Kinetic models of CO oxidation on gold nanoparticles

Surface Science 630 (2014) 286–293

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Kinetic models of CO oxidation on gold nanoparticles Vladimir P. Zhdanov Competence Centre for Catalysis, Chalmers University of Technology, S-41296 Göteborg, Sweden Boreskov Institute of Catalysis, Russian Academy of Sciences, Novosibirsk 630090, Russia

a r t i c l e

i n f o

Article history: Received 11 July 2014 Accepted 25 August 2014 Available online 1 September 2014 Keywords: Catalytic CO oxidation Reaction mechanism Reaction orders Role of the support Mean-field kinetic equations

a b s t r a c t Despite 27 years of experimental and theoretical studies, the mechanism and kinetics of CO oxidation on gold nanoparticles are still open for debate. One of the key features of this reaction is that the reaction turnover frequency rapidly drops with an increasing particle size presumably due to a crucial role of a small number of sites located at the perimeter of nanoparticles. This factor limits the applicability of the conventional meanfield kinetic models implying that the reaction steps occur in a Langmuir overlayer. To clarify this aspect, the conventional kinetics are herein compared with those calculated in the opposite limit implying the reaction to occur on kinetically independent pairs of sites. The results predicted by the models of these two categories are found to differ if the reaction itself is rapid compared to other steps. In the practically interesting case when the reaction is slow, the results are similar. The analysis of different reaction schemes indicates that for the low-temperature reaction regime the apparent reaction orders can be explained assuming cooperative CO and O2 adsorption at different sites. In addition, the scale of the apparent pre-exponential factor for this reaction has been rationalized on the basis of the conventional transition-state theory. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Interpretation of the kinetics of heterogeneous catalytic reactions is customarily based on kinetic models implying that an adsorbed overlayer is infinite like it takes place on single-crystal surfaces. The corresponding most advanced mean-field (MF) and Monte Carlo treatments rely now often on the mechanistic steps and energetic parameters obtained by using density functional theory (DFT) [1,2]. In particular, this approach was used to describe CO oxidation on Pt, Pd, and RuO2 surfaces [3,4]. In the case of catalyst nanoparticles, the DFTbased models can also be successfully employed [5,6]. The multitude of reaction pathways including those occurring on the interface between nanoparticles and a support makes, however, often useful the use of coarse-grained models focused on different reaction channels with their specifics in order to interpret the apparent reaction orders. A good example here is CO oxidation on gold nanoparticles. Gold is usually catalytically inactive. Nm-sized gold particles may, however, be active in some reactions as was first shown by Haruta and co-workers in their seminal study of oxidation of 1 vol.% CO in air at 1 atm and temperatures from 300 K down to 200 K [7]. Since then, this reaction has been actively studied experimentally and theoretically in many groups as described in reviews [8–13] (for other heterogeneous and homogeneous reactions, see [14,15]). Referring primarily to these reviews and recent studies, its key features can be outlined as follows.

E-mail address: [email protected].

http://dx.doi.org/10.1016/j.susc.2014.08.025 0039-6028/© 2014 Elsevier B.V. All rights reserved.

The reaction kinetics, explored customarily under steady-state conditions at CO and O2 pressures of 10–50 and 10–200 mbar [10], are qualitatively different at temperatures below and above 330 K [8,16]. The apparent activation energy is typically 20–30 kJ/mol in the former case and nearly negligible (≃2 kJ/mol) in the latter case. While the reports on the temperature dependence of the reaction rate are numerous, the detailed data on the apparent reaction orders are still limited. At T b 330 K, a general trend for both reactants is that the reaction is first-order at low pressures and becomes to be zeroorder with increasing pressure [11]. At T N 330 K, the reaction order in O2 is reported to be 0.5 [13]. The low-temperature reaction regime (at T b 330 K) is highly structure-sensitive [8]. The turnover frequency (TOF), i.e., the reaction rate per exposed Au atom rapidly drops with increasing gold nanoparticle size above 4 nm. For nanoparticles with sizes from 1.5 nm up to 10 nm, the reaction rate is approximately proportional to the number of Au atoms at the perimeter of nanoparticles [16], i.e., the sites at the catalyst–support interface seem to play an important role [11]. The infrared-kinetic measurements also indicate that the reaction occurs primarily at this interface [17]. The corresponding sites can be inhibited when the Au is preoxidized [18]. The structure sensitivity of the high-temperature reaction regime (at T N 330 K) is lower [8], and the reaction rate is approximately proportional to the total number of exposed Au atoms [16]. On the other hand, there are clear indications that the catalyst–support interface or, more specifically, surface lattice oxygen close to the Au nanoparticles can play a key role also at these temperatures [13].

