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Cellular automaton approach to the NOCO reaction on a catalytic surface
21 July 1995
ELSEVIER
CHEMICAL PHYSICS LETTERS Chemical Physics Letters 241 (1995) 185-188
Cellular automaton approach to the N O - C O reaction on a catalytic surface Hui-yun Pan a, Hai Jun Wang
b
a Department of Chemistry, Zhengzhou University, Zhengzhou 450052, People's Republic of China b Institute of Solid State Physics, Sichuan Normal University, Chengdu 610066, People's Republic of China
Received 31 March 1995; in final form 23 May 1995
Abstract The NO-CO reaction on a catalytic surface is studied with the aid of the cellular automaton model of Chopard and Droz. Proper stoichiometry is taken into account in an appropriate way. A mean-field analysis of this model is given, with the results indicating that a first-order kinetic phase transition from the CO * - N * poisoned phase to the reactive phase always occurs, so that a steady reactive phase necessarily exists, independent of the type of lattice. The conclusion that for the NO-CO surface reaction the square lattice cannot support a steady reactive phase seems to be a finite size effect.
1. Introduction The N O - C O reaction on a catalytic surface is important to the elimination of poisonous gases emitted from automobiles. In recent years, Yaldram and Khan [1,2], also Brosilow and Ziff [3], made a Monte Carlo study o f this reaction on a single-crystal catalyst. They carried out numerical simulations on square and hexagonal lattices, and show that, for the case of a hexagonal lattice, a steady reactive state is achieved and this state exists between two critical concentrations of CO (or NO), while for the square lattice, no steady reactive state occurs. Thus they conclude that the type o f lattice is crucial to the evolution of the system towards a steady reactive state. In this Letter, we give a mean-field analysis of this model with the results indicating that for the N O - C O reaction on a catalytic surface a steady reactive state always exists, independent of the type of lattice. The conclusion that the square lattice
cannot support a reactive state seems to be a finite size effect.
2. The model The lattice model of a surface reaction on a catalyst was first introduced by Ziff, Gulari and Barshad to describe the oxidation of carbon monoxide on a catalytic surface [4]. In their model there are two elementary steps: adsorption and reaction. For the N O - C O reaction considered the two steps are: a CO monomer (A) strikes a lattice site with probability y and is adsorbed to form CO* if the site is empty, while a NO dimer (DB) strikes a site with probability 1 - y and is adsorbed to form an O* ( B * ) and a N * ( D * ) if the selected site and a randomly chosen nearest neighbour are both empty. A * reacts with a neighbouring B * to form a CO 2 molecule which desorbs immediately leaving behind
0009-2614/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0009-2614(95)00618-4
H. Pan, J. Wang / Chemical Physics Letters 241 (1995) 185-188
186
two empty sites, while two nearest D * react to form a N 2 molecule and desorb: The ZGB model of the NO-CO reaction may be formulated as follows: A(g) + * ~ A * ,
with probability y,
(la) 3. Mean-field analysis
DB(g) + 2 * --*D* + B * , with probability 1 - y,
(lb)
A* + B * ---)ABq" + 2 . ,
for nearest neighbours, (lc)
D* + D * ---) D2 ~" + 2 * ,
for nearest neighbours, (10)
where * represents an empty site, 2 . represents a nearest * pair, and A * , B* and D* represent adsorbed A, B and D, respectively. For simplicity of mathematical treatment and the possibility of following the dynamic evolution of the microscopic degrees of freedom, the cellular automaton (CA) version [5] of the ZGB model is preferred. In the CA model the time evolution of site states is given by the following rules: A~V,
ifMNNB,
A ~ A,
otherwise,
B~V,
ifMNNA,
B ~ B,
otherwise,
D~V,
ifMNND,
D ~ D,
otherwise,
V ~ A,
with probability y,
V ~ C,
otherwise,
C ~ B or D with equal probabilities, C ~ V,
otherwise,
[5,6], and it will surely yield satisfactory results for the NO-CO surface reaction as well.
