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Langmuir 1993,9, 2748-2154
Twofold Description of Porous Media and Surface Structures: A Unified Approach To Understand Heterogeneity Effects in Adsorption and Catalysis V. Mayagoitia,’ F. Rojas, and I. Kornhauser Departamento de QuEmica, Universidad Autdnoma Metropolitana, Iztapalapa, Apartado Postal 55-534, Mgxico 13, D.F. 09340, Mgxico Received September 29,1992. In Final Form: December 21,199P A review is made about a general development that allows the morphology of heterogeneous structures such as adsorbent surfacesand porous media to be described analyticallyand represented digitally. Possible applicationsof this development include capillary condensationand evaporation,textural determinations, physical adsorption, activated chemisorption, and reaction kinetics in the adsorbed phase.
Introduction Some treatments have been found to possess a character so general as to be applicable to many phenomena, seemingly unrelated at first sight. Examples of these are the percolation theory and fractal approach. In recent times it has also become evident that our “twofold description” can deal with many subjects in different fields, ita usefulness consisting in the possibility to describe intricate topological properties in terms of a twofold diagram and a very simple correlation function. In particular, it has already been applied to (i) porous media,13 and capillary phenomena taking place within them, such as capillary condensation and evaporation,@ and textural determinations;’ (ii) heterogeneous adsorbent surfaces,8 physical adsorption equilibria and surface diffusion? and surface characterizati0n;’O (iii) activated chemisorption on heterogeneous surfaces;ll and (iv) kinetics of reaction in the adsorbed phase.I2 Among other subjects recently explored we can cite the description of the morphology of aggregates resulting from the sol-gel transition,13 and that of p01ymers.l~ Our approach consists in a deep understanding of the morphology of the heterogeneous structure in which a given a Abstract published in Advance ACSAbstracts, August 15,1993.
(1)Mayagoitia, V.; Komhauser, I. In Principles and Applications of Pore Structural Characterization; Haynes, J. M., Rossi-Doria, P., Eds.; Arrowsmith: Bristol, 1985;p 15. (2)Mayagoitia,V.; Cruz, M. J.;Rojas, F. J.Chem. Soc.,Faraday Trans. 1 1989,85,2071. (3)Cruz, M. J.; Mayagoitia,V.; Rojas, F. J.Chem. Soc.,Faraday Trans. 1 1989,85,2079. (4)Mayagoitia,V.;Rojas,F. InFundamentalsof AdsorptionZfiLiapis, A. I., Ed.; The Engineering Foundation: New York, 1987;p 391. (6) Mayagoitia, V.; Rojas, F.; Kornhauser, I. J.Chem. SOC., Faraday Trans. 1 1988,84,785. (6)Mayagoitia, V.; Gilot, B.; Rojas, F.; Kornhauser, I. J. Chem. SOC., Faraday Trans. 1 1988,84,801. (7)Mayagoitia, V.; Rojas, F. In Fundamentals of Adsorption ZZfi Mersmenn, A. B., Scholl, S. E., Eds.; The Engineering Foundation: New York, 1991; p 563. (8)Mayagoitia, V.; Rojas, F.; Pereyra, V. D.; Zgrablich, G. Surf. Sci. 1989,221,394. (9)Mayagoitia,V.;Rojas, F.; Riccardo, J. L.; Pereyra, V. D.; Zgrablich, G. Phys. Rev. E 1990,41,7150. (10)Zgrablich, G.; Riccardo, J. L.; Pereyra, V. D.; Ramlrez-Cuesta, A.; Mayagoitia, V.; Rojas, F. Actas del XZZ Simposio Zberoamericano de Catdlisis (Proceedings of the XZZZberoamerican Symposium of Catalysis); Imtituto Brasileiro de Petr6leo: Rfo de Janeiro, 1990; p 265. (11)Mayagoitia, V.; Kornhauser, I. In Fundamentals of Adsorption ZV; Suzuki, M. Ed.; The Engineering Foundation: New York, submitted for publication. (12)Mayagoitia,V.; Rojas, F. Actas del XZZSimposioZberoamericano de Catdlisis (Proceedings of the XZZ Zberoamerican Symposium of Catalysis); Instituto Brasileiro de Petr6leo: RLo de Janeiro, 1990;p 242. (13)Mayagoitia, V.; Domhguez, A.; Rojae, F. J. Non-Cryst. Solids 1992,147& 148,183. (14)Kuznetaova, G. B.; Mayagoitia, V.; Kornhauser, I. Znt. J. Polym. Mater., in press.