V.P. Zhdanov / Surface Science 630 (2014) 286–293

Oxygen and CO may induce restructuring of gold nanoparticles as shown on CeO2 by changing O2 and CO pressures in the range from 10−6 to 10−2 bar and 10−10 to 10−4 bar, respectively [19]. Catalytic gold nanoparticles are more stable on TiO2, Fe2O3, and NiO compared to Al2O2 and SiO2 [8]. For active gold nanoparticles, the metal–support interaction is expected to be important, and there are various experimental indications of the role of this factor [12]. At room temperature, on the other hand, the TOFs measured at Al2O2, SiO2 and TiO2 are nearly equal [8]. The reaction can be influenced by moisture positively [16] and negatively [20]. The experimental findings outlined above were complemented by theoretical studies. In particular, the energetics of some of the likely reaction steps were analyzed on Au clusters by using DFT [17,18,21–23]. Typically (except [23]), the focus was on the reaction of a single pair of reactants including adsorbed CO and O2 or O. The reaction was usually considered to occur on gold, and the support was not treated explicitly [21–23]. In particular, the catalytic activity of Au was compared with those of other metals [21]. The ability of the metal atoms to activate reactants was shown to depend on the coordination number of the active metal site [21]. The calculations performed on an Au55 cluster indicate that the high-spin states may play an important role in the reaction [22]. There are also indications that the reaction may occur via a three-molecular pathway including CO interaction with one of O atoms forming an O 2 molecule and simultaneous interaction of another O atom with another CO with the formation eventually of two CO 2 molecules [23]. On the Au–TiO2 interface, the reaction of CO with O2 and O was shown to take place primarily at dual Au–Ti sites [17,18]. The experimental studies of the steady-state kinetics of CO oxidation on gold nanoparticles were reviewed in [10]. A few related conventional MF kinetic models have been proposed [24–27]. All these models imply that the reaction occurs in an unlimited uniform adsorbed overlayer with one or two types of sites, and as a rule the CO2 formation is assumed to occur via the Langmuir–Hinshelwood steps (unlimited means that the catalyst surface is so large that the boundary is negligible). The simplest expressions for the dependence of the reaction rate on CO and O2 pressures were proposed in [24]. The case of two types of sites was discussed with emphasis on the analogy with the Michaelis–Menten model [25]. The Eley–Rideal-type steps were included [26]. The effect of H2O on the reaction kinetics was recently discussed [27]. Despite the available experimental and theoretical studies, the understanding of the kinetics of CO oxidation on gold nanoparticles is far from complete. Our present analysis of a few generic kinetic models of this reaction concerns three interconnected aspects: (i) As already noticed, the available MF models imply that the adsorbed overlayer is uniform and unlimited. The experiments indicate, however, that the reaction often takes place primarily at the perimeter of nanoparticles with the size smaller or about 4 nm. The number of sites at the perimeter of such particles is small, and these sites are expected to be energetically heterogeneous. Under such circumstances, one cannot exclude that a more adequate description of the reaction kinetics can be done assuming the reaction to occur on independent pairs of adsorption sites (or small patches containing a few sites). This limit of the kinetics of heterogeneous catalytic reactions is rarely treated in the literature [28]. In the context of the reaction under consideration, it has not been analyzed. We show the specifics of this limit. (ii) The experimental studies of the kinetics were primarily focused on the temperature dependence of the reaction rate (see, e.g., the corresponding data collected in [8,29]). Although the apparent reaction orders in CO and O2 are rarely reported, there are clear indications (see, e.g., [30]) that in the low-temperature

287

regime both orders are close to zero provided the reactant pressures are not too low. Our goal is to identify the conditions of realization of this situation. (iii) We discuss whether the experimentally observed kinetic features can be quantitatively interpreted by using the reaction rate constants with the pre-exponential factors estimated by employing the conventional transition-state theory. Mechanistically, CO oxidation may occur on gold nanoparticles via molecular or dissociative O2 adsorption. In our presentation, the corresponding reaction schemes are described in Sections 2 and 3, respectively. The reaction rate constants are discussed in Section 4. Focusing on the reaction orders and taking the specifics of the reaction under consideration into account, we use the kinetic equations implying a few adsorption sites or a few types of adsorption sites. For comparison, some of the results are derived for the situations when a surface contains one type of sites. Lateral interactions between adsorbed particles are not taken into account. The latter approximation is often reasonable if reaction occurs on a few sites (as expected for gold nanoparticles) and/or if the reactant coverages are high or low. In principle, lateral interactions can be included into the treatment (as described in [31]). This is, however, beyond our present goals. As the first mandatory step in the analysis of reaction kinetics, the predictions of the models without lateral interactions should be scrutinized under steady-state conditions. This is the aim of our work. 2. Schemes with molecular O2 adsorption 2.1. Reaction steps Reactions schemes with molecular O2 adsorption are widely implied in the experimental studies and DFT calculations mentioned in the Introduction. The conventional scheme of this category include reversible adsorption of CO and O2, CO þ S1 ⇌ CO  S1;

ð1Þ

O2 þ S2 ⇌ O2  S2;

ð2Þ

reaction between adsorbed CO and O2, CO  S1 þ O2  S2→CO2 þ O  S2 þ S1;

ð3Þ

and subsequent reaction between adsorbed CO and O, CO  S1 þ O  S2→CO2 þ S1 þ S2;

