170
Langmuir 1996, 12, 170-182
A New Molecular Probe Method To Study Surface Topography of Carbonaceous Solid Surfaces† W. Rudzin´ski* and K. Nieszporek Department of Theoretical Chemistry, Maria Curie-Skłodowska University, Lublin 20-031, Poland
J. M. Cases, L. I. Michot, and F. Villieras Laboratoire Environment et Mine´ ralurgie, Institut National Polytechnique de Lorraine, Ecole Nationale Superieure de Ge´ ologie de Nancy U.A. 235 du CNRS, 54501 Vandoeuvre Cedex, France Received October 21, 1994. In Final Form: June 12, 1995X Adsorption on a heterogeneous surface of flexible molecules composed of a number of mers, like n-alkanes for instance, is strongly affected by topography of surface adsorption sites. Therefore, fitting to experimental adsorption isotherms, theoretical expressions developed for multisite-occupancy adsorption on heterogeneous surfaces characterized by random and patchwise topography should make it possible to distinguish whether a studied solid surface is characterized by random or by patchwise topography. For certain fundamental but also practical reasons, we took into consideration carbonaceous adsorbents. Following the general approach to multisite-occupancy adsorption published recently by Rudzin´ski and Everett, we have developed here a generalized form of the Dubinin-Astakhov isotherm equation, taking into account surface topography and the interactions between the adsorbed molecules. The developed isotherm equation has been used successfully by us to study the nature of surface topography of a number of carbonaceous adsorbents.
Introduction It is now generally realized that the actual (really existing) solid surfaces are energetically heterogeneous to a more or lesser extent.1,2 In terms of localized adsorption, it means the variation of adsorption energy when going from one to another site across the surface. The generally accepted quantitative measure of the energetic heterogeneity of the actual solid surfaces is the differential distribution of a number of adsorption sites among the corresponding values of adsorption energy. This function, used usually in its form normalized to unity, is called “the adsorption energy distribution”. That quantitative information is sufficient for a complete thermodynamic description of adsorption equilibria in the systems where one adsorbed molecule occupies one adsorption site and no interactions exist between adsorbed molecules (Langmuir model). However, in the systems where the interactions between adsorbed molecules cannot be ignored, or when one adsorbed molecule occupies more than one site, another important physical factor comes into play. This is the way in which adsorption sites characterized by different adsorption energies are distributed on a heterogeneous solid surface. In other words, this is the topography of a heterogeneous solid surface. Yang and co-workers3-5 and Zgrablich and co-workers6-10 have shown that even in the * Author to whom the correspondence should be addressed. † Presented at the symposium on Advances in the Measurement and Modeling of Surface Phenomena, San Luis, Argentina, August 24-30, 1994. X Abstract published in Advance ACS Abstracts, January 1, 1996. (1) Jaroniec, M.; Madey, E. Physical Adsorption on Heterogeneous Solids; Elsevier: Amsterdam, 1988. (2) Rudzin´ski, W.; Everett, D. H. The Adsorption of Gases on Heterogeneous Surfaces; Academic Press: New York, 1992. (3) Kapoor, A.; Ritter, J. A.; Yang, R. T. Langmuir 1989, 5, 1118. (4) Kapoor, A.; Yang, R. T. AIChE J. 1989, 35, 1735. (5) Kapoor, A.; Yang, R. T. Chem. Eng. Sci. 1990, 45, 3261. (6) Pereyra, V.; Zgrablich, G.; Zhdanov, V. P. Langmuir 1990, 6, 691. (7) Pereyra, V.; Zgrablich, G. Langmuir 1990, 6, 118. (8) Mayagoitia, V.; Rojas, F.; Pereyra, V.; Zgrablich, G. Surf. Sci. 1989, 221, 394.
0743-7463/96/2412-0170$12.00/0
Figure 1. A schematic representation of random topography of an energetically heterogeneous surface composed of two kinds of adsorption sites, represented by white and black circles.
simplest case of Langmuirian adsorption the surface topography affects strongly the surface diffusion of adsorbed molecules. So far, two extreme models of surface topography have, almost exclusively, been considered in theoretical works on adsorption. They are shown schematically in Figures 1 and 2. The first one was the “random” model, introduced in literature by Hill.11 It assumes that adsorption sites characterized by different adsorption energies are distributed over a solid surface completely at random. In other words, no spatial correlations exist between adsorption sites having the same energy of adsorption. A schematic picture of that kind of surface topography is shown in Figure 1. (9) Mayagoitia, V.; Rojas, F.; Riccardo, J. L.; Zgrablich, G. Phys. Rev. 1990, B41, 7150. (10) Riccardo, J. L.; Pereyra, V.; Zgrablich, G.; Royas, F.; Mayagoitia, V.; Kornhauser, I. Langmuir. (11) Hill, T. L. J. Chem. Phys. 1949, 17, 762.
© 1996 American Chemical Society
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Figure 2. A schematic representation of patchwise topography of an energetically heterogeneous surface composed of two kinds of adsorption sites.
Figure 3. A schematic representation of a mediate surface topography when the heterogeneous surface is composed of two kinds of adsorption sites.
The other extreme model of surface topography was the “patchwise” model, introduced to literature by Ross and Olivier.12 It assumes, that adsorption sites having the same adsorption energy are grouped on a heterogeneous surface into “patches”. These patches are large enough so that the states of an adsorption system in which two interacting molecules are adsorbed on different patches could be neglected. Thus, the adsorption system can be considered as a collection of independent subsystems, being only in a material and thermal contact. On the contrary, adsorption systems exhibiting random surface topography should be considered as a thermodynamic entity. A schematic picture of a patchwise topography is shown in Figure 2. The theoretical description of one-site-occupancy adsorption equilibria for both patchwise and random topography was elaborated in the works by Steele,13 Pierotti and Thomas,14 Rudzin´ski et al.,15-18 and Zgrablich and co-workers.19,20 Of course, it goes without saying that the actual solid surfaces will exhibit a surface topography “mediate” between patchwise and random. That kind of surface topography is shown schematically in Figure 3 and is sometimes called the “partially correlated” surface. Theoretical solutions for adsorption on surfaces characterized by the “mediate” surface topography have been developed by Tovbin21,22 and by Rippa and Zgrablich.23 Every theoretical description of the systems characterized by “mediate” surface topography will be based on a concept of a quantitative measure of the degree of spatial correlations between adsorption sites characterized by the same adsorption energy. However, no methods have been known until very recently to determine quantitatively such a degree of
spatial correlations. Even the judgement of whether an adsorption system exhibits patchwise or random topography was difficult for the majority of investigated adsorption systems. Although different expressions are obtained for adsorption equilibria or surface diffusion on surfaces characterized by different topographies, a practical discrimination between various topographies is difficult. The obtained equations contain a number of unknown parameters, the computer elucidation of which is difficult at the present accuracy of adsorption measurements. Thus, experimental studies of one-site-occupancy adsorption do not allow at present determination of the surface topography by a suitable computer analysis of experimental data. On the contrary experimental and numerical studies of multisiteoccupancy adsorption seem to create a hope for even a quantitative determination of the degree of the spatial correlations between adsorption sites having similar adsorption properties. A first method to determine quantitatively the degree of these spatial correlation has been proposed recently by Rudzin´ski and Everett in their monograph “Adsorption of Gases on Heterogeneous Surfaces”.2 It is based on some ideas of multisite-occupancy adsorption on heterogeneous solid surfaces launched by Marczewski et al.24 and by Nitta et al.25 It requires knowledge of adsorption isotherms of the first few members of a hydrocarbon homologous series, measured at one temperature, but at low surface coverages. From these data one can estimate the so-called “topography factor”, which is equal to zero for patchwise topography and to unity for random topography. Here, we are going to propose another edition of this method which requires a knowledge of only one adsorption isotherm of a molecule composed of at least two identical segments. Then we are going to show how the modified Rudzin´ski-Everett method can be applied to surfaces characterized by other than the Gaussian-like adsorption energy distributions. In particular, we will focus our attention on the adsorption energy distribution leading to the Dubinin-Astakhov isotherm equation. That equation is known to represent well experimental isotherms of adsorption by activated carbons. So, we will focus our attention here on the application of this modified method to study the surface topography of various activated carbons.
