Molecular Processes on Heterogeneous Solid Surfaces - American

Langmuir 1996, 12, 129-138

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Molecular Processes on Heterogeneous Solid Surfaces† G. Zgrablich*,‡ Centro Regional de Estudios Avanzados, Gobierno de la Provincia de San Luis, Casilla de Correo 256, 5700 San Luis, Argentina

V. Mayagoitia and F. Rojas Departmento de Quı´mica, Universidad Auto´ noma Metropolitana, Me´ xico D.F., Me´ xico

F. Bulnes, A. P. Gonzalez, M. Nazzarro, V. Pereyra, A. J. Ramirez-Pastor, J. L. Riccardo, and K. Sapag Departmento de Fı´sica, Facultad de Ciencias Fı´sico-Matema´ ticas y Naturales, Universidad Nacional de San Luis, Chacabuco y Pedernera, 5700 San Luis, Argentina Received November 7, 1994. In Final Form: April 5, 1995X

The dual site-bond model (SBM) to describe heterogeneous surfaces with different energetic topographies is reviewed and applied to the study of molecular processes on solid surfaces such as adsorption, surface diffusion, reactions, and diffusion limited aggregation. It is found that all these processes are strongly affected by the energetic topography, and the observed behaviors are discussed.

1. Introduction Real solid surfaces involved in a great number of phenomena, for example, adsorption, catalysis, corrosion, films growth, etc., are heterogeneous. We are interested in describing the interaction of gases with such surfaces. In this case, heterogeneity means that, if a probe molecule is moved on a plane parallel to the solid surface, the interaction energy between the molecule and the solid varies in a stochastic way from point to point. This stochastic potential is an energy surface in a threedimensional space [x,y,U(x,y,z0)], where x and y are the spatial coordinates on the solid surface and U(x,y,z0) is the potential energy seen by the probe molecule at the point (x,y) and at a distance z0 from the solid surface such that U(x,y,z) is minimum. A rigorous treatment of any molecular process occurring on such an heterogeneous surface should consider the molecules as immersed in the bivariate continuum stochastic field E(x,y) ≡ -U(x,y,z0). This is a very difficult task which could only be attempted in the framework of computer simulation with the aid of powerful supercomputers. Realistic computer simulations of different kinds of solids are already being carried out and their interaction with gas molecules is being studied by molecular dynamics.1,2 Our approach is to simplify the problem by concentrating our attention on certain elements of the adsorptive energy surface which are more relevant for the behavior of molecular processes. These elements are the adsorptive “sites”, i.e., the deeper wells in the surface U(x,y,z0), and the “bonds”, i.e., the saddle points through which a molecule passes when jumping from each site to a nearestneighbor one following the minimum energy path. † Presented at the symposium on Advances in the Measurement and Modeling of Surface Phenomena, San Luis, Argentian, August 24-30, 1994. ‡ Also at the Departamento de Fı´sica, Universidad Nacional de San Luis, San Luis, Argentina. X Abstract published in Advance ACS Abstracts, January 1, 1996.

(1) Bakaev, V. A. Surf. Sci. 1988, 198, 571. (2) Bakaev, V. A.; Steele, W. A. J. Chem. Phys. 1993, 98, 9922.

0743-7463/96/2412-0129$12.00/0

Even though these elements occupy finite size regions, we make another approximation by considering the surface a regular lattice of sites connected by bonds and by considering the behavior of adsorbed molecules in the framework of lattice-gas theory. The problem now is how to obtain a statistical description of sites and bonds in such a way as to be able to generate different energetic topographies with a minimum number of parameters. One possible description, known as the generalized Gaussian model (GGM),3-5 was developed by considering a bivariate Gaussian distribution for pairs of site energies determined by a correlation function between the energy B and that of a site at R B +b r as of a site, ES, at position R

〈ES(R B ) ES(R B+b r )〉 ) σ2e-(1/2)(r/r0)