ð4Þ

where S1 and S2 are the surface sites. An example of a more specific scheme includes steps (1) and (2) combined with a concerted step of CO2 formation [23], 2ðCO  S1Þ þ O2  S2 → 2CO2 þ 2S1 þ S2:

ð5Þ

The analysis of the schemes above depends on whether sites S1 and S2 are equivalent and whether they form a large array of interconnected sites and can be treated as an unlimited adsorbed overlayer (Sections 2.2, 2.3 and 2.4) or independent pairs of sites (Section 2.5). 2.2. Competitive adsorption In kinetic models of heterogeneous catalytic reactions, adsorption sites are often assumed to be identical (in our context, this means S1 ≡ S2), each site is considered to be able to adsorb only one atom or molecule, and the adsorbed layer is assumed to be unlimited. Such models, implying competitive adsorption of reactants, can be formalized in terms of coverages of adsorption sites by adsorbed species. For

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V.P. Zhdanov / Surface Science 630 (2014) 286–293

Scheme (1)–(4), the conventional MF equations for the corresponding CO, O2, and O coverages, θA, θB, and θC, are as follows dθA =dt ¼ ka P A ð1−θA −θB −θC Þ−kd θA −r 1 θA θB −r 2 θA θC ;

ð6Þ

dθB =dt ¼ κ a P B ð1−θA −θB −θC Þ−κ d θB −r 1 θA θB ;

ð7Þ

dθC =dt ¼ r 1 θA θB −r2 θA θC ;

ð8Þ

where PA and PB are the reactant pressures, and ka, kd, κa, κd, r1 and r2 are the adsorption, desorption, and reaction rate constants. Under steady-state conditions, Eq. (8) yields θC ¼ ðr 1 =r 2 ÞθB :

ð9Þ

and one has θA ¼

κ d κ a PB ; r 1 ð2κ a P B −ka P A Þ

ð21Þ

θB ¼

κ a P B −ka P A =2 ; κ d þ ðκ a P B −ka P A =2Þð1 þ r 1 =r 2 Þ

ð22Þ

W≃

κ d ka P A : 2κ d þ ð2κ a P B −ka P A Þð1 þ r 1 =r 2 Þ

ð23Þ

If kaPA → 2κaPB, the surface is nearly free of reactants (θA ≪ 1 and θB ≪ 1), and the reaction rate is W ¼ κ a P B ¼ ka P A =2:

Using this relation, Eqs. (6) and (7) can be rewritten as ka P A ½1−θA −ð1 þ r1 =r2 ÞθB  ¼ kd θA þ 2r 1 θA θB ;

ð10Þ

κ a P B ½1−θA −ð1 þ r 1 =r2 ÞθB  ¼ κ d θB þ r 1 θA θB :

ð11Þ

These equations can easily be solved analytically. The corresponding expressions are, however, cumbersome and not instructive. For our goals, it is sufficient to show what happen in two important limits of slow and rapid reaction, respectively. If the reaction is slow, one can calculate θA and θB neglecting r1 θA θB in the right-hand parts of Eqs. (10) and (11). This results in

ð24Þ

Looking at Eq. (14), one can notice that if one of the terms in the denominator dominates, the reactions orders are + 1 for one reactant and − 1 for another reactant. According to Eqs. (18) and (23), the reaction orders are close to + 1 for one reactant and zero for another reactant. These examples and a more detailed general analysis show that Eqs. (6)–(8) cannot predict the orders close to zero simultaneously for both reactants. The latter is, however, observed for low-temperature CO oxidation on gold [30]. Thus, the model with one type of sites does not seem to be acceptable (at least for the low-temperature reaction regime) despite its often use in DFT calculations. 2.3. Cooperative adsorption

ka P A =kd ; θA ¼ 1 þ ka P A =kd þ ð1 þ r1 =r 2 Þκ a P B =κ d

ð12Þ

κ a P B =κ d : 1 þ ka P A =kd þ ð1 þ r 1 =r 2 Þκ a P B =κ d

ð13Þ

θB ¼

The reaction rate is accordingly given by W ¼ r 1 θA θB ¼

r 1 ðka P A =kd Þðκ a P B =κ d Þ : ½1 þ ka P A =kd þ ð1 þ r 1 =r 2 Þκ a P B =κ d 2

ð14Þ

If the reaction is rapid, the situation depends on ratio of the reactant pressures. If kaPA N 2κaPB, the surface is primarily covered by CO, i.e., θB ≪ θA, and Eqs. (10) and (11) are reduced to

Cooperative adsorption implies the existence of different sites for different species. Here, we analyze the case when CO adsorbs on one sites (S1) and O2 and O adsorb on the other sites (S2), and the reaction occurs via steps (1)–(4). For this model, the conventional MF equations are as follows dθA =dt ¼ ka P A ð1−θA Þ−kd θA −r 1 θA θB −r 2 θA θC ;

ð25Þ

dθB =dt ¼ κ a P B ð1−θB −θC Þ−κ d θB −r 1 θA θB ;

ð26Þ

dθC =dt ¼ r 1 θA θB −r 2 θA θC ;

