Langmuir 1998, 14, 3847-3857
3847
Physical Adsorption in Micropores: A Condensation Approximation Approach Vladimir Kh. Dobruskin* st. Aiala 21, Beer-Yacov, 70300, Israel Received November 5, 1997. In Final Form: February 4, 1998 An approach to adsorption in micropores is developed on the basis of the condensation approximation method. In the case of micropores with half-width less than 0.8 nm, a volume filling of the single micropore and a two-dimensional (2D) condensation on its walls occur at the same critical pressure, pc. From this point of view, the statistical mechanical theories of adsorption on homogeneous surfaces considering lateral interactions may be taken to be a starting point to the description of physical adsorption in micropores. The increased well depth, *, in micropores causes the 2D condensation on their surface to occur at a smaller pc compared with a nonporous surface. As a consequence, the overall adsorption isotherm is determined by the distribution of micropore walls over adsorption energies resulting from a random distribution of micropore widths, d. Proceeding from a model of heterogeneity and the normal distribution of micropore widths, the log-normal distributions of micropore volumes over * and pc are obtained, and an occupied adsorption volume as a function of outer pressures is found. This approach has been successfully applied to seven benzene adsorption isotherms on four species of active carbons. It provides a correct description of equilibrium data in a wide temperature range, leads to reasonable distribution function of * and d, and gives values of an average micropore size that are close to the experimental one.
Introduction The important problem in adsorption on microporous solids is a description of the micropore filling process and an evaluation the parameters that characterize this process quantitatively. The theory of the volume filling of micropores was proposed by Dubinin and co-workers1 on semiempirical grounds in the late 1940s and then has been reviewed several times from 1960 to 1990.2-7 In the Dubinin-Astachov (DA) theory of single-gas adsorption, which is the more general version of the DubininRadushkevich (DR) theory, the degree of micropore filling, θ (fractional adsorption), expressed in terms of the limiting adsorption volume, W0, and the occupied adsorption volume, W,
θ ) W/W0
(1)
is related to the adsorption potential A, the parameters E and n, and the affinity coefficient β,
[ (βEA ) ]
θ ) exp -
n
adsorption potential is the work of adsorption that compresses the ambient gas until it condenses
A ) RT ln ps/p
where p is the partial pressure in the gas phase and ps is the saturated pressure of the liquid adsorbate. The experimental studies have resulted in a number of useful approximations, the Stoeckli relationship10 being the most important
d h (nm) ) 12/E (kJ/mol)
The values of E and n are calculated with respect to the standard gas with β0 ) 1, which is considered to be a probe of energy heterogeneity that provides information about adsorption sites. According to Polanyi,8,9 the * E-mail:
[email protected]. (1) Dubinin, M. M.; Radushkevich, L. V. Dokl. Akad. Nauk. SSSR 1947, 55, 331. (2) Astakhov, V. A.; Dubinin, M. M.; Romankov, P. G. In Adsorbents, their Preparation, Properties and Application; Dubinin, M. M., Plachenov, T. G., Eds.; Nauka: Leningrad, 1971; p 92. (Russian). (3) Dubinin, M. M. In Adsorption-Desorption Phenomena; Ricca, F., Ed.; Academic Press: London, 1972; pp 3-18. (4) Astakhov, V. A.; Dubinin, M. M.; Mosharova, L. P.; Romankov, P. G. TOXT 1972, 6, 343. (5) Dubinin, M. Chem. Rev. 1960, 235. (6) Dubinin, M.; Stoeckli, H. F. J. Colloid Interface Sci. 1980, 75, 34. (7) Stoeckli, H. F. Carbon 1990, 28, 1 (8) Polanyi, M. Verh. Dtsch. Phys. Ges. 1914, 16, 1032. (9) Polanyi, M. Trans. Faraday Soc. 1932, 28, 316.
(4)
where d h is the average half-width of the corresponding micropore system. It is generally accepted11,12 that experimental adsorption isotherms, θ(,T,p), represent an average over all values of the adsorption energies, , existing on the gas-solid interface
θ(T,p) ) (2)
(3)
∫Ωθ(,T,p) f() d
(5)
where θ(T, p) is the single-pore (individual or “local”) isotherm for the adsorption sites exhibiting an adsorption energy , f() denotes the density distribution function of , and Ω is the integration region over all possible adsorption energies. Rudzinski and Everett11 proposed a theoretical basis for the Dubinin theory in the framework of integral transforms by averaging the Langmuir local isotherm. Dubinin, Stoekli, Jaroniec, et al. proposed methods to determine the pore size (PSD) and energy distributions for microporous sorbents from adsorption experiments on the base of the integral transformation.12-14 These methods have principal drawbacks: they use either (10) Stoeckli, H. F. Adsorpt. Sci. Technol., Special Issue 1993, 10. (11) Rudzinski, W.; Everett D. H. Adsorption of Gases on Heterogeneous Surface; Academic Press: New York, 1992. (12) Jaroniec, M.; Madey, R. Physical Adsorption on Heterogeneous Solids; Elsevier: Amsterdam, 1988; pp 45-46. (13) Dubinin, M. M. Dokl. Akad. Nauk SSSR 1984, 275, 1442. (14) Stoeckli, H. F. Carbon 1990, 28, 1.
