Langmuir 1992,8, 1518-1531
1518
Feature Article Adsorption and Surface Diffusion on Generalized Heterogeneous Surfaces J. L. Riccardo, M. A. Chade, V. D. Pereyra, and G. Zgrablich’ Instituto de Znvestigaciones en Tecnologi’a Qui’mica, Universidad Nacional de Sun Luis-CONICET, Casilla de Correos 290,5700 Sun Luis, Argentina Received August 20,1991. In Final Form: January 9,1992 A complete study of adsorption, surface diffusion, clustering, and percolation of adsorbed gases on heterogeneoussurfacesis presented in such a way as to stress the importanceof adsorptiveenergy topography (i.e. correlation between adsorptive energies of different sites). Theoretical models as well as simulation techniques are developed in order to describe these phenomena on a wide variety of topographies. The characterization of surfaces through combined use of adsorption and surface diffusion experiments is discussed. 1. Introduction
The role of surface characteristics in many processes of practical importance is a topic of increasing interest in surface science. Adsorption, surface diffusion, and reactions on catalysts are some of the phenomena which are strongly dependent upon surface structure. Most materials, with or without high specific areas, have heterogeneous surfaces and it is of substantial interest to attempt a complete characterization of such heterogeneity. In the last 40 years physical adsorption has been used for determining energetic properties of heterogeneous substrates, but a satisfactory solution to this problem is still Adsorption has often been used to get adsorptive energy distributions from experimental isot h e r m ~ . ~Diffusion -~ on homogeneous surfaces has also been widely studied and has demonstrated to be specially sensitive to the presence of ordered adsorbed phase^.^-^ Also the many efforts made to study diffusion on twodimensional heterogeneous media show that surface migration is strongly affected by surface heterogeneity.’O-’* It has already been pointed out6J9-22that, in addition to adsorptive energy distribution, adsorptive energy
* To whom correspondence should be addressed at CREA, Centro Regional d e Estudios Avanzados, Gobierno de SanLuis, Argentina. (1) Steele, W. A. The Interaction of Gases with Solid Surfaces; Pergamon Press: New York, 1974; p 195. (2) Steele, W. A,, Fundamentala ofddsorption; Myers, A. L., Belfort, G., Eds.; United Engineering Trustees, Inc.: New York, 1984; pp 743. (3) Roes, S.; Olivier, J. P. On Physical Adsorption Interscience; New York, 1964. (4) OBrien, J. A.; Myers, A. L. J. Chem. SOC.,Faraday Trans. 1 1985, 81,351. (5) Jaroniec, M.; Briuer, P. Surface Sci. Rep. 1986, 6, 65. (6) Murch, G. E.; Nowick, A. S. Diffusion in Crystalline Solids; Academic Press: New York, 1984. (7) Murch, G. E. Philos. Mag. A 1981,43, 871. (8)Sadiq, A.; Binder, K. Surf. Sci. 1983, 130, 348. (9) Naumoveta, A. G.; Vedula, Yu. S. Surf. Sci. Rep. 1985,4, 365. (10) Kirchheim, R.; Stolz, U. Acta Metall., 1987, 35, 281. (11) Kirchheim, R. Acta Metall. 1987, 35,271. (12) Mak, C. H.; Andersen, H. C.; George, S. M. J. Chem. Phys. 1988, 88 (6), 4052. (13) Haus, J. W.; Kehr, K. W. Phys. Rep. 1987, 150, 263. (14) Alexander, S.;Bernaaconi,J.; Schneider,W.; Orbach, R.Reu.Mod. Phys. 1981,53, 175. (15) Pereyra, V.; Zgrablich, G.; Zhdanov, V. P. Langmuir 1990,6,691. (16) Pereyra, V.; Zgrablich, G. Langmuir 1990, 6, 118. (17) Havlin, S.; Ben-Avraham, D. Ado. Phys. 1987. 36. 695, and references therein. 0743-746319212408-1518$03.OO/O
topography must be considered as well to characterize surface heterogeneity through its interactions with an adsorbed gas. Until now two different models, corresponding to two limiting topographic cases, have been widely used to obtain the energy distribution from experimental isotherms namely, the patchwise heterogeneous surface, proposed by Ross and Olivier? and the independent adsorptive sites model, due to Hill.23 According to the first one, the surface is composed of a collection of macroscopic homogeneous patches in such a way that all sites within agiven patch have the same adsorptive energy but different patches have a given energy distribution. In the independent sites model, each adsorption site has a randomly distributed energy which is totally independent of that of any other site. These models hardly represent the great majority of real surfaces but it must be noticed that they have allowed a practical interpretation of experimental data for a quantity of real ads or bent^.^^^^ A more general description for adsorption on arbitrary heterogeneous surfaces was introduced through the generalized Gaussian model (GGM) developed in ref 20. This formulation is based on the fact that energies of different sites will be, ingeneral, somewhat related. This dependence is to be considered in a statistical sense and it is formulated through a correlation function which is defined in terms of a characteristic correlation length for a given surface. In the above framework, the former models appear as two limiting topographic cases. Highly correlated surfaces (very large correlation length) correspond to the homotatic patches picture while uncorrelated (or random) heterogeneous surfaces (null correlation length) correspond to the independent sites description. A vast set of intermediate topographies can be characterized by finite correlation lengths and they appear as the most interesting and realistic cases.22 However the fact that (18) Pereyra, V. Ph.D. Thesis, Universidad Nacional de San Luis, Argentina, 1988. (19) Steele, W. A. The Gas-Solid Interface; Flood, E. Alison, Ed.; Dekker: New York, 1967; Vol. I. (20) Ripa, P.; Zgrablich, G. J. Phys. Chem. 1975, 79,2118. (21) Zgrablich, G.; Pereyra, V.; Ponzi, M.; Marchese, J. AIChE J. 1986, 32, 1158. Pereyra, V.; Zgrablich, G.Surf. Sci. 1989, 209, 512. (22) Riccardo, J. L.; Pereyra, V.; Rezzano, J. L.; Rodriguez Saa, D.; Zgrablich, G. Surf. Sci. 1989, 204, 289. (23) Hill, T. L. J. Chem. Phys. 1949, 17,520. (24) Sircar, S.; Myers, A. L. Surf. Sci. 1988, 205, 353.
