Langmuir 1985, 1, 443-452 the characterization and catalytic properties of a variety of highly dispersed bimetallic entities, it may be anticipated that the above program when applied to such systems will lead to a more detailed understanding of hydrocarbon conversion over metal alloys/clusters. Having noted these reservations, it still remains that the experimental studies on the hydrogenolysis/dehydrogenation reactions cited above lead to general conclusions
443
on the importance of reactive-site clusters on reaction efficiency. In the same spirit, it may be hoped that the details of the reaction mechanism in each case do not invalidate the general theoretical conclusions that follow from our study of the distribution and concentration of active sites, although, to be sure, they will certainly refine the understanding of the problem once they are incorporated into the model.
Influence of Structure on Reaction Efficiency in Surface Catalysis. 2. Reactivity at Terraces, Ledges, and Kinks G. Joseph Staten,+Matthew K. Musho,t and John J. Kozak* Department of Chemistry and Radiation Laboratory,$ University of Notre Dame, Notre Dame, Indiana 46556 Received September 11, 1984. I n Final Form: January 25, 1985 In this paper we continue our development of a lattice-based theory of reaction efficiency on catalytic surfaces. The specific problem dealt with in this paper is the role of lattice imperfections in influencing the efficiency of diffusion-controlled reactions on such surfaces. We consider a set of reaction centers distributed on a surface on which there are terrace, ledge, and kink sites and calculate numerically using our Markovian (implicit function) approach the changes in the efficiency of the process (as monitored by calculating the average walklength ( n ) )in three distinct situations: (1)we consider a single reaction center imbedded on a surface (of up to 121 sites) and located first at a terrace site, then at a ledge site, and finally at a kink site of the lattice, (2) we consider a single reaction center (at a terrace vs. ledge vs. kink site) in competition with a whole set of reaction centers distributed uniformly over the surface of the support, and (3) we consider a single reaction center (again at a terrace/ledge/kink site) in competition with a set of competing reaction centers but where the latter are positioned at the ledge sites of the lattice, with the remaining sites of support assumed to be neutral (or nontrapping) sites. We introduce simple arguments (in which the trapping (or reaction) probability is correlated with the ligation number) to allow us to compare directly the interplay between entropic and energetic factors in influencing the overall efficiency of the reaction-diffusion process. The possible relevance of these calculations to the experimental studies of Somorjai and co-workers in which turnover number was studied as a function of step density and kink density for hydrogenation and hydrogenolysis reactions of hydrocarbons on clean platinum surfaces is brought out and discussed.
I. Introduction The physical problem we wish to address in this paper concerns the role of terrace vs. ledge vs. kink sites in influencing the efficiency of reaction-diffusion processes on surfaces. In the preceding paper1 we portrayed the surface of a pure metal (Cu) or metal alloy (Cu/Ni) as a perfect array of hexagonally close-packed metal atoms; the migration of the reactant was assumed to proceed via “hollow-to-hollow”jumps and the reaction-diffusion process was studied on the dual lattice (v = 3) to this physical array of (surface) coordination number v = 6. In order to characterize more conveniently reactions a t terrace/ ledge/kink sites, we choose here to consider processes on surfaces for which the atoms are packed together in square-planar symmetry with v = 4. Notice that any one of the terrace sites in Figure 1is characterized by a connectivity u = 4, whereas the site labeled T in Figure l a has a valency u = 3, the site labeled T in Figure l b is of valency
’ *
Permanent address: Magnavox Corporation, Fort Wayne, IN 46808. Permanent address: Miles Laboratories, Elkhart, IN 46614. #The research described herein was supported in part by the Office of Basic Energy Sciences of the Department of Ehergy. This is Document NDRL-2633 from the Notre Dame Radiation Laboratory.
v = 4, and the site labeled T in Figure ICis of valency v = 5. All labeled sites in the Figure 1 code the physical
locations of the atoms comprising the surface being studied. Now, notice that if one assumes “hollow-to-hollow” jumps on the terraces of any of these lattices, the number of directions in which the diffusing adatom can migrate is four; since a terrace is effectively a “perfect” squareplanar lattice, its dual will also be a square-planar lattice (v = 4). Consider now the possible motion of an adatom at or on a ledge or in the vicinity of a kink, keeping in mind that we consider here only steps of monatomic height. For such structures we impose no gravitational “preference” for an atom situated a t the top vs. the bottom of a step. Otherwise, as stressed by Beiiard,2 “this may suggest for example, that preferential adsorption at surface steps will occur at the bottom of steps, whereas in reality there may be cases where the strongest binding is at the top”. To bypass a criticism of lattice (or ball and stick) models of reactive surfaces, Beiiard suggests that such models should be viewed “upside down”. Given this, it is evident that all “hollow-to-hollow” transitions on our surfaces, Figure 1, should be characterized by 4-fold degrees of freedom; in (1) Politowicz, P. A.; Kozak, J. J., preceding paper in this issue. (2) “Adsorption on Metal Surfaces”:Beiiard, J., Ed.: Elsevier: Amsterdam, 1983.
