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Langmuir 1985, 1 , 443-452 443the characterization and catalytic properties of a varietyof highly dispersed bimetallic entit ies, it may be antici-pated that the above program when applied t o such sys-tems will lead to a more detailed understanding of hy-drocarbon conversion over metal alloys/clusters.Having noted these reservations, it still remains that theexperimental studies on the hydrogenolysis/dehydroge-nation reactions cited above lead to general conclusions

on the importance of reactive-site clusters on reactionefficiency. In the same spirit, it may be hoped that thedetails of the reaction mechanism in each case do notinvalidate the general theoretical conclusions that followfrom our study of the distribution and concentration ofactive sites, although, to be sure, they will certainly refinethe understanding of the problem once they are incorpo-rated into the model.

Influence of Structure on Reaction Efficiency in SurfaceCatalysis. 2. Reactivity at Terraces, Ledges, and KinksG. Joseph Staten,+Matthew K. Musho,t and Jo hn J. Kozak*

Department of Chemistry and Radiation Laboratory,$ University of Notre Dame,Notre Dame , Indiana 46556Received Septe mb er 11, 1984. I n Final Form: Janua ry 25, 1985

In this paper we continue our development of a lattice-based theory of reaction efficiency on catalyticsurfaces. The specific problem dealt with in this paper is the role of lattice imperfections in influencingthe efficiency of diffusion-controlled reactions on such surfaces. We consider a set of reaction centersdistributed on a surface on which there are terrace, ledge, and kink sites and calculate numerically usingour Markovian (implicit function) approach the changes in the efficiency of the process (as monitored bycalculating the average walklength ( n ) )n three distinct situations: (1)we consider a single reaction centerimbedded on a surface (of up to 121 sites) and located first a t a terrace site, then a t a ledge site, and finallyat 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 se t 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 setof competing reaction centers but where the latter are positioned at the ledge sites of the lattice, with theremaining 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 comparedirectly the interplay between entropic and energetic factors in influencing the overall efficiency of thereaction-diffusion process. The possible relevance of these calculations to the experimental studies ofSomorjai and co-workers in which turnover number was studied as a function of step density and kinkdensity for hydrogenation and hydrogenolysis reactions of hydrocarbons on clean platinum surfaces is broughtout and discussed.

I. IntroductionThe physical problem we wish to address in this paperconcerns the role of terrace vs. ledge vs. kink sites in in-fluencing the efficiency of reaction-diffusion processes onsurfaces. In the preceding paper1 we portrayed the surfaceof a pure metal (Cu) or metal alloy (Cu/Ni) as a perfectarray of hexagonally close-packed metal atoms; the mi-gration of the reactant was assumed to proceed viahollow-to-hollow umps and the reaction-diffusion pro-cess was studied on the dual lattice (v =3) to this physicalarray of (surface) coordination number v =6. In order tocharacterize more conveniently reactions at terrace/ledge/kink sites, we choose here to consider processes onsurfaces for which the atoms are packed together insquare-planar symmetry with v =4. Notice that any oneof the terrace sites in Figure 1 s characterized by a con-nectivity u =4, whereas the site labeled T in Figure l a hasa valency u =3, the site labeled T in Figure l b is of valency ermanent address: Magnavox Corporation, Fort Wayne, I N* Permanent address: Miles Laboratories, Elkhart, I N 46614.#T he research described herein was supported in part by theOffice of Basic Energy Sciences of the Department of Ehergy. This

is Document NDRL-2633 from the Notre Dame Radiation Labora-tory.

46808.

0743-7463/8512401-0443$0l.50/0

v =4, and the site labeled T in Figure IC s of valency v=5 . All labeled sites in the Figure 1 code the physicallocations of the atoms comprising the surface being stud-ied. Now, notice that if one assumes hollow-to-hollowjumps on the terraces of any of these lattices, the numberof directions in which the diffusing adatom can migrateis four; since a terrace is effectively a perfect square-planar lattice, its dual will also be a square-planar lattice( v =4). Consider now the possible motion of an adatomat or on a ledge or in the vicinity of a kink, keeping in mindthat we consider here only steps of monatomic height. Forsuch structures we impose no gravitational preferencefor an atom situated a t the top vs. the bottom of a step.Otherwise, as stressed by Beiiard,2 this may suggest forexample, that preferential adsorption at surface steps willoccur at the bottom of steps, whereas in reality there maybe cases where the strongest binding is at the top. Tobypass a criticism of lattice (or ball and stick) models ofreactive surfaces, Beiiard suggests that such models shouldbe viewed upside down. Given this, it is evident tha t allhollow-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: Am-sterdam, 1983.0 1985 American Chemical Society