0 1992 American Chemical Society
Generalized Heterogeneous Surfaces the mathematical formulation of the GGM relies on the calculation of gas-solid virial coefficients makes the model impractical for high surface coverage analysis. Surface diffusion is an even more complex phenomenon than adsorption given that activation energy for diffusion, i.e. the energy barrier to be overcome by a molecule to jump from one site to another, must be specified in addition to adsorptive site energies. Simple models25 assume that activation energy can vary from site to site independently of any adsorptive energy structure. It is reasonable to think that these barriers cannot vary freely but on the contrary they should be rather expected to depend strongly on adsorptive energy of sites since “sites” and “barriers” should be viewed just as appropriate elements of a continuum adsorptive energy surface in three dimensions. In a recent attempt to investigate the influence of adsorption energy distribution on the collective surface diffusion, it has been proposed that activation energies for jumps between two nearest neighbor sites must be somewhat related to adsorption energies of both sites.15J6 Results show that the diffusion coefficientis quite sensitive to the activation energy dispersion. Simple adsorptive energy distributions, like uniform, Gaussian, and lognormal functions, were used but topological correlations were not taken into account in these calculations. Indeed, the existence of correlations between adsorptive energies provokes changes in the structure displayed by the set of barriers, i.e. their relative positions. In addition, for high surface coverage, not only the activation energy distribution but also the presence of many migrating molecules has to be considered. This last effect is usually associated with the mean number of empty nearest neighbors (the so-called vacancy factor), which is intrinsically related to the adsorptive energy topography. In this way the effect of adsorptive energy correlations on the adsorption isotherm is transmitted also to the behavior of thediffusion coefficient. An alternative description of heterogeneous substrates, i.e. the dual site-bond description, taking into account two basic elements on the surface, namely, sites and bonds (energy saddle points be\ween two nearest neighbor sites) has been recently developed.2628 Therein the need of introducing a bond energy distribution in addition to the site energy distribution was stressed in order to obtain a deeper insight into surface phenomena. The key assumption was that these two distributions cannot be, in general, independent since the energy of a given bond must be higher or at least equal to that of the two connecting sites (construction principle). A remarkable result emerging from this model is that adsorption (and naturally also surface diffusion) depends on the bond distribution through constraints on site energy structure induced by the construction principle. We wish to briefly clarify here that the GGM and the dual site-bond (DS-B)models are complementary ones. In the DS-Bmodel a correlation between a bond and ita two connecting sites is established by the construction principle; this correlation is then transmitted to other elements in the lattice of sites and bonds so that site energies (aswell as bond energies) are finally also somehow correlated. For the GGM a correlation function is given for site energies while activation energies (consequently (25)Klaften, J.; Silbey, R. Surf. Sci. 1980,92,393. (26)Mayagoitia, V.;Rojas, F.; Pereyra, V.; Zgrablich, G . Surf. Sci. 1989,221,394. (27)Mayagoitia, V.;Rojae, F.; Riccardo, J. L.; Pereyra, V.; Zgrablich, G. Phys. Reu. B 1990,41,7150. (28)Cruz, M. J.; Mayagoitia, V.; Rojas, F. J. Chem. Soc., Faraday Tram. I, in press.
Langmuir, Vol. 8, No. 6,1992 1519 bond energies) are functionally related to site energies. In this way neither one of the models contains the other. The aim of the present work is, on one hand, to reformulate the GGM in terms of Fermi-Dirac statistics and mean random field approximations for an adsorbed interacting lattice gas in order to get manageable adsorption isotherms as well as to observe correlation effects on the entire coverage range. Adsorption heats and the development of clusters of adsorbed molecules are studied through Monte Carlo simulation techniques. On the other hand, our interest is focused on how the diffusion coefficient is influenced by the adsorptive energy topography under the assumption of a particular relation between activation energies and adsorptive energies and of a generalized adsorptive energy distribution. Section 2 is devoted to a description of adsorption on arbitrary heterogeneous surfaces. In section 2.1 a generalized mean field lattice gas isotherm is developed. A reasonable approximation for adsorbate-adsorbate interactions is proposed in section 2.2. Theoretical results are discussed in section 2.3. In sections 2.4 and 2.5, the simulation of arbitrary heterogeneous surfaces is explained and simulated isotherms as well as adsorbed phase thermodynamics are analyzed. Clustering and percolation of adsorbed molecules are the subjects of section 2.6 and a general discussion is presented in section 2.7. In section 3 surface diffusion is analyzed and section 4 is devoted to a discussion of how a heterogeneous surface can be characterized from experimental data on adsorption and surface diffusion of gases. Finally, general conclusions are given in section 5. 2. Adsorption
A standard procedure is currently used to obtain adsorptive energy distributions from experimental gas adsorption data. This is performed by solvingthe well-known integral equation5124
where B(p,T) is the mean surface coverage a t chemical potentialp and temperature T, 8(k,T,e)is a local coverage for a site with adsorptive energy t, and f ( c ) represents the adsorptive energy probability density suitably normalized. Equation 1 contains three functions, two of which are a priori unknown while the third one can be determined from experiments so that clearly there is not a unique solution to eq 1. A variety of model isotherms for 8 are available, namely, Langmuir or Jovanovic for noninteracting localized gas, Fowler-Guggenheim for interacting localized gas, Volmer for noninteracting mobile adsorbate, Hill-de Boer for interacting mobile adsorbate, BET for multilayer adsorption, et^.^^^^^^^ Although qualitative methods for selecting a local isotherm, either from the shape of experimental isotherm30 or from the chemical nature of the surface,31have been proposed, they hardly have general applicability. Surfaces with the same f ( 4 might exhibit quite diverse detailed structures like the energy profiles displayed in Figure 1. Therein homotactic patches and random energy profiles are shown in parts a and c, respectively. A third intermediate situation is shown in part b, where energies of neighbor sites are neither equal nor abruptly different (29)Gregg,S. J.;Sing,K. S. W. Adsorption, Surface AreaandPorosity; Academic Press: New York, 1982. (30)Briuer, P.; House, W. A.; Jaroniec, M. Thin Solid Films 1986, 123,245. (31)Rudzinski, W.; Lajtar, L.;Patrykiejew, A. Surf.Sci. 1977,67,195.
Riccardo et al.
1520 Langmuir, Vol. 8, No. 6, 1992 M
e
7f = z e i n i
+ Cuijninj
i=l
1#I
where ti is the adsorptive energy at site “i”, Uij is the and interaction energy between a molecule located at “in another located at “ j ” , and ni is the occupation number of site i (ni = 0 or 1). This Hamiltonian can be rewritten as
s = CMn i [ e i+ C u i j n j l i#i
i=l
where the expression in brackets is the interaction field seen by a particle at site “i”. We can make a generalized mean field approximation in which a particle at site “in sees a mean field
Ei = ti + CVijOi
Figure 1. Adsorptive energy variation along some coordinatex
on the surface for different topographies.