2

(1)

where σ is the dispersion of the site energy distribution and r0 a correlation length. In this description, bond energies are not allowed to vary at random but they are obtained in a deterministic way in terms of site energies.6,7 The parameter r0, the correlation length, describes different energetic topographies in such a way that for r0 ) 0 we have a completely random surface but as r0 increases surfaces with a higher spatial correlation degree are obtained and in the limit r0 f ∞ a surface composed by a collection of macroscopic periodic homogeneous patches is approached. A more recent description,8-10 known as the dual sitebond model (SBM) because sites and bonds are treated (3) Ripa, P.; Zgrablich, G. J. Phys. Chem. 1975, 79, 2118. (4) Riccardo, J. L.; Pereyra, V.; Rezzano, J. L.; Rodriguez Sa´a, D. A.; Zgrablich, G. Surf. Sci. 1988, 204, 289. (5) Riccardo, J. L.; Chade, M.; Pereyra, V.; Zgrablich, G. Langmuir 1992, 8, 1518. (6) Pereyra, V. D.; Zgrablich, G. Langmuir 1990, 6, 118. (7) Pereyra, V.; Zgrablich, G.; Zhdanov, V. P. Langmuir 1990, 6, 691. (8) Mayagoitia, V.; Rojas, F.; Pereyra, V.; Zgrablich, G. Surf. Sci. 1989, 221, 394. (9) Mayagoitia, V.; Rojas, F.; Pereyra, V.; Zgrablich, G. Phys. Rev. 1990, B41, 7150. (10) Riccardo, J. L.; Pereyra, V.; Zgrablich, G.; Rojas, F.; Mayagoitia, V.; Kornhauser, I. Langmuir 1993, 9, 2730.

© 1996 American Chemical Society

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equal, is based on the joint distribution

F(ES,EB) ) FS(ES)FB(EB)φ(ES,EB)

(2)

where FS(ES) [FB(EB)] is the probability of finding a site [bond] with energy between ES and ES + dES [EB and EB + dEB], F(ES,EB) is the probability of finding a pair of connected sites and bonds with those energies and φ(ES,EB) is a correlation function which carries the useful information on the energetic topography. There could be many ways of defining the correlation, or structure, function φ(ES,EB), corresponding to the many ways in which bonds and sites of different energies could be assigned to each other. However, if we allow for the maximum degree of randomness in such an assignation process, compatible with the rule that the energy of a site must be greater than that of any of its connected bonds, then we obtain8

{

S(E ) dS/(B - S)] ∫S(E )

exp[-

φ(ES,EB) )

S

B

for ES g EB

B(ES) - S(ES) for ES < EB

0

(3)

where

B(E) )

∫0E FB(EB) dEB g S(E) ) ∫0E FS(ES) dES

(4)

The main parameter on which φ(ES,EB) depends is Ω, the overlapping between sites and bonds distributions, FS and FB. In the special case where FS and FB are uniform distributions

{ {

1/∆ 0

for s e ES e s + ∆ otherwise

1/∆ FB ) 0

for b e EB e b + ∆ otherwise

FS )

(5)

(6)

then a simple expression can be obtained for φ(ES,EB)

φ(ES,EB) ) exp[-R(ES,EB)Ω/(1 - Ω)]/(1 - Ω) (7) where

{

R(ES, EB) ) (ES - s)/(b + ∆ - s) 1 (ES - EB)/(b + ∆ - s) (b + ∆ - EB)/(b + ∆ - s)

for EB e s; ES e b + ∆ for EB e s; ES > b + ∆ for EB > s; ES e b + ∆ for EB > s; ES > b + ∆

In this description it is found that the spatial correlation between the adsorptive energies at two points separated a distance r decays as:

〈E(R B ) E(R B+b r )〉 ∼ e-r/r0

(8)

where now r0 ≈ Ω/(1 - Ω). Different energetic topographies can be generated in this model by changing FS and FB and, in particular, Ω. Figure 1 represents schematically some of them by showing for each FS, FB, and Ω (left side) how the energy along a direction x on the solid surface changes (right side). It is important to distinguish between two particular kinds of surfaces: a trap surface (the one where all bond energies have the same energy while site energy changes with position) and a barrier surface (all site energies are the same while bond energies change with position).