phenomenon is supposed to take place. This preliminary step, unavoidable since the structure of the substrate plays a determinant role on the development of subsequent processes,has been however completely neglected by many researchers. It is not at all sufficient to consider an adsorption-energy or a pore-size distribution, but the more delicate aspect of how these adsorption energies or pore sizes are topologically distributed throughout the system, i.e., the aspect of “morphology” is of prime importance. Indeed, morphology plays an equal or even a more important role than the mere distributions of energy or size, as is the case for capillary condensation and evaporation in porous media: different morphologies lead to distinct mechanisms of vapor-liquid transitions and hysteresis shapes to such an extent that a classification can be established for porous materials in terms of morphology, rather than pore sizes.5 This contribution is a review of the works that we have been publishing during the past seven years. In the fist part a somewhat abstract procedure about the foundations of the twofold description is presented; afterward a more specific treatment is outlined for each topic related to surfaces and porous media, as well as some adsorption and catalytic phenomena taking place in them.
Foundations of the Twofold Description In order to obtain a convenient statistical description of many complex systems, it is proposed to perform the following steps. To decompose the system into a collection of two kinds of interrelated elements, “sites” and ‘bonds”, each having a different function in the network that must be clearly established. To recognize a “metric”,or property such as size, number of repetitive units, energy, etc., that characterizes each element. The nature of this metric must be the same for both sites and bonds. To point out, from the very definitions of site and bond, a “construction principle”, an obvious and constant inequality in the metric for each site compared with those of ita corresponding bonds. Networks observing this principle are termed as ‘self-consistent”. To propose a distribution of this metric for each kind of element. The 2-fold distribution must be set in a number of elements basis and normalized. If the distributions of sites and bonds are inappropriate or their overlap is considerable, then the construction principle risks being violated. In order to avoid the occurrence of this event, two “self-consistency laws” arise:
0743-1463/93/2409-2748$04.00/00 1993 American Chemical Society
Heterogeneity Effects in Adsorption and Catalysis (i) the first law deals with restrictions imposed on the distributions as a whole in order to have a permitted collection of sites and bonds, and (ii) the second law is of a local character and prevents the union of elements that could violate the construction principle. A balance of sites and bonds when performing a selfconsistent assignation of the metric to directly related elements allows a ready determination of the function expressing the correlation of metrics throughout the network. This function is similar for different types of networks. When the randomness in this assignation of the metric to the elements is raised to a maximum, self-consistency remaining as the only restriction, the structure is termed “verisimilar”,or that having the most expected morphology in the absence of other particular information. Conversely, a confrontation between the verisimilar model and experiment permits an assessment of the physicochemical constraints that, acting during the formation of the network, lower the randomness. The application of this function, either by analytical (probabilistic) or by digital (Monte Carlo) methods, renders evidence of a “segregation effect” of the metric throughout the network; i.e., usually networks are nonfully-random media, but there exhibit regions of reunited elements with extreme values of the metric. This description renders more detailed-dualinformation about the distribution of the metric, but most of all gives precious information about the morphology, or the precise sequence of values of the metric throughout the network, statistically expressed. The term twofold description is related to the consideration of two kinds of elements, as is the case for structures in adsorption and catalysis that we will deal with, but the treatment could also be multivariate, as for arboreous aggregates.l3
Porous Networks Model of the Porous Network. This is the originall and perhaps the clearest application of our method. Sites are holes, antrae, or cavities while bonds are capillaries, passages, or merely windows communicatingsites. These elements alternate inevitably to form the porous network. Each bond delimits two sites, while the connectivity of a site is given by the number of bonds meeting into it. A regular structure, from a topological point of view, is that having all sites with the same connectivity. It is possible to treat both regular and irregular networks by means of our development. The size of a site is approximated as the radius of the sphere inscribed within the cavity, while the size of a bond is the radius of the inscribed circle at the minimum crosssection of the passage. Then, in spite of the very complex geometry that these entities could possess, sites are idealized as hollow spheres while bonds are hollow cylinders. Real capillary networks are characterizedby a restriction that is absent in other kinds of networks, this instituting the construction principle for this particular case: a site can never be smaller in size than any of its delimiting bonds. Overlap between site- and bond-size distributions becomes then an important parameter controlling the morphology of the network. Structures with a negligible degree of overlap allow element sizes to distribute fully at random throughout the network, since the construction principle could never be violated under this circumstance. On the other hand, when overlap is considerable topological-sizecorrelations arise in such a way that there form
Langmuir, VoZ. 9, No. 10,1993 2749
Figure 1. Representation of a porous network with c = 4, calculated from a Monte Carlo method. Sites and bonds are represented as circles and cylinders, respectively. The largest elements have no filling pattern, elements of intermediate size are hatch crossed, and the smallest entities are filled. The size segregation effect is evident.
