Nanostructure effects on the kinetics and deactivation at reactions over multifunctional catalysts

Nanostructure effects on the kinetics and deactivation at reactionsover multifunctional catalysts

Yahia Alhamed a, Krassimira Kumbilieva b,n, Abdulrahim AlZahrani a, Lachezar Petrov a,c

a Chemical and Materials Engineering Department, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Kingdom of Saudi Arabiab Institute of Catalysis, Bulgarian Academy of Sciences, “Acad. G. Bonchev” str., block 11, Sofia 1113, Bulgariac SABIC Chair of Catalysis, Chemical and Materials Engineering Department, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Kingdom of Saudi Arabia

H I G H L I G H T S

� Catalyst deactivation rate dependson the size of supported metal par-ticles.

� Nanosized metal particles are highlysensitive for poisoning and deactiva-tion.

� Multi-centered and single activesites differ in vulnerability level oncoking.

� Coking lowers probability for siteensembles catalyzing structure-sensi-tive routes.

� Active sites edging coke havereduced potential to favorstructure-sensitive steps.

G R A P H I C A L A B S T R A C T

a r t i c l e i n f o

Article history:Received 13 July 2013Received in revised form3 October 2013Accepted 20 October 2013Available online 28 October 2013

Keywords:Catalyst deactivationKineticsMathematical modelingSupported multifunctional catalystsProbability modelsNanostructure effects

a b s t r a c t

This study brings into focus some nanostructure-size effects on the deactivation kinetics at reactions oversupported multifunctional catalysts. The active phase dispersion as nano-sized islands on the supportpredetermines diversities in the action of active sites depending on their location and structure. In viewof this, the applied approach assumes participation of active site types differing by coordination,configuration and contribution to various reaction routes. The suggested model concerns regularitiesassociated with the availability of active surface atoms in proper arrangements which facilitate structure-sensitive reactions. Problems linking the vulnerability of active sites with their geometry and structureare put to discussion. Furthermore, the model relates the probabilities for action of different site types tothe size of active-phase islands. The effect of site blockage on the probabilities for existence of multi-siteactive centers (catalytic clusters) facilitating structure-sensitive reactions is explored. The performedanalysis points out that, when matter concerns reactions facilitated by multi-centered active sites, twodistinct reasons can be specified, by virtue of which coke species may affect the activity of catalyticclusters: (i) canceling the action of partially or totally coke-covered cluster configurations and(ii) reducing the capability of the adjacent active atoms to construct multi-centered active configurations.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Detailed study of bi- and multifunctional supported catalystsgains in importance in view of their wide application in industrialprocesses. Most of these processes are accompanied by cokeformation. The presence of active sites of different propertiesand nature, facilitating networks of mutually dependent reaction

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n Correspondence to: Institute of Catalysis, Bulgarian Academy of Sciences,“Acad. G. Bonchev” str, block 11, Sofia1113, Bulgaria. Tel.: þ359 2 885 4298;fax: þ359 2 971 2967.

E-mail addresses: [email protected] (K. Kumbilieva),[email protected] (L. Petrov).

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routes, provokes essential challenges to forecasting the overalleffects. For that matter, it is of importance the development ofkinetic models reflecting how the contribution of distinct types ofsites to the catalytic action is influenced by the diversities in theiractivities. At the same time, when facing the problems concerningthe deactivation of multifunctional catalysts, it should be accoun-ted that active sites differing by nature, configuration, adsorptionability, contribution to different reaction routes (see, e.g. Yang andSatterfield, 1983; Topsøe et al., 1989, 1996; Ozkan et al., 1994;Furimsky and Massoth, 1999), may be vulnerable in different wayby the actual deactivation-responsible factors. To this end, thoroughexamination is required to gain better understanding on variousaspects of the phenomena. Adequate models considering the con-stituent effects are needed for finding resources to reduce the harmfulconsequences from the deterioration of the catalyst properties.Notwithstanding this necessity, such models are still in deficiency.

The well-known classification of Boudart (1969) specifies twomajor classes of heterogeneous catalytic processes, specificallystructure-insensitive and structure-sensitive reactions. The rate ofstructure-insensitive reactions depends on the surface concentrationof the active-metal atoms, and is not significantly influenced by thesizes of the moiety containing the active component. As shown in aseries of works (e.g., Biloen et al., 1977, 1979; Broekhoven et al.,1985; Cortright and Dumesic, 1994; Ribeiro et al., 1994), one or twoadjacent surface atoms can facilitate structure-insensitive reactions.Conversely, the rate of structure-sensitive reactions can be markedlychanged when changes of metal dispersion, crystal planes or defectstructures take place. Typical examples are hydrogenolysis, isomer-ization, polymerization, cracking, coke formation. These reactionsare facilitated by the so-called catalytic clusters (Sinfelt, 1973, 1977,Slinkin, 1981) – principally, ensembles involving several (M) atomsof the active metal in proper configuration for multiple-site adsorp-tion (Sinfelt, 1973, 1977; Anderson, 1973; Biloen et al., 1977, 1979;Slinkin, 1981; Laine, 1983; Broekhoven et al., 1985, Ho, 1988;Meitzner et al., 1988; Perot, 1991; Cortright and Dumesic, 1994;Ribeiro et al., 1994; Jablonski et al., 1999; Bendarova et al., 2002).

Bond (1985) advanced observation and theoretical arguments thatcertain peculiarities in the behavior of supported catalysts can beinterpreted as evidence for the availability of more than one type ofactive sites, each being characterized by individual configurations ofthe constituent surface atoms and specificity of the kinetic parameters.He points in his working hypothesis that catalysts may change fromfavoring structure-sensitive reactions (e.g. hydrogenolysis) to favoringstructure-insensitive reactions (e.g. dehydrogenation) in result ofalterations (including poisoning or coke formation) which reduce theaverage size of the entities of the active metal.

Furthermore, the dispersion of the active phase of catalystsas nano-structures on the support gives rise to diversities inthe properties of active sites depending on their geometry and location.

The differences in the coordination of sites arising from theirlocation (internal or interfacial) within the island structures, may callforth differences in catalytic properties and stability of action. Theactive sites located on the interface of the active phase with thesupport are coordinately unsaturated, and may be hence classified assimilar to the pattern of the crystalline edges and corners. Theanalysis performed in Yang and DePristo (1994) points out that thefree energy of the internal atoms depends on the size of the island,contrary to the energy values characterizing edge and corner sites.The internal and the interfacial sites may facilitate different reactionroutes. In general, the routes catalyzed by various types of sitesshould follow different kinetic regularities. The geometric factorsmay result in ligand effects. Problems associated with the ratio ofinternal and interfacial active sites gain in actuality in view of thegrowing interest for elucidating the action of supported Au catalysts(e.g., Haruta et al., 1993; Bond and Tompson, 1999; Cortie and Lingen,2002). Following the understanding that interfacial sites may exhibit

unique activity (Somorjai, 1992), various models have been advanced,aimed to explain the catalyst behavior with due regard for theproperties of internal and interfacial active sites.

Recently, Parmon (2007, 2010) adduced thermodynamic con-siderations that if the number of active centers contained in a givennanostructure is smaller than a certain critical value Ncr � 100, thiscan modify the adsorption and catalytic properties of the comprisedactive metal atoms. On applying the thermodynamic analysisderived (Parmon, 2010) and the Temkin theory (Temkin, 1979) forreaction kinetics exhibited on nonuniform catalyst surfaces, Murzinput to discussion the impact of size effects on the kinetics of severalcatalytic processes (Murzin, 2009, 2010a, 2010b). Qualitative ana-lysis and numerical calculations evidenced that the kinetic regula-rities can vary depending on the size of the nanostructures onwhich the reactions are proceeding.