As is well known, mean-field analyses [6,7] are useful to the study of kinetic phase transitions in a surface reaction model, even a one-site approximation may provide valuable results, at least in the case of the first-order phase transition. In this approximation the surface coverages 0A, 0B, 0D, 0 C and 0 v satisfy the following equations: d0A dt
1 = YOv - --0A0 B, n
(3a)
d0 a 1 1 1 . . . . 0~ - ~ 0 A 0 B, dt 2 n
(3b)
d0 D 1 1 1 . . . . 0~- -0g, dt 2 n n
(3c)
(2a)
d0 c d--~ = (1 - y ) O v - 0c,
(30)
(2b)
0v = 1 -0 A - 0a - 0 D - 0c,
(3e)
(2c)
(2d) if MNN C, (2e)
where V represents an empty state and the state C describes the conditional occupation of the site by an atom B or D according to Chopard and Droz [5], a MNN pair denotes a matching nearest-neighbour pair, and was first introduced by Ziff et al. [6] to take into account strict stoichiometry in BD-adsorption, ABreaction and D - D combination. The time evolution rules (2a)-(2e) fix the state of a site at time t + 1 according to the states of this site and its nearest neighbours at time t. The CA model works successfully for the CO-O 2 reaction on a catalytic surface
where n is the coordination number of the lattice (n = 4 for a square lattice, n = 6 for a hexagonal lattice and so forth). The factors of ~1 in the first terms on the right-hand side of Eqs. (3b) and (3c) are due to the rule (2e): C evolves to B or D with equal probabilities for MNN C; Eq. (3e) means that the sum of coverages of different species (including C and V) is unity. These equations represent the change of the system in one cycle of the CA, and they may be readily derived on the basis of the time evolution rules (2a)-(2e). In Eqs. (3a)-(3e), we set dOc/dt equal to zero and obtain the steady state solutions, 2ny 0 c = -1- -, y
(4a)
2ny 0v = (1 _ y ) 2 ,
(4b)
H. Pan, J. Wang/Chemical Physics Letters 241 (1995) 185-188
v~-ny 0D
1-y
,
(4c)
O.,A=b+_(b2-O~) 1/2, b = ½(1 - 0 D - 0c - 0 v ) ,
(4d)
where 0 a / 0 A corresponds to the positive/negative root, respectively. These solutions are valid only if b>~ 0D, or _1(1
v/2 ny
2ny
2
1 -y
1 -y
2ny )>~--,~/2ny (1 _ y ) 2
1 -y
that is, only if y satisfies the following equality or inequality: (1 + 2n + 3v~-
n)y e-
(2 + 4n + 31/2
+ 1/> 0.
n)y (5)
Solving the above equality, we get Y = Ys = { (2 + an + 31/2- n) - [ ( 3 4 + 24v~-)n2 + 8n] 1/2} × [ 2 ( 1 + 2 n + 3v/2- n)] -1 ,
(6)
where only the negative root is retained, as y must take a value less than unity. It can be easily shown that for y < Ys, the above inequality is satisfied for arbitrary coordination number n, and so y~ is the spinodal point. The existence of a spinodal point for arbitrary n means that a first-order kinetic phase transition occurs at a certain value of y somewhat smaller than Ys, independent of the type of lattice. At this point some numerical results may be helpful to understand how the value of ys changes with the change of n; Eq. (6) yields different y~ for different n: with no match requirement, n - 1, y~ = 0.1055; for a honeycomb lattice, n = 3, y~ = 0.0385; for a square lattice, n = 4, ys = 0.0292; for a hexagonal lattice, n = 6, y~ = 0.0197. A comparison between the above results shows that y~ decreases with an increase in n; this is due to the fact that as n increases the probability of the occurrence of a matching nearest-neighbour C pair decreases, and as a consequence the adsorption probability of NO dimer decreases, hence y~ decreases as well. On the basis of the above analysis one may draw the following conclusion: for the NO-CO reaction
187
on a catalytic surface, a steady reactive phase always exists, independent of the type of lattice; in other words, the steady reactive phase is not an artifact of the type of lattice, at least for the case of the NO-CO surface reaction considered. The suggestion that the square lattice cannot support such a reactive phase seems to be a finite size effect.