regions constituted by large elementsalternated with other smaller elements. Figure 1shows, from a Monte Carlo calculation for a medium degree of overlap, a structure depicting such alternated regions. A t the limit when overlap tends to completion, there emerge extended “homotattic” regions of homogeneous size for both sites and bonds, and in order to find two elements with a considerable difference in size, it is necessary to travel extensively across the network. Now, a medium degree of overlap appears to be the rule rather than the exception for natural and modified porous solids, so that consideration of size correlations between neighboring elements is unavoidable. The studies made by several researchers on capillary condensation and evaporation, imbibition, mercury intrusion, immiscible displacement, etc. should be considered as valid only for the case of a fully-random structure. In particular, the model in which cylinders (bonds) of totally random sizes join together to form the network is completely unrealistic, since site sizes are subjected to severe constraints while verisimilitude would require that randomness be shared alike between sites and bonds. In the model in question site sizes are completely disregarded, while in fact they are quite important for the structuralization of the network. Application to Capillary Condensation and Evaporation. Even though the adsorption potential has a great importance on this matter, for the sake of simplicity it will not be considered here. The Kelvin equation15provides the critical mean radius of curvature, Rcr, as a function of the relative vapor pressure. Such a radius and the meniscus shape determine (15) Thomson, W.Philos. Mag. 1871,42,448.
2750 Langmuir, Vol. 9,No. 10,1993
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together the critical size of the element for a phase transition to occur,5*6J6the most delicate factor being the latter one. As a hemispherical meniscus is twice as curved as a cylindrical one of the same radius, a factor of 2 is present all along the discussion of capillarity. Classification of porous materials within five types5 according to the relative positions of their sits- and bondsize distributions leads to a better understanding of the morphological aspects of the porous medium as well as to an assessment of the different mechanismsarising during capillary condensation and evaporation. For each one of these types of materials, relevant characteristics can be recognized in their hysteresis loops. Interactions between sites and bonds are present during capillary condensationas well as during evaporation. Sites "opened at several poles", a novel conceptof our treatment, are the counterpart of what is very popular for bonds: "bonds opened at both ends". These characteristics drastically control the behavior of phase transitions. Let us assume, for the sake of simplicity, that bonds lead directly to the free vapor phase. During condensation then, a sequential filling, according to their size, can arise for them, in accordance with their cylindrical geometry. In this way, bonds fill on their own. On the other hand, it is impossible for a site to fill on its own unless the c surrounding bonds have been previously filled with condensate. The requirement to fulfill in this case is clear: all these bonds must possess radii in such a way that r l , r2, r, C R/2, R being the site radius. If several bonds remain unfilled, the meniscus located in the pore lacks continuity, and its advancement to fill completely the site with condensate is impossible, even if this element is in a saturated state. In the event of only one of the c bonds being devoid of condensate, the meniscus can still advance straightforwardly into the site, filling not only the site but also the remaining empty bond. Now, if one of the bonds possesses a radius rl equal to that of the site, this fills reversibly only if r2, ... rc C R / 2 . For any other case, condensation in the site is controlled by the largest bond among the remaining r2 to re. Anyway, a hysteretic behavior would always be inherent to bonds labeled as r2, r,. Condensationand evaporation in porous networks obey the above arguments, but might be complicated because of the possibility of the existence of many different menisci paths within the network. Quinn and McIntosh17 were the first to stress the importance of this pore-blocking effect during evaporation. Everett18 and Barkerlg gave an explanation of the fundamental aspects of it. More recently, assisted and hindered transitions arising from cooperative effects all along the network have been pointed out by Morioka and Kobayashi,2O and by Mayagoitia et al.18 Cooperative behavior during condensation seems to be the rule rather than the exception. In summary, these phenomena obey the following rules: Condensation in a site of size Rs takes place, in principle, when Rs I Rcr. However, at least c - 1of its bonds must be already filled with condensate, in order to guarantee the advancement of the vapor-liquid meniscus.
