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Twofold Description of Topological Disordered Surfaces† Vicente Mayagoitia,*,‡ Fernando Rojas,‡ Isaac Kornhauser,‡ E. Ancona,‡ Giorgio Zgrablich,§,| and Roberto Jose´ Faccio| Departamento de Quı´mica, Universidad Auto´ noma MetropolitanasIztapalapa, Apartado Postal 55-534, Me´ xico 13, D.F., 09340 Me´ xico, Centro Regional de Estudios Avanzados (CREA), 5700 San Luis, Argentina, and Departamento de Fı´sica, Universidad Nacional de San Luis, 5700 San Luis, Argentina Received September 1, 1994. In Final Form: July 28, 1995X The “dual” theory is applied to describe the structure of adsorbent surfaces consisting of by several types of adsorption sites, each kind bearing a given connectivity (i.e., possessing a given number of delimiting bonds or energy barriers between sites), and the principal conclusion is that, for most of adsorbent surfaces, concomitantly with an energy segregation effect (sites and bonds group together forming alternated regions of very high and very low adsorption energies), a connectivity segregation effect can arise too (connectivity distributes throughout the network sensibly obeying adsorption energy correlations).
Introduction An adsorbent surface can be conveniently described as a network of two kinds of alternated elements: adsorption sites (or minima in the adsorption potential) and bonds (potential energy barriers between sites). While particles preferentially adsorb on sites, the function of bonds is to structuralize the morphology of the surface, these entities then becoming important in determining the equilibrium and dynamic behavior of the adsorbed phase.1,2 Recently,3 a treatment has been proposed for disordered structures, in which geometric heterogeneity (the distance between adsorption sites, a parameter controlling the strength of lateral adsorbate interactions, is no longer constant but can exhibit a random character) is taken into account. Indeed, many surfaces may also display a strong topological heterogeneity, which means that the connectivity, C (the number of delimiting bonds leading to first-order neighboring sites), changes from site to site throughout the network. The task in this contribution is to describe adsorbent surfaces of such kind. Indeed, what a molecule would “see” when approaching the surface is illustrated in Figure 1, where black circles are adsorption sites and the white space is unavailable for adsorption. These clusters of adsorption sites could correspond to chemically eligible regions of the surface for adsorption (such as deposits of an active phase on a catalytic surface), while the atoms lying at the white zone are not suitable. Sites located deep in the adsorption region are well connected (with C ) 6 as maximum for this example), while those along the boundaries of the cluster are scarcely surrounded by other sites. Some sites stand isolated, in which case adsorption, but not surface migration, is possible. For the sake of simplicity, geometrical heterogeneity will be omitted in the subsequent treatment (note that according to Figure 1, it is possible to represent topological † Presented at the symposium on Advances in the Measurement and Modeling of Surface Phenomena, San Luis, Argentina, August 24-30, 1994. ‡ Universidad Auto ´ noma MetropolitanasIztapalapa. § Centro Regional de Estudios Avanzados. | Universidad Nacional de San Luis. X Abstract published in Advance ACS Abstracts, January 1, 1996.
(1) Mayagoitia, V.; Rojas, F.; Pereyra, V. D.; Zgrablich, G. Surf. Sci. 1989, 221, 394. (2) Mayagoitia, V.; Rojas, F.; Riccardo, J. L.; Pereyra, V. D.; Zgrablich, G. Phys. Rev. B 1990, 41, 7150. (3) Benegas, E. I.; Pereyra, V. D.; Zgrablich, G. Surf. Sci. 1987, 187, L647.
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Figure 1. Topologically irregular (while geometrically regular) surface network of adsorption sites (in black).