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Figure 1. Configuration of sites fo r th e largest terraced latticeconsidered in this paper. T h e reaction center (trap) s designatedT and is situated at a site of valency v =3 (a),4 (b), or 5 (c).particular, even for hollows at the bottom or top of aledge, or for a hollow in the vicinity of a kink, the con-nectivity (valency) of the reaction space remains v =4.What will change in the problem for an adatom in one ofthe latter configurations (i.e., a ledge or kink vs. a terracehollow) s the number of first, second, ...nearest neigh-bors, and this, of course, changes the energetics in thevicinity of a step or kink.In order to be able to distinguish and then to charac-terize energetic differences among terrace/ledge/ kink sitesin terms of a simple convention, we shall shift our pointof view back to the atoms comprising the actual surface.That is, rather than assume a hollow-to-hollowdiffusionmechanism of a species in the vicinity of a reaction center,we shall assume that the reactant jumps from one physicalsurface site (an atom) to another in its random walkmovement. Thus, the number of degrees of freedomavailable to a reactant at a given point in i t s trajectoryacross the surface will be exactly that of the atomic siteon the underlying lattice ( v =4 for a terrace or ledge site,Y =3 or v =5 for a kink site.) Since the coordinationnumber n, of atoms at particular sites of the lattice canbe enumerated by inspection, an electronic factor canbe associated with the number ni f missing bonds fora surface atom in a lattice characterized, in bulk, by cubicsymmetry ( v =6). As will be demonstrated in section IV,

by this device we shall be able to quantify the interplaybetween entropic and energetic factors in reaction-diffu-sion processes on surfaces with imperfections.11. Formulation

To highlight the importance of lattice imperfections ininfluencing the efficiency of reaction-diffusion processeson low-dimensional surfaces, we construct three lattices.In Figure 1we display the configuration of sites for thelargest lattice considered in this paper, the lattice withedgelength 1 =11. The central t rap is designated T andis situated a t a site of valency v =3 (Figure l a) , valencyv =4 Figure lb) , or valency v =5 (Figure IC).It is im-portant to emphasize that although we have presentedeach of these cases as a terraced structure, each setting ofv is also consistent with a strictly two-dimensional con-figuration of sites, with a pronounced distortion of thelattice about tbe reaction center T (for the valencies v =3 and 5) . The manner in which a vertical ordering can beincorporated in the formulation of the model as it standswill be described in section V.Consider next the trajectory of the diffusing coreactant.In general, the site-to-site motion of the particle can beinfluenced by a variety of potential effects, e.g., potentialsmay induce jumps between non-nearest-neighbor sites onthe lattice (see ref 3) or may bias the motion of the particlein the vicinity of the reaction center (see ref 4). In thispaper, however, since we wish to expose clearly the con-sequences of introducing geometrical constraints or im-perfections (see above), we consider here particle motioncharacterized by nearest-neighbor, unbiased random walksonly.As regards boundary conditions, we refer the reader tothe preceding paper for a description of periodic vs. con-fining boundary conditions. In implementing the Markovchain theory6 for the underlying lattice-statistical problem,one finds tha t when one considers a single trap (or, moregenerally, a set of traps) positioned a t a centrosymmetricsite (or, a set of sites symmetrically positioned with respectto the boundary of the system), there is no formal dif-ference in the structure of the Markovian equations writtendown describing these two boundary conditions. Thus,with respect to the two boundary conditions studied in thispaper (viz., periodic vs. confining), since the reaction centerT is positioned a t the center of the lateral face (the ydirection in the coordinate system defined in the Figure1) for each (odd) lattice considered, imposition of eitherboundary condition leads to the same results with respectto that coordinate direction. Slight differences can arise,however, when one compares results generated when pe-riodic vs. confining boundary conditions a re imposed inthe x direction (again see Figure 1 or the alignment of thecoordinate axes), this because a slight symmetry breakingis introduced when one considers the cases v =3 and 5.However, as will be documented later in this paper, thenumerical differences which arise in the calculation of (n)are so slight (at least for lattices of edgelength 1>9) thatthe results reported in th is paper may be considered rel-evant t o the following two physical situations: (1)a setof sites with traps (i.e., a unit cell defined by one of theconfigurations represented in Figure 1)periodically rep-licated through all space or (2) a finite cluster of sites,compartmentalized in the sense that a diffusing coreactantis (passively) reflected at the boundary of the cluster (viaimplementation 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.


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Influence of Structure on Reaction Efficiency.28oo.r a

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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 bythe 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 distributedsymmetrically 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. ResultsThe principal aim of th e preceding paper was to dem-onstrate how constellations of reaction centers, competingwith a central trap, could influence the efficiency of theunderlying reaction-diffusion process. This pursuit is alsoof central importance in the present s tudy but here, inorder to develop systematically an understanding of theeffeds brought in by considering connectivities other than

u =4 around the central trap T, we consider first and insome detail the simplest case of a single t rap positionedat site T (see Figure l ) ,with all other lattice sites regardedas nontrapping or neutral sites.In Figure 2 we display the average number (n) of stepsrequired for trapping on a terraced lattice as a functionof system size (as calibrated by the edgelength 1) . Spe-cifically, the N - 1 background sites are assumed to bestrictly nontrapping (neutral) and distributed symmetri-cally about the target molecule, the lat ter positioned at asite of valency u =3,4, r 5. The cases illustrated in thisfigure correspond to four settings of the reaction parameters, where s scales th e degree of reversibility of the reactionat the central trap (see the discussion in section I1 of thepreceding paper). Thus, Figure 2a documents the changes

