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Langmuir 1993,9, 2730-2736
Characterization of Energetic Surface Heterogeneity by a Dual Site-Bond Model J. L. Riccardo,. V. Pereyra,? and G. Zgrablich* Imtituto de Investigaciones en Tecnologh Quimica, INTEQUI, C.C. 290, 5700 Sun Luis, Argentina, and Centro Regional de Estudios Avanzados, Gobierno de la Provincia de Sun Luis, C.C. 256, 5700 Sun Luis, Argentina
F. Rojas, V. Mayagoitia, and I. Kornhauser Departamento de Qutmica, Universidad Autbnoma Metropolitana, Itztapalapa, A P 55-534,Mhxico 13D.F.,Mhxico Received November 5,199.P A dual site-bond model for the description of correlated heterogeneous surfaces is presented. This model assumes the surface as a network of adsorptive "sites" connected by "bonds" (saddle point energies). Site and bond energies are statistically described by their energy distribution functions and the adsorptive energy surface by the joint site-bond energy distribution which is expressed in terms of a correlation function. The latter is determined by allowing the maximum randomness degree admitted by the "construction principle". This principle states that the potential at an adsorptivesite must be deeper than that of any bond connected to that site. Energy correlations naturally arise when site-bond distributions overlap. A topologicalcharacterizationof these correlationsthrough an energycorrelationlength is proposed. This correlation length appears as a meaningful physical parameter. A transition between two welldifferentiated morphologies of correlated elementa upon variation of the overlapping degree of site-bond energy distribution is found. This fact is analyzed in terms of cluster morphology and percolation theory. A simulation procedure to simulate adequately a site-bond heterogeneous surface is developed.
Introduction Adsorption on heterogeneous surfaces is a phenomenon of great theoretical and practical importance. In the last two decades research has been mainly oriented to develop methods to get quantitative information about the energetic nature of the surface from sets of equilibrium adsorption isotherms.14 These methods, necessarilymake use of the available models for the local and the overall adsorption isotherm. However, the choice of a given local isotherm equation implies strong assumptions about the real nature of the absorbent. Up to the present most studies of adsorption on solid surfaces have assumed one of the three followingmodels of a heterogeneous surface:(a) patchwise model, the surface is assumed to be a collection of macroscopic patches, each patch containing sites of a given energy coming from a distribution; (b) random model, sites of equal energy are scattered totally at random over the surface without any spatial correlation; (c) medial model, a mixture of the two preceding ones, some adsorption sites of equal energy are grouped in large patches and the remaining ones are distributed at random in between the existing patches. Although the appropriate local isotherm could be inferred from a given overall isotherm in some special cases, t Present addrw: Johannw GutenbergUniversist Mainz, Institut ftir Phyeik, Postfach 3980, Staudingerweg 7, D-6600 Mainz 1, Germany. 0 Abstract published in Advance ACSAbstracts, August 16,1993. (1) Jaroniec, M.; BraUer, P. Surf. Sci. Rep. 1986,6, 65. (2) Roes, 5.;Olivier,J. P. On Physical Adsorption; Interscience: New York, 1964. (3) O'Brien, J. A.; Myers, A. L. J. Chem. SOC., Faraday Trans 1 1986,
81, 351. (4) Bratier, P.; Szombathely, M. V.; Heuchel, M.; Jaroniec, M. In Fundamentals of Adsorption; American Institute of Chemical Engineers: New York, Mersmann, A. B., Scholl, 9. E., Eds.; Germany, 1989; D r
166.
(5) Steele, W. A. J. Phys. Chem. 1963, 67, 2016. (6) Pierotti, R. A.; Thomas,H. E. Tram.Faraday Soc. 1973,70,1725. (7) Rudzinski, W.; Lajtar, L.; Patrykiejew, A. Surf. Sci. 1977,67,195. (8) Jaroniec, M.; Patrykiejew, A.; Borbwko, M. Prog. Surf. Membr. Sci. 1981, 14, 1.