4. Remarks In conclusion some general remarks may be made: (1) The cellular automaton model of a surface reaction on a catalyst is easily adaptable to numerical simulations as well as mathematical analyses, besides its intrinsic capability to follow the dynamic evolution of microscopic degrees of freedom of the reaction system and its merit of taking into account local fluctuations and spatial correlations of the surface reaction species, as other lattice models. (2) For lattice models much can be learned from mean-field analyses even in the one-site approximation, especially the problem of whether a steady reactive phase exists or not for a concrete reaction may be resolved in this approximation. Once the existence of the spinodal point is established, the first-order kinetic phase transition from a poisoned phase to the reactive phase necessarily occurs. (3) In this Letter we include a variable coordination number in the mean-field analyses to facilitate the study of the influence of the type of lattice on the existence of a steady reactive phase; for the NO-CO reaction no such influence has been found. (4) For a certain coordination number n (e.g. n = 6) details of the reactive phase of the NO-CO system may be obtained from the set of equations derived in this Letter, among others, Eqs. (4a)-(4d) yield the coverages of various species as a function of y, qualitatively in agreement with simulation results.
Acknowledgement We thank the referee for valuable comments. We also acknowledge support from the National Natural Science Foundation of China and the Natural Science Research Foundation of Henan Province.
188
H. Pan, J. Wang / Chemical Physics Letters 241 (1995) 185-188
References [1] K. Yaldram and M.A. Khan, J. Catal. 131 (1991) 369. [2] K. Yaldram and M.A. Khan, J. Catal. 136 (1992) 279. [3] B.J. Brosilow and R.K. Ziff, J. Catal. 136 (1992) 275.
[4] R.M. Ziff, E. Gulari and Y. Barshad, Phys. Rev. Letters 56 (1986) 2553. [5] B. Chopard and M. Droz, J. Phys. A 21 (1988) 205. [6] R.M. Ziff, K. Fichthorn and E. Gulari, J. Phys. A 24 (1991) 3727. [7] R. Dickman, Phys. Rev. A 34 (1986) 4246.
ELSEVIER
CHEMICAL PHYSICS LETTERS Chemical Physics Letters 241 (1995) 185-188
Cellular automaton approach to the N O - C O reaction on a catalytic surface Hui-yun Pan a, Hai Jun Wang
b
a Department of Chemistry, Zhengzhou University, Zhengzhou 450052, People's Republic of China b Institute of Solid State Physics, Sichuan Normal University, Chengdu 610066, People's Republic of China
Received 31 March 1995; in final form 23 May 1995
Abstract The NO-CO reaction on a catalytic surface is studied with the aid of the cellular automaton model of Chopard and Droz. Proper stoichiometry is taken into account in an appropriate way. A mean-field analysis of this model is given, with the results indicating that a first-order kinetic phase transition from the CO * - N * poisoned phase to the reactive phase always occurs, so that a steady reactive phase necessarily exists, independent of the type of lattice. The conclusion that for the NO-CO surface reaction the square lattice cannot support a steady reactive phase seems to be a finite size effect.
1. Introduction The N O - C O reaction on a catalytic surface is important to the elimination of poisonous gases emitted from automobiles. In recent years, Yaldram and Khan [1,2], also Brosilow and Ziff [3], made a Monte Carlo study o f this reaction on a single-crystal catalyst. They carried out numerical simulations on square and hexagonal lattices, and show that, for the case of a hexagonal lattice, a steady reactive state is achieved and this state exists between two critical concentrations of CO (or NO), while for the square lattice, no steady reactive state occurs. Thus they conclude that the type o f lattice is crucial to the evolution of the system towards a steady reactive state. In this Letter, we give a mean-field analysis of this model with the results indicating that for the N O - C O reaction on a catalytic surface a steady reactive state always exists, independent of the type of lattice. The conclusion that the square lattice
cannot support a reactive state seems to be a finite size effect.