(12) Ross, S.; Olivier, J. P. On Physical Adsorption; Interscience Publishers, Inc.: New York, 1964. (13) Steele, W. A. J. Phys. Chem. 1963, 67, 2016. (14) Pierotti, R. A.; Thomas, H. E. Trans. Faraday Soc. 1974, 70, 1725. (15) Rudzin´ski, W.; Lajtar, L. J. Chem. Soc., Faraday Trans. 2 1981, 77, 153. (16) Rudzin´ski, W.; Baszynska, J. Z. Phys. Chem., Leipzig 1981, 262, 533. (17) Rudzin´ski, W.; Jagiello, J. J. Low Temp. Phys. 1981, 45, 1; 1982, 48, 307. (18) Rudzin´ski, W.; Jagiello, J.; Grillet, Y. J. Colloid Interface Sci. 1992, 87, 478. (19) Ripa, P.; Zgrablich, G. J. Phys. Chem. 1979, 79, 2118. (20) Riccardo, L.; Pereyra, V.; Rezzano, J. L.; Rodriguez Saa, D. A.; Zgrablich, G. Surf. Sci. 1988, 204, 289. (21) Tovbin, Y. K. Dokl. Akad. Nauk SSSR 1977, 235, 641. (22) Tovbin, Y. K. Zh. Fiz. Khim. 1982, 56, 686. (23) Rippa, R.; Zgrablich, G. J. Phys. Chem. 1975, 79, 2118.
(24) Marczewski, W. A.; Derylo-Marczewska, A.; Jaroniec, M. J. Colloid Interface Sci. 1986, 109, 310. (25) Nitta, T.; Kuro-Oka, M.; Katayama, T. J. Chem. Eng. Jpn. 1984, 17, 45. (26) Rudzin´ski, W.; Jagiello, J. J. Low Temp. Phys. 1982, 48, 307.
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Theory We consider the single-component, multisite-occupancy adsorption on both patchwise and random solid surfaces.2 For the reader’s convenience we repeat here briefly certain principles from the work by Marczewski et al.,24 which will be essential for our further considerations. So, let us assume that every admolecule adsorbs on the surface in such a way that each “mer” interacts strictly with a single site and in the same way. The total adsorption energy of a chosen admolecule is, n,
˜ i(l) ∑ l)1
(i ) 1, 2, ..., W)
(7)
n ) nj˜
(8)
and
(1)
where the subscript l denotes the local “mer” position in an admolecule, ˜ i(l) is the interaction energy of the lth “mer” with the ith type of adsorption site, and W is the number of site types. When the molecule size equals the site size (n ) 1)
˜ 1 ) ˜ i(1) ) ˜ i
(2)
In our further consideration the accent tilde (˜) will be used to denote a quantity related to one mer. We define next χn(n) as the differential distribution of the number of the surface configurational states of an adsorbed molecule, into corresponding values of n. This is simply the generalization of the “adsorption energy distribution” for multisite-occupancy adsorption. For practical purposes, it is useful to consider the effect of n on kth central moment µkn of the adsorption energy distribution χn(n),
µkn )
n ) n˜ i
For the considered patchwise surface topography
n
n )
large to neglect the number of sites placed in the patch boundaries. This means that if a given site is of Si type, then its neighbor is also of Si type. Thus, in the case of the patchwise surface
∫∆ (n - n)kχn(n) dn n
(3)
where ∆n is the physical domain of n. The adsorption energy distribution χn(n) is now the differential distribution of the number of the surface (configurational) states of an adsorbed molecule into the corresponding values of n. The second central moment µ2 is related to the variance σ, 2
µ2 ) σ
(4)
µkn ) nkµ˜ k1
The most important effect of n on χn(n) will, of course, be that on the second central moment, i.e. the effect of n on the variance σn. From eq 8 we have
σn ) nσ˜ 1 ) (µ2n)1/2
(5)
shows whether χn(n) is symmetrical (β*1 ) 0), a left hand widened (β*1 < 0), or a right hand widened (β*1 > 0) function. The measured quantity is the number of adsorbed molecules Nt, which, in terms of our theoretical treatment, is to be related to the following average θt
Ntn θt(p,T) ) ) M
∫Ω θ(n, p, T)χn(n) dn
(6)
where M is the total number of adsorbed sites, Ω is the physical domain of n, and θ(n,p,T) is the fractional occupancy of the configurational surface states, characterized by the adsorption energy n. This is simply the generalization of the “local adsorption isotherm” known widely in the theories of one-site-occupancy adsorption on heterogeneous solid surfaces. Let us consider first the patchwise model where adsorbent surface is divided into energetically homogeneous patches, which are sufficiently
(10)
It may also easily be shown that, for patchwise topography considered for the moment,
1 χn(n) ) χ1(˜ 1) n
(11)
From eqs 10 and 11 it follows that the average energy and the width of the distribution function increase with the increasing number of adsorption sites which are occupied by an adsorbed molecule. However, the other parameters determining the shape of the distribution function χn(n), like the skewness β*1 for instance, remain for the patchwise topography invariant of n. A somewhat different situation occurs in the case of random surface topography. In this case
χn+1(n+1) )
n+1 ) n + j˜
(12)
∫∆ χn(n)χ1(n+1 - n) dn
(13)
n
where
(n+1 - n) ∈ ∆˜
(14)
Further,
whereas higher central moments are related to the form of χn(n). For instance, the skewness β*1,
β*1 ) µ3/σ3
(9)
µ2n ) nµ˜ 21,
i.e.
σ2n ) nσ˜ 21
(15)
µ3n ) nµ˜ 31
(16)
2 µ4n ) nµ˜ 41 + 3µ˜ 21 n(n - 1)
(17)
β*1n ) β˜ *11/n1/2
(18)
but
and
Thus, in contrast to patchwise topography, in the case of random surface topography increasing the value of n affects also the shape of the adsorption energy distribution. With the increasing value of n the distribution function χn(n) becomes more and more similar to a Gaussian function. In the case of the patchwise surface topography, θ(n,p,T) can then be just one of the equations used to represent multisite-occupancy adsorption on homogeneous solid surfaces. At the same time, there was no paper proposing an expression for the “local adsorption isotherm” θ(n,p,T)
Solid Surface Topography
Langmuir, Vol. 12, No. 1, 1996 173
for the surfaces characterized by random surface topography, until Nitta published a first solution for that problem25 in 1984. Below we give a brief sketch of his considerations, for the reader’s convenience. When Nt molecules are adsorbed on a heterogeneous surface, the system partition function Q(Nt,M,T), may be written as t Q ) qN s
{
˜ ij
}
∑ g(Nt,M,{Nij}) exp ∑i ∑j Nij kT
{Nij}
which minimizes the Helmholtz energy of the system (or maximizes ln Q) with these constrans. The chemical potential, spreading pressure, internal energy, etc. are derived by the differentiation of ln Q with respect to Nt, M, or T. Then, the method of undetermined multipliers is used to determine the distribution {Nij}. From the condition of the maximum with respect to variable {Nij}, the relations are obtained,
( ) Nij
(19)
-ln
Nij ∑ j)1
Mi -
where ij is the energy of adsorption of jth group (mer) on ith kind of sites. Nij is the number of adsorption-pairs of sites i and a group j; {Nij} is a distribution of these adsorption pairs, the subscript i0 being used for the empty site, g(Nt,M,{Nij}) is a combinatorial factor, expressing the number of the distinguishable ways of distributing the Nt molecules on M sites under the condition of a special distribution {Nij}; qs is the product of internal and vibrational partition functions of an adsorbed molecule. As emphasized by Nitta, it is difficult to find a rigorous expression for g(Nt,M,{Nij}) by taking into consideration the mutual correlations between neighboring sites and neighboring groups in a molecule. A simple expression is obtained by assuming that all pairs of site-mer {ij} are independent, under the constraints imposed by the distribution {Nij}. That assumption and the procedure used by Nitta are similar to the quasi-chemical approximation presented by Guggenheim for moleculemolecule interactions. The expression for g(Nt,M,{Nij}) derived in this way by Nitta reads W
ln g(Nt,M,{Nij}) ) ln g0(Nt,M) -
( ) Nij!