x +Y [XU]*-

in the efficiency of reaction, as gauged by the value of (n),when there is a 25% probability that the above reactionwill proceed a t once to completion with formation of theproduct Z (or, conversely, a 75% probability that thediffusing coreactant (X) , upon confronting the targetmolecule (Y), forms an excited-state complex but one tha teventually falls apart with regeneration of the species X,which subsequently resumes its random walk on the lat-tice). Similarly, in Figure 2, parts b, c , and d record thebehavior observed as a function of system size for thesettings s =0.50,0.75, and 1.00,respectively, with the latterchoice (s =1.0)corresponding to the case where the speciesX reacts with Y irreversibly upon first encounter, formingthe product Z . The behavior of the reaction-diffusionsystem as a function of a continuously varying reactionparameter s can be studied explicitly provided we specifya given lattice. The results of such a study are presentedin Figure 3 where, for a lattice characterized by the ed-gelength 1 = 11, the reaction efficiency is studied as afunction of s. In this figure our results are displayed asa function of (1- s) , a survival probability for a walkeron a lattice characterized (here) by a single, centrosym-metric trap.Given the background of data reported in Figures 2 and3, we cannow begin to explore the consequencesas regards


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446 Langmuir, Vol. 1 , No. 4, 985 Staten, Musho, and KozakThe collection of results displayed in Figure 2 providesthe point of departure for ou r discussion. There it is seenthat for a lattice of a given size (here ranging from one ofedgelength 1 =5 to one of I =11) the number of near-est-neighbor routes to a central target molecule is criticalin determining the efficiency of a reaction-diffusion pro-cess. Our calculations show that there exist alreadyquantitative differences in the value of ( n ) ven for thesmallest lattice considered (viz., the 5 X 5 lattice) and tha t(in an absolute numerical sense) these differences tend to

increase with increase in the spatial extent of the system.Moreover, we find tha t these differences persist when onechanges the factor s which gauges the effectiveness oftrapping at the active site. On a relative basis, the trendsfound are the following: (1)For a given setting of s, th epercent difference between the v =3 and 5 results de-creases with increase in the lat tice size (as calibrated bythe edgelength 1 in the figures). (2) For a given setting of1, the percent difference between the u =3 and 5 resultsdecreases with increase in the degree of irreversibility ofthe reaction (asstudied by increasing the magnitude of theparameter s). These trends document the importance ofshort-range (here cage) effects in influencing reaction-diffusion processes. The cage effect studied here, whicharises from a strictly geometrical restriction (or enhance-ment) in the number of nearest-neighbor channels linkingthe active site to the remainder of the reaction space ofthe system, differs from the short-range (chemical/cage)effect identified in our earlier s t ~ d y . ~n that work, westudied a situation wherein immediate access to the targetmolecule (situated on a regular lattice with no imperfec-tions) was mandated once a nearest-neighbor site wasreached in the random motion of the diffusing coreactant.Taken together, these studies quantitate the importanceof factors controlling the behavior of the coreactant in theimmedia te vicinity of the active site.Whereas the dependence of the average walklength onthe parameter s was displayed in Figure 2 for discretevalues (viz., s =0.25, 0.50, 0.75, and 1.0) for lattices ofvarious sizes, we can also calculate for a given lattice thewalklength ( n ) s a continuous function of s. Thus, inFigure 3 profiles of the walklength ( n ) s. the independentvariable (1- s) are given, where the quantity (1- s) is asurvival (o r escape) probability of a reactant migrating ona lattice (here, an 11 X 11 lattice) with a target moleculesurrounded by -120 nontrapping sites but linked to itsenvironment by three, four, or five channels. From theresults displayed in this figure, it is seen that the separationbetween results calculated for (nearest-neighbor channels)u =4 vs. 3 is greater than that for u =4 vs. 5, especiallyin the range s


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Langmuir, Vol. 1,N o . 4, 1985 447

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lattices subject to different boundary conditions or char-acterized by different valencies; that is, the reaction-dif-fusioh process became kinetically controlled. As is evidentfrom the results displayed in Figure 4, a similar result isfound here since, even when the number of nearest-neighbor channels to the target molecule is changed (Av=fl vis-&vis the host lattice of valency v =4), a changethat produced marked changes in ( n ) when only a singlereaction center was present (see Figure 2), the resultscalculated for (n) end to coalesce for values of p in thevicinity of 3%. In fact, it is seen clearly (compare Figure4a for which s =0.25 vs. Figure 4d where s =1.0) thatwhen one studies this effect as a function of s, the smallerthe value of s, the smaller the value of p that is effectivein erasing distinctions between results calculated for the(nearest neighbor) valencies v =3 vs. 4 vs. 5. Moreover,notice tha t regardless of the setting of s (compare Figures3 and 4) there is a precipitous drop in the calculated valueof (n)when one "turns onn chemically the -120 back-ground sites of the lattice. In fact, orders of magnitudemay separate the values calculated for ( n ) ,depending onthe value of s, when the background is characterized bya mere 5% reactivity.It is plain from the preceding discussion tha t the resultsrecorded in Figure 3 vs. those in Figure 4 describe two,