the determination of the adsorption energy distribution is sometimes incorrect due to the limited character of the known models. Surfaces with short range correlations in adsorption energies are not represented by any of the cited models and consequently cannot be suitably characterized. A more general description for surfaces with intermediate topographies was introduced by Ripa and Zgrablich (the generalized Gaussian m ~ d e l ) The . ~ essential feature of this formulation is that the topography of the adsorptive potential can be characterized by a correlation length on the surface, which is a measure of the size of surface regions with highly correlated site energies. In the framework of this description cases a and b correspond to the two limiting topographic cases of infinite and null correlation length, respectively. Recently the influence of short range correlations (finite correlations lengths) on monolayer localized adsorption has been studied.lO These models of the heterogeneous surface have a common feature; they are site descriptions, i.e., they tacitly recognize the adsorptive sites as the main elements of the surface potential field. However, if any of these models for heterogeneous surfaces is used to study, for instance, the collective surface diffusion, saddle point energies (or activation energies for jumps) must be introduced.lOJl It has already been pointed out that a proper description of a heterogeneous surface must take into account at least two basic elements of the adsorptive field, namely, sites (adsorption energy minima) and bonds (saddle point) energies between nearest-neighbor adsorption sites).I2 In other words, besides the usual "site" energy distribution (9) Ripa, P.; Zgrablich, G. J. Phys. Chem. 1975, 79, 2118. (10) Riccardo, J. L.;Pereyra, V.; Rezzano,J. L.; Saa,D. R.; Zgrablich, G. Surf. Sci. 1989,204,289. Riccardo, J. L. Ph.D. Thesis, Universidad Nacional de San Luis, San Luis, 1991. Riccardo, J. L.;Chade, M.;Pereyra, V.; Zgrablich, G. Langmuir 1992,8, 1518. (11) Bulnes, F.; Riccardo, J. L.; Zgrablich, G.; Pereyra, V. Surf. Sci. 1992,260, 304-310. (12) Mayagoitia, V.; Rojas, F.; Pereyra, V.; Zgrablich, G. Surf. Sci. 1989, 22, 394.
0743-746319312409-2730$04.00/0 0 1993 American Chemical Society
Characterization of Energetic Surface Heterogeneity function, a “bond” energy distribution function must be incorporated. The arrangement of bond energies on the surface cannot be, in general, independent of that of site energies since every bond must have an energy lower (in absolute value) than that of the two connected sites. When the site and bond energy distribution overlap, this fundamental constraint, called construction principle, leads naturally to the appearance of energy correlations in such a way that there cannot exist a surface with a weaker correlation degree than this. The correlation can be stronger or with certain special features due to particularities during formation and modification of the surface, or to the chemical nature itself. In section 1of this work we briefly review the reasoning line followed to obtain the analytical description of the heterogeneous surface (included its topography) from the existence of sitebond distribution and by observing the construction principle. Section 2 is devoted to develop a methodology to simulate properly a dual site-bond surface. This includes the introduction of a statistical relaxation method to simulate strongly correlated heterogeneoussurfaces. In section 3, analysis and characterization of energy correlations are made. Special emphasis is used to establish a conceptual relation between the dual description and the generalized Gaussian model. Finally, a brief discussion about characterization of a heterogeneous surface by complementary use of adsorption and surface diffusion is given in section 4. 1. Dual Site-Bond Description
The basis of the dual description is that bonds (saddle point energies between adsorption sites) are as important as sites for defining properly the heterogeneous surface due to their influence on surface phenomena. Sites as well as bonds, observed independently on the adsorption field, are statistically characterized by their energy probability density functions; Fs(E) and FB(E)for sites and bonds, respectively. We denote S(E) and B(E) the respective distribution functions
Langmuir, Vol. 9, No. 10,1993 2731