s
∑ ∑ ln N*! i)0 j)0
(20)
ij
where g0 is the combinatorial factor for a homogeneous surface and the asterisk stands for the random distribution corresponding to the homogeneous surface. The expression used to represent g0 was
g0(Nt,M) )
(21)
where ζ is the constant relating to the flexibility and symmetry number of a molecule and n is the number of sites occupied by a molecule, given by the sum of all nj’s (j ) 1, 2, ..., n). Substituting eqs 20 and 21 into eq 19 and replacing the summation by the maximum term on the right-hand side of eq 19, one obtains the following expression for ln Q, W
s
∑ ∑ i)0 j)0
(
ln
+ Nij!
)
Nij˜ ij
N*ij!
kT
(22)
There are constraint conditions for the numbers of adsorption pairs {Nij} W
Nij - njNt ) 0 ∑ i)1
kT
(i ) 1, ..., W; j ) 1, ..., s) where βj is the Lagrange multiplier relating to a group j originated from eq 24. From the condition of the maximum term with respect to {Nij}, Nitta arrives at the following expression for the chemical potential of the adsorbed molecules, µs,
µs
( ) () ( ) ( ) [( ) ] ∂ ln Q
) -ln(qsζ) + ln
)kT
∂Nt
Nt
-
M
M,T
n ln
M - nNt
njNt
s
-
M
∑ j)1
nj ln
nj ln ∑ j)1
+
M - nNt Nij
s
ij
-
Mi -
Nij ∑ j)1
kT
(23)
θt ) nNt/M
(27a)
the fraction of sites occupied; (27b)
the fraction of sites occupied by segment of type j; θij ) Nij/Mi
(i ) ..., W; j ) 1, ..., s)
(27c)
the fraction of sites of type i occupied by segments of type j. The problem of evaluating θt is not trivial, and Nitta proposed a complicated numerical procedure, in which only a discrete distribution of adsorption energy can be applied. However, that problem becomes easy when all the segments of an admolecule are of the same kind. This, for instance, is the case of hydrocarbon homologous series, the adsorption of which by activated carbons is so frequently studied experimentally. Nitta’s equation (26) takes then the following form,
µs ) -ni - kT ln(qsζn) - (n - 1)kT ln θt + nkT ln
(24)
The problem to be solved may now be expressed as that of determining the adsorption-pairs distribution {Nij}
(26)
To evaluate θt, three kinds of surface coverages, θt, {θtj}, and {θij} are to be defined
s
Nij - Mi ) 0 ∑ j)1
(25)
) βj
θtj ) njNt/M (j ) 1, ..., s)
M! ζNt Nt!(M - nNt)! M(n-1)Nt
ln Q ) Nt ln qs + ln g0 +
ij +
θi (28) 1 - θi
where
θi ) Ni/M
(29)
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Rudzin´ ski et al.
and
ni ) n1i
(30)
When the surface becomes homogeneous, i.e., θi f θt, eq 28 reduces to the well-known Flory’s isotherm for multisite-occupancy adsorption on homogeneous solid surfaces
µg ) µs ) -n - kT ln(qsζn) + nkT ln
θ (1 - θ)n
(31)
The form of Nitta’s equation (30) might suggest, at the first sight, that while evaluating θt defined in eq 6, one has to accept also the “patchwise” relation between χ1(1) and χn(n). Such a conclusion has, in fact, been drawn by Rudzin´ski and Everett who decided to insert the chapter on multisite-occupancy adsorption shortly before their monograph was in print. However, a more careful analysis of Nitta’s approach must bring one to the conclusion that eq 30 does not imply such a “patchwise” relation between χ1(˜ 1) and χn(n) in the case of random surface topography. ni is to be interpreted only as the sum of the adsorption energies of all segments. Thus, when all the segments are identical, Nitta’s assumption that all the pairs {i,j} are independent is true only if ith values are independent. This means a lack of spatial correlations between ith values of various sites what is true only in the case of random topography. It means, Nitta’s expression (28) is valid for surfaces having random topography. The averaging of θi in eq 31 with respect to the dispersion of ni defined in (30) can be carried out easily by applying the Rudzin´ski-Jagiello2 method. Thus,
θt )
∫
∞ nc
χn(n) dn + corr
(32)
Figure 4. Effect of r and E on the form of the adsorption energy distribution χn(n) calculated from eq 39, by assuming that 0n ) 0. (A) The value of r ) 2: En/kT ) 1 (1), En/kT ) 2 (2), En/kT ) 3 (3). (B) The value of En ) 2kT: r ) 1 (1), r ) 2 (2), r ) 3 (3).
equation for (p) nc takes then the following form,
(
(p) nc ) -kT ln
where nc is found from the condition
( ) ∂2θ ∂2n
)0
(33)
n)nc
and corr is a correction term which can, safely, be neglected in the case of strongly heterogeneous surfaces. After performing the differentiation (33) in eq 28, we arrive at the following expression for nc,
(r) nc
s
) -µ - kT ln(qsζn) - (n - 1)kT ln θt
[ ]
pΛ3 µg ) ln kT qgkT
(35)
eq 34 takes the following form
(r) nc ) -kT ln(nK′p) - (n - 1)kT ln θt
(36)
χn(n) )
K′ ) qsζΛ /qgkT
(37)
Λ is the thermal Broglie wavelength and qg is the internal molecular partition function of the molecules in the bulk gas phase. In the case of patchwise topography, θi and ni in eq 33 are to be identified with θ and n in eq 31. The obtained
r(n - 0n)r-1 (En)r
{ [ ]}
n - 0n exp En
r
(39)
where 0n is the lowest value of the adsorption energy n on a given heterogeneous surface, En is related to the width of that function, and r governs its shape. It is shown in Figure 4. Stoeckli27 showed that this function represents the energetic heterogeneity of the surfaces of activated carbons, the adsorption on which is described by the Dubinin-Astakhov isotherm equation. This equation is obtained by putting n ) 1 in the expressions for nc, and then inserting nc into eq 32. Doing so, we obtain
{ [ ]} { [
where 3
(38)
In the case of multisite-occupancy adsorption the growing value of n may affect, in general, both the spread (the second central moment µ2n) and the shape (µ3n and higher central moments) of the function χn(n). One of the simplest analytical approximations that is flexible enough to reproduce such behavior is the following function
(34)
where the superscript “r” refers to random topography. After replacing µs by µg of an ideal gas,
)
n1+n/2 K′p (n1/2 + 1)n-1
θt ) exp -
c - 0 E1
r
) exp -
]}
kT p0 ln E1 p
r
(40)
where (27) Stoeckli, H. F.; Lavanchy, A.; Krachenbuchl, F. In Adsorption at the Gas-Solid and Liquid-Solid Interface; Rouquerol, I., Sing, K. S. W., Eds.; Elsevier: Amsterdam, Oxford, New York, 1982; p 201.
