Twofold Description of Topological Disordered Surfaces†

Vicente Mayagoitia,*,‡ Fernando Rojas,‡ Isaac Kornhauser,‡ E. Ancona,‡Giorgio Zgrablich,§,| and Roberto Jose Faccio|

Departamento de Quımica, Universidad Autonoma MetropolitanasIztapalapa, ApartadoPostal 55-534, Mexico 13, D.F., 09340 Mexico, Centro Regional de Estudios Avanzados

(CREA), 5700 San Luis, Argentina, and Departamento de Fısica, Universidad Nacional deSan 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 typesof adsorption sites, each kind bearing a given connectivity (i.e., possessing a given number of delimitingbonds or energy barriers between sites), and the principal conclusion is that, formost of adsorbent surfaces,concomitantlywith an energy segregation effect (sites andbonds group together formingalternated regionsof very high and very low adsorption energies), a connectivity segregation effect can arise too (connectivitydistributes throughout the network sensibly obeying adsorption energy correlations).

Introduction

An adsorbent surface can be conveniently described asanetworkof twokinds of alternated elements: adsorptionsites (or minima in the adsorption potential) and bonds(potential energy barriers between sites). While particlespreferentially adsorb on sites, the function of bonds is tostructuralize themorphology of the surface, these entitiesthen becoming important in determining the equilibriumand dynamic behavior of the adsorbed phase.1,2

Recently,3 a treatmenthasbeenproposed fordisorderedstructures, inwhichgeometricheterogeneity (thedistancebetween adsorption sites, a parameter controlling thestrength of lateral adsorbate interactions, is no longerconstant but can exhibit a random character) is takeninto account. Indeed, many surfaces may also display astrong topological heterogeneity, which means that theconnectivity, C (the number of delimiting bonds leadingto first-order neighboring sites), changes from site to sitethroughout the network. The task in this contribution isto describe adsorbent surfaces of such kind.Indeed,what amoleculewould “see”whenapproaching

the surface is illustrated in Figure 1, where black circlesare adsorption sites and the white space is unavailablefor adsorption. These clusters of adsorption sites couldcorrespond to chemically eligible regions of the surfacefor adsorption (such as deposits of an active phase on acatalytic surface), while the atoms lying at thewhite zoneare not suitable. Sites located deep in the adsorptionregion are well connected (with C ) 6 as maximum forthis example), while those along the boundaries of thecluster are scarcely surroundedby other sites. Some sitesstand isolated, in which case adsorption, but not surfacemigration, is possible.For the sake of simplicity, geometrical heterogeneity

will be omitted in the subsequent treatment (note thataccording toFigure1, it is possible to represent topological

heterogeneity without introducing geometric heterogene-ity, so that thesemorphologiesmayexist in the realworld),even if in the most general case a surface should displayall kinds of energetic, geometric, and topological hetero-geneities.Since the treatment will remain within the frame of

our site and bond theory of heterogeneous adsorbentsurfaces, we will first present a synthesis of our previouswork about ordered lattices, necessary to understand thesubsequent original development for the case of variableconnectivity.

Ordered Lattices

The statistical treatment termed as the “site and bondtheory” or “dual theory” has been principally applied todescribe the morphology of (i) heterogeneous adsorbentsurfaces,1,2 and their implications in physical adsorptionequilibria, surfacediffusion, and surface characterization,

† Presented at the symposium onAdvances in theMeasurementandModeling of Surface Phenomena, SanLuis, Argentina, August24-30, 1994.

‡ Universidad Autonoma MetropolitanasIztapalapa.§ Centro Regional de Estudios Avanzados.| Universidad Nacional de San Luis.X Abstract published inAdvanceACSAbstracts,January1, 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.

Figure1. Topologically irregular (while geometrically regular)surface network of adsorption sites (in black).

