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Study of damage spreading in dimer-dimer irreversible surface reaction models
ELSEVIER
Physica A 215 (1995) 451 460
Study of damage spreading in dimer-dimer irreversible surface reaction models Ezequiel V. A l b a n o Instituto de Int~estigaciones Fisicoquimicas Te6rieas y Aplicadas (INIFTA), Faeultad de Ciencias Exactas, Unit~ersidad Nacional de La Plata, Suc. 4, Casilla de Correo 16, (1990) La Plata. Argentina Received 12 September 1994
Abstract
The spreading of globally distributed damage was studied in two variants of an irreversible dimer dimer surface reaction model of the type 12A2 + B2 ~ B2A. The first model (model M l) neglects desorption of B2. On the contrary, the second version (model M3) takes this process into account. Both models exhibit irreversible (kinetic) phase transitions (IPT) from a reactive state with sustained production of B2A molecules to surface poisoned states without B2A production. Model M1 has a zero-width reaction window with a single critical point at which damage spreading is observed. For model M3 the damage also spreads and exhibits an abrupt discontinuity at the critical point where a continuous IPT from the poisoned state to the reactive regime is observed. Damage healing was only observed when M3 can be mapped onto a simple adsorption-desorption process of B2. Close to the chaotic-frozen transition the order parameter critical exponent was found to be fl ~ 1.075 _+ 0.019.
1. Introduction
Recently a growing interest in the theoretical investigation of microscopic irreversible reaction models has developed (see e.g. Refs. [1-29]). Models exhibiting nonequilibrium phase transitions are of particularly great interest in the chemical, physical and biological sciences. Examples of these models include, single component reaction systems such as the contact model [1], the A-model [6], the BK-model [-9], the DR-model [28], etc., and multi-component reaction process such as the monom e r - m o n o m e r surface reaction models [3,4,8,10,11,20,21], the ZGB-model [2,3,5,7,12-20,22], and several variants of the d i m e r - d i m e r reaction model [23-27], etc. Some d i m e r - d i m e r surface reaction models were inspired by the catalytic oxidation of hydrogen [23-27]. They exhibit irreversible phase transitions and their critical behavior has been studied by means of Monte Carlo simulations, finite-size scaling techniques and a mean-field approach [23 27]. The reaction scheme studied in this 0378-4371/95/$09.50 :~; 1995 Elsevier Science B.V. All rights reserved SSDI 0 3 7 8 - 4 3 7 1 ( 9 4 ) 0 0 2 9 9 - 1
E.V. Albano/Physica A 215 (1995) 451 460
452
work is based upon the well known Langmuir Hinshelwood mechanism for which the reactants must be adsorbed on the catalytic surface, so [23-27] A2(g) + 2(*) ~ 2A(a),
(la)
kl
Bz(g) + 2(*) ~- 2B(a),
(lb)
A(a) + B(a) --. AB(a) + (*),
(lc)
AB(a) + B(a) ~ B2A(g ) + 2(*),
(ld)
kz
where (*) denotes a vacant site on the catalyst surface, while (a) and (g) indicate adsorbed and gas phase species, respectively. Also, kl and k2 are rate constants for Bz-desorption and adsorption, respectively. I have very recently shown that damage spreading introduces a new kind of dynamic critical behavior in some irreversible reaction processes such as the monomer-monomer surface reaction model [29], the ZGB-model [29, 30] and a modified version of the ZGB-model that includes the Eley-Rideal mechanism [30]. The damage spreading problem consists in first taking a steady state configuration of the system {a A} and creating, at t = 0, an initial damage D(0) in that configuration (this procedure gives a second configuration {aB}) [29-39]; for a review see Ref. [40]. The time evolution of both configurations is then investigated using the same dynamics and the damage spreading is characterized by calculating the Hamming distance between configurations, defined by 1
N
D(t) = ~ ~ ]a~(t) - a~(t)l,
(2)
i=1
where N is the number of sites in the system. Physically D(t) measures the fraction of sites for which both configurations are different. Starting with a small D(0)-value, D(t) will evolve asymptotically to zero in the "frozen phase", whereas it will tend to a finite value different from zero in the "chaotic phase" 1-29-40]. While the study of damage spreading in systems exhibiting reversible phase transitions has received much attention [31-40], a similar study of systems undergoing irreversible transitions is still in its infancy [29, 30]. Within this context, the aim of the present work is to investigate the spreading of damage in two variants of the dimer-dimer surface reaction model denoted by M1 and M3, respectively, in Ref. [23]. The manuscript is organized as follows: in Section 2 the dimer-dimer surface reaction scheme is described and the theoretical background on damage spreading is discussed; Section 3 is devoted to the presentation and discussion of the results. The conclusions are outlined in Section 4.