or lower than, E. However, even though the first law is obeyed, a bond of energy EB might incorrectly be linked to a site of energy E, C EB in building up the surface. The second law is of local character and avoids this inconsistency Second law 4(Es,EB) = 0 for E , C E , For E , 1 EB it is found12J3that the function 4 has the following expression:
It is worth nothing that eq 2 leads to the surface with the maximum degree of randomness allowed by the construction principle, we refer to this as the most “verisimilar” surface that can be obtained from Fs and FB,i.e. the surface which admits the greatest variety of statistical representations. The functionsFs and FBcan be arbitrary functions with the only restriction of fulfiiing the fiist law. If Fs and FBare assumed to be uniform functions, 4 can be analytically calculated.12 Thus, for
Fofor S, IE IS,
[
Fs 0 otherwise
(3)
for B, IE I B , and using eq 2, we obtain
where I is the overlap (common area of FSand FB)and cp a function defined by
(E, - S , ) / ( B , - S,) for E , S S,, E , IB, for EB IS,, E , > B , - E B ) / ( B ,- S,) for E, > S,, E, IB,
S ( E ) = J O E ~ , ( ~ d/ ) ~ ’ (5)
B(E) = J E ~ B ( ~ / ) However, by merely knowing these functions nothing can be said about the energetic topography of the complete surface. The way of describing the alternation of sites and bond energies over the surface is to introduce, at least, the joint distribution of nearest neighbor site-bond energies F ( E ~ , E B )This . function can be derived from the knowledge of F d E ) and FB(E), and the application of the construction principle: the energy of a given site is higher than, or at least equal to, the energy of any of ita delimiting bonds. Conversely, the energy of a given bond is lower than, or at most equal to, that of the two connected sites. Since Fs(E) and F&) can overlap (Le., there are bonds with energy higher than that of some sites, but these cannot be connected on the surface), F(Es&) is not simply the product of Fs(Es) and FB(EB)but must be written as (1) F(Es,EB) = Fs(E,) FB(EB)~ ( E s , E B ) where $(Es,EB)is a correlation function that will be derived by employing the construction principle. Two self-consistent laws must be established to fulfill the construction principle: First law B(E)1 S(E) for every E This law ensuresthat there are enough bonds with energy equal to,or lower than, E to link sites of energy equal to,
Figure 1 roughly represents, by energies profiles, the effect of the overlapping degree I on the surface energy topography. If site and bond distributions do not overlap, the energy of each element can be assigned independently giving a site-bond heterogeneous surface exhibiting a totally random character (Figure la). As the overlap between distributions increases, many sites have energy lower than those of certain bonds, but these elements cannot be together on the surface. Thus, the overlap leads naturally to the reunion of sites and bonds with similar energies corresponding to a correlated site-bond surface (Figure lb). This segregation effect is complete when the overlap approaches the maximum value (Figure IC). 2. Simulation of the Surface The simulation of the surface consists of building the surface through a sequential assignment of site and bond energies according to the joint probability density function F ( E ~ , E B ) We . briefly summarize here the Markovian process, already described elsewhere.“ To illustrate the way in which the energies are assigned to the elements and the branching procedure followed to (13)Mayagoitia, V.;Rojee, F.; Riccardo, J. L.; Pereyra, V.; Zgrablich, G. Phye. Rev. B 1990,41,7160-7165. (14)Mayagoitia, V.;Rojee, F.; Kornhauser, 1.; Zgrablich, G.; Pereyra, V. International Conference on Gee Separation and Purification,Texaa (1991),in prese.
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2732 Langmuir, Vol. 9, No.10, 1993
rnd = J:Fs(E)
I
I
E>
I1
Fh S4
:
I
I
s, I
Y
dE
where md is a random number with a normalized uniform distribution. Once the site energy has been fiied, the right and down bonds are constrained, so that their energies E Band ~ E B ~ must be sampled from amodified conditional distribution given by where the integrand of (7) is obtained as follows
While the site energies of the first row and first column are being fixed, each site is only constrained by the next previous bond, then for sites belonging to the blocks at the boundaries the conditional distribution of site energy must be used
I x] Figure 1. One-dimensional representations of the adsorptive energy along a direction x on the surface (right-hand side) for site-bond distributions with different degrees of overlapping I (left-hand side).