ð27Þ

ka P A ð1−θA Þ ¼ kd θA þ 2r 1 θA θB ;

ð15Þ

where θ A is the CO coverage of sites S1, θ B and θ C are the O 2 and O coverages of sites S2, and the other designations are as in Eqs. (6)–(8). Under steady-state conditions, Eq. (27) again [cf. Eq. (9)] yields

κ a P B ð1−θA Þ ¼ r 1 θA θB :

ð16Þ

θC ¼ ðr 1 =r 2 ÞθB :

ð28Þ

Substituting this relation into Eqs. (25) and (26) results in The solution of these equations yields θA ¼

ka P A −2κ a P B kd κ a P B ; ; θB ¼ kd þ ka P A −2κ a P B r 1 ðka P A −2κ a P B Þ

W ¼ r 1 θA θB ¼

kd κ a P B : kd þ ka P A −2κ a P B

ka P A ð1−θA Þ ¼ kd θA þ 2r1 θA θB ;

ð29Þ

ð17Þ

κ a P B ½1−ð1 þ r 1 =r 2 ÞθB  ¼ κ d θB þ r 1 θA θB :

ð30Þ

ð18Þ

According to Eq. (29), θB is expressed via θA as θB ¼

If kaPA b 2κaPB, the surface is primarily covered by oxygen, i.e., θA ≪ θB, Eqs. (10) and (11) are reduced to ka P A ½1−ð1 þ r1 =r2 ÞθB  ¼ 2r 1 θA θB ;

ð19Þ

κ a P B ½1−ð1 þ r1 =r2 ÞθB  ¼ κ d θB þ r 1 θA θB ;

ð20Þ

ka P A ð1−θA Þ−kd θA : 2r 1 θA

ð31Þ

Substituting Eq. (31) into Eq. (30) yields 2

r 1 ðka P A þ kd ÞθA þ r 1 ð2κ a P B −ka P A ÞθA þ ðka P A þ kd Þ½κ d þ κ a P B ð1 þ r 1 =r 2 ÞθA −ka P A ½κ d þ κ a P B ð1 þ r 1 =r 2 Þ ¼ 0:

ð32Þ

V.P. Zhdanov / Surface Science 630 (2014) 286–293

This equation can be represented as

Using Eqs. (36) and (37), the reaction rate can be represented as

2

θA þ aθA −b ¼ 0;

ð33Þ

with a ¼ ½κ d þ κ a P B ð1 þ r1 =r 2 Þ=r 1 þ

2κ a P B −ka P A ; ka P A þ kd

ð34Þ

W¼

ka P A ½κ d þ κ a P B ð1 þ r 1 =r 2 Þ ; r 1 ðka P A þ kd Þ

ð35Þ

and accordingly θA = − a/2 + (a2/4 + b)1/2. Using this expression, one can obtain θB via Eq. (31) and θC via Eq. (28), and then the reaction rate, W = r1θAθB, can be calculated. The expression derived above for θA, θB, θC, and W are exact and can easily be used. To illustrate the model predictions, it is convenient to employ dimensionless rate constants obtained by the normalization of the rate constants to an arbitrary chosen rate constant. In particular, Fig. 1 shows how the reaction rate and reactant coverages may depend on r 1 and the reactant pressures. To clarify the specifics of the kinetics predicted, it is instructive to analyze explicitly (as in Section 2.2) two limiting situations when the reaction is slow and rapid, respectively. If the reaction is slow reaction compared to CO adsorption, the second term in the right-hand part of Eq. (29) can be neglected, and one has ka P A : kd þ ka P A

θA ¼

ð36Þ

r 1 ka P A κ a P B : ðkd þ ka P A Þ½κ d þ ð1 þ r 1 =r 2 Þκ a P B þ r 1 ka P A =ðkd þ ka P A Þ

r 1 ka P A κ a P B : ðkd þ ka P A Þ½κ d þ ð1 þ r 1 =r 2 Þκ a P B 

ð39Þ

If the reaction is rapid and kaPA N 2κaPB, the surface is primarily covered by CO, i.e., θB ≪ θA, and Eq. (30) is reduced to κ a P B ¼ r 1 θA θB :

ð40Þ

The right-hand part of this equation defines the reaction rate, i.e., W ¼ r 1 θA θB ¼ κ a P B :

ð41Þ

The CO coverage can be obtained by substituting Eq. (40) into Eq. (29), θA ¼

ka P A −2κ a P B : kd þ ka P A

ð42Þ

Substituting the latter expression into Eq. (40) yields θB ¼

κ a P B ðkd þ ka P A Þ : r 1 ðka P A −2κ a P B Þ

ð43Þ

If the reaction is rapid and kaPA b 2κaPB, the CO coverage is low, θA ≪ θB, and Eq. (29) is reduced to ka P A ¼ 2r 1 θA θB ;

Substituting this expression into Eq. (30) results in

ð38Þ

If the reaction is slow also compared to O2 adsorption, the term r1kaPA/ (kd + kaPA) in the denominator of the latter expression can be neglected, and it is reduced to W¼

b¼

289

ð44Þ

and accordingly κ a PB θB ¼ : κ d þ ð1 þ r 1 =r 2 Þκ a P B þ r 1 ka P A =ðkd þ ka P A Þ