Kinetic analysis of processes realized with participation of differenttypes of active sites, are still scarce. The considerations correspondingto the theory of real adsorbed layer (Roginsky, 1960; Kiperman, 1979;Rutkin and Petersen, 1979; Temkin, 1986; Boudart, 1986) requiresophisticated mathematical processing, especially in concern of med-ium surface coverage. Themodels frequently considered are developedwithin the frames of the conventional Langmuir–Hinshelwoodkinetics, by tacit consent ignoring the effects reasoned by surfacenon-uniformity. Yet, under conditions of medium coverage, effectsconditioned by non-uniformities of the catalyst surface may outgrowsome of the postulates for the ideal adsorbed layer. The uncertaintywhether it is convenient to apply the related approximations imposesan essential theoretical barrier for deriving kinetic models consideringthe participation of sites with different properties.

The problem gains in complicity when the effects of catalystdeactivation have to be considered. The reaction networks of manycomplex processes involve routes resulting in formation of speciesdeteriorating the catalyst activity. Coke formation may be con-sidered as the most common reason for catalyst deactivation. Thegeneration of coke precursors is a structure-sensitive reaction,requiring multi-centered catalytic clusters.

Deeper insight on these phenomena requires linking theproblems of nonstationary or quasi-stationary reaction proceedingand problems associated with diversities in the properties andvulnerability of the different types of active sites.

Murzin (2002) suggested an approach for the kinetic description ofprocesses occurring through a surface collision of species adsorbed ontwo distinct sites of different nature. The model proposed in Liberkovaet al. (2002) describes the activity and selectivity of a Pt/SnO2 catalystin terms of apolar and polarized types of sites, characterized bydifferent deactivation profiles.

Facing the necessity to account for effects called forth by non-uniformity of the catalyst surface, special models have to bederived for processes involving active sites of different action,properties, structure and vulnerability. Given the fact that cokeformation is a structure sensitive reaction, it is both of academicand practical interest the development of models focused on theparticipation of various active site types in processes accompaniedby catalyst deactivation.

2. Approach

In this study, we shall try to model the geometric factorsdeterminative for the availability of active sites facilitatingstructure-sensitive and structure-insensitive reactions, and forthe related coke-caused variations of the catalyst action. Accord-ingly, it is desirable to distinguish the contribution of active sitetypes characterized by different structure and properties.

In view of avoiding discrepancy with the Hinshelwood–Langmuirkinetics, we developed an original approach (Kumbilieva and Petrov,

Y. Alhamed et al. / Chemical Engineering Science 105 (2014) 77–9178

2011; Kumbilieva et al., 2011) assuming the following approximation:given that n types of active sites contribute to the catalytic action, onecan consider the catalyst surface containing n co-existing idealadsorbed layers, each of which is characterized by own intrinsicparameters. Then the function of the overall catalyst activity atmoment t can be defined as a vector sum of the “individual activity”aj functions related to the action of the particular types of activesites:

aðtÞ ¼ ∑n

j ¼ 1ajðtÞ ð1Þ

As in most multifunctional catalytic systems two or three typesof active sites are expected to realize noticeable contribution to thecatalytic performance, we limit the analysis to three types of sites.The use of this approach in our previous works (Kumbilieva et al.,2006, 2009) served adequately to describe and interpret theobserved peculiarities in the performance of particular processes:xylene combustion over Pd and isobutane dehydrogenation overPt catalysts, by considering the participation respectively of twoand of three types of active sites. More detailed information on thereaction peculiarities and the related model considerations can befound in the original studies. The relations following from thepostulated models proved to be in agreement with the regularitiesestablished from experimental data. This gives us reason to expectthat such an approach can turn fruitful for analyzing the peculia-rities of other catalytic systems overflowing the restrictions of theideal adsorbed layer.

As mentioned above, the active phase of most supportedmultifunctional catalysts is dispersed on the carrier in the formof nano-sized entities containing a finite number of surface atomsof the active metal. In the further rendering we shall designatethese entities as “active-phase islands”. We shall denote by N0 thenumber of surface atoms of the active metal comprised within anactive-phase island on the fresh catalyst. In the following, the0 superscripts will be used to designate that matter refers variablesconcerning the fresh catalyst. To account for the consequences ofcoke formation, we shall denote by Nc the number of active surfaceatoms covered by coke species at a given current stage of the process,and by N the number of the remaining uncovered activesurface atoms.

By the term “catalytic cluster” we shall designate an ensembleof several (viz. M in number) neighboring active surface atomsproperly located to facilitate a structure-sensitive reaction. Natu-rally, the specific value of M determining the number of atomsconstructing a given catalytic cluster depends on the concretereaction under concern, and on the crystallographic facet of thesupport. Furthermore, the active phase fragmentation in the formof islands on the support preconditions diversity in the coordina-tion, and accordingly the catalytic properties of the internal andinterfacial active sites. On this account, it appears reasonable todistinguish three types of active sites: interfacial active sites,internal single active sites, and complex active sites – catalyticclusters, constituted by M single sites in proper configuration.

We shall denote as Y-type the interfacial active sites locatedalong the island boundaries; as Z-type – the sites located insidethe active phase islands; and as X-type – ensembles involving adefinite number (M) neighboring surface atoms in proper configura-tion (catalytic clusters). Respectively, we can specify the variables NY ,NZ , and NX , each of which is relevant to the current number of sitesof the corresponding type available within an active-phase island(consistently, N0

Y , N0Z , and N0

X for the fresh catalyst).For reactions proceeding under kinetic control, we can specify

the activity function as

a¼ azþaY þax ð2Þ

Here az, aY and ax stand for the “individual activity” functionsdetermining the contribution to the reaction performance of thesites designated above as Z-type, Y-type, and X-type. Each of thesefunctions may be considered proportional to the current number(Nz , NY and Nx) of active sites of the corresponding type:

azpNz; aY pNY ; axpNx ð3Þ

When studying the catalyst deactivation, the initial conditionsfor the reaction system are associated with absence of coke on thefresh catalyst:

NY ¼N0Y ; NZ ¼N0

z ; NX ¼N0x at Nc ¼ 0 ð4Þ

When estimating the number NX of catalytic clusters within anactive phase island, in view of the finite number of surface metalatoms involved within a particular active phase nanostructure(roughly between 10 and 3000), it is reasonable to take intoconsideration the probability μM that (M) surface atoms are properlylocated to construct a catalytic cluster. The values of the probabilityμM may vary between 1 and 0: 0rμMr1. Therefore, the number NX

of catalytic clusters within an active phase island will depend on thenumber Nof available single sites, the number (M) of surface atomsconstructing the cluster, and the probability μM :

Nxp1MμMN ð5Þ

In respect of the fresh catalyst, we have as initial condition:

N0x p

1Mðμ0MÞN0 ð6Þ

On modeling the changes of the reaction system caused byblockage of active sites, we accept that the key deactivation-responsible factor matches with the number (Nc) of surface sitesblocked by coke or tightly adsorbed species. It is reasonable toassume that the number Nz of internal single active sites willdecrease simultaneously with increase of Nc .

�dNz ¼ kz dNc ð7Þ

As for the number NY of interfacial active sites, the trend oftheir decrease (see Eq. (8)) may be weaker (viz. kY okz) providingthat overlap of coke onto the support takes place. The functionf thðNcÞ is supposed to be stepwise and to take account of the factthat coke overlap on the support may result in certain coketolerance effect at the early stages of some processes (Gray et al.,1995; Richardson et al., 1996; Diez et al., 1999).

�dNY ¼ kY f th ðNcÞdNc ð8Þ

Under the assumption that a certain threshold amount of cokeaccumulated will guard against coke overlap, the function f thðNcÞis supposed to change stepwise. In the simplest case

f thðNcÞ ¼ 0 for NcoNthresholdc ð9aÞ

f thðNcÞ ¼ 1 for NcZNthresholdc ð9bÞ

In respect of the changes concerning the active catalytic clusters(active sites of X-type), we would like to bring the attention to theμM term. The essential point is that, as the number of blocked sitesincreases, the probability μM may as well change more or lessnoticeably in the course of deactivation. Considering (5), we come

Y. Alhamed et al. / Chemical Engineering Science 105 (2014) 77–91 79

to regularities of the form:

�dNx ¼ kx1M

μMdNdNc

þNdμMdNc

� �dNc ð10Þ

With regard to (3c), the relation (10) represents the coke-caused decrease of the catalyst activity in regard of structure-sensitive reactions, which are realized through the participation ofmulti-site intermediates. The probability μM for availability of suchintermediates can be assumed constant only in case when itsdecrement in the course of coke formation dμM=dNc

� �is small

enough to satisfy the condition

NdμMdNc

� �⪡ μM

dNdNc

� �:

Otherwise, the deactivation model has to consider that theprobability μM for availability of (M) single sites in proper proxi-mity and configuration may decrease in different modes, stipulat-ing various effects on the selectivity in regard of the structure-insensitive and structure-sensitive reactions.