All energies for the elements of first row first column can be assigned by using eqs 6,7, and 9. The situation of sites in the bulk is somehow different since they are always constrained by two already settled connecting bonds. Then, the site energy is obtained through where maX(EBl,&) denotes the bigger of these energies, and Fs(E/EslnEBz) designates the proability for a given ~ site, with two connecting bonds having energies E B and EBZ,to have and energy in the interval (E,E + dE). The analytical expression of Fs(E/EBlnEBz) is
FS(EIEBlnEB2) = FS(E)
4@$RI)
d(E&W)
Jms&(EB,Eap)Fs(E) 4(E$B1)
I
I
w First
1
Elementary Block
1
Column
Figure 2. Schematic representation of a square lattice of adsorptive sites connected by saddle points (bonds). Sites are represented by circlea and saddle points by channelsconnecting them. An elementary block, constituted by a site (S),the right bond (RB),and down bond (DB),is separatelyrepresentedwithin the square. Blocks are drawn in full line at the boundaries,and in dashed line at the bulk of the lattice. connect them, we use a simple square lattice. Figure 2 shows a portion of the lattice corresponding to the top left corner, where “sites”are represented by circles and “bonds” by channels connecting circles. We named as “the elementary unit” the term constituted by a site and ita right and down bonds since the lattice can be constructed by successive branching of this unit. The sequence of assignation can be divided into two main stages: (1)energy assignment to the blocks at the top and left boundaries; (2) energjrassignment to the blocks in the bulk of the matrix. From now on, we assume that the energy interval out of which the corresponding probability density function vanishes is (S1,Sz) for sites and (&&) for bonds. The energy assignation to the first row blocks begins by the top left corner block. The energy E8 for the site is obtained from ita energy distribution Fs(E) through
(11) throughout this work uniform density functions for site and bond energies have been used; in this case then, the definite integrals of eqs 7,9, and 10 can be analytically calculated. Several heterogeneous lattices of 100 X 100 sites (2 X 100 X 100 bonds) have been generated for different sitebond distributions. For low I I 0.4 the energy distributions on the simulated lattices show a good statistical agreement with the theoretical ones; i.e. a good agreement between simulated and theoretical momenta is encountered up to the fourth moment and the site-bond frequency functions of the simulated set of energies do not display any distortions compared to the theoretical ones. However for intermediate and high overlap (II 0.51, the statistical parameters evaluated on the simulated lattice differ, sometimes considerably, from the expected ones, and the frequencies of site and bond energies show that there is an overpopulation of highly energetic sites and bonds, at the expense of the less energetic ones (see Table I and Figure 3). In fact, small statistical fluctuations occurring in generating bonds energies are amplified in the step for bulk sites assignation, since the site energy is directly conetrained by the “maximum” energy of the two preceding bonds. 2.1. Statistical Relaxation Procedure. When the overlapping degree is high, the surface obtained by simulation through the self-consistent dual description of the previous section is far from the desired one. To correct
Langmuir, Vol. 9, No. 10, 1993 2733
Characterization of Energetic Surface Heterogeneity Table I
characteristics of the theoretical energy distributionsa lattice 1: Si = 2, Sz = 4, B1 0.4, Bz 2.4, Z 0.2 theoretical values simulated values lattim 2: S1 2, S2 4, B1 0.8, Bz = 2.8,Z = 0.4 theoretical values simulated values lattice 3 S1 = 2,Sa = 4, B1 = 1.6, BZ= 3.6,Z = 0.75 theoretical values simulated values (BR) simulated values (AR) a
All energies are expressed in arbitrary units.
PI,
PI I Ea
PZ
ra
P4
3.000 2.999
0.333 0.336
0.000 0.002
0.200 0.202
1.400 1.400
0.333 0.330
0.000 0.001
0.200 0.196
3.000 3.006
0.333 0.330
0.000 0.003
0.200 0.196
1.800 1.802
0.333 0.330
0.000 0.003
0.200 0.197
3.000 3.374 3.001
0.333 0.181 0.333
0.000
0.200 0.099 0.199
2.500 2.789 2.499
0.333 0.274 0.333
0.000 0.093 0.000
0.200 0.192 0.200
0.054 0.001
rz, pa, and p4 denote the f i t four momenta of the sits-bond distributions Fs(Es) and
I = 0.75. Also the site-bond histograms result completely .
1.01
satisfactory as shown in Figure 3.