(a)

(b)

ð45Þ

(c)

1.0

θA , θB , W

0.1

θA

0.6

θB

0.4 0.2

W 0.0

0.0

-1

-1

θA , θB , W

0.8

θA

0.2

W ¼ r 1 θA θB ¼ ka P A =2:

0.5

θB

0.3

θA , θB , W

ð37Þ

θA

0.4 0.3

θB

0.2 0.1

W

W

0.0

-2

-3

-2

-1

0

Log10( r1 )

1

2

Log10( W )

Log10( W )

Log10( W )

-1

-2

-3

-2

-1

0

Log10( ka PA )

1

2

-2

-2

-1

0

1

2

Log10( ka PA )

Fig. 1. Mean-field reaction kinetics under steady-state conditions in the case of cooperative CO and O2 adsorption according to Eqs. (25)–(27) solved exactly [Eqs. (32)–(35)]: adsorbate coverages and reaction rate are shown as a function of (a) r1 for kaPA = 0.5 and κaPB = 1, (b) kaPA for r1 = 1 and κaPB = 1, and (c) κaPB for r = 1 and kaPA = 0.5 (the other parameters were chosen as: r2 = r1 and kd = κd = 1). Panel (a) illustrates the transition from slow to rapid reaction kinetics. Panel (b) shows the change of the reaction order in CO from 1 to zero with increasing CO pressure. Panel (c) exhibits similar results for O2.

290

V.P. Zhdanov / Surface Science 630 (2014) 286–293

Substituting Eq. (44) into Eq. (30) results in θB ¼

κ a P B −ka P A =2 : κ d þ κ a P B ð1 þ r 1 =r 2 Þ

ð46Þ

With this expression, Eq. (44) yields θA ¼

ka P A ½κ d þ κ a P B ð1 þ r1 =r2 Þ : r 1 ð2κ a P B −ka P A Þ

ð47Þ

If the reaction is slow, the model presented in this subsection predicts [see Fig. 1 or expressions (38) and (39)] that the reaction orders in CO and O 2 may simultaneously be close to zero, as observed in experiments [30], provided the reactant pressures are sufficiently high. A simplified version of this model was earlier proposed in [25].

Under such circumstances, as already noted in the Introduction, a more adequate description of the reaction kinetics may consist in assuming the reaction to occur on independent pairs of adsorption sites. In the framework of this model, CO and O2 adsorption can also be competitive or cooperative, i.e., the two sites, S1 and S2, can be identical or different. In analogy with the MF treatments presented, the cooperative adsorption on different sites seems to be preferable. Focusing on this case and assuming the reaction to occur via steps (1)–(4), we can describe the reaction kinetics in terms of the probabilities P 00 , P 0B , P 0C , P A0 , P AB , and P AC , that (i) the sites S1 and S2 are free, [(ii) and (iii)] site S1 is free and site S2 is occupied by O2 or O, and [(iv)–(vi)] site S1 is occupied by CO and site S2 is free or occupied by O2 or O, respectively. By definition, the sum of the probabilities is equal to unity, P 00 þ P 0B þ P 0C þ P A0 þ P AB þ P AC ¼ 1:

ð53Þ

2.4. Concerted reaction

2

ka P A ð1−θA Þ ¼ kd θA þ 2rθA θB ;

(a)

P0C

ð48Þ

2

κ a P B ð1−θB Þ ¼ κ d θB þ rθA θB ;

ð49Þ

ð50Þ

ð51Þ

-2

The reaction orders predicted by these expressions are the same as those given by Eqs. (38) and (39) provided that the reactant pressures are sufficiently high. If the O2 pressure is low while the CO pressure is high, the reaction orders are the same as well. If the CO pressure is low, the reaction order in CO is 2 according to Eq. (51) [or Eq. (52)] and 1 according to Eq. (38) [or Eq. (39)]. 2.5. Cooperative adsorption on pairs of sites The MF models described in Sections 2.2–2.4 imply that the catalyst surface is uniform, the array of adsorbed sites is large, and adsorbate diffusion is rapid. At gold nanoparticles with the size smaller or about 4 nm, the CO2 formation takes, however, often place primarily at the perimeter. The number of sites at the perimeter of such particles is small, and this area is expected to be energetically heterogeneous.