It can be concluded, considering (3a)–(3c), (7), (8) and (10), thatthe deactivation regularities relevant to the structure-insensitiveand structure-sensitive reactions may be quite different.

This aspect of the phenomena is not sufficiently explored. Inthis connection, we shall extend the foregoing approach, bringingto focus the influence of coke formation on the probabilities foroccurence of configurations of active ensembles facilitating actualstructure-sensitive reactions.

3. Probability models

In view of the finite number of surface active atoms comprisedwithin a separate active-phase nanostructure, it is reasonable,when estimating the number NX of involved catalytic clusters, totake into account the probability μM for availability of M properlylocated active surface atoms. It is an essential point that in thecourse of coke build, the probability μM may decrease in differentmodes, stipulating various effects on the selectivity in regard ofthe structure-insensitive and structure-sensitive reactions.

When coke or poisoning species affect a surface atom consti-tuting a given catalytic cluster, the event puts out of action theentire active ensemble. The particular fact that change in the stateof only one surface atom can essentially influence the specificaction of a group of M integrated atoms imposes the necessityto examine the phenomena on atomic level. At the same time,the remaining unblocked atoms participating in the terminatedconfiguration partially keep their potential to contribute to thestructure-sensitive route of the process via participation in othercluster configurations.

The system may come to a point of terminating the activity forstructure-sensitive reactions if the number and density ofunblocked metal atoms contained in the active-phase island dropsto a critical level at which the probability thatM of them may be inproper proximity and configuration to form the catalytic clustersapproaches zero.

Evidently, the regularities concerning the change of activitytowards structure-sensitive reactions are complex, and need care-ful examination of the mechanism specificities and the relatedprobability factors. Accordingly, it appears expedient to link thealteration of catalyst activity for structure-sensitive reactions withthe coking-caused alterations of the probabilities for existence ofcatalytic clusters configured by M atoms. In view of this, we shallapply some of the principles and considerations of the probability

calculus in an attempt to gain understanding on these regularitiesby modeling the processes on atomic level.

If we define the function characterizing the current activity asthe ratio of the current reaction rate to the rate on the freshcatalyst, the initial conditions can be set as

aZ ¼ 1 NZ ¼N0Z for Nc ¼ 0

aY ¼ 1 NY ¼N0Y for Nc ¼ 0

aX ¼ 1 NX ¼N0X for Nc ¼ 0

When modeling the coke-caused decrease of the activityfunctions, we shall consider that coke build up occurs via succes-sive appending of surface species tightly adsorbed on multi-centered X-type sites (catalytic clusters). It can therefore beassumed stepwise character of the growth of coke formations.When matter concerns a nanosized active-phase island, containinga finite number of active sites, the stepwise type of the depen-dences can hardly be neglected. Accordingly, it is more correct torepresent these changes in terms of finite differences:

�ΔNz ¼ kzðΔNcÞ with initial condition : NZ ¼N0Z for Nc ¼ 0

ð11Þ

�ΔNY ¼ kY f th ðNcÞðΔNcÞ with initial condition : NY ¼N0Y for Nc ¼ 0

ð12Þ

�ΔNx ¼ kx1M

μMΔNΔNc

þNΔμMΔNc

� �ΔNc ð13Þ

with initial condition: N¼N0; NX ¼N0X ; μM ¼ 1 for Nc ¼ 0

In concern of the finite value of the increment ðΔNcÞ, it is areasonable approximation to assume ðΔNcÞ ¼M, under the sugges-tion that each step of coke growth portion is realized throughincorporating a surface form bound to an M-centered ensemble.

N, ΔN, Nz and ΔNz are most frequently linearly related to Nc.The concern is to determine the changes of the probabilityfunctions ðμMÞ in the course of coke build. These functions mayevolve in different way for types of sites requiring different surfaceensembles. Accordingly, the reaction routes may be affected indifferent manner.

On deriving our set of models, we shall give special considera-tion to the following parameters characterizing a given catalyticsystem:

� The number (M) of surface atoms of the active metal construct-ing a catalytic cluster capable to facilitate a given structure-sensitive reaction. As particular examples, the cases of M¼3, 4,and 6 will be considered.

� The number of single active atoms comprised within an active-phase island;

� The crystallographic facet of the catalyst surface. More speci-fically, we shall inspect {1 1 1} and {1 0 0} facets, as far as theseare the facets typically occurring on the surfaces of supportedcatalysts.

3.1. Study cases at {1 1 1} facet

Because of the hexagonal lattice arrangement, each surface activeatom (A) is surrounded by six nearest neighbors, which are capableto participate in the configuration of a catalytic cluster in partnershipwith A. Let us consider an arbitrary site A and its neighbors denotedas “a1”, “a2”, “a3”, “a4”, “a5”, and “a6” (see Fig. 1). A three-centeredcluster (M¼3) can be constituted via one-of-six possible configura-tions, which involve the triangles A a1 a2, A a2 a3, A a3 a4, A a4 a5,A a5 a6, and A a1 a6. Let us denote by uA12, uA23, uA34, uA45, uA56, uA61


the probabilities for realization of these variants. It is reasonable toassume as an arbitrary unit the probability U0

A that one of thesevariants can be realized on the fresh catalyst.

U0A ¼ ðu0

A12þu0A23þu0

A34þu0A45þu0

A56þu0A61Þ ¼ 1 ð14Þ

In terms of probability calculus, the possible configurationswith the participation of A can be classified as equiprobable andmutually exclusive events, as far as A can be engaged in no morebut one of them. In accordance to the calculus laws of chances, wecan ascribe to each of the six u0

Aij probabilities the value of (1/6).Regarding the overall surface of an active-phase island on the

fresh catalyst, we can write

U0overall ¼

1

N0 ∑N0

1U0

j ¼ 1 ð15Þ

In case the catalyst surface is influenced by coke formation, theprobability for realization of cluster configurations can be equatedto the average of the variables Uj, characterizing the capability ofthe distinct atoms to be constituents of catalytic clusters.

Uoverall ¼1

N0 ∑N

1Uj ð16Þ

where N stands for the current number of single sites, which arenot blocked; hence

N¼N0�Nc ð17Þ

If we denote by s the fraction of unblocked sites, it can bedefined as

s¼ N

N0 ¼N0�Nc

N0 ð18Þ

The probability UA a given atom A to be constituent of acatalytic cluster represents the conditional probability the properconfiguration to be realized with some of the neighboring pairs,under the event that A itself is not covered by coke. Assuming theprobability A not to be covered by coke equal to the fraction ofunblocked sites s, from probability calculus laws we obtain:

UA ¼ sðuA12þuA23þuA34þuA45þuA56þuA61Þ ð19Þ

Each of the variables uA12, uA23, uA34, uA45, uA56, uA61 comes toreflect the conditional probability that both atoms participating in

the concerned pair are not covered by coke. According to theprobability conjunction law, these variables can be specified asfollows:

uA12 ¼ u0A12γ1γ2 ð20aÞ

uA23 ¼ u0A23γ2γ3 ð20bÞ

uA34 ¼ u0A34γ3γ4 ð20cÞ

uA45 ¼ u0A45γ4γ5 ð20dÞ

uA56 ¼ u0A56γ5γ6 ð20eÞ

uA61 ¼ u0A61γ6γ1 ð20fÞ

The values of the γk functions change step-wise as follows:

γk ¼ 1 if the kth site is not blocked ð21aÞ

γk ¼ 0 if the kth site is blocked ð21bÞ

On substituting (20a)–(20f) in Eq. (19), we obtain

UcA ¼ sðu0

A12γ1γ2þu0A23γ2γ3þu0


A56γ5γ6þu0A61γ6γ1Þ

ð22Þ

For the fresh catalyst, s¼ 1, and in agreement with (21a), allthe stepwise functions are considered equal to unity: γ0k ¼ 1; thusEq. (22) becomes identical with (14).