- -

rather extreme situations. In Figure 3 we considered onlya single, chemically active target molecule embedded inan array of neutral sites, whereas in Figure 4 we assumedthe entire array of background sites could compete chem-ically with the target molecule. Both from the standpointof considering (eventually) the problem of crystal growthand rom the considerations of the following section (whereproblems relating to heterogeneous catalysisare discussed),it is desirable to consider a situation intermediate betweenthese two extremes.We now draw the reader's attention to the results ofFigure 5. These results pertain to a situation where onlythe atoms or molecules comprising the ledge of the lattice(see Figure 1)can compete chemically with the moleculesituated at the centrally disposed active site, the remainingterrace sites of the lattice are assumed to remain chemi-cally inert. The results calculated for this intermediatecase are reminiscent of those displayed in Figure 4, butthere are some interesting differences. First, as regardssimilarities, one notices in both calculations an effectivecoalescence of results calculated for given settings of ( s , p ) ,but here differences persist over a somewhat wider rangeof values of p characterizing he ledge trapping probability;convergence occurs here for percentages somewhat lessthan 10% (as opposed to th e 5% figure noted in Figure


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448 Langmuir, Vol. 1 , N o. 4, 1985 Staten, Musho, and Kozak

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Figure 5. Plot of the average number ( n ) f steps required fo r trapping on an 1 =11 terraced lattice (see Figure 1)vs. a survivalprobability (1- ) against ledge trapping. Considered here is the case of a reaction center (target molecule), situated at a siteof valencyv =3 (dashed line),4 (solid line),or 5 (dotted line) and characterizedby a reaction (trapping) probabilitys. The background sitesare assumed to be distributed symmetrically about the target molecule;of these, the ledge sites can, with probability p , interact withand trap the random walker, while the remaining terrace sites are neutral (i.e., p =0). The diffusing coreactant is assumed to encounterconfining boundary conditions in the x direction and confining/periodic boundary conditions in the y direction. Displayed here arethe 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 indeeda reduction in the magnitude of ( n ) s one "turns on" theledge trapping probability, the effect is not as pronounced(quantitatively) as tha t found when al l background sitesare activated.There is, however, an interesting effect that appears inFigure 5 hat is not present in the Figure 4. One finds,for each setting of s considered, a "crossover" in the profilesgenerated for the case v =3 vs. 4. For example, in casec of Figure 5 where s =0.75, for (1- p ) n the vicinity of0.97 (or p - 3%), the dashed (v =3) and solid ( v = 4)curves intersect. In the range p 3%, thecurve for v =3 is intermediate between the v =4 and 5curves. The reason for this behavior can be seen byviewing, once again, the lattices displayed in Figure 1.There it will be seen that although the number of sitesalong each edge is 11 for each of the three lattices, thelattices corresponding to v =3 and 5 have two more ledgesites than is the case for v =4. These two additional ledgesites, when placed in competition with the central t rap T,enhance the possibility tha t trapping of the diffusing co-reactant will occur; in fact, both the value of ( n ) nd thesetting of (1- ) at which the crossover occurs are sensitive

to th is difference in the number of active ledge sites.Although heretofore we have chosen to display the re-sults of this study (and the preceding one) in graphicalform, it is of interest to tabulate the numerical results forthe "intermediate" case described above. This we havedone in Tables 1-111, since in this format a further pointcan be made concerning the boundary conditions subjectto which the calculations reported in this paper wereperformed. In particular, when one is dealing with an Idlattice with a centrosymmetric reaction center (and noimperfections) the use of periodic or confining boundaryconditions leads to exactly the same results (see our earlierdescription of these two boundary conditions). When onebreaks the symmetry of the reaction space, whether bymoving the target molecule off center or by introducingimperfections (defects), the results calculated assumingthese two choices of boundary condition can differ, withthe differences gradually diminishing with increase inlattice size. For the lattices of edgelength 11displayed inFigure 1, the disposition of the trap will be exactly cen-trosymmetric with respect to the coordinate directiondesignated x , but will be slightly off center with respectto the coordinate direction y for the cases u =3 and 5.Thus, the results registered in Tables 1-111 quantify, as


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Influence of Structure on Reaction Efficiency. 2 Langmuir, Vol. 1,N o. 4, 1985 449Table I. Comparison of Results for ( n ) or Two C hoices ofBoundary C ondition for the Case Y =3 O

(1 - P) ( f l ) I b (n)rrC A( f l )0.900.910.920.930.940.950.960.970.980.990.900.910.920.930.940.950.960.970.980.990.900.910.920.930.940.960.960.970.980.990.900.910.920.930.940.950.960.970.980.99

s =0.2573.03 73.2278.10 78.2884.32 84.5092.16 92.33102.32 102.49116.03 116.19135.54 135.70165.53 165.67217.53 217.65329.84 329.91

s =0.5070.58 70.7575.11 75.2880.61 80.7887.44 87.5996.13 96.28107.57 107.71123.32 123.45146.39 146.49183.39 183.47252.45 252.50s =0.7568.65 68.8172.79 72.9477.77 77.9283.89 84.0491.58 91.71101.53 101.66114.92 115.03133.91 134.00162.94 163.01212.84 212.87s =1.067.08 67.2470.93 71.0875.53 75.6781.13 81.2688.10 88.2297.00 97.12

