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Cooperative Sequential Adsorption of k-mers on Heterogeneous Substrates† C. Zuppa and M. Ciacera Departamento de Matema´ ticas, Universidad Nacional de San Luis, Chacabuco 917, 5700 San Luis, Argentina
G. Zgrablich* Laboratorio de Ciencias de Superficies y Medios Porosos y Centro Latinoamericano de Estudios Ilya Prigogine, Universidad Nacional de San Luis, Chacabuco 917, 5700 San Luis, Argentina Received September 28, 1998. In Final Form: January 12, 1999
The cooperative sequential adsorption of dimers and trimers on heterogeneous substrates with random and patchwise (chessboard and strip) topographies is studied by Monte Carlo simulation. Strong influence of the topography is found on the adsorption kinetics, measured through the dependence of the sticking coefficient with the surface coverage, and the jamming coverage. In particular, an unexpected behavior of the jamming coverage is found for dimers and trimers on a random topography, which, unlike the case of dimers in one dimension, differs from that corresponding to a homogeneous surface.
1. Introduction Cooperative sequential adsorption (CSA) processes play an important role in a variety of problems, ranging over chemical reactions on polymer chains and catalyst surfaces, surface growth phenomena, attachment of molecules on biological polymers, and even the behavior of ecological and sociological systems. A very complete review of the field up to 1993 is given in the classical paper by Evans.1 Further developments can be found in refs 2-7. Perhaps the most widely known sequential adsorption process is the chemisorption of molecules on metal surfaces, a problem of great importance in catalysis.8 Surfaces are generally heterogeneous. Even for a perfect crystal we may have adsorption sites of different energies for a given molecule (on top sites or sites with different coordination numbers). Moreover, on an imperfect crystal, † 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) Evans, J. W. Rev. Mod. Phys. 1993, 65, 1281. (2) Ben-Naim, E.; Krapivsky, P. L. J. Phys. A: Math. Gen. 1994, 27, 3575. (3) Pereyra, V.; Albano, E.; Duering, E. Phys. Rev. E 1993, 48, R3229. Monetti, R. A.; Albano, E. V.; Pereyra, V. J. Chem. Phys. 1994, 100, 5378. Linares, D.; Pereyra, V. Phys. Rev. E 1996, 54, 617. (4) Rodgers, G. J.; Singh, P.; J. Phys. A.: Math. Gen. 1995, 28, 767. (5) Lee, J. W. J. Phys. A.: Math. Gen. 1996, 29, 33. (6) Stacchiola, D. J.; Eggarter, T. P.; Zgrablich, G. J. Phys. A.: Math. Gen. 1998, 31, 185. (7) Stacchiola, D. J.; Ciacera, M.; Zuppa, C.; Eggarter, T. P.; Zgrablich, G. J. Chem. Phys. 1998, 108, 1730. (8) Somorjai, G. A. Chemistry in Two Dimensions; Cornell University Press: Ithaca, NY, 1981. (9) Rudzinski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: New York, 1992. (10) Rudzinski, W.; Steele, W. A.; Zgrablich, G. Equilibria and Dynamics of Gas Adsorption on heterogeneous Solid Surfaces; Elsevier: Amsterdam, 1997. (11) Riccardo, J. L.; Chade, M. A.; Pereyra, V. D.; Zgrablich, G. Langmuir 1992, 8, 1518. (12) Sales, J. L.; Zgrablich, G. Phys. Rev. B 1987, 35, 9520; Surf. Sci. 1987, 187, 1. (13) Lombardo, S. J.; Bell, T. Surf. Sci. 1988, 206, 101.
sites at terraces, kinks, and vacancies provide very different adsorption energies to a molecule. This heterogeneity is even stronger in nanoscopic metal particles such as those dispersed on a catalyst support. The importance of the substrate heterogeneity in equilibrium and nonequilibrium adsorption and desorption gas-solid processes has been widely demonstrated in the literature.9-15 Despite this, most work on CSA is connected with homogeneous lattices or continua. Among the few works dealing with heterogeneous substrates we may mention some exact results for monomers with nearest-neighbor exclusion and dimers in 1-dimensional lattices2,3,6,7,16 and some Monte Carlo simulation results for dimers in 2-dimensional lattices.7,17 It is to be noticed that for 2-dimensional lattices series expansions with hierarchy truncation exist for homogeneous substrates, while for the heterogeneous case Monte Carlo simulation is the only effective method. Our scope in this work is to study the effects of substrate heterogeneity, characterized by simple spatial distributions (topography) of two kind of sites (strong and weak), on the sequential adsorption of dimers and trimers. In particular, we are interested in determining the way in which the sticking coefficient S(θ), as a function of the site coverage θ, and the jamming coverage θJ, i.e., the saturation coverage assuming that each site can be covered by at most one particle, are affected by heterogeneity. A comparative analysis will give us an idea of those properties which will be dependent on the k-mer size. In section 2, we define the model and outline the Monte Carlo simulation method. In sections 3 and 4 we give the results for dimers and trimers, respectively, analyzing the observed behavior. Conclusions are finally given in section 5. (14) Sales, J. L.; Un˜ac, R. O.; Gargiulo, M. V.; Bustos, V.; Zgrablich, G. Langmuir 1996, 12, 95. (15) Uebing, C.; Pereyra, V.; Zgrablich, G. Surf. Sci. 1996, 366, 185. (16) Oliveira, M. J.; Tome, T. Phys. Rev. E 1994, 50, 4523. (17) Kozak, E.; Lajtar, L.; Patrykiejew, A.; Sokolowski, S. Physica A 1993, 198, 345.