2. The model The lattice model of a surface reaction on a catalyst was first introduced by Ziff, Gulari and Barshad to describe the oxidation of carbon monoxide on a catalytic surface [4]. In their model there are two elementary steps: adsorption and reaction. For the N O - C O reaction considered the two steps are: a CO monomer (A) strikes a lattice site with probability y and is adsorbed to form CO* if the site is empty, while a NO dimer (DB) strikes a site with probability 1 - y and is adsorbed to form an O* ( B * ) and a N * ( D * ) if the selected site and a randomly chosen nearest neighbour are both empty. A * reacts with a neighbouring B * to form a CO 2 molecule which desorbs immediately leaving behind
0009-2614/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0009-2614(95)00618-4
H. Pan, J. Wang / Chemical Physics Letters 241 (1995) 185-188
186
two empty sites, while two nearest D * react to form a N 2 molecule and desorb: The ZGB model of the NO-CO reaction may be formulated as follows: A(g) + * ~ A * ,
with probability y,
(la) 3. Mean-field analysis
DB(g) + 2 * --*D* + B * , with probability 1 - y,
(lb)
A* + B * ---)ABq" + 2 . ,
for nearest neighbours, (lc)
D* + D * ---) D2 ~" + 2 * ,
for nearest neighbours, (10)
where * represents an empty site, 2 . represents a nearest * pair, and A * , B* and D* represent adsorbed A, B and D, respectively. For simplicity of mathematical treatment and the possibility of following the dynamic evolution of the microscopic degrees of freedom, the cellular automaton (CA) version [5] of the ZGB model is preferred. In the CA model the time evolution of site states is given by the following rules: A~V,
ifMNNB,
A ~ A,
otherwise,
B~V,
ifMNNA,
B ~ B,
otherwise,
D~V,
ifMNND,
D ~ D,
otherwise,
V ~ A,
with probability y,
V ~ C,
otherwise,
C ~ B or D with equal probabilities, C ~ V,
otherwise,
[5,6], and it will surely yield satisfactory results for the NO-CO surface reaction as well.
As is well known, mean-field analyses [6,7] are useful to the study of kinetic phase transitions in a surface reaction model, even a one-site approximation may provide valuable results, at least in the case of the first-order phase transition. In this approximation the surface coverages 0A, 0B, 0D, 0 C and 0 v satisfy the following equations: d0A dt
1 = YOv - --0A0 B, n
(3a)
d0 a 1 1 1 . . . . 0~ - ~ 0 A 0 B, dt 2 n
(3b)
d0 D 1 1 1 . . . . 0~- -0g, dt 2 n n
(3c)
(2a)
d0 c d--~ = (1 - y ) O v - 0c,
(30)
(2b)
0v = 1 -0 A - 0a - 0 D - 0c,
(3e)
(2c)
(2d) if MNN C, (2e)
where V represents an empty state and the state C describes the conditional occupation of the site by an atom B or D according to Chopard and Droz [5], a MNN pair denotes a matching nearest-neighbour pair, and was first introduced by Ziff et al. [6] to take into account strict stoichiometry in BD-adsorption, ABreaction and D - D combination. The time evolution rules (2a)-(2e) fix the state of a site at time t + 1 according to the states of this site and its nearest neighbours at time t. The CA model works successfully for the CO-O 2 reaction on a catalytic surface
where n is the coordination number of the lattice (n = 4 for a square lattice, n = 6 for a hexagonal lattice and so forth). The factors of ~1 in the first terms on the right-hand side of Eqs. (3b) and (3c) are due to the rule (2e): C evolves to B or D with equal probabilities for MNN C; Eq. (3e) means that the sum of coverages of different species (including C and V) is unity. These equations represent the change of the system in one cycle of the CA, and they may be readily derived on the basis of the time evolution rules (2a)-(2e). In Eqs. (3a)-(3e), we set dOc/dt equal to zero and obtain the steady state solutions, 2ny 0 c = -1- -, y
(4a)
2ny 0v = (1 _ y ) 2 ,
(4b)
H. Pan, J. Wang/Chemical Physics Letters 241 (1995) 185-188
v~-ny 0D
1-y
,
(4c)
O.,A=b+_(b2-O~) 1/2, b = ½(1 - 0 D - 0c - 0 v ) ,
(4d)
where 0 a / 0 A corresponds to the positive/negative root, respectively. These solutions are valid only if b>~ 0D, or _1(1
v/2 ny
2ny
2
1 -y
1 -y
2ny )>~--,~/2ny (1 _ y ) 2
1 -y
that is, only if y satisfies the following equality or inequality: (1 + 2n + 3v~-
n)y e-
(2 + 4n + 31/2
+ 1/> 0.