108.80 108.89125.14 125.22149.33 149.38188.76 188.79

0.190.180.180.170.170.160.160.140.120.070.170.170.170.150.150.140.130.100.080.050.160.150.150.150.130.130.110.090.070.030.160.150.140.130.120.120.090.080.050.03

OTerraced lattice ( 1 = 11) with a central, target molecule (Y,s)and competing (ledge) reaction centers (p). bReactant subject toperiodic boundary conditions in the x direction an d periodic/con-fining boundary conditions in the y direction (see Figure 1).Reactant subjec t to confining boundary conditions in the x direc-tion and periodic/confining boundary conditions in the y direction.well, the extent of the differences found upon imposingthe two different sorts of boundary conditions in the ydirection. As 1s seen from the data, the differences in thecalculated values of ( n ) essentially evaporate in the limitp - (the limiting case of a single, more-or-less centrallydisposed trap T) and amount to about one step when theledge sites are activated.Thus, the results recorded in Tables 1-111 quantify nu-merically (1 ) the crossover point in the v =3 vs. 5 profilesin the vicinity of p - 3% for ledge trapping and (2 ) thenear insensitivity of the results calculated for ( n ) to theassumption of finite (confining boundary conditions) vs."infinite" (periodic boundary conditions) clusters when theedgelength of the unit lattice is I =11. Taken together,these results document the delicate interplay betweenchemical factors influencing the fate of a diffusing reactant(as reflected in the studies of competitive trapping) andthe strictly geometrical factors (system size, nature of theboundaries, nearest-neighbor channels) characterizing the

Table 11. Comparison of R esults for (n ) for Two Choicesof Bound ary Condition for the Case Y =4'

0.900.910.920.930.940.950.960.970.980.990.900.910.920.930.940.950.960.970.980.990.900.910.920.930.940.950.960.970.980.990.900.910.920.930.940.950.960.970.980.99

s =0.2577.17 78.0482.29 83.1588.55 89.3896.36 97.17106.39 107.17119.74 120.47138.39 139.06116.37 166.86212.50 212.96304.09 304.35

s =0.5072.99 73.7477.34 78.0782.58 83.2788.99 89.6597.05 97.67107.46 108.03121.44 121.94141.21 141.62171.30 171.59222.66 222.79s =0.7569.94 70.6173.78 74.1278.36 78.9683.90 84.4690.75 91.2799.44 99.90110.82 111.21126.38 126.69148.95 149.15184.62 184.719 =1.067.62 68.2271.10 71.6775.21 75.7580.16 80.6686.21 86.6693.78 94.18

103.53 103.87116.58 116.84134.92 135.08162.59 162.66

0.870.860.830.810.780.730.670.590.460.260.750.730.690.660.620.570.500.410.290.130.670.640.600.560.520.460.390.310.200.090.600.570.540.500.450.400.340.260.070.16

OTerraced lattice (1 =11) with a central, target molecule (Y,s)and competing (ledge) reaction centers ( p ) . bReactant subject toperiodic boundary conditions in the x direction and periodic/con-fining boundary conditions in the y direction. Reactant subject toconfining boundary Conditions in the I: direction and periodic/confining boundary conditions in the y direction.reaction space on which the reaction-diffusion processtakes place.

V. Relevance to Heterogeneous CatalysisIn this section we wish to display the possible relation-ship between our model calculations and experimentalstudies in which the catalytic effectiveness of 1qw vs. highMiller index surfaces has been studied. As a point ofreference, we shall focus on the studies of Somorjai andco-workers' on dehydrogenation (of cyclohexene to benzeneand of cyclohexane to benzene) and hydrogenolysis (ofcyclohexane to n-hexane) in which turnover number w asstudied as a function of step density and kink density onclean 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.


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450 Langmuir, Vol. 1,No. 4 , 1985 Staten, Musho, and Kozaka ledge site from a terrace site, however, is the effectiveelectronic or ligating character of an atom positioned atthat site. For example, a site buried in the bulk of a crystalpossessing cubic symmetry will be six-coordinated. In thissymmetry, an atom situated a t a terrace si te will be five-coordinated, with the missing nearest neighbor resultingin electronic unsaturation at that site. Fo r definiteness,let us use the notation n, to denote the overall coordinationnumber of a given site and the symbol nl to denote thenumber of unsatisfied bonds (consistent with a givensymmetry). In this notation, then, a terrace atom wouldbe coded u =4, n , =5, and nl =1. The ledge site denotedT in Figure l b would be coded u =4, , =4, nd nl =2,while the ledge site labeled 61 in that figure would be codedu =4, n , =6, nl =0. From this last example, it is clearthat the coordination number n , of a step site may be lessor greater than a simple terrace site, depending on thelocation of the step site.

The characterization of surface kink sites in Figures la,cfo r u = 3 and 5 proceeds in a manner similar to thatoutlined above for the case v =4. In particular, the siteslabeled T and 61 in Figure l a are both of valency u =3and, although they occupy (apparently) different positionson the lattice as drawn, they are topologically equivalentsites. The energetic characterization of the two sites isquite different, however; inspection of Figure l a shows thatfor site T, u =3, n , =6, and nl=0 while for site 61 we findu =3, n , =3, and nl =3. Similarly, in Figure IC,site Twould be characterized by the specifications u =5, n, =5, and nl =1while site 62 would be coded u =5, n , =6,and n l =0.

