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Langmuir 1997, 13, 1104-1117
An Improved Derivative Isotherm Summation Method To Study Surface Heterogeneity of Clay Minerals† F. Villie´ras,*,‡ L. J. Michot,‡ F. Bardot,‡ J. M. Cases,‡ M. Franc¸ ois,‡ and W. Rudzin´ski§ Laboratoire Environnement et Mine´ ralurgie, ENSG and URA 235 du CNRS, BP 40, 54 501 Vandoeuvre les Nancy cedex, France, and Department of Theoretical Chemistry, Maria Curie-Sklodowska University, Pl. Marii Curie-Sklodowskiej 3, Lublin 20-031, Poland Received November 6, 1995. In Final Form: April 8, 1996X Due to their crystallochemical properties, clay minerals feature different types of structural surfaces which have their own adsorption energy distribution. To study that type of surface heterogeneity, the DIS (derivative isotherm summation) method has been developed by us. Now, a modified version of the DIS method is derived by using the Jagiełło-Rudzin´ski approach and assuming that the local energy distributions are represented by the Dubinin-Asthakov distributions. Two different types of isotherms equations are used, one to describe adsorption in micropores and another one for describing adsorption on external surfaces. The derivatives of experimental adsorption isotherms with regard to ln(p/ps) are simulated by combinations of the derivatives of corresponding local adsorption isotherms. This best fit provides information on adsorption capacity of the local existing domains, on the symmetry of their energy distribution function, and on the parameters characterizing the lateral interactions in each adsorption domain. Using the new equations for the local isotherm derivatives allows now to simulate very accurately derivatives of experimental adsorption isotherms, obtained by using our high-resolution quasi-equilibrium volumetric technique. This was proven in the case for three different well characterized clay minerals: a structural microporous one (palygorskite) and two nonporous lamellar ones (kaolinites). The obtained parameters allow a description of the adsorption energy distribution of their different surfaces, their textural parameters, as well as the energy distrubution in micropores for the microporous samples. In addition to the experimental adsorption isotherms, the related experimental heats of adsorption were employed as a second independent source of information about the energetic heterogeneity of the studied clay minerals. Using the parameters determined from adsorption isotherms, the corresponding isosteric heats of adsorption were calculated and compared with experimental values. The simultaneous good fit of the experimental isotherm derivatives and of the experimental heats of adsorption was a solid check for the correctness of the determined parameters characterizing the adsorption energy distributions and the lateral interactions between the adsorbed molecules.
Introduction The fact that the really existing surfaces are energetically heterogeneous is now generally recognized by the scientists investigating adsorption at the solid/gas1,2 and solid/solution1-4 interface. This surface heterogeneity is a fundamental feature of the real solid surfaces. It can be due to different crystal faces, local crystalline disorder, surface roughness, or the presence of impurities. It may also have its source in the presence of pores of various sizes and shapes. A large variety of experimental methods has been developed for studying more or less directly the structure of solid surfaces. In the case of clay minerals, optic and electronic microscopy provide proof of their macroscopic heterogeneity. On the local scale, atomic force microscopy provides very impressive images showing evidence of surface heterogeneity. However, all these methods yield only auxiliary information. * To whom correspondence should be addressed: fax, (33) 83 57 54 04; e-mail,
[email protected]. † Presented at the Second International Symposium on the Effects of Surface Heterogeneity in Adsorption and Catalysis on Solids held in Poland/Slovakia, September 4-10, 1995. ‡ Laboratoire Environnement et Mine ´ ralurgie. § Maria Curie-Sklodowska University. X Abstract published in Advance ACS Abstracts, September 15, 1996. (1) Jaroniec, M.; Madey, R. Physical Adsorption on Heterogeneous Solids; Elsevier: Amsterdam, 1988. (2) Rudzin´ski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: London, 1991. (3) Cases, J. M.; Mutaftschiev, B. Surf. Sci. 1968, 9, 57. (4) Cases, J. M. Bull. Mine´ ral. 1979, 102, 684. (5) Cases, J. M.; Cunin, P.; Grillet, Y.; Poinsignon, C.; Yvon, J. Clay Miner. 1986, 21, 55.
S0743-7463(95)01008-0 CCC: $14.00
The most essential question concerns the nature of surface heterogeneity (gas-solid interactions) with respect to a given adsorbate molecule. Indeed, surface heterogeneity cannot be considered as an absolute feature as it is related to a given adsorption system, i.e., a given adsorbent/adsorbate system. It has been observed that some surfaces can appear to be nearly homogeneous for a certain adsorbate and heterogeneous when another molecule is adsorbed. This, for instance, is the case of the 5A molecular sieve which is homogeneous with respect to O2 molecules and heterogeneous for the adsorption of N2 molecules.5 In the case of kaolinite, the N2 and Ar adsorption enthalpy are very different6 and nitrogen cannot be used as a probe of morphological properties of this mineral. Therefore, the nature of surface heterogeneity must always be related to the type of adsorbed molecules, i.e., to the subsequent normal adsorbateadsorbent interactions. In this respect, essential information about surface heterogeneity can be obtained by applying methods commonly referred to as “molecular probe methods”. A molecular probe method is simply a theoretical/ numerical analysis of a certain adsorption quantity (characteristics) measured in the course of adsorption of a chosen adsorbate. In particular, valuable information about surface heterogeneity can be obtained from calorimetric experiments such as low-temperature gas adsorption microcalorimetry (LTAM), where the measured quantity is the differential molar enthalpy of adsorption |∆adsh|.7 The plot of |∆adsh| versus surface coverage θ is (6) Miller, G. W.; Knaebel, K. S.; Ikels, K. G. AIChE J. 1987, 33, 194. (7) Rouquerol, J. Calorime´ trie d’adsorption aux basses tempe´ ratures; CNRS Publ.: Paris, 1972.
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related to the distribution of adsorption energy. This method was successfully applied in the qualitative and quantitative description of heterogeneous adsorbents.5,8-10 Rudzin´ski and his collaborators11-15 have demonstrated that a simultaneous analysis of an experimental adsorption isotherm and of the related heat of adsorption curve (vs surface coverage) may lead to a new level of understanding of the important role of surface heterogeneity in various adsorption systems. Three years ago, we developed in our laboratory a new method to study the surface energetic heterogeneity called the derivative isotherm summation (DIS) method which follows that investigation strategy.16 The present paper proposes a new edition of the DIS method, aimed to eliminate certain shortcomings of the previous method. Before presenting the basic ideas of that new method, we will briefly comment on the position of our DIS method among other commonly used methods for studying surface heterogeneity. Usually, adsorption experiments are performed at such a temperature that adsorption on a heterogeneous surface can be treated in terms of localized adsorption. This was shown for instance in Bakaev’s computer simulation of real oxide surfaces.17 In the case of patchwise topography, the common periodicity of the gas-surface potential on homogeneous patches or domains will create tendencies to localized adsorption. Furthermore, the geometrical disorder which is the most common source of the energetic heterogeneity also causes energy barriers to appear for two-dimensional translation across the surface. Thus, the surface energetic heterogeneity is usually considered as a dispersion of the number of the local gas-solid potential minima, called the “adsorption sites”, among the value of the gas-solid potential at these minima taken with a reverse sign and called the “adsorption energy”. The majority of the methods designed to study surface heterogeneity is based on the assumption that the spectrum of adsorption energies in a given gas/solid system is suffiently dense to be represented by a continuous function. This function is the differential distribution of the number of adsorption sites among the corresponding values of adsorption energy , usually applied in its form normalized to unity
∫Ωχ() d ) 1
(1)
where Ω is the physical domain of . The experimental monitored average “total” fractional coverage of all the adsorption sites at a temperature T and pressure p, θt(p,T), is then represented by
θt(p,T) )
∫Ωθ(,p,T) χ() d
(2)
where θ(,p,T) is the “local” isotherm, i.e., the fractional coverage of adsorption sites having adsorption energy (8) Rudzin´ski, W.; Jagiełło, J.; Grillet, Y. J. Colloid Interface Sci. 1982, 87, 478. (9) Grillet, Y.; Cases, J. M.; Franc¸ ois, M.; Rouquerol, J.; Poirier, J. E. Clays Clay Miner. 1988, 36, 233. (10) Cases, J. M.; Grillet, Y.; Franc¸ ois, M.; Michot, L.; Villie´ras, F.; Yvon, J. Clays Clay Miner. 1991, 39, 233. (11) Rudzin´ski, W.; Baszynska, J. Z. Phys. Chem. 1981, 262, 33. (12) Rudzin´ski, W.; Partyka, S. J. Colloid Interface Sci. 1982, 89, 25. (13) Rudzin´ski, W.; Michalek, J.; Partyka, S. J. Chem. Soc., Faraday Trans. 1 1982, 78, 2361. (14) Rudzin´ski, W.; Charmas, R.; Partyka, S. Langmuir 1991, 7, 354; Colloids Surf. 1993, 70, 111. (15) Lajtar, L.; Narkiewicz-Michalek, J.; Rudzin´ski, W.; Partyka, S. Langmuir 1993, 9, 3174. (16) Villie´ras, F.; Cases, J. M.; Franc¸ ois, M.; Michot, L. J.; Thomas, F. Langmuir 1992, 8, 1789. (17) Bakaev, V. A. Surf. Sci. 1988, 198, 571.