0743-7463/8512401-0443$0l.50/0 0 1985 American Chemical Society
Staten, Musho, and Kozak
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Figure 1. Configuration of sites for the largest terraced lattice considered in this paper. The reaction center (trap)is designated T and is situated at a site of valency v = 3 (a), 4 (b), or 5 (c). particular, even for “hollows” a t the bottom or top of a ledge, or for a “hollow” in the vicinity of a kink, the connectivity (valency) of the reaction space remains v = 4. What will change in the problem for an adatom in one of the latter configurations (i.e., a ledge or kink vs. a terrace “hollow”)is the number of first, second, ... nearest neighbors, and this, of course, changes the energetics in the vicinity of a step or kink. In order to be able to distinguish and then to characterize energetic differences among terrace/ledge/ kink sites in terms of a simple convention, we shall shift our point of view back to the atoms comprising the actual surface. That is, rather than assume a “hollow-to-hollow”diffusion mechanism of a species in the vicinity of a reaction center, we shall assume that the reactant jumps from one physical surface site (an atom) to another in its random walk movement. Thus, the number of degrees of freedom available to a reactant at a given point in its trajectory across the surface will be exactly that of the atomic site on the underlying lattice (v = 4 for a terrace or ledge site, Y = 3 or v = 5 for a kink site.) Since the coordination number n, of atoms a t particular sites of the lattice can be enumerated by inspection, an “electronic factor” can be associated with the number niof “missing” bonds for a surface atom in a lattice characterized, in bulk, by cubic symmetry (v = 6). As will be demonstrated in section IV,
11. Formulation To highlight the importance of lattice imperfections in influencing the efficiency of reaction-diffusion processes on low-dimensional surfaces, we construct three lattices. In Figure 1 we display the configuration of sites for the largest lattice considered in this paper, the lattice with edgelength 1 = 11. The central trap is designated T and is situated at a site of valency v = 3 (Figure la), valency v = 4 (Figure lb), or valency v = 5 (Figure IC). It is important to emphasize that although we have presented each of these cases as a terraced structure, each setting of v is also consistent with a strictly two-dimensional configuration of sites, with a pronounced distortion of the lattice about tbe reaction center T (for the valencies v = 3 and 5). The manner in which a vertical ordering can be incorporated in the formulation of the model as it stands will be described in section V. Consider next the trajectory of the diffusing coreactant. In general, the site-to-site motion of the particle can be influenced by a variety of potential effects, e.g., potentials may induce jumps between non-nearest-neighbor sites on the lattice (see ref 3) or may bias the motion of the particle in the vicinity of the reaction center (see ref 4). In this paper, however, since we wish to expose clearly the consequences of introducing geometrical constraints or imperfections (see above), we consider here particle motion characterized by nearest-neighbor, unbiased random walks only. As regards boundary conditions, we refer the reader to the preceding paper for a description of periodic vs. confining boundary conditions. In implementing the Markov chain theory6 for the underlying lattice-statistical problem, one finds that when one considers a single trap (or, more generally, a set of traps) positioned at a centrosymmetric site (or, a set of sites symmetrically positioned with respect to the boundary of the system), there is no formal difference in the structure of the Markovian equations written down describing these two boundary conditions. Thus, with respect to the two boundary conditions studied in this paper (viz., periodic vs. confining), since the reaction center T is positioned a t the center of the lateral face (the y direction in the coordinate system defined in the Figure 1) for each (odd) lattice considered, imposition of either boundary condition leads to the same results with respect to that coordinate direction. Slight differences can arise, however, when one compares results generated when periodic vs. confining boundary conditions are imposed in the x direction (again see Figure 1for the alignment of the coordinate axes), this because a slight symmetry breaking is introduced when one considers the cases v = 3 and 5. However, as will be documented later in this paper, the numerical differences which arise in the calculation of ( n ) are so slight (at least for lattices of edgelength 1 > 9) that the results reported in this paper may be considered relevant to the following two physical situations: (1)a set of sites with traps (i.e., a unit cell defined by one of the configurations represented in Figure 1)periodically replicated through all space or (2) a finite cluster of sites, compartmentalized in the sense that a diffusing coreactant is (passively) reflected at the boundary of the cluster (via implementation of confining boundary conditions). (3) Musho, M. K.; Kozak, J. J. J. Chem. Phys. 1983, 79,1942. (4) Musho, M. K.; Kozak, J. J. J. Chem. Phys. 1984, 80, 159.