The numerical specifications aid down in the precedingparagraph have been developed in order to separate andcontrast two distinct ideas, one entropic and one energetic.The valency u , as noted several times already, codes theconnectivity of the lattice (or a particular site); i.e., it codesthe number of ways of reaching a target site from theimmediate (nearest-) neighbors of that site. In the presentcontext, it is a measure of the accessibility of a given siteto a diffusing reactant and controls the catalytic efficiencyof the process (see Figures 2-5), all other things (energetics,concentration of reaction centers) being held constant. Onthe other hand, the factor identified as nl in the above isreflective of the electronic unsaturation (ligating ability)at a particular site; it is, in effect, a parametrization of thestrength of interaction at tha t site. In order to study theinterplay between these two factors, one entropic ( u ) andone energetic ( n l ) , uppose we associate different settingsof 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 ) or all possible combinations of {v,nl)or lattices withedgelength 1 =5, 7, 9, and 11 is presented in Table IV.We now recast the data presented in Table IV into aform in which contact can be made with the sort of resultsreported by Somorjai et aL7 in their studies of hydrocarbonconversion over platinum. As in the preceding paper, weargue that since ( n ) s the average number of steps re-quired for trapping then, given a mean jump time for thestepwise motion of the reactant, (n)-I is proportional toa turnover number. In the hydrocarbon conversion ex-periments, Somorjai estimates the number of surfaceplatinum atoms to be 1.5 X l O I 5 atoms. Accordingly, weconvert the concentration of traps in our model to aneffective site density. Thus, on an 11X 11 lattice, whena single ledge site is considered, the concentration relativeto the total cluster (of 121 sites) would be 1 /121 =0.00826;

Table 111. Comparison of Results for ( n ) for TwoChoicesof Boundary Condition for the Case Y =5 O

0.900.910.920.930.940.950.960.970.980.990.900.910.920.930.940.950.960.970.980.990.900.910.920.930.940.950.960.970.980.990.900.910.920.930.940.950.960.970.980.99

s =0.2571.34 71.5375.98 76.1781.63 81.8288.66 88.8597.66 97.85109.58 109.77126.10 126.32150.57 150.82190.50 190.85267.33 267.98

s =0.5067.14 67.3471.04 71.2475.71 75.9381.43 81.6588.57 88.8197.75 98.02109.99 110.31127.12 127.53152.81 153.39195.60 196.60s =0.7564.19 64.4167.62 67.8671.70 71.9576.63 76.9082.70 82.9990.35 90.70100.32 100.73113.82 114.36133.17 133.92163.18 164.38s =1.062.01 62.2665.13 65.3968.80 69.0973.21 73.5278.58 78.9385.28 85.69

93.86 94.35105.25 105.89121.12 121.98144.72 146.03

0.190.190.190.190.190.190.220.250.350.650.200.200.220.220.240.270.320.410.581.000.220.240.250.270.290.350.410.540.751.200.250.260.290.310.350.410.490.640.861.31

Terra ced lattice ( 1 = 11)with a central, target molecule (Y,s)and competing (ledge) reaction centers @). bReacta nt subject t operiodic boundary conditions in the I direction and periodic/con-fining boundary conditions in the y direction. cReactant subject toconfining boundary conditions in the I direction and periodic/confining boundary conditions in the y direction.activity (turnover number) with increase in the concen-tration of step or kink atoms, and we wish to explorewhether our lattice-based calculations also reveal differ-ences in reaction efficiency when such surface imperfec-tions are considered.We begin by referring to Figure 1 and consider theFigure l b first. Although, for convenience of illustration,we have drawn the surface with a step or ledge, it is evidentthat the surface is topologically equivalent to a flat, planarsurface; imagine the lattice to be a rubber sheet andnotice tha t a simple diffeomorphic distortion [pulling thesheet in opposite directions from the top (1-12-23-34-45-56-66-77-88-99-110) and bottom (11-22-33-44-55-66-76-87-98-109-120)] converts the figure into an 11 X 11square-planar configuration with a centrosymmetric trapT. Thus, topologically, the site labeled T is equivalent toany of the terrace sites; there are four nearest-neighborpaths to the site T; i.e., the valency u of site T as well asall of the remaining (120) sites is 4. What can distinguish


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Influence of S tructure on Reaction Efficiency. 2 Langmuir, Vol. 1 , N o . 4, 1985 451ledge site 4(T) (see Table V) produces a higher calculatedturnover number than the terrace s ite but the ledge site4(61) yields a lower turnover number than this (same)terrace site. A similar distinction can be seen upon com-paring the v =3 kink sites, 3(T) and 3(61), against a terracesite and the v =5 kink sites, 5(T ) and 5(62), against thesame terrace site. These calculations show convincinglythat the overall catalytic activity of a particular site restson a delicate balance between energetic and entropicfactors, i.e., between the ligating ability of a particular site(as epresented in the present study by nl)and its acces-sibility to a diffusing coreactant (as monitored here bydistinguishing among the possible number of channels vto th e active site). If, as stressed recently by Cardillo:absorbates sample all defects on the underlying lattice,then the sensitivity to structure documented in the abovecalculations may be taken as a calibration of the relativeimportance of terrace vs. ledge vs. kink sites in influencingthe kinetics of surface reactions.