10.1021/la981338y CCC: $18.00 © 1999 American Chemical Society Published on Web 04/02/1999
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Figure 2. Dimer jamming coverages for random topography.
total number of animals ) T ) T1 + ... + TN total number of free animals ) F ) F1 + ... + FN The main problem is how to choose during the adsorption process the next animal in order to correctly reflect the adsorption kinetics. Through the central limit theorem, it is clear that the nonnormalized rate at which the next animal can be filled at each stage corresponding to a coverage θ is
ν(θ) )
( )(
)
F ζ1F1 + ... + ζNFN T F
Index n of next adsorbing animal has a probability distribution
Figure 1. Dimer normalized sticking coefficients as a function of coverage for random topography: (a) pA ) 0.2; (b) pA ) 0.4; (c) pA ) 0.8.
N
q(n) ) (ζnFn/F)S-1
S)
∑ζ F /F n
n
n)1
2. CSA Process and Monte Carlo Simulation Method The model and method described here, even if applied to the case of dimers and trimers, is valid for any kind of lattice animals (k-mers). We consider an L × L square lattice (L ) 500 in our simulations) of sites and a random rain of k-mers falling on the sites parallel to the surface. The probability that a k-uple of sites (i1, i2, ..., ik) is wet by a k-mer per unit time interval dt is ζi1, i2, ..., ik ) (Ri1 Ri2 ... Rik)1/k, where Rl is the strength of site l. Cooperativity arises from the fact that a k-uple of sites which is wet by the rain can only be occupied if all its sites are empty. Occupation is irreversible. In general, as the order of the k-mer increases (and even for k ) 2), the simulation process gets slower and slower as the jamming coverage is approached. Therefore a smart method is necessary to obtain reliable results in finite computing times. The basic idea for the simulation method is as follows. We form at the beginning a list of all possible animals that could be in principle accommodated on the lattice for each site (i.e., all those k-mers which would cover the given site) and a list of free animals (i.e., those in conditions to be filled). Each animal is characterized by an adsorption probability ζn obtained through the appropriate function of the strengths of the sites belonging to it. The set of probabilities will be finite (ζn; n ) 1, ... N). Then, for each probability ζn we have a list An of animals and a list Bn of free animals, respectively, with that probability, such that, if Tn and Fn are the numbers of animals and free animals in An, then
Taking all this into account, the next animal is chosen by (a) selecting a random index n with probability distribution q and (b) selecting at random a free animal in Bn. For (a), a random number R uniformly distributed in (0,1) is generated; then if n-1
∑ k)0
n
q(k) < R e
∑ q(k);
q(0) ) 0
k)0
the index n is chosen. For (b), we simply take the value of a discrete random number uniformly distributed in [1,N]. Then, each adsorption step is accomplished by choosing a free animal as described above, adsorbing it on the lattice and updating the coverage. The list of free animals is then updated by eliminating all animals corresponding to each one of the newly occupied sites. At this point the sticking coefficient is computed through
S(θ) dθ/dt ) ) ν(θ)/ν(0) S(0) dθ/dt|t)0 The jamming coverage θJ is computed at the end of the process, when no more free animals are available. The whole procedure is repeated a number M of times, so that a convenient precision is obtained in the calculation of mean values. In our simulations, a statistical error of (0.0005 in θJ was obtained with M ) 500. For random topographies the calculations are repeated for 25 different realizations of the lattice. This is enough to ensure good statistics, given that the lattice is considerably big (500 × 500). We considered heterogeneous substrates with two kind of sites, strong A sites with RA ) 1 and weak B sites with RB ) a < 1.