S0743-7463(97)01211-0 CCC: $15.00 © 1998 American Chemical Society Published on Web 06/16/1998
3848 Langmuir, Vol. 14, No. 14, 1998
empirical relationship (eq 4 or its versions) or arbitrarily chosen kernels of the transformations (the DR or the DA equations) or an arbitrary energy distribution function. The most promising theories of physical adsorption in micropores are statistical ones, which need not assume the existence of a distinct adsorbed phase, but require a knowledge of the interaction potential for the fluidsorbent and fluid-fluid interactions. These studies15-18 provide information about individual micropore isotherms. PSD of porous carbon is obtained by a fitting of model isotherms and a distribution function of pore widths to experimental uptakes. Current achievements in this field concern adsorption of simple molecules on porous solids. In this work we present a method for the treatment of adsorption experiments based on the condensation approximation (CA) approach initiated by Roginski19,20 and then intensively developed in many papers.11,12 In the present version, the micropore filling is regarded as an evolution of the two-dimensional condensation, which occurs on micropore walls at the critical condensation pressures. This approach provides a correct description of equilibrium data in a wide temperature range, leads to a reasonable energy distribution function, and gives the value of an average micropore size that is close to the experimental one. 1. Model of a Local Adsorption Isotherm Cooperative Effects in Micropores. Gregg and Sing21-24 suggested that a monolayer formation on the pore walls significantly enhances the adsorption affinity in the pore core (the volume of the pore remaining after the adsorbed layer(s) is formed on micropore walls) and, thus, complete pore filling occurs. They refer to this phenomenon as the cooperative mechanism involving the interaction between adsorbate molecules. Our intention now is to show that the Gregg and Sing suggestion is valid for narrow micropores. In recent decades a very large number of experimental and theoretical publications have appeared concerning adsorption in slitlike micropores.25-28 Everett and Powl presented of the most detail calculations of potential energy profiles for atoms in slitlike pores and of the enhancement of the depth of the potential energy wells compared with those for adsorption by a single surface.27 They reported the results of calculations corresponding to two models, namely, of adsorption between two single lattice planes and between two semi-infinite slabs of solid. The calculations were based on the Lennard-Jones potential for the interaction between single atomic or molecular species of x and y (15) Lastokie, C.; Gubbins K. E.; Quirke, N. J. Phys. Chem. 1993, 97, 4786-4796. (16) Lopez-Ramon, M. V.; Jagiello, J.; Bandosz, T. J.; Seaton, N. A. Langmuir 1997, 13, 4435. (17) Gusev, V. I.; O’Brien, J. A.; Seaton, N. A. Langmuir 1997, 13, 2815. (18) Gusev, V. I.; O’Brien, J. A. Langmuir 1997, 13, 2822. (19) Roginski, S. Z. Dokl. Akad. Nauk SSSR 1944, 45, 61. (20) Roginski, S. Z. Adsorption and Catalysis on Heterogeneous Surface. Izv. Akad. Nauk SSSR 1949, (Russian). (21) Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity; Academic Press: London, 1982. (22) Carlott, P. J. M.; Roberts, R. A.; Sing, K. S. W. Carbon 1987, 25, 59. (23) Sing, K. S. W. Carbon 1987, 25, 155. (24) Rao, M. B.; Jenkins, R. G. Carbon 1986, 236. (25) Crowell, A. D. J. Chem. Phys. 1954, 22, 1397. (26) Steele, W. A. J. Chem. Phys. 1955, 59, 57. (27) Everett, D. H.; Powl, J. C. J. Chem. Soc., Faraday Trans. 1976, 72, 619. (28) Nicholson, D. J. Chem. Soc., Faraday Trans. 1996, 92, 1.
Dobruskin
[( ) ( ) ]
xy ) 4xy*
σxy r
12
-
σxy r
6
(6)
where r is the distance between the nuclei of the atoms, xy* is the depth of the potential energy minimum, and σxy is the distance at which xy ) 0. The asterisk (*) denotes the potential energy minimum. The total energy of interaction was assumed to be the pairwise sum of the interatomic potential functions
)
∑j jx(rj)
(7)
where the subscript j refers to the jth atom of the solid and rj refers to the distance from the external atom to the jth atom. As a result the following equation (the 10-4 potential) was obtained for interactions, cx,), between a molecule and two parallel carbon lattice planes whose nuclei are a distance 2d apart, the molecule being a distance z from the central plane
[( ) ( ) ( ) ( )]
cx,)(z) ) 4πmcx*σcx2
σcx 10 1 σcx 10 + 5 d+z d-z σcx 1 σcx 4 + 2 d+z d-z
4
(8)
where m is the number of interacting centers per unit area of the lattice plane and subscripts c and x denote atoms of a carbon lattice and adsorbate, respectively. In the case of semi-infinite slabs, the solid-fluid interactions are well described by the Steele 10-4-3 potential29
[
cx,)(z) ) 2πmcx*σcs2∆ × σcx 10 σcx 10 σcx 4 σcx 1 2 + + 5 d+z d-z 2 d+z d-z σcx4 1 1 + 3∆ (d + z + 0.61∆)3 (d - z + 0.61∆)3
{( ) ( ) } {( ) ( ) }
{
}
4
]
(9)
where ∆ ) 0.340 nm is the distance between the basal planes. The typical potential functions exhibit a characteristic double minimum corresponding to the position of the equilibrium separation between the adsorbate and a graphite wall. As a pore is narrowed, the potentials of each wall increasingly overlap and minima eventually merge to give a single central potential well. Let us envision the hypothetical situation when surfaces of both walls are covered by monolayer films of adsorbed molecules and the next molecule is being adsorbed on the film surface (Figure 1). We designate the molecules of the first and the second layers by 1 and 2, respectively. Our intention now is to calculate the potential energy of molecule 2. We can split the sum in eq 7 into two parts as follows
)
∑j j2(rj) + ∑l l2(rl)
(10)
where the summations in the first term extend over interactions with all carbon atoms and the summations in the second term extend over interactions with all film atoms. In doing so, we split the calculation into two parts and, in fact, represent the interaction of molecule 2 as a sum of interactions in two slits, the second being formed by the parallel planes of monolayer films. If nuclei of two (29) Steele, W. A. J. Phys. Chem. 1977, 82, 817.
Physical Adsorption in Micropores
Langmuir, Vol. 14, No. 14, 1998 3849 Table 1. The Lennard-Jones Parameters for Interactions of Noble Gases substance
σxx, nm
σcx, nm
cx/k, K
xx/k, K
Ar Kr Xe
0.340 0.360 0.410
0.340 0.350 0.375
57.8 66.6 79.5
120 171 221
close packing at the distance of minimum energy. For a two-dimensional adsorbed film, one could choose c ) 6, and the site separation distance q ) 21/6σxx.31 In this case for a molecule on the surface surrounded by a regular hexagonal array, the area per molecule, ω, is found as31
ω) Figure 1. A slit pore model: dashed circles, carbon atoms; solid circles, atoms of adsorbate.