207Langmuir 1996, 12, 207-210

0743-7463/96/2412-0207$12.00/0 © 1996 American Chemical Society

and (ii) porous media,4 treating such topics as capillarycondensationandevaporation,5 textural determinations,6etc. It is of course the terminology dealing with surfaceswhich 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 thesurface (at an infinite distance from the surface, E ) 0).Then sites correspond to theminima ofEwhile bonds arelocated at every saddle point of potential energy betweeneach pair of neighboring sites. The “adsorption energy”will be represented as ε ) |E|.A “construction principle” originating from the very

definition of “sites” and “bonds” states that “The energyof a site, εS, is greater or at most equal to the energy, εB,of any of its own delimiting bonds.” This principle playsa central role in the description of heterogeneous sur-faces.1,2If FS(ε) and FB(ε) are the normalized energy distribu-

tion functions of ε for sites and bonds, then S(e) and B(e)represent, respectively, the probabilities for a site and abond to have values of ε lower than or at most equal toe:

F(εS∩εB), the probability density for the joint event offinding a given site of energy εS and concurrently a givenone of its bonds with an energy εB, can be expressed as:

where φ(εS,εB) is a correlation function to be explainedafterward.In order to fulfill the construction principle, two self-

consistency lawsmustbe observed. The first one concernsa general relationship between the overall distributions:

since for a given site distribution, enough bonds of lowenergies 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 εS and εB forcontiguous elements:

Conversely, for the correct condition of having εS g εB,it has been found1,2 that the function φ can be written as

Consequently, fora sitewithacertainadsorptionenergyεS, the conditional probability density to find the energyεB for a given one of its bonds is

As the overlap between distributions becomes consider-able,manysiteshave εvalues smaller than thoseof certainbonds (these bonds of course cannot be thedelimiting onesof such sites), and in order for the construction principleto remainvalid, an “energy segregation effect” arises. Thisphenomenon is the result of a favorable influence whichpromotes the reunion of elements of similar energies (aswas already visualized by Ripa and Zgrablich7). Whenoverlap is nearly complete, as the segregation effect is thedominant factor, there appear “homotattic” patches(whereinall elementspossess the sameenergy), themodelof Ross and Olivier8 being appropriate at this limit.Nevertheless, it is worth while to point out that in thiscase of almost complete overlap, the energy values forboth sites and bonds within each one of the homogeneousdomains become about the same, this favoring mobilerather than localized adsorption.Thus, a proper assessment of the surface topology of

heterogeneous adsorbents is only possible by consideringthe twofold energy distribution.

Disordered Lattices

Muchmore information is needed to dealwith irregularnetworks. It will be assumed that all bonds delimit twosites,while a site of the ith typepossessesCibonds. Thereexists a fraction xi(ε) of sites of energy ε belonging to theith type. xi could correspond to a discrete distribution,e.g., Poisson’s, in such a way that the mean connectivityof sites of energy ε is

C(ε) ) ∑i xi(ε) Ci (7)

and

∑i xi(ε) ) 1 (8)

Themean connectivity of thewhole network is given by

Ch ) ∫0∞C(ε) FS(ε) dε (9)

We first consider that the number of bonds (or, moreprecisely,half-bonds),NB(e), belonging toall sitesof energylower than e is

NB(e) ) NS∫0e C(ε) FS(ε) dε (10)

For the particular value e ) ∞ we find

NB(∞) ) NS∫0∞ C(ε) FS(ε) dε ) NSCh (11)

where NS is the overall number of sites of the network.Considering that each site of the ith type possesses Cihalf-bonds, a new relevant quantity, S′ is defined as theratio NB(e)/NB (e ) ∞):

S′(e) ) ∫0e{C(ε)/Ch } FS(ε) dε (12)

S′ represents the minimal fraction of bonds requiredto link all sites of assorted connectivities equal or lowerthan e.The first lawmust now bemodified to take into account

the possible dependence of C(ε) with ε. Let us considera given energy e. The fraction of sites (or bonds) havingan energy e or lower, S(e) (or B(e)), is still in accordancewith eq 1. However for irregular networks, it is S′(e)instead of S(e) that should be compared with B(e), the

(4) Mayagoitia, V.; Kornhauser, I. In Principles and Applications ofPore 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. InFundamentals ofAdsorption II; Liapis,A. I., Ed.; The Engineering Foundation: New York, 1987; p 391.Mayagoitia, V.; Rojas, F.; Kornhauser, I.J.Chem.Soc., FaradayTrans.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.; TheEngineeringFoundation: NewYork, 1991; p 563.