-2

-1

0

1

2

Log10( r1 )

(b) 0.3

pA , pB , W

ð52Þ

PA0

0.1

-3

If the reaction is slow also compared to O2 adsorption, the term r(kaP2A/(kd + kaPA)2 in the denominator can be neglected, and one has r ðka P A Þ2 κ a P B : ðkd þ ka P A Þ2 ½κ d þ κ a P B 

P0B

-1

2

W¼

0.2

0.0

The reaction rate, defined by W = rθ2A θB, is then represented as r ðka P A Þ κ a P B
: W¼ ðkd þ ka P A Þ2 κ d þ κ a P B þ rðka P A Þ2 =ðkd þ ka P A Þ2

P00

Log10( W )

κ a PB : κ d þ κ a P B þ r ðka P A Þ2 =ðkd þ ka P A Þ2

0.3

PAB (or PAC )

where r is the rate constant of step (5), and the other designations are as Sections 2.2 and 2.3. In the most relevant case when the reaction is slow compared to CO adsorption, the second term in the right-hand part of Eq. (48) can be neglected, and the CO coverage is given by Eq. (36). Substituting Eq. (36) into Eq. (49) yields θB ¼

0.4

Probabilities

Taking into account that the above-introduced MF model with cooperative adsorption can be used to explain the observed reaction orders, it is instructive to show what happens if the adsorption and desorption steps [(1) and (2)] are kept as described while the two sequential reaction steps [(3) and (4)] are replaced by the concerted step (5). The corresponding steady-state MF equations are as follows

pB pA

0.2

0.1

W 0.0

-2

-1

0

1

2

Log10( r1 ) Fig. 2. Reaction kinetics under steady-state conditions as a function of r1 in the case of cooperative CO and O2 adsorption on pairs of sites according to Eq. (54) with kaPA = 0.5, κaPB = 1, kd = κd = 1, and r2 = r1. Panel (a) shows the reaction rate and pair probabilities. Panel (b) exhibits the reaction rate and probabilities (solid lines) that site 1 is occupied by CO, pA ≡ P A0 þ P AB þ P AC , and site 2 is occupied by O2, pB ≡ P 0B þ P AB. For comparison, the dashed line show the MF results calculated according to Eqs. (25)–(27) with the same parameters.

V.P. Zhdanov / Surface Science 630 (2014) 286–293

corresponds to atomic oxygen), satisfying the following kinetic equations

The kinetic equations for the probabilities are as follows dP 00 =dt ¼ kd P A0 þ κdP 0B þ r 2 P AC −ðka P A þ κP B ÞP 00 ; dP 0B =dt ¼ κ a P B P 00 þ kd P AB −ðκ d þ ka P A ÞP 0B ; dP 0C =dt ¼ kd P AC þ r 1 P AB −ka P A P 0C ; dP A0 =dt ¼ ka P A P 00 þ kd P AB −ðkd þ κ a P B ÞP A0 ; dP AB =dt ¼ ka P A P 0B þ κ a P B P A0 −ðkd þ κ d þ r1 ÞP AB ; dP AC =dt ¼ ka P A P 0C −ðkd þ r 2 ÞP AC ;

ð54Þ

ð55Þ

All the rate constants are here as in Eqs. (6)–(8). Our calculations indicate (see, e.g., Fig. 2) that the steady-state kinetics predicted by Eqs. (54) and (25)–(27) are similar. If the reaction is slow compared to adsorption, the kinetics are nearly identical. If the reaction is rapid, there are, however, some deviations. The most notable one is that in this limit Eqs. (25)–(27) predict that the surface is primarily covered either by CO or by oxygen. In contrast, Eq. (54) predicts that there are non-negligible probabilities to find CO or oxygen on a pair of sites. 3. Dissociative O2 adsorption

¼ r 1 P B0 þ r 2 P 0B −κ a P B2 P 00 ; ¼ r 2 P BB −r 1 P B0 −κ 12 P B0 þ κ 21 P 0B ; ¼ r 1 P BB −r 2 P 0B þ κ 12P B0 −κ 21 P 0B ; ¼ κ a P B2 P 00 − r 1 þ r 2 P BB ;

ð59Þ

Depending on the ratio between κ12, κ21 and other rate constants, Eq. (58) can describe various situations. If the jumps of oxygen atoms between the sites are negligible, i.e., κ12 = κ21 = 0, and the system is close to steady state, we obtain "

=

P 00 ¼ ðr 1 þ r2 Þ

r þ r 2 r 2 r 1 1 þ 1 þ þ κ a P B2 r 1 r 2

κ a P B2

!# ;

! r 1 þ r2 r 2 r 1 ; 1þ þ þ κ a P B2 r 1 r 2

= −1

−1

P B0 ¼ r 2 r 1

P 0B ¼ r 1 r 2

ð60Þ

ð61Þ

=

1þ

! r1 þ r 2 r2 r 1 ; þ þ κ a P B2 r1 r 2

ð62Þ

=

1þ

! r1 þ r 2 r2 r 1 ; þ þ κ a P B2 r1 r 2

ð63Þ

=

W ¼ κ a P B2 P 00 ¼ ðr 1 þ r2 Þ ð56Þ

1þ

! r 1 þ r2 r 2 r 1 : þ þ κ a P B2 r 1 r 2

ð64Þ

According to this expression for the reaction rate, we have W ≃ κ a P B2

Depending on the specification of adsorption sites, this scheme can be used in different situations. For example, all the sites may be located on gold near the gold–support interface. Another example is when the sites S1 and S3 are on gold while the sites S2 and S4 are on the support. In general, the number of kinetically connected sites may be large, i.e., they may form an array. Alternatively, the O2 dissociation and subsequent reaction steps may occur on kinetically independent pairs of sites. Focusing on the latter case, we consider that the Langmuir– Hinshelwood step is relatively slow and accordingly CO adsorption and desorption are close to equilibrium. In this case, we can use the effective reaction rate constants, r1∗ and r2∗, including the CO coverages of sites S3 and S4. Employing the Langmuir expressions for these coverages, we have r 1 ¼ r1 ka1 P A =ðkd1 þ ka1 P A Þ; r 2 ¼ r2 ka2 P A =ðkd2 þ ka2 P A Þ;