Solid Surface Topography
Langmuir, Vol. 12, No. 1, 1996 175
ln p0 ) -ln K′ -
0 kT
(41)
The experimental adsorption isotherms are analyzed usually by using the following linearized form of the Dubinin-Astakhov isotherm,
( )[ ]
kT ln Nt ) ln M E1
r
p0 ln p
39. In doing so we arrive at the following equation for θt,
{[
θt ) exp -
[
]
(44)
0n n1+n/2 K′ ln p ) -ln 1/2 kT (n + 1)n-1 0
(42)
(28) Kaminsky, R. D.; Monson, P. A. Microscopic Models for Adsorption Equilibrium in Heterogeneous Solids. Abstracts of the Fundamentals of Adsorption conference held in Kyoto 1992; p 156. (29) Bojan, M. J.; Vernov, A. V.; Steele, W. A. Langmuir 1992, 8, 901. (30) Gubbins, J. K.; Lastowsky, K. Langmuir 1993. (31) Astakhov, V. A.; Dubinin, M. M.; Romakov, P. G. Theor. Osn. Khim. Techn. 1969, 3, 292. (32) Dubinin, M. M. Chem. Rev. 1960, 60, 235. (33) Dubinin, M. M. Chemistry and Physics of Carbon; Walker, P. L., Ed.; Marcel Dekker: New York, 1966; Vol. 2, p 51. (34) Dubinin, M. M. Adsorption-Desorption Phenomena; Ricca, Ed.; Academic Press: New York, 1972; p 3. (35) Dubinin, M. M.; Astakhov, V. A. Molecular Sieve Zeolites II; Adv. Chem. Ser. 102; American Chemical Society: Washington, DC, 1971; p 69. (36) Eguchi, Y. Kagaku Kojo 1969, 13 (9), 45. (37) Eguchi, Y. J. Jpn. Petrol. Inst. 1970, 13, 106. (38) Kawazoe, K.; Astakhov, V. A.; Kavai, T.; Eguchi, Y. Kagaku Ko´ gaku 1971, 35, 1006. (39) Kawazoe, K.; Kawai, T. Seisan Kenkyu 1973, 25, 513. (40) Kawazoe, K.; Kawai, T.; Eguchi, Y.; Itoga, K. J. Chem. Eng. Jpn. 1974, 7, 158. (41) Nakahara, T.; Hirata, M.; OHmori, T. J. Chem. Eng. Jpn. 1975, 20, 195.
(43)
r
where
r
where Nt is the measured adsorbed amount and M is the monolayer capacity expressed in the same units. Choosing properly the parameters p0 and r, one should get the linear relationship between ln Nt and [ln p0/p]r. Concerning the parameter p0, it was commonly assumed to be the saturated vapor pressure of adsorbate at temperature T. Such a choice reflects the classical view on adsorption in porous materials as a graduate filling of micropores by liquid-like adsorbed phase. In this picture of adsorption, n ) kT ln p0/p is the value of the “adsorption potential” which causes the liquefaction of adsorbate molecules in an empty pore at the pressure p. The function (39) would mean the distribution of micropore volume among the value of that adsorption potential, the minimum value of which is zero. Thus, in the model of the “micropore filling”, we have a collection of subsystemssmicropores which are only in thermal and material contact. The adsorption in these micropores proceeds in an ideally stepwise fashion described by eq 32, in which now c ) kT ln p0/p. The present computer simulations show that the state of the adsorbate molecules in micropores is much different from that of the molecules in the bulk liquid,28-30 so the “micropore filling” cannot be identified with the bulk condensation. Thus, p0 cannot be identified generally with the saturated pressure value. The applicability of the Dubinin-Astakhov equation is generally demonstrated. First such papers were published by Dubinin and co-workers31-35 but soon new impressive experimental evidence for the wide applicability of the Dubinin-Astakhov equation came from the works published by Japanese scientists.36-41 Thus, while applying the new approach to multisite adsorption on carbonaceous adsorbents, we will represent χn(n) by the function given in eq 39. The generalized form of the Dubinin-Astakhov equation for the case of patchwise surface topography is obtained by inserting (p) nc defined in eq 38, into eq 32, when χn(n) is represented by the function defined in eq
]}
kT p0 ln En p
In a similar way we obtain the expression for θt in the case of random topography, by inserting (r) nc defined in eq 36 into eq 32,
{[
θt ) exp -
p0 kT ln (n-1) En pθ t
]} r
(45)
where
0n ln p ) -ln(nK′) kT 0
(46)
Thus, in the case of patchwise topography, the experimental data should be well correlated by the linear plot
[ ]
ln Nt vs ln
p0 p
r
(47)
While this linear regression is carried out, the two parameters p0 and r are to be adjusted by computer. Then in the case of random topography, eq 43 suggests the following linear regression to be made,
ln Nt ) ln
( )[
M kT n En
r
ln
p0 pθ(n-1) t
]
r
(48)
in which the three parameters p0, r, and M are to be adjusted by computer. Now let us remark that in the case of patchwise topography the variance En estimated from fitting eq 43 to the experimental data should fulfill the condition En ) nE1, whereas for the random topography En ) n1/2E1. Surface topography affects also the way in which the interactions between adsorbed molecules influence the adsorption on heterogeneous surfaces. As a starting point we have to consider the contribution to the surface chemical potential µs. Even when the simplest mean-field (Bragg-Williams) approximation is accepted, a simple expression for µsin is obtained only if one neglects the “excluded interactions” by the existing chemical bonds between the adsorbed segments. While accepting this simplification, we have
µsin ) zu12θ
(49)
where z is the number of the nearest neighbors adsorption sites, decreased by the average number of chemical bonds between the adsorbed mer and the admolecule. Further, µ12 is the interaction energy between two segments adsorbed on neighboring adsorption sites. As the condition (33) is fulfilled when θ ) 1/2, (p) nc takes the more general form,
(
(p) nc ) -kT ln
)
n1+n/2 K′p - 12zu12 (n1/2 + 1)n-1
(50)
and the final equation (43) is still valid, except that p0 is
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to be interpreted now as follows,
] ( )
[
ln p0 ) -ln
0n 1 zu12 n1+n/2 K′ 2 kT kT (n1/2 + 1)n-1
(51)
Thus, in the case of patchwise topography and strong surface heterogeneity the experimental data will still be correlated by the linear plot (47), even in the presence of interactions between adsorbed molecules. In the case of random surface topography, the interactions between adsorbed molecules will be a source of a new configurational term in the isotherm equation. Now, the starting point is eq 28 which we will write as follows,
µs ) -ni - kT ln(qsζn) - (n - 1)kT ln θt + θi + zu12θt (52) nkT ln 1 - θi Accordingly, the generalized function (r) nc takes the following form, s (r) nc ) -µ - kT ln(qsζn) - (n - 1)kT ln θt - zu12θt (53)
or, in another form
(r) nc ) -kT ln(nK′p) - (n - 1)kT ln θt - zu12θt
(54)
Thus, the generalized form of eq 48 taking into account interactions between adsorbed molecules reads
ln Nt ) ln
( )[ r
M kT n En
ln
p0 pθ(n-1) t
-
zu12 θ kT t
]
r
(55)
where p0 is still given by eq 46. To obtain the expression for Nt in the case of mediate surface topography, we modify eq 55 in the way suggested in the monograph by Rudzin´ski and Everett2