but they are rather smoothly varying according to some statistical correlation degree. This kind of profile was found in simulations of the interaction of a probe molecule with an amorphous substrate which was modeled as a random closed packed assembly of spheres.32It is expected that adsorption isotherms, adsorption heats, and adsorbate clustering will have a distinct behavior in each case. Even though correct criteria for selecting adequate local isotherms should exist, only those limiting cases displayed in parts a and c of Figure 1 can be treated through eq 1. Inversion of eq 1 would give an incorrect f ( ~ for ) any intermediate topography. 2.1. Mean Random Field (MRF) Formulation of t h e Lattice Gas Isotherm. A general formulation, as we mentioned, has been already proposed through the generalized Gaussian model,20where it was emphasized that a multivariate energy distribution must be used in order to describe the intricate surface heterogeneity coming from nonregular arrangements of bulk atoms, impurities defects, and the complex chemical nature of the surface itself. Statistical correlations in this model can be appropriately characterized through a correlation length ro which is the distance between sites whose adsorptive energies are strongly interdependent. Then, for example, for very large correlation length, ro >> a, where a is the lattice constant, the surface separates into macroscopic homogeneous patches; for null correlation length, ro = 0, nearest neighbor sites are uncorrelated and its energies may often take very different values giving rise to a random heterogeneous surface, while for short correlation length, ro = a, significant adsorptive energy fluctuations are not observed within small distances. For practical purposes this model has to be reformulated in terms of Fermi-Dirac statistics in order to obtain a manageable isotherm equation. We assume the surface to be represented by a lattice of M (M a) adsorptive sites, which is exposed to an ideal gas phase and the system reaches equilibrium at chemical potential g and temperature T. The Hamiltonian for the adsorbed phase can be written as
-
~~
(32) Bakaev, V. A. Surf. Sci. 1988,198, 571.
where Oj = ( n j )is the mean occupation number at site “ j ” , but this field is different for each site. By the way, we notice that Ei is a random mean field due to the random character of the adsorptive energy ti. With the mean field Hamiltonian M
7f‘ = c n i E i i=l
the grand partition function for the system is given by
z=
e-B(”H’-Nd states
=
e-Btnl(E1-~)+nz(Ez-rr)+...l n1,nz,...
where B = l/kT and N = E n;. This, as well known, leads to the Fermi-Dirac statistics for the mean occupation number of a single particle state “i”
i = 1, ...,M
-
We recall that, as Ei is a function of all Oj (for j # i), eq 3 actually represents a system of M (M m) coupled transcendental equations. Now a heterogeneous surface is characterized by a multivariate density distribution f M (€1, ...,E M ) ,which describes the statistical ensemble of surfaces. For a given member of such an ensemble, i.e. for a given choice of adsorptive energy values €1, ..., CM, we can define a local coverage (4)
and consequently the mean surface coverage 8 will be given by the mean over the total ensemble
8=J
... J f M ( e 1 , ...,E M ) O(t1, ...,
dt1 ... dtM
(5)
Equations 3 to 5 constitute the formal mean field solution for adsorption on heterogeneous lattices. In this way the adsorption isotherm could be in principle numerically calculated. The problem of adsorption of gases on heterogeneous surfaces is equivalent to the problem of ferromagnetism in a spatially inhomogeneous magnetic field for which, in spite of the great effort dedicated by many researches,
Generalized Heterogeneous Surfaces
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Langmuir, Vol. 8, No. 6,1992 1521
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interaction A8 with the rest outside ro. It has been assumed that the mean coverage on the shadowed area cannot considerably differ from that on AN. Outside ro adsorptive sites can have, in average, all possible adsorptive energies in such a way that its mean coverage corresponds to the mean coverage on the whole surface, 8. Thus the isotherm equation is now determined by
e: =
(9)
Figure 2. A square lattice of adsorption sites where Nth order mean field approximation zones are shown.
solutions of the type shown here are the best attainable ones up to now. Our generalized mean field approximation, eq 2, where the mean field is different for each site, is simple enough to allow a numerical solution without losing the influence of energy topography. 2.2. Further Simplifications Due to Short Range Interactions. Difficulties arising in solving a huge set of coupled equations like eqs 3-5, where all pair interactions are explicitly accounted for, must be eluded if a feasible solution of the isotherm equation is to be achieved. Further simplifications can be introduced due to the short range molecular interactions. An appropriate mean field approximation for adsorbateadsorbate interactions should not ignore the local characteristics of adsorptive energies and also the fact that the main contribution will come from the nearest neighbor sites. Since the correlation length determines an area around a given site spanned by sites with highly correlated energies while sites outside that region will have energies rather independent from that of the central site, mean field equations should distinguish the regions r < ro and r > ro. The following ideas are illustrated in Figure 2. Three regions around a site i can be differentiated: a nearest neighbor region AN where detailed interactions will be considered; the region of sites "j" not belonging to AN but included in the circle of radius ro;the region outside this circle. The effective mean field seen from site "i" can then be written as
where
i = 1,....N Several approximation degrees can be now investigated: (a) The first-order approximation ( N = 11, where no detailed adsorbate-adsorbate interactions are involved, leads to two well-known local adsorption isotherms for interacting lattice gases: 0(€,8) =
exp[-p(e + - p)] 1+ exp[-@(c+ ~8 - p)]
(10)
for ro = 0 Hill's isotherm, and
for ro
-
a
Fowler-Guggenheim's isotherm. Consequences of using eqs 10 and 11 on the determination of f ( t ) through the We inverse problem have been widely analyzed,4~5~24~30~31~33 only wish to stress here how they originate from the same isotherm equation for two limiting correlation length values. (b) The second-order approximation ( N = 2) is a more interesting one inasmuch as explicit pair interactions are taken into account and site correlation effects may be investigated. In this case we have the pair of coupled equations
+ xoe + A8 - p)] 1+ exp[-p(el + U1202+ x06 + - p)I exp[-@(c, + U1201+ X08 + A8 - p)I 0, = 1 + exp[-@(e, + U1,Ol + xoe + he - p ) 1
o1 =
Equation 6 means that interaction between a molecule at site "i" and the remaining adsorbate is represented by three additive interaction terms: a detailed interaction with ( N - 1) closest sites belonging to AN, a mean field interaction A00 with the region r I ro excluding AN (shadowed area in Figure 2), and another mean field
exp[-@(tl + U,,6,
Local coverage on a pair of sites with energies given by = (e1 + 0 2 ) P and the overall isotherm is o(t1,C2,~)
(12)
(€1, €2)
is
(13)
(33) Brauer, P.;Jaroniec, M. J . Colloid Interface Sci. 1985,105,183.
Riccardo et al.
