Stochastic model of reaction rate oscillations during CO oxidation over zeolite-supported catalysts

Chemical Engineering Science 58 (2003) 4797 – 4803

www.elsevier.com/locate/ces

Stochastic model of reaction rate oscillations during CO oxidation over zeolite-supported catalysts Nikolai V. Peskova;∗ , Marina M. Slinkob , Nils I. Jaegerc a Department

of Computational Mathematics & Cybernetics, Moscow State University, Moscow 119899, Russia of Chemical Physics, Russian Academy of Science, Kosygina Str. 4, Moscow 117334, Russia c Institut f) ur Angewandte und Physikalische Chemie, FB 2, Universit)at Bremen, PF 330440, Bremen 28334, Germany b Institute

Received 20 June 2002; received in revised form 10 December 2002; accepted 12 December 2002

Abstract The particle size e7ect on the oscillatory behaviour during CO oxidation over zeolite-supported Pd catalysts is simulated with the help of a deterministic point model and a stochastic mesoscopic model. The point model is developed on the basis of Sales, Turner and Maple (STM) model, which is modi8ed to consider the e7ects of the oxidation of the Pd bulk upon the catalyst activity. It is demonstrated that the deterministic point model can simulate the main properties of regular reaction rate oscillations. The stochastic model is based on the developed point model and simulates the reaction by a Markovian chain of elementary transitions, which correspond to changes in numbers of atoms and molecules of reagent species on the surface of Pd particle due to elementary steps of reaction. The stochastic model explains the role of statistical 9uctuations and correlations in the reaction dynamics on the surface of an nm-sized catalyst particle. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Kinetics; Particle; Zeolites; Mathematical modeling; Oscillations

1. Introduction Particle size e7ects in heterogeneous catalysis a7ecting yield and selectivity are well known and have been extensively studied in recent years (Che & Benett, 1989; Bond, 1991; Bukhtiyarov & Slinko, 2001). It was shown experimentally (Henry, 2000) and theoretically (Zhdanov & Kasemo, 2000) that essential di7erences in the catalytic activity exist between an extended surface and an nm-sized cluster. The main goal of the present paper is to model the particle size e7ects on the oscillatory behaviour of CO oxidation reaction over zeolite supported Pd catalysts. Reaction rate oscillations that occur during the oxidation of CO over Pd catalysts have been examined in great detail both over single crystal surfaces under UHV conditions (Basset & Imbihl, 1990) and over supported catalysts at atmospheric pressure (BBocker & Wicke, 1985; Jaeger, MBoller, & Plath, 1986; Slinko, Jaeger, & Svensson, 1989; ∗

Corresponding author. E-mail addresses: [email protected] (N. V. Peskov), [email protected] (M. M. Slinko), [email protected] (N. I. Jaeger). 0009-2509/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2002.12.003

Jaeger, Liauw, & Plath, 1996). Recently, we reported on the e7ect of the particle size upon the dynamic behaviour of the oxidation of CO over zeolite-supported palladium catalysts (Slin’ko, Ukharskii, Peskov, & Jaeger, 2001a, b). The stochastic mathematical model was developed, demonstrating that this e7ect might be related to the in9uence of intrinsic 9uctuations upon the dynamic behaviour due to the small number of reactant molecules involved in the case of very small particles in comparison with an extended catalytic surface (Peskov, Slinko, & Jaeger, 2002). The stochastic model was based on the model for the CO oxidation over a Pd (1 1 0) single crystal face that considered lateral interactions on the surface (Hartmann, Krischer, & Imbihl, 1994). However, it is well known, that the morphology of an nm-sized particle includes various single crystal planes and the role of edges and corners is essential. It is very diHcult to simulate lateral interactions on such a complex surface. To avoid this problem, we propose here a new model based on a modi8ed STM model originally proposed for the description of oscillatory behaviour of CO oxidation over polycrystalline Pt, Pd and Ir catalysts and which did not consider lateral interactions (Sales, Turner, & Maple, 1982). It will be shown that the developed model can reproduce characteristic details of

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the observed e7ects of the particle size on the dynamic behaviour of the reaction system.