equal to . The “total” fractional coverage θt(p,T) is defined as equal to Nt(p,T)/M, where Nt(p,T) is the experimentally measured adsorbed amount and M is the total number of the adsorption sites on the investigated solid surface, expressed in the same units as Nt(p,T). The aim of the numerous papers dealing with energetic surface heterogeneity can be stressed as follows: provided that the analytical form of θ(,p,T) governing the local adsorption on different adsorption sites is known and that Nt(p,T) is measured experimentally, how can χ() be calculated. From a purely mathematical point of view, the problem formulated in this way means simply solving a Fredholm integral equation of the first kind.1,2 Mathematical monographs provide various proposals for solving such an equation, but only few of those proposals have been used to calculate χ(). This is due to the fact that Nt(p,T) in the physical problem is given by a discrete set of experimental values which inhere also experimental errors. This fact creates a new challenge for the solution which was dealt with in various ways by various authors. So, various theoretical papers have offered specific solutions when the kernel θ(,p,T) takes the analytical form of a certain adsorption isotherm equation. The literature concerning this problem is very large, but some relevant reviews and monographs which have already been published can be helpful for the readers wishing to enter this area.1,2,18-22 A convenient classification of these methods has been proposed by Re,23 who distinguishes the following three groups of methods: 1. “Analytical methods” first introduced by Sips24,25 in which both the “local” and the “total” adsorption isotherms are given by analytical expressions. 2. “Numerical methods”, which range from simple optimization of a parametrized form of χ() to more accurate methods which explicitly take into account the ill-posed character of the Fredhom integral equation. That difficulty is manifested by the fact that even small changes in the data of Nt(p,T) cause large changes in the calculated function of χ().26-30 Some new ideas concerning the solution of that problem have been recently proposed by Jagiełło.31 3. “Approximate methods”, based on accepting a certain approximation for the local isotherm θ(,p,T).8,32-37 According to Cerofolini and Re,22 the solution methods can also be classified as “global” and “local”. The global (18) House, W. A. Colloid Science; Everett, D. H., Ed.; Specialist Periodical Reports; Royal Society of Chemistry: London, 1983; Vol. IV, p 1. (19) Jaroniec, M. Adv. Colloid Interface Sci. 1983, 18, 149. (20) Jaroniec, M.; Bra¨uer, P. Surf. Sci. Rep. 1986, 6, 65. (21) Sircar, S.; Myers, A. L. Surf. Sci. 1988, 205, 353. (22) Cerofolini, G. F.; Re, N. Riv. Nuovo Cimento 1993, 16 (7), 1. (23) Re, N. J. Colloid Interface Sci. 1994, 166, 191. (24) Sips, R. J. Chem Phys. 1948, 16, 490. (25) Sips, R. J. Chem Phys. 1950, 18, 1024. (26) Vos, C. H. V.; Koopal, L. K. J. Colloid Interface Sci. 1985, 105, 183. (27) Ros, S.; Morrison, I. D. Surf. Sci. 1975, 52, 103. (28) Sacher, R. S.; Morrison, I. D. J. Colloid Interface Sci. 1979, 70, 153. (29) House, W. A. J. Colloid Interface Sci. 1978, 52, 256. (30) Von Szombately, M.; Brauer, P.; Jaroniec, M. J. Comput. Chem. 1992, 13, 17. (31) Jagiełło, J. Langmuir 1994, 10, 2778. (32) Harris, L. B. Surf. Sci. 1968, 10, 128. (33) Cerofolini, G. F. Thin Solid Films 1974, 23, 129. (34) Jaroniec, M.; Rudzin´ski, W.; Sokolowski, S.; Smarzewski, R. Colloid Polym. Sci. 1975, 253, 164. (35) Hsu, C. C.; Wojciechowski, B. W.; Rudzin´ski, W.; Narkiewicz, J. J. Colloid Interface Sci. 1978, 67, 292. (36) Nederlof, M. M.; Van Riemsdijk, W. H.; Koopal, K. J. J. Colloid Interface Sci. 1990, 135, 410. (37) Jagiełło, J.; Ligner, G.; Papirer, E. J. Colloid Interface Sci. 1990, 137, 128.