heterogeneity without introducing geometric heterogeneity, so that these morphologies may exist in the real world), even if in the most general case a surface should display all kinds of energetic, geometric, and topological heterogeneities. Since the treatment will remain within the frame of our site and bond theory of heterogeneous adsorbent surfaces, we will first present a synthesis of our previous work about ordered lattices, necessary to understand the subsequent original development for the case of variable connectivity. Ordered Lattices The statistical treatment termed as the “site and bond theory” or “dual theory” has been principally applied to describe the morphology of (i) heterogeneous adsorbent surfaces,1,2 and their implications in physical adsorption equilibria, surface diffusion, and surface characterization, © 1996 American Chemical Society
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and (ii) porous media,4 treating such topics as capillary condensation and evaporation,5 textural determinations,6 etc. It is of course the terminology dealing with surfaces which will be used now. First, it is necessary to define the alternated elements conforming the adsorbent network, “sites” and “bonds”: the local adsorption potential, E, varies throughout the surface (at an infinite distance from the surface, E ) 0). Then sites correspond to the minima of E while bonds are located at every saddle point of potential energy between each pair of neighboring sites. The “adsorption energy” will be represented as ) |E|. A “construction principle” originating from the very definition of “sites” and “bonds” states that “The energy of a site, S, is greater or at most equal to the energy, B, of any of its own delimiting bonds.” This principle plays a central role in the description of heterogeneous surfaces.1,2 If FS() and FB() are the normalized energy distribution functions of for sites and bonds, then S(e) and B(e) represent, respectively, the probabilities for a site and a bond to have values of lower than or at most equal to e:
S(e) )
∫0eFS() d;
B(e) )
∫0eFB() d
(1)
F(S∩B), the probability density for the joint event of finding a given site of energy S and concurrently a given one of its bonds with an energy B, can be expressed as:
F(S∩B) ) FS(S) FB(B) φ(S,B)
(2)
where φ(S,B) is a correlation function to be explained afterward. In order to fulfill the construction principle, two selfconsistency laws must be observed. The first one concerns a general relationship between the overall distributions:
first law
B(e) g S(e) for every e
(3)
since for a given site distribution, enough bonds of low energies must be available to link such collection of sites. The second law holds locally in order to avoid the existence of an inconsistent pair of values of S and B for contiguous elements:
φ(S,B) ) 0 for S < B
second law
(4)
Conversely, for the correct condition of having S g B, it has been found1,2 that the function φ can be written as
(
∫B(B( )) BdB - S)
exp φ(S,B) )
S
B
B(B) - S(B)
(5)
Consequently, for a site with a certain adsorption energy S, the conditional probability density to find the energy B for a given one of its bonds is (4) Mayagoitia, V.; Kornhauser, I. In Principles and Applications of Pore Structural Characterization; Haynes, J. M., Rossi-Doria, P., Eds.; Arrowsmith: Bristol, 1985; p 15. Mayagoitia, V.; Cruz, M. J.; Rojas, F. J. Chem. Soc., Faraday Trans. 1 1989, 85, 2071. Cruz, M. J.; Mayagoitia, V.; Rojas, F. J. Chem. Soc., Faraday Trans. 1 1989, 85, 2079. (5) Mayagoitia, V.; Rojas, F. In Fundamentals of Adsorption II; Liapis, A. I., Ed.; The Engineering Foundation: New York, 1987; p 391. Mayagoitia, V.; Rojas, F.; Kornhauser, I. J. Chem. Soc., Faraday Trans. 1 1988, 84, 785. Mayagoitia, V.; Gilot, B.; Rojas, F.; Kornhauser, I. J. Chem. Soc., Faraday Trans. 1 1988, 84, 801. (6) Mayagoitia, V.; Rojas, F. In Fundamentals of Adsorption III; Mersmann, A. B., Scholl, S. E., Eds.; The Engineering Foundation: New York, 1991; p 563.