1522 Langmuir, Vol. 8,No. 6, 1992
In order to study the influence of the correlation length ro on adsorption isotherm, we use a bivariate Gaussian distribution fz(e1,tz) given by fz(tl,t2)
= [ ( 2 d 2detHI-'l2 X
where the elements of the covariance matrix Hij are defined by
Hij = ( ( t i - ; ) ( e j - i ) )= (kT8)'C(rij)
(16)
Here kT8 (we have introduced the "heterogeneity temperature'' Ts)has the usual meaning of variance and the correlation function C(rij) can be expressed in terms of the correlation length ro, as done in ref 20, as
--) ]
0
C(rij)= exp[- 2( 1 r 2
[
(:)-I2-
(:)"I
0
5 exp
This choice is somewhat arbitrary and clearly other correlation functions might be used. 2.3. Model Results. Realistic adsorbate-adsorbate interactions must be considered if correlation length effects on equilibrium adsorption isotherms are to be observed. We have used here a Lennard-Jones pair interaction potential Uij defined by U(rij)= -4kTgg
5
(18)
where kTgg(we have introduced the "interaction temperature'' T g g )and u have the usual meaning of minimum potential value and zero potential separation, respectively. Adsorption isotherms were calculated, through eq 14, for interacting and non-interacting adsorbates on a heterogeneous square lattice for various correlation length values. All distances are referred to the lattice spacing constant. The short range of the potential allowed us to neglect interaction with adsorbate molecules which are beyond third nearest neighbors. The computational method is based on the numerical integration in eq 14 for a fixed value of the chemical potential. For each point in the integration region, i.e. for a given pair of values (el, tz) the system of equations (12) is numerically solved using a two-dimensional NewtonRaphson procedure. Some typical isotherms predicted by the MRF model are shown in Figure 3. In general we see that for a surface with a higher correlation length, adsorption is stronger at low coverage (0 < 0.5) and weaker at high coverage (6 > 0.5). This correlation length effect is highly coupled to adsorbate-adsorbate interactions; in fact it is very weak in part a for T,,/T = 0.5 and much strong in part b for T,,/T = 1,while other parameters are the same, and it is null for noninteracting adsorbates (not shown). On the contrary, a higher adsorptive energy dispersion (higher heterogeneity) smooths out the correlation length effect as can be seen from part c, with a small dispersion, T , / T = 0.7, and from part d, with a higher dispersion, T s / T = 1.5,for the same values of the remaining parameters. Figure 3c is particularly interesting since a steplike isotherm appears for ro = 0 as a result of strong lateral interactions and low heterogeneity, denoting a sudden nucleation process which is not present for larger correlation lengths.
-
10
[*/"j 1 0 '
Figure 3. Adsorption isotherms predicted by the MRF model for three correlation lenizths: rn = 0: - - -, r, = 1: ro = 2. (a) T = 400 K, T,/T 2 0.5, T./T = 1.5;.(b)T = 406 K, T -/T = 1, TJT = 1.5; ( c ) T = 500 K, T,,/T = 1, T,/T = 0.7; (d) = 500 K, T,/T = 1, T,/T = 1.5.
-.
- - -.
!f
Finally, the usual temperature effect (due to the fact that we have not used here an adimensional adsorptionenergy), lower adsorption for higher temperature, can be observed by comparing parts b and d. The coupling between adsorbate-adsorbate interactions and correlation length of adsorptive energies can be used to explain the main feature of these adsorption isotherms. In fact, for a correlated surface a highly adsorptive site is likely to be surrounded by other similar sites and this, combined with attractive lateral interactions, enhances adsorption at low coverage in comparison to an uncorrelated surface. On the contrary, at high coverages only low-energy sites remain uncovered, which for a correlated surface are likely to be surrounded by other similar sites, resulting a weaker adsorption than in the case of uncorrelated surfaces. However, the physical meaning of the observed behavior is much richer and a complete analysis can be made only after studying the thermodynamic, clustering, and percolation properties of the adsorbate for differently correlated surfaces, which c a be obtained through Monte Carlo simulation. 2.4. Correlated Surface SimulationTechnique. As a complement to the MRF model we used the Monte Carlo simulation method in order to test theoretical predictions as well as to complete the study of the adsorbed phase thermodynamic, clustering, and percolation properties. The first step was to simulate a surface with adsorptive energies obeying the correlation function given by eq 17, over which adsorption was to be performed, by assigning to adsorptive sites forming an L X L = M square lattice energies sampled from the bivariate Gaussian distribution whose probability density is given by eq 15. To make such an assignment, circles of radius equal to the correlation length ro centered on randomly chosen sites were considered and the energies for sites belonging to each circle were generated through
5 = Ff(c',r)
(19)
where Ff(&,r)is the conditional probability that, given a
Generalized Heterogeneous Surfaces Ln C
Langmuir, Vol. 8, No. 6, 1992 1523
w 09-7
.-
08. 07.
06-
-6
-5
-4
-3
-2 0
-I
1
2
3
4
5
6
7
8
9
2
r
Figure 4. Comparison between the correlation function of simulated surfaces (points) and theoretical ones (full lines) for severalvalues of the correlation length r,: BR, before relaxation; AR, after relaxation. central site with energy e, one finds a site with energy Sf a distance r apart, and 5 is a random number uniformly distributed in the range (0,l).This conditional probability is obtained from f2(el,e2) through
(20) Values of E for the central sites can be sampled from a Gaussian distribution provided that geometrical centers of circles are separated at least by a distance 2ro. Any site which has been already assigned an energy value is Ymarked”and ita energy is not changed in this part of the procedure. When no more unmarked circles of radius ro could be found (at this point the surface has been covered up in a coarse-grained sense and only smaller regions or interstitial sites between circles are still unmarked) the search is started again with a smaller radius and this goes on until all sites have been marked. Indeed, after ending the above described procedure, the resulting lattice shows a correct correlation between the energies of sites belonging to the same circle but the overall correlation function is far from the desired theoretical one due to border effects between circles. A kind of relaxation method is then started where pairs of sites are randomly selected and an energy replacement for one of them is attempted by sampling a new energy from the conditional probability. The change is made if it goes in the direction of minimizing the difference between the actual correlation function and the theoretical one. Results of this procedure are shown in Figure 4 where a satisfactory agreement is obtained after the relaxation method is applied. It must be noticed that a different method, based on the generation of a Markovian sequence through a multivariate probability distribution, has been proposed in ref 32 tosimulate a correlated lattice. However, very large
1
2
3
4
5
6
1
8
9
rx~k~,]~ io’
Figure 5. Comparisonbetween Monte Carlosimqlated isotherms (points) and MRF model predictionswith N = 2 (solid lines):0, r, = 0,. , ro = 1; A r, = 2. (a) TplJT = 0.5. (b) T,/_T = 1. The broken line represents the homogeneous case with c -3 kcal/ mol. lattice sizes are needed with such a method in order to minimize severe anisotropy problems. The circle-filling and relaxation method described above is fast and easy to use and the optimum lattice size is only a function of the correlation length. Periodic boundary conditions have been used for the generation of the correlated surface, the same which will be used below in the simulation of all surface processes. 2.5. Simulation of Adsorption Isotherms and Thermodynamics. Once the surface has been generated by the procedure explained above, the usual Monte Carlo method in the grand canonical ensemble is followed to simulate adsorption isotherm^.^^^^^ All of the pointa in Figures 5 and 6 have been obtained by averaging over 100 configurations on a 60 X 60 square lattice after discarding 50 initial configurations. Between succeesive configurations, 10 Monte Carlo steps per lattice site were made. All isotherms correspond to a temperature T 400 K and to a surface characterized by ;lkT = -3.7, T,/T = 1.85, and u = 2-lI6(measured in the lattice spacing units) but with different topographies. Simulationresults are in excellent agreementwith model calculations at all coverages for weakly interacting adsorbates (Figure 5) and up to 6 = 0.65 for strongly interacting adsorbates (Figure 61,indicating that the MRF
-
(34) Binder, K . Monte Carlo Methods in Statistical Physics: Springer-Verlag: Berlin, 1986. (35) Nicholson, D.; Parsonage, N. G. Computer Simulation and the Statistical Mechanics of Adsorption; Academic Press: London, 1982.