When modeling the coke formation process, it is worth noti-cing the following points. As a coke precursor is supposed to arisevia polymerization of species tightly adsorbed on catalytic clusters,it is reasonable to assume that an arising coke precursor wouldcover at least two adjacent ensembles involving 2M single sites (or6 sites for the particular case M¼3). Thus, in terms of the types ofactive sites specified above, we accept that an emerging cokeprecursor initially blocks two X-type sites (or, equally, (2M) singleZ-type sites); its further build up is realized in a stepwise mode viaincorporating at each step new M-centered catalytic clusters.On this account, the analysis considers cases of coke precursors/molecules involving 2, 3, 4, 5, 6, 7, … catalytic clusters, each ofwhich consists of M single sites. It is reasonable to suppose thatthe catalytic system will be sensitive to the stepwise mode of cokebuild up, particularly for the cases when the total number of activesites comprised in the active-phase island is of the range of severalhundred or less.

Suppose that a given atom (A) is not blocked, but an adjacentcoke molecule covers one of its nearest neighbors (e.g. a1). As faras each of the 6 neighbors of A is involved in two of the sixpossible for A cluster configurations, it follows that 2 of the6 possible configurations for A to form a catalytic cluster have tobe ruled out. Thus the capability of A to form cluster configurationsis affected. As 2-of-6 chances are excluded, the probability attrib-uted to this site to form catalytic clusters is reduced by 2/6. In theparticular pattern, the condition (21b) implies γ1 ¼ 0 whileγ2 ¼ γ3 ¼ γ4 ¼ γ5 ¼ γ6 ¼ 1, and it follows from (20a), (20f) and (18)the relations:

uA12 ¼ 0; uA61 ¼ 0;

UAð1Þ ¼ sðu0A23γ2γ3þu0


A56γ5γ6Þ ¼ s 46

� �¼ sF1ð23Þ

In case coke species cover a pair of the neighbors of A (e.g., a1 anda2, see Fig. 2b), in obedience to (21b) we have to account γ1 ¼ 0 andγ2 ¼ 0, while γ3 ¼ γ4 ¼ γ5 ¼ γ6 ¼ 1. Hence uA12 ¼ 0;uA23 ¼ 0;uA61 ¼ 0:

Fig. 1. Study case: hexagonal lattice arrangement {1 1 1}. Provisional pattern of anarbitrary active surface atom (A) and its nearest neighbors capable to configure acatalytic cluster in partnership with A.


Consequently, three of the possible six configurations have tobe excluded, and we obtain (24) for the current chance that A willparticipate in a catalytic cluster:


A45γ4γ5þu0A56γ5γ6Þ ¼ s 3

6

� �¼ sF2 ð24Þ

For the case of 3 coked neighbors of A (e.g., a1, a2, and a3, seeFig. 2c), we get accordingly:

γ1 ¼ 0; γ2 ¼ 0; γ3 ¼ 0 while γ4 ¼ γ5 ¼ γ6 ¼ 1;

uA12 ¼ 0; uA23 ¼ 0; uA34 ¼ 0; uA61 ¼ 0;


A56γ5γ6Þ ¼ sð26 Þ ¼ sF3 ð25Þ

For the case of 4 coked neighbors of A (e.g., a1, a2, a3, and a4,see Fig. 2d), similar analysis brings to the following relations:

γ1 ¼ 0; γ2 ¼ 0; γ3 ¼ 0; γ4 ¼ 0 while γ5 ¼ γ6 ¼ 1;

uA12 ¼ 0; uA23 ¼ 0; uA34 ¼ 0; uA45 ¼ 0; uA61 ¼ 0;

UAð4Þ ¼ sðu0A56γ5γ6Þ ¼ sð16 Þ ¼ sF4 ð26Þ

The practical chances a given atom to remain non-coveredbeing surrounded by 5 coke-covered neighbors (e.g., a1, a2, a3, a4,and a5) are negligible, but for the sake of model completeness wetake into account the theoretical chance, considering F5 ¼ 0, whichfollows from the relations:

γ1 ¼ γ2 ¼ γ3 ¼ γ4 ¼ γ5 ¼ 0;

uA12 ¼ uA23 ¼ uA34 ¼ uA45 ¼ uA56 ¼ uA61 ¼ 0; UAð5Þ ¼ 0 ð27Þ

It follows from the stated considerations that one can distin-guish two distinct reasons by virtue of which coke species mayreduce the activities of the catalytic clusters: (i) on account ofcoke-covered parts of the cluster configurations; (ii) in conse-quence of reducing the chances of the adjacent active atoms toconstruct cluster configurations. It becomes evident on comparing(23)–(26) that the capability of each active atom to facilitatestructure-sensitive interactions is related to the number of blockedneighboring surface atoms. This provides reasons to categorize theunblocked single active sites according to the number of theirnearest neighbors blocked by coke or poison species. Correspond-ingly, we shall specify as distinct groups the sites with one, two,three, four, five, and none blocked neighbors. Each group can becharacterized by the value of the probability factor Fk accountingthe chances for contribution to structure-sensitive reactions of theactive surface atoms with (k) blocked neighbors. Specifically, forthe case of three-centered clusters (M¼3) and {1 1 1} hexagonallattice, as evident from (23)–(26), the Fk probability factors takethe following values:

F0 ¼ 1; F1 ¼23

� �; F2 ¼

12

� �; F3 ¼

13

� �; F4 ¼

16

� �; F5 ¼ 0 ð28Þ

Let L1, L2, L3, L4, L5, and Lf denote respectively the number ofsingle sites with one, two, three, four, five, and none blockedneighbors. Lf stands for the number of single active sites, whichare not adjacent to coke species; while their capability to partici-pate in catalytic clusters is not affected by proximity of blocking

Fig. 2. Study case: 3-centered active sites over {1 1 1} lattice. Patterns of coke species (black and gray circles) which do not cover a considered surface active atom (A), butcover one (a), two (b), three (c), four (d) of its nearest neighbors (gray circles).


agents, they are characterized by a probability factor F0 ¼ 6=6¼ 1.The values of these numbers vary in the course of coke build up, anddepend on the size and geometry of the coke molecules, and on thelattice specificity. For example a coke molecule covering two three-centered catalytic clusters, is surrounded by 12 adjacent uncoveredactive sites (see Fig. 3a): 6 of them (denoted as B1, B2, B3, B4, B5, andB6) can be classified as active sites with one coke-covered neighbor,characterized by probability factor F1 ¼ 2=3 the other six (denoted asD1, D2, D3, D4, D5, and D6) – as active sites with two coke-covered

neighbors, characterized by probability factor F2 ¼ 1=2. Dependingon the mode of further build up, a coke molecule covering threethree-centered catalytic clusters, may present several different geo-metric forms, which are characterized by different number of thesurrounding uncovered atoms, and of the atoms contiguous to one,two, or three coke-covered sites (see Fig. 3b–d).

Similar line of arguments can be used to model the cases M¼4and 6, providing some differences associated with the arrange-ment peculiarities.

Fig. 3. Study case: 4-centered active sites over {1 1 1} lattice. Patterns of coke species covering two (a) or three (b)–(d) catalytic clusters. Depending on the geometric formof the coke formation, the surrounding uncovered atoms may be contiguous to one (B1–B8), two (D1–D9), or three (T1, T2) coke-covered sites.