0.28 0.24

2. Experimental methods and observations The e7ect of the size of the palladium crystallites on the activity and on the dynamic behaviour of the catalytic reaction has been studied under shallow bed conditions in a continuous stirred tank reactor. The activity and the dynamic behaviour of the system have been analysed under similar experimental conditions for pre-oxidised catalysts with the same Pd loading, of 0.05%, the same surface area of 15:1 cm2 but di7erent size of the Pd particles. For catalyst A the size of Pd particles was equal to 10 nm, while for catalyst B it was equal to 4 nm. The details of experimental procedure and preparation of catalysts can be found in Slin’ko et al. (2001a, b). The catalytic activity of Pd zeolite catalysts as well as their dynamic behaviour greatly depends upon pre-treatment and particle size. Fig. 1 depicts the comparison of the catalytic activity of catalysts A and B at 478 K and 0.5% CO inlet concentration (CO partial pressure, PCO = 500 Pa) following various pre-treatments. The dynamic behaviour of catalysts A and B are shown after they were oxidised for 12 h in 9owing synthetic air at 633 K (oxidised catalysts) and after treatment with a reactant mixture containing 1% CO in 0.45

4 nm, reduced catalyst

0.40

4nm, oxidised catalyst 0.35 0.30

CO2 concentration, vol.%

0.25 0.20 0.15 0.10 1000

1200

0.20

1400

1600

1800

2000

10 nm, reduced catalyst

0.15

0.10

10 nm, oxidised catalyst 0.05

0.00 0

500

1000

1500

2000

2500

3000

3500

Time, s

Fig. 1. Comparison of catalytic activity of catalysts with 4 nm (catalyst B) and 10 nm-sized Pd particles (catalyst A) at 478 K and 0.5% CO inlet concentration (PCO = 500 Pa) following various pre-treatment conditions.

CO2 Concentration, vol.%

0.20 0.16 2000

2100

2200

2300

2400

2500

3100

3200

3300

3400

3500

12100

12200

12300

12400

12500

0.28 0.24 0.20 0.16 3000 0.28 0.24 0.20 0.16 12000

Time, s Fig. 2. Evolution of regular oscillations in the case of catalyst A with time on stream at 503 K and 0.3% CO inlet concentration (PCO = 300 Pa).

synthetic air at 478 K (reduced catalysts). Fig. 1 shows that catalyst B with 4 nm particles displays a much higher activity. Moreover, under the chosen experimental conditions oscillatory behaviour can only be observed for pre-oxidised catalysts, while no oscillations could be detected in the case of the reduced catalyst. Catalyst A with 10 nm particles shows under the same experimental conditions a contrasting behaviour, i.e., only the reduced catalyst produces oscillations, while the reaction proceeds in a steady state over the pre-oxidised catalyst. It can also be observed that there is a much smaller di7erence between the activity of the reduced and the oxidised catalyst B in comparison with catalyst A. At higher temperatures oscillatory behaviour over catalyst A is observed over both the oxidised and reduced catalyst, while catalyst B generates oscillations only after preliminary oxidation. The di7erence is that the region of oscillations for catalyst B with smaller Pd particles is larger compared to catalyst A. The other main di7erence is that only chaotic oscillations could be observed over catalyst B, while in the case of the pre-oxidised catalyst A regular oscillations were found to develop in a narrow range of CO inlet concentrations between 0.3 and 0.32 vol% (300 ¡ PCO ¡ 320 Pa) at 503 K. Fig. 2 shows for catalyst A sections from a 8 h time series at 503 K and 0.3% CO inlet concentration (PCO = 300 Pa). It can be seen how amplitude, period and activity increase with time on stream, while the waveform slowly changes with the system spending more time in the high activity state. Initially, the pre-oxidised catalyst A displayed very low activity and it took some time for the appearance of oscillations. The observed changes are attributed to the establishment of a more reduced state of the catalyst. Another di7erence between catalysts A and B can be observed in the transient periods during which a stationary

N. V. Peskov et al. / Chemical Engineering Science 58 (2003) 4797 – 4803

oscillatory state is established over the pre-oxidised catalysts. A slow increase of the reaction rate with time can be assigned to a slow reduction of the catalyst and the size of the Pd particles was found to be of a signi8cant in9uence. The transient period is shorter in the case of 4 nm particles as compared to 10 nm particles.

4799

0.5

0.4

w

0.3

0.2

ws, 10 nm Oscillations

3. Mathematical model 0.1

ws , 4 nm

3.1. Point model for the reaction on a catalyst particle 0.0

The oxidation of CO over Pd catalysts has been studied under normal and low pressures conditions (Turner, Sales, & Maple, 1981a, b; Engel & Ertl, 1978). In an excess of oxygen the reaction proceeds via a Langmuir–Hinshelwood (LH) mechanism, including the following steps: k

CO + M ⇔1 MCO; k−1

400

600

800

PCO [Pa]

Fig. 3. The region of oscillations in model (2) with the values of parameters given in the text. The lines ws are the steady state fractions of bulk oxide for 4 and 10 nm-sized particles calculated from Eq. (3).