ln Nt ) ln
)[
(
M kT - 1-t/2 n n E1
r
ln
p0 t(n-1)
pθt
zu12 1 kT 2
1-t
()
θtt
]
r
(56)
In eq 56, t ∈ (0,1) is the topography randomness factor. When t f 0, eq 56 reduces for the one developed for patchwise topography, whereas when t f 1, eq 56 reduces to eq 55 developed for fully random surface topography. In eq 56, p0 is given by the following expression,
[
ln p0 ) -ln
]
0n K′ kT (n1/2 + 1)t(n-1) n1+(n/2)t
(57)
Discussion Equation 56 should be applicable to all adsorption systems in which the actual spectrum of adsorption energies can be well approximated by the “smooth” onemodal functions χn(n) shown in Figure 4. However, for a number of reasons, carbonaceous adsorbents seem to be the best objects to verify the applicability of eq 56 in order to determine the topography of a heterogeneous solid surface. This is because the nature of the topography of carbonaceous adsorbents is known from direct observations by using scanning tunneling microscopy (STM). The
structure of graphite has been extensively studied, and it would be far too long to mention even the most interesting papers. Also the topography and atomic structure of some glassy carbons42 and carbon fibers43 have been successfully imaged by STM. Then, two years ago Donnet and Custodero44 reported for the first time a successful application of STM to study the topography of activated carbon blacks. Their STM studies confirm basically the generally accepted model of the surface of activated carbons as composed of graphite-like ordered zones, joined by less ordered zones on which surface functional groups are located. Somewhat surprising in their STM experiment was an apparent 0.4 × 0.25 nm2 rectangular network of different graphite-like layers oriented more or less toward one direction on a grain scale. This contradicts the views expressed first by Hirsh and Diamond,45 and next commonly accepted, that the number of the graphite-like layers increases from 2 to 4-5 with the increasing diameters of these layers. It was also believed that the diameters of these layers varied typically between 0.6 and 3 nm. Meanwhile, the domains observed in the STM image obtained by Donnet and Custodero were very regular in a size of about 5 nm. Only one N330 carbon black was studied in their experiment, and it goes without saying that the (average) dimensions of the graphite-like domains, as well as the degree of their mutual ordering will depend upon the nature (preparation) of a carbon black sample. These obvious experimental facts create the following test for the applicability of our equation (56), when verified by studying isotherms of multisite-occupancy adsorption by activated carbons. Namely, while studying adsorption by various carbon samples and adsorption in different adsorption regimes, one should observe correlations between the parameters E1 and t. The decreasing tendency in E1 caused by the growing adsorption on the graphite-like homogeneous domains, should be accompanied by decreasing values of the parameter t due to the growing patchwise like character of that surface topography. To check this prediction, we took into consideration the experimental isotherms of methane and ethane adsorption on the activated carbon BPL, reported by Reich et al.,46 and on two other carbons M-30 and M-38 reported by Nitta et al.47 The BPL carbon adsorbent has been a popular adsorbent used in various adsorption processes, and therefore frequently studied in theoretical papers. Then, it should also be added that the data reported by Reich et al. cover a wide range of temperatures which makes them a very attractive material for theoretical analysis. The data reported by Nitta et al. contain many, carefully measured points, which is also essential for a reliable evaluation of unknown parameters, when they are found by fitting theoretical expressions to the experimental points. The carbon adsorbents M-30 and M-38 were developed by Osaka Gas Co., Ltd., for the natural gas storage. They had an unusually high surface area close to the theoretical highest surface area of carbon adsorbents built up of hypothetical single graphite layers, 2630 m2/g. (42) Miyamoto, T.; Kaneko, R.; Miyake, S. J. Vac. Sci. Technol. B 1991, 9, 1336. (43) Magonov, S. N.; Cantow, H. J.; Donnet, J. B. Polym. Bull. 1990, 23, 552. (44) Donnet, J. B.; Custodero, E. Carbon 1992, 30, 813. (45) Hirsch, P. B. Proc. R. Soc. London 1954, A226, 143. (46) Reich, R.; Ziegler, W. T.; Rogers, K. A. Ind. Eng. Chem. Process Des. Dev. 1980, 19, 336. (47) Nitta, T.; Nozawa, M.; Kida, S. J. Chem. Eng. Jpn. 1992, 25, 176.
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Langmuir, Vol. 12, No. 1, 1996 177
Table 1. Values of the Parameters in Eq 56 Obtained When Fitting by Computer Equation 56 to the Experimental Data for Methane and Ethane Adsorption on the Carbon BPL, Reported by Reich et al.46 a adsorbate
p0 [psia]
M [mg mol/gC]
kT/E1
r
methane (212 K) ethane (212 K)
1991 581.4 (1960) 1800 581.1 (1988) 1242 575.3 (1675)
8.39 14.82 (15.08) 6.95 13.32 (13.94) 5.63 11.39 (12.25)
0.196 0.225 (0.2012) 0.270 0.299 (0.259) 0.380 0.392 (0.323)
2.08 2.47 (2.67) 1.82 1.79 (2.05) 1.51 1.72 (1.883)
methane (260 K) ethane (260 K) methane (301 K) ethane (301 K) a
t
zu12/kT
0.99 0.99)
(0.305)
0.98 (0.96)
(0.106)
0.72 (0.92)
(0.044)
The values in parentheses are those obtained by accepting in our computer exercises the values of zu12/kT, listed in the last column. Table 2. Values of the Parameters in Equation 56 Obtained When Fitting by Computer Equation 56 to the Experimental Data for Methane and Ethane Adsorption at 298.15 K, on the Two Activated Carbons M-30 and M-38, Reported by Nitta et al.47 a adsorbate-adsorbent
p0 [kPa]
M [mol/kg]
kT/E1
r
methane-M-30 ethane-M-30
8726 4905 (4809) 9680 7342 (5924)
18.60 37.28 (36.08) 15.43 41.85 (39.31)
0.493 0.464 (0.461) 0.434 0.412 (0.442)
1.29 1.52 (1.595) 1.47 1.78 (1.517)
methane-M-38 ethane-M-38 a
t
zu12/kT
0.69 (0.65)
(0.151)
0.87 (0.99)
(0.179)
The values in the parentheses are those obtained by accepting in our computer exercises the values of zu12/kT, listed in the last column.
Figure 5. Agreement between theory (eq 56) and experiment in the case of methane and ethane adsorption on the carbon BPL at 212 K, studied experimentally by Reich et al.46 The white circles are the experimental values of ln Nt for methane adsorption, plotted as the function [ln(p0/p)]r by using the parameters collected in Table 1. The black circles are the experimental values of ln Nt for ethane adsorption, plotted as ))]r, calculated by using the set of the the function [ln(p0/(pθt(n-1) t best-fit parameters obtained by assuming that zu12/kT ) 0. The solid lines are the theoretical values of ln Nt, predicted by eq 56.