A four-centered cluster (M¼4) can be constituted via one-of-12possible configurations, six of which involve tetragons with firstnearest neighbors (Aa1a2a3, Aa2a3a4, Aa3a4a5, Aa4a5a6, Aa5a6a1,and Aa6a1a2), and the other six consisting of tetragons involv-ing second neighbors (Aa1d7a2, Aa2d8a3, Aa3d9a4, Aa4d10a5,Aa5d11a6, and A a6 d12 a1, see Fig. 4). Due to the participation ofsecond neighbors in the potentially possible configurations for 4-centered catalytic clusters, the capability of (A) to participate in suchclusters may be affected if some of the second neighbors get blocked.The thread of reasoning brings to the following estimations. Given theatom (A) is not blocked, but an adjacent coke molecule covers one ofits nearest neighbors (e.g. a1), either five or six of the 12 possibleconfigurations have to be excluded, depending on whether the cokespecies cover at the same time one or two of the second neighbors(e.g., d7, d12, see Fig. 5a and b). Thus the probability factor F1attributed to this site to form catalytic clusters may get valuesð7=12Þ or ð6=12Þ. In case coke species cover a pair of the nearestneighbors of A (e.g., a1 and a2), we have to account that 7 or 8 of the12 possible configurations have to be excluded, depending onwhetherthe coke species cover at the same time one or two of the secondneighbors (e.g., d7, d8, see Fig. 5c and d). Accordingly, the probabilityfactor F2 attributed to this site to form catalytic clusters would equalð5=12Þ or ð4=12Þ. In case coke species cover three of the nearestneighbors of A (at that unavoidably together with two or three secondneighbors) (e.g., a1, a2, a3, d7, d8, d12, see Fig. 5e), 8 or 9 of thepossible six configurations have to be disaffirmed. When coke speciescover four of the nearest neighbors, (together with three or four of thesecond neighbors) of A (e.g., a1, a2, a3, a4, d7, d8, d9, d12 see Fig. 5f),the chances that this atom will participate in a catalytic cluster arereduced to 1/12. It can be summarized in concern of four-centeredcatalytic clusters, that proximity of coke species brings to the followingvalues of the probability factors characterizing the capability of activesites to form the clusters:

F0 ¼ 1; F1 ¼1324

� �; F2 ¼

924

� �¼ 3

8

� �; F3 ¼

724

� �; F4 ¼

112

� �; F5 ¼ 0

ð29ÞWhen matter concerns six-centered clusters (M¼6) needed to

facilitate structure-sensitive interaction, the values ascribed to theprobability factors characterizing the capability of active sites toform the clusters are as follows:

F0 ¼ 1; F1 ¼47

� �; F2 ¼

27

� �; F3 ¼

17

� �; F4 ¼ F5 ¼ 0 ð30Þ

It should be not skipped from consideration the particular case ofcatalytic clusters involving interfacial active atoms. Accounting thegeometric considerations, the probability an interfacial edge site toparticipate in cluster configuration can be equated to the probabilityfactor characterizing internal active atoms with two blocked nearestneighbors. The probability an interfacial corner site to participate incluster configuration can be equated to the probability factor char-acterizing internal active atoms with three blocked nearest neighbors.

Altogether, we can estimate in the abovementioned terms theaverage of the current probabilities, characterizing the capabilityof the distinct atoms to be constituents of catalytic clusters underconditions of coke formation as follows:

Uaver ¼1

N0s Lf F0þL1F1þðL2þNedgeÞF2�

þðL3þNcornerÞF3þL4F4þL5F5¼ 1

N0sψ conv ð31Þ

The value of the Uaver function changes with the increase of Nc andwith the linked change of the Lk values in the course of coke build up.

The term ψ conv ¼ fLf F0þL1F1þðL2þNedgeÞF2þðL3þNcornerÞF3þL4F4þL5F5g accounts for the conditional probability the properconfiguration to be realized with some of the neighboring sites.It should be noted that this term is different for the differentvariants of cluster configurations. The related values of the Fkprobability factors are specific for the different numbers of surfaceatoms constituting a catalytic cluster. Hence, the particular prob-abilities μM for availability of catalyst clusters configured by Matoms may follow different dependences for different M values. Inrespect of the cases considered for {1 1 1} lattice, with due regardfor (28)–(31), we obtain:

3.1.1. Study case M¼3

μ3 ¼1

N0s Lf þð23 ÞL1þð12 ÞðL2þNedgeÞþð13 ÞðL3þNcornerÞþð16 ÞL4� ð32Þ

Accordingly, taking into account the relations (6), (17), (18) and(32), we can estimate the expected current number of three-centered catalytic clusters available on the active-phase islandcomprising ðN0

ZÞ single sites, Nc of which covered by coke species:

NX ¼ ð13 Þ1

N0s Lf þð23 ÞL1þð12 ÞðL2þNedgeÞþð13 ÞðL3þNcornerÞþð16 ÞL4� ðN0�NcÞ ð33Þ

The values of L1,L2,L3,L4 and Lf depend on the concrete values ofN0 and the current Nc value.


μ4 ¼1

N0s Lf þð1324 ÞL1þð38 ÞðL2þNedgeÞþð 724 ÞðL3þNcornerÞþð 112 ÞL4� ð34Þ

The expected current number of catalytic clusters available onan active-phase island can be estimated after the relation:

NX ¼ ð14 Þ1

N0s Lf þð1324 ÞL1þð38 ÞðL2þNedgeÞ�

þð 724 ÞðL3þNcornerÞþð 112 ÞL4ðN0�NcÞ ð35Þ


μ6 ¼1

N0s Lf þð47 ÞL1þð27 ÞðL2þNedgeÞþð17 ÞðL3þNcornerÞ� ð36Þ

The expected current number of catalytic clusters available onan active-phase island can be estimated after the relation:

NX ¼ ð17 Þ1

N0s Lf þð47 ÞL1þð27 ÞðL2þNedgeÞþð17 ÞðL3þNcornerÞ� ðN0�NcÞ

ð37ÞFig. 4. Study case: 4-centered active sites over {1 1 1} lattice. Provisional pattern ofan arbitrary active surface atom (A) and its nearest first (a1–a6) and second(d7–d12) neighbors capable to configure a catalytic cluster in partnership with A.


The first multiplier in Eq. (37) is set to 1=7Þ�instead of ð1=6Þ

because, when realizing six-centered adsorption of a molecule onhexagonal lattice, actually seven single sites come to be comprisedwithin the surface formation.

3.2. Study cases at {1 0 0} planar facet

The essential discrepancy with the hexagonal lattice models isassociated with the specific arrangement of the tetragonal lattice. Each

Fig. 5. Study case: 4-centered active sites over {1 1 1} lattice. Patterns of coke species (black and gray circles) which do not cover a considered surface active atom (A), butcover some of its nearest first and second neighbors (gray circles).


surface active atom (A) is surrounded by four first nearest neighbors,and four second nearest neighbors which are capable to participate inthe configuration of a catalytic cluster in partnership with A. Let usconsider an arbitrary site A and its first nearest neighbors (denoted as“a1”, “a2”, “a3”, “a4”) and second nearest neighbors (denoted as “d5”,“d6”, “d7”, and “d8”) (see Fig. 6). A three-centered cluster (M¼3) canbe constituted via one-of-eight possible configurations, which involvethe triangles Aa1d5, Ad5a2, Aa2d6, Ad6a3, Aa3d7, Ad7a4, Aa4d8, andAd8a1. Let us denote by uA15, uA52, uA26, uA63, uA37, uA74, uA48, uA23 anduA81 the probabilities for realization of these variants, and assume asan arbitrary unit the probability U0

A that one of these variants can berealized on the fresh catalyst:

U0A ¼ ðu0

A15þu0A52þu0

A25þu0A63þu0

A37þu0A74þu0

A48þu0A81Þ ¼ 1 ð38Þ

Likewise the case of hexagonal lattice, the possible configura-tions with the participation of A can be classified as equiprobableand mutually exclusive events, as far as A can be engaged in nomore than one of them. Therefore, we can ascribe the value ofð1=8Þ to each of the eight u0

Ajk probability functions.The subsequent analysis follows the chain of argument stated

for the case of hexagonal lattice, and for the sake of shortness weshall not repeat it in detail. The specificity to be noted concerns thephysical meaning of the parameters L1, L2, L3, L4 and L5. L1 standsfor the number of active single sites which flank the corners ofcoke species out of perpendicular (or second nearest neighbors ofcoke; see, e.g., the sites marked as s1 in Fig. 7). L2 stands for thenumber of active single sites which have one coke-covered firstnearest neighbor and one second nearest neighbor (see, e.g., thesites marked as s2 in Fig. 7). L3 stands for the number of activesingle sites which have two second and one first nearest neighborscovered by coke (see, e.g., the sites marked as s3 in Fig. 7). L4 standsfor the number of active single sites which have two second andtwo first nearest neighbors covered by coke (see, e.g., the sitesmarked as s4 in Fig. 7). L5 stands for the number of active singlesites, which have three second and two first nearest neighborscovered by coke (see, e.g., the sites marked as s5 in Fig. 7).