parallel carbon lattice planes are a distance 2d1 apart, two parallel walls of the internal pore are a distance
2d2 ) 2d1 - 0.340 - βσxx
(11)
apart (Figure 1), where 0.340 nm is the effective diameter of a carbon atom, σxx is the distance for pair atom-atom interactions at which xx ) 0, and β is the coefficient that determines the collision distance of a molecule with the surface because of adsorption forces. This value was discussed by Everett27 and Steele,30and we accept the recommended value of β ) 0.84.27 Since interactions in the both external and internal pores are determined by eq 8, the potential energy minimum of molecule 2, 2*, may be calculated as a sum of two similar terms
[(
2* ) 4πm1cx*σcx2
(
σcx 1 5 d1 + z2*
σcx 1 2 d1 + z2* 4πm2xx*σxx2
[(
) ( 4
+
) ( 10
+
σcx d1 - z2*
σcx d1 - z2*
)] 4
)
10
-
+
) ( ) ( ) ( )]
10 10 σxx σxx 1 + 5 d2 + z2* d2 - z2* 4 4 σxx σxx 1 + (12) 2 d2 + z2* d2 - z2*
Here σcx and σxx are the distances at which cx and xx are equal to zero, m1 ) 38.2 and m2 are the numbers of interacting centers per unit area (nm2) of the first (carbon) micropore and the second internal pore, respectively, xx* is the potential energy minimum for the interaction between molecules of adsorbate, and z2* is the equilibrium coordinate of molecule 2. The second term in eq 12 is equal to the well depth for interactions between molecule 2 and two lattice planes of the internal micropore, xx,)*. Unfortunately, little is known about the adsorbed film density. Rudzinski and Everett11 described the situation in this field as follows: “There has been a good deal of discussion concerning the state of adsorbed molecules in the case of adsorption in pores, but the general feeling is growing that the picture of localized adatoms may be a good starting point for theoretical consideration.” So we presume that (i) walls of the internal pore form the lattice with the structure determined by the graphite lattice, (ii) the number of nearest neighbors, c, is equal to that for close packing, and lattice dimensions are also those for (30) Steele, W. A. J. Phys. Chem. 1978, 7, 817-821.
x3 1/6 2 x3 (2 ) σxx2 ) 3 σxx2 2 x4
(13)
1 0.9165 ) ω σ 2
(14)
and
m2 )
xx
The ratio of the well depths of molecule 2 to that of molecule 1, R, is derived from eqs 8 and 12
R)
2*(z2*) cx,)*(z1*)
(15)
where z1* and z2* are the equilibrium coordinates of molecules in positions 1 and 2 and the well depth of molecule 1, 1*, is equal to cx,) at z1*. The problem of great interest for us is to find a range of micropore halfwidths for which
Rg1
(16)
The filling of pores for which R g 1 is quit different compared with those for which R < 1. When R < 1, the adsorption energy of a molecule located at specific sites in the first layer is greater than the energy of a molecule located in the second or higher layers. Such adsorption behavior is adopted in the theories of a multilayer adsorption. If the cooperative mechanism described by Gregg and Sing was operative, we would expect that R g 1.24 This would indicate that the adsorption affinity in the core is larger than that for the adsorption in the first layer resulting in pore filling.24 Therefore, eq 16 defines the approximate boundary between the pores in which a layer-by-layer adsorption takes place and those in which only a volume condensation occurs. The Lennard-Jones parameters27,31,32 of some rare gases used in the potential calculation are shown in Table 1, and the results of calculations are presented in Table 2. These data show that there is the limit slit half-width, dlimit, so that
2* > 1*
for d1 < dlimit
(17)
For example, these values are equal to about 0.7 and 0.8 nm for argon and xenon, respectively. Within these separations, decreasing of the interaction energy because of a greater distance of molecule 2 from a graphite surface is superimposed by the enhancement of interactions in the narrowed second pore. Figure 2 illustrates the (31) Steele, W. A. The Interaction of Gases with Solid Surface; Pergamon Press: Oxford, 1974. (32) Miitland, G., C.; Rigby, M.; Smith, E. B.; Wakeham, W. A. Intermolecular Forces. Their Origin and Determination; Clarendon Press: Oxford, 1981.
3850 Langmuir, Vol. 14, No. 14, 1998
Dobruskin
Table 2. Calculation of the Interaction Energy in Micropores d2, nm
d2/σxx
d1, nm
d1/σxx
z2*, nm
cx,)*/k, K
Argon, m2 ) 7.93 atoms per 0.000 -987 ( 0.275 -974 ( 0.585 -968
cx,)/k, K nm2
xx,)*/k, K
2*/k, K
R
-234 -154 -134
-829 -523 -445
-1063 -677 -579
1.077 0.695 0.598
0.340 0.440 0.540
1.000 1.294 1.588
0.653 0.753 0.853
1.920 2.214 2.508
0.360 0.460 0.560
1.000 1.278 1.556
0.681 0.781 0.881
1.892 2.170 2.449
Krypton, m2 ) 7.07 atoms per nm2 0.000 -1277 -302 ( 0.256 -1191 -201 ( 0.552 -1183 -174
-1182 -758 -640
-1484 -959 -815
1.161 0.806 0.688
0.410 0.450 0.510 0.610
1.000 1.098 1.243 1.488
0.752 0.792 0.852 0.952
1.835 1.932 2.078 2.322
Xenon, m2 ) 5.45 atoms per nm2 0.000 -1663 -469 0.000 -1650 -382 ( 0.214 -1637 -317 ( 0.482 -1625 -275
-1527 -1352 -1023 -847
-1996 -1735 -1340 -1122
1.200 1.051 0.818 0.690
Figure 2. Energy of interactions of xenon inside the carbon micropore with d1 ) 0.752 nm, cx,)/k , and inside the related internal micropore with d2 ) 0.410 nm, xx,)/k, as a function of z/r0. The well depth of xenon inside the carbon micropore is equal to cx,)*/k ) -1662.8 and decreases to cx,)/k ) -468.8 at z2 ) 0. But when molecule 2 is situated at z2 ) 0, the decreasing of cx,) is superimposed by the interaction energy xx*/k ) -1527.2 with the walls of the narrowed internal micropore and | cx,) + xx,)*| > |cx,)*|.
Figure 3. Ratio of the well depths of xenon molecules adsorbed on the xenon monolayer film to that on the carbon surface as a function of the micropore half-width.
interactions of xenon inside the carbon micropore with d1 ) 0.752 nm and inside the related internal “xenon” micropore with d2 ) 0.410 nm. The total energy of interactions of molecule 2, 2*/k ) -1996 K, is equal to the sum of energies of interactions with a carbon slit, cx,)/k ) -468.8 K, and with the related “xenon” slit, xx,)*/k ) -1527.2 K, and is less than that of molecule 1, cx,)*/k ) -1662.8 K; that is, |2*| ≡ |cx,) + xx,)*| > |1*| ≡ |cx,)*|. The region d1 < dlimit correspondents to separations that provide the occurrence of molecule 2 in positions where a double minimum of the second micropore merges to give a single central potential well, which is maximized to roughly double its depth for a planar surface. Figure 3 demonstrates the dependence of R via d1 in the case of xenon adsorption. It shows that dlimit ) 0.8 nm.