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methods give the distribution function in the whole range of Ω but require knowledge of the global isotherm in the whole physical range of pressure values p ∈ (0,+∞). The “local” methods give the distribution function only in a limited range of adsorption energies but require knowledge of the global (total) isotherm only in a limited pressure domain. Usually, analytical and numerical solutions are of global character whereas approximated methods are of local character. The new method presented here is a hybrid between analytical and approximate methods. The latter methods have frequently been used and discussed for a number of reasons. One of their advantages lies in the stability of the obtained analytical solutions. The other one is the simplicity of the expressions for the distribution function. Finally the computation of χ() in a certain domain of requires only the knowledge of Nt(p,T) in a certain domain of p. Of all the approximate methods, the condensation approximation (CA)32 is probably the most popular and commonly used. This was also the first method ever used for calculating χ(). It appeared first in the papers by Roginski at the end of the 1930s.2 That method has its origin in some fundamental features of the functions used as the kernel θ(,p,T) in the integral equation (2). Namely, one essential feature of the kernel θ(,p,T), regardless of the isotherm equation used (Langmuir, Bragg-WilliamsTemkin, Hill-de Boer, etc.), is that in the hypothetical limit T f 0, it always degenerates into the step function θc(,p,T)
|
0 for < c(p,T) θc(,p,T) ) 1 for g c(p,T)
∫∞(p,T)χ() d c
(4)
The position of the “condensation” step c is chosen in such a way that θc(,p,T) could approximate best the true kernel θ(,p,T). Appropriate theoretical considerations showed that the step should be located at the value of at which the function θ() exhibits an inflection point. Therefore, c(p,T) can be determined by using the following relation:
( ) ∂2θ ∂2
)0
(5)
)c
Let θ(,p,T) be the Langmuir isotherm
( kT) θ(,p,T) ) 1 + Kp exp( ) kT Kp exp
(6)
where k is the Boltzmann constant and K a slightly temperature dependent constant. Condition 5 leads then to the following expression for the function c(p,T)
c ) -kT ln(Kp)
dθt ) -χc(c) dc
(7)
Other isotherm equations will lead to somewhat different
(8)
At constant temperature
dc ) -kT d ln(p)
(9)
so
χc(c) )
1 ∂θt(ln(p)) kT ∂ ln(p)
(10)
In the hypothetical limit T f 0, χc(c) f χ(). At finite temperatures, χc(c) will deviate from the true function χ(). Nevertheless, χc(c) must resemble the true adsorption energy distribution. By using the Rudzin´ski-Jagiełło method,31,37 one can calculate the exact function χ() from the function χc(c). In the method developed in our laboratory, the calculated function χc(c) is used merely as crude information about the nature of surface heterogeneity. More or less distinct peaks on χc(c) are ascribed to certain groups of similar adsorption sites. The experimentally measured adsorbed amount Nt(p,T) is then represented by n
Nt(p,T) )
(3)
This function is frequently referred to as the “condensation” isotherm because in the case when (attractive) lateral interactions exist, the step may be related to a twodimensional condensation. At finite but not too high temperatures, the local isotherm will still be stepwiselike. Therefore, it is possible within a certain degree of accuracy to replace θ(,p,T) by θc(,p,T) in eq 2. This leads to
θt(p,T) )
expressions for c(p,T). However, the leading term will always be -kT ln(p). Equation 4 leads to
Miθi(i,p,T) ∑ i)1
(11)
( )
(12)
Accordingly
∂Nt
n
) ∂ ln(p)
∂θi
Mi ∑ ∂ ln(p) i)1
In our DIS method, the quantities Mi are not obtained by fitting Nt(p,T) in eq 11 but by fitting ∂Nt/∂ ln(p) in eq 12. In order to determine the number of similar adsorption sites on the investigated solid surface, the experimental curve ∂Nt/∂ ln(p) is first inspected in order to determine the number n. The computer exercises consist then of choosing the best sets {M, K, } to fit the experimental curve ∂Nt/∂ ln(p) by a linear combination of the derivatives ∂θi/∂ ln(p). At first sight, this procedure may appear as crude compared to a variety of sometimes sophisticated methods proposed in the literature. However, the complexity of many of those methods arises from the need to consider the effect of experimental errors on the calculated function χ(). In our case, this problem was solved by building an experimental setup providing about 2000 experimental isotherm data points measured with high accuracy. The experimental noise was further eliminated through an appropriate electronic low-pass filter. Furthermore, almost all the sophisticated methods proposed so far have one questionable point. Namely, the adsorption mechanism is always assumed to be the same in all microscopic surface domains. This means that θ(,p,T) in eq 2 is the same for all adsorption sites. When the heterogeneous surface is considered as a set of homogeneous patches, the various patches are simply considered as being characterized only by different values of . This assumption can be severely questioned as it is physically reasonable to assume that different patches can be characterized, for instance, by different lateral interaction parameters. Next, in the case of microporous adsorbents, the adsorption in micropores and the mul-
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Langmuir, Vol. 13, No. 5, 1997 1107
tilayer adsorption on the external surfaces of mesopores must be described by different isotherm equations. The above questioned simplifications, together with the experimental errors typical for standard adsorption measurements, may sometimes result in serious errors in the calculated function which could hardly be suspected by looking at the sophisticated mathematical-numerical treatment. Furthermore, the calculated functions were rarely used to calculate other adsorption characteristics for comparing them with other independent adsorption experimentssthe heat of adsorption, for instance. In our case, the related isosteric heat of adsorption, Qst, is also calculated and compared with values independently measured by suitable calorimetric equipment. The theoretical heat of adsorption is calculated as follows:38
st
( ) ( )
(13)
)
(14)
∂θi
n
Mi qist ∑ ∂ ln(p) i)1
Qt (θt) )
∂θi
n
Mi ∑ ∂ ln(p) i)1 where
(
qst ) -k
∂ ln(p) ∂(1/T)
T
T
θ
It is then compared with the heat of adsorption, |∆adsh|, measured in an independent calorimetric experiment. We finish our considerations in the introductory section by demonstrating the advantages of fitting the ∂Nt/∂ ln(p) experimental curve instead of the commonly used procedure to fit Nt values. Figure 1 reveals these advantages in the case of the adsorption of nitrogen at 77 K on a palygorskite outgassed at 100 °C under a residual pressure of 10-4 Pa. The ∂Nt/∂ ln(p) curve (Figure 1c) exhibits four more or less overlapping peaks which must correspond to four various groups (domains) of similar adsorption sites on the surface. This conclusion could hardly be drawn from the observation of the experimental Nt curves shown in parts a and b of Figure 1. Of course, such a conclusion is difficult to obtain from standard adsorption experiments39 as the numerical differentiation of the data with respect to ln(p) may result in a heavy noise. The experimental accuracy of the adsorption measurements is then a prerequesite for the application of our method. The above described method was successfully applied to analyze various adorbent/adsorbate systems in terms of surface heterogeneity. The data interpretation was confirmed in many cases by an independent knowledge of the crystallographical structure of the adsorbent and its evolution upon chemical or thermal treatments.16,40 Nevertheless, some difficulties were encountered in fitting properly some experimental ∂Nt/∂ ln(p) curves by using our classical DIS method demonstrating, thus, the need for some further improvement of this method. Theory 1. The Principles and Shortcomings of the Classical DIS Method. Before presenting our new improved DIS method, it seems necessary to repeat for the reader’s convenience some of its principles described in our previous publication. In that method, the different adsorption (38) Ross, S.; Oliver, J. P. On Physical Adsorption; Wiley-Interscience: New York, 1964. (39) Villie´ras, F.; Michot, L. J.; Didier, F.; Ge´rard, G. Submitted to Langmuir. (40) Michot, L. J.; Villie´ras, F.; Franc¸ ois, M.; Yvon, J.; Le Dred, R.; Cases, J. M. Langmuir 1994, 10, 3765.
Figure 1. Nitrogen adsorption at 77 K on palygorskite outgassed at 100 °C under a residual pressure of 10-4 Pa: (a) adsorbed quantity vs p/ps; (b) adsorbed quantity vs ln(p/ps); (c) derivative of the adsorbed quantity with regard to ln(p/ps), plotted vs ln(p/ps).
domains were assumed to be homogeneous. In this case the condensation approximation is not needed and classical adsorption isotherms describing adsorption on a homogeneous surface, such as Langmuir and BET equations, along with their extensions taking into account lateral interactions between admolecules, were used to describe adsorption in these domains.16 The Langmuir isotherm can be written in the following form
θ)
Ceu 1 + Ceu
(15)
where C ) Kps and u ) ln(p/ps)
∂θ Ceu ) θ(1 - θ) ) ∂u (1 + Ceu)2
(16)
u u ∂2θ Ce (1 - Ce ) ) ) θ(1 - θ)(1 - 2θ) ∂u2 (1 + Ceu)3
(17)
According to eq 17 at the inflection point, ∂2θ/∂u2 ) 0, the
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Villie´ ras et al.