SITE
z 0.64
W
L u 0.2
I-
0
2
3
5
4
ENERGY
Figure 3. Histogram6 of sitebond energies for a simulated square lattice of l(r (2 X l(r bonds). Full lines and broken lines
represent the sitebond energy probability densities before relaxation (BR) and after relaxation (AFt),respectively.
statistical deviations and spurious energy correlations, a relaxation procedure is then started. The procedure is baaed on the following reasoning: if one small local correction can be made on the energetically distorted lattice by applying a given criterion for replacing some unsuited energies, then the correction can be propagated throughout the surface and amplified by successive application of this criterion. The procedure followed to relax an elementary block and the criterion adopted can be summarized as follows: (i) Accumulate as histograms the actual energy frequencies for sites and bonds of the original simulated lattice. (ii) Take the first lattice site of energy Es and generate a new energy Es' for it, constrained by the two preceding bonds (provided that periodic boundary conditions are used every site has the top and left bonds already settled). If the actual frequency for Es' is lower than that for Es, then replace ES by Es' and update the site energy histogram. (iii) Take the right bond of energy EB and generate a new energy EB' for it constrained by the previous site energy. If the actual frequency for EB' is lower than that for&, then replace EBby EB' and update the bond energy histogram. (iv) Repeat (iii) for the down bond. (v) Repeat from step ii for each elementary block of the fmt row, second row, etc. until ending. We call "one relaxation step" (Rs)to this sequence of steps. The result of the statisticalrelaxation can be appreciated in Table I where an excellent agreement for the four first momenta is obtained after 3RS for I = 0.6, and 200Rs for
3. Characterization of Topological Correlations A better characterization of surface heterogeneity should into account not only the mere determination of the simplest statistical parameters of the adsorptive energy distribution, like the mean energies and variances, but also a deeper knowledge of the magnitude and morphology of correlations. It has already been stressed that the influence of site energy correlations is by no means negligible in adsorption of interacting gases on heterogeneous surfaces.1° Topological correlations are expected to play a major role in the development of surface processes such as adsorption and surface diffusion in adsorption separation,14 and it has been recently claimed that this would be valid also for chemisorpti~n.~~ In this section we will analyze the site and bond energy correlations arising from the simulation of heterogeneous surfaces through the self-consistent dual site-bond description. It has been noted before that for a given overlap I , if the construction principle is to be observed, a fraction of sitebond energy pairs cannot exist on the surface. Energy correlations become, naturally, stronger as the overlap I increases. In the dual description it can be said that energy correlations are characterized (or associated to) the overlapping degree I. However this is a vague assertion since no idea about the size and shape of surface regions containing elementa with similar energies can be obtained from this parameter. As already pointed out,l0 this information is crucial to deal properly with lateral interactions in lattice gases adsorbed on correlated heterogeneous surfaces. One question to be answered here is: how do site (bond) energy correlations decay with the site-site (bond-bond) distance on the surface for a give overlap? The correlation function between sites, bonds, and sitebond energies is defined, in general, as follows:
with x y = S or B and r = p, - the separation distance among the pair of elementsconsidered. The brackets (...) represent the statistical average over all seta of surface energies. The decay of the site-site correlation functin P ( r )with the site-site distance r was calculated on simulated square lattices of lo4 sites and 2 X 10.' bonds for several values of I. (16)Rudzinski, W. Private Communication (1991).
Riccardo et al.
2734 Langmuir, Vol. 9,No. 10, 1993 Chart I
-- -
system variable property transition connectivity of occupied sites finite cluster infinite cluster simple lattices site percolation p (site occupation probability) lattice gas 6 (mean surface coverage) local density dilute condensed adsorbed phase the heterogeneous surface in I (overlap of energy distributions energy correlations (morphology of dilute condensed correlations cluster of sites and bonds having similar energies)
Ln C(r1 A
+
1-01
X 1.03
-7-
A
1.04
0.1
0.5
I Figure 5. Correlationlength ro (measured in latticeunits) versus the overlap degree I between site-bond distributions. Crosses and points represent results obtained by analyzing only one simulated lattice for each I. Squareswere obtainedby averaging over one hundred lattices for each I. The full line represents the theoretical prediction for the dependence of ro upon I made in ref 10 (RoQ = 1/(1 - Z). OVERLAP
-
the fact that for I 0, ro is overestimated due to small fluctuations of energies appearing in simulating the lattice; here the lattice size becomes almost irrelevant). The squares represent an important aspect of Figure 5. By averaging the correlation function over about one hundred lattices for every I, it is concluded that ro does not change with I for I I 0.5. Since in this range ro is smaller than one lattice constant and given that we know exactly that ro = 0 for I = 0, then the following hypothesis can be ventured: the topological correlation length ro of a dual site-bond heterogeneous surface is null for I I I, and finite for I > I,, where I, is between 0.4 and 0.6. The critical overlap I, defines a threshold between two clearly differentiated topologies of correlated energies: (a) below I, the elements with similar energies, although in some way correlated, are not aggregated in compact zones (the idea of correlation length becomes meaningless); (b) above I, a site (or bond) of a given energy is very likely surrounded by other elements having similar energies forming a "lake" adequately characterized by a finite correlation length. We name this critical change as the "dilute correlation" "condensed correlations" transition and in the next section we will interpret this behavior in the framework of percolation theory. 3.1. Percolation of Correlations. The aim of this section is to understand the topblogical transition from dilute correlations to condensed correlations (null corfinite correlation length) based on rection length percolation theory.