The resultant reckoning gives the following dependences.

3.2.1. Study case M¼3Probability μ3 for occurrence of three-centered catalytic clus-

ters within an active-phase moiety comprising N0 single sites,

Nc of which covered by coke species:

μ3 ¼1

N0s Lf þð34 ÞL1þð58 ÞL2þð12 ÞðL3þNedgeÞþð38 ÞL4þð14 ÞL5� ð39Þ

Accordingly, the expected current number of three-centeredcatalytic clusters available on an active-phase island can beestimated after the relation:

NX ¼ ð13 Þ1

N0s Lf þð34 ÞL1þð58 ÞL2þð12 ÞðL3þNedgeÞ�

þð38 ÞðL4þNcornerÞþð14 ÞL5ðN0�NcÞ ð40Þ

The values of Lf ,L1, L2, L3, L4 and L5 depend on the concretevalue of N0 and Nc .

3.2.2. Study case M¼4Probability μ4 for occurrence of four-centered catalytic clusters

within an active-phase moiety comprising N0 single sites, Nc ofwhich covered by coke species:

μ4 ¼1

N0s Lf þð34 ÞL1þð12 ÞðL2þL3þNedgeÞþð14 ÞðL4þL5þNcornerÞ�

ð41ÞThe expected current number NX of catalytic four-centered

clusters available on an active-phase island can be estimatedaccordingly:

NX ¼ ð14 Þ1

N0s Lf þð34 ÞL1þð12 ÞðL2þL3þNedgeÞþð14 ÞL4þL5Þ�

N0�Nc

�

ð42Þ

3.2.3. Study case M¼6Probability μ6 for occurrence of six-centered catalytic clusters

within an active-phase moiety comprising N0 single sites, Nc ofwhich covered by coke species:

μ6 ¼1

N0s Lf þð34 ÞL1þð12 ÞðL2þL3þNedgeÞþð14 ÞðL4þL5Þ� ð43Þ

The expected current number NX of six-centered catalyticclusters available on an active-phase island can be estimatedaccordingly:

NX ¼ ð16 Þ1

N0s Lf þð34 ÞL1þð12 ÞðL2þL3þNedgeÞþð14 ÞL4þL5Þ� ðN0�NcÞ

ð44Þ

4. Results and discussion

For the case of hexagonal lattice {1 1 1}, the active-phaseislands were modeled as flat entities of hexagonal form. The island

Fig. 6. Study case: tetragonal lattice arrangement {1 0 0}. Provisional pattern of anarbitrary active surface atom (A) and its first (a1–a4) and second (d5–d8) nearestneighbors capable to configure a catalytic cluster in partnership with A.

Fig. 7. Pattern of coke species over {1 0 0} lattice. Depending on the geometricform of the coke species, the surrounding uncovered atoms (designated as s1, s2, s3,s4 and s5) may be contiguous to different number of coke-covered sites (blackcircles).


can be imagined as embraced by six edges, each containing (b)surface atoms of the active metal. Accordingly, it can be estimatedthe number of interfacial single sites N0

Y ¼ 6b; 6 of which specifiedas corner sites, and ðN0

Y �6Þ – as edge sites.

N0Y ¼ 6b¼NcornerþNedge ð45aÞ

Ncorner ¼ 6; Nedge ¼ 6b�6 ð45bÞ

As mentioned above, the initial number N0Z of internal single

active sites is in linear dependence on the area of the island(Sisland). Expressed in terms of b, we obtain for the case ofhexagonal lattice:

N0z ¼ βconvSisland�N0

Y ¼ ð1:5ffiffiffi3

pÞβconvb2�6b ð46Þ

The coefficient βconv accounts for the influence of the convexityof the active-phase particles on the actual number of surfaceatoms comprised within the island.

Simulation were performed on the grounds of these assump-tions for the cases of catalytic clusters constituted by three (M¼3),four (M¼4), and six (M¼6) contiguous surface atoms, for values ofb equal to 6; 10; 15; 20; and βconv ¼ 1. Correspondingly, consider-ing (45) and (46), the initial number of interfacial and internalsingle sites on the fresh active-phase island can be determined asfollows:

N0z ¼ 57; N0

Y ¼Nedge ¼ 30; Ncorner ¼ 6 at b¼ 6 ð47aÞ

N0z ¼ 200; N0

Y ¼Nedge ¼ 54; Ncorner ¼ 6 at b¼ 10 ð47bÞ

N0z ¼ 494; N0

Y ¼Nedge ¼ 84; Ncorner ¼ 6 at b¼ 15 ð47cÞ

N0z ¼ 919; N0

Y ¼Nedge ¼ 114; Ncorner ¼ 6 at b¼ 20 ð47dÞ

The initial potential number of catalytic clusters (X-type sites)relevant to the fresh catalyst is assumed to be N0

x ¼ ððN0z

þðNedgeÞÞ=MÞμ0M , and to drop down in the course of coke forma-tion, together with the decrease of the number of the unblockedsingle active sites and the probability function μM , in accordancewith the dependences (32)–(37).

The plots in Fig. 8 show patterns of the simulated decreaseof the μM function assessing the current probability for occurrenceof three-centered catalytic clusters with respect to the consideredcases. The calculated variations of the analogous functions corre-sponding to four-centered and six-centered catalytic clustersfollow similar trends.

As can be seen from these plots, the probability for occurrenceof catalytic clusters sharply decreases in the very beginning ofcoking, and noticeably differs from a proportional-to-Nc pattern.The deviation is due to considering the effect of reducing thechances of unblocked, but adjacent to coke active centers toparticipate in cluster configurations. Therefore, these effects arefar not negligible, and their feasible consequences have to be takeninto account in predictive modeling of structure-sensitive reac-tions accompanied by coke formation.

According to our simulation results, in the case of involving 100or less single active sites involved, the drop of the probabilityfunction is drastic, and practically does not slow down. In the casesof active-phase entities containing more than 500 surface activeatoms, the initial drop of the probability function is smootherand flattens on attaining a certain level of poisoned sites. It isnot surprising that the effects become more pronounced for thecase of active-phase nanostructures involving small number ofsingle active sites, and soften on increasing the size of active-phaseislands.

For closer look on the point, we shall bring the attention toFig. 3b–e. For example, a coke precursor lumping together 3 surfaceintermediates tightly bound to three-centered active sites (atM¼3),will put out of action 9 single active sites by reason of direct coking;and in addition appreciably reduce the potential of 15 adjacent sitesto facilitate structure-sensitive interactions. Accordingly, the initialappearance of as though small coke precursor will result in far moredramatic drop of the probability for existence of multi-centeredactive sites within active-phase islands comprising 100 single sites,comparably to entities involving 500 or more active surface atoms(as evident on comparing the curves in Fig. 8).

The important consequence of these probability-for-cluster-configuration constraints is that, depending on the active-phasedispersion, the probability factor may appreciably influence thealterations of the catalyst activity towards structure-sensitive reac-tions. On substituting in Eq. (13) the results obtained for the changesof the probability factor ðΔμMÞ in the course of coke formation, wecan estimate the coke-caused variation of the functions characteriz-ing the catalyst activity for structure-sensitive reactions. The plot inFig. 9 shows the prognosticated variation of the activity forstructure-sensitive reactions facilitated by multi-centered ensembleson {1 1 1} facets, simulated for cases of nanostructures involvingdifferent initial number of single sites. Not surprisingly, these curvesresemble by shape those relevant to the ðΔμ3Þ vs. Nc relations.

For the case of tetragonal lattice {1 0 0}, the active-phaseislands can be modeled as flat entities of quadratic form. Then,for (b) surface atoms of the active metal contained in each of thefour edges, the number of interfacial single sites is N0

Y ¼ 4b.Accordingly, ðN0

Y �4Þ of these are specified as edge sites, and4 – as corner sites.