Figure 1. One-dimensional representation of different topographies obtained in the SBM.

We claim that a great majority of molecular processes occurring on heterogeneous surfaces are strongly affected by energetic topography. Some of these processes, i.e., adsorption and surface diffusion, have been deeply investigated by using the GGM,5,7 while only some isolated results have been presented on the basis of the SBM. Our purpose here is to give a unified view of the effects of energetic topography on surface processes like adsorption, surface diffusion, surface reactions, and diffusion limited aggregation (DLA), by using the SBM. All results presented here will be based on Monte Carlo simulation of each process considered. These processes are simulated on correlated site-bond surfaces which are generated according to the procedure described in detail in ref 10. Figure 2 is a color representation of two such surfaces corresponding to Ω ) 0.7 (a) and Ω ) 0.9 (b), respectively. The white color represents regions with deepest (strongest) adsorptive potential, while the blue color, on the other end of the scale, represents regions with shallowest (weakest) adsorptive potential. It can be observed how the mean patch size (of similar adsorption sites), or the correlation length r0, increases noticeably when Ω goes from 0.7 to 0.9. 2. Adsorption Here we must distinguish between adsorption of monomers and dimers, or higher order mers, where we intend for monomers those molecules occupying a single adsorption site, for dimer a molecule occupying two nearest neighbor adsorption sites, and so on. This distinction is relevant due to the fact that for monomers, adsorption is sensitive to the surface energetic

Molecular Processes on Heterogeneous Solid Surfaces

Langmuir, Vol. 12, No. 1, 1996 131

Figure 2. Color representation of two surfaces with different topographies: (a) Ω ) 0.7; (b) Ω ) 0.9. Lighter zones represent deeper adsorptive potential, darker ones correspond to weaker adsorptive potential. Uniform site and bond distributions were used, eqs 5 and 6, with ∆ ) 5 kcal/mol on a square lattice.

topography only if molecules interact with each other, while for dimers, or higher order mers, adsorption is

significantly affected by the energetic topography even if lateral interactions are negligible.

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Figure 3. Adsorption isotherm and isosteric heat of adsorption for three surfaces with different topographies. Adsorbateadsorbate interactions are described by a Lennard-Jones potential. The dimensionless temperature is kT/U0 ) 1, where U0 is the depth of the intermolecular potential.

Figure 4. Adsorption isotherms of dimers on surfaces with two kinds of sites S1 and S2 with energies ES1 and ES2 forming patches with mean size r0. Relative abundances of sites are f1 and f2, respectively.

Figure 3 shows the behavior of adsorption for monomers interacting through a Lennard-Jones potential on surfaces with different topographies, where adsorption isotherms (coverage θ versus pressure p) are shown in (a) and isostheric heat of adsorption, qst, versus coverage curves in (b). The morphology of adsorbate clusters is essential to explain the observed behavior. For a random heterogeneous surface, Ω ) 0, at low coverage molecules occupy the strongest adsorptive sites, but these are preferently surrounded by weak sites so that adsorption and attractive lateral interaction do not

Zgrablich et al.

Figure 5. Tracer diffusion coefficient (a) and jump correlation factor (b) for surfaces with different topographies.