following relations hold32,41
by
(18)
1 + eu + B(-eu + 6e2u - e3u) + B2(e3u + e4u) ) 0 (27)
where the asterisk refers to the inflection point. When Nt(p) is a single Langmuir isotherm, the position of the inflection point corresponds to the peak maximum on the curve. The value of the first derivative at maximum is given by:
The derivative of the BET isotherm for C ) 1000 exhibits both a maximum and a minimum. A real solution of eq 27 exists if P*/Ps e 0.0718 or u* e -2.63. The C constant corresponding to the maximum position can be evaluated by calculating the lower solution of eq 27
Ceu* ) 1 and θu)u* ) 1/2
∂θ ∂u
( )
)
u)u*
1 4
(19)
The monolayer capacity M is thus given by the product 4(∂Nt/∂u)u)u*. The possible lateral interaction between adsorbed molecules was taken into account by using the BWT (Bragg-Williams-Temkin) isotherm
θ)
Ceaθ eu ω with a ) aθ u kT 1 + Ce e
(20)
valid at temperatures where the mean-field approximation can still be applied (the local correlations between adsorbed molecules can be neglected). Appropriate theoreticalnumerical investigations show that eq 20 can be applied safely except for temperatures close to the critical temperature. The derivatives of eq 20 can be written as
θ(1 - θ) ∂θ ) ∂u 1 - aθ(1 - θ)
(21)
∂2θ θ(1 - θ)(1 - 2θ) ) ∂u2 [1 - aθ(1 - θ)]3
(22)
From eq 22, one can deduce easily that42
θu)u* )
1 and 2
(∂u∂θ)
) u)u*
1 4-a
(23)
The determination of the position u* of the maximum and a correct adjustment of the parameter “a” to reproduce the shape of the derivative of an experimental isotherm make it possible to determine the constant C and the theoretical height of the derivative. Then, the adjustment of the theoretical and of the experimental heights yields directly the monolayer capacity. When the affinity of the adsorbate for the surface is such that C < 1000, the adsorption in the second layer starts effectively even before the first layer is completed. This influences the shape of the isotherm, and a correction for the multilayer formation has to be applied. The BET theory was then used. The BET equation and its derivatives read:
θ)
Ceu (1 - e )[1 + (C - 1)eu] u
[1 + (C - 1)e2u]Ceu ∂θ ) ∂u (1 - eu)2[1 + (C - 1)eu]2
(24)
(25)
∂2θ ) {Ceu[1 - (C - 2)eu + 6(C - 1)e2u + ∂u2 (C - 1)(C - 2)e2u + (C - 1)2e4u]}/ {(1 - eu)3[1 + (C - 1)eu]3} (26) With B ) C - 1, the relation between C and u* is given (41) Cerofolini, G. F. Surf. Sci. 1971, 24, 391. (42) Cerofolini, G. F. Thin Solid Films 1975, 26, 53.
B)C-1) eu(1 - 6eu + e2u) - [e2u(1 - eu)2(1 - 14eu + e2u)]1/2 2e3u(1 + eu) (28) Maxima are easier to determine and energetic constants should then be calculated by using eq 28. The value (∂θ/ ∂u)u)u* is then calculated by substituting the value of C obtained from eq 28 in eq 21. The final treatments are the same as for the Langmuir model. In the case of nonnegligible lateral interactions, the following modification of the BET equation was used:
θ)
Ceaθeu (1 - e )[1 + (Ceaθ -1)eu] u
(29)
Ceaθeu + θeu[1 - (Ceaθ - 1)(1 - 2eu)] ∂θ ) ∂u (1 - eu){1 + (Ceaθ(1 + aθ) - 1]eu} - aCeaθeu (30) Rudzin´ski and Everett2 have shown that eq 29 can be developed rigorously using the methods of statistical thermodynamics only if θ appearing in the term eaθ is identified with the surface coverage in the first adsorbed layer θ1. When the lateral interactions are neglected in the second and subsequent adsorbed layers, it can be shown that
θ1 )
Ceaθ1eu [1 + (Ceaθ1 - 1)eu]
(31)
For statistical coverages θ < 2, the concentration in the first adsorbed layer is usually much higher than in the second and subsequent layers. Equation 31 can then be applied, i.e.
θ ) θ1/1 - eu
(32)
For higher surface coverages, application of eqs 31 and 32 may lead to a certain underestimation of the role of the lateral interactions. Then, though eq 29 is not fully rigorous, we decided to apply this equation, which must have resulted in a certain overestimation of the role of the lateral interactions. Therefore, the estimation of parameter a in our method was likely lower than its actual value. Experimental values of a and C can be determined step by step: (1) a and C are fixed, θ is computed from eq 29. (2) ∂θ/∂u is then calculated from eq 30 and the position of the maximum is determined. If this position is different from u*, C is corrected and tested in the same way. (3) If the shape of the experimental isotherm obtained, after C adjustment, differs from the shape of the experimental isotherm, the lateral interaction parameter a is modified and C recalculated. The structure of the numerical program and some computational details were described in our first publication.16 Examples of successful applications of this method to study the surface heterogeneity of various clay minerals were presented. In the case of the studied adsorbents, the adsorption on different domains of their surfaces can
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Langmuir, Vol. 13, No. 5, 1997 1109
Figure 2. Nitrogen adsorption at 77 K on palygorskite, the result of fitting the experimental isotherm derivative (the solid bold line) by combination of appropriate derivatives corresponding to Langmuir-Temkin and BET isotherm equations (the thin line), using the best fit parameters collected in Table 1. The dotted lines are the theoretical derivatives (∂θi/∂ ln p) multiplied by Mi, and the solid thin line is their sum (∂Nt/∂ ln p) defined in eq 12.
follow different adsorption mechanisms. That makes most of the methods proposed to study surface heterogeneity useless in this case. However, in some cases such as the adsorption of argon on sepiolite,6 this method did not lead to satisfactory results. The same problem is encountered in the case of the adsorption of nitrogen at 77 K on palygorskite outgassed at 100 °C. Figure 2 shows the experimental derivative and the best fit which we could achieve. The experimental derivative ∂Nt/∂ ln(p) exhibits two distinct maxima and two shoulders. It was fitted using five Langmuir and two BET isotherm derivatives. While trying to fit the experimental derivative ∂Nt/∂ ln(p/ps), the vertical step observed around ln(p/ps) ) -15 could not be reproduced properly. It was then decided to choose the parameters of domain 1 collected in Table 1 so that the integral area under the theoretical curve could be the same as of the experimental one. Figure 3 shows the comparison between the experimental heat of adorption drawn versus surface coverage and the theoretical one, calculated by using the parameters obtained by fitting the experimental isotherm derivative (Table 1) and eq 13. The shape of the experimental heat curve is more or less reproduced, but the agreement between the two curves is not satisfactory. This is a consequence of the poor simulation of the experimental derivative ∂Nt/∂ ln(p/ps) which can never be fitted by a linear combination of theoretical derivatives ∂θ/∂ ln(p/ps) having symmetrical bell-like shapes obtained for the applied “local” adsorption isotherms (Langmuir, BET, ‚‚‚).
Figure 3. Nitrogen adsorption at 77 K on palygorskite, comparison of the experimental (‚‚‚) and the theoretical (s) heat of adsorption calculated from eq 13 by using the using the derivatives (∂θi/∂ ln p) and the qsti functions corresponding to Langmuir-Temkin and BET isotherms.16 The parameters used are in that calculation and those collected in Table 1.
This DIS method also fails to reproduce experimental curves ∂Nt/∂ ln(p/ps) where ∂Nt/∂ ln(p/ps) is more or less parallel to the ln(p/ps) axis over a wide region of values. Such behavior results from a broad distribution of adsorption energies which can hardly be represented by a linear combination of Dirac delta functions. Consequently, the experimental curve cannot be represented by a linear combination of derivatives ∂θ/∂ ln(p/ps), corresponding to homogeneous solid surfaces. An example of such behavior is presented in Figure 4. 2. The Principles of the Modified DIS Method. Further difficulties were also encountered in fitting the curves ∂Nt/∂ ln(p/ps) in the domains of high relative pressures. These difficulties can be related to certain approximations encountered in the classical BET equation. For a better understanding, we will repeat briefly the statistical development of the BET equation. We start by writing the expression for the grand partition function for the classical BET model
Ξ ) (1 + y)M y)
(33)
j1eµ/kT
(34)
1 - j∞eµ/kT
where j1 and j∞ are the molecular partition functions for the primarily and secondarily adsorbed molecules and M is the number of adsorption sites. The adsorption isotherm N h (the average number in our adsorption system) is given by
Table 1. Nitrogen Adsorption at 77 K on Palygorskitea domain i
model
position of maximum (ln(p/ps))
energetic constant C
lateral interaction ω
Mi (cm3‚g-1)
specific surface area of domain i (m2‚g-1)
1 2 3 4 5 6 7
Temkin Temkin Temkin Temkin Langmuir BET BET
-14.64 -14.13 -12.92 -11.66 -9.44 -6.43 -2.89
414 500 432 900 123 000 47 100 12 500 500 13.2
3.4kT 2.3kT 2.4kT 1.8kT 0kT 0.4kT 1kT
11.5 10.3 2.5 1.8 5.8 16.4 16.7
50.4 45.1 11.0 8.0 25.2 71.8 73.2
a DIS parameters obtained when fitting the experimental derivative (∂N /∂ ln(P/P )) by a linear combination (12) of the theoretical t s derivatives (∂θi/∂ ln p) corresponding to Langmuir-Temkin and BET isotherms, used to represent the local adsorption in various surface domains, assumed to be energetically homogeneous.