Langmuir, Vol. 1, No. 4, 1985 445
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Figure 2. Plot of the average number (n) of steps required for trapping on a terraced lattice vs. the system size, as calibrated by the edgelength (1). Considered here is the case of a reaction center (target molecule), situated at a site of valency v = 3 (triangles), 4 (squarea), or 5 (circles)and characterized by a reaction (trapping)probability s. The N - 1background sites are assumed to be distributed
symmetrically about the target molecule. The diffusing coreactant is assumed to encounter confining boundaries in the x direction (see Figure 1) and confining/periodic boundary conditions in the y direction. Displayed here are the following four cases: (a) s = 0.25, (b) 0.50, (c) 0.75, (d) 1.00 (deep trap). 111. Results The principal aim of the preceding paper was to demonstrate how constellations of reaction centers, competing with a central trap, could influence the efficiency of the underlying reaction-diffusion process. This pursuit is also of central importance in the present study but here, in order to develop systematically an understanding of the effeds brought in by considering connectivities other than u = 4 around the central trap T, we consider first and in some detail the simplest case of a single trap positioned a t site T (see Figure l),with all other lattice sites regarded as nontrapping or neutral sites. In Figure 2 we display the average number ( n ) of steps required for trapping on a terraced lattice as a function of system size (as calibrated by the edgelength 1). Specifically, the N - 1 background sites are assumed to be strictly nontrapping (neutral) and distributed symmetrically about the target molecule, the latter positioned at a site of valency u = 3,4,or 5. The cases illustrated in this figure correspond to four settings of the reaction parameter s, where s scales the degree of reversibility of the reaction x + Y [XU]* z a t the central trap (see the discussion in section I1 of the preceding paper). Thus, Figure 2a documents the changes
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in the efficiency of reaction, as gauged by the value of (n), when there is a 25% probability that the above reaction will proceed at once to completion with formation of the product Z (or, conversely, a 75% probability that the diffusing coreactant (X), upon confronting the target molecule (Y), forms an excited-state complex but one that eventually falls apart with regeneration of the species X, which subsequently resumes its random walk on the lattice). Similarly, in Figure 2, parts b, c, and d record the behavior observed as a function of system size for the settings s = 0.50,0.75, and 1.00, respectively, with the latter choice (s = 1.0) corresponding to the case where the species X reacts with Y irreversibly upon first encounter, forming the product Z. The behavior of the reaction-diffusion system as a function of a continuously varying reaction parameter s can be studied explicitly provided we specify a given lattice. The results of such a study are presented in Figure 3 where, for a lattice characterized by the edgelength 1 = 11, the reaction efficiency is studied as a function of s. In this figure our results are displayed as a function of (1- s), a survival probability for a walker on a lattice characterized (here) by a single, centrosymmetric trap. Given the background of data reported in Figures 2 and 3, we can now begin to explore the consequences as regards
446 Langmuir, Vol. 1, No. 4,1985
Staten, Musho, and Kozak
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Figure 3. Plot of the average number ( n )of steps required for trapping on an 1 = 11terraced lattice (seeFigure 1)w. the survival
probability, (1- s). We consider a single reaction center (target molecule), situated at a site of valency u = 3 (dashed line),4 (solid line),or 5 (dotted line) and characterized by a reaction (trapping) probability s. The background sites are assumed to be distributed symmetrically about the target molecule. The diffusing coreactant is assumed to encounter confining boundaries in the x direction and confining/periodic boundary conditions in the y direction. reaction efficiency of introducing additional reaction centers which may compete with the central trap T. We consider reaction-diffusion processes taking place on lattices characterized by edgelength 1 = 11. To proceed, we introduce a second reaction parameter p (0 < p < 1) defined as the probability that the diffusing species X may undergo a reaction a t any one of the N - 1 sites surrounding the central trap. We then consider the interplay between reaction at the central trap (which we continue to parametrize by the reaction probability s) vs. reaction a t the N - 1 surrounding sites (scaled uniformly by the parameter p ) , all as a function of the valency v characterizing the connectivity of the lattice in the immediate vicinity of the central trap. Specifically,we plot in Figure 4 the average walklength ( n )vs. a survival probability (1 - p ) against background trapping for each of four settings of the reaction parameters, viz., s = 0.25, 0.50, 0.75, and 1.00, as displayed in Figure 4, parts a, b, c, and d, respectively. A further study of the effectiveness of background trapping was carried out, one in which the ledge (or step) sites only (e.g., sites 115-104-93-82-71,50-39-28-17-6and sites 116-105-94-83-72-51-40-29-18-7in Figure la) were permitted to compete with the central, target molecule T. The results of this study, displayed in Figure 5, quantify the dependence of the average walklength ( n ) on the survival probability (1- p ) against ledge trapping, for each of the four settings of the reaction parameters (viz., s = 0.25,0.50,0.75,and 1.0) studied in Figure 4. We emphasize that in this series of calculations the remaining terrace sites are assumed to be strictly nontrapping (or neutral) sites characterized by the probability p = 0. We now take up a discussion of the results presented in Figures 2-5 and comment on their possible implications. IV. Interpretation of Results The discussion of the results reported in section I11 will be in two parts. In this section we shall focus on the implications of the results obtained vis-&vis some general problems of reaction-diffusion theory. Then, in the following section, we shall attempt to display the relevance of these results to problems of reactivity on solid surfaces.