Riccardo et al.
1524 Langmuir, Vol. 8, No. 6,1992
0
05
e
Figure 6. Integral heat of adsorption as a function of mean surface coverage.
01
02 03
06
05
06 0 7
0 8 09
1
si
Figure 7. Differential molal adsorption heat as a function of mean surface coverage.
Table I. Symbol Convention for Figure 6 to 11
model is a good approximation to describe adsorption on correlated heterogeneous surfaces unless the system is close to an adsorbate phase transition. For the thermodynamics of adsorptionl~~ we focused our attention on the integral molal adsorption heat H and on the differential molal adsorption heat q d and entropy s. However some other quantities like the mean lateral interaction and the mean adsorbate-surface interaction energies have been calculated as well. The integral heat H can be easily calculated in the canonical ensemble, by fixing the coverage 8 and the temperature T, through
(21)
t 0
where E a is the total adsorbate energy for a given configuration at coverage The differential molal heat q d and entropy s are obtained from
e.
(22)
(23)
Finally, by assuming an ideal gas phase and a liquidlike adsorbed phase, the spreadingpressure 4 can be calculated as a function of p by the relation
Results for thermodynamic quantities are shown in Figures 6 to 8, with the symbol convention given in Table I and compared to the ideal (non interacting) adsorbate
05
B
Figure 8. Differential molal adsorption entropy as a function of mean surface coverage.
behavior (solid line). The effect of energy correlations is to enhance the integral adsorption heat (Figure 6). This effect is stronger at low coverage and for higher lateral interactions. The differential molal heat of adsorption (Figure 7) behaves in a more interesting way: at low 8, q d is enhanced for a surface with a larger correlation length, while at high 8 the opposite effect occurs, just as in the adsorption isotherms. Differential entropy behavior deserves some special comments. Entropy changes with coverage may be regarded as a configurational entropy change plus a thermal entropy change. Configurational entropy is proportional to the chemical potential (at fixed temperature) of the adsorbate and thermal entropy to the transferred heat upon differential changes of the adsorbed amount. It can be seen from Fig-urea 5 and 7 that ro has opposite effects on p and Qd. For 8 < 0.5, p decreases with ro; however the variation in q d is stronger and, as a result,
Generalized Heterogeneous Surfaces
Langmuir, Vol. 8, No. 6, 1992 1525
the differential entropy decreases with the adsorptive energy correlation. The situation is completely reversed for 8 I0.5. An overall consistent explanation will emerge after studying the behavior of adsorbate clusters. 2.6. Adsorbate Cluster Morphology a n d Percolation. This section is concerned with the influence of energetic topography (say energetic correlation) on the adsorbed phase morphology from the point of view of percolation theory. As is well known, lattice and continuum percolation concepts have been successfully applied to a huge variety of problems due to their versatility and simpli~ity.3~-~~ Actually we are interested in some features of percolation theory like cluster size distribution, mean cluster size, and percolation threshold, which are expected to be dependent on the adsorptive energy topography. The morphology of a noninteracting adsorbed phase at low temperature (low enough as to ensure localized adsorption) should reflect the structure of adsorptive energy distribution. In fact, as temperature approaches zero, the probability P(e) for a gas molecule to stick on a site of energy e becomes a steplike function
P(r) =
1 0
t f
I
Figure 9. Mean number of unitary clusters per lattice site for a noninteracting adsorbate for different correlation lengths: 0 , ro = 0;+, ro = 1; 0 , ro = 2. The full line represents the case of a homogeneous surface.
lo-'lo
ifrIp otherwise
where for a given coverage 8 the cutoff energy ~1 can be obtained by solving (26)
In what follows the usual meanings for connectedness and cluster are used. Occupied nearest neighbor sites are considered to be directly connected by a bond, and a cluster is formed by a set of filled sites in such a way that each member of the set is connected to at least another one in the same set. Furthermore the critical concentration, or the percolation threshold, is defined by the appearance of at least one cluster connecting two opposite lattice sides (provided the lattice is sufficiently large). To obtain the cluster size distribution n, (the mean number of clusters of size s per lattice site) and the mean cluster size S(8),Monte Carlo simulations in the canonical ensemble (T and 8 fixed) on square lattices with adsorptive energies given by the bivariate Gaussian distribution for several values of ro were performed and a clustercounting algorithm recognizing periodical boundary conditions (similar to Stauffer's algorithm38) was used. Surfaces were generated through the procedure described in section 2.5 and simulations were carried out at low temperature, say ;IkT = 15, instead of zero temperature, to allow several MC configurations to be sampled from the same lattice. The heterogeneity parameter corresponds to TJT = 1.85. Each point drawn in Figures 9 and 10 has been obtained by averaging 200 MC configurations with 20 MCS per lattice sites between successive configurations to ensure that the effects of statistical correlations are minimized. The behavior of the unitary cluster distribution nl (Figure 9) for noninteracting adsorbates reveals clearly the influence of the correlation length. For ro = 0 the (36) Zallen, R.The Physics of Amorphous Solids; Wiley-Interscience: New York, 1983.
(37) Deustscher, G.;Zallen, R.; Adler, J. Percolation Structure and Processes,AnnakrofTheIsraelPhysica1Society:Israel Physical Society: Jerusalem, 1983; Vol. 5. (38) Stauffer, D. Introduction to Percolation Theory; Taylor and Francis: London, 1985. (39) Balberg, I. Philos. Mag. B 1987,56 (6), 991. (40) Wagner, N.;Balberg, I. J. Stat. Phys. 1987, 49 (112). (41) Balberg, I. Phys. Rev. Lett. 1987,59 (12), 1305.