n)y (5)
Solving the above equality, we get Y = Ys = { (2 + an + 31/2- n) - [ ( 3 4 + 24v~-)n2 + 8n] 1/2} × [ 2 ( 1 + 2 n + 3v/2- n)] -1 ,
(6)
where only the negative root is retained, as y must take a value less than unity. It can be easily shown that for y < Ys, the above inequality is satisfied for arbitrary coordination number n, and so y~ is the spinodal point. The existence of a spinodal point for arbitrary n means that a first-order kinetic phase transition occurs at a certain value of y somewhat smaller than Ys, independent of the type of lattice. At this point some numerical results may be helpful to understand how the value of ys changes with the change of n; Eq. (6) yields different y~ for different n: with no match requirement, n - 1, y~ = 0.1055; for a honeycomb lattice, n = 3, y~ = 0.0385; for a square lattice, n = 4, ys = 0.0292; for a hexagonal lattice, n = 6, y~ = 0.0197. A comparison between the above results shows that y~ decreases with an increase in n; this is due to the fact that as n increases the probability of the occurrence of a matching nearest-neighbour C pair decreases, and as a consequence the adsorption probability of NO dimer decreases, hence y~ decreases as well. On the basis of the above analysis one may draw the following conclusion: for the NO-CO reaction
187
on a catalytic surface, a steady reactive phase always exists, independent of the type of lattice; in other words, the steady reactive phase is not an artifact of the type of lattice, at least for the case of the NO-CO surface reaction considered. The suggestion that the square lattice cannot support such a reactive phase seems to be a finite size effect.
4. Remarks In conclusion some general remarks may be made: (1) The cellular automaton model of a surface reaction on a catalyst is easily adaptable to numerical simulations as well as mathematical analyses, besides its intrinsic capability to follow the dynamic evolution of microscopic degrees of freedom of the reaction system and its merit of taking into account local fluctuations and spatial correlations of the surface reaction species, as other lattice models. (2) For lattice models much can be learned from mean-field analyses even in the one-site approximation, especially the problem of whether a steady reactive phase exists or not for a concrete reaction may be resolved in this approximation. Once the existence of the spinodal point is established, the first-order kinetic phase transition from a poisoned phase to the reactive phase necessarily occurs. (3) In this Letter we include a variable coordination number in the mean-field analyses to facilitate the study of the influence of the type of lattice on the existence of a steady reactive phase; for the NO-CO reaction no such influence has been found. (4) For a certain coordination number n (e.g. n = 6) details of the reactive phase of the NO-CO system may be obtained from the set of equations derived in this Letter, among others, Eqs. (4a)-(4d) yield the coverages of various species as a function of y, qualitatively in agreement with simulation results.
Acknowledgement We thank the referee for valuable comments. We also acknowledge support from the National Natural Science Foundation of China and the Natural Science Research Foundation of Henan Province.
188
H. Pan, J. Wang / Chemical Physics Letters 241 (1995) 185-188
References [1] K. Yaldram and M.A. Khan, J. Catal. 131 (1991) 369. [2] K. Yaldram and M.A. Khan, J. Catal. 136 (1992) 279. [3] B.J. Brosilow and R.K. Ziff, J. Catal. 136 (1992) 275.
[4] R.M. Ziff, E. Gulari and Y. Barshad, Phys. Rev. Letters 56 (1986) 2553. [5] B. Chopard and M. Droz, J. Phys. A 21 (1988) 205. [6] R.M. Ziff, K. Fichthorn and E. Gulari, J. Phys. A 24 (1991) 3727. [7] R. Dickman, Phys. Rev. A 34 (1986) 4246.