Condensation in a bond of size Rg occurs independently if Rg I Rcr/2,or in an assisted manner, provided that at least one of the delimiting sites has been filled by liquid and& IRcr. Starting from a state of complete saturation, invasion of vapor to any element requires a continuous path of vapor from the bulk vapor phase to the entrance of such an element. For bonds, an additional requirement is that RB 1 Rcr. Consideration of all these mechanisms leads to the certainty that morphology, or the precise sequencein which element sizes distribute throughout the network, "...controls, ...to a major extent, ...the condensation-evaporation characteristics".21 Application to Textural Determinations. The analysis of pore structure from adsorption hysteresis data should render the twofold size distribution of sites and bonds, from which, in principle, any textural property could be estimated. Much effort has been made to describe porous media by means of several parameters. Among them, the socalled "pore-size distribution" could help substantially to reach this goal, since other important properties, such as for instance the "cumulative" specific surface area, can be calculated from it. Wheeler22proposed an estimation of the pore-size distribution from a mass balance of the condensate leaving the porous structure, along the descending boundary curve of capillary evaporation (DBC) or "desorption branch". The first calculations were carried out by Shull,23and a major development was achieved by Barrett et al. in 1951.24 Since then, a great number of refinements have been introduced by many researchers. Initially, it was thought that the DBC would render better results than those from the ascending boundary curve of capillary condensation (ABC)or "adsorption branch". This choice seems reasonable, if only because the hemispherical menisci present in the DBC are more stable than the cylindrical ones in the ABC. The latter distinction constitutes the basis of the "delayed meniscus theory" of Foster26 and Cohan.26 However, Cranston and Inkley27 proved that, in many cases, the ABC gives better results than the DBC, taking as criterion of consistency the close agreement between the BET and the cumulative surface areas. At that time, however, it became evidentl7 that a pore-blocking effect during capillary evaporation would appreciably distort the results of the pore-size distribution obtained from the DBC, hence providing further support to the election of Cranston and Inkley, and from then on, the percolation process consisting in vapor invading the originally liquid-filled structure has been studied by several methods.28 The contemporary task on this subject, in very general terms, is the following: to determine how either the ABC or the DBC or both can be properly analyzed, Le., how menisci geometries and cooperative phenomena should be introduced to obtain the pore-size distribution curve. A surprisingly elegant and concise answer to the above quest is reached when the idea of the twofold size distribution is introduced, at least for the simplest types of porous materials.7 For type I (a case of zero overlap, where the size
(16) Mayagoitia, V.; Rojas, F.; Kornhauser, 1. J. Chem. SOC., Faraday Trans. 1 1985,8I, 2931. (17) Quinn, H. W.; Mc Intosh, R. In Surface Actiuity; Schuiman, J. H., Ed.; Butterworthe. London, 1957; Vol. 2, p.122. (18) Everett, D. H. In Structure and Properties of Porous Materials, ColstonPapers;Everett,D. H., Stone,F. S.,E&.; Butterworh London, 1958; Vol. 10, p 117. (19) Barker, J. A. In Structure and Properties of Porous Materials, ColetonPapers;Everett,D.H., Stone,F. S.,Eds.;Butterworh London, 1958; Vol. 10, p 125. (20) Morioka, Y.; Kobayashi, J. J. Chem. SOC. Jpn. 1979,2, 157.
(21) Everett,D.H.In TheSolid-GasZnterface;Flood,E.A.,Ed.;Marcel Dekker: New York, 1967; Vol2, p 1083. (22) Wheeler, A. Discussed at the American Association for the Advancementof Science,Conference in Catalysis at Gibson laland, 1945. 1948, 70, 1405. (23) Shull, C. G. J.Am. Chem. SOC. (24) Barrett, E. P.; Joyner, L. G.; Halenda, P. P. J. Am. Chem. SOC. 1951, 73, 373. (25) Foeter, A. G. Trans. Faraday SOC.1932,28,645. (26) Cohan,L. H. J. Am. Chem. SOC.1944,66,98. (27) Craneton, R. W.; Inkley, F. A. Ado. Catal. 1957,9, 143. (28) Mason, G . Powder Technol. 1984,39,21.
...
...
Heterogeneity Effectsin Adsorption and Catalysis distributions lie very far from each other; the smallest site is more than twice the size of the largest bond, the hysteresis loop being extremely broad), hemispherical menisci must be considered for both ABC and DBC. There is no reason to expect any coincidence between the information drawn from the ABC (the site-size distribution) and the DBC (the bond-size distribution). For type I11(overlapremains null, but now distributions come very close, so that the size of the biggest site is less than twice the size of the smallest bond; examples of these solids are model globular carbon samples;29because of intense cooperativeeffecta which arise during condensation and evaporation, hysteresis loops of these materials are narrow and both ABC and DBC are steep), both curves give the bond-size distribution, and there should be a coincidence of results. The site-size distribution can no longer be found from the boundary curves, but perhaps it could be estimated from scanning data, which are selective with respect to site sizes (see ref 6, Figure 212). For type V, (overlap tends to completion; in this highly segregated structure there develop homotattic ensembles, where all elements have the same size, behaving as independent domains), both the ABC and the DBC lead to the unique size distribution. The method of Barrett et al. is valid only for this case. Finally, let us propose some guidance to researchers. A general analysis of adsorption data to determine the textural characteristics of a given solid is so difficult to perform that, from a practical point of view, the following procedure is recommended. First, it is necessary to decide, according to the characteristics of the hysteresis loop, which type of porous structure the solid more closely resembles. Second, the analysis should be undertaken accordingto the particular equations which correspond to such a type. Additional information arising from other sources can be very useful to ascertain the value of c and to obtain some other qualitative inferences.