(1a) by the system of equations:

k2

O2 + 2M⇒2MO; k

3 MCO + MO⇒CO 2 + 2M;

(1b)

x˙ = PCO k1 (1 − x − y) − k−1 x − k3 (1 − w)2 xy − k5 xz;

(1c)

y˙ = PO2 k2 (1 − x − y)2 (1 − w − z)2 − k3 (1 − w)2 xy

where M denotes vacant sites on the metal surface, MCO and MO adsorbed CO molecules and oxygen atoms, respectively. The authors of the STM model assumed that a slow formation and removal of subsurface oxygen is responsible for the oscillatory behaviour and two additional steps were added to the LH mechanism (Sales, Turner, & Maple, 1982): k

4 MO + Mv ⇒M v O + M;

k

5 Mv O + MCO⇒CO 2 + Mv + M;

(1d) (1e)

where Mv O and Mv indicate subsurface oxygen atoms and free sites in the subsurface layer. The experimental results presented in Section 2 show that the dynamic behaviour of the experimentally observed oscillations depend greatly upon the pre-treatment of the catalyst and on the degree of oxidation of Pd particles. To consider the e7ects of Pd bulk oxidation we introduce in the model a new parameter w; 0 6 w 6 1, denoting the fraction of the bulk Pd oxide. The mathematical model is developed based on the following assumptions: (i) Subsurface oxygen and bulk oxide have no e7ect on CO adsorption and desorption. (ii) O2 can adsorb only on a pair of neighbouring free Pd surface sites with no bulk oxide and subsurface oxygen underneath. (iii) The surface reaction between chemisorbed CO and O can occur only over non-oxidised Pd. (iv) The limiting oxygen capacity of the subsurface layer is assumed to be 1 − w. The variation of surface coverage with CO (x), the surface coverage with O (y), the concentration of the subsurface oxygen Mv O (z) at 8xed value of the fraction of the bulk oxide (w) can be described in the mean 8eld approximation

− k4 y(1 − w − z); z˙ = k4 y(1 − w − z) − k5 xz:

(2)

The variables have a physical meaning as long as all of them are not negative and x + y; z 6 1. Fig. 3 shows the oscillatory region of system (2) in the plane (PCO ; w), obtained at PO2 = 13 332 Pa and with the following set of rate constants: k1 = 2:25;

k−1 = 100;

k2 = 0:75;

k3 = 4000;

k4 = 0:02;

k5 = 0:019:

(The dimension of k1 and k2 is (Pa s)−1 , the others k have the dimension s−1 .) At the border of the region the oscillations appear via the Andronov–Hopf bifurcation. One can see that with increasing PCO the oscillations appear 8rst on the oxidised particles at w = 0:4 and are absent on the fully reduced catalyst particle. This is in agreement with the experimental observation that at 503 K the catalyst had to be oxidised 8rst in order to obtain reaction rate oscillations. The experimental data, presented in Section 2 demonstrate the slow variation of the catalyst activity due to the process of the slow reduction of the catalyst. To simulate the slow evolution of the degree of the bulk oxidation w, we propose a simple heuristic model of the bulk Pd oxidation and reduction based on the following assumptions: (i) The process of oxidation involves the entire bulk of a particle and the rate of oxidation is proportional to the oxygen pressure PO2 . (ii) The process of reduction can occur only on the surface of the particle due to a negligible CO diffusion into Pd. The rate of this process is supposed to be proportional to PCO and to a dimensionless parameter s that

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3.2. Stochastic model of the reaction on a catalyst particle

300 0.4

0.2

Reaction rate

100

w

200

0.0

0 1000

1500

2000

300 0.4 200 0.2

100

0.0

0 2000

2500

3000

Time [s]

Fig. 4. Time series of the reaction rate R (solid line, left scale) and concentration of the bulk oxide w (dashed line, right scale) obtained by solving model (2)–(3) starting with the initial value of w0 = 1 for a 10 nm particle at PCO = 400 Pa.

means the surface/volume ratio. For simplicity, we de8ne the numerical value for s as 1=rp , where rp is the radius of the Pd particle, that is, s = 0:5 for a 4 nm-sized particle and s = 0:2 for a 10 nm-sized particle. Therefore, the evolution of w can be described by the linear equation w˙ = PO2 k6 (1 − w) − k−6 PCO sw