E.V. Albano/Physica A 215 (1995) 451 460
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2. Simulation details and theoretical background 2.1. The dimer-dimer reaction models The dimer dimer reaction models were simulated on square lattices of side L = 100 lattice units using periodic boundary conditions. Damage spreading was studied in two variants of the d i m e r - d i m e r process, namely models M1 and M3 using the notation of Ref. [23]. In model M 1 a site, say site 1, of the catalytic surface is initially selected at random. Then the simulation proceeds as follows: (i) If site 1 is occupied by A(a) the trial ends; (ii) if site 1 is empty a nearest neighbor INN) site, say site 2, is also selected at random. If site 2 is occupied the trial ends because there is no place for dimer adsorption. Otherwise, if site 2 is also empty a dimer, either A2 or B2, is adsorbed. So B2 (A2) is selected at random with probability Y (1 - Y), where Y is the mole fraction of B2 in the gas phase. When a dimer becomes adsorbed its six NN sites must be examined in order to account for the reactions described by Eqs. ( l c d ) . These reactions are assumed to take place only when the involved species are adsorbed on NN sites. Finally, (iii) if site 1 is occupied by B(a) a N N site, say again site 2, is selected at random. If site 2 is occupied the trial ends, otherwise the B(a) is allowed to move from site 1 to site 2 to represent diffusion. After that, three N N sites of site 2 (site 1 is now vacant) must be examined in order to account for a possible reaction event, as described above in (ii). Model M1 neglects B2-desorption, so kl = ~ and k2 = 0 in Eq. (lb). Model M3 involves Bz-desorption, so Eq. (lb) with k2 = ~, must be accounted for either after the adsorption of a Bz-dimer or after the diffusion of B(a). Further assumptions involved in the algorithms employed in this work are the following: AB(a) is formed on the site occupied by A(a) while the site corresponding to B(a) is vacated. When more than one N N of type B(a) is found around a newlyadsorbed A(a), one of them is selected at random in order to form an AB(a), but this intermediate reacts immediately with one of the (randomly selected) remaining B(a) to form BzA(g). Random selection of B(a) is only relevant when the number of NNs is three. On the other hand, for a newly-adsorbed B(a), there could be A(a), B(aj and AB(a) NNs. In all cases the reaction path is selected at random. The Monte Carlo time Unit (t) involves L 2 trials, so each site of the lattice will be visited once on average during each time unit. Additional simulation details can be found in earlier publications [23 27].