N0Y ¼ 4b¼NcornerþNedge ð48aÞ

Ncorner ¼ 4; Nedge ¼ 4b�4 ð48bÞ

In point of the initial number N0Z of internal single active sites,

for the case of tetragonal lattice:

N0Z ¼ βconvb

2�4b ð49Þ

Simulation calculations were performed on the grounds of theseassumptions for the cases of catalytic clusters constituted by three(M¼3), four (M¼4), and six (M¼6) contiguous surface atoms, forvalues of b equal to 10; 16; 24; 32; βconv ¼ 1. Correspondingly,considering (48) and (49), the initial number of interfacial and

Fig. 8. Decrease of probabilities for performance of structure-sensitive reactionsfacilitated by 3-centered catalytic clusters depending on the coking of singleactive sites for the case of {1 1 1} lattice and different initial number N0 of thesingle active sites comprised within the active-phase island: curve 1 – N0¼93,curve 2 - N0¼260, curve 3 – N0¼585, and curve 4 – N0¼1040.


internal single sites on the fresh active-phase island can be deter-mined as follows:

N0Z ¼ 60; Nedge ¼ 36; Ncorner ¼ 4 at b¼ 10 ð50aÞ

N0Z ¼ 192; Nedge ¼ 60; Ncorner ¼ 4 at b¼ 16 ð50bÞ

N0Z ¼ 480; Nedge ¼ 92; Ncorner ¼ 4 at b¼ 24 ð50cÞ

N0Z ¼ 896; Nedge ¼ 124; Ncorner ¼ 4 at b¼ 32 ð50dÞ

The initial potential number of catalytic clusters (X-type sites)relevant to the fresh catalyst is assumed to be N0

X ¼ ððN0Zþ

NedgeÞ=MÞμ0M , and to drop down in the course of coke formation,together with the decrease of the number of the unblocked singleactive sites and the probability function μM , in accordance with thedependences (32)–(37).

The plots in Fig. 10 show patterns of the simulated decrease ofthe μM function assessing the current probability for occurrence ofthree-centered catalytic clusters for the considered cases. Thepatterns relevant to the probabilities for arising of four-centeredand six-centered catalytic clusters follow similar trends.

The plots in Fig. 11 show the prognosticated variation of theactivity for structure-sensitive reactions facilitated by three-centeredensembles on {1 0 0} facets, simulated for cases of nanostructuresinvolving different initial number of single sites.

The estimated variations of the analogous functions correspond-ing to structure-sensitive reactions facilitated by four-centered orsix-centered catalytic clusters follow similar trends.

Conclusions based on any other provisional shape or values ofβconv will differ by specific coefficients, depending on the assumedshape, but will not change the logics of the analysis.

Strictly speaking, the cases described above are relevant tohomogeneous structure of the considered facets. At the same time,the nanostructures of practical importance contain various crystal-lographic defects, such as kinks, edges, corners, lattice damages, etc.Each particular defect conditions changes in the number of thesurrounding active neighbors. The corresponding effect can bemodeled through variation of the values of the Lk parameters. Forexample, if a given defect eliminates one, two or three active surfaceatoms, the probability factors characterizing the neighboring sites areto be assigned respectively the values of the probability factors

characterizing internal active atoms with one, two, or three blockedneighbors.

The above-stated results are indicative that the deteriorationfactor may affect in different aspects and extent the various types ofactive sites in relation with their geometric configuration. A con-siderable number of catalytic systems involve structure-insensitivesteps, which are accompanied by competing or harmful structure-sensitive reactions. In concern of processes such as hydrodesulphur-ization (HDS), hydrodenitrogenation (HDN) and others it is wellrecognized (Laine, 1983; Yang and Satterfield, 1983; Ho, 1988;Topsøe et al., 1989; Perot, 1991, 1996; Ozkan et al., 1994; Furimskyand Massoth, 1999) that diverse pathways of the reaction networkare possible, depending on whether adsorption through the loneelectron pair of the heteroatom, or via the aromatic π bond will takeplace. As far as the latter mode of adsorption demands a catalyticcluster involving at least six surface atoms in proper configuration, itbecomes clear the substantial role of the geometric factor for the

Fig. 9. Variation of catalyst activity for structure-sensitive reactions facilitated by3-centered catalytic clusters depending on the coking of single active sites for thecase of {1 1 1} lattice and different initial number N0 of the single active sitescomprised within the active-phase island: curve 1 – N0¼93, curve 2 – N0¼260,curve 3 – N0¼585, and curve 4 – N0¼1040.

Fig. 10. Decrease of probabilities for performance of structure-sensitive reactionsfacilitated by 3-centered catalytic clusters depending on the coking of single activesites for the case of {1 0 0} lattice and different initial number N0 of the singleactive sites comprised within the active-phase island: curve 1 – N0¼100, curve2 – N0¼256, curve 3 – N0¼576, and curve 4 – N0¼1024.

Fig. 11. Variation of catalyst activity for structure-sensitive reactions facilitated by3-centered catalytic clusters depending on the coking of single active sites for thecase of {1 0 0} lattice and different initial number N0 of the single active sitescomprised within the active-phase island: curve 1 – N0¼100, curve 2 – N0¼256,curve 3 – N0¼576, and curve 4 – N0¼1024.


availability of these nanostructures in regard to the selectivitybetween competing structure-sensitive and structure-insensitivereactions. Furthermore, the different types of sites may differ invulnerability to various reasons of deactivation. As widely known,carbon deposits are supposed to affect in different way the activityof single sites favouring structure-insensitive dehydrogenation andthe activity of larger catalytic clusters facilitating cracking andisomerisation (see, e.g., Hughes, 1984; Hegedus and McCabe, 1984;Oudar and Wise, 1985; Butt and Petersen, 1988).

In this concern, a thorough examination should consider that theoverall catalyst activity is determined by the individual activitiescharacterizing the distinct site types. In the common case, each typeof sites supplies its own contribution to the catalytic performance.Both the apparent activity and selectivity may be governed by theratio of these specific contributions. In the event that active sites ofdifferent types are affected in different extent by the deterioratingfactors, this will result in different regularities of the apparentdeactivation kinetics.

Hence, the complexity of such catalytic systems is associatednot only with the variety of interactions and species involved inthe reaction network, but also with the deactivation functionsrelevant to each site type in the course of the process.

At first thought, it may seem reasonable to assume that theactivity of the multi-centered sites should decrease synchronouslywith the increase of the number of active surface atoms covered bycoke, inasmuch as a catalytic cluster would be disabled if one ormore of the constituting surface metal atoms are covered by coke.Following such an assumption, structure-sensitive and structure-insensitive reactions should follow similar trends of rate decrease inthe course of coke formation. But when matter concerns reactionsoccurring on nanostructures containing finite number of active sites,such a conclusion is rather questionable. The deactivation model hasto consider that deactivation-caused decrease of the number ofunblocked sites may affect as well the probability for occurrence ofproper multi-centered configurations catalyzing the structure-sensitive reactions. Specifically, the term ½NðdμM=dNcÞ� in Eq. (10) isto appreciate such an effect. The physical sense of neglecting this termis equivalent to the condition that the number of active catalyticclusters, and hence the activity towards structure-sensitive reactions,should decrease linearly versus the increase of coke coverage. Whenthe magnitude of this term is accountable, the decrease of activitytowards structure-sensitive reactions should follow trends similar tothe patterns shown in Figs. 9 and 11. Actually, such pattern of activityfall (specified by Wheeler as “non-selective deactivation”) is oftenobserved in experiment and manufacture. The abundance of suchphenomena in practical catalytic processes is inarguable. It is of nosmall importance the part of these processes which fit the frameworkstated by the suggested model – deactivation-accompanied structure-sensitive reactions proceeding over multifunctional catalysts withactive phase dispersed as nano-sized islands on carrier. Certainly, thedeactivation kinetics of each concrete process is to be determined onthe basis of assessing the specific characteristics of the system –

extent of active phase dispersion, crystallographic facets and latticedefects, kinetic characteristics of the rivaling and coke formationreactions, etc. These are to give rise to quantitative specificities.However the particular differences, the aim of the present discussionis to reason the necessity of a common approach which would be anapproximation step grounded on physically based forecast.