ð58Þ

P 00 þ P BB þ P B0 þ P 0B ¼ 1:

P BB ¼ 1

The conventional scheme of CO oxidation on Pt, Pd or Rh includes dissociative O2 adsorption, reversible CO adsorption, and the Langmuir– Hinshelwood reaction between adsorbed O and CO [3,32]. In the corresponding kinetic models, the sites for O and CO adsorption are usually considered to be the same, and the Langmuir–Hinshelwood step is assumed to be rapid. For gold nanoparticles, the conventional scheme may also be applicable especially at relatively high temperatures [33]. Here, however, as already noted, the sites for adsorption are likely to be different and the Langmuir–Hinshelwood step may be slow. One of the minimal reaction schemes of this category includes four different types of adsorption sites: O2 þ S1 þ S2→O  S1 þ O  S2; CO þ S3 ⇌ CO  S3; CO þ S4 ⇌ CO  S4; O  S1 þ CO  S3 → CO2 þ S1 þ S3; O  S2 þ CO  S4 → CO2 þ S2 þ S4:

dP 00 =dt dP B0 =dt dP 0B =dt dP BB =dt

where κa is the rate constant of dissociative adsorption of O2, and κ12 and κ21 are the rate constants of jumps of oxygen atoms between the sites. The balance equation for these probabilities is

and the reaction rate is defined as W ¼ r 1 P AB :

291

ð57Þ

where r1, r2, ka1, ka2, kd1 and kd2 are the reaction, adsorption and desorption rate constants. The oxygen adsorption on sites S1 and S2 can then be described by using four probabilities, P 00 ; P BB ; P B0 and P 0B (the subscripts characterize the occupation of the sites; B

ð65Þ

provided that κ a P B2 ≪r 1 þ r 2 , and   −1 −1 W ≃ ðr1 þ r 2 Þ= 1 þ r 2 r 1 þ r 1 r2

ð66Þ

provided that κ a P B2 ≫r 1 þ r 2 . If the jumps of oxygen atoms between the sites are rapid and the sites are energetically equivalent, i.e., κ12 = κ21, the model yields   2κ a P B2 þ r 1 þ r 2 ;

=

P 00 ¼ ðr 1 þ r2 Þ

P BB ¼ κ a P B2

  2κ a P B2 þ r 1 þ r 2 ;

=

ð68Þ

P B0 ¼ P 0B ¼ P BB =2;

W ¼ κ a P B2 P 00 ¼

κ a P B2 ðr 1 þ r 2 Þ 2κ a P B2 þ r 1 þ r 2

ð67Þ

ð69Þ

:

ð70Þ

292

V.P. Zhdanov / Surface Science 630 (2014) 286–293

If one of the sites is on gold while the other site is on the support, the sites may be highly non-equivalent. One of the physically reasonable models of this category implies that the reaction occurs primarily on site S1 and the O jumps from this site to site S2 are not favorable. This case can be described assuming r2∗ = 0 and κ12 = 0. With this specification, Eqs. (58) are reduced to dP 00 =dt dP B0 =dt dP 0B =dt dP BB =dt

¼ r 1 P B0 −κ a P B2 P 00 ; ¼ κ 21 P 0B −r 1 P B0 ; ¼ r1 P BB −κ 21 P 0B ; ¼ κ a P B2 P 00 −r 1 P BB ;

ð71Þ

and under steady-state conditions we have P 00 ¼ r 1

h

a B2

P BB ¼ P B0 ¼ 1

P 0B ¼ r 1



=κP

i

;

   2 þ r 1 =κ a P B2 þ r1 =κ 21 ;

=

h

=κ

2 þ r 1 =κ a P B2 þ r 1 =κ 21

 21

2 þ r 1 =κ a P B2 þ r 1 =κ 21

=

W ¼ κ a P B2 P 00 ¼ 1

i

ð72Þ

ð73Þ

;

  2=r 1 þ 1=κ a P B2 þ 1=κ 21 :