2.2. Damage spreading The simulations were started with empty lattices and allowed to run for 2 × 103 time units to reach the steady state. Then the damage was created and its spreading was monitored as the time t, after damage initiation, increases. This can be achieved by following the dynamics of the damaged and the undamaged configurations simultaneously. For this purpose, the crucial idea is to apply the same sequence of random
454
E.V. Albano / Physica A 215 (1995) 451-460
numbers in the algorithm in order to produce the same dynamics on the configurations {aT}. This procedure requires special care because for each adsorption-reaction trial, a different number of random numbers is frequently needed to follow the dynamics of the configurations {o-~}. So, in order to keep the synchronism, a set of random numbers {ri}, i = 1, . . . , M , with M just large enough to account for all possible situations, was generated before starting the trial. Then, during the trial, the random numbers were used sequentially for the same purpose, e.g. to choose the incoming species, to select the neighboring site, etc. In Eq. (2) D(t) is defined in terms of O-, (a spin variable which usually takes two values). For the models studied in this work the sites of the lattice may be V, A, B and AB when they are vacant, or occupied by A, B and AB-species, respectively. So, all contributions to the Hamming distance given by V-A, V-B, V-AB, A-B, A-AB and B-AB are taken to be equal to unity in the evaluation of Eq. (2). According to the discussion in the introduction, the dynamic behavior of the system is called "chaotic" if D(t) takes a finite value for long times if D(0) -~ 0 [40]. On the contrary, the damage is healed in the "frozen" phase, (D(z) = 0 for sufficiently large r), this means that during their respective time evolutions both configurations become identical. Since it is necessary to work with finite, but small, D(0)-values it is necessary to take the limit D(0) --, 0 to obtain reliable results. However, this tedious work can be avoided if three configurations {o'A}, {0"B} and {ac}, such a s DAB(0 ) = DBc(0)= ½DAc(0) = s, are considered [29, 40]. Then
D(t) = DAB(t) + Dac(t) -- OAc(t)
(3)
is a very good approximation to D(0) ~ 0. In all cases considered in the present work the initial damage was created by changing a fraction s = 0.1 of the occupied sites at random. So, we are interested in studying the spreading of a global initial damage distributed throughout the system, in contrast with another approach which only considers the spreading of a local initial damage [40].
3. Results and discussion
3.1. Results from model M1 Fig. 1 shows the phase diagram of model M 1. As follows from Fig. 1, model M 1 has a single critical point at Yc = 2/3, such that for Y < Yc the surface becomes poisoned by a binary mixture of {A(a)+ AB(a)}, while for Y > Yc the surface becomes saturated by B(a). Exactly at Y -- Y¢, sustained reaction with B2A-production was found in the simulations. However, it is expected that in the t --* ~ limit the reactive state will vanish and the surface will ultimately become poisoned with either a binary mixture of {A(a)+ AB(a)} or B(a) [23-27]. The poisoned state with the binary mixture of {A(a) + AB(a)} is jammed, that is the surface cannot be fully covered due to the blocking effect of the previously adsorbed species towards dimer adsorption.
E.V. Albano/Physica A 215 (1995) 451 460 1.00
I
f
455
I
0.75
0
0.50
N
7
v"
v"
"~
0.25
x~ 0.00 0.00
--" 0.50
"0.25"
0.75
1.00
Y Fig. 1. Plot of the surface c o v e r a g e 0 of the a d s o r b e d species A, AB and B, versus Y for m o d e l M 1.
0.8
0.6
0.4
0.2
0.0 0
I
i
f
J
1000
2000
3000
4000
5000
t Fig. 2. Plot of D(t) versus t t a k e n at Yc = z for model M1.
Within this phase the total coverage with {A(a)+ AB(a)} is close to 0.907 (the jamming coverage for the random dimer filling problem in two dimensions [41,42]). In the case of model M1 we have investigated the spreading of damage just at criticality. Fig. 2 shows a plot of D(t) versus t. It was found that the damage spreads
E, V. Albano / Physica A 215 (1995) 451-460
456
very fast. This finding is in contrast to results of the monomer-monomer model where, just at the single critical point Y¢ = ½, the damage grows according to a single power law O(t) oz t ~ (6 ~ 0.28) [29]. The growth of the damage is consistent with the fact that the final state of the surface will be a poisoned one. In order to explain this result, three possible scenarios must be taken into account: (i) if the different configurations used to evaluate the damage evolve towards the poisoned state with B D(ov)= 0, (ii) if the configurations evolve towards differently poisoned states D ( ~ ) = 1, and (iii) if the final poisoned states correspond to the binary mixture D ( ~ ) > 0 since the states are jammed.
3.2. Results from model M3
Fig. 3 shows the phase diagram of model M3 which exhibits a continuous irreversible transition between a poisoned (jammed) state with {A(a) + AB(a)} and a stationary regime with BzA-production at the critical point Y~ -~ 0.7014 [23-25]. For Y > Y¢ the damage spreads and after some time, usually 20 < t < 50, it reaches a stationary value as it is shown in Fig. 4. For Y < Yc, D > 0 since the poisoned state is jammed and the damage created in the poisoned state can not be healed out completely. Fig. 5 shows a plot of the stationary value of the damage {D(t ~ ~)} versus Y. From the figure it follows that D has a discontinuity just at the critical point Y¢. This result strongly suggests that the damage undergoes a first order transition
0.6
A
R,O 0.4
...............".......