The values of the parameters which we have found by fitting our eq 56 to the experimental isotherms of methane and ethane adsorption reported by Reich et al. and by Nitta et al. are collected in Tables 1 and 2. The computer exercises were carried out twice. First we fixed the value of the parameter zu12/kT as equal to zero, i.e., we neglected the interactions between the adsorbed molecules. The results of these best-fit exercises are shown in Figure 5. While considering methane adsorption, one can see an excellent linear correlation of the experimental data given by eq 56, and the accompanying assumption of the negligible role of the interactions in the adsorbed phase, represented by the assumption that zu12/kT ) 0. On the
contrary, in the case of ethane adsorption, one can see small but systematic positive deviations from the linear plot. A similar situation can be seen in the case of methane and ethane adsorption at the two higher temperatures, 260 and 301 K, so, we do not show these results. Rudzin´ski and Everett have demonstrated2 (Figure 6.2 in their book) that one of the possible sources of such deviations may be an incorrect choice of p0. It may happen quite frequently because of the common practice to replace p0 by the saturated pressure value ps., Later on, we will discuss the physical conditions at which the equality p0 ) ps may be justified. As in our paper p0 was correctly chosen as a best-fit parameter, we ascribed the systematic positive deviations from linearity in the case of ethane adsorption as being due to neglecting effects of the interactions between adsorbed molecules. The maximum adsorbed amount of methane studied experimentally by Reich et al.46 was, at that temperature 8.11 mg mol/gC, whereas in the case of ethane it was 7.39 mg mol/gC. Nevertheless, in the terms of multisite-occupancy adsorption the surface coverage of sites by ethane molecules was almost twice larger than that by methane molecules. As well-known, the interaction effects between adsorbed molecules play more and more important role with the increasing surface coverage. So, in the case of ethane adsorption we repeated our best-fit exercises trying to find the zu12/kT value by making more linear the plot
[
ln Nt vs ln
p0 pθt(n-1) t
-
zu12 1 kT 2
1-t
()
θtt
]
r
The best-fit parameters found in this way are those listed in parentheses in Table 1. Figure 6 shows this plot in the case of ethane adsorbed by BPL at 212 K. One can see a better linear correlation in Figure 6, compared to that observed in Figure 5. The parameter r has now a somewhat higher value than in the case of methane which is in agreement with the fundamentals of multisite-occupancy adsorption on heterogeneous solid surface. (The growing number of sites occupied by one adsorbed molecule results not only in the growing variance but also into the growing symmetry of the adsorption energy distribution.) The parameter t is still close to unity,
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Of course, it is still the same carbon surface. The changes in the parameter values found for different adsorption regimes (temperatures, surface coverages) are due to different values of these parameters characterizing the surface areas which are covered preferentially at different adsorption regimes. Activated carbons are characterized by a certain dispersion of the dimension x of their micropores. Dubinin and co-workers,48,49 and Stoeckli and co-workers50-52 investigating scattering of X-rays by carbocenous adsorbents, found the following relation between the parameter En of benzene and the micropore dimension x,
x ) 13.0/En Figure 6. Linear correlation of the isotherm of adsorption of ethane on carbon BPL at 212 K, offered by eq 56. The black circles are the experimental points for ln Nt plotted against )) - (zu12/kT)(1/2)1-tθtt]r by using the parameters [ln(p0/(pθt(n-1) t within parentheses collected in Table 1. The solid lines are the theoretical values calculated from eq 56 by using the same set of the parameters.
Figure 7. Relationships between the parameters kT/E1, r, and t collected within the parentheses in Table 1.
indicating thus a random topography of the surface sites covered in this range of (experimental) surface coverages. This is still another reason why the function χ2(2) calculated from χ1(1) by eq 13 should be more Gaussianlike than χ1(1); i.e., it should be characterized by a higher value of r than χ1(1). We face a similar situation in the case of ethane adsorption at the two higher temperatures 260 and 301 K, so, we do not show here the corresponding graphical illustrations. In view of these findings we treat the parameters within the parentheses in Tables 1 and 2 as more reliable ones. However, the differences between the parameters within the parentheses and without parentheses mean simply certain corrections. In both cases one can see clear interrelations of the various parameters listed in Tables 1 and 2. Figure 7 shows some of them. According to the main idea of the present publication, in the case of carbonaceous adsorbents, the correctness of the theory can be verified through the relation between the parameters E1 and t. The data presented in Figure 7 prove the self-consistency of our argument. According to our present knowledge of the nature of the surface of activated carbons, the decreasing value of E will indicate a more homogeneous surface due to a growing role of the homogeneous graphite-like domains. At the same time it must mean a growing patchwise character of the covered surface, characterized by the decreasing value of the randomness parameter t. It can easily be seen in Figure 7.
(58)
A similar relation was reported by MacEnaney53,54 on the basis of his SAX studies. The studies of benzene adsorption showed also decreasing tendency of r with a decreasing value of En. That kind of relationship can also be seen in Table 1 and in Figure 7. In other words, larger pores will be characterized by smaller values of r. It has been known for a long time that strongly activated carbons exhibiting large micropores are characterized by low values of r close to unity. All that has been said above results in a logical consistent picture of the adsorption by activated carbons. We will stress it as follows: The different adsorption sites are distributed on the walls of micropores having different dimensions. There are, however, correlations between the micropore dimension x, and the degree of the energy dispersion (variance), represented by eq 58. The smaller the pore, the more “diffuse” the spectrum of adsorption energies, and a more symmetrical right hand widened distribution of adsorption energies, characterized by the growing value of the parameter r. Provided that the smallest pores are formed by less ordered zones of carbon atoms, the spatial distribution of the adsorption in the small micropores will have a random character, characterized by t-values close to unity. Such correlations between the adsorption parameters E1, r, and t are observed in Table 1 (Figure 6). It should, however, be realized that these parameters represent certain average values for the experimentally investigated range of surface coverages. The gradual filling of adsorption sites in the sequence of decreasing adsorption energies assumed by neglecting the corr term in eq 32 does not mean the gradual filling of pores in the sequence of increasing pore dimensions x. Provided thus, that the smallest pores are filled first, pores having larger dimensions will already be partially occupied before the smaller pores are completely covered. This explains an interesting phenomenon observed in Table 1. While looking at the tables of the experimental data for methane and ethane adsorption, reported by Reich et al.,46 we can see that the investigated (and analyzed by us) range of surface coverages decreases with the increasing temperature from 8.11 mg mol/g C of methane and 7.39 mg mol/g C of ethane at 212.7 K to 4.57 and 5.5 mg mol/g C at temperatures 301.4 K. Provided, thus, that the filling of pores had proceeded gradually in the sequence of (48) Dubinin, M. M. Characterisation of Porous Solids (held at Neuchatel in 1987); Gregg, S. J., Sing, K. S. W., Stoeckli, H. F., Eds.; Society of Chemical Industry: London, 1979. (49) Dubinin, M. M.; Plavnik, G. M.; Zavarina, E. D. Carbon 1964, 2, 261. (50) Stoeckli, H. F. Chimia 1974, 28, 728. (51) Stoeckli, H. F.; Morel, D. Chimia 1980, 34, 502. (52) Stoeckli, H. F. Carbon 1990, 28, 1. (53) McEnaney, B. M. Carbon 1987, 25, 69. (54) McEnaney, B.; Mays, T. J. in COPS II Conference, Alicante, May 1990.