Proceeding from Steele’s approximation30 for a slit pore formed by the two semi-infinite slabs, one obtains analogous expressions for a well depth of molecule 2 and for a ratio Rs. In this case, the interaction summed over all basal planes converges rapidly even for relatively longrange attractive terms. In the region of the minimum energy, the entire contribution of all planes excluding the surface is less than 15% of the total.33 The results of calculations of Rs for noble gases do not substantially differ from those in the case of micropores formed between the single planes; nevertheless, they give slightly less values of dlimit. Since a single layer of graphite has a surface area, s, of 1314 m2 g-1 (2628 m2 g-1 if both sides are counted), Everett and Powl suggested that walls of real microporous carbons with s ≈ 1000-1200 m2 g-1 would consist of some two layers, and the 10-4 potential was more realistic.27 Examining the adsorption potential within the pore core, Rao and Jenkins24 concluded that the existence of adsorbed layers does not enhance the adsorption potential for volume filling of the pore core to occur. They did not treat interactions with monolayer films as those within a micropore and underestimated the film contributions in the total interaction energy. Model of Adsorption Behavior in Micropores. Irrespective of either perfectly mobile or completely localized models of adsorption on the perfect surface are applied, the attractive lateral interaction energy must be involved to predict an adsorption behavior.31 The exhausting discussion on this problem was given by Steele.31 In the simplest Fowler-Guggenheim (FG)34 approach, the lateral interaction is taken into account by assuming that the total interaction energy is the same for all configurations of atoms on the adsorption sites and is equal to cw, where c is the average number of nearest-neighbor sites and w is the lateral interaction energy. The isotherm curve passes through metastable and unstable regions, and in this region it is approximated by a straight line normal to the p-axis; this line is plotted according to Maxwell’s equal-area rule. The above straight line represents a region where a two-dimensional condensation (that is, a phase transition) occurs. Honig’s approach based on the methodology of order-disorder theory, allowed the possibility to turn to the contributions of different parameters in the surface condensation.35 He showed that for the hexagonal lattice with c ) 6 not only nearest-neighbor but next-nearest-neighbor interactions should be taken into account, and the effect is by no means negligible. When these interactions are taken into ac(33) Steele, W. A. Carbon 1987, 25, 17. (34) Fowler, R. H. Proc. Camb. Philos. Soc. 1936, 32, 144. (35) Honig, J. M. In The Solid-Gas Interface; Flood, E A., Ed.; Mir: Moscow, 1970; p 316 (Russian edition).
Physical Adsorption in Micropores
Langmuir, Vol. 14, No. 14, 1998 3851
count, θ increases rapidly with the pressure and the cooperative effect sets at a lower surface coverage and terminates at a higher surface coverage in the range
possess a strongly heterogeneous surface, the CA should be accurate enough, as it involves less error the broader the distribution.11
0.1 < θ < 0.9-0.95
2. Adsorption by the Heterogeneous Micropore System
(18)
Somewhat similar results are predicted by theories based on van der Waals approach, a quasi-chemical approximation and a model of exact lattice gas.11,12,31,36,37 Keeping in mind models of the adsorption behavior with lateral interactions and the cooperative mechanism, the adsorption process in an individual micropore can be describe as follows. As the pressure is increased from zero, adsorption begins to occur on the pore walls. When a micropore surface coverage, θ, comes to the critical value at the critical condensation pressure, pc, the surface condensation initiates. This process continues at the same pressure until a point is reached when R ≈ 1. At this point, before a completion of the first layer formation, the adsorption process “is energetically as favorable for an adsorbate molecule to exist between the monolayers of adsorbate in the center of the pore, as it is to complete the monolayer coverage.”38 The volume condensation starts and results in the complete micropore filling. Hence, a volume filling of the single micropore and a 2D-condensation on its walls occur at the same critical pressure. From this point of view, the statistical mechanical theories of adsorption on homogeneous surfaces considering lateral interactions may act as a starting point to the description of physical adsorption in micropores. To render the problem tractable, we propose to replace the actual isotherm for the individual micropore wall by the simpler step function
{
0 for p < pc θ(p) ) 1 for p > pc
(19)
as accepted in the condensation approximation method. Somewhat similar isotherms in the micropore region with abrupt filling are predicted by the nonlocal density functional theory.15 Accuracy of this approach increases at sufficiently low temperatures in the subcritical region when the abrupt condensation jump is observed; at the supercritical temperatures the filling transitions tend to be less distinct. The problem of estimating the errors connected with the use of the CA for representation of the monolayer adsorption was discussed by Harris.39,40 In the case of adsorption in micropores, the average force field acting on the adsorbed molecule from the surrounding molecules is enhanced by the proximity of the fluid layer on an opposing wall. It is likely that a 2-D condensation in micropores be set even at a lower surface coverage and terminate at a higher surface coverage than that given by an inequality.18 Since inside a micropore a condensation covers also a free space between monolayers of adsorbate in the center of pore, it contributes more in an overall adsorption process than in the case of only monolayer adsorption. As a consequence, a maximum possible error of the CA method is reduced. Rudzinki and Everett11 pointed out that “in most cases of adsorption on heterogeneous surfaces, the effects of surface heterogeneity are probably much stronger than the small errors due to above approximate solutions.” Since porous carbons usually (36) Bakaev, V. A.; Steele, W. A. J. Chem. Phys. 1993, 98 (12), 9922. (37) Rudzinski, W.; Laitar, L. J. Chem. Soc., Faraday Trans. 1981, 77, 153-158. (38) Marsh, H. Carbon 1987, 25, 49. (39) Harris, L. B. Surf. Sci. 1968, 10, 129. (40) Harris, L. B. Surf. Sci. 1969, 13, 377.