~
~
BaA
""V 0.0 0.50
=
=
Cl
0.75
1.00
Y Fig. 3. Plot of the surface coverage 0 of the a d s o r b e d species A, AB a n d B a n d the rate of B2A-production (R), versus Y for m o d e l M3.
E.V. Albano/'Physica A 215 (1995) 451 460
45?
when the parameter Y is finely tuned in order to irreversibly change the state of the system from the reactive regime (Y > Yc) to the poisoned phase (Y ~< Yc). The observed discontinuous transition of D just at Yc is in dramatic contrast with the continuous irreversible phase transition from the poisoned to the reactive state, which
0,2
--
i
I
I
I
i 4O
i 60
i 8O
0.1
V.V"
0.0
v
i 20
100
t Fig. 4. P l o t s o f D(t) v e r s u s t for m o d e l M 3 for t w o v a l u e s o f Y. • : Y = 0.85 a n d V : Y = 0.95.
1
l
T
0.6
0.4
\ \
0.2
\
\
0.0 0.7
0.8
0.9
Y Fig. 5. P l o t o f D(I ---, ~ ) v e r s u s Y for m o d e l M 3 .
E.K Albano/Physica A 215 (1995) 451 460
458
I
t
10 -1
10 - 2
10 -3
10 -3
I
I
10-2
10 -1
AY Fig. 6. Plot of D(t ~ ~ ) versus AY = YH - Y for m o d e l M3. The s t r a i g h t line with slope fl = 1.075 was o b t a i n e d by m e a n s of a least s q u a r e fit of the data.
the system undergoes at the same critical point. This novel behavior, which is in contrast to results obtained for the ZGB model and the m o n o m e r - m o n o m e r model where the damage undergoes smooth transitions [29], also indicates that damage spreading introduces a new kind of dynamic critical phenomena in the variant M3 of the dimer-dimer model. As shown in Fig. 5 the damage becomes healed just in the limit Yn = 1. For this Y-value model M3 corresponds to a simple adsorption-desorption process of B2 which obviously belongs to the "frozen" phase. Since the damage itself is the natural order parameter, it may be anticipated that the following behavior should hold
D(t
~
oo)~(Y
H -
Y)a,
(4)
close to the chaotic-frozen transition, where fl is the order parameter critical exponent. Fig. 6 shows that a log-log plot of D versus A Y = YH -- Y exhibits a linear behavior which allow us to determine fl by means of a least square regression, e.g. fl - 1.075 _+ 0.019, where the error bars merely reflect the statistical error.
4. Conclusions The spreading of damage was studied in two versions of a dimer-dimer reaction model in two dimensions. In the case of model M 1 the damage spreads very fast just at criticality. Model M3 exhibits a richer behavior since a discontinuity of the damage is
E.V. Albano/Phvsica A 215 (1995) 451 460
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observed just at the critical point corresponding to the (standard) continuous irreversible phase transition from the poisoned state to the reactive regime. This behavior is consistent with the fact that the poisoned state is jammed, however the understanding of the discontinuity deserves further studies. In the limit Y = 1 damage healing was observed and the corresponding order parameter critical exponent of the chaoticfrozen transition was found to be/3 =~ 1.075.