Adsorbent Surfaces Model of the Adsorbent Surface for Physisorption. Over the past years models of adsorptive surfaces have been stated in a rather crude manner, attention being exclusively focused on the energy distribution of the traditional adsorption sites. No recognition at all has been given to a second kind of surface elements: the adsorption bonds. These entities, which are the maxima or saddle points lying in the pathway between each pair of neighboring sites, are however certain to have a very strong influence on the energetictopology of the adsorbent surface if only because they are the links that contiguous elements share together. In reference to previous attempts to model heterogeneous surfaces that have been reported in the literature, it can be pointed out that all these proposals could be accommodated within one of the followinglines of thought with respect to the energetics of adsorption: The adsorption energy is randomly distributed throughout the surface, so that there are no energy correlations between neighboring sites. Conversely, there may arise very strong energetic correlations between contiguous sites in such a way that the surface structuralizes as an ensemble of homotattic domains (i.e., a collection of homogeneous substructures each constituted by elementswith practically the same energy), a model proposed by Ross and Oli~er.3~ (29) Everett, D. H.; Rojaa, F.J . Chem. SOC.,Faraday Trans. 1 1988, 84, 1456. (30) Ross,S.;Olivier,J. P. On Physical Adsorption; Interscience: Ne* York, 1964.
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Langmuir, Vol. 9,No. 10, 1993 2751
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Figure 2. Variation of the adsorption potential energy of sites along a linear trajectory across the adsorptive network. Calculations were made from a preestablished distribution of adsorptive sites and bonds. A more realistic case perhaps is one in which a certain degree of correlation is introduced between energies of adjacent sites, a situation developed by Ripa and Zgrablich.31 Having reflected on all these previous attempts and recognizing the correlating function performed by the adsorption bonds, it can now be established that a satisfactory model of a heterogeneous adsorbent surface should necessarily include two basic features: (i) consideration, along with the energy distribution function for the adsorption sites, of a second energy distribution function for the adsorption bonds and (ii) compliance to two self-consistency laws, both emerging from the fundamental condition establishing that any site must be in a deeper energy level than any of its surrounding bonds. The abovefeatures allowthe morphology of topologically correlated structures to be described as adsorbent surfaces with the highest degree of verisimilitude (likelihood). A self-consistentpicture of a heterogeneous surface must then fulfill the following construction principle: "the absolute value of the adsorption energy of a site is greater or at least equal to the absolute value of the adsorption energy of any of its own delimiting bonds". The principle plays an outstanding role in the description of heterogeneous surfaces. Fulfillment of the construction principle throughout the network promotes the apparition of an energy segregation effect,this effect being the result of a directional action in which elements of resembling energies meet; zones of these characteristics alternate successively across the network. An extreme situation when the segregation effect has its maximum intensity is when the overlap between the energy distributions is nearly complete. Thus, only a consideration of the twofold energy distribution allows a proper assessment of the surface topology of heterogeneous adsorbents. The analytical aspects of this description have already been developed elsewhere.8~9 Simulation of heterogeneous surfaces by means of Monte Carlo methods provides the evidence foreseen about the existence of (i) the energy segregation effect, stronger as the overlap between the site- and bondsize distributions increases, and (ii) the periodical alternance in the morphology of the network. The usefulness of Monte Carlo (digital) methods is remarkable, since the analytical treatment of extended regions of the substrate is quite impossible. Figure 2 illustrates the variation of the adsorption potential energy of sites along a linear trajectory across an adsorbent surface; a Monte Carlo simulation method based on a preestablished twofold distribution was used to perform the calculations. Of particular interest in this kind of representation would be the apparition of the segregation effect a t high values of the overlap. The morphology of the adsorptive surface exerts an enormous influence upon the structuralization of the adsorbed phase and surface transport. (31)Ripa, P.;Zgrablich, G. J. Phys. Chem. 1976,79, 2118.
Mayagoitia et al.
2752 Langmuir, Vol. 9,No. 10,1993
from site
Figure 3. Potential energy curve for chemisorption on a site. e. is the adsorption energy,e, is the activation energy of adsorption, and ea is the activation energy of desorption.
