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Langmuir 1999, 15, 5990-5996
Energetic Topography Effects on Surface Diffusion† F. Bulnes, F. Nieto, V. Pereyra, and G. Zgrablich* Laboratorio de Ciencias de Superficies y Medios Porosos and Centro Latinoamericano de Estudios Ilya Prigogine, Universidad Nacional de San Luis, Chacabuco 917, 5700 San Luis, Argentina
C. Uebing Max-Planck-Institut fu¨ r Eisenforschung, D-40074 Du¨ sseldorf, Germany Received September 28, 1998. In Final Form: March 4, 1999 The chemical diffusion coefficient D for noninteracting particles on heterogeneous surfaces with different energetic topographies is studied by Monte Carlo simulation. Two kinds of topographies are considered: strongly correlated patches and weakly correlated amorphous surfaces. Topography effects are clearly identified due to the fact that no adsorbate-adsorbate interactions are competing. These effects are strong and depend on the trap-barrier feature and on the spatial correlation length for adsorptive energy. The dependence of D on the temperature T does not follow, in general, the Arrhenius law. The relation between the values of D at coverage near 1 and near 0 appears as a useful tool to characterize the energetic topography.
1. Introduction Surface diffusion of adsorbed particles on heterogeneous surfaces has been studied for a long time and still presents several aspects which are not satisfactorily understood.1 The growing interest in this problem resides not only in the challenge of theoretical unsolved problems but also in the role surface diffusion plays in important technological processes, like the growth of thin films, the tailoring of nanoscopic surface structures, etc.2,3 One of the aspects requiring a better understanding is the way in which the energetic topography affects surface diffusion. In fact, even though it has been shown that many molecular surface processes, and in particular surface diffusion, are indeed dependent on the way adsorptive energy is spatially distributed on the surface (topography),4 a systematic study showing how the main characteristics determining the topography act on the migration of adsorbed particles has not been undertaken so far. The present work intends to be a starting step in this direction. Unlike adsorption, where lateral interactions, which are necessary in order to get adsorption isotherms sensitive to the topography, compete with topography in such a way as to obscure its effects,5 surface diffusion is found * Corresponding author. Fax: 54-2652-425109. Phone: 54-2652436151. E-mail:
[email protected]. † Presented at the Third International Symposium on Effects of Surface Heterogeneity in Adsorption and Catalysis on Solids, held in Poland, August 9-16, 1998. (1) Zgrablich, G. In Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces; Rudzinski, W., Steele, W. A., Zgrablich, G., Eds.; Elsevier: Amsterdam, 1997; pp 373-450. (2) Barabasi, A. L.; Stanley, H. E. Fractal Concepts in Surface Growth; Cambridge University Press: New York, 1995 and references therein. Meakin, P. Phys. Rep. 1993, 235, 189. Halpin-Healy, T.; Zhang, Y. C. Phys. Rep. 1995, 254, 215. (3) Yang, M. X.; Gracias, D. H.; Jacobs, P. W.; Somorjai, G. Langmuir 1998, 14, 1458. (4) Zgrablich, G.; Mayagoitia, V.; Rojas, F.; Bulnes, F.; Gonza`lez, A. P.; Nazarro, M.; Pereyra, V.; Ramı`rez Pastor, A. J.; Riccardo, J. L.; Sapag, K. Langmuir 1996, 12, 129. (5) Ciacera, M.; Zuppa, C.; Zgrablich, G.; Riccardo, J. L.; Steele, W. A. Surf. Sci. 1996, 356, 257.