-
-
Characterization of Energetic Surface Heterogeneity
Langmuir, Vol. 9,No. 10,1993 2735
(a 1
I = 0.2
(b) 1 = OX
82
52
E
(d1
I = 0.75
Figure 6. Site and bond energyfrequencyfunctionsfor different degrees of overlapping ((a) I = 0.2; (b) I = 0.4; (c) I = 0.6; (d) I = 0.75). For each value of I, the region hatched to the left represents the fraction I of bonds belonging to the overlapping region; the region hatched to the right representsan equal fraction of higher energetic sites (white circles in Figure 7).
As is well-known,all percolation processes have a general common characteristic: some "property" of a given "system" changes suddenly for certain values of the variables defining the state of the system. A comparison between typical percolation transitions is made in Chart I. We can summarize the correlation percolation process as follows: our "system" is the heterogeneous surface itself (the heterogeneous site-bond lattice); the surface is built up from the site and bond energy distributions as explained before. The shape of distributions is held constant throughoutbut the overlap can be varied. In consequence, a set of heterogeneous surfaces can be obtained with energy overlapping degree varying from 0 to 1. Energy topography changes under variation of I. To observe topological character of energy correlations, the "property" to be observed is the morphology of correlated energy clusters. This means to look at the morphology of clusters constituted by sites and bonds having similar energies. This morphology is intended to be characterized, when possible, by a correlation length on the surface. A sudden change of morphology is found
Figure 7. Square lattice of circles and channels of different sizes representing a heterogenous surface of adsorptive sites connected by bonds (saddle points of the adsorptive energy surface). Bigger circles correspondto higher energeticsites.White sites and white bonds in photographs a-d (top to bottom) correspond, respectively, to the hatched regions of Figure 6a-d (hatched to the left for bonds and hatched to the right for sites).
in a narrow range of I,causing the null *finite correlation length transition. Although we have considered the
2736 Langmuir, Vol. 9, No. 10, 1993
correlationfunction between site energies CSS(r),a similar behavior is expected if Cgg(r) is analyzed. To visualize energy clusters, the simulated heterogeneoussurfaces corresponding to the distributions given in Figure 6 have been represented by a square lattice of circles connected by channels, as shown in Figure 7, the circles represent adsorptive sites and the channels the saddle point between two nearest neighbor sites. The sizes of circles and channels are proportional to site and bond energies so that bigger circles are the deeper adsorptive sibs (highly adsorptive sites) and the bigger channels are the higher energetic bonds (considered in absolute values of energies). Four different lattices corresponding to different values of I are shown. For a given overlap I, a fraction of the higher energetic bonds and sites are illuminated (the hatched regions of Figure 6). Thus, the set of energies of the white elements is S, I E, I23, (14) S, - Z(S2 - 5’1) 5 E, 5 S2 for every I On one hand this set of bonds contains the more correlated ones since these are directly constrained by the construction principle to link lower energy sites. On the other hand, the way in which aggregation of correlated elements proceeds for high I can be disclosed by illuminating simultaneously the higher energetic sites. In Figure 7 (photographs taken from the computer screen) the boundaries of elements are red and the sites and bonds with energies within the hatched regions of Figure 6 are white. For low I (I= 0.2) (see Figure 7a), white sites and bonds are either isolated or forming very ramified small clusters (we refer to a cluster as a group of directly or indirectly connected white elements). The clusters look like “lattice animals” and clearly no finite correlation length can be associated to these. We say that we have dilute conditions. By increasing the overlapping degree to I = 0.4, we can see that the correlated energy elements form somehow more compact clusters (see Figure 7b). On one hand it is expected that the clusters will grow by an increase of the fraction illuminated. On the other hand it is observed that the clusters have evolved from a ramified to somehow more compact general form (which does not occur €or random percolation). However an important amount of sites and bonds remain isolated or in small clusters and consequently a correlation length may not probably be associated to the whole white phase. When