Model of the Adsorbent Surface for Chemisorption. With respect to chemisorption there has always been a profound interest in correlating the energy of adsorption (e,) with the activation energy of adsorption (e,), both quantities being useful to account for phenomena such as heterogeneous catalysis, surface characterization, or surface reactive separations. However, it should be pointed out that no common relationship could be found for the wide variety of heterogeneous adsorbents as each one displays a particular chemical nature. In order to describe and characterize the solid surface and predict equilibrium and the rate of activated chemisorption, a general method (based on a statistical development that resembles very closely that already applied above for physisorption) is proposed to assess the correlation between the two incumbent parameters. First of all, the two quantities in question should be identified unambiguously. Second, the foundations of a “twofold” description (i.e., in terms of a dual distribution of energies) should be set: a construction principle and two selfconsistency laws allow the structuralization of a selfconsistent and verisimilar network representing the chemisorbent surface. Even if lateral interactions between sites were primarily considered as absent, there arises an interesting kind of correlation between them, owing to the activation energies they share together. Finally, the consequences of such correlations on the behavior of activated chemisorption and desorption would be briefly discussed. The adsorbent surface is visualized again as an interconnected network constituted by sites and bonds. A dualenergy distribution is required once more as this conditions the topological structuralization of the adsorption energy throughout the surface. Chemisorbed entities are considered as localized in well-defined sites, each of these being characterized by a pair of values of the adsorption energy and the activation energy of adsorption. For each site there also exists another important parameter related to the latter one, the activation energy of desorption ( E d , in such a way that €8 = e, + e, (see Figure 3). A construction principle, arising from the very definitions of ea and €6, is stated as e, 5 e8 for every site. Any chemisorbent network should observe this principle and if so is termed self-consistent. Otherwise it would be devoid of physical reality. Again, in order to fulfill the construction principle, two self-consistency laws must be observed. (i) The first law concerns a general relationship between the overall distributions of the energy of adsorption and the activation energy of desorption, and guarantees a sufficient provision
of e, values as to have a proper apportionment of them among all of the E6 values. (ii) A second law arises which holds locally for each site and avoids the union of an inconsistent pair of values of e, and €8. This law supplies the condition of sufficiency complementary to that of necessity imposed in the first law. The model of the surface outlined here has been termed verisimilar, i.e., that constructed from a minimum number of constraints (the construction principle being the only restriction introduced), or that expected to be the closest in appearance to a real system, in the lack of other information (i.e., additional constraints, especially those relative to the particular physicochemical nature of the surface associated with the history of its preparation or any other relevant aspect of it). The construction principle is based on a relationship between €69 the whole, and ea, the fraction (this principle could have been equally applied to €6 and e, instead of ea). Nevertheless, the e, distribution curve could be readily estimated from the latter two. Chemisorption. On the basis of the model of the adsorbent surface for chemisorption discussed in the preceding section, the following ideas about the evolution of this process can now be introduced. In general during a chemisorption process, throughout the surface there would appear (i) favorable sites for adsorption, Le., those with a high e, and a low e, and (ii) unfavorable sites for adsorption, i.e., those bearing a low and a high e,, so that only a fraction, perhaps minimal, of sites would be active. However, special cases of energy distributions may lead to the occurrence of sites with (iii) a low e, and a low 6. and (iv) a high e, and a high e.. If the energy barriers associated with bonds are low enough, a cooperative mechanism may proceed molecules chemisorb first in sites of type iii; afterward they may migrate to sites of type iv. For desorption, molecules may migrate from sites of type iv to sites of type iii and then desorb. This time all sites would be active. In order to prepare the field for the discussion of reaction kinetics in the adsorbed state in the forthcoming section, some important aspects relative to the description of the chemisorbed phase when multicomponentadsorbed phase is present will be dealt with in brief. In particular it will be treated as the case of the activated adsorption of a binary mixture of componentsA and B over a solid surface. For the sake of simplicity it will be assumed that lateral adsorbate interactions are negligible. The justification for this assumption is that, thinking in terms of a localized chemical adsorption, there are strong interactions in the form of chemical bonds between the adsorbent and the adsorbate, and then the lateral interactions, if not related with the appearance of additional chemical bonds, are of lesser importance. For the localized chemisorption of a binary mixture two extreme cases can be visualized. Noncompetitive adsorption. There are two types of sites, a and 8, for the adsorption of A and B, respectively. Each type possesses its own energy distribution. Every bond is the energy barrier separating sites of different types; thus, there exists a unique distribution of bonds which is common to both types a and 8. The energy correlations arising between neighboring sites can be estimated by means of the statistics between the energy distribution for each type of site and the energy distribution of bonds. Results of this statistical procedure such as the probability of having a molecule A adsorbed on a site a and a molecule B adsorbed on an adjacent site 8 are crucial for the development of reaction rate expressions in the adsorbed state.