Model of Carbon Heterogeneity. Models of carbon porous structure have been intensively discussed.41-44 Since detailed information concerning the nature and spatial distributions of the heterogeneity is not available at present, we assume that variations in the adsorption energy * due to overlapping of the potential of the opposite walls overshadow random variations of other parameters. This means that within the bases the average energy is approximately constant, but bases of micropores with different d are characterized by different values of cx,)*, that is, cx,)* ) f (d). According to this model, the total micropore surface is considered to be composed of micropore bases (walls) that are regarded unisorptic areas, totally independent of each other. Adsorbate lateral interactions are allowed only between molecules on the same base. Since there are a lot of factors affecting the process of active carbon preparation, it is likely that the central limit theorem of the probability theory45 can be applied to the resultant effect. That is, the distribution of the micropore widths could be modeled by the normal density function, fN (d,µd,σd)45,46
fN (d,µd,σd) )
{ (
)}
1 1 d - µd exp 2 σd σdx2π
2
(20)
where µd and σd are the mathematical expectation and the standard deviation of d, respectively. Note that fN is a symmetrical function of d and extends to -∞. Random variables in our case (the micropore widths) are positive. However, if the parameters µd and σd are such that the probability of obtaining negative realizations of d is negligibly small, this condition can approximately be accommodated: as long as d , µd, the Gaussian decays so rapidly that no appreciable errors are introduced by using the unrealistic limits. The probability of the occurrence of micropores with half-widths less than d, Pr(x 1. As the pore width is reduced beyond d/r0 < 1, the repulsive parts of the opposing wall potential begin to increase, until the entire solid-fluid slit potential becomes repulsive. For micropores with d/r0 < 0.84, the pore space is inaccessible to adsorbate and no adsorption occurs. This repulsive part of the entire solid-fluid slit potential is not taken into account by eq 26. An after effect of a such simplification will be discussed further. For the range of the most physical interest d/r0 > 1, the logarithm of (* - 0* ) may be described by the linear function of d
(
ln(* - 0*) ) ln 0* + k 1 -
)
d r0
(27)
The value of (* - 0* ) represents an excess of adsorption energy in micropores with respect to an open graphite surface. The adsorption energy for a hypothetical ideal graphite surface can be calculated theoretically, as was demonstrated by Kiselev and co-workers.47 Critical Condensation Pressure of the Micropore Volume Filling. In the case of monolayer adsorption, the important problem in the CA method of finding the relation p ) pc() was examined by Harris and Cerofolini.39,40,48,49 The comprehensive discussion of this problem was given by Rudzinski and Everett.11 Our intention now is to find pc() for the micropore volume filling. We proceed with our treatment from the simplest FG isotherm. But the same result for pc() may be obtained from the much more realistic quasi-chemical approximation after more algebra. The condensation pressure on a nonporous surface is obtained from the FG isotherm as follows11
pc ) K exp[(- + 2kTc)/kT]
(28)
where Tc ) | cw/4k | is the critical temperature for a 2D condensation and ) * is the adsorption energy. The constant K has the following detailed form
K)
(2πm)3/2(kT)5/2 qg(T) h3ga(T)
(29)
(47) Avgul, N. N.; Kiselev, A. V.; Poshkus, D. P. Adsorption of Gases and Vapors on the Homogeneous Surfaces; Chemistry: Moscow, 1975 (Russian). (48) Cerofolini, G. F. J. Low Temp. Phys. 1971, 24, 391. (49) Cerofolini, G. F. Surf. Sci. 1972, 6, 473.
Physical Adsorption in Micropores
Langmuir, Vol. 14, No. 14, 1998 3853
Here m is the mass of the adsorbate molecule, h is Planck’s constant, and gg and ga are the partition functions for the internal degrees of freedom of the gas and of the adsorbed species, respectively. When ) *0, eq 28 defines the critical pressure for a graphite surface, pc0
pc0 ) K exp[(-0* + 2kTc)/kT]
(30)
According to the accepted model, micropore walls are considered to be unisorptic areas. Hence, a lateral interaction energy may be regarded to be identical for all walls, that is, cw ) constant. Since * ) f(d), each of the micropore walls may be characterized by its own critical pressure determined by *. Substituting eq 26 into eq 28 and keeping in mind eq 30, after some algebra, one arrives at
pc0 ln ) pc
[(
0 exp k 1 -
)]
d r0
kT
(31)
(
)
( )
pc0 d ) ln 0* + k 1 pc r0
Y ) ωX + τ
(33)
where ω and τ are constants, and the variable X is distributed according to a normal model, Y is also distributed according to a normal model46 with the expected value and the standard deviation
µY ) ωµX + τ;
σY ) |ω|σX
(34)
2. If random variables Z and Y are related by
Y ) ln Z
(35)
and the transformed variable Y is distributed according to a normal model, Z is distributed according to a lognormal model45
2
(36)
Y ) ωd + τ
(37)
where
ω)-
k and τ ) k + ln 0 r0
(38)
and Y denotes either ln(* - 0*) or ln(RT ln pc0/p). Applying the first theorem to the linear function of a normal-distributed random variable d, one obtains that Y follows the normal distribution with parameters
µy ) -
k µ + k+ ln 0 r0 d σy )
(32)
An adsorption energy in eq 32 is expressed in units that refer to 1 mole of a gas. Equations 31 and 32 establish a relation between the critical pressure of a 2-D condensation on micropore walls and the micropore width. Since critical pressures of a 2-D condensation and micropore filling pressures are in a one-to-one correspondence, eqs 31 and 32 determine the micropore filling pressure as a function of a micropore width. Distribution Function of Critical Pressures and Adsorption Energies. The increased well depth in micropores causing a 2-D condensation on their surface occurs at a lower pressure compared with the nonporous surface. Our intention now is to derive the distribution of critical condensation pressures for the carbon sample whose micropore widths and fractional adsorption volumes are distributed according to the normal model. This function in conjunction with the CA approach will provide the calculation of the adsorption isotherm. Since * and pc are functions of a random variable d, the theorems of transformations of distribution functions should be used. These theorems resulting from the general relationship (eq 25) specify:45,46 1. If random variables X and Y are related by the linear function
)}
where the parameters µ and σ refer to the expected value and the standard deviation, respectively, of the related normal model, that is, µ ) µY and σ ) σY. The values of ln(* - 0*) (eq 27) and ln(RT ln pc0/p) (eq 32) are related to the micropore half-width by the same linear function
It follows from eq 31 that the logarithm of RT ln pc0/p may be represented by the same linear function of d as the logarithm of (* - 0* ) (eq 27)