cooperate with each other. This makes the adsorption isotherm rise slowly while the heat of adsorption presents the typical decrease due to heterogeneity. At high coverage, however, due to the fact that adsorbate clusters are very ramified, weak adsorption sites which still remain empty are likely to be surrounded by adsorbed molecules, qst increases due to lateral interactions, and the adsorption isotherm is enhanced. For a highly correlated surface, Ω ) 0.9, on the contrary, for which the energetic topography is close to a patchwise one, at low coverage a molecule occupying a strong site is likely to be surrounded by other molecules occupying neighbor strong sites. This makes adsorption and attractive lateral interactions cooperate with each other resulting in an increasing qst with an enhancement of the adsorption isotherm. At high coverages, however, empty weak sites are likely surrounded by other empty weak sites, which form “lakes” in the adsorbate, resulting in a decrease in qst and a strong attenuation in the adsorption isotherm. Let us go now to the case of adsorption of dimers. For simplicity we shall discuss the adsorption of symmetric dimers (i.e., composed of two identical particles) on a surface with only two types of sites, with adsorption energies ES1 and ES2, exhibiting different topographies. This problem has been treated in the context of mean field approximations;11-13 however here we shall use Monte Carlo simulation to expose more accurately the behavior of adsorption isotherms and to be able to obtain the results for intermediate topographies. These kinds of correlated surfaces with discrete distributions of sites (and bonds) can be obtained in the context of the SBM through a limiting procedure. For example, the surface under consideration with sites of energies ES1 and ES2 with


Langmuir, Vol. 12, No. 1, 1996 133

Figure 6. (a) Chemical diffusion coefficient for surfaces with different topographies. (b) Uniform site distribution, FS, and the modified bond distribution, FB.φ, from which bonds are sampled for three values of site energy: 1 S s, 2 S s + ∆/4, 3 S b + ∆.

occurs for 1 < r0 < 2. This behavior is of great importance for the use of dimers and higher order mers as probe molecules to characterize the energetic topography of heterogeneous surfaces, as pointed out by Rudzinski14 in his contribution to this symposium. 3. Surface Diffusion We shall analyze first the simpler case of tracer diffusion and then the chemical diffusion coefficients. The diffusion coefficient of a tracer particle is defined as

D* ) lim tf∞

〈R2(t)〉

(9)

2at

where R2(t) is the mean square displacement of the particle after a time t and a is the lattice constant. Another fundamental quantity in studying tracer diffusion is the “jump correlation factor”, f, given by Figure 7. Density of A particles as a function of time for surfaces with different topographies.

relative abundances, say f1 ) 0.6 and f2 ) 0.4, respectively, can be obtained by considering initially a uniform site distribution between energies ES1 and ES2 (and a convenient distribution for bonds), then with a convenient overlapping Ω to obtain a desired correlation length r0, a correlated surface is generated in the usual way and, finally, all site energies in the interval [ES1,ES1 + 0.6(ES2 - ES1)] are set equal to ES1 and the rest are set equal to ES2. Figure 4 shows simulated adsorption isotherms for symmetric dimers on such bimodal surfaces with patchwise topographies with mean patch size r0 ranging from 0 (random surface) to ∞. As can be seen, adsorption is sensitive to the topography even though no adsorbateadsorbate interactions have been considered. It is interesting to note that even the qualitative behavior of the adsorption isotherm changes with the topography. In fact, for a random surface (r0 ) 0) we see that a three-step isotherm, corresponding to the dimer adsorption energies 2ES2, ES1 + ES2, and 2ES1, is clearly obtained. As r0 increases, the adsorption energy ES1 + ES2 occurring only at the interface between site 1 and site 2 patches becomes less and less frequent and the three-stepped form is transformed in a two-stepped one and this transformation (11) Nitta, T.; Kuro-Oka, M.; Katayama, T. J. Chem. Eng. Jpn. 1984, 17, 45. (12) Nitta, T.; Yamaguchi, T. J. Chem. Eng. Jpn. 1992, 25, 420. (13) Marczewski, A. W.; Derylo-Marczewska, A.; Jaromiec, M. J. Colloid Interface Sci. 1986, 109, 310.