0
10
20
30
S Figure 10. Cluster size distribution for different correlation lengths and different coverage values, corresponding to a noninteracting adsorbate. variation of nl with 8 is the same as in the case of an homogeneous surface, while for ro > 0, nl is considerably lower even at relatively low coverage. As a complement, Figure 10showsthat the influenceof the correlation length on the typical exponential fall of the cluster size distribution is stronger for low coverage (8 = 0.2) than for high coverage (8 = 0.58). Both figures show that the clustering of adsorbate is enhanced by a larger correlation length. Snapshots of MC configurationsrepresent the adsorbate for different situations in Figures 11through 14. At low 8 (Figure 11)for a noninteracting adsorbate, occupied sites are highly dispersed for a random surface (a) while there is a considerable amount of island formation for a correlated surface (b and c), reflecting the fact that for ro 2 1a highly adsorptive site is very probably surrounded by other highly adsorptive sites. If we go now to high coverages, Figure 13, we see that the unoccupied sites (white regions) are quite disperse for ro = 0 (a) while larger white islands are present for larger ro (b and-c); i.e. we have the dual behavior compared with the low 8 behavior. The effect of attractive interactions, Figure 14, is to increase the adsorbate island formation for all surfaces. On comparison of Figures 12c and 14a it can be seen that a noninteracting adsorbate on a correlated surface may look like an interacting one on a random surface. Mean cluster size S in Figure 15 (also named percolation susceptibility) indicates that correlation forces the ad-
Riccardo et al.
1526 Langmuir, Vol. 8,No. 6,1992
Figure-11. Snapshots of noninteracting adsorbate at T = 100 K and 0 = 0.3: (a) TO = 0; (b) ro = 1; (c) T O = 2. sorbate to percolate at lower coverages. A quite interesting result is, perhaps, the dependence of the percolation threshold on the correlation length ro displayed in Figure 16. The percolation threshold decreases monotonically from 8, = 0.59 for ro = 0 down to 8, 0.5 for ro m , The value of 8, = 0.59 is well understood given that for ro = 0 different adsorptive energies are randomly distributed on the lattice so that the critical density should be the wellknown equal probability random percolation threshold p c = 0.59275.38 The lowest attainable critical density 8, = 0.5 can be explained on the basis of well-established continuum percolation arguments. For a lattice with very large correlation length, sites with equal adsorptive energy group together in large compact circles. Patches differing in energy are randomly arranged on the lattice and border 03 limit. Low-temperature effects vanish in the ro adsorption proceeds by filling from the higher to lower energy circles. In the thermodynamic limit (lattice size L =) this lattice percolation problem can be mapped onto a continuum one, like that depicted in Figure 17, where the lattice has been replaced by a two-dimensional continuum sheet and clusters transformed in black spots. Now if the adsorptive energy distribution exhibits homogeneity on a macroscopic scale, isotropic correlations and symmetry around the mean adsorptive energy, as the bivariate Gaussian distribution f i ( e l , e z ) does, then black domains at coverage_8are topologicallyequivalent to white domains at (1 - 0) leading to the well-known twodimensional random continuum percolation threshold 8, = 0.5.3738 Points corresponding to intermediate values of ro in Figure 16should be considered as only qualitatively correct
-
-
-
Figure 12. Snapshots of noninteracting adsorbate at T = 100 K and 0 = 0.56: (a) ro = 0, (b) ro = 1,(c) ro = 2. The bigger cluster has been darkened.
due to finite size effects. However, the two extreme cases just discussed should be regarded as exact. The effect of ro on the percolation threshold is illustrated in the snapshots of Figure 12 which are taken at 8 = 0.56,just below the random lattice percolation threshold: (a) ro = 0, the uncorrelated surface is far from percolating; (b) ro = 1, the system is near critical, where a small amount of bonds added would provoke percolation; (c) ro = 2, the system is well above the percolation threshold, and several connected paths lead throughout the lattice.
-
Since site occupation probability depends upon temlimit the adperature, it is expected that in the T sorbed phase ignores adsorptive energy correlations and feels the lattice as homogeneous. Thereby the vertical line at 8 = 0.59will separate nonpercolating and percolating phases in Figure 16. Snapshots showing this effect are depicted in Figure 18 where it can be seen that adsorbates on surfaces with ro = 0 and FO = 2 look very similar if temperature is raised. Finally, we wish to mention that, even if critical exponents will not be modified by adsorptive energy correlation, the behavior of several quantities will be sensitive to it below 8,. In particular, for the cluster radius, which behaves like38
where s is the number of lattice points belonging to the cluster and
Generalized Heterogeneous Surfaces
Langmuir, Vol. 8, No.6, 1992 1527
Figure 13. Snapshots of noninteracting adsorbate at T = 100 K and 0 = 0.65: (a) ro = 0; (b) ro = 1;(c) ro = 2. The bigger cluster has been darkened.
Figure 14. Snapshots of an adsorbate with attractive nearestneighbor interactions of -0.5 kcal/mol, at T = 100 K: (a) ro = 0; (b) ro = 1; (c) ro = 2. The biggest cluster has been darkened.
Ln
S 9 8
-
adsorptive energy correlation will modify dffar below critm. icality from df = 1.56 for ro = 0 to df = 2 for ro 2.7. Interpretationof the Adsorption Behavior for acorrelated Surface. We are now in a position to discuss the adsorption process in light of cluster morphology on correlated surfaces, as a complement to the analysis performed in section 2.3. At low coverage, Figure 11, adsorption on a correlated surface proceeds through a nucleation and growth process on small islands of highly adsorptive energies, while these energetic sites are highly disperse in an uncorrelated surface. In this way, for a correlated surface, a cooperative action between the correlation length of adsorptive energies and attractive adsorbate-adsorbate interactions results in an enhancement of the differential heat of adsorption and the adsorbed amount. At high coverage the situation is reversed, Figure 13, empty sites (white regions corresponding to low adsorptive energy) form islands for a correlated surface and are highly disperse for an uncorrelated one. In this way, for an uncorrelated surface an empty site has a considerable probability of being surrounded by one or more adsorbed molecules resulting thus in an enhancement of the effect of lateral interactions which produce then higher differential adsorption heat and adsorbed amount.
7
6
5 4
3 2 1
0 -1
-2
01
0.2 0.3 0.6 0.5
0.6 0.7 0.8 0.9
B
Figure 15. Mean cluster size behavior for a noninteracting adsorbate as a function of the mean coverage: 0 , ro = 0;0,ro = 2. The hatched strip represents the region where finite size effects become considerable.
3. Surface Diffusion 3.1. A Model for the Activation Energy. Surface diffusion of adsorbed molecules has a predominant influence on several gas-solid processes, such as thermal
Riccardo et al.
1528 Langmuir, Vol. 8, No. 6,1992 00
CI.1
I
Non-Percolation Region
I
II
r0
2
f
II
I I I II
1
I I
0.5
I I
I I
U
X 0
Figure 16. Percolation threshold as a function of ro (right scale). The equivalent scale on the left is measured in terms of the correlation function at r = a, the lattice spacing. Dots ( 0 )indicate the mean coverage at which the spanning cluster appears and the right dashed line passing through them separates the percolation region from the nonpercolation one. The left dashed line stands for the dual behavior, i.e. percolation of unoccupied sites. I
I
Figure 17. Transition from lattice percolation to continuum percolation.