F(B/S) ) FB(B) φ(S,B)
(6)
As the overlap between distributions becomes considerable, many sites have values smaller than those of certain bonds (these bonds of course cannot be the delimiting ones of such sites), and in order for the construction principle to remain valid, an “energy segregation effect” arises. This phenomenon is the result of a favorable influence which promotes the reunion of elements of similar energies (as was already visualized by Ripa and Zgrablich7). When overlap is nearly complete, as the segregation effect is the dominant factor, there appear “homotattic” patches (wherein all elements possess the same energy), the model of Ross and Olivier8 being appropriate at this limit. Nevertheless, it is worth while to point out that in this case of almost complete overlap, the energy values for both sites and bonds within each one of the homogeneous domains become about the same, this favoring mobile rather than localized adsorption. Thus, a proper assessment of the surface topology of heterogeneous adsorbents is only possible by considering the twofold energy distribution. Disordered Lattices Much more information is needed to deal with irregular networks. It will be assumed that all bonds delimit two sites, while a site of the ith type possesses Ci bonds. There exists a fraction xi() of sites of energy belonging to the ith type. xi could correspond to a discrete distribution, e.g., Poisson’s, in such a way that the mean connectivity of sites of energy is
∑i xi() Ci
C() )
(7)
and
∑i xi() ) 1
(8)
The mean connectivity of the whole network is given by
∫0∞C() FS() d
C h )
(9)
We first consider that the number of bonds (or, more precisely, half-bonds), NB(e), belonging to all sites of energy lower than e is
∫0e C() FS() d
NB(e) ) NS
(10)
For the particular value e ) ∞ we find
∫0∞ C() FS() d ) NSCh
NB(∞) ) NS
(11)
where NS is the overall number of sites of the network. Considering that each site of the ith type possesses Ci half-bonds, a new relevant quantity, S′ is defined as the ratio NB(e)/NB (e ) ∞):
S′(e) )
∫0e{C()/Ch } FS() d
(12)
S′ represents the minimal fraction of bonds required to link all sites of assorted connectivities equal or lower than e. The first law must now be modified to take into account the possible dependence of C() with . Let us consider a given energy e. The fraction of sites (or bonds) having an energy e or lower, S(e) (or B(e)), is still in accordance with eq 1. However for irregular networks, it is S′(e) instead of S(e) that should be compared with B(e), the (7) Ripa, P.; Zgrablich, G. J. Phys. Chem. 1975, 79, 2118. (8) Ross, S.; Olivier, J. P. On Physical Adsorption; Interscience: New York, 1964.
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available fraction of eligible bonds. Instead of eq 3, we have
first law
B(e) g S′(e)
for every e (13)
in such a way that for a given energy distribution of sites, there must be enough bonds of low energies to link such sites of multiple demanding connectivities. In order to visualize how sites and bonds could be interconnected to form the network, it is convenient to follow a procedure that has been outlined elsewhere.1 The method is now generalized to treat irregular networks: sites are sorted from lower to higher energies, precisely in this order, and half bonds are assigned to them. Bonds must obviously possess a lower or at most equal energy than the site in question. The randomness of this assignation is then raised to a maximum, this maximum being only conditioned by observation of the construction principle. This site and bond balance allows one to obtain a network which is (i) self-consistent (i.e. complying with the construction principle) and (ii) the “most verisimilar” (corresponding to the maximum randomness allowed, then possessing the maximal entropy, and leading to the maximum number of configurations). Equation 5 is generalized to
φ′(S,B) )
∫B( )
dB B - S′ B(B) - S′(B)
(
exp -
B(S) B
)
(14)
Note that φ′ follows the same form as φ, this generalization merely undertaken by a reinterpretation of one of the terms inside φ (i.e. S′, which takes into account variable connectivity). The conditional probability density for a site of a given adsorption energy S to find the energy B for a precise one of its bonds now becomes as
F(B/S) ) FB(B) φ′(S,B)
(15)
As noted before, when the overlap between distributions is considerable, there arises an energy segregation effect. In the event of connectivity increasing with , the energy segregation effect should concomitantly lead to a “connectivity segregation effect”, since the former signifies the reunion of elements with bigger energies, and these would precisely adopt higher connectivities. Discussion The treatment of irregular networks differs from that of regular substrata only if C() depends effectively on . For example, an inspection of a Zachariassen model of a glass immediately reveals that this is the case. This is the structure used by Benegas, Pereyra, and Zgrablich3 to treat geometric heterogeneity. Instead, here we are interested in topological aspects. In the Zachariassen model, there extends an irregular lattice constituted by rings of variable size, depending on the number of participating siloxane elements. It is very clear that the number of neighboring rings of a given ring (i.e. its connectivity) is determined by the number of its oxygen atoms. If this arrangement is considered as a surface structure, each ring could be visualized as an adsorption site. Then, the bigger the rings are, the bigger is the connectivity of the adsorption site. Now, the relationship between the size of the ring and is in some way cumbersome to establish, but suppose that small molecules adsorb at the centers of the rings. If the ring is big, the molecule can penetrate and will be important. Conversely, for small rings will be low
Figure 2. The limiting allowable distribution of the highest possible bond energies (the line with positive slope) compared with its corresponding uniform site-energy distribution. Connectivity increases from 2 to 6 in going from the lowest to the highest site energies (see text).