ln RT ln
{ (
1 1 ln z - µ exp 2 σ zσx2π
fLN(z,µ,σ) )
(39)
k σ r0 d
(40)
Applying the second theorem and substituting Y either by ln(* - 0*) or by ln(RT ln pc0/p), we shall obtain the log-normal distribution of Z, where Z is either (* - 0*) or (RT ln pc0/p), respectively
fLN(*-0*,µy,σy) ) 1 (* - 0*)σyx2π
(
fLN RT ln
{ (
exp -
)
pc0 ,µ ,σ ) pc y y
)}
1 ln(* - 0*) - µy 2 σy
{(
1 1 exp 2 pc0 RT ln σ x2π pc y
ln RT ln
2
)}
pc0 - µy pc
σy
(41)
2
(42)
If one designates
Ac ) RT ln
pc0 pc
(43)
eq 42 may be transformed to
fLN(Ac,µy,σy) )
{ (
)}
1 1 ln Ac - µy exp 2 σy Acσyx2π
2
(44)
Adsorption Isotherm. According to the CA approach (eq 19), all micropores for which pc < p will be filled by the condensed adsorbate at given pressure p. Therefore, the fraction of occupied micropore should be calculated by means of the CDF, FLN(Ac,µy,σy), assosiated with a density function given by eq 44
FLN(h,µy,σy) )
1 σyx2π
∫0h A1c exp
{ ( -
)}
1 ln Ac - µy 2 σy
2
dAc (45)
3854 Langmuir, Vol. 14, No. 14, 1998
Dobruskin
where h ) RT ln pc0/p. This CDF determines a micropore fraction for which RT ln pc0/pc e h. The latter is valid for all micropore walls for which pc > p, that is, for unoccupied micropores. Hence, the CDF (eq 45) is equal to the micropore fraction (1 - θ), which at given p is free from adsorbate. Because of this, θ is expressed as follows
θ)1-
1 σyx2π
∫0h A1cexp
{ ( -
)}
1 ln Ac - µy 2 σy
2
dA
(46)
Integral 46 cannot be solved in closed form but is readily available from standard normal tables45 after transforming FLN to FN
1-θ)
∫-∞(ln A - µ )/σ c
y
y
{ } 2
1 z exp dz 2 x2π
(47)
The quantile of the CDF gives the value of q ) (ln Ac µy)/σy at which (1 - θ) reaches its value. The quantile is effectively the inverse of the CDF; for example, the median is given by a quantile of 0.5. The treatment of a set of experimental pairs of values (a, p), where a is an experimental uptake, can be produced as follows. After a is transformed to θ ) a/a0, where a0 is the uptake corresponding to the complete micropore filling, a set of quantiles corresponding to (1 - θ) is found from the standard normal tables. Let us suppose that a critical condensation pressure for a nonporous surface pc0 is known and RT ln pc0/p can be calculated. If experimental data are described by eq 46, the plotting of quantiles via ln Ac would result in the straight line q ) (ln Ac - µy)/σy. The slope, 1/σy, and the intercept, -µy/σy, of this line may be found by the least-squares method. Proceeding from the calculated values of σy and µy, the distribution of the fractional volumes over (* - 0*) is described by eq 41. If the values of the well depth 0* for a nonporous surface and the Lennard-Jones diameter are known, a distribution of micropore widths is found by combining eqs 39, 40, and 20. The value of 0* is equal to the energy of adsorption on a graphite surface at the limit of zero amount adsorbed. According to the present approach, adsorption energies in micropores are calculated with respect to the adsorption on a nonporous surface, which may be considered to be the reference point for our treatment. One of the important parameters in eqs 43 and 46 is the condensation pressure on a nonporous surface. It is obvious that eq 3 can be rewritten as
A ) RT ln ps/p ) RT ln ps/pc0 + RT ln pc0/p
(48)
In doing so, we represent the adsorption potential in micropores as a sum of the adsorption potential for a nonporous surface and the work of compression Ac ) RT ln pc0/p. If pc0 is close to the saturated pressure, Ac may be approximated by A and no appreciable errors are introduced by using the saturated pressure instead of pc0; otherwise, Ac is less than A. The problem is now to estimate a critical condensation pressure on an active carbon surface. The parameters of porous structure are usually calculated with respect to benzene, which is taken to be the standard gas. The most thorough measurement of benzene adsorption on a graphitized carbon black Sterling FT (2700°) had been presented by Pierotti and Smalwood.50 But a kink in the submonolayer region that could be attributed to the 2D (50) Pierotti, R. A.; Smalwood, R. E. J. Colloid Interface Sci. 1966, 22, 469.
Figure 5. Quantiles q via ln A for AG-3P (the line 1), SKT-3 (the line 2), and ART (the line 3) active carbons. The straight lines 1 and 2 are described by equations q ) -3.55805 + 1.53564 ln A and q ) -5.2854 + 1.7499 ln A, respectively.
condensation had not been observed. The kink at p ≈ 50 mm Hg (p/ps ) 0.75) may be explained by a multilayer adsorption. An analysis of adsorption isotherms for benzene on carbon black31,47 shows that for this system lateral interactions are negligible. Steele supposes that the most likely explanation for the behavior of the benzene-graphitized carbon black system is that these adsorbed molecules lie flat on the surface and the most energetic configurations do not contribute to the lateral interactions. This point of view correlates with the study of aromatic hydrocarbons carried out by Williams.51 Nonbonded potential parameters for C‚‚‚C, C‚‚‚H, and H‚‚‚H interactions derived from crystal structures and properties of aromatic hydrocarbons provided an interesting partition of crystal lattice energy into components. In benzene, the C‚‚‚C net potential energy is strongly negative, while the H‚‚‚H net potential energy proved to be slightly positive. The latter configurations mostly contribute to the lateral interaction when molecules lie flat on the surface. In our treatment of the benzene adsorption, we accept, to a first approximation, that for benzene-active carbon systems a critical condensation pressure on an active carbon surface pc0 is close to the saturated pressure ps. Further studies are desirable in the case of other adsorbates. 3. Comparison with Experimental Data The methods under discussion were applied to seven benzene adsorption isotherms. The active carbon (AC) samples employed are the following: AG-3P produced by a vapor-gas activation from coal feedstock; ART and SKT-3 prepared by a chemical activation of peat; SAU produced by a thermal decomposition of polymer starting material. These commercial active carbons have been intensively studied in Russian publications.52,53 It has been shown that all of them are microporous carbons with small mesopore surfaces. For carbons employed, the limiting adsorption, a0, may be taken to be equal to the adsorption at a relative pressure corresponding to the beginning of the hysteresis loop (p/ps ) 0.17). The equilibrium adsorption values plotted as quantiles via ln A are shown in Figures 5 and 6. The main conclusion from Table 4 and Figures 5 and 6 is that experimental data are well described by eq 46 in the wide range of equilibrium relative pressures with relative deviations (51) Williams, D. E. J. Chem. Phys. 1966, 45, 3770. (52) Lukin, V. D.; Novoselski, A. B. Cyclic Adsorption Processes; Khimia: Leningrad, 1989 (Russian). (53) Buturin, G. M. Highporous carbonaceous materials; Khimia: Moscow, 1976 (Russian).