(3)

which indicates an exponential approach of w towards the steady state value ws =(1+ps)−1 , where p=PCO =PO2 , and  = k−6 =k6 . The graphs of ws (PCO ) calculated with  = 500 are also shown in Fig. 3. The chaotic nature of the oscillatory behaviour presented in Section 2 and identi8ed in Slin’ko et al. (2001a, b), is apparently the result of interaction between many Pd particles via the di7usion of reagents in the pores of the zeolite that cannot be reproduced by our point model, describing the process only on a single Pd particle. In addition, as it will be shown below, statistical 9uctuations play an important role in the reaction dynamics on the surface of an nm-sized particles. Therefore, in this stage of the mathematical modelling our goal is to simulate only qualitatively the main features of the experimental observations. The values of parameters were taken from the single-crystal data and were 8tted to simulate the reaction rate and the periods of the oscillations. Fig. 4 shows the development of the reaction rate oscillations in time on a 10 nm-sized particle at PCO = 400 Pa obtained by model (2)–(3) at k−6 = 1:5 × 10−5 (Pa s)−1 , k6 = 3:0 × 10−8 (Pa s)−1 , and with the initial value for w; w0 = 1. It can be seen that the general picture of changes in the waveform and the period of the oscillations is qualitatively similar to the experimental data presented in Fig. 2. As in the experiment the reduction of the preliminary oxidised catalyst slowly proceeds in time. This process corresponds to a decrease of w and leads to the increase of the average reaction rate and the transformation of the waveform of the reaction rate oscillations from “spike up” to “spike down”.

Assuming that the number of adsorption centres on the particle surface is equal to the number of surface atoms, this number varies for regular octahedra from Ns ∼ = 400 to 3000 while the particle size grows from 4 to 10 nm. In the oscillatory regime of the reaction the numbers of adsorbed CO molecules and O atoms, NCO and NO , and the number of subsurface oxygen atoms NO∗ , can vary in the range from 0 to Ns . At any moment of time these numbers can be regarded as a random values with unknown probability distribution P(NCO ; NO ; NO∗ ; t). At 8nite and not too large values of Ns statistical 9uctuations of random numbers of atoms can play a signi8cant role and Eqs. (2) may become incorrect. In order to take into account a 8nite number of atoms participating in the reaction and their statistical 9uctuations as well as their mutual correlations the following Markovian model is proposed. The model has three discrete variables NCO , NO , and NO∗ with restrictions, NCO + NO 6 Ns ; NO∗ 6 (1 − w)Ns . The acceptable transitions in the Markovian model correspond to the elementary steps in the reaction scheme: CO adsorption–desorption : NCO → NCO ± 1;

(1a)

O2 adsorption : NO → NO + 2;

(1b)

CO + O reaction : (NCO ; NO ) → (NCO − 1; NO − 1);

(1c)

O → O∗ transition : (NO ; NO∗ ) → (NO − 1; NO∗ + 1); (1d) CO + O∗ reaction : (NCO ; NO∗ ) → (NCO − 1; NO∗ − 1): (1e) The CO oxidation on the catalyst particle is simulated as a random sequence of acceptable transitions under standard conditions, which implies that multiple transitions are forbidden; any transition can occur at an arbitrary moment of time (Markovian model with continuous time) and is instantaneous. Each transition has its own probability that depends only upon the current state of the model (Markovian property). Instead of transition probability for each event we de8ne the transition probability per unit time, or transition rate , and then the probability of transition during the suHciently short time interval St will be equal to St. For compatibility with model (2) the transition rates are de8ned as follows: 1 (N) = k1 PCO (Ns − NCO − NO ); −1 (N) = k−1 NCO ;
2 1 NO ∗ 2 (N) = k2 PO2 1 − ws − (Ns − NCO − NO ) Ns Ns ×(Ns − NCO − (NO + 1));

N. V. Peskov et al. / Chemical Engineering Science 58 (2003) 4797 – 4803

5 (N) = k5

NCO NO∗ ; Ns

(4)

where N is the triple {NCO ; NO ; NO∗ }, which de8nes the current state of the model. Under the conditions described the time evolution of the state probability distribution P(N; t) will be ruled by the master equation dP(N; t) =− dt

5 

i (N)P(N; t)

i=−1

+

5 

i (N(i) )P(N(i) ; t);