The collection of results displayed in Figure 2 provides the point of departure for our discussion. There it is seen that for a lattice of a given size (here ranging from one of edgelength 1 = 5 to one of I = 11) the number of nearest-neighbor routes to a central target molecule is critical in determining the efficiency of a reaction-diffusion process. Our calculations show that there exist already quantitative differences in the value of ( n ) even for the smallest lattice considered (viz., the 5 X 5 lattice) and that (in an absolute numerical sense) these differences tend to increase with increase in the spatial extent of the system. Moreover, we find that these differences persist when one changes the factor s which gauges the effectiveness of trapping at the active site. On a relative basis, the trends found are the following: (1)For a given setting of s, the percent difference between the v = 3 and 5 results decreases with increase in the lattice size (as calibrated by the edgelength 1 in the figures). (2) For a given setting of 1, the percent difference between the u = 3 and 5 results decreases with increase in the degree of irreversibility of the reaction (as studied by increasing the magnitude of the parameter s). These trends document the importance of short-range (here cage) effects in influencing reactiondiffusion processes. The cage effect studied here, which arises from a strictly geometrical restriction (or enhancement) in the number of nearest-neighbor channels linking the active site to the remainder of the reaction space of the system, differs from the short-range (chemical/cage) effect identified in our earlier s t ~ d y .In~ that work, we studied a situation wherein immediate access to the target molecule (situated on a regular lattice with no imperfections) was mandated once a nearest-neighbor site was reached in the random motion of the diffusing coreactant. Taken together, these studies quantitate the importance of factors controlling the behavior of the coreactant in the immediate vicinity of the active site. Whereas the dependence of the average walklength on the parameter s was displayed in Figure 2 for discrete values (viz., s = 0.25, 0.50, 0.75, and 1.0) for lattices of various sizes, we can also calculate for a given lattice the walklength ( n )as a continuous function of s. Thus, in Figure 3 profiles of the walklength ( n )vs. the independent variable (1- s) are given, where the quantity (1- s) is a survival (or escape) probability of a reactant migrating on a lattice (here, an 11 X 11 lattice) with a target molecule surrounded by -120 nontrapping sites but linked to its environment by three, four, or five channels. From the results displayed in this figure, it is seen that the separation between results calculated for (nearest-neighbor channels) u = 4 vs. 3 is greater than that for u = 4 vs. 5, especially in the range s < 0.2. This would seem to suggest that a decrease (Av = -1) in the number of nearest-neighbor channels to the active site, relative to the overall connectivity of the host lattice (here, v = 4), is of greater consequence than a defect that permits an increase (Av = +1) in the number of such channels. Attention will be drawn to this point (i.e., the sensitivity to nearest-neighbor channels, Au = -1) in a later paragraph. We now discuss the results presented in Figure 4. There we admit the possibility that all the sites surrounding the target molecule, sites previously assumed to be passive (or “neutral”) chemically, can now with probability p compete with the central trap. In two of our earlier studies where “perfect” square-planar5 or hexagonal6lattices were considered, it was found that a 5% probability of reaction at these adjacent sites effectively erased distinctions between ( 5 ) Walsh, C. A.; Kozak, J. J. Phys. Reu. B 1982, 26, 4166. (6) Politowicz, P. A.; Kozak, J. J. Phys. Rev. B 1983, 28, 5549.
Langmuir, Vol. 1, No. 4, 1985 447
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lattices subject to different boundary conditions or characterized by different valencies; that is, the reaction-diffusioh process became kinetically controlled. As is evident from the results displayed in Figure 4, a similar result is found here since, even when the number of nearestneighbor channels to the target molecule is changed (Av = fl vis-&vis the host lattice of valency v = 4), a change that produced marked changes in ( n ) when only a single reaction center was present (see Figure 2), the results calculated for ( n )tend to coalesce for values of p in the vicinity of 3%. In fact, it is seen clearly (compare Figure 4a for which s = 0.25 vs. Figure 4d where s = 1.0) that when one studies this effect as a function of s, the smaller the value of s, the smaller the value of p that is effective in erasing distinctions between results calculated for the (nearest neighbor) valencies v = 3 vs. 4 vs. 5. Moreover, notice that regardless of the setting of s (compare Figures 3 and 4) there is a precipitous drop in the calculated value of (n)when one "turns onn chemically the -120 background sites of the lattice. In fact, orders of magnitude may separate the values calculated for ( n ) ,depending on the value of s, when the background is characterized by a mere 5% reactivity. It is plain from the preceding discussion that the results recorded in Figure 3 vs. those in Figure 4 describe two,
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0.98
1.0
SURVIVAL PROBABILITY (LEDGE TRAPPING)(I-p)
0.90
0.92
0.94
0.96
0.98
1.0
SURVIVAL PROBABILITY (LEDGE TRAPPING) ( I-p)