Physical Adsorption in Micropores
Langmuir, Vol. 14, No. 14, 1998 3855
Figure 6. Quantiles q via ln A for SAU active carbon: (filled polygons), 293 K; (ellipses), 313 K; (filled triangles), 353 K; (rhombus), 413 K.
Figure 7. Distribution of the benzene adsorption energies for SKT-3 and AG-3P.
Table 4. Comparison of Experimental and Calculated Isotherms of Benzene Adsorption on AG-3P at 293 K a, θ mmol/g (experiment)
ps/p 4.78 × 10-5 8.17 × 10-5 1.25 × 10-4 2.38 × 10-4 5.88 × 10-4 9.68 × 10-4 1.20 × 10-3 2.07 × 10-3 3.35 × 10-3 4.52 × 10-3 6.78 × 10-3 9.53 × 10-3 1.80 × 10-2 2.99 × 10-2 5.24 × 10-2 8.27 × 10-2 0.17
0.48 0.53 0.60 0.72 0.96 1.12 1.18 1.40 1.61 1.77 2.00 2.22 2.66 3.06 3.60 4.11 5.12
0.094 0.104 0.117 0.141 0.188 0.219 0.231 0.273 0.314 0.346 0.391 0.434 0.520 0.598 0.703 0.787 1.000
ln A
q θ (1 - θ) (calculated)
3.188 3.133 3.089 3.012 2.897 2.828 2.796 2.712 2.631 2.577 2.499 2.428 2.281 2.146 1.972 1.804
1.318 1.262 1.189 1.077 0.887 0.776 0.737 0.602 0.483 0.397 0.278 0.167 -0.049 -0.247 -0.533 -0.797
0.091 0.105 0.119 0.143 0.187 0.216 0.231 0.272 0.315 0.345 0.390 0.432 0.522 0.604 0.702 0.785
Table 5. The Parameters of Active Carbons parameters of eqs 39, 40, 46 active carbons
µY
σY
AG-3P ART SKT-3 SAU
2.317 3.029 3.020 3.141
0.651 0.583 0.572 0.410
parameters of eq 2
µd/r0 σd/r0
µAc, σAc, E, kJ/mol kJ/mol kJ/mol
1.34 1.17 1.17 1.14
12.541 9.114 13.271 1.641 24.511 15.080 24.430 2.000 24.136 15.001 24.728 2.000 27.345 10.759 28.459 3.000
0.16 0.15 0.14 0.10
n
not exceeding 1-1.5% for AG-3P and 5% for SKT-3, ART, and SAU. Only for ps/p < 10-5 do deviations increase. It is a necessary condition that plotting of the isotherms in coordinates of q ) ln(Ac - µy)σy via ln A should give a straight line, but it is not sufficient. The slopes and intercepts of the straight lines give values of µy and σy, which are determined only by the porous structure and independent of temperature. The experimental uptakes for the benzene adsorption at 293, 323, 353, and 413 K on the polymer-based SAU active carbon are plotted in Figure 6. Special consideration should be given to the fact that eq 46 correctly describes the temperature dependence of adsorption and slopes and intercepts of the straight lines are independent of temperature. An analysis of micropore structures and adsorption properties, with respect to benzene at 293 K, is given in Table 5. The expected values and the standard deviations of micropore half-widths were calculated from eqs 39 and 40. The value of 0* (eq 26) was taken as 40 kJ/mol proceeding from the energy of a benzene interaction with a graphite surface at the limit of zero amount adsorbed.47
Figure 8. Distribution of the micropore volume over reduced pore half-widths for SKT-3 and AG-3P active carbons.