1110 Langmuir, Vol. 13, No. 5, 1997
Villie´ ras et al.
qst ) Q0 - Q′1
[1 - k′(p/ps)]2
(43)
1 + (C - 1)[k′(p/ps)]2
where
Q0 ) -kT2
∂ ln(k′/ps) ∂T
(44)
∂ ln C ∂T
(45)
is the heat of condensation
Q′1 ) kT2
Figure 4. Derivative of Ar adsorption isotherm measured at 77 K on FeOOH sample outgassed at 25 °C under a residual pressure of 10-4 Pa.
(δ δµln Ξ)
N h ) kT
When the state of the molecules adsorbed in the second and higher layers is the same as that of the molecules in the saturated vapor, then k′ ) 1, and limp/psf1qst ) Q0 ) heat of liquefaction. However, in systems where k′ is smaller than unity, qst cannot be interpreted as the heat of liquefaction even if (p/ps) is close to unity. We are now going to develop the most essential improvement of the DIS method: instead of representing χ() by a linear combination of Dirac delta functions.
Mi
n
)
T,M
Mj1eµ/kT [1 - j∞eµ/kT][1 - j∞eµ/kT + j1eµ/kT]
χ() )
(46)
(35) we will use a combination of χi() functions n
We assume the ideal gas approximation
µ ) µ0 + kT ln(p)
δ( - i) ∑ i)1 M
χ() ) (36)
Mi
χi() ∑ i)1 M
(47)
such that the integration and write j1 in a form showing its dependence on the adsorption energy
∫∞χi() d
j1 ) j10e/kT
could be done analytically, and a function χi() be sufficiently flexible in shape. The θBET() function under the integral in eq 2 will be taken as the modified BET isotherm defined by eq 38. This proposal can be justified as follows. The surfaces of many solids, and of clay minerals in particular, feature different faces which have their own chemical properties and heterogeneities. The adsorption centers in a given surface domain are expected to have similar but not identical properties. Thus, the distribution of adsorption energies on a certain face cannot be represented generally by a Dirac delta function. The spectrum of adsorption energies on a given face will be more or less diffuse and may not be symmetrical. We will write the integral in eq 2 as
(37)
Using this notation, we can write eq 35 in the following form33,43
θ)
N h M
) (1 + P)
KrPe/kT 1 + KrPe/kT
(38)
where
Kr ) j10/j∞ P)
p/ps 1 - k′(p/ps)
(39)
(48)
c
∫Ωθ1() χ() d
θt(p,T) ) (1 + P)
(40)
(49)
where and µ0/kT
k′ ) j∞pse
(41)
KrPe/kT
(49a)
1 + KrPe/kT
The integral in eq 49 can be evaluated using the Rudzin´ski-Jagiełło approach
The comparison of eqs 38 and 24 yields
C ) Kre/kT
θ1 )
(42)
2
θ1t ) The isosteric heat of adsorption corresponding to the modified BET equation (38) reads (43) Waksmundzki, A.; Rudzin´ski, W.; Sokolowski, S.; Jaroniec, M. Pol. J. Chem. 1974, 48, 1741.
∫Ωθ1() χ() d ) -X(c) - Π6 (kT)2(∂χ ∂ ))
(50)
c
where
X() )
∫χ() d
(51)
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Langmuir, Vol. 13, No. 5, 1997 1111
and where c is found from the condition
( ) ∂2θ1 ∂2
)0
(52)
)c
Therefore, after differentiating θ1 in eq 49a, we obtain
c ) -kT ln(KrP)
θt ) -(1 + P) X(c)
(54)
For any function χ(), the derivatives ∂θ/∂ ln(p/ps) to be used to fit the experimental ∂Nt/∂ ln(p/ps) are calculated by using eq 54
∂θt ∂ ln p
T
) -(1 + P)χ(c)
( ) ∂c ∂ ln p
T
( )
- X(c)
∂P ∂ ln p
T
(55)
For the function c defined in eq 53, we have
[ ] ∂c ∂ ln p
T
[∂ ∂Pln p]
) -kT(P + 1)
(56)
) P(P + 1)
(57)
T
Finally, we comment on the possible effect of the interactions between adsorbed molecules. As already mentioned, multiplying C in the BET equation by eaθ will result in an overestimation of the effects due to lateral interactions, whereas multiplying C by eaθ1 will result in a certain underestimation of their role. The latter procedure, however, has a rigorous thermodynamic background as it takes into account the lateral interactions between the molecules adsorbed in the first layer. Therefore, in the present improved edition of the DIS method, it was chosen to accept that second possibility. Neglecting the lateral interactions in the second and higher layers will probably lead to estimated values of the interaction parameters somewhat higher than the actual ones. While carrying out the integration in eq 51 and taking into account the interactions between the adsorbed molecules, the expression for c obtained from eq 52, will depend on the topography of a heterogeneous surface, i.e., the way in which various adsorption sites are distributed on the heterogeneous solid surface. Usually two extreme situations are considered: (i) The “patchwise” topography model. Sites with identical adsorption properties are grouped at the heterogeneous surface into patches which leads to the following expression for c:30
c ) -kT ln(KrP) -
[ ] ∂c ∂ ln p
(53)
Jagiełło et al.37 have shown that except for cases where the variance of the derivative ∂θ1/∂ is smaller at least by 10% of that of the function χ(), the correction term on the right hand side of eq 51 can usually be neglected. Therefore, for many probable physical situations, it is possible to write
[ ]
a random topography. We will demonstrate it in our future publication. The introduction of lateral interactions into the new edition of our method only changes the form of the derivatives in eq 55, used to fit the experimental derivative ∂Nt/∂ ln(p/ps).