(7) Ripa, P.; Zgrablich, G. J. Phys. Chem. 1975, 79, 2118.(8) Ross, S.;Olivier, J. P.OnPhysicalAdsorption; Interscience: New

York, 1964.

S(e) ) ∫0eFS(ε) dε; B(e) ) ∫0eFB(ε) dε (1)

F(εS∩εB) ) FS(εS) FB(εB) φ(εS,εB) (2)

first law B(e) g S(e) for every e (3)

second law φ(εS,εB) ) 0 for εS < εB (4)

φ(εS,εB) )exp(-∫B(εB)B(εS) dB

B - S)B(εB) - S(εB)

(5)

F(εB/εS) ) FB(εB) φ(εS,εB) (6)

208 Langmuir, Vol. 12, No. 1, 1996 Mayagoitia et al.

available fraction of eligible bonds. Instead of eq 3, wehave

first law B(e) g S′(e) for every e (13)

in such away that for a given energy distribution of sites,there must be enough bonds of low energies to link suchsites of multiple demanding connectivities.In order to visualize how sites and bonds could be

interconnected to form the network, it is convenient tofollowaprocedure that has been outlined elsewhere.1 Themethod is now generalized to treat irregular networks:sites are sorted from lower to higher energies, preciselyin this order, and half bonds are assigned to them. Bondsmust obviously possess a lower or at most equal energythan the site in question. The randomness of thisassignation is then raised to a maximum, this maximumbeing only conditioned by observation of the constructionprinciple.This site and bond balance allows one to obtain a

network which is (i) self-consistent (i.e. complying withthe construction principle) and (ii) the “most verisimilar”(corresponding to themaximumrandomnessallowed, thenpossessing the maximal entropy, and leading to themaximum number of configurations). Equation 5 isgeneralized to

φ′(εS,εB) )exp(- ∫B(εB)B(εS) dB

B - S′)B(εB) - S′(εB)

(14)

Note that φ′ follows the same form as φ, this generali-zation merely undertaken by a reinterpretation of one ofthe terms insideφ (i.e.S′,which takes intoaccountvariableconnectivity).The conditional probability density for a site of a given

adsorption energy εS to find the energy εB for a precise oneof its bonds now becomes as

F(εB/εS) ) FB(εB) φ′(εS,εB) (15)

Asnotedbefore,when theoverlapbetweendistributionsis considerable, there arises an energy segregation effect.In the event of connectivity increasing with ε, the energysegregation effect should concomitantly lead to a “con-nectivity segregation effect”, since the former signifiesthe reunion of elements with bigger energies, and thesewould precisely adopt higher connectivities.

Discussion

The treatment of irregular networks differs from thatof regular substrata only if C(ε) depends effectively on ε.For example, an inspection of a Zachariassen model of aglass immediately reveals that this is the case. This isthe structure used by Benegas, Pereyra, and Zgrablich3to treat geometric heterogeneity. Instead, here we areinterested in topological aspects. In the Zachariassenmodel, there extends an irregular lattice constituted byrings of variable size, depending on the number ofparticipating siloxane elements. It is very clear that thenumber of neighboring rings of a given ring (i.e. itsconnectivity) is determined by the number of its oxygenatoms. If this arrangement is considered as a surfacestructure, each ring could be visualized as an adsorptionsite. Then, the bigger the rings are, the bigger is theconnectivity of the adsorption site.Now, the relationship between the size of the ring and

ε is in some way cumbersome to establish, but supposethat small molecules adsorb at the centers of the rings.If the ring is big, the molecule can penetrate and ε willbe important. Conversely, for small rings ε will be low

because themolecule interactswith fewer elements of thesubstratum. Then the energy segregation effect cor-respondingly involves a “connectivity segregation effect”,since the former signifies the reunion of elements withthe biggest energies, and these have precisely the highestconnectivities.Thisneweffectmustbe indeedof theutmost importance