ð74Þ

ð75Þ

Depending on the ratio between r1∗, κ a P B2 and κ21, the model identifies   three qualitatively different situations. If r 1 ≪ min κ a P B2 ; κ 21 , the whole reaction is limited by the Langmuir–Hinshelwood step, and

exponential factor for this step. The latter pre-exponential factor can in turn be estimated by using the transition-state theory as ν≃

kB T F  ; h F

ð79Þ

where F⁎ and F are the partition functions in the activated and ground states. Taking into account that kBT/h ≃ 1013 s−1, one can conclude that the ratio F⁎/F should be about 10−9 in order to explain the scale of the measured apparent pre-exponential factor. The lowest value of F⁎/F can be obtained assuming the motion in the activated state to be constrained, i.e., F∗ ≃ 1, while the partition functions for the ground state to be large, e.g., about 104 for CO and 105 for O2. Such high values of the partition functions are physically feasible [31,34] (especially for the concerted scheme discussed in Section 2.4) despite the fact that usually F is considered to be smaller. In particular, adsorbed CO and O2 molecules may have 2D translational and rotational partition function. For each molecule, the product of these functions may be of the order of 105 [31], and accordingly one cannot exclude that the scale of the ratio F⁎/F is about 10−9. Our conclusion that the transition-state theory allows one to rationalize the scale of the measured apparent pre-exponential factor for CO oxidation on gold nanoparticles contrasts that in Ref. [35]. One of the reasons of the discrepancy is that the authors of Ref. [35] interpreted the reaction conversion in terms of the rate constants of two reaction steps (desorption and reaction itself). To interpret the conversion, one should explicitly describe the reactor performance. It was not, however, done. In addition, the scale of various partition functions was not scrutinized. 5. Conclusion

W ≃ r1 =2:

ð76Þ

If κ a P B2 ≪ minðr1 ; κ 21 Þ, the reaction is limited by O2 adsorption, and W ≃ κ a P B2 :

ð77Þ

  If κ 21 ≪ min r 1 ; κ a P B2 , the reaction is limited by O jumps from site S2 to site S1, and W ≃ κ 21 :

ð78Þ

4. Kinetic parameters One of the aspects of the kinetics of CO oxidation on gold nanoparticles is in the interpretation of the corresponding apparent Arrhenius parameters. Concerning this aspect and focusing on the low-temperature reaction regime (at T b 330 K), we repeat (cf. Sections 1 and 2) that the experiments indicate that in this case the apparent reaction orders in CO and O2 may be low or, more specifically, close to zero (see, e.g., [30]). The corresponding TOF measured at T = 273 K is typically 0.01–0.1 s−1 [29]. The apparent activation energy is slightly different in different studies. Using Ea = 28 kJ/mol as a representative value [25], we obtain that the scale of the apparent pre-exponential factor is ν = (0.2 − 2) × 104 s−1. According to the models (Sections 2 and 3), the nearly zero apparent reaction orders in CO and O2 can be explained assuming CO and O2 adsorption to be cooperative and the reaction is slow. Irrespective of the details of the corresponding models, the scale of TOF is expected be determined in this case by the rate constant of reaction between adsorbed CO and O2 or O [see, e.g., Eq. (39) or (52) at kaPA ≫ kd and κaPB ≫ κd, and Eq. (55) or (76)]. Thus, the experimentally measured apparent pre-exponential factor can be identified with the pre-

We have analyzed in detail a few generic kinetic models of CO oxidation on gold nanoparticles with emphasis on the structure sensitivity, apparent reaction orders in CO and O2, and apparent Arrhenius parameters. High structure sensitivity of this reaction is often attributed to a crucial role of a small number of sites located at the perimeter of nanoparticles. In this limit, the conventional concept of an unlimited adsorbed overlayer may be not applicable. To clarify the limits of applicability of the kinetic models based on this concept, we have compared the corresponding kinetics (Section 2.3) with those calculated in the opposite limit implying that the reaction occurs on kinetically independent pairs of sites (Section 2.5). The results predicted by the models of these two categories have been found to differ if the reaction itself is rapid compared to other steps. In the practically interesting case when the reaction is slow, the results are similar. These findings justify the use of the conventional models. Our analysis indicates that for the low-temperature reaction regime the apparent reaction orders can be explained assuming CO and O2 adsorption to occur cooperatively at different sites. This aspect is usually neglected in DFT calculations performed under conditions corresponding to the low-coverage case. Concerning the apparent Arrhenius parameters, we conclude that the scale of the apparent pre-exponential factor for the lowtemperature regime can be explained in terms of the conventional transition-state theory. Although the models and analysis presented and the conclusions drawn clarify various aspects of the kinetics of CO oxidation on gold nanoparticles, we finally note that this area is still far from exhausted. One of the extensions of our treatment may, e.g., include lateral interactions. The models without lateral interactions (like those used in our analysis) are known to tend to predict simple (e.g., rational) apparent reaction orders. With lateral interactions, the apparent reaction orders may be arbitrary. Although for the interpretation of the experiments mentioned in the Introduction this does not seem to be crucial, the models with lateral interactions can potentially be useful. Another

V.P. Zhdanov / Surface Science 630 (2014) 286–293

extension is to analyze in more detail what may happen on the catalyst– support interface. This very broad subject is still poorly understood. In this context, we may notice that the analytical treatments (like those in our analysis) of the kinetics of reactions occurring on a small number of different catalytic sites with lateral interactions and with participation of the support are expected to be hardly possible. In such situations, the kinetic Monte Carlo simulation may be efficient [2–4,36].

[12] [13] [14] [15]

Acknowledgments

[20] [21]

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