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Figure 5. Dimer normalized sticking coefficients for chessboard patches topography: (a) patch size ) 2; (b) patch size ) 5. Figure 3. Dimer normalized sticking coefficients for strip topography: (a) strip size ) 1; (b) strip size ) 5.
Figure 6. Dimer jamming coverages for chessboard patches topography as a function of patch size.
Figure 4. Dimer jamming coverages for strip topography as a function of strip size. Three topographies were studied for dimers and trimers CSA, random distributions with different A and B concentrations pA and pB ) 1 - pA, respectively, strips of A and B sites of different size d, and chessboard patches of A and B sites of different size
d. These two last topographies are of particular interest in view of recent developments in the fabrication of model supported catalysts with determined nanostructures by means of lithographic technology.18 Now we go on to present and discuss our results.
3. Dimers We start with a random mixture of strong A sites, with RA ) 1 and concentration pA, and weak B sites, with RB (18) Yang, M. X.; Gracias, D. H.; Jacobs, P. W.; Somorjai, G. A. Langmuir 1998, 14, 1458.
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Figure 7. Straight trimer normalized sticking coefficients for different topographies.
) a < 1 and concentration pB ) 1 - pA. Results for the normalized sticking coefficient as a function of coverage and the corresponding jamming coverage are shown for values of a ranging from 0.5 to 0.0001 in Figure 1a for pA ) 0.2, Figure 1b for pA ) 0.4 and Figure 1c for pA ) 0.8. As a decreases, S(θ) shows three distinct regimes corresponding to the filling of AA, AB, and BB pairs, respectively, for small and intermediate values of pA. In particular, as pA approaches 0.5, the coverage ranges for the three regimes have similar extensions. For high values of pA the behavior corresponding to a homogeneous lattice, independent of a, is recovered. The jamming coverage as a function of the concentration of weak sites, pB, for different values of a, is shown in Figure 2. As we can see, for large values of a the jamming coverage is approximately constant while for much smaller values it increases slightly with decreasing pB. This is an unexpected result, since exact results in 1-dimensional lattices predict no change in the jamming coverage for random distributions.6,7 It seems that, in two dimensions, the random inclusion of a small amount of weak sites forces the dimers to adsorb in such a way as to make the filling more efficient. We consider now the lattice topography being determined by alternating strips of strong A and weak B sites of width (size) d, which we denote by std. In the case of st1, when a ) 0, we have a one-dimensional adsorption process on the strips of strong sites and the jamming coverage will be given by θJ ) 1/2(1 - exp(-2)).
Figure 8. Broken trimer normalized sticking coefficients for different topographies.
For small positive values of a, dimers will adsorb first on strips of strong sites reaching a coverage close to the jamming coverage for a homogeneous chain θ1 = 1/2(1 exp(-2)) = 0.43. In a second step, dimers will adsorbe in crossed positions filling the remaining empty strong sites up to θ2 = 0.57. Finally the remaining segments of the strips of weak sites will be partially filled with an upper limit given by the value corresponding to the onedimensional chain. The final jamming coverage should then be close to the value
θJ =
1 1 + (1 - exp(-2)) = 0.93 2 2
Figure 3a shows the simulation results for st1 in concordance with the above analysis, where the three different filling regimes can clearly be appreciated for small values of a. Obviously as a increases to 1 the homogeneous behavior is recovered. The jamming coverage optimizes for st2. In fact, in this case the strips of strong sites will again fill completely while the strips of weak sites have a higher flexibility to optimize their coverage. For st3, however, holes begin to appear in the strips of strong sites and θJ again decreases reaching the value for a homogeneous lattice for wide enough strips. Figure 3b shows the behavior for strips of
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Figure 9. Jamming coverages for straight and broken trimers for different topographies.