‘ 3
r,
I
2
Figure 19. Adsorptive energy variation along a direction x on the surface.
teractions was i n ~ e s t i g a t e d . ’ 0 ~ 1 1 ~Here, ~ 5 ~ ~we~ ~are ~ ~interested in discussing surface diffusion in close connection with adsorption features, focusing our attention on how adsorptive energy topography influences the collective diffusion coefficient,a problem which has not been treated in the literature. There are two main factors connecting surface diffusion and adsorption processes. On one hand, ~ J ~it Jis ~reasonable to it was already pointed O U ~ ~ that propose that energy barriers for jumps between two sites cannot be, generally speaking, independent of their adsorptive energies. Therefore adsorptive energy correlations will influence jumping probability correlations. On the other hand, we expect adsorptive energy topography to be crucial in modifying the vacancy factor, say, the availabilityof empty sites surrounding a fiied site to which a jump might be performed. It can be added that changes in local concentrations caused by energy correlations should greatly affect the dynamics of migration. In what follows we will describe the diffusion process on a two-dimensionalarray of adsorptive sites. As assumed before, sites can be singly occupied and heterogeneity is introduced through the Gaussian bivariate distribution, eq 15, where different topographies are characterized by the correlation length ro. Nearest-neighbor and nextnearest-neighborlateral interactions are taken into account as given by the Lennard-Jones potential. Diffusion proceeds by means of activated jumps to empty nearestneighbor sites. It has been recently proposed15 that the variation in activation energy for a jump from a site labeled “i” to a nearest-neighbor site “j”, due to heterogeneity, could be expressed as a weighted average of their adsorptive energies ti, tj as follows (see Figure 19) = (a- l)(ti - ;)
+
-i)
(28)
where 0 5 a I 1/2 is a structure parameter taking into account the influence of destination site on the jump. If we denote with :t the saddle point energy (bond energy) between sites i and j , its variation AtY is given by Figure-18. Snapshots of noninteracting adsorbate at T = 600 K and 0 = 0.56. The biggest cluster has been darkened.
desorption, growth of evaporated films on surfaces, or reactions on catalysts, and this deserves to be studied on its own. Diffusion of noninteracting and interacting adsorbates on uniform surfaces has been throughly d i s c ~ s s e d 6 - 9 and J ~ ~recently ~ ~ ~ ~ ~diffusion on heterogeneous surfaces at finite coverage has been dealt with where the simultaneous influence of heterogeneity and lateral in___
~
~~
..
ActJ
-
( E ~ J ;b)
CX(C~
- ;)
+ a(tj- ;)
These concepts are illustrated in Figure 19. After a short calculation it can be seen that statistical measures of activation energy, like mean value and covariance,are given by i-j ia = (ea
)
= (i- ib)
_____
(42)Bassett, D.W. In Surface Mobilities on Solid Materials; Binh, V. T., Ed.; Wiley: New York, 1983. (43)Gomer,R. Rep. Prog. Phys. 1990,53, 917.
(29)
The corresponding equations for bonds are
(30)
Generalized Heterogeneous Surfaces
Langmuir, Vol. 8, No. 6, 2992 1529
I
Figure 20. Adsorptive energy profiles along a direction x on the surface for ro = 0 and different values of a,from top to bottom; a = 0, a = 111, a = ‘12. 0
I
1.1
0
1.2
.
t
2 1
0
1
1,z-1
I
I
I
IC1
I
I
1
I
Figure 21. Same as Figure 20 for PO = 1.
.. (
- tb)(e:”
(et’)
-
-
+ ea) - i b ) ) = CY2kT:[C(rik) + C(ril) + ib
= (e
(31)
C(rj;.k)+ C(rjJ1 (32) The influence of the structure parameter on the diffusion coefficient in the limiting case of randomly distributed adsorptive energies (ro= 0) has been already in~estigated.’~ Now we intend to study how energy topography influences diffusion by using relation 28. A wide variety of surface profiles can be obtained by combining different values of the correlation length ro and of the parameter a as shown in Figures 20,21, and 22. When ro = 0 (Figure 20) we have for a = 0 an adsorptive energy profile like that of a random traps lattice (RTL), a lattice of traps with randomly distributed depths, while for a = 0.5 the heterogeneous potential is a continuous random perturbation (CRP) of the homogeneous periodical potential (the dashed curve of Figure 19). When ro = a (Figure 21), trap depths vary smoothly for CY = 0 along the surface due to statistical correlations, originating a correlated traps lattice (CTL), while for a = 0.5 the heterogeneous potential is a continuous correlated perturbation (CCP) characterized by both bond and site energies smoothly varying. In the extreme case of ro 03 (Figure 22) equal depth traps are
-
grouped in very large patches for a = 0, while for a = 0.5 we have homogeneous perturbations on very large lattice extensions. 3.2. The Collective Diffusion Coefficient. For the calculation of the collective diffusion coefficient the procedure used in ref 15 can be easily generalized to include heterogeneity. In this way the net flux of molecules from row 1 to row 2 of a given lattice with coordination number z , as shown in Figure 23, can be expressed in the general form J=
-[ vefp2exp(p/kT)-k T
1 ap
: x
a8
where the expression between brackets is the collective diffusion coefficient, with veff = v eXp(-€b/kT), u being the jump-attempts frequency, p the chemical potential, and C = 0/a2the adsorbate concentration. is the probability of sites “1” and “2” to be empty with their environment marked by index i (which refers to specific adsorptive energy as well as to the site occupation states of a given configuration around them) and Wi* is the lateral interaction of the activated complex (the molecule on top of the saddle point separating sites 1 and 2) with its environment i. Aeb1*2 denotes the saddle point energy variation due to heterogeneity as given by eq. 29. Pooli
Riccardo et al.
1530 Langmuir, Vol. 8, No. 6,1992
v g
1 I
bk\
06-
0
.
0
0
0
04
~
0 0
\ 0
Figure 24. Surface diffusion coefficient for a noninteracting adsorbate, T IT = 0: -, ro = 0; - - -, ro = 1; ro = 2. (a) a = 0; (b) a
=gf/q;
(c) a =
-
02 -
0a
d
0 00
0-
a,
'/2.
05
-
e
Figure 26. Monte Carlo results for the vacancy factor (mean number of nearest-neighborempty sites per occupied site) as a function of mean coverage. Symbols are as in Table I. The solid line represents the well-known behavior of a-noninteracting adsorbate on a homogeneous surface, V = 1 - 8.