Acknowledgements This work was supported by the Consejo Nacional de Investigaciones Cientificas y T6cnicas (CONICET) de la Republica Argentina. The Alexander von Humboldt Foundation (Germany), the Volkswagen Foundation (Germany) and Fundaci6n Antorchas (Argentina) are greatly acknowledged for the provision of valuable equipment.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]
T.M. Liggett, Interacting Particle Systems (Springer Verlag, Heidelberg, 1985). R.M. Ziff, E. Gulari and Y. Barshad, Phys. Rev. Lett. 56 11986) 2553. P. Meakin and D.J. Scalapino, J. Chem. Phys. 87 (1987) 731. A. Sadiq and K. Yaldram, J. Phys. A (Mat. Gen.) 21 (1988) L207. B. Chopard and M. Droz, J. Phys. A (Math. Gen.) 21 (1988) 205. R. Dickman and M. Burschka, Phys. Lett. A 127 (1988) 132. K. Yaldran and A. Sadiq, J. Phys. A (Math. Gen.) 22 (1989) 1925. K. Fichthorn, E. Gulari and R.M. Ziff, Phys. Rev. Lett. 63 (1989) 1527. D.A. Browne and P. Kleban, Phys. Rev. A 40 (1989) 1615. D. ben-Avraham, S. Redner, D. Considine and P. Meakin, J. Phys. A. (Math. Gen.) 23 (1990) L613. D. ben-Avraham, D. Considine, P. Meakin, S. Redner and H. Takayasu, J. Phys. A (Math. Gen.) 23 t1990) 4297. 1. Jensen and H.C. Fogedby, Phys. Rev. A 42 (1990) 1969. D. Considine, H. Takayasu and S. Redner, J. Phys. A. (Math. Gen.) 23 (1990) L1181. E.V. Albano, J. Phys. A (Math. Gen.) 23 (1990) L545. E.V. Albano, Phys. Rev. B Rapid Commun. 42 (1990) 10818 E.V. Albano, Surface Sci. 235 (1990) 351. E.V. Albano, J. Chem. Phys. 94 (1991) 1499. J.W. Evans and M.S. Miesch, Phys. Rev. Lett. 66 (1991) 833. J.W. Evans and M.S. Miesch, Surface Sci. 245 (1991) 401. J.W. Evans, Langmuir 7 (1992) 2514. E.V. Albano, Phys. Rev. Lett. 69 (1992) 656. J. Mai, A. Casties and W. von Niessen, Chem. Phys. Lett. 196 (1992) 358. E.V. Albano, J. Phys. A (Math. Gen.) 25 11992) 2557; Corrigendum 26 (1993) 3667. E.V. Albano, J. Stat. Phys. 69 (1992) 643. A. Maltz and E.V. Albano, Surf. Sci. 277 (1992) 414. K. Yaldran, K.M. Khan, N. Ahmed and M.A. Khan, J. Phys. A (Math. Gen.) 26 11993) 2663 E.V. Albano, Physica A 198 (1993) 503. 1. Jensen and R. Dickman, Phys. Rev. E 48 (1993) 1710. E.V. Albano, Phys. Rev. Lett. 72 (1994) 108. E.V. Albano, Phys. Rev. E 50 (1994) 1129.
460 [31] [32] [33] [34] [35] [36] [37] [38] [39] [40]
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H.E. Stanley, D. Stauffer, J. Kert6sz and H.J. Herrmann, Phys. Rev. Lett. 59 (19871 2326. G. Le Ca~Sr, Physica A 159 (1989) 329. A. Coniglio, L. de Arcangelis, H.J. Herrmann and N. Jan, Europhys. Lett. 8 (1989) 315. H.J. Herrmann, Physica A 168 (1990) 516. O. Golinelli and B. Derrida, J. Phys. A (Math. Gen.) 22 (1989) L939. A.M. Mariz, J. Phys. A (Math. Gen.) 23 (19901 979. A.M. Mariz, A.M.C. de Souza and C. Tsallis, J. Phys. A tMath. Gen.) 26 t19931 L1007. I.A. Campbell, Europhys. Lett. 21 (1993) 959. G.M. Zhang and C.Z. Yang, Europhys. Lett. 22 (19931 505. H.J. Herrmann, Simulation of random growth processes, in: The Monte Carlo method in condensed matter physics, Vol. 71, Ed. K. Binder (Springer Verlag, Heidelberg, 1992) p. 93. [41] J.W. Evans, D.R. Burgess and D.K. Hoffman, J. Chem. Phys. 79 (1983) 5011. [42] J.W. Evans and R.S. Nord, Phys. Rev. B 31 (19851 1759; R.S. Nord and J.W. Evans, J. Chem. Phys. 93 (1990) 8397.