when lateral interactions of adsorbed molecules arepresent, eq 15 allows a quantitative assessment of itsstrength. Even in the absence of these interactions, thiseffect may have important consequences upon otherrelevant aspects about the dynamics and equilibrium ofthe adsorbed phase, e.g. surface diffusion.As an example of the interesting changes introduced by

topologically disordered networks, we present an ex-tremely simple and illustrative case: suppose that thesite-energy distribution is uniform and that the con-nectivity C(ε) is an increasing linear function of the siteenergy, in such a way that from the lowest to theuppermost limits of site energies, connectivity increasesfrom 2 to 6, Figure 2 (note the connectivity axis).If connectivity were kept constant throughout all sites,

the limiting allowable distribution of the highest possiblebond energies would correspond to the same distributionas for sites, i.e. a caseof complete overlapwouldbepresent.This arbitrary situation is very interesting to analyze inthe light of variable connectivity. Now since the require-ments for bonds vary according to C(ε), the relationshipgiven by eq 3 for ordered lattices B(e) g S(e) no longerapplies, and then the first law B(e) g S′(e) for disorderedlattices, eq 11, is the one that is observed. Sites of lowenergies demand few bonds, so that the limiting distribu-tion of bonds still good enough to link thewhole collectionof sites, is givenby the linewithpositive slope representedin Figure 2, corresponding to B(e) ) S′(e). In this novelcase, complete overlap is in some way overcome. ShouldC(ε) decrease with ε, the antagonistic effect would occur,preventing a too high degree of overlap.Keeping the same site-energy distribution and C(ε)

situation as in the previous example, it is possible to

Figure 2. The limiting allowable distribution of the highestpossible bond energies (the line with positive slope) comparedwith its corresponding uniform site-energy distribution. Con-nectivity increases from 2 to 6 in going from the lowest to thehighest site energies (see text).

Topological Disordered Surfaces Langmuir, Vol. 12, No. 1, 1996 209

analyze straightforwardly the implications of variableconnectivity on topological energy correlations. Now thebond-size distribution will be considered the same as forsites (complete overlap). If thenetworkwere topologicallyregular, by virtue of eq 5 every bond should possess thesame energy as the sites it is linking, εB ) εS (homottaticpatches). However, in the present case, complete overlapwould be no longer a limiting situation of completecorrelation between εS and εB and then eq 13 allows aspan of values of εB for a given value εS. Figure 3 depictsF(εB/εS) as a function of εB for a constant value of εS

corresponding to the mean value of ε. The uniform site-bonddistribution and F(εB/εS) curves are bothnormalized.Future work should be directed to the prediction of

adsorption isotherms, the study of the structuralizationof the adsorbed phase, and surface diffusion in the caseof topological heterogeneity, as has already been done3for geometric heterogeneity. These are complementaryaspects of more complex networks. The whole field ofadsorption in lattice surfaces (coverage in terms ofadsorption site potential, lateral interactions, multisiteoccupancy by r-mers, kinetics of adsorption, temperatureprogrammed desorption, etc.) is going to be renewed forstructures 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 unavoidableand ever present, Monte Carlo representations of adsorp-tion-site trees and networkswill surely take amuchmoreimportant role. Intricate structures couldperhapsbe onlyvisualized by means of digital models. Then, reality willbe discerned from fantasy, since only those morphologiesable to have a representation in space are endowed of thepossibility to exist in the real world.

Conclusions“Connectivity segregation” is a fundamental property,

as important as “energy distribution” or “energy segrega-tion”, for the simulationand characterization of adsorbentsurfaces possessing a high degree of topological hetero-geneity. It is unavoidable to take into account thisessential characteristic displayed by many surfaces.

Acknowledgment. This work was supported andmade possible by the National Council of Science andTechnologyofMexico (CONACyT)underProjectNo. 5387-E, aswell as under the JointResearchProject: “Catalisis,Fisicoquımica de Superficies e Interfases Gas-Solido” byCONICET (Argentina) and CONACyT (Mexico).

LA9407065

Figure 3. F(εB/εS) as function of εB for a constant value of εScorresponding to the mean value of ε. The uniform site-bonddistribution and F(εB/εS) curves are both normalized.

210 Langmuir, Vol. 12, No. 1, 1996 Mayagoitia et al.