Heterogeneity Effects in Adsorption and Catalysis Competitive adsorption. Each site can fix either A or B, but the twofold energy distribution can just be referred in terms of the adsorption energy of component A. The adsorption energy distribution of sites (based solely on substance A) together with the energy distribution of bonds determine the topology of the adsorption potential throughout the network. The transformation of the adsorption energy of B in terms of A can be made as follows. The probability density to have an adsorption energy, e ~ , for component B, is calculated by making reference to the energy EA* that would correspond to the same site if it were occupied by A, in accordance with a “mixing rule”, as for instance of a Polanyi type. Nevertheless, it must be recognized that applications of this rule are usually made under a context that certainly involves forces not as specific as in this case. Energy correlations between neighboring sites can then be calculated by means of a statistical procedure based on the dual energy distribution. The surface coverage of the ith substance can be expressed in terms of its own partial pressure and adsorption equilibrium constant together with the partial pressures and equilibrium constants of the other components. The global coverage of each species on the adsorbent can be obtained from an average of the degrees of coverage over sites with a different adsorption energy throughout the surface. If the lateral interaction is negligible, then the adsorption of every species is independent, except for the aspects of surface competitiveness and stoichiometry. Finally it is important to say that any equilibrium state of the surface could be described in detail by means of a “domain complexion diagram”, which indicates the amounts of each surface species adsorbed over sites with different adsorption potential energies, for given values of temperature and partial pressures of the diverse components. The diagram would be analogous to a representation previously proposed for capillary processes in porous media.sJ3 In this case, however, if it is considered that chemisorption takes place only on sites, the bond complexion diagram would be empty. Reaction Kinetics in the Adsorbed Phase. The principles relative to the kinetics of reactions occurring in the adsorbed phase can be stated on the grounds oF2 (i) a description of the topology of the adsorption potential over the catalytic surface and (ii)a probabilistic assessment of the abundance of different chemical species adsorbed on adjacent sites, this proximity of the reactive species being a necessary condition for the reaction to take place. In spite of the good agreementfound frequently between experiment and adsorption models that assume homogeneous surfaces (e.g., Langmuir’s), the efforts to confront experimental data with models such as that of Freundlich, demonstrate the enormous interest of the surface heterogeneity. Multiple problems arise with respect to this quality: kinetics over some sites may follow one mechanism, while over other sites there may arise a different one, as suggested by H a l ~ e y In . ~this ~ treatment, however, this complexity will be omitted. In the restrained field where chemisorption can be considered as localized over some well-defined potential wells (sites) on the surface, the foundations of the treatment outlined here seem absolutely clear: (i) the establishment of a construction principle of the adsorptive network, as it has just been made in a previous section, and (ii) the development of a probabilistic approach relative to the occupancy by reactive species of neighboring sites. As said before, no matter which physicochemical processes had lead to the structuralization of a particular ~~
(32) Halsey, Jr., G.D.J. Chem. Phys. 1949, 17, 758.
Langmuir, Vol. 9,No. 10,1993 2753 surface, the construction principle must be fulfilled. Even more, for any reaction sensitive to the particular surface structure, the kind of calculations involved in the probabilistic approach developed in (ii) are essential. On the subject of surface reactions several mechanisms have become classical, notably that of Langmuir-Hinshelwood. The aim here is to set up more specific mechanisms compatible with our particular model of the heterogeneous adsorbent surface. Any method proposed to deal with the kinetics of surface reactions should involve the following steps: (i) a description of the topology of the adsorptive potential, (ii) an assessment of the chemisorbed phase, and (iii) an establishment of the mechanisms leading to the reaction rate expressions. In the present treatment the adsorbed statewill be rather visualized as a localized one. The model, however, could be extended to treat nonlocalized adsorption. Smoluc h ~ w s k has i ~ ~issued interesting ideas in this direction, but this point will not be pursued any further. Monomolecular reactions though occurring in accordance with the heterogeneity of the adsorptive network, are not affected by the precise structure of the surface. The same is valid with processes following the Rideal mechanism. They correspond to “easy”reactions.% These cases are of no concern in this work. Conversely the “difficult*reactions, involving the participation of several sites in the reaction mechanism, are sensitive to the topological correlation of energies on the surface. Thus, the following distinction can be established. Topological properties are those which depend on a twofold distribution of energies. Nontopological properties, conversely, depend solely on a unique distribution. To treat easy reactions, it is just necessary to consider the lone energy distribution of sites while a bimolecular reaction, for example,requires consideration of a twofold energy distribution. Given the limited extension of this work, a bimolecular reaction will be the only case to deal with. It will be assumed that the adsorption of the diverse components (substances A and B)has already attained the equilibrium, and that the surface reaction constitutes the rate-limiting step. It is classical to consider the rate expression of a bimolecular reaction A + B C as R = keA8B where R and k are, respectively, the rate and the rate constant of the global reaction and eA and flB are the mean coverages of species A and B over the surface. However, this rate expression can be perfected if the topology of the adsorption potential is considered. Two extreme cases are now discussed. Noncompetitive adsorption. It is convenient to define an “active site”. This term will refer to a site of a certain type possessing a first-order neighboring site of the other type. Let us consider that for the reaction to occur around a site a,this must be occupied by a reactive molecule A; furthermore, at least one of its ca first-order neighbors (sites of type 8) must be occupied by a molecule of B, where ca is the connectivity of sites a, i.e., the number of sites 8 surrounding a site a. The same reasoning is applicable to sites 8. The joint probability of these two events leads directly to the statement of the local reaction rate per active site a or fl in terms of the coverages and the corresponding connectivity. The global reaction rate per site a or 0 is then calculated from an average over all possible values of the adsorption energy for the type of site in question.