Physica A 215 (1995) 451 460
Study of damage spreading in dimer-dimer irreversible surface reaction models Ezequiel V. A l b a n o Instituto de Int~estigaciones Fisicoquimicas Te6rieas y Aplicadas (INIFTA), Faeultad de Ciencias Exactas, Unit~ersidad Nacional de La Plata, Suc. 4, Casilla de Correo 16, (1990) La Plata. Argentina Received 12 September 1994
Abstract
The spreading of globally distributed damage was studied in two variants of an irreversible dimer dimer surface reaction model of the type 12A2 + B2 ~ B2A. The first model (model M l) neglects desorption of B2. On the contrary, the second version (model M3) takes this process into account. Both models exhibit irreversible (kinetic) phase transitions (IPT) from a reactive state with sustained production of B2A molecules to surface poisoned states without B2A production. Model M1 has a zero-width reaction window with a single critical point at which damage spreading is observed. For model M3 the damage also spreads and exhibits an abrupt discontinuity at the critical point where a continuous IPT from the poisoned state to the reactive regime is observed. Damage healing was only observed when M3 can be mapped onto a simple adsorption-desorption process of B2. Close to the chaotic-frozen transition the order parameter critical exponent was found to be fl ~ 1.075 _+ 0.019.
1. Introduction
Recently a growing interest in the theoretical investigation of microscopic irreversible reaction models has developed (see e.g. Refs. [1-29]). Models exhibiting nonequilibrium phase transitions are of particularly great interest in the chemical, physical and biological sciences. Examples of these models include, single component reaction systems such as the contact model [1], the A-model [6], the BK-model [-9], the DR-model [28], etc., and multi-component reaction process such as the monom e r - m o n o m e r surface reaction models [3,4,8,10,11,20,21], the ZGB-model [2,3,5,7,12-20,22], and several variants of the d i m e r - d i m e r reaction model [23-27], etc. Some d i m e r - d i m e r surface reaction models were inspired by the catalytic oxidation of hydrogen [23-27]. They exhibit irreversible phase transitions and their critical behavior has been studied by means of Monte Carlo simulations, finite-size scaling techniques and a mean-field approach [23 27]. The reaction scheme studied in this 0378-4371/95/$09.50 :~; 1995 Elsevier Science B.V. All rights reserved SSDI 0 3 7 8 - 4 3 7 1 ( 9 4 ) 0 0 2 9 9 - 1
E.V. Albano/Physica A 215 (1995) 451 460
452
work is based upon the well known Langmuir Hinshelwood mechanism for which the reactants must be adsorbed on the catalytic surface, so [23-27] A2(g) + 2(*) ~ 2A(a),
(la)
kl
Bz(g) + 2(*) ~- 2B(a),
(lb)
A(a) + B(a) --. AB(a) + (*),
(lc)
AB(a) + B(a) ~ B2A(g ) + 2(*),
(ld)
kz
where (*) denotes a vacant site on the catalyst surface, while (a) and (g) indicate adsorbed and gas phase species, respectively. Also, kl and k2 are rate constants for Bz-desorption and adsorption, respectively. I have very recently shown that damage spreading introduces a new kind of dynamic critical behavior in some irreversible reaction processes such as the monomer-monomer surface reaction model [29], the ZGB-model [29, 30] and a modified version of the ZGB-model that includes the Eley-Rideal mechanism [30]. The damage spreading problem consists in first taking a steady state configuration of the system {a A} and creating, at t = 0, an initial damage D(0) in that configuration (this procedure gives a second configuration {aB}) [29-39]; for a review see Ref. [40]. The time evolution of both configurations is then investigated using the same dynamics and the damage spreading is characterized by calculating the Hamming distance between configurations, defined by 1
N
D(t) = ~ ~ ]a~(t) - a~(t)l,
(2)
i=1
where N is the number of sites in the system. Physically D(t) measures the fraction of sites for which both configurations are different. Starting with a small D(0)-value, D(t) will evolve asymptotically to zero in the "frozen phase", whereas it will tend to a finite value different from zero in the "chaotic phase" 1-29-40]. While the study of damage spreading in systems exhibiting reversible phase transitions has received much attention [31-40], a similar study of systems undergoing irreversible transitions is still in its infancy [29, 30]. Within this context, the aim of the present work is to investigate the spreading of damage in two variants of the dimer-dimer surface reaction model denoted by M1 and M3, respectively, in Ref. [23]. The manuscript is organized as follows: in Section 2 the dimer-dimer surface reaction scheme is described and the theoretical background on damage spreading is discussed; Section 3 is devoted to the presentation and discussion of the results. The conclusions are outlined in Section 4.