The average (...) is taken over a statistical ensemble of surfaces with a bivariate distribution function. To observe the effects of the adsorptive energy topography, we must use at least the MRF approximation of second order ( N = 2). Thus, the diffusion coefficient becomes D(8) = vefp2exp(p/kT)- 1 acc R(8) (34) kT a8 where R(8) is given by
-
R(8) =
JJr,
(t 1, t2) exp{-a[(e,
(e2 - ;)I
-
- t) +
- W*8)(1- 01)(1 - 8,) de, de2
and Wi* in eq 31 has been replaced, for the sake of simplicity, by a mean interaction W*8. Here 0, 8, el, and 8 2 have the same meaning as in the formulation of adsorption. Since they must satisfy isotherm equations, el, 82, and 8 are obtained by solving the coupled equations 12 to 14, where 8 and p are known. All calculations were made for T = 400 K, elkT = -3.7, and T,/T = 1. Activated complex mean interaction was taken as W* = 0 given that its influence is merely that of a constant multiplying factor enhancing (decreasing) D(8) for attractive (repulsive) energy. We have preferred to represent D(@/D,(O) vs 8, where Do(0) is the value of D(8) for 8 = CY = ro = 0 , in order to compare D(8) for several surfaces with different values of ro and a. Results are shown in Figures 24 and 25 for noninteracting and interacting adsorbates, respectively. The behavior of the vacancy factor shown in Figure 26 is now another fundamental piece for the understanding of the effect of adsorptive energy topography on the surface migration process. An interesting case is that of Figure 24a, which represents an ideal adsorbate on a lattice of traps. Activation energy here is, except an additive constant, equal to the absolute value of the adsorptive energy; consequently the-influence of the adsorptive energy topography on D(8) can be separately investigated. For a RTL (ro = 0 ) as well as CTL (ro = l), D(8) steadily increases with 8 due to the fact
that activation energy decreases as adsorbed molecules cover less energetic sites (lessdeep traps). Howevercurves for ro = 1, 2 show a much faster increment for 0 > 0.2 reaching much higher values at 8 = 1. In fact, from Figure 13 we see that at high coverage, for ro = 0, migrating molecules move in a very intricate region (white areas), while for ro > 0 white areas form larger islands so that the net flux through a given line is clearly enhanced. The effect of the vacancy factor in this case is not important due to the absence of lateral interactions. As CY increases (Figure 24b,c) the barrier to be overcome in a jump depends not only on the starting site energy but also on the destination site energy in such a way that barriers are lowered for very energetic starting sites (low coverage) and raised for less energetic starting sites (high coverage) giving rise to an increment (decrement) of diffusion at low (high) coverages in addition to the general behavior described for CY = 0. The symmetry of curves for CY = 0.5, showinga maximum a t 8 = 0.5 is a consequence of the symmetrical role played by the two sites participating in a jump on determining the activation energy and the assumed symmetry of the site energy distribution. A more complex situation rises in the case of an interacting adsorbate (Figure 25) where for a given structure of activation energies (given CY)a qualitatively distinct behavior is produced by finite correlation length. The minimum and maximum displayed in parts a and b for ro = 1,2, in contrast with ro = 0, is a manifestation of the vacancy factor which becomes important when lateral interactions are present. The competing effects of decreasing activation barriers as 8 increases and the strong decrease in the vacancy factor as ro increases yield the minimum a t low 8 for a = 0 and rg = 1,2. The maximum at high 0 may be understood by considering the mobility of vacancies instead of molecules. For CY = 0.5, Figure 25c, the effect of decreasing activation barriers as 8 increases is not sufficiently strong and the minimum and maximum disappear. In addition to this we have an overalldecreasing factor due to attractive lateral interactions. In conclusion, surface diffusion is strongly affected by adsorptive energy correlations through induced correlations on activation barriers as well as through the influence of adsorbate cluster morphology. 4. Characterization of Heterogeneous Surfaces The problem of the energetic characterization of an heterogeneous surface through its interaction with the
Generalized Heterogeneous Surfaces molecules of a gas can be analyzed from a new perspective when complementary use is made of adsorptive and surface diffusion. The same surface may look quite different to different probe molecules,24so that the problem becomes more difficult since complete sets of experimental data are not available at present for the same gas-solid system. Physical adsorption has been largely used in pursuing this objective, and so many efforts have been devoted to develop computational algorithms to obtain adsorptive energy distributions from equilibrium adsorption data.5 However these methods are rather sensitive to the model assumed in describing the adsorption process and only simple statistical parameters can be obtained. Adsorption data analysis on its own would be insufficient if structure parameters like the correlation length are to be specified. On one hand, correlation length effects on adsorption isotherms are so closely related to lateral interaction that weakly interacting adsorbates would hardly be appropriate to disclose energetic topography. On the other hand, for strongly interacting adsorbates, we would not be able to identify the energetic topography from a set of experimental adsorption isotherms. At most we could assume a given topography and try to fit theoretical isotherms to experimental ones. However several different topographies would give a satisfactory fitting unless other parameters, like for example Tggand T,, were independently determined. With respect to surface diffusion the situation changes given that D(6)seems to be strongly modified by correlation length and activation energy structure even for noninteracting molecules. If a complete set of adsorption isotherms and diffusion coefficient measures were made on the same sample using a weakly interacting adsorbate, then something about ro and a could be known. For instance, mean value and dispersion of adsorptive energy distribution could be obtained from adsorption isotherms through the already available methods and then ro and CY could be determined by fitting the set of surface diffusion data. For strongly interacting adsorbates the problem becomes
Langmuir, Vol. 8, No. 6,1992 1531 harder but we think that qualitative differences appearing in adsorption isotherms, adsorption heats, and diffusion coefficient may be useful to get some idea about the energetic structure. Nevertheless for a number of gassolid systems, specially for chemisorption, interaction energies can be independently estimated through LEED and then a combined analysis of adsorption and diffusion data might be applied. Effects of ro and CY on adsorption, desorption, and nucleation kinetics are being worked out in order to investigate how a surface can be characterized through a wider set of intrinsically connected phenomena. 5. Conclusion
A general formulation of adsorption and surface diffusion of gases on heterogeneous surfaces has been developed which is capable of taking into account the energetic structure of adsorptive sites through a correlation length ro and a bond characteristic parameter a. Model results are in good concordance with Monte Carlo simulations at moderate adsorbate-adsorbate interactions. Moreover, Monte Carlo simulation allows the study of adsorbate cluster morphology and percolation threshold which are strongly influenced by ro. The correlation length appears as an adequate physically meaningful parameter characterizing the adsorptive energy structure. Adsorption is quite sensitive to ro only for interacting adsorbates; the effect of adsorptive energy structure is stronger for stronger lateral interactions. Surface diffusion is much more sensitive to the structure parameters ro and CY, even for noninteracting adsorbates. An important conclusion is that adsorptive energy correlation induces not only quantitative but also qualitative differences in adsorption isotherms and diffusion coefficient. Another important point is that complementary interpretation of adsorption equilibrium and surface diffusion data seems to be a promising way toward a better characterization of heterogeneous surfaces, and the design of appropriate complete sets of experiments is encouraged.