Solid Surface Topography
Langmuir, Vol. 12, No. 1, 1996 179
Figure 8. θ1t and θ2t functions defined in eqs 59 and 60, plotted as functions of the dimensionless temperature (kT/eu).
increasing pore dimensions x, the numerically estimated (average) values of the parameters E1, r, t should increase with the increasing temperature at which the experiment was carried out. Meanwhile, one can see an inverse tendency. This phenomenon is due to a redistribution of the adsorbed molecules from smaller to larger pores, caused by the increasing adsorption temperature. To understand it better, let us imagine the following experiment: We keep the equilibrium bulk pressure constant, then change the temperature of adsorption system, and see, how the coverage of micropores having different dimension x changes. For the sake of simplicity, we consider onesite-occupancy adsorption and two kinds of micropores; a narrow micropore characterized by r ) 2 and E1 ) 1 eu (energy unit), and a wide micropore characterized by r ) 1, and E1 ) 1 eu. We establish still the pressure value in such a way that ln p0/p ) 1 and assume that K′ and 0 are the same for both pores. Then, the temperature dependence of the surface coverage of the small micropores θ1t will be given by
{ [ ]}
θ1t ) exp -
kT 1 eu
2
(59)
whereas for the large micropores
{ [ ]}
kT θ2t ) exp 1 eu
(60)
Figure 8 shows θ1t and θ2t as the function of temperature T. We can see there a preferential adsorption in small pores (r ) 2) at low temperatures and a reverse tendency at higher temperatures. So far, we have not discussed the M values found by computer, listed in Tables 1 and 2. These are some apparent, temperature dependent values, which may acquire their physical meaning only in the systems with a narrow distribution of micropore dimensions. As for the parameter p0 it cannot, in general, be identified with the saturation pressure of the bulk adsorbate. An exception would be activated carbons with a small amount of mesopores, in which the adsorption mechanism is believed to be somewhat different. (Possibility of polymolecular adsorption etc.) Then from the condition,
lim θt ) 1
(61)
pfp0
one arrives at the equality p0 ) ps. Nevertheless, such an assumption was commonly made in the papers on the
application of DR and DA equations to adsorption by activated carbons. That assumption has some historical roots. Namely, the early stage of the application of DR equation was influenced strongly by Polanyi’s view on adsorption as a condensation of the adsorbate on the surface to a liquid-like state. That state was assumed to be the bulk liquid one, so the energy of adsorption was put equal to kT ln(ps/p). The adsorption in pores was viewed as a gradual condensation in micropores in the sequence of decreasing adsorption energies. The value of that adsorption energy was related to the micropore dimension x. And this was the idea lying behind the early applications of the DR equation to determine the pore size distribution. Substantial progress toward the application of DR equation to determine the pore size distribution is due to the experimental and theoretical works by Dubinin,55 Stoeckli,56 Jaroniec, and co-workers.57-60 The adsorption in micropores is no longer considered as a condensation process, but simply an adsorption process which can be described by the DR equation, the parameter En being related to micropore dimension x, through the relation 58. The experimentally measured adsorbed volume Vt was assumed to be represented by the following average
[
p kT ln s ∫0∞ exp{- βE p 0
Vt ) V0
]} 2
f(E0) dE0
(62)
where E0 is the En value for benzene, V0 is the total volume of all micropores, and β is the “affinity” parameter when other than benzene adsorbate is used in the adsorption experiment. The additional conditions are
f(E0) g 0
∫0∞f(E0) dE0 ) 1
and
(63)
A particularly simple form of this equation was proposed by Jaroniec,57-61 who applied γ-type distribution to represent f(E0). Equation 62 takes then the following form,
Vt ) V0
[ ( )] q
q+
(m+1)
A β
2
(64)
where q and m are the parameters of the γ distribution showing a large degree of flexibility for different values of the parameters, m and q. Of course, the assumption that p0 ) ps, and that r ) 2 for all micropores must affect to some extent the calculated pore size distribution. Perhaps it might be more reasonable to identify p0 with the pressure value corresponding to the beginning of the hysteresis loop. This is the pressure at which micropores are believed to be filled, and the adsorption starts in mesopores. (So the condition (61) is fulfilled.) Now let us consider yet the intriguing problem, why the p0 parameter, found by computer while fitting experimental data measured in different temperatures, decreases with temperature. At present, our explanation is as follows: The increasing temperature causes redistribution of the molecules from small to larger pores. This is the experi(55) Dubinin, M. M.; Stoeckli, H. F. J. Colloid Interface Sci. 1980, 75, 34. (56) Stoeckli, H. F. J. Colloid Interface Sci. 1977, 59, 184. (57) Jaroniec, M.; Piotrowska, J. Monatsh. Chem. 1986, 7, 117. (58) Jaroniec, M.; Madey, R.; Lu, X.; Choma, J. Langmuir 1988, 4, 911. (59) Jaroniec, M.; Madey, R. J. Chem. Soc., Faraday Trans. 2 1988, 84, 1139. (60) Jaroniec, M.; Madey, R. J. Phys. Chem. 1989, 93, 5225. (61) Lu, X.; Jaroniec, M.; Madey, R. Langmuir 1991, 7, 173.
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Table 3. Comparison of the Experimentally Determined Similarity Coefficient β ) En/E0 Where E0 Is the Value of the Parameter E1 Defined in Equation 88, Estimated for n-Butane, and the β-Values Calculated from Parachor Ratio, and from the Molar Volume Ratioa activated carbon PA
alkane
n-butane propane ethane BPL n-butane propane ethane Columbia n-butane propane ethane
temp saturation pressurea characteristic exptl similarity similarity coefficient similarity coefficient (T), K (p0), (mmHg) energy (E0), kJ/mol coefficient β from parachor ratio from molar volume ratio 273 273 273 273 273 273 298 298 298
755 3205 13640 755 3205 13640 1713 6045 21170
23.3 18.3 17.1
1 0.73 0.55 1 0.81 0.67 1 0.85 0.71
1 0.8 0.6 1 0.8 0.6 1 0.8 0.6
1 0.9 0.7 1 0.9 0.7 1 0.9 0.7
a The presented data were taken from Tables III and IV in the paper by Lu et al.61 The Parachor ratio and volume ratio were calculated by putting equal unity their values for n-butane.
mental fact determined by best-fit exercises, showing that the parameter r decreases with temperature. The adsorption potential in larger pores is (on average) lower than in smaller pores. In terms of Polanyi’s theory, that adsorption potential is given by the term kT ln(ps/p). The adsorption potential in larger pores is (on average) lower than in smaller pores. In terms of Polanyi’s theory, that adsorption potential is given by the term kT ln(ps/p). The concept of p0 is a generalization of the meaning of ps parameter for the case when the state of the adsorbed molecules is not identical with that of the saturated vapor pressure. Thus, in spite of growing T, smaller adsorption potential can be due only to smaller ps (or p0) values. Then Lu et al.61 proposed that one should use n-butane rather than benzene as the reference substance.60 While applying their eq 64 to three activated carbons, PA, BPL, and Columbia, they arrived at the conclusion that these carbons possess relatively small structural heterogeneity, in accordance with the earlier findings reported by Lee et al.62 Jaroniec and co-workers measured next the adsorption of ethane, propane, and butane on these three activated carbons, and applied eq 43 to correlate the experimental adsorption isotherms, by assuming that p0 ) ps, and r ) 2. The parameters which they obtained in this way are collected in Table 3. The parameter β means simply En/E4, where n is the number of the CHx mers in the alkane molecule. As eq 43 is valid for patchwise topography, β should be equal to n/4. Such a dependence of β on n can be seen in Table 3 only for the case of the PA activated carbon. In the case of BPL and Columbia carbons, β changes with n as the function n1/2/41/2 rather, indicating, thus, random surface topography. This is shown in Figure 9. The results discussed above seem to suggest the following conclusion: The surface topography has so strong an effect on multisite-occupancy adsorption, that even if one takes a certain version of DA equation not exactly corresponding to a particular case of surface topography, the estimated parameters En will “feel” that surface topography. This is a very encouraging observation concerning the possibility of distinguishing between various surface topographies. Although the form of the DA equation applied by Jaroniec and co-workers to study the adsorption of n-alkanes by the BPL carbon is somewhat different from the DA equation used in our present study, both these works suggest a random-like topography rather than the surface adsorption sites of that activated carbon. The randomness topography factors t estimated by us for BPL, and collected in Table 1, are closer to unity rather (62) Lee, T. V.; Huang, J. C.; Rothstein, D.; Madey, R. Carbon 1984, 22, 493.