(5)

i=−1

where N(i) denotes the state, from which the state N can be reached via one ith transition. Eq. (5) is in fact a large system, the dimension of which is equal to the number of all di7erent states of the Markovian model. Although this system could be solved numerically we shall not do it here, but instead generate the trajectories of the Markovian process with the help of the Monte Carlo algorithm. The evolution equations for the average values of the variables can be derived from Eq. (5). For example, the equations for x(t) = NCO =Ns have the form T 2 xy − k5 xz x˙ = k1 PCO (1 − x − y) − k−1 x − k3 (1 − w) −

1 [k3 (1 − w) T 2 (NCO − NCO )(NO − NO ) Ns2

− k5 (NCO − NCO )(NO∗ − NO∗ )]; where the angular brackets a denote the mathematical expectation of a random variable a. The equations for the average values of the stochastic model consist of all the terms of the deterministic equations (2) and additional correlation terms. In the limit Ns → ∞ the correlation terms are assumed to vanish, but at 8nite Ns they can play an essential role in the dynamics of the system. Thus the stochastic model could be regarded as an expansion of the corresponding point model for the 8nite surface of a small particle. The trajectories of the Markovian process governed by Eq. (5) are generated with the help of one of the Monte Carlo algorithms with continuous time described by Gillespie (1992). Let t  be the time of sequential change in the state of the system. The next time t  = t  + St of the state change is determined as follows. The interval St between two successive events (the expectation time) is considered to be a random number with the Poisson probability distribution (t  ) exp(−(t  )), where (t  ) = 1 (t  ) + −1 (t  ) + 2 (t  ) + 3 (t  ) + 4 (t  ) + 5 (t  ) is

calculated for the state held in the system between t  and t  . Therefore, St is de8ned by the formula St = −ln(t )=(t  ), where t is the standard random number uniformly distributed in the interval (0; 1). The event, which occurs at time t  , is determined with the help of the other standard random number e uniformly distributed in the interval (0; (t  )). If e ¡ 1 (t  ) then the CO adsorption has to occur, else if e − 1 (t  ) ¡ −1 (t  ) then the CO desorption has to occur, and so on. 3.3. Results of stochastic simulations In this section, we present the results of the stochastic simulation of the reaction on 4 and 10 nm-sized particles. The set of parameter values de8ned for the point model (2) was used in the simulations. In addition it was supposed that w = ws . The results are presented in the form of time series of the reaction rate per one surface site. More exactly, each reading of the reaction rate time series is R(tk ) = (r3; k + r5; k )=(Ns St), where St = tk − tk−1 = 0:1 s, r3; k and r5; k are the numbers of elementary events of reaction CO + O and CO + O∗ , that occur during the time St, respectively. 4 nm-sized particle: The oscillations are observed in the interval 200 ¡ PCO ¡ 866 Pa, while in the point model the interval of oscillations is 464 6 PCO 6 736 Pa. The significant enlargement of oscillatory interval compared to the mean-8eld model is due to noise-induced oscillations, which arise in the outer vicinities of the Andronov–Hopf bifurcation points. Fig. 5 shows the examples of noise-induced and kinetic oscillations in the reaction rate over a 4 nm particle. At PCO =303 Pa the noise-induced oscillations look like fast transitions from a high active state to a low active state that occur at random moments of time, while at PCO = 733 Pa the system spends more time in low activity state and occasionally performs short jumps into the high activity state. Kinetic oscillations in the middle of the oscillatory interval of the point model (2) at PCO = 600 Pa are strongly corrupted by statistical 9uctuations and this fact might be a

(a)

PCO = 330 Pa

(b)

PCO = 600 Pa

(c)

PCO = 730 Pa

400 200 0

Reaction Rate

NCO NO ; Ns
 NO ∗ NO ; 4 (N) = k4 1 − ws − Ns

3 (N) = k3 (1 − ws )2

4801

400 200 0 400 200 0 0

100

200

300

400

500

Time [s]

Fig. 5. Time series of noise-induced (a), (c) and kinetic (b) reaction rate oscillations produced by the stochastic model for a 4 nm particle.