〈

f) 1+

2

n

n-m

∑∑ n m)1 k)1

〉

b r k·r bk+m a2

(10)

where b ri is the displacement vector corresponding to jump “i” and n is the total number of jumps performed by the particle in time t. D* and f are related through

D* ) lim tf∞

[ ] na2f 2dt

(11)

where d is the dimensionality of the space. Figure 5 shows the behavior of D*/D∞* (D∞* being the limit of D* at infinite temperature) and f for different topographies. D*/D∞* varies between two extreme boundaries, Figure 5a, the higher one corresponding to random barriers and the lower one corresponding to random traps. This is the manifestation of two fundamental features, the “trap feature” and the “barrier feature”, which are at the basis of the diffusion process on any surface. The “trap feature” consists in the fact that very few deep traps are sufficient to restrict the motion of the particle for long times, independently of how many shallower traps are present and where are they situated. Evidently, a correlated traps surface will behave just in the same fashion. On the other hand the jump correlation factor f, Figure 5b, will be constant for traps, since all jumping directions from a trap will be totally equiprobable. The “barrier feature”, (14) Rudzinski, W.; Nieszporek, K. Langmuir. Submitted for publication.

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Figure 8. DLA cluster grown on a homogeneous surface. Different colors indicate different arrival times.

Figure 9. DLA cluster grown on a surface with Ω ) 0.9 at βE ha ) 1.

instead, consists in the fact that the particle can find alternative walks to go around a higher barrier by choosing a jump in another direction with a lower activation barrier. This makes D*/D∞* grather than the value corresponding to a homogeneous surface. However in the case of correlated barrier surface, low barriers start to group together, being necessarily surroundend by higher bar-

riers, and this plays the role of “multisite trap” and then for sufficiently high correlation the trap feature predominates. This can also be observed for a general correlated site-bond surface, where D*/D∞* is below the value corresponding to a homogeneous surface. It is interesting to note the behavior of the jump correlation factor, Figure 5b, for barrier or general site-bond surfaces, showing a strong decrease as the temperature T decreases due to the flip-flop effect, consisting in a particle jumping forward and backward alternatively over a low barrier between two sites which are bound by higher barriers. Another important feature is that, for a general site-bond, D*/D∞* approaches the random trap behavior as Ω increases, which can be understood by observing that as Ω approaches unity, the surface becomes equivalent to a collection of extended random traps. An interesting complementary results has been obtained recently15 for barrier surfaces, showing that in twodimensional lattices the tracer diffusion does not depend on the degree of disorder of barriers, i.e., does not depend on the dispersion of the distribution of barrier heights. Going now to the chemical diffusion coefficient D(θ), this can show very complex structure as a function of coverage θ when particles interact between them.16 For that reason we restrict our analysis to the case of noninteracting particles. Figure 6 shows the behavior of D(θ)/D(0) for site-bond surfaces with different correlation degrees. The interesting fact is that D(θ)/D(0) shows a (15) Argyrakis, P.; Milchev, A.; Pereyra, V.; Kehr, K. W. Phys. Rev., in press. (16) Murch, G. E. Philos. Mag. 1981, 43A, 871.


Langmuir, Vol. 12, No. 1, 1996 135

Figure 10. DLA cluster on a Ω ) 0.9 surface at βE h a ) 1.

maximum for Ω > 0 and this maximum moves to lower values of θ as Ω increases, Figure 6a. To understand this behavior, we must analyze Figure 6b, representing the site distribution FS and three modified distributions FBφ(ES,EB) from which bonds corresponding to sites with energies s, s + ∆/4, and s + ∆/2, respectively, are sampled. As coverage increases, sites with lower binding energy are being occupied by adsorbed molecules. For a fixed Ω, the mean energy of bonds connected to sites of a given energy is closer to the site energy when it is high (low coverage) and farther when it is low (high coverage), as long as ES > b + ∆. Therefore, for Ω ) 0, while the mean site energy is always decreasing with coverage, the mean bond energy is constant and, as a result, the activation energy for migration continuously decreases with θ and D(θ)/D(0) presents no maximum. For Ω > 0, on the contrary, the mean bond energy is constant as long as ES > b + ∆ and then starts to decrease, compensating for the decrease in Es. This produces a minimum in the activation energy for migration and then a maximum in D(θ)/D(0) which shifts to lower values of θ as Ω increases. 4. Surface Reactions We chose for our analysis one of the simplest reactions, the annhilation reaction