Figure 9. Dependence of the β values on the number n of the CHx mers in adsorbate molecule for the ethane (O), propane (4), and butane (0) adsorption on the carbon BPL at 273 K and for the adsorption of ethane (b), propane (2), and butane (9) on the Columbia carbon at 298 K. The values of β are those reported in Table 3 in the paper by Lu et al.61 and collected here in Table 3.
than to zero. Also the values of the parameter r estimated by us are not far from 2. However, there are also certain differences too. For instance the value of the parameter p0 estimated by us is much lower than the saturation pressure ps. Then, eq 56 has been developed with the purpose to estimate more or less quantitatively the degree of the spatial correlations between surface sites having similar adsorption properties. So, no surprise, that our computer analysis of the adsorption isotherms of the n-alkanes reveals more physical details. One can deduce from Table 1 that in spite of the small geometric heterogeneity of the pores of BPL, the wider pores may have topography much different from the fully random topography of the small pores. This is because at lower temperatures, when the small pores are preferentially covered, the estimated parameter t is close to unity. At higher temperatures, in which the redistribution of adsorbate goes toward increasing adsorption in wider pores, the estimated values of the randomness parameter t fall to 0.7. While using n-alkanes to estimate the surface topography of activated carbons, one should not ignore the role of the interactions between the adsorbed molecules. The “classical” DR and DA equations ignore that role, so much of the data reported so far in literature should be treated with caution. Stoeckli and co-workers63 first generalized the DR classical isotherm by taking into account interactions between adsorbed molecules. (63) Morel, D.; Stoeckli, H. F.; Rudzin´ski, W. Surf. Sci. 1982, 114, 85.
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Langmuir, Vol. 12, No. 1, 1996 181
Our computer exercises show that even at the present standard accuracy of adsorption measurements, application of eq 56 should make it possible to distinguish between various surface topographies. This paper, however, still does not offer a procedure for determining the pore size distribution. However, the results presented in this paper show that the determination of the pore size distribution of activated carbons will involve the necessity of considering the problem of surface topography. It will also concern very promising methods applying the density functional theory and computer simulations of adsorption in pores to determine the pore size distribution.28,64-68 So far, most of the theoretical results has been obtained for idealized micropores having regular geometry and energetically homogeneous walls. The slitlike pores were most frequently considered. The calculated (simulated) isotherms predict condensation in (micro)pores to occur at pressures much lower than the bulk saturation pressure. The above mentioned theoretical isotherms should be applicable to analyze adsorption by strongly activated carbons. During the activation the lowest-density amorphous carbon burns out first, and small micropores of irregular structure appear. Further burning out causes less perfect small-size crystallites to burn out, and as a result big slitlike micropores are formed. The overall adsorption isotherm Nt(p) is then usually well correlated by the Freundlich equation.69 Thus, Ehrburger-Dolle have applied the theoretical results by Lastoskie et al. to study the porous structure of such carbons.70 They wrote the derivative (dVt/dL), where L ) 2x, as follows
dVt ∝ L(kT/E)γ-1 dL
( )( )
dVt dVt dp ) dL dp dL
Vt ) V0
() p ps
kT/E
γ = 12
D)3-
kT γ E
(66)
(67)
Provided, thus, that the local adsorption in micropore is represented by a step function having the step at the pressure, at which Lastoskie’s isotherm is discontinuous, we obtain (64) Seaton, N. A.; Walton, J. P. R. B. Carbon 1989, 27, 6. (65) Lastoskie, C.; Gubbins, K. E.; Quirke, N. J. Phys. Chem. 1993, 97, 4786. (66) Balbuena, P. B.; Gubbins, K. E. Langmuir 1993, 9, 1801. (67) Lastoskie, C.; Gubbins, K. E.; Quirke, N. Langmuir 1993, 9, 2693. (68) Kaminsky, R. D.; Maglara, E.; Conner, W. C. Langmuir 1994, 10, 1556. (69) Carrasco-Mavin, F.; Lopez-Ramon, M. V.; Moreno-Castilla, C. Langmuir 1993, 9, 2758. (70) Ehrburger-Dolle, F. Langmuir 1994, 10, 2052.
(69)
In the case of CH2Cl2 adsorption on the carbon AGB where the overall adsorption isotherm was perfectly correlated by the Freundlich isotherm, the estimated value of (kT/E) was 0.43. Equation 69 yields then a nonphysical value D ) -2.26. To a certain extent this nonphysical result must be due to the fact that the actual pore filling is not perfectly gradual as has been assumed in the above consideration. Another reason is that the “local” adsorption in micropores may not be exactly represented by Lastoskie’s computed isotherms, due to a certain energetic heterogeneity of the pore walls. This was assumed implicitly by EhrburgerDolle saying that “it is realistic to describe the local isotherm corresponding to the filling of pores of size Lc by a power law pkT/E (E′ * E)”. In the other words they assume the local isotherm to be also a Freundlich equation, always associated with adsorption on heterogeneous surfaces. With such an assumption Ehrburger-Dolle arrived at some reasonable estimation of the fractal dimension D of activated carbons
D)3-
8 E0
(70)
the overall adsorption on which is correlated by the Freundlich equation. The above relation is formally similar to the one
D ) 6.44 -
The micropore filling was assumed at the beginning as perfectly gradual, and the derivative (dp/dL) was calculated from the relation between micropore dimension L and the condensation pressure in pores having the dimension reported by Lastoskie et al.65 Ehrburger-Dolle approximated it by the following power law,
p ∝ Lγ,
but in terms of fractal dimension D71
(65)
and the derivative (dVt/dp) was calculated from the Freundlich overall isotherm,
(68)
74 E
(71)
obtained by Jaroniec et al.72 for the case of activated carbons, where overall adsorption is correlated by the Dubinin-Radushkevich equation. One may, therefore, follow the argumentation by Ehrburger-Dolle and say that it seems realistic to describe the local adsorption in micropores by DR equation. As the narrow pores formed by amorphous carbon must have even more energetically heterogeneous walls, it is obvious that computer simulations of adsorption in such pores must take into account the energetic surface heterogeneity. First computations of that kind have already been done by Kaminsky,28 Steele and co-workers,29 and in the Faculty of Chemistry of UMCS in Lublin.73 The combined geometric-energetic heterogeneity of pore walls will create various local minima in the gassurface potential function, which may well be distributed three-dimensionally. These local minima will, obviously, be separated by various energy barriers hindering free translation. This must justify using the model of localized adsorption as a convenient theoretical tool. It should, however, be clearly stated that adsorption in micropores may not be fully localized. In such a case, the finite dimensions of pores will create effects which are similar to those created by surface energetic heterogeneity. This phenomenon has long been known in adsorption theories.2 Thus, the parameter E estimated from the experimental (71) Pfeifer, P.; Avnir, D. J. Chem. Phys. 1983, 79, 3558. (72) Jaroniec, M.; Gilpin, R. K.; Choma, J. Carbon 1993, 31, 325. (73) Sokolowski, S.; Partykiejew, A. Private information.
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isotherms may sometimes overestimate the energetic heterogeneity of pore walls. Computer simulations also emphasize the necessity of taking into account attractive interactions between the molecules adsorbed in pores. Hence our idea to ascribe the positive deviations from the linear DA plots at higher pressures (surface coverages) to interactions between the adsorbed molecules. The relations between the parameters E0, r, and t deduced in this paper are obviously approximate, but together with the previously discovered relations between E0 and the micropore dimension x (of L), they may appear
Rudzin´ ski et al.
useful in developing new, more precise methods for determining the pore size distribution of activated carbons from the experimental adsorption isotherms. Acknowledgment. The present study is a part of the research project carried out thanks to the financial support from the State Committee for Scientific Research, Project No. 2 P303 06305. W. Rudzin´ski wishes to express his thanks to INPL for the grant which made his extended visit to the University of Nancy possible. LA9408275