VI. DiscussionThe influence of structure on reactivity, as explored inthe preceding contribution, was found to depend criticallyon the number and configuration of reaction centers de-fining a multiplet or ensemble. While these particularstructural effects are undoubtedly of great importance theyare not the only ones thought to play a role in affectingthe efficiency of reaction-diffusion processes on a cata-lytically active surface. Of importance as well is thepresence of defects and surface reconstruction, factorsunder intense investigation today.1 Accordingly, in thispaper we have relaxed the assumption tha t the surface ofthe support is an idealized d =2 surface and have con-sidered explicitly the role of terrace vs. ledge vs. kink sitesin affecting the reactivity of the system. Then, in the spiritof the preceding paper, we have explored the consequencesof assuming configurations of reaction sites in competitionwith each other and have studied the net effect of thesesites on the reaction efficiency.In brief, two limiting cases and one intermediate casewere studied in this paper. In the first case, we consideredonly a single reaction center positioned at a site of valency

v = 3 vs. 4 vs. 5 and calculated the lattice-statisticalquantity ( n ) as a function of increasing system size. Wefound that for a given trapping probability (s) at the re-action center, the percent differences calculated for ( n )for the valencies v =3,4, and 5 persisted but decreasedwith increase in lattice size, whereas for a given lattice sizethe percent difference decreased as the reaction assumedmore and more of an irreversible character (i.e., as s- ).A second limiting case, tha t for which the target site wasembedded in a lattice all sites of which could compete withthe target, was studied and found to yield results similarto those reported in our earlier studies on square-planarand hexagonal lattices in d = 2; viz., assuming a - 5 %reactivity at the N - 1 site surrounding the central trapeffectively erased distinctions among lattices (or here sites)of different valencies in the same dimension. Finally, weconsidered an intermediate case, that fo r which only theledge sites competed with the target molecule, the lattersituated at a central site characterized by a valency v =3, 4,or 5. Here we found th at the consequences of con-sidering the presence of a banded ensemble of competing

Table IV . Comparison of Resu lts for (n for Lattices witha Single. Cen trally Positioned Reaction Center b , s )5 x 5

7 x 7

9 x 9

11x 11

3

4

5

3

4

5

3

4

5

3

4

5

654365436543654365436543654365436543654365436543

012301230123012301230123012301230123012301230123

0.250.500.751.000.250.500.751.000.250.500.751.000.250.500.751.000.250.500.751.000.250.500.751.000.250.500.751.000.250.500.751.000.250.500.751.000.250.500.751.000.250.500.751.000.250.500.751.00

147.7378.4755.3743.82106.4056.4039.7331.4089.3647.7933.9327.00

294.04160.85116.4494.22218.15120.1587.4971.15154.6294.0971.3761.37

495.08276.66203.82167.39372.99210.99156.99129.99309.82178.71134.99113.13752.13427.16318.78264.58572.20330.20249.54209.20476.19281.12216.08183.56

when multiplied by 1.15 X 1015 atoms, this gives the sitedensity, 1.24 X 1013atoms/12, the estimate listed in TableV.The data recorded in Table V (constructed from thedata in Table IV asdescribed above) allow one to comparethe relative effectiveness of terrace, ledge, and kink siteswhen both accessibility and ligating ability (respectivelyentropic and energetic effects) are taken into account. Ifwe regard the terrace site as portraying the perfecttwo-dimensional, planar surface, with the data on ledgeand kink sites reflecting the influence of imperfectionson the catalytic process (as evealed in the studies of So -morjai et al. on high Miller index surfaces or in studieson the consequences of surface reconstruction and theselvedge effect8), hen ou r calculations show that step andkink sites may be more or less effective than terrace sitesdepending on the location of the former. For example, the

(8) See: Forty, A. J. Con temp . Ph ys . 1983, 24, 271.

(9 )Cardillo, M. J. Langmuir 1986, 1, . See also: Cardillo, M. J.Springer S er. Chem. Phys. 1982,20, 149.(10) See especially contributions by: Roelofs, L. D. Springer Ser.Ch em. Ph ys . 1982,20, 219. Lagally, M. G. Springer Ser. Chem. Phys.1982,20, 281.