A+Af0

5. Diffusion Limited Aggregation (DLA)

A popular example of such a reaction could be the associative desorption of hydrogen on metals. The normal asymptotic behavior of the density of A particles, ρA, is

ρA(t) tf∞ f t-d/2

(for d e 2)

by Torney and McConnell17,18 for one-dimension and subsequently generalized by Toussaint and Wilczek19 for d-dimensions and for the reaction A + B f 0. This kind of reaction has also been investigated for geometrically heterogeneous media, such as percolation fractals in two and three dimensions, and a superuniversal asymptotic behavior was found20 whatever the initial arrangement of particles could be. We performed Monte Carlo simulations for a onedimensional correlated site-bond surface with three kinds of initial conditions: (a) random initial conditions (RIC), where particles are initially distributed at random; (b) correlated initial conditions (CIC), where particles are initially in thermodynamical equilibrium with the substrate; (c) grouped initial conditions (GIC), where particles are initially grouped together occupying a compact part of the lattice. We found that for RIC and GIC the kinetics is independent of the surface topography even in the intermediate regime. For CIC, however, we find an intermediate regime with a different kinetics, Figure 7, which extends over greater times as Ω increases. For high enough values of Ω, see, for example, Ω ) 0.99, the intermediate regime lasts long enough to consider that the reaction is completed (the density is of the order of 10-4) for all practical purposes. The case of GIC for Ω ) 0.99 is also shown for comparison.

(12)

where d is the dimensionality of the space considered. In absence of an external field, exact analytical results for the dependence of the density on the time was obtained

The DLA model was introduced by Witten and Sander,21 after pioneering work by Finegold22 focused to a biological (17) Torney, D. C. J. Chem. Phys. 1983, 79, 3606. (18) Torney, D. C.; McConnell, H. J. Phys. Chem. 1983, 87, 1441; Proc. R. Soc. London 1983, A387, 147. (19) Toussaint, D.; Wilczek, F. J. Chem. Phys. 1983, 78, 2642. (20) Meakin, P.; Stanley, H. E. J. Phys. A: Math. Gen. 1984, 17, L173. (21) Witten, T. A.; Sander, L. M. Phys. Rev. Lett. 1981, 47, 1400. (22) Finegold, L. X. Biophys. J. 1979, 35, 783.

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Figure 11. DLA cluster on a Ω ) 0.9 surface at βE h ) 0.5.

Figure 12. DLA cluster on a Ω ) 0.9 surface at βE h a ) 0.05.

application, and has stimulated growing interest in the study of a variety of nucleation and growth processes since then.23,24 In the Witten and Sander model DLA clusters are formed from the growth of a seed particle, located at the (23) Stanley, H. E.; Ostrowski, N. On Growth and Form, NATO, Vol. 100 (1986).

origin; individual particles are launched uniformly from a launching circle, sufficiently big, and they perform activated random walks until they stick to the growing cluster. The DLA clusters formed in this way are fractal (24) Meakin, P. In Computer Simulation Studies in Condensed Matter; Landau, D. P., Mon, K. K., Schutller, H. B., Eds.; Springer Verlag: Berlin, 1988.


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Figure 14. Variation of the exponent γ with βE h a for surfaces with different topographies.

Figure 13. Histogram of sites occupied by the DLA cluster on a Ω ) 0.9 surface at different values of βE h a.