ω 2
(58)
(ii) The “random” topography model assuming that different adsorption sites are distributed randomly on a heterogeneous surface. This leads to
c ) -kT ln(KrP) - ωθ1t
(59)
In the applications of our approach presented here, the investigated solid surfaces can be considered as composed of patchwise-like domains. It is then logical to accept the random topography on each patch. Nevertheless, our approach can also be applied to surfaces characterized by
T
) -kT(P + 1) + ωχ(c)
[ ] ∂c ∂ ln p
T
(60)
or after the solution
[ ] ∂c ∂ ln p
-kT(P + 1) 1 - ωχ(c)
)
T
(61)
In order to complete the set of the equations needed to fit both the experimental derivative ∂Nt/∂ ln(p/ps) and the related heat of adsorption measured in an independent calorimetric experiment, it is yet necessary to choose an analytical approximation for χ(). As already mentioned, that function should be sufficiently flexible to reproduce the shapes of the peaks observed in the experimental curves ∂Nt/∂ ln(p/ps). Thus, it should be generally a Gaussian-like function. The case of nitrogen adsorption on palygorskite (Figure 1) suggests that it must be able to reproduce left hand widened Gaussian curves as well as right hand widened ones. Furthermore, this function appears in expressions 55 and 61 together with its integral X(c). To avoid long time-consuming numerical integrations, the expression of χ() should be defined so that its integral could be expressed by a simple elementary function. Finally, the choice of an analytical expression for χ() should be dictated by physical arguments. These three conditions led us to choose the following expression to represent χ() ) (Mi/ M)χi(i):
χi() )
ri( - i0)ri-1 (Ei)
ri
0
ri
e-[(-i )/Ei]
(62)
The integral form of that function X() reads n
θt )
Mi
θit ∑ i)1 M
(63a)
where
θit ) (1 + P)
∫Ωθ1i() χi() d ) -(1 + P) Xi(ci) ci ) -kT ln(KriP) 0
Xi() ) -e-[(-i )/Ei]
(63b) (63c)
ri
(63d)
0
In the above equations, is the minimum value of the adsorption energy and E is the variance of χ(). The parameter r governs the shape of that function. It is a Gaussian-like function right-hand widened for r < 3 and left-hand widened for r > 3. Its variation with r is presented in Figure 5. For the purpose of convenience and clarity, we will drop for the moment the subscript “i” in our expressions. In the case of the Langmuir model of adsorption, when c ) -kT ln(Kp), eq 54 leads to the following explicit expression for θt(p,T) 0
r
0
r
θt(p,T) ) e-[(c- )/E] ) e-[(kT/E)ln(p /p)]
(64)
which is the well-known Dubinin-Asthakov isotherm equation. For r ) 2, this equation becomes the Dubinin-
1112 Langmuir, Vol. 13, No. 5, 1997
Villie´ ras et al.
Figure 5. Effect of the parameter r on the shape of the adsorption energy distribution χ() defined in eq 62. The other parameters are: E/kT ) 5 and ° ) 0.
Raduskhevich isotherm. In eq 64
ln p0 ) -ln K -
0 kT
(65)
Equation 64 can be extended to take into account the effect of lateral interactions by replacing
( )
kT P0 ln E P
by ∆ )
( )
kT ωθ P0 ln E P E
In practical applications of the DA (Dubinin-Asthakov) equation p0 is commonly identified with the saturated vapor pressure ps. This approximation may be justified in some cases. However in general p0 cannot be identified with ps and must be treated as a best-fit parameter. In the case of monolayer adsorption the derivative can be written as
kT r-1 [∆] θ E ) rω T 1[∆]r-1θ E
[ ] ( ∂θt ∂ ln p
r
(66)
)
Figure 6 presents the evolution of [∂θt/∂ ln p]T with changing parameters r, E, and ω. For multilayer adsorption, the derivative [∂θt/∂ ln p]T takes the following form
[ ] ∂θt ∂ ln p
) T
[
where
]
kT E
ω r-1 -∆r 1r∆ e E
()
0
and the height of this maximum. To that purpose, we calculate ∂2θt/∂ ln(p)2 from eq 67. Let ∆′, f1, f1′, f2, and f2′ denote the following functions:
∆′ )
( ) + P(P + 1) e
(P + 1)2r∆r-1
Figure 6. Evolution with changing parameters of the theoretical derivatives (∂θi/∂ ln p) corresponding to the DubininAsthakov adsorption isotherms and given by eq 66: (a) evolution with r when E/kT ) 10, ω ) 0; (b) evolution with E/kt when r ) 3, ω ) 0; (c) evolution with ω when r ) 3, E/kT ) 10.
-∆r
( )
∂∆ ) ∂ ln p T ∂θt kT ω 1 (P + 1) E E P + 1 ∂ ln p
(67)
( ) [[ ]
f1 ) (P + 1)2r∆r-1
0
c - kT ln Kr + kT ω ∆) )ln P - θ1t (68) E E E E The evolution of the derivative with r, E, and ω is presented in Figure 7. Let A° denote the following quantity 0
A° ) kT ln Kr +
(69)
Provided that χ() is a one-modal function, the fitting of the experimental derivative ∂Nt/∂ ln(p/ps) by the function ∂θt/∂ ln(p/ps) given in eq 67 will involve the adjustment of the following parameters: M, r, p0, kT/E, ω/E, A°, k′. The number of free parameters can be reduced to five by studying the position of the maximum of ∂Nt/∂ ln(p/ps)
f2 ) 1f′1 )
(
)
∂f1 ∂ ln P
(
r-1 -∆r
e
]
- θtP (70) (71) (72)
)
T
( ) ) () r∆r-2
f′2 )
(Eω)r∆
(kTE)
T
∂f2 ∂ ln P
kT (P + 1)2[2P∆ + (r - 1)∆′] (73) E
) -r
T
r ω r-2 ∆ ∆′[(r - 1) - r∆r]e-∆ (74) E
The derivative ∂2θt/∂ ln(p)2 then takes the following form
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Langmuir, Vol. 13, No. 5, 1997 1113
carry the subscript “i”, and the conditions for the position and height of the maximum of ∂Nit/∂ ln(p) are to be employed. Finally, from eqs 14, 55, and 64, one obtains the expressions for qist to be inserted into eq 13. For the model of monolayer adsorption, we have
[ ( )]
qist ) Qi(0) + ωiθ1t,i + Ei ln
1 θ1t,i
1/ri
(77)
where
Qi(0) ) -k
[ ()
]
i0 ∂ ln Kri + 1 kT ∂ T
(78)
whereas for multilayer adsorption model, we obtain
(
){
p qist ) Qi(0) + 1 - k′i Qi(1) + ωiθ1t,i + ps
[ ( )] }
Ei ln
1 θ1t,i
1/ri
(79)
where
Qi(0) ) -k
Figure 7. Evolution with changing parameters of the theoretical derivatives (∂θi/∂ ln p) corresponding to the multilayer extension of the Dubinin-Asthakov adsorption isotherms and given by eq 67: (a) evolution with r when E/kT ) 10, ω ) 0; (b) evolution with E/kt when r ) 3, ω ) 0; (c) evolution with ω when r ) 3, E/kT ) 10.
( ) [ ∂2θt
)
2
∂ ln P
T
f′1f2 - f′2f1 (f2)2
]
r
+ P(P + 1)(2P + 1) e-∆ -
[
]
f1 r + P(P + 1) r∆r-1∆′e-∆ (75) f2
The condition ∂2θt/∂ ln(p)2 ) 0 thus leads to the following equation for the value of P* at which the maximum will occur on the experimental curve ∂Nt/∂ ln(p/ps)
[f1f2 + P(P + 1)(f2)2]r∆r-1∆′ ) 0 (76) Now, for the general (common) case of multimodal adsorption energy distribution, χ() the experimental curve ∂Nt/∂ ln(p) has to be fitted by the linear combination
∂Nt
n
) ∂ ln p
∑ i)1
Mi
( ) ∂θit
∂ ln p
(80)
It is to be emphasized that, from the modified statistical development of the BET equation (38), it follows that Q0 must not be equal to the heat of liquefaction of the adsorbate, even when k′ is close to unity. The smaller the value of k′i, the smaller the tendency to multilayer formation. It should, however, be pointed out that one cannot arrive at equations for monolayer adsorption as the limit k′ f 0 in the equations for multilayer adsorption. The statistical derivation of Langmuir and BET equations is based on partially different assumptions. It is now time to emphasize that the new improved DIS method allows us to decompose the experimental derivative M[∂θt/∂ ln(p/ps)] into the individual contributions Mi[∂θit/∂ ln(p/ps)] corresponding to the different domains of the surface. That numerical decomposition also provides us with the values of the heterogeneity and interaction parameters for each of these domains which represents the most essential information about a given heterogeneous surface. Now it is the time to realize that although the parameters Mi, ri, p0i, Ei, and ωi have correctly been determined, the function χ() ) (Mi/M)χi(i) is not the exact adsorption energy distribution. As the function ∂Nt/∂ ln(p) is related to the condensation approximation for χ()
∂Nt
f′1f2 - f′2f1 + P(P + 1)(2P + 1)(f2)2 -
( ) ()
ps ∂ ln 1 k′i ∂ T
∂ ln p
n
) Mχc() )
Miχic() ∑ i)1
(81)
we decomposed in fact the condensation approximation for χ(). Somebody wishing to obtain an exact shape of the adsorption energy distribution may calculate it from its present condensation approximation along the lines outlined by Rudzin´ski, Jagiełło, and Everett.2,8,37
(76a)
where ∂θit/∂ ln(p) is given either by expression 66 or by expression 67 in which the parameters r, P0, E, and ω will
Experimental Section Methods. The experimental setup used for obtaining the lowpressure isotherms was previously described.16,39,40 It allows collection of between 2000 and 3500 experimental data points
1114 Langmuir, Vol. 13, No. 5, 1997
Villie´ ras et al.