Figure 5. Plot of the average number ( n )of steps required for trapping on an 1 = 11 terraced lattice (see Figure 1) vs. a survival probability (1- p ) against ledge trapping. Considered here is the case of a reaction center (targetmolecule), situated at a site of valency v = 3 (dashed line), 4 (solid line), or 5 (dotted line) and characterized by a reaction (trapping) probability s. The background sites are assumed to be distributed symmetrically about the target molecule; of these, the ledge sites can, with probability p , interact with and trap the random walker, while the remaining terrace sites are neutral (i.e., p = 0). The diffusing coreactant is assumed to encounter confining boundary conditions in the x direction and confining/periodicboundary conditions in the y direction. Displayed here are the following four cases: (a) s = 0.25, (b) 0.50, (c) 0.75, (d) 1.00 (deep trap). 4 and ref 5 and 6). Furthermore, whereas there is indeed a reduction in the magnitude of ( n )as one "turns on" the ledge trapping probability, the effect is not as pronounced (quantitatively) as that found when all background sites are activated. There is, however, an interesting effect that appears in Figure 5 that is not present in the Figure 4. One finds, for each setting of s considered, a "crossover" in the profiles generated for the case v = 3 vs. 4. For example, in case c of Figure 5 where s = 0.75, for (1- p ) in the vicinity of 0.97 (or p 3%), the dashed (v = 3) and solid (v = 4) curves intersect. In the range p < 3% the v = 3 curve yields the largest values of ( n )whereas, for p > 3%, the curve for v = 3 is intermediate between the v = 4 and 5 curves. The reason for this behavior can be seen by viewing, once again, the lattices displayed in Figure 1. There it will be seen that although the number of sites along each edge is 11 for each of the three lattices, the lattices corresponding to v = 3 and 5 have two more ledge sites than is the case for v = 4. These two additional ledge sites, when placed in competition with the central trap T, enhance the possibility that trapping of the diffusing coreactant will occur; in fact, both the value of ( n )and the setting of (1- p ) at which the crossover occurs are sensitive
-
to this difference in the number of active ledge sites. Although heretofore we have chosen to display the results of this study (and the preceding one) in graphical form, it is of interest to tabulate the numerical results for the "intermediate" case described above. This we have done in Tables 1-111, since in this format a further point can be made concerning the boundary conditions subject to which the calculations reported in this paper were performed. In particular, when one is dealing with an Id lattice with a centrosymmetric reaction center (and no imperfections) the use of periodic or confining boundary conditions leads to exactly the same results (see our earlier description of these two boundary conditions). When one breaks the symmetry of the reaction space, whether by moving the target molecule off center or by introducing imperfections (defects), the results calculated assuming these two choices of boundary condition can differ, with the differences gradually diminishing with increase in lattice size. For the lattices of edgelength 11displayed in Figure 1,the disposition of the trap will be exactly centrosymmetric with respect to the coordinate direction designated x , but will be slightly off center with respect to the coordinate direction y for the cases u = 3 and 5. Thus, the results registered in Tables 1-111 quantify, as
Influence of Structure on Reaction Efficiency. 2 Table I. Comparison of Results for ( n )for Two Choices of Boundary Condition for the Case Y = 3 O (1 - P) (fl)Ib (n)rrC A(fl) s = 0.25
Langmuir, Vol. 1, No. 4, 1985 449 Table 11. Comparison of Results for (n ) for Two Choices of Boundary Condition for the Case Y = 4' s = 0.25
0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99
73.03 78.10 84.32 92.16 102.32 116.03 135.54 165.53 217.53 329.84
73.22 78.28 84.50 92.33 102.49 116.19 135.70 165.67 217.65 329.91
0.19 0.18 0.18 0.17 0.17 0.16 0.16 0.14 0.12 0.07
0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99
77.17 82.29 88.55 96.36 106.39 119.74 138.39 116.37 212.50 304.09
78.04 83.15 89.38 97.17 107.17 120.47 139.06 166.86 212.96 304.35
0.87 0.86 0.83 0.81 0.78 0.73 0.67 0.59 0.46 0.26
0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99
s = 0.50 70.58 75.11 80.61 87.44 96.13 107.57 123.32 146.39 183.39 252.45
70.75 75.28 80.78 87.59 96.28 107.71 123.45 146.49 183.47 252.50
0.17 0.17 0.17 0.15 0.15 0.14 0.13 0.10 0.08 0.05
0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99
s = 0.50 72.99 77.34 82.58 88.99 97.05 107.46 121.44 141.21 171.30 222.66
73.74 78.07 83.27 89.65 97.67 108.03 121.94 141.62 171.59 222.79
0.75 0.73 0.69 0.66 0.62 0.57 0.50 0.41 0.29 0.13
0.90 0.91 0.92 0.93 0.94 0.96 0.96 0.97 0.98 0.99
s = 0.75 68.65 68.81 72.79 72.94 77.77 77.92 83.89 84.04 91.58 91.71 101.53 101.66 114.92 115.03 133.91 134.00 162.94 163.01 212.84 212.87
0.16 0.15 0.15 0.15 0.13 0.13 0.11 0.09 0.07 0.03
0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99
s = 0.75 69.94 70.61 73.78 74.12 78.36 78.96 83.90 84.46 90.75 91.27 99.44 99.90 110.82 111.21 126.38 126.69 148.95 149.15 184.62 184.71
0.67 0.64 0.60 0.56 0.52 0.46 0.39 0.31 0.20 0.09
0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99
s = 1.0 67.08 67.24 70.93 71.08 75.53 75.67 81.13 81.26 88.10 88.22 97.00 97.12 108.80 108.89 125.14 125.22 149.33 149.38 188.76 188.79
0.16 0.15 0.14 0.13 0.12 0.12 0.09 0.08 0.05 0.03
0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99
67.62 71.10 75.21 80.16 86.21 93.78 103.53 116.58 134.92 162.59
9
= 1.0 68.22 71.67 75.75 80.66 86.66 94.18 103.87 116.84 135.08 162.66
0.60 0.57 0.54 0.50 0.45 0.40 0.34 0.26 0.16 0.07
OTerraced lattice (1 = 11) with a central, target molecule (Y,s) and competing (ledge) reaction centers (p). bReactant subject to periodic boundary conditions in the x direction and periodic/confining boundary conditions in the y direction (see Figure 1). Reactant subject to confining boundary conditions in the x direction and periodic/confining boundary conditions in the y direction.