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452 Langmuir, Vol. 1, No. 4, 1985 Staten, M u sh o , and KozakTable 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) X 1.5 X 10l6 atoms)site U nl

terrace 4 1 latticesite densityturnover no.site densityturnover no.site densityturnover no.site densityturnover no.site densityturnover no.site densityturnover no.site densityturnover no.

ledge 4(T) 2 lattice

ledge 4(61) 0 1attice

kink(s) 3(T) 0 lattice

kink(s) 3(61) 3 lattice

kink(s) 5(T) 1 lattice

kink(s) 5(62) 0 lattice

1 =111.24 x 10133.03 x 10-31.24 x 10134.01 x 10-31.24 x 10131.74 x 10-31.23 x 10131.33 x 10-31.23 x 10133.78 x 10-31.23 x 10133.56 x 10-31.23 x 10132.10 x 10-3

1 =11

E =11

1 =11

1 =11

1 =11

1 =11

reaction centers were not quite as extreme as for the casewhere al l N - 1background sites were activated. Yet, evenhere, since the efficiency of the overall process changeddramatically (increased) when < l o% of the backgroundledge sites were activated, we argued that these data werein qualitative accord with the conclusions reached by So-morjai and Cardillo and their co-workers concerning thecritical influence of ledges (and other defects) on reactivity,conclusions drawn from their experimental studies on lowvs. high Miller index surfaces.In order t o provide a more detailed characterization ofthe effects found at terrace vs. ledge vs. kink sites, calcu-lations were performed in which the various settings of thereaction parameter s were placed in correspondence withthe number of free bonds nlcharacterizing the electronicunsaturation at particular sites of the Iattice. When takenin conjunction with the valency u (describing the numberof channels linking the particular site to the remaining N- 1 sites of the lattice), it was found that the pair ( n p )provided a convenient way of organizing the energetic andentropic factors governing the efficiency of reaction a t thegiven site. When both (nbvJ ere considered, we found thatcertain defect (ledge, kink) sites were more effective thanterrace sites in leading to reaction bu t that there were caseswhere terrace sites remained more effective than ledge orkink sites in enhancing the efficiency of the underlyingprocess. These calculations dramatize, we believe, thedelicate interplay between energetic and entropic factorsin influencing the efficiency of reaction on a catalyticallyactive surface with defects. They suggest tha t althoughthe presence of defects (ledges, kinks) may certainly alterthe turnover number determined experimentally, the netalteration should probably be interpreted as a trade offof effects; tha t is, relative to processes on clean low Millerindex surfaces, certain defects characterized by a given (nl,u)may enhance while other types {nl,v)may suppress theconversion to products, with the net effect of these latticeimperfections reflected in t he (ensemble-averaged) turn-over number determined experimentally.As in the previous study, it is important to evaluatecritically the approach taken in this paper in assessing the

1 = 91.85 x 10134.74 x 10-31.85 x 10131.85 x 10132.68 x 10-31.83 x 10132.02 x 10-31.83 x 10135.97 x 10-31.83 x 10135.60 x 10-31.83 x 10133.23 x 10-3

1 = 96.37 X1 = 9

1 = 9

1 = 9

1 = 9

1 = 9

1 = 73.06 x 10138.32 x 10-33.06 x 10133.06 x 10134.58 x 10-33.00 x 10133.40 x 10-33.00 x 10133.00 x 1013

3.00 x 10136.47 x 10-3

1 = 711.43 X1 = 7

1 = 7

1 = 710.61 X1 = 710.63 X1 = 7

1 = 517.73 X1 = 525.17 X1 = 5

6.00 x 1013

6.00 x 10136.00 x 10139.40 x 10-35.77 x 10136.77 x 10-35.77 x 1013

5.77 x 1013

5.77 x 101311.19 x 10-3

1 = 5

1 = 522.82 X1 = 520.92 X1 = 5

difference in reactivity a t steps, ledges, and kinks. First,although the distribution of reactive sites in our model wa sdesigned to handle situations of increasing complexity(moving from a consideration of single sites to domains ofactive sites distributed over the surface), it cannot beclaimed that any one of these configurations representsa particular physical system; most catalytic processes ofinterest occur on highly defect surfaces of supportedcrystallites in which the support-crystallite interaction canexert a considerable influence on the efficiency of thecatalytic process. Furthermore, chemical interactions (bothadsorbate-adsorbate and adsorbate-substrate) may be sostrong that adsorbates will not readily diffuse across certainplanes, as shown by studies of Hz n W using field ionmicroscopy;l such constraints are only weakly implied bythe class of boundary conditions employed in our study.Again, it is evident from the above discussion (and fromremarks in section VI of the preceding paper) that aprincipal goal of future studies must be to fold into ourmodel the results of electronic structure calculations onabsorbate-metal atom clusters in order to place thespecification of the site trapping probability on a rigorousbasis. Furthermore, it would be very instructive to com-pare the kinetic consequences of the sort of model de-veloped here with calculations of the rate constant basedon transition state (or other) theories, as have been re-viewed, for example, by Cardillo and Tully.I2 Thesegeneralizations of our approach are under developmentbut, as regards the present contributions, it may be hopedthat our calculations provide some insight into the relativeimportance of chemical interaction vs. physical diffusionon surfaces with multiplets of active sites or surfaces withdistinguishable terrace, ledge, and kink sites of differingcatalytic activity.

(11)We a r e thankful to the referee for bringing this study to ourattention; see: Roberts, M. W.; McKee, C. S. Chemistry of the Metal-Gas Interface; Clarendon Press: Oxford, 1978. See, in particular, thediscussion of the work of Folman, M.; Klein, R. S u rf . Sci. 1968,l , 430cited in section 8.3.(12)See, for example, the references cited in: Tully, J. C.; Cardillo,M. J. Science (Washington,D.C.)1984,223, 445.

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