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(33) Smoluchowski, R.Memoria del VI Congreso de la Academia Nacional dezngenierfa (Proceedings ofthe VZSymposium oftheNational Academy of Engineering of Mezico); MBxico, 1980; p 231. (34) Boudart, M. Adu. Catal. 1969,20, 153.
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2754 Langmuir, Vol. 9, No.10,1993
Competitive adsorption. The site of interest possesses an adsorption energy E Afor the species A, and an energy Q for the species B. Such a site may be occupied by A, B, or C in diverse forms (C may be fixed on one or on several sites),or could be empty. In order to be active, the site must be occupied by A or B. If the site contains A, there must be at least one molecule of the species B fixed to one of its c first-order neighbors, where c, the connectivity of the adsorbent network, is assumed constant. The same argument is applied to the case in which B is the adsorbed component on the site of interest. The joint probability of the former two events can be easily set up, leading to an expression for the local reaction rate per site in terms of the coverage of the site where A adsorbs, 6A(EA), the coverage of the neighboring site with B, ~ ( E B ) and c. The global reaction rate per site is obtained by performing an average over all possible values of the adsorption energy assumed by the site of interest. The final rate expressions will reflect, of course, the expected topological information, and will be coupled with the surface stoichiometry and the adsorption isotherm of each species at every adsorption energy. The rate expressions in terms of the observed partial pressures could be obtained from all these relationships. The surface rate equations as a function of the surface coverages lead to several limiting cases of interest, finding kinetics of first, second, or complex order, according to the relative abundance of every one of the components present in the adsorbed phase. It will only be mentioned in the following simple result: if there is an associative A2, at the two extreme cases of overlap reaction 2A between site and bond distributions and when 6A 0, the following results are obtained for the reaction rate in the global adsorptive network
-
-
~ ( 8 ) ~ zero overlap R={
-
&(e2)
for complete overlap
where 8 is the mean surface coverage of adsorbed A molecules, O2 is the mean value of the square of this coverage, and within k,the rate constant, is involved the connectivity of the adsorption network. A study of the influence of the surface structure upon the order of reaction has been sketched. The effect of such a structure on the rate constant k, something that seems very interesting to pursue, should also be investi-
gated. The applications of calculations such as those outlined in this section are practically of an unlimited extent.
,
Discussion Everett36 refers to the work of Rojas at Bristol.% Rojas, following the ideas of Bligh and Everett, succeeded to produce a model porous material having a remarkable regularity, consisting in a rhombohedral-packedstructure of monosized, nonporous spherical particles having a diameter in the range 100-200 nm. The adsorptiondesorption isotherms of benzene vapor at 25 OC were determined. The experimentalascending scanningcurves were compared with those calculated by using the theory presented in this work. The agreement was excellent. Many more confrontations with experimental data would be necessary to validate further this theory: several capillary processes in well-defined solids corresponding to the five types of porous structures we have proposed should be analyzed. However, special care must be taken during the acquisition of these data. Recently, EverettF7 by confronting experimental data from several of the most prestigious laboratories in the world, relative to the adsorption and desorption boundary curves of nitrogen on the same series of silica gels, perceived serious discrepancies among the results of the different sources, far exceeding the experimental precision of the measurements. In the light of this founding, one is tempted to wonder if any quantitative confrontation with experiment in the actual state of the art could validate any theory. Conclusions The twofold treatment described in this review allows a convenient understanding of the morphology of heterogeneous structures where a sorption or catalytic phenomenon takes place. Consideration of the processes occurring in heterogeneous substrates simulated on the basis of the above description reveals the existence of several unusual effects and mechanisms. Acknowledgment. This work was supportedand made possible by The Ministery of Public Education of MBxico (SEP). (36) Everett, D. H. In Principle8 and Applications of Pore Structural Characterization; Haynes, J. M., h i - D o r i a , P., Eds.; Arrowsmith Bristol, 1985; p 66. (36) Rojaa, F. Ph.D. Thesis, Bristol University, Brietol, 1982. (37) Everett, D. H. In Principles and Applications ofPore Structural Chracterization; Haynes, J. M., h i - D o r i a , P., E&.; h o w s m i t h Bristol, 1985; p 143.