because the molecule interacts with fewer elements of the substratum. Then the energy segregation effect correspondingly involves a “connectivity segregation effect”, since the former signifies the reunion of elements with the biggest energies, and these have precisely the highest connectivities. This new effect must be indeed of the utmost importance when lateral interactions of adsorbed molecules are present, eq 15 allows a quantitative assessment of its strength. Even in the absence of these interactions, this effect may have important consequences upon other relevant aspects about the dynamics and equilibrium of the adsorbed phase, e.g. surface diffusion. As an example of the interesting changes introduced by topologically disordered networks, we present an extremely simple and illustrative case: suppose that the site-energy distribution is uniform and that the connectivity C() is an increasing linear function of the site energy, in such a way that from the lowest to the uppermost limits of site energies, connectivity increases from 2 to 6, Figure 2 (note the connectivity axis). If connectivity were kept constant throughout all sites, the limiting allowable distribution of the highest possible bond energies would correspond to the same distribution as for sites, i.e. a case of complete overlap would be present. This arbitrary situation is very interesting to analyze in the light of variable connectivity. Now since the requirements for bonds vary according to C(), the relationship given by eq 3 for ordered lattices B(e) g S(e) no longer applies, and then the first law B(e) g S′(e) for disordered lattices, eq 11, is the one that is observed. Sites of low energies demand few bonds, so that the limiting distribution of bonds still good enough to link the whole collection of sites, is given by the line with positive slope represented in Figure 2, corresponding to B(e) ) S′(e). In this novel case, complete overlap is in some way overcome. Should C() decrease with , the antagonistic effect would occur, preventing a too high degree of overlap. Keeping the same site-energy distribution and C() situation as in the previous example, it is possible to
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corresponding to the mean value of . The uniform sitebond distribution and F(B/S) curves are both normalized. Future work should be directed to the prediction of adsorption isotherms, the study of the structuralization of the adsorbed phase, and surface diffusion in the case of topological heterogeneity, as has already been done3 for geometric heterogeneity. These are complementary aspects of more complex networks. The whole field of adsorption in lattice surfaces (coverage in terms of adsorption site potential, lateral interactions, multisite occupancy by r-mers, kinetics of adsorption, temperature programmed desorption, etc.) is going to be renewed for structures such as that depicted in Figure 1. There, instead of developing analytical treatments in which some kind of Bragg-Williams approximation that, even thought impossible to justify, is almost unavoidable and ever present, Monte Carlo representations of adsorption-site trees and networks will surely take a much more important role. Intricate structures could perhaps be only visualized by means of digital models. Then, reality will be discerned from fantasy, since only those morphologies able to have a representation in space are endowed of the possibility to exist in the real world. Figure 3. F(B/S) as function of B for a constant value of S corresponding to the mean value of . The uniform site-bond distribution and F(B/S) curves are both normalized.
analyze straightforwardly the implications of variable connectivity on topological energy correlations. Now the bond-size distribution will be considered the same as for sites (complete overlap). If the network were topologically regular, by virtue of eq 5 every bond should possess the same energy as the sites it is linking, B ) S (homottatic patches). However, in the present case, complete overlap would be no longer a limiting situation of complete correlation between S and B and then eq 13 allows a span of values of B for a given value S. Figure 3 depicts F(B/S) as a function of B for a constant value of S
Conclusions “Connectivity segregation” is a fundamental property, as important as “energy distribution” or “energy segregation”, for the simulation and characterization of adsorbent surfaces possessing a high degree of topological heterogeneity. It is unavoidable to take into account this essential characteristic displayed by many surfaces. Acknowledgment. This work was supported and made possible by the National Council of Science and Technology of Mexico (CONACyT) under Project No. 5387E, as well as under the Joint Research Project: “Cata´lisis, Fisicoquı´mica de Superficies e Interfases Gas-So´lido” by CONICET (Argentina) and CONACyT (Me´xico). LA9407065