objects with a well-determined fractal dimension Df ≈ 1.72 ( 0.02.23 Figure 8 shows one such cluster containing 50 000 particles, where different colors represent different arrival times. It can be seen that it is very rare to find late particles (brown) together with much earlier ones (for example green), which means that the cluster grows in a quite time-ordered fashion from inside to outside. DLA has not been studied before for heterogeneous substrates. We have simulated DLA on surfaces with three correlation degrees: Ω ) 0, 0.7, and 0.9 at four temperatures, such that βE h a ) 3, 1, 0.5, and 0.05. The surfaces were obtained under the same conditions as for Figure 2. Figure 9 shows the cluster obtained for Ω ) 0.9 and βE h a ) 1 (low temperature). Its general aspect is that it is less dense than the cluster of Figure 8. Figures 10-12 represent DLA clusters for a highly correlated surface, Ω ) 0.9, at three temperatures βE h a ) 1, 0.5, and 0.05, respectively, overimposed on the color representation of the adsorptive surface. We see that at low temperature, βE h a ) 1, Figure 10, the cluster has grown practically following the deeper adsorptive energy zones (clear colors) crossing only very seldom the darker regions. This intuitively leads to think that the fractal dimension of the cluster should be smaller than that corresponding to a normal DLA cluster. As temperature increases the particles have enough energy to jump over higher barriers and then more and more branches crossing darker regions can be found (see Figure 12), which should result in an increasing in the fractal dimension. The argument about the region occupancy can be quantitatively established through Figure 13, showing the histograms of the frequency of occupation of sites of given energy at different temperatures by particles in the cluster. It could be argued that if the cluster is left to grow for a large enough time, then maybe the less occupied regions should finally fill up with additional branches; however, the time-ordered growth character discussed above (Figure 8) makes this possibility negligible. We measured the fractal dimension in our DLA clusters, by using the relation

N ∼ RDf

(13)

where N is the number of particles in the cluster within a distance R from the center (R must be greater than a few lattice spacing and smaller than the cluster size). Results, shown in Table 1, are consistent with our intuitive arguments exposed above; i.e., the fractal dimension seems to decrease when Ω increases and increases when the temperature increases (βE h a decreases). However these results should be considered only as a “probable tendency” due to the fact that finite size effects were not measured in our simulation, due to insufficient computer power. Another exponent which reveals an important change in the cluster properties with the energetic topography is the one arising from the correlation function

C(r) ∼ 〈ρ(r′)ρ(r + r′)〉

(14)

where ρ(r) is equal to 1 if a site is occupied and 0 if it is empty. It is found that C(r) behaves according to the power law

C(r) ∼ r-γ

(15)

The exponent γ is represented as a function of βE h a for different topographies in Figure 14, showing a different behavior of clusters grown on strongly correlated surfaces. 6. Conclusions The SBM makes possible the study of the effects of the adsorptive energy topography on many relevant surface molecular phenomena. Its advantage is that different topographies can be generated by varying one single parameter, Ω. We have studied adsorption, surface diffusion, surface reaction, and DLA as relevant phenomena which are involved in many processes of industrial importance. Adsorption is clearly and strongly affected by energetic topography as it is already well known. We have explained in detail how adsorption isotherms are affected, in terms of adsorbate cluster morphology. Adsorption of dimers, and higher order mers, appears as a promising method for the characterization of surface topography. Another important alternative for this seems to be surface diffusion, due to the very different behaviors of two extreme topographies, i.e, traps and barriers. It seems that the behavior of the tracer diffusion coefficient for a general surface is determined by the amount of “trap feature” and of “barrier feature” present for a given topography, the “trap feature” being more dominant. Reaction kinetics, we studied the simplest case of the annihilation reaction, is also affected by the topography,

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although not in the asymptotic regime but in the intermediate one. Reactions are slowed down when the topography becomes more correlated. For very highly correlated surfaces the intermediate regime lasts long enough for the reaction to be considered completed to any practical purpose. Finally DLA clusters are clearly affected in their morphology due to the fact that branches crossing over high-energy barriers are very unprobable. This intuitively leads to think that the fractal dimension should decrease for increasing correlation in the topography and should increase when temperature increases for a fixed Ω. This,

Zgrablich et al.

in fact, seems to be the tendency with the fractal dimension measured on simulated clusters, but further simulations showing the importance of finite size effects are necessary to confirm the results. Acknowledgment. Parts of the present work were developed in the framework of a cooperation program between CONICET of Argentina and CONACYT of Me´xico. LA9408782

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