Table 2. Nitrogen Adsorption at 77 K on Palygorskitea domain
model
position of maximum (ln(p/ps))
r
E/kT
ln(p0/ps)
lateral interaction ω
adsorbed volume (cm3‚g-1)
equivalent specific surface area (m2‚g-1)
1 2 3 4 5 6
DA DA DA DA MDA MDA
-14.73 -14.13 -12.99 -10.99 -6.50 -2.72
14 12.5 10 4.5 2.5 1.9
1.73 3.66 3.25 10.03 6.23 3.25
-13.0 -10.5 -9 -1.5 0 0
0kT 0kT 0kT 0kT 2.6kT 1.2kT
12.7 7.2 2.7 9.1 15.6 17.2
55.5 31.5 11.7 39.8 68.4 75.3
a The set of the parameters obtained by using our modified DIS method (eq 12), in which the Dubinin-Asthakov equation (eq 66) is applied to adsorption in micropores (DA), and its multilayer extension (eq 67) is applied to external surfaces (MDA).
Figure 8. Nitrogen adsorption at 77 K on palygorskite: best fit (s) of the experimental isotherm derivative obtained by using our modified DIS procedure in which the Dubinin-Asthakov equation is applied to describe adsorption in micropores (DA), and its multilayer extension is applied to describe adsorption on external surfaces (MDA). The parameters used in this fit are collected in Table 2. The experimental curve is in bold. The broken lines (‚‚‚) are the composite derivatives (∂θi/∂ ln p) multiplied by Mi. for relative pressures lower than 0.15. As in the first version of the DIS method the fitting procedure is not automatic but is carried out by successive adjustments through an interactive step by step graphic display described in ref 16. Materials. Three different adsorbents were used in this study, a microporous clay mineral, palygorskite, and two different samples of a lamellar phyllosilicate, kaolinite, which feature different basal and lateral faces, each face having its own energy distribution. The palygorskite studied here comes from the Montagne de Reims, France, and was supplied by B.R.G.M. (Orle´ans, France). Its mineralogical purity is >93%. The major impurities are quartz (4%), anorthite (0.8%), calcite (0.6%), anatase (0.5%), and mica (1%).10 The two samples of kaolin are referred to as FU7 and MM. Kaolin FU7 comes from the Charentes deposit (France) and was supplied by AGS (France). It is poorly crystallized with a specific surface area measured by the BET-argon method of 47 m2‚g-1. Its lateral faces account for ≈12% of the total surface.5 Kaolin MM comes from the M&M mine in south-central Georgia (USA). It was treated and provided by P. O’Day from Stanford University. It is well crystallized and its specific surface area measured by the BET-nitrogen method is equal to 13.3 m2‚g-1.44
Results and Discussion Adsorption on Palygorskite. Palygorskite belongs to a family of hydrous silicate minerals. It is made up of long ribbons of talc-like layers, one ribbon being linked to (44) O’Day, P.; Parks, G. A.; Brown, G. E., Jr. Clays Clay Miner. 1994, 42, 337.
Figure 9. Nitrogen adsorption at 77 K on palygorskite: comparison of the experimental (b) and calculated (s) qst curves. The calculated curve was obtained by using the parameters collected in Tables 2 and 3. The dotted lines (‚‚‚) are the heat constributions to Qst, Miqsti(∂θi/∂ ln p)/∑Mi(∂θi/∂ ln p) from adsorption in various domains.
the next by inversion of SiO4 tetrahedra along a set of Si-O-Si bonds. Structural channels 13.7 × 6.4 Å in size are parallel to the fiber length and form intrafibers micropores. The association of fibers generates an interfiber microporosity. The external surface area of the fibers, as measured by the Harkins and Jura method, is equal to 63 m2‚g-1.10 Low-temperature adsorption microcalorimetry (LTAM) experiments (Figure 3)10 reveal three zones assigned to the intrafiber microprosity, interfiber microporosity, and external surfaces, respectively. The volume of gas adsorbed in the micropores depends on the outgassing procedure (temperature and vacuum).10 The amount of gas adsorbed in the third zone corresponds to surface areas of 65 and 55 m2‚g-1, as measured by nitrogen and argon, respectively. The third zone can then be safely assigned to the external surface area of the fibers. Figure 1a presents the experimental nitrogen adsorption isotherm obtained after outgassing palygorskite at 100 °C during 15 h under a residual pressure of 10-4 Pa. Its shape is typical for a microporous adsorbent. Table 1 collects the parameters obtained for the different domains assumed to be homogeneous, i.e., obtained by using our classical DIS procedure. The corresponding fit is shown in Figure 2. In all following sections, the domains will be numbered according to increasing ln(p/ps) value. The amount of the gas adsorbed in the two first domains (21.8 cm3‚g-1) corresponds to the filling of the intrafiber micropores as observed by LTAM (23.9 cm3‚g-1). The amount of nitrogen adsorbed in domains 3, 4, and 5 (10.1 cm3‚g-1) corresponds to the filling of the interfibermicropores as observed by LTAM (15.4 cm3‚g-1) whereas domain 6 likely corresponds to the external surface area
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Langmuir, Vol. 13, No. 5, 1997 1115
Table 3. Nitrogen Adsorption at 77 K on Palygorskitea domain
1
2
3
4
5
6
Q(0) (J‚mol-1) Q(1) (J‚mol-1)
18200 0
17000 0
16400 0
13500 0
6900 1500
6900 1500
a The Q(0) and Q(1) values used to simulate the experimental Qst curves by the theoretical functions (eqs 77-79) corresponding to DA and MDA equations.
of the fibers (16.43 cm3‚g-1 by DIS and 15.0 cm3‚g-1 by LTAM). The presence of a very low energy domain (domain 7) probably means condensation in large interfiber micropores not revealed by LTAM experiments. Figure 3 shows the comparison of the experimental heat of nitrogen adsorption at 77 K on palygorskite, with the theoretical one calculated from eq 13, by using the theoretical derivatives (∂θi/∂ ln p)T and qsti functions corresponding to Langmuir-Temkin and BET “local” isotherms.16 As previously said, the shape of the experimental curve is more or less reproduced by the calculated one, but the curves are not superimposed. In the case of our modified DIS method, eqs 66 and 67, the number of the adjustable parameters needed to perform the decomposition of an experimental isotherm derivative is much larger than in the previous version of the DIS method. It was then necessary to add some constraints reducing the number of possible decompositions. The constraints that we fixed were based on some physical considerations. The value of r decreases with decreasing ln(p/ps) whereas the value of p0 increases.2 For multilayer adsorption, i.e., for adsorption on external surfaces, p0 was taken equal to ps. Figure 8 displays the simulated isotherm derivative obtained by using the parameters collected in Table 2. One can see a definite enhancement of the agreement between the experimental and simulated isotherms compared to what was obtained when we used the first version of the DIS procedure (Figure 2). The vertical step around ln(p/ps) ) -15 can now be reproduced, and distinguishing only six domains is sufficient to fit the experimental isotherm derivative. The physical interpretation of these domains remains the same as in the case of the decomposition done by using the first (classical) version of our DIS procedure. Domains 1 and 2 correspond to the intrafibermicropores (19.9
Figure 10. Nitrogen adsorption at 77 K on palygorskite, the relation between the best fit parameters Q(0) and ln(p0/ps), found while using our modified DIS procedure to fit the experimental isotherm derivative and the related heat of adsorption.