The parameter of k was assigned to be 4 as it was found for monatomic gases. The expected values, µLN ) µAc, and the variance, σLN2 ) σAc2, of the log-normal random variable, that is, of the adsorption potential, were calculated by the expressions45
{
}
σY2 µLN ) exp µY + 2
(49)
σLN2 ) exp{2µY + σY2}(exp{σY2} - 1)
(50)
Note that µY and σY in eqs 39-40 are the expected value and the variance of the related normal model and not those of the log-normal model. The average adsorption heat (0* + µAc) for the benzene adsorption on SKT-3 and AG-3P are expected to be ≈64.2 and ≈52.5 kJ/mol. Figure 7 depicts the log-normal density functions corresponding to distributions of the adsorption energies for benzene adsorption on SKT-3 and AG-3P. The distributions of fractional volumes over reduced pore widths are illustrated by Figure 8. Both of these distribution functions should be regarded as giving the distributions for an effective porous material, in which all of the heterogeneity of the real AC is attributed to a distribution of micropore sizes. This effective porous material is approximated by the system of independent micropores of identical volumes, each micropore being within the confines of ideal graphite planes. Equations 39 and 40 show that r0 exerts strong influence on µd and σd. For adsorption of monatomic gases, r0 is the distance at which the energy of interactions with a lattice is equal to zero. Since for nonspherical polyatomic molecules multicenter potential models are employed,43 it is better in this case to accept the experimental value of r0. When estimating the average micropore sizes, Stoekli10 takes the critical dimension of benzene to be
3856 Langmuir, Vol. 14, No. 14, 1998
equal to 0.41 nm. If we accept r0 ) 0.41 nm, µd values of SKT-3 and SAU are found to be 0.48 and 0.46 nm, in good agreement with the values of 0.49 and 0.43 nm given by eq 4. Accuracy of the Model. Equation 46 is applicable for a benzene adsorption in the range of p/ps ≈ 1 × 10-5 to 1 × 10-1. The main cause of deviations is the approximate form of eq 26, which disregards the repulsion forces at d/r0 < 1. For micropores in the range of 0.84 < d/r0 < 1, as the pore width is reduced, there is a rapid rise in filling pressures. This approximation makes its appearance in overestimating the contributions of the smallest micropores in the total amount adsorbed at the lowest p and in underestimating their contributions at high pressures. As noted, for micropores with d/r0 < 0.84, the pore space is inaccessible to adsorbate and no adsorption occurs. The problem of the cutoff below d/r0 ) 0.84 might be addressed by adjusting the normalization factor in eq 20. We could consider the more rigorous truncated distribution fN*(d,µd,σd), which is determined as follows54
fN*(d,µd,σd) ) fN(d,µd,σd)/[1 - FN(0.84,µd,σd)], d/r0 > 0.84 (51) d/r0 e 0.84 0,
{
where FN(0.84,µd,σd) is the normal CDF at d/r0 ) 0.84. Since the values of [1 - FN(0.84,µd,σd)] at µd and σd from Table 5 are 0.99-0.999, the error due to application of eq 20 is less than that induced by the approximate relationship given by eq 26. The more accurate description of the experimental data may be obtained by the numerical solution on the ground of the explicit eq 8 as an alternative to the approximate form given by eq 26. There is another cause of deviation of the experimental and calculated data. According to our model, the micropore and their walls may be only in two states: either to be free from adsorbate at p < pc or to be filled with adsorbate at p > pc. Hence, in the low-pressure region eq 46 does not take into account the submonolayer adsorption and does not converge to the Henry isotherm limit. Further results will be published in due course, and it is hoped to gain more information on the contribution of micropores with d/r0 < 1 and on the initial sections of the isotherms. Nomenclature A ) adsorption potential Ac ) work of compression (eq 43) a ) experimental uptake a0 ) uptake corresponding to the complete micropore filling c ) number of nearest neighbors 2d ) separation between two carbon planes in eq 8 2d1 ) separation between two carbon planes (eq 11) 2d2 ) separation between walls of the internal pore (eq 11) dlimit ) limit slit half-width d/r0 ) reduced half-widths d h ) average half-width of the micropore system in eq 4 E ) parameter of eq 2 FN, FLN ) normal and log-normal cumulative functions, respectively fN, fLN ) normal and log-normal density functions, respectively f() ) density distribution function of in eq 5 gg, ga ) partition functions in eq 29 (54) Burington, R. S.; May, D. C. Handbook of Probability and Statistics with Tables; McGraw-Hill: New York, 1970; p 80.
Dobruskin h ) Planck’s constant h ) RT ln pc0/p in eq 45 K ) constant in eq 28 k ) Boltzmann’s constant k ) constant in eq 26 m ) mass of the adsorbate molecule m ) number of interacting centers per unit area of the lattice plane m1, m2 ) numbers of interacting centers per unit area of the carbon and internal micropores, respectively N ) total number of micropores in the carbon sample in eq 22 n ) parameter of eq 2 Pr (x < d) ) probability of the occurrence of micropores with half-widths less than d p ) partial pressure in the gas phase pc ) critical condensation pressure pc0 ) critical pressure for a graphite surface ps ) saturated pressure q ) quantile of a cumulative distribution function q ) site separation distance for a surface close packing in eq 13 R ) ratio of the well depths of a molecule 2 to that of a molecule 1 R ) gas constant r0 ) (sections 2 and 3) ≡ σcx (section 1) r ) distance between the nuclei of the atoms in eq 6 rj ) distance from the external atom to the jth atom s ) wall area T ) absolute temperature Tc ) critical temperature for a 2D condensation v ) volume of the individual micropore W0 ) limiting adsorption volume W ) occupied adsorption volume w ) lateral interaction energy x and y ) atomic or molecular species in eq 6 X, Y, Z ) random variables Y ) denotes either ln(* - 0*) or ln(RT ln pc0/p) Z ) denotes either(* - 0* ) or (RT ln pc0/p) z ) distance of a molecule from the central plane z1*, z2* ) equilibrium coordinates of molecules in the positions 1 and 2, respectively β ) coefficient that determines the collision distance of a molecule with the surface β ) affinity coefficient in eq 2 ∆ ) 0.340 nm is the distance between the basal planes ) energy of interaction, adsorption energy * ) (sections 2 and 3) ≡ cx,)*(section 1) cx,) ) energy of interaction between a molecule and two parallel carbon lattice planes xx,) ) energy of interaction between a molecule and two walls of the internal micropore 1* ≡ cx,)* ) potential energy minimum (the well depth) of the molecule 1 2* ) potential energy minimum (the well depth) of the molecule 2 cx,)* ) well depth for interactions between a molecule and two parallel carbon lattice planes cx,)* ) (section 1) ≡ * (sections 2 and 3) xx* ) depth of the potential energy minimum for a single pair of molecules of adsorbate xx,)* ) well depth for interactions between a molecule and two walls of the internal micropore xy* ) depth of the potential energy minimum for a single pair x and y 0* ) well depth for adsorption on a graphite surface ∆ ) an excess of adsorption energy in micropores with respect to an open graphite surface µd ) mathematical expectation of d µY, µX ) mathematical expectation of random variables X and Y, respectively µLN, µAc ) mathematical expectation of log-normal variable and Ac, respectively θ ) fractional adsorption volume
Physical Adsorption in Micropores σcx ) distances at which cx ) 0 σ xx ) distance at which xx ) 0 σxy ) distance at which xy ) 0 σd ) standard deviation of d σY, σX ) standard deviation of random variables X and Y, respectively σLN, σAc ) standard deviation of log-normal variable and Ac, respectively τ ) constant defined in eq 38 Ω ) integration region over all possible adsorption energies
Langmuir, Vol. 14, No. 14, 1998 3857 ω ) constant defined in eq 38 ω ) area per molecule in eq 13
Acknowledgment. The author is grateful to Professor David Avnir of the Herbrew University of Jerusalem for fruitful discussion of results and to Dr. S. D. Kolosentzev and Dr. Yu. Ustinov of the Technological Institute of St. Petersburg for kindly supplying experimental adsorption data. LA971211T