to have such a sensitivity even for noninteracting particles. This is a great advantage, and for this reason we restrict our present study to the case of null adsorbateadsorbate interactions, to put forward with greater clarity the pure topographic effects. In section 2, we discuss which are the main characteristics determining the energetic topography of a heterogeneous surface by analyzing the adsorptive energy surface of a simple model heterogeneous solid. In section 3, the simulation method to obtain diffusion coefficients, in particular the chemical diffusion coefficient, is outlined. Then, in sections 4 and 5, results on the effects of the energetic topography are obtained and analyzed. Final conclusions are given in section 6. 2. Main Characteristics Determining the Topography To base our analysis on a well-defined simple system, let us consider a heterogeneous solid consisting of a regular crystal of atoms A (for example an hcp crystal) where a small fraction is substituted by impurity atoms B. We move a probe atom P on the (X,Y) surface of the crystal; the probe interacts with atoms A and B with a LennardJones potential,
UPS(r) ) -4PS
[( ) ( ) ] σPS r
6
-
σPS r
12
where S stands for the substrate atom, A or B, and and σ are the usual energy depth and particle diameter parameters, respectively. At each point i ) (X,Y), the total interaction energy of the probe atom is calculated as a function of Z by summing up all pairwise interactions with the substrate’s atoms within a cutoff distance rc ) 4σPS.
E(X,Y,Z) )
∑
UPS(rij)
rij e rc
Then, by finding the minimum in the coordinate Z, we obtain the equilibrium height Z0 and the adsorptive
10.1021/la981348z CCC: $18.00 © 1999 American Chemical Society Published on Web 05/07/1999
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Figure 1. Energy surface landscape for a hcp crystal with 20% of impurity atoms with -E0PA/kB ) 160 K and -E0PB/kB ) 480 K; darker regions represent stronger adsorptive energy, while brighter ones correspond to weaker adsorptive energy. In the top panel we present a general view, while the lower panel shows an enlarged picture of a small portion of the surface. The adsorption sites and the bonds between sites are clearly visible.
energy at position (X,Y) on the surface. What we get in this way is the adsorptive energy surface seen by the probe atom, defined as E(X,Y,Z0) ) minZ {E(X,Y,Z)}. Figure 1 shows this energy surface for a crystal with 20% of impurity atoms with PA/kB ) 160 K, PB/kB ) 480 K, and σPS ) 0.35 nm; darker regions represent stronger adsorptive energy, while brighter ones correspond to weaker adsorptive energy. In the top side, we see a general view where significant correlation is seen to be present, in the sense that strong adsorptive regions appear to be quite larger than one lattice size despite the low density of impurity atoms, reflecting the fact that the probe atom interacts with many atoms of the substrate at once. As a first rough approximation, the energy surface could be considered as a collection of irregular patches of different strengths. However, the energetic topography shows a quite greater complexity, and such a picture could lead to
an oversimplified model not reflecting important behaviors in molecular processes occurring on the surface. The cuts on the borders of the sample give the adsorptive energy profiles along X and Y directions, reinforcing the idea of a high complexity. The problem is how to model in a simple and still realistic way such a complex behavior. In other words, which are the characteristic (and relevant) quantities necessary to construct simple models capable of reproducing in a statistical sense the main topographic features? In the bottom side of Figure 1 we can see an enlarged picture of a small region of the adsorptive energy surface. There we can distinguish two fundamental elements1,10,11: sites, or energy wells where an adsorbed (6) Gomer, R. Rep. Prog. Phys. 1990, 53, 917. (7) Gomer, R. Surf. Sci. 1973, 38, 373. Chen, J. R.; Gomer, R. Surf. Sci. 1979, 79, 413. DiFoggio, R.; Gomer, R. Phys. Rev. B 1982, 25, 3490. Morin, R. Surf. Sci. 1985, 155, 187.
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Figure 2. Site (S) and bond (B) energy distributions (lefthand side) and the corresponding one-dimensional profile along the x direction (right-hand side): (top) random trap model; (middle) random barrier model; (bottom) general site-bond model.