OTerraced lattice (1 = 11) with a central, target molecule (Y,s) and competing (ledge) reaction centers ( p ) . bReactant subject to periodic boundary conditions in the x direction and periodic/confining boundary conditions in the y direction. Reactant subject to confining boundary Conditions in the I: direction and periodic/ confining boundary conditions in the y direction.
well, the extent of the differences found upon imposing the two different sorts of boundary conditions in the y direction. As 1s seen from the data, the differences in the calculated values of ( n ) essentially evaporate in the limit p 0 (the limiting case of a single, more-or-less centrally disposed trap T) and amount to about one step when the ledge sites are 'activated. Thus, the results recorded in Tables 1-111 quantify numerically (1) the crossover point in the v = 3 vs. 5 profiles in the vicinity of p 3% for ledge trapping and (2) the near insensitivity of the results calculated for ( n ) to the assumption of finite (confining boundary conditions) vs. "infinite" (periodic boundary conditions) clusters when the edgelength of the unit lattice is I = 11. Taken together, these results document the delicate interplay between chemical factors influencing the fate of a diffusing reactant (as reflected in the studies of competitive trapping) and the strictly geometrical factors (system size, nature of the boundaries, nearest-neighbor channels) characterizing the
reaction space on which the reaction-diffusion process takes place.
-
-
V. Relevance to Heterogeneous Catalysis In this section we wish to display the possible relationship between our model calculations and experimental studies in which the catalytic effectiveness of 1qw vs. high Miller index surfaces has been studied. As a point of reference, we shall focus on the studies of Somorjai and co-workers' on dehydrogenation (of cyclohexene to benzene and of cyclohexane to benzene) and hydrogenolysis (of cyclohexane to n-hexane) in which turnover number was studied as a function of step density and kink density on clean platinum surfaces. In certain of these investigations, Somorjai et al.' noticed a remarkable difference in catalytic (7) For a comprehensive summary of this work, see: Somorjai, G . A. "Chemistry in Two Dimensions: Surfaces"; Cornel1 University Press: Ithaca, NY, 1981.
Staten, Musho, and Kozak
450 Langmuir, Vol. 1, No. 4 , 1985 Table 111. Comparison of Results for ( n ) for Two Choices of Boundary Condition for the Case Y = 5 O s = 0.25
0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99
71.34 75.98 81.63 88.66 97.66 109.58 126.10 150.57 190.50 267.33
71.53 76.17 81.82 88.85 97.85 109.77 126.32 150.82 190.85 267.98
0.19 0.19 0.19 0.19 0.19 0.19 0.22 0.25 0.35 0.65
0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99
s = 0.50 67.14 67.34 71.04 71.24 75.71 75.93 81.43 81.65 88.57 88.81 97.75 98.02 109.99 110.31 127.12 127.53 152.81 153.39 195.60 196.60
0.20 0.20 0.22 0.22 0.24 0.27 0.32 0.41 0.58 1.00
0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99
s = 0.75 64.19 64.41 67.62 67.86 71.70 71.95 76.63 76.90 82.70 82.99 90.35 90.70 100.32 100.73 113.82 114.36 133.17 133.92 163.18 164.38
0.22 0.24 0.25 0.27 0.29 0.35 0.41 0.54 0.75 1.20
0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99
s = 1.0 62.01 62.26 65.13 65.39 68.80 69.09 73.21 73.52 78.58 78.93 85.28 85.69 93.86 94.35 105.25 105.89 121.12 121.98 144.72 146.03
0.25 0.26 0.29 0.31 0.35 0.41 0.49 0.64 0.86 1.31
“Terraced lattice (1 = 11) with a central, target molecule (Y,s) and competing (ledge) reaction centers @). bReactant subject to periodic boundary conditions in the I direction and periodic/confining boundary conditions in the y direction. cReactant subject to confining boundary conditions in the I direction and periodic/ confining boundary conditions in the y direction.