At larger number of active sites comprised within the active-phase island, the curves relevant to the fall of activity deviate lessnoticeably from a proportional-to-Nc pattern.

In summary, when modeling the coke-caused deactivation ofcatalysts favoring structure-sensitive reactions, it is not a preciseapproximation to assume the activity for these reactions propor-tional to the degree of coke-coverage. At least, three additionaleffects have to be accounted:

1. The active sites which are in proximity to coke species exhibitreduced potential to facilitate structure-sensitive reactions;

2. As far as the generation of coke precursors requires multi-centered site configurations, special terms appreciating thefeasible consequences of the previous point have to be includedin the reaction model of any process accompanied by cokeformation;

3. The importance of the probability factor can hardly be neglec-ted when matter concerns the selectivity of processes in whichstructure-sensitive interactions give rise to diverse reactionpathways.

The authors believe that the stated approach can be applied toestimate the alterations in the activity for structure-sensitivereactions of actual multifunctional catalysts, through solving theset of Eqs. (11)–(13) by use of concrete, experimentally substan-tiated values for the kZ , kY , kX , and other eventually includedcoefficients. The data for estimating the initial values N0

Y and N0Z of

the interfacial and the internal surface active atoms comprisedwithin an arbitrary active-phase island can be figured out on thegrounds of the dispersion and crystallographic characteristicsof the catalyst sample. The value M specifying the number ofcontiguous surface active atoms constituting the required catalyticcluster depends on the configuration of the adsorbed forms of thespecies providing for a given structure-sensitive reaction. In casecoke overlap on the support provides for coke tolerance effect,experimental data about the amount of coke accumulated whilethe coke tolerance effect is evidenced can be used for estimatingthe parameters characterizing the f thðNcÞ function.

Definitely, for thorough understanding of the phenomenadiscussed, it should be not underestimated the significance ofthe variety of ligand effects. As the matter stands, however, theseare problems of separate concern. The objective of the presentstudy is to distinguish the contribution of potential coke formationeffects preconditioned by purely geometric factors.

The obtained results might be helpful for the design ofsupported metal catalysts. By use of the proposed approach inrespect of a particular catalytic process, it is possible to model andfind the optimal metal particles size for established chemicalcomposition of the catalyst. It is possible to achieve essentialimprovements of the catalyst performance by suitable control ofthe process parameters in the course of catalyst synthesis. Hence,better understanding of the catalyst properties on atomic levelmay be instrumental in the optimal catalyst design.

For example, it is widely accepted that highly dispersed metalcatalysts are preferable for the purpose to achieve larger numberof surface atoms participating in the catalytic process. It followsfrom the results shown in Figs. 9 and 11 that very small metalparticles are at the same time quite vulnerable. In a numberof cases, the activity and stability of catalysts with very small metalparticles may turn actually to depend on the size of the reactingmolecules. Therefore, in order to maintain sufficiently high activityof the catalyst and in the same time keep it stable, we need toknow how to prepare catalysts with optimal metal particlesdispersion. The optimal particles size will depend on the reactionmechanism and on the multiplicity of the chemical reactionsproceeding on the catalyst surface. For single route reactionsmono modal particle size distribution might be optimal, but formultiroute catalytic processes, especially in oil processing indus-try, the multimodal distribution will offer better activity.

It is a well known fact that the preparation of optimal catalystsdepends on a variety of parameters, the values of which always area result of compromise solutions. The proposed approach cannotsolve the general problems arising during the catalyst design, butcan be a useful tool in final polishing of the catalyst properties.


5. Conclusions

As the performance of structure-sensitive reactions is facili-tated by specific ensembles of properly located atoms of the activemetal (defined as catalytic clusters), it is of key importance to gainbroader understanding on the regularities determining the avail-ability of catalytic clusters on the active phase. The dispersion ofthe active phase of supported catalysts as nanosized islands on thecarrier brings up the question about the impact of size effects onthe kinetics of structure-sensitive reactions. When estimating thecontribution of multiple M-centered active sites, it is reasonable totake into account the probability μM for availability of the needednumber of surface atoms properly located to construct a catalyticcluster. The problem gains in significance for processes accompa-nied by poisoning or coke formation. In view of the finite numberof active metal atoms contained in an active-phase island, itshould be taken into account that deactivation-caused decreaseof their number may affect as well the probability for occurrenceof catalytic clusters favoring the structure-sensitive interactions.The performed analysis points out that, when matter concernsreactions facilitated by multi-centered active sites, two distinctreasons can be specified by virtue of which coke species may affectthe activities of the catalytic clusters: (i) canceling the action ofpartially or totally coke-covered cluster configurations; and(ii) reducing the capability of the adjacent active atoms to con-struct cluster configurations. Both phenomena are conductive todepressing the chances of the surface active atoms to participate instructure-sensitive reactions. These effects appear to be morepronounced for the case of active-phase nanostructures involvingsmall number of single active sites, and soften on increasing thesize of active-phase islands. The simulation results indicate that incase the active-phase islands involve less than 1000 single sites,the influence of the probability factors are not negligible and haveto be taken into account for the sake of adequacy. The problem ofvulnerability of these entities in the course of coke formation is ofspecial importance for the process selectivity.

Nomenclature

A arbitrary surface active atoma1, a2, a3, a4, a5, a6– nearest first neighbors of the active surface

atom A under considerationaðtÞ time-dependent function of the overall catalyst activity

at a given moment tajðtÞ “individual activity” function related to the action of the

jth distinct type of active sitesaXðtÞ, aY ðtÞ, aZ ðtÞ activity functions related to the action respec-

tively of X-, Y-, Z-type sitesB1, B2, B3, B4, B5, B6 denote active sites with one coke-covered

neighborb the number of surface atoms of the active metal con-

stituting an edge of active-phase islandD1, D2, D3, D4, D5, D6 denote active sites with two coke-covered

neighborsd7, d8, d9, d10, d11, d12 second neighbors of the active surface

atom A under considerationFk probability factor accounting the chances for contribu-

tion to structure-sensitive reactions of the active surfaceatoms with (k) blocked neighbors

kX , kY , kZ rate coefficientsLk the number of single sites with (k) blocked neighborsLf the number of single active sites, which are not adjacent

to coke speciesM the number of active surface atoms in the configuration

of a catalytic cluster facilitating a given structure-sensitive reaction

N the number of active surface atoms comprised within anactive-phase island

N0 the number of active surface atoms within an active-phase island on the fresh catalyst

Nc the number of active surface atoms covered by cokespecies

Ncorner the number of corner interfacial single sitesNedge the number of edge interfacial single sitesNX current number of X-type sites available within an

active-phase islandNY current number of Y-type sites available within an

active-phase islandNZ current number of Z-type sites available within an

active-phase islandUA the probability ascribed to the surface active atom A to

participate in catalytic clusterUoverall the probability for realization of cluster configurations

within an active-phase islanduAjk the probability that an arbitrary active site A and its

neighbors aj and ak can form a catalytic clusterX-type active sites ensembles involving a definite number (M)

neighboring surface atoms in proper configuration (cat-alytic clusters)

Y-type active sites interfacial active sites located along the active-phase-island boundaries

Z-type active sites single active sites located inside the active-phase islands

Greek letters

βconv coefficient accounting the influence of the convexity ofthe active-phase particles on the actual number of sur-face atoms comprised within an active-phase flat island

γk; γ2; γ3; γ4; γ5; γ6 stepwise function taking the value of 0 if the kthsite is covered by coke

μM the probability that (M) surface atoms are properlylocated to construct a catalytic cluster

s fraction of unblocked active sitesψ conf the term accounting the conditional probability for

realization of configuration proper for a catalytic cluster

Acknowledgment

This work was funded by the Deanship of Scientific Research(DSR), King Abdulaziz University, Jeddah, under Grant no. (135-001-D1433). The authors, therefore, acknowledge with thanks DSRtechnical and financial support.

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