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N. V. Peskov et al. / Chemical Engineering Science 58 (2003) 4797 – 4803 (a)

PCO = 330 Pa

(b)

PCO = 470 Pa

(c)

PCO = 625 Pa

400 200

Reaction Rate

0 400 200 0 400 200 0 0

100

200

300

400

500

Time [s]

Fig. 6. Time series of noise-induced (a), (c) and kinetic (b) reaction rate oscillations produced by the stochastic model for a 10 nm particle. The dashed line in the panel (b) shows the oscillations produced by the point model under the same conditions.

reason, why regular oscillations have not been observed on the catalysts containing 4 nm particles. 10 nm-sized particle: The oscillations in the stochastic model are observed in the interval of about 266 ¡ PCO ¡ 665 Pa while in the point model (2) the oscillatory region is 368 6 PCO 6 630 Pa. The shrinking of the interval of the CO pressure where oscillations occur (in comparison with 4 nm particle) is in agreement with the experimental data. The examples of noise-induced (a, c) and kinetic (b) stochastic oscillations are shown in Fig. 6. While the pressure PCO increases through the oscillatory region the shape of the oscillation changes similar to the previous case of the 4 nm particle, but the oscillations reveal a more regular character and their amplitude is smaller. The kinetic oscillations on separated particles could probably be smoothed via the gas-phase di7usion yielding regular oscillations of the global reaction rate. For comparison purposes, the reaction rate oscillations obtained from the point model for 467 Pa are shown as a dashed line in Fig. 6. It can be seen that the stochastic and the deterministic oscillations have similar waveforms, but the formers have a shorter (average) period than the latter. During the presented time interval there are about 20 stochastic cycles and about 16 deterministic cycles. 4. Discussions and conclusions The paper presents two models designed to simulate the particle size e7ect that has been observed during the study of the CO oxidation over zeolite-supported Pd catalysts. One of the most interesting problems was to understand the possible reason for the experimental observation that catalyst B with 4 nm particles produced only chaotic oscillations, while catalyst A with 10 nm Pd particles can generate regular oscillations under appropriate experimental conditions. In Section 3.1 a new deterministic point model in mean 8eld approximation was developed to simulate the reaction rate oscillations observed during the CO oxidation in dependence

on the size of the Pd particles. Many experimental trends could be successfully modelled, e.g., the dependence of the catalytic activity and the waveform of the oscillations upon the particle size and the pre-treatment of the catalyst. The higher activity of the smaller particles can be explained by the attainment of a more reduced state of the Pd in smaller particles in the course of the reaction. Such a situation occurs, because the rate of the bulk reduction of Pd was assumed to be inversely proportional to the particle size. In this case, the smaller particles reach a less oxidized state during the reaction resulting in a higher activity. The stochastic model developed in this paper reveals a signi8cant role of internal 9uctuations during heterogeneous catalytic reactions on an nm-sized supported metal particles. The results of the simulations with the stochastic model shown in Figs. 5 and 6 demonstrate the large di7erence in the oscillatory behaviour of the reaction rate for particles of various sizes. For the same parameters and experimental conditions the region of oscillations for 4 nm size particles is much larger than for 10 nm particles due to the much larger e7ect of internal 9uctuations upon the small particles. The comparison of oscillations for 4 and 10 nm particles demonstrates, that 4 nm-sized particles produce more complex and irregular oscillations in a larger region of CO partial pressures compared to 10 nm particles in accordance with the experimental data. The cause of the drastic increase of the oscillatory region for the catalyst with the smallest particles is connected to the presence of noise-induced oscillations, i.e. oscillations that can be produced only by the stochastic model and that are absent in the deterministic limit. Recently, Reichert, Starke, and Eiswirth (2001) studied the role of internal 9uctuations on the dynamic behaviour of CO oxidation over low-index single-crystal Pt surfaces via a stochastic model based on the same frame as our model. It was demonstrated that internal 9uctuations became essential only for very small surface cells, such as 8eld emitter tips with 20×200 A area (40 nm2 ). For larger single-crystal surfaces the internal 9uctuations are averaged out and do not e7ect the dynamic behaviour of the system. The surface area of 10 nm particles is about 300 nm2 but still it could be shown that 9uctuations can strongly a7ect the reaction dynamics. So one could conclude that the e7ect of 9uctuations depends also upon the type of the model and the values of parameters. Finally, we note that the stochastic model developed in this paper is able to simulate the oscillatory behaviour only on one catalyst particle. The mathematical modelling of the collective behaviour of an array, consisting of many such particles will be the subject of future work. Acknowledgements The authors acknowledge the 8nancial support from the Russian Fund of Fundamental Researches (grant 00-03-32125) and INTAS (grant 99-1882).

N. V. Peskov et al. / Chemical Engineering Science 58 (2003) 4797 – 4803

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