activity (turnover number) with increase in the concentration of step or kink atoms, and we wish to explore whether our lattice-based calculations also reveal differences in reaction efficiency when such surface imperfections are considered. We begin by referring to Figure 1 and consider the Figure l b first. Although, for convenience of illustration, we have drawn the surface with a step or ledge, it is evident that the surface is topologically equivalent to a flat, planar surface; imagine the lattice to be a rubber “sheet” and notice that a simple diffeomorphic distortion [pulling the sheet in opposite directions from the top (1-12-23-34-4556-66-77-88-99-110) and bottom (11-22-33-44-55-66-7687-98-109-120)] converts the figure into an 11 X 11 square-planar configuration with a centrosymmetric trap T. Thus, topologically, the site labeled T is equivalent to any of the terrace sites; there are four nearest-neighbor paths to the site T; i.e., the valency u of site T as well as all of the remaining (120) sites is 4. What can distinguish
a ledge site from a terrace site, however, is the effective electronic or ligating character of an atom positioned at that site. For example, a site buried in the bulk of a crystal possessing cubic symmetry will be six-coordinated. In this symmetry, an atom situated a t a terrace site will be fivecoordinated, with the missing nearest neighbor resulting in electronic “unsaturation” at that site. For definiteness, let us use the notation n, to denote the overall coordination number of a given site and the symbol nl to denote the number of unsatisfied bonds (consistent with a given symmetry). In this notation, then, a terrace atom would be coded u = 4, n, = 5, and nl = 1. The ledge site denoted T in Figure l b would be coded u = 4,n, = 4,and nl = 2, while the ledge site labeled 61 in that figure would be coded u = 4, n, = 6, nl = 0. From this last example, it is clear that the coordination number n, of a step site may be less or greater than a simple terrace site, depending on the location of the step site. The characterization of surface kink sites in Figures la,c for u = 3 and 5 proceeds in a manner similar to that outlined above for the case v = 4. In particular, the sites labeled T and 61 in Figure l a are both of valency u = 3 and, although they occupy (apparently) different positions on the lattice as drawn, they are topologically equivalent sites. The energetic characterization of the two sites is quite different, however; inspection of Figure l a shows that for site T, u = 3, n, = 6, and nl = 0 while for site 61 we find u = 3, n, = 3, and nl = 3. Similarly, in Figure IC,site T would be characterized by the specifications u = 5, n, = 5, and nl = 1 while site 62 would be coded u = 5, n, = 6, and nl = 0. The numerical specificationslaid down in the preceding paragraph have been developed in order to separate and contrast two distinct ideas, one entropic and one energetic. The valency u, as noted several times already, codes the connectivity of the lattice (or a particular site); i.e., it codes the number of ways of reaching a target site from the immediate (nearest-) neighbors of that site. In the present context, it is a measure of the accessibility of a given site to a diffusing reactant and controls the catalytic efficiency of the process (see Figures 2-5), all other things (energetics, concentration of reaction centers) being held constant. On the other hand, the factor identified as nl in the above is reflective of the electronic unsaturation (ligating ability) at a particular site; it is, in effect, a parametrization of the strength of interaction at that site. In order to study the interplay between these two factors, one entropic ( u ) and one energetic (nl),suppose we associate different settings of the parameter s in our model with the nl. Specifically, we shall identify the setting s = 1.0 with the value nl = 3, s = 0.75 with nl = 2, s = 0.50 with nl = 1, and finally s = 0.25 with nl = 0. A tabulation of the average walklength ( n )for all possible combinations of {v,nl)for lattices with edgelength 1 = 5, 7, 9, and 11 is presented in Table IV. We now recast the data presented in Table IV into a form in which contact can be made with the sort of results reported by Somorjai et aL7 in their studies of hydrocarbon conversion over platinum. As in the preceding paper, we argue that since ( n ) is the average number of steps required for trapping then, given a mean jump time for the stepwise motion of the reactant, (n)-I is proportional to a turnover number. In the hydrocarbon conversion experiments, Somorjai estimates the number of surface platinum atoms to be 1.5 X l O I 5 atoms. Accordingly, we convert the concentration of traps in our model to an effective site density. Thus, on an 11 X 11 lattice, when a single ledge site is considered, the concentration relative to the total cluster (of 121 sites) would be 1/121 = 0.00826;
Influence of Structure on Reaction Efficiency. 2 Table IV. Comparison of Results for (n) for Lattices with a Single. Centrally Positioned Reaction Center b , s ) 5 x 5
7 x 7
6 5 4 3
0
4
3
2 3
0.25 0.50 0.75 1.00
147.73 78.47 55.37 43.82
6 5 4 3
0 1 2 3
0.25 0.50 0.75 1.00
106.40 56.40 39.73 31.40
5
6 5 4 3
0 1 2 3
0.25 0.50 0.75 1.00
89.36 47.79 33.93 27.00
3
6 5 4 3
0 1 2 3
0.25 0.50 0.75 1.00
294.04 160.85 116.44 94.22
4
6 5 4 3
0 1 2 3
0.25 0.50 0.75 1.00
218.15 120.15 87.49 71.15
6 5 4 3
0 2 3
0.25 0.50 0.75 1.00
154.62 94.09 71.37 61.37
3
6 5 4 3
0 1 2 3
0.25 0.50 0.75 1.00
495.08 276.66 203.82 167.39
4
6 5 4 3
0 1 2 3
0.25 0.50 0.75 1.00
372.99 210.99 156.99 129.99
5
6 5 4 3
0 1 2 3
0.25 0.50 0.75 1.00
309.82 178.71 134.99 113.13
3
6 5 4 3
0 1 2 3
0.25 0.50 0.75 1.00
752.13 427.16 318.78 264.58
4
6 5 4 3
0 1 2 3
0.25 0.50 0.75 1.00
572.20 330.20 249.54 209.20
5
6 5 4 3
0 1 2 3
0.25 0.50 0.75 1.00
476.19 281.12 216.08 183.56
5
9 x 9
11 x 11
1
1
when multiplied by 1.15 X 1015atoms, this gives the site density, 1.24 X 1013atoms/12, the estimate listed in Table V. The data recorded in Table V (constructed from the data in Table IV as described above) allow one to compare the relative effectiveness of terrace, ledge, and kink sites when both accessibility and ligating ability (respectively entropic and energetic effects) are taken into account. If we regard the terrace site as portraying the “perfect” two-dimensional, planar surface, with the data on ledge and kink sites reflecting the influence of “imperfections” on the catalytic process (as revealed in the studies of Somorjai et al.’ on high Miller index surfaces or in studies on the consequences of surface reconstruction and the selvedge effect8),then our calculations show that step and kink sites may be more or less effective than terrace sites depending on the location of the former. For example, the (8) See: Forty, A. J. Contemp. Phys. 1983, 24, 271.
