinet. Katal. 1971, 12, 473. (81) Halliday, D.; Resnick, R. Physics; John Wiley: New York, 1967; Section 17-3.

Adsorption Compression - American Chemical Society

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