the overlap is increased to I = 0.55 very large and compact clusters appear surrounding smaller unilluminated regions (corresponding to low energetic sites and bonds that are also very correlated) (see Figure 7c). Because the elements of similar energies are now closely grouped, it becomes meaningful to use a topological correlation length for characterizing energy correlations. It is worth noting that a critical overlap about IC = 0.6 agrees with the prediction of percolation theory for the criticalparameter of a correlated site percolation problem. We call this topological change a dilute-condensed correlation transition because of the resemblance of this picture with the mixing of bubbles and condensed vapor in the coexistence region of the liquid-vapor transition. For high overlap (I = 0.741, similar elements join in quasi-homogeneous macroscopic patches and the segregation effect becomes evident (see Figure 7d). 4. Characterization of Heterogeneous Surfaces by
Complementary Use of Surface Phenomena Physical adsorption has been extensively used for characterizing the heterogeneous surface from a purely
Riccardo et al.
energetic view point.’“ Severalcomputationalalgorithms are available to get the simpler statistical parameter from equilibrium adsorption data. However the analysis of adsorption data would be insufficient to obtain some quantitative information about energy correlations, like for example the correlation length. The effect of short range site energy correlations on adsorption isotherms and on collective surface diffusion of interacting lattice gases on heterogeneous surfaces has been recently investigated in the framework of the generalized Gaussian model.1° Adsorption isotherms are affected by correlation only for strongly interacting adsorbates; therefore weakly interacting adsorbates would not be appropriate to reveal the topography of the adsorptive field. On the contrary surface diffusion is highly sensitive to energy correlation even for noninteracting adsorbates. Theoretical adsorption isotherms and the surface diffusion coefficient have also recently been calculated in the frame of the dual description. Both phenomena are strongly influenced by the overlap of site and bond distributions. It is possible, in principle, to think that site and bond distributions could be obtained by fitting simultaneously sets of adsorption isotherms and diffusion coefficient measure for the same gas-solid system. Quantitative informationabout the overlap and correlation length could be then extracted. However such complete experimental data are not available at present. Conclusion A statistical self-consistent dual model for deecribing the heterogeneous surface has been presented. This description is based on two main elements which can be distinguished in the adsorptiveenergy surface representing the gas solid interaction: sites and bonds. The statistical properties of the whole site-bond energy distribution are completely described by a site-bond correlation function 4. This function depends on the process by which the surface has been formed. The caee of a surface constructed from site-bond energy distributions without any additional restriction to that naturally imposed by the construction principle was analyzed. A statistical relaxation procedure is necegsary to simulate adequately the heterogeneous surfaces when there exist strong energy correlations on the surface. This dual description shows a transition in the morphology of Correlated energies clusters when the overlap reaches a critical value, IC = 0.5. Above this threshold correlated energies are grouped in close clustera and the energetic topography can be properly characterized by an energy Correlation length. Below le,correlationare “dilute” without any characteristiclength. It is expected that these different topographies will induce distinct behaviors in all surface processes and, particularly, in adsorption, surface diffusion and surface reaction. Further studies on the magnitude and characteristic of energy correlations and their influence on surface phenomena are necessary. Acknowledgment. This work was supported, and made possible, by the Cooperation Program between CONACyT (National Council of Science and Technology of Mexico) and CONICET (National Council of Science and Technology of Argentina) under the Joint Project “Cattllisis,FisicoquImica de Superficiese Interfasea GasS6lido” being executed at the Universidad Authnoma Metropolitana Itztapalapa and the Instituto de Investigaciones en Tecnologfa Quimica, Universidad Nacional de San Luis. Our special thanks to D. Flores Esnayra, BSc, for his assistance in the programming work.