cm3‚g-1 estimated in our modified DIS procedure instead of 23.9 cm3‚g-1 by LTAM). Domains 3 and 4 correspond to interfibermicropores (11.8 cm3‚g-1 instead of 15.4 cm3‚g-1 by LTAM). Domain 5 corresponds to the external surface of the fibers (15.7 cm3‚g-1 instead of 15.0 cm3‚g-1 by LTAM) whereas domain 6 likely corresponds to condensation phenomena in mesoporosity. In all cases it seems, in the case of our experiments, that the outgassing was slightly less efficient than in the LTAM experiment; whatever the decomposition used, the amount adsorbed in the micropores is inferior. Figure 9 shows the fit of the experimental isosteric heat of adsorption by the theoretical one calculated from eqs 13, 77, and 79 for the decomposition presented in Figure 8 and the corresponding parameters collected in Table 2. The calculated qst curve is very close to the experimental one. The horizontal branch at low surface coverage is slightly shorter in the theoretical curve as a result of the lesser amount of accessible micropores. The curve was obtained using Q(0) and Q(1) values displayed in Table 3.
Table 4. Argon Adsorption at 77 K on FU7 Kaolinitea domain
model
position of maximum (ln(p/ps)
energetic constant C
lateral interaction ω
Mi (cm3‚g-1)
equivalent specific surface area (m2‚g-1)
1 2 3
BET BET BET
-11.1 -7.4 -3.8
67700 1600 41
0kT 0kT 0.4kT
0.25 1.07 9.71
0.9 4.0 36.0
a
The DIS parameters obtained by appling our classical DIS method. Table 5. Argon Adsorption at 77 K on FU7 Kaolinitea
domain
model
position of maximum (ln(p/ps))
r
E/kT
lateral interaction ω
adsorbed volume (cm3‚g-1)
specific surface area (m2‚g-1)
Q(0) (J‚mol-1)
Q(1) (J‚mol-1)
1 2
MDA MDA
-7.60 -3.70
2.8 2.3
8.04 4.08
1.4kT 1.15kT
1.44 9.74
5.3 36.1
5600 5600
3000 1800
a The parameters used to simulate the experimental isotherm derivative and the related experimental heat of adsorption, by equations corresponding to the multilayer extension of the Dubinin-Astakhov equation.
Table 6. Argon Adsorption at 77 K on MM Kaolinitea domain
model
position of maximum (ln(p/ps))
r
E/kT
lateral interaction ω
adsorbed volume (cm3‚g-1)
specific surface area (m2‚g-1)
Q(0) (J‚mol-1)
Q(1) (J‚mol-1)
1 2 3 4
MDA MDA MDA MDA
-10.05 -6.59 -4.50 -2.43
5.2 3.3 3 2.0
10.4 6.42 4.05 3.03
0.2kT 1.7kT 1.85kT 0.95kT
0.14 0.42 1.50 0.97
0.5 1.6 5.6 3.6
5200 5200 5200 5200
0 0 200 2300
a The parameters used to simulate the experimental isotherm derivative and the related experimental heat of adsorption, by equations corresponding to the multilayer extension of the Dubinin-Astakhov equation.
1116 Langmuir, Vol. 13, No. 5, 1997
Figure 11. Argon adsorption at 77 K on kaolin FU7 outgassed at 120 °C under a residual pressure of 10-4 Pa, fitting experimental isotherm derivative (the thin solid line) by using the multilayer extension of the Dubinin-Asthakov equation to describe adsorption on external surfaces (MDA). The parameters obtained in that fit are displayed in Table 5. The experimental curve is in bold, whereas the dotted lines show the composite derivatives Mi(∂θi/∂ ln p).
Figure 12. Argon adsorption at 77 K on kaolin FU7, comparison of experimental (b) and calculated (s) heats of adsorption. The calculated curve was obtained by using the parameters displayed in Table 5. The dotted lines are the heat contributions from the two surface domains.
The obtained best fit parameters Q(0) and ln(p0/ps) (Tables 2 and 3) exhibit a dependence which appears to be linear (Figure 10). That interesting interrelation surely deserves further study. Adsorption on Kaolin. The adsorption of argon at 77 K on kaolin FU7 outgassed at 100 °C under a residual pressure of 10-4 Pa was already analyzed in terms of the classical version of our DIS procedure.16 Three BET local isotherms were used there for decomposing the experimantal isotherm derivative. The parameters corresponding to this decomposition are collected in Table 4. The first two domains were assigned to the adsorption on lateral faces whereas the third domain was assigned to the adsorption on basal surfaces. It then yielded an aspect ratio corresponding to 12% of lateral faces.5 The present fit of the experimental isotherm derivative obtained by using our modified DIS procedure is shown
Villie´ ras et al.
Figure 13. Argon adsorption at 77 K on kaolin MM outgassed at 120 °C under a residual pressure of 10-4 Pa, simulation (s) of the experimental derivative isotherm using the multilayer extension of the Dubinin-Asthakov equation applied to external surfaces (MDA). The parameters used are displayed in Table 6. The experimental curve is in bold and the dotted lines are the composite derivatives Mi(∂θi/∂ ln p).
Figure 14. Argon adsorption at 77 K on kaolin MM, comparison of experimental (b) and calculated (s) heats of adsorption. The calculated curve was obtained by using the parameters displayed in Table 6. The dotted lines are the heat contributions from the various surface domains.
in Figure 11. Only two domains are distinguished now while decomposing the experimental curve. The parameters used in this simulation are presented in Table 5. The first domain can be assigned to the lateral faces and the second one to the basal faces. The aspect ratio then corresponds to 12.8% of lateral faces. In Figure 12, the theoretical Qst curve calculated by using the parameters displayed in Table 6, is compared with the experimental data points. The fit seems to be fairly good except for very low surface coverages. In the case of kaolin MM, the experimental isotherm derivative was fitted by distinguishing four domains, the adsorption in which was described by the multilayer versions of Dubinin-Astakhov isotherms (Figure 13). The parameters obtained in that way are collected in Table 6. If domains 1 and 2 are assigned to the lateral faces and domains 3 and 4 to basal faces, the aspect ratio corresponds to 18.5% lateral faces. As this kaolin is well crystallized, it is to be expected finding a lower lamellarity than in
Surface Heterogeneity of Clay
kaolin FU7, which is poorly crystallized.5 The simulation of the experimental heat of adsorption is displayed in Figure 14. The agreement between the two curves seems to be very good. Conclusion The modified derivative isotherm summation procedure appears to be a promising method for studying energetic surface heterogeneity of various solids, in particular highly heterogeneous solids such as clay minerals. It is more realistic as it takes into account the random nature of energy distribution of surface domains considered previously as a collection of homogeneous patches. The Dubinin-Asthakov energy distribution is assumed to describe the local dispersion of adsorption energies. It allows simulation of asymmetric peaks that increase the
Langmuir, Vol. 13, No. 5, 1997 1117
accuracy of the decomposition of an experimental isotherm derivative. However, it also increases the number of adjustable parameters making it compulsory to introduce some further constraints based on physical considerations which seem to be justified by the comparison between experimental and simulated Qst curves. Our modified procedure has been tested by studying samples having well-known structural and textural properties. Acknowledgment. The authors wish to acknowledge Dr. Y. Grillet for providing us the LTAM results. W. Rudzin´ski, wishes to express his thanks to INPL for providing the grant making his extended visit to Professor Cases’ laboratory possible. LA9510083