Langmuir, Vol. 1, No. 4, 1985 451 ledge site 4(T) (see Table V) produces a higher calculated turnover number than the terrace site but the ledge site 4(61) yields a lower turnover number than this (same) terrace site. A similar distinction can be seen upon comparing the v = 3 kink sites, 3(T) and 3(61), against a terrace site and the v = 5 kink sites, 5(T) and 5(62), against the same terrace site. These calculations show convincingly that the overall catalytic activity of a particular site rests on a delicate balance between energetic and entropic factors, i.e., between the ligating ability of a particular site (as represented in the present study by nl)and its accessibility to a diffusing coreactant (as monitored here by distinguishing among the possible number of channels v to the active site). If, as stressed recently by Cardillo: absorbates sample all defects on the underlying lattice, then the sensitivity to structure documented in the above calculations may be taken as a calibration of the relative importance of terrace vs. ledge vs. kink sites in influencing the kinetics of surface reactions.
VI. Discussion The influence of structure on reactivity, as explored in the preceding contribution, was found to depend critically on the number and configuration of reaction centers defining a multiplet or ensemble. While these particular structural effects are undoubtedly of great importance they are not the only ones thought to play a role in affecting the efficiency of reaction-diffusion processes on a catalytically active surface. Of importance as well is the presence of defects and surface reconstruction, factors under intense investigation today.1° Accordingly, in this paper we have relaxed the assumption that the surface of the support is an idealized d = 2 surface and have considered explicitly the role of terrace vs. ledge vs. kink sites in affecting the reactivity of the system. Then, in the spirit of the preceding paper, we have explored the consequences of assuming configurations of reaction sites in competition with each other and have studied the net effect of these sites on the reaction efficiency. In brief, two limiting cases and one intermediate case were studied in this paper. In the first case, we considered only a single reaction center positioned at a site of valency v = 3 vs. 4 vs. 5 and calculated the lattice-statistical quantity ( n ) as a function of increasing system size. We found that for a given trapping probability (s) at the reaction center, the percent differences calculated for ( n ) for the valencies v = 3,4, and 5 persisted but decreased with increase in lattice size, whereas for a given lattice size the percent difference decreased as the reaction assumed more and more of an irreversible character (i.e., as s 1). A second limiting case, that for which the target site was embedded in a lattice all sites of which could compete with the target, was studied and found to yield results similar to those reported in our earlier studies on square-planar and hexagonal lattices in d = 2; viz., assuming a -5% reactivity a t the N - 1 site surrounding the central trap effectively erased distinctions among lattices (or here sites) of different valencies in the same dimension. Finally, we considered an intermediate case, that for which only the ledge sites competed with the target molecule, the latter situated at a central site characterized by a valency v = 3, 4,or 5. Here we found that the consequences of considering the presence of a banded ensemble of competing
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(9) Cardillo, M. J. Langmuir 1986, 1, 4. See also: Cardillo, M. J. Springer Ser. Chem. Phys. 1982,20, 149. (10) See especially contributions by: Roelofs, L. D. Springer Ser. Chem. Phys. 1982,20, 219. Lagally, M. G. Springer Ser. Chem. Phys. 1982,20, 281.
452 Langmuir, Vol. 1, No. 4, 1985
Staten, Musho, and Kozak
Table V. ComDarison of Turnover Numbers for Lattice with a Sin&. Centrally Positioned Reaction Center ( V a l ) turnover no. (-(n)-’) vs. site density (-(1/12) site terrace ledge ledge kink(s) kink(s) kink(s) kink(s)
U
4 4(T) 4(61) 3(T) 3(61) 5(T) 5(62)
X
1.5 X 10l6 atoms)
nl 1
2
0 0 3 1
0
lattice site density turnover no. lattice site density turnover no. 1attice site density turnover no. lattice site density turnover no. lattice site density turnover no. lattice site density turnover no. lattice site density turnover no.
1 = 11 1.24 x 1013 3.03 x 10-3 1 = 11 1.24 x 1013 4.01 x 10-3 E = 11 1.24 x 1013 1.74 x 10-3 1 = 11 1.23 x 1013 1.33 x 10-3 1 = 11 1.23 x 1013 3.78 x 10-3 1 = 11 1.23 x 1013 3.56 x 10-3 1 = 11 1.23 x 1013 2.10 x 10-3
reaction centers were not quite as extreme as for the case where all N - 1background sites were activated. Yet, even here, since the efficiency of the overall process changed dramatically (increased) when