intermediate size, st5. We can also see that as the strip size increases, the intermediate regime disappears and the kinetics shows only the two extreme well-differentiated regimes. The behavior of θJ as a function of d for different values of a is given in Figure 4. We see that for very large d the jamming coverage approaches the value for a homogeneous surface, and then as d decreases, its value increases, being maximum at d ) 2, and decreases again for d ) 1. This behavior is stronger for smaller a values. Finally we consider a chessboard topography of square patches of size d, which we denote by cd, of strong A and weak B sites. For c1 patches, θJ will have obviously two values: zero for a ) 0 and =0.91, the value for a homogeneous lattice, for a > 0. For c2, θJ will have the value 1/2 for a ) 0. For 0 < a , 1, strong patches will fill completely in a first step and then weak patches will do the same, resulting in θJ ) 1. As a increases further θJ decreases but will always be higher than for any other topography considered here for the same value of a, Figure 5a. As we take larger patches, c3, holes start to appear in the strong patches and θJ decreases with respect to c2. In addition, a third, intermediate, filling regime appears corresponding to adsorption on AB pairs, Figure 5b. For even larger patches the third regime gradually disappears and the homogeneous value of the jamming coverage is recovered. The variation of θJ as a function of d for different a values is given in Figure 6, where the strong difference in the filling efficiency of c2 with respect to other patch sizes can be appreciated. 4. Trimers In the case of CSA of trimers we have a more complex situation, given that trimers may present two kinds of
lattice animals: straight trimers or broken trimers. We consider the adsorption process for three cases: pure straight, pure broken, and a 50%-50% mixture of both kinds, on surfaces with the same topographies as for dimers. Figure 7 shows the behavior of the sticking coefficient for straight trimers. For a random distribution, with a high concentration of weak sites, we see that four filling regimes appear for very small values of a, corresponding to the filling of AAA, AAB, ABB, and BBB triplets. These different regimes gradually wash off as a increases or as pB approaches the extreme values 0 or 1 (not shown). For a strip topography, only three filling regimes are clearly distinguished for d ) 2 and very small values of a. At the beginning trimers adsorb on strong strips, parallel to it (AAA triplets), then, in a second stage, they will adsorb on ABB triplets perpendicular to the strips, filling the holes left in strong strips (pairs of AA sites perpendicular to the strip will be almost unexistent), and finally BBB triplets on the weak strips will be filled. The three regimes still appear for d ) 1, but only two regimes survive for d > 2 (not shown). For chessboard patches topography, as it can be readily understood, three regimes appear for d g 3, while ony two are present for smaller patch size. The behavior of the sticking coefficient for broken trimers, shown in Figure 8, is quite similar, with the difference that four regimes can be observed for strips (not shown) and chessboard patches, with d > 2. The adsorption of a mixture of straight and broken trimers presents a behavior which is intermediate between those corresponding to the pure species. The behavior of the jamming coverage for trimers presents a great variety of interesting features, Figures 9 and 10. A first, and also unexpected result as in the case
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minimum appears for higher pB values in the case of straight trimers than for broken ones. For the chessboard patches topography there is a maximum filling efficiency for d ) 3 for all species and a minimum one for d ) 2 (straight) and d )1 (broken and mixture). In the case of strips the maximum filling efficiency depends both on the species and on the value of a. The results for chessboard patches and strips can be readily understood on the base of an analysis similar to the one already carried out for dimers.
Figure 10. Jamming coverages of a 50%-50% mixture of trimers for different topographies.
of dimers, is that for random distributions the jamming coverage depends on the concentration of weak sites, for straight trimers as well as for broken ones and for a mixture. In general, θJ increases as pB decreases and may reach slightly higher values compared to the ones corresponding to homogeneous surfaces, 0.847 for straight and 0.834 for broken trimers. It appears that the formation of small clusters of weak sites increases the filling efficiency and that for smaller a values this works the better the smaller are the clusters. However, this behavior differs substantially above the percolation threshold for weak sites, where a minimum is produced (below the value corresponding to an homogeneous surface), and this
5. Conclusions We have given a comparative study of the CSA of dimers and trimers on heterogeneous 2-dimensional substrates composed by two kind of sites, strong and weak, and characterized by different topographies, random, strips, and chessboard patches. In general, heterogeneity affects appreciably the adsorption kinetics, measured by the sticking coefficient, showing different filling regimes as a function of coverage, in all cases studied. The jamming coverage for dimers is affected by a random topography, in contrast to exact results for 1-dimensional chains.6,7 For all topographies we find that the filling efficiency is always greater than for an homogeneous surface, being maximum for strips and chessboard patches of size 2, as would be expected. For trimers we have a much richer behavior, depending on the shape of the trimer and the topography. The jamming coverage is slightly affected by the concentration of weak sites and by the relative strength of weak and strong sites for a random topography. In general the filling efficiency increases with decreasing concentration of weak sites, but it can reach values smaller than those corresponding to a homogeneous surface above the percolation threshold of weak sites. For the chessboard patches and strip topographies, we find differences with the behavior of dimers, since we may have either greater or smaller jamming coverages than those corresponding to a homogeneous surface, depending on the shape of the trimer and the topography. As a final remark, the unexpected effect observed for random distibutions reveals a complex interplay between the shape and extent of the k-mer and the size distribution of weak and strong sites, which deserves further investigation. Acknowledgment. This work was partially supported by CONICET (Argentina). The authors acknowledge Vı´ctor Pereyra for helpful and enlightning discussions. LA981338Y