Figure 3. Random barrier model and the flip-flop effect corresponding to a very high backward jump probability (i.e., to a correlation factor f < 1).
molecule will spend a much greater time, and bonds, or energy saddle points through which a molecule will most probably jump to migrate from one site to another. A site may be connected to several bonds, and in general, we may have a variable connectivity on the surface, while a bond is always connected to two sites. Our assumption is that we do not lose relevant information by disregarding the behavior of the energy at all other (X,Y) positions. In fact, sites are the places where adsorbed particles will spend most of the time and their energy play a fundamental role in the adsorptiondesorption equilibrium, while bonds (this widely accepted terminology comes from percolation theory) act as connecting channels between sites. Bond energy, together with that of sites, determines the jumping probability for particle diffusion assumed as an activated process. Then, a minimum realistic statistical description of the topography will require three functions: the site and bond energy distributions and the spatial correlation function for these elements. Different forms of the site and bond distributions and different correlation’s functions are able to generate a variety of topographies. The principal features characterizing a topography are the trap-barrier feature, which is determined by the forms of the site and bond distributions, and the correlation length, determined by the correlation function. Trap-Barrier Feature. It is usual in the terminology of surface diffusion to refer to a surface characterized by (8) Uebing, C.; Pereyra, V.; Zgrablich, G. Surf. Sci. 1996, 366, 185. (9) Uebing, C.; Gomer, R. Surf. Sci. 1995, 331, 930. (10) Mayagoitia, V.; Rojas, F.; Pereyra, V.; Zgrablich, G. Surf. Sci. 1989, 221, 394. (11) Mayagoitia, V.; Rojas, F.; Riccardo, J. L.; Pereyra, V.; Zgrablich, G. Phys. Rev. B 1990, 41, 7150.
Bulnes et al.
Figure 4. Snapshot of the bivariate trap surfaces. Deep traps with adsorption energy D are denoted by filled squares. Shallow traps with adsorption energy S are denoted by empty squares: (a) ordered chessboard structure with size d ) 4 and (b) random structure of the same nonoverlapping patches.
a constant adsorptive energy with potential wells at certain given locations as a trap surface, and to one with localized potential barriers as a barrier surface.15 However, these two are extreme cases of the great variety of surfaces that can be created. We introduce the term trap-barrier feature to characterize this variety. Figure 2 illustrates how the site and bond distributions influence this feature. On the left-hand side these distributions are shown, while on the right-hand side the resulting energy profile along a direction X is plotted. In the extreme cases where the bond distribution or the site distribution is a Dirac δ distribution, we have a trap or a barrier surface, respectively. In general, we will have a mixed behavior, or a general site-bond profile generated by arbitrary site and bond distribution functions. Trap and barrier surfaces appear then as particular cases of a site-bond surface. Trap and barrier topographies affect the diffusion of adsorbed particles in different ways, mainly due to two effects: (a) the “flip-flop” effect (see Figure 3), which makes the jump correlation factor different from 1 for a barrier surface (being 1 for a trap surface) and (b) the “pass by” effect, due to the fact that in a barrier surface a particle has the possibility to pass by a high-energy bond by choosing an alternative jump through a lower barrier. The trap feature is a dominating one, since, as we can see in Figure 2, for a barrier surface there will be always an effect of “extended traps” between high barriers. In other words, in a surface with random barriers, the potential between two high-energy barriers separated by several lattice spacing will, to a certain extent, act on surface diffusion like an effective trap. Correlation Length. For a macroscopically homogeneous and isotropic surface, we define the spatial correlation function for adsorptive energy as
C(B) l ) 〈E(τ b)E(τ b + B)〉 l
(1)
where b τ andBare l vectors on the (X,Y) plane and 〈...〉 stands for the statistical average over the surface. This correlation usually decays with a characteristic length, l0, called the correlation length, i.e., the typical distance of high spatial correlation. This is another characteristic of the topography which, as we shall see, affects surface diffusion. (12) Riccardo, J. L.; Chade, M. A.; Pereyra, V. D.; Zgrablich, G. Langmuir 1992, 8, 1518. (13) Riccardo, J. L.; Steele, W. A. J. Chem. Phys. 1996, 105, 9674. (14) Riccardo, J. L.; Steele, W. A.; Ramı´rez Cuesta, A.; Zgrablich, G. Langmuir 1997, 13, 1064. (15) Haus, J. W.; Kehr, K. W. Phys. Rep. 1987, 150, 263.
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Figure 7. Arrhenius plot of the chemical diffusion coefficient for random (d ) 1) and ordered patches (d ) 1 and d ) 5) topographies at a fixed surface coverage, θ ) 0.5.
Figure 5. (a) Normalized chemical diffusion coefficient DKG/ D°, obtained by the Kubo-Green method. (b) Jump diffusion coefficient DJ/D° as a function of coverage, θ.
Figure 8. Representation of the site and bond probability density functions FS(E) and FB(E). The overlapping is given by the parameter Ω.
where B J and F are the particle flux and density, respectively. This coefficient has a rich behavior for different topographies, since it depends on the surface coverage, θ, and furthermore it can be accurately measured by FIM, FEM, and STM techniques.6,7 As already pointed out, to show in an undisturbed way the pure effects of topography, we consider the diffusion coefficient for noninteracting particles. It has already been shown8 that, when spatial correlation is present, it is convenient to obtain the chemical diffusion coefficient via the Kubo-Green method,
DKG ) (TF)DJ
(2)
where TF is the thermodynamic factor given by
TF ) Figure 6. DKG(1)/DKG(0) as a function of patches size d for three different temperatures. Squares represent the results for ordered patches, while circles represent the results for random patches.
3. Monte Carlo Simulation of Diffusion Coefficient We focus our analysis on the chemical diffusion coefficient, D, defined through Fick′ s law,
B J ) -D∇F(r b)
(
) [
[∂µ/kBT] ∂ ln θ
)
T
]
〈(δN)2〉 〈N〉
-1
(3)
where µ is the chemical potential and DJ is the jump diffusion coefficient defined as
DJ ) lim tf∞
1 4tN
N
〈(
∆r bi)2〉 ∑ i)1
(4)
Details of the Monte Carlo simulation of diffusion coefficient are given in refs 9 and 16; here we briefly outline (16) Bulnes, F.; Pereyra, V.; Riccardo, J. L. Phys. Rev. E 1998, 58, 86.
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Figure 9. Snapshot of the surface energy topography generated by the dual site-bond model and for different values of the parameter Ω. As the overlapping between the density distribution function is increased, the surface becomes more correlated.
Figure 10. (a) Normalized values (with respect to zero coverage) and (b) nonnormalized values of the chemical diffusion coefficients calculated by the Kubo-Green method as a function of coverage for different correlation lengths.
the procedure. The TF is obtained by simulating the adsorption equilibrium in the grand canonical ensemble (eq 3), either by deriving the adsorption isotherm or, equivalently, by computing the particle number fluctuations in an area A containing 〈N〉 particles. For the jump diffusion coefficient we compute the mean square of the sum of the particles displacements (eq 4). The basic step is a jump of an adsorbed particle from a filled initial site to a nearest-neighbor vacant site. The jump occurs with a probability
PJ ) exp(-Ea/kBT)
(5)
where Ea is the activation energy given by the difference between the energy of the bond connecting the two sites
Figure 11. (a) Temperature dependence of the normalized chemical diffusion coefficients for a fixed value of the overlapping parameter Ω ) 0.7. (b) Arrhenius plot of the normalized chemical diffusion coefficients at a fixed coverage, Ω ) 0.5, for two values of the overlapping parameter Ω ) 0.1 and 0.7.
and the energy of the initial site. At each step a particular jump is selected from all possible ones for all adsorbed particles, from distribution (5), and the time elapsed from the previous jump is obtained from the corresponding Poisson distribution.16 In this way we can speed up the calculations, even at low temperature, and keep a low statistical error (