E.V. Albano/Physica A 215 (1995) 451 460
453
2. Simulation details and theoretical background 2.1. The dimer-dimer reaction models The dimer dimer reaction models were simulated on square lattices of side L = 100 lattice units using periodic boundary conditions. Damage spreading was studied in two variants of the d i m e r - d i m e r process, namely models M1 and M3 using the notation of Ref. [23]. In model M 1 a site, say site 1, of the catalytic surface is initially selected at random. Then the simulation proceeds as follows: (i) If site 1 is occupied by A(a) the trial ends; (ii) if site 1 is empty a nearest neighbor INN) site, say site 2, is also selected at random. If site 2 is occupied the trial ends because there is no place for dimer adsorption. Otherwise, if site 2 is also empty a dimer, either A2 or B2, is adsorbed. So B2 (A2) is selected at random with probability Y (1 - Y), where Y is the mole fraction of B2 in the gas phase. When a dimer becomes adsorbed its six NN sites must be examined in order to account for the reactions described by Eqs. ( l c d ) . These reactions are assumed to take place only when the involved species are adsorbed on NN sites. Finally, (iii) if site 1 is occupied by B(a) a N N site, say again site 2, is selected at random. If site 2 is occupied the trial ends, otherwise the B(a) is allowed to move from site 1 to site 2 to represent diffusion. After that, three N N sites of site 2 (site 1 is now vacant) must be examined in order to account for a possible reaction event, as described above in (ii). Model M1 neglects B2-desorption, so kl = ~ and k2 = 0 in Eq. (lb). Model M3 involves Bz-desorption, so Eq. (lb) with k2 = ~, must be accounted for either after the adsorption of a Bz-dimer or after the diffusion of B(a). Further assumptions involved in the algorithms employed in this work are the following: AB(a) is formed on the site occupied by A(a) while the site corresponding to B(a) is vacated. When more than one N N of type B(a) is found around a newlyadsorbed A(a), one of them is selected at random in order to form an AB(a), but this intermediate reacts immediately with one of the (randomly selected) remaining B(a) to form BzA(g). Random selection of B(a) is only relevant when the number of NNs is three. On the other hand, for a newly-adsorbed B(a), there could be A(a), B(aj and AB(a) NNs. In all cases the reaction path is selected at random. The Monte Carlo time Unit (t) involves L 2 trials, so each site of the lattice will be visited once on average during each time unit. Additional simulation details can be found in earlier publications [23 27].
2.2. Damage spreading The simulations were started with empty lattices and allowed to run for 2 × 103 time units to reach the steady state. Then the damage was created and its spreading was monitored as the time t, after damage initiation, increases. This can be achieved by following the dynamics of the damaged and the undamaged configurations simultaneously. For this purpose, the crucial idea is to apply the same sequence of random
454
E.V. Albano / Physica A 215 (1995) 451-460
numbers in the algorithm in order to produce the same dynamics on the configurations {aT}. This procedure requires special care because for each adsorption-reaction trial, a different number of random numbers is frequently needed to follow the dynamics of the configurations {o-~}. So, in order to keep the synchronism, a set of random numbers {ri}, i = 1, . . . , M , with M just large enough to account for all possible situations, was generated before starting the trial. Then, during the trial, the random numbers were used sequentially for the same purpose, e.g. to choose the incoming species, to select the neighboring site, etc. In Eq. (2) D(t) is defined in terms of O-, (a spin variable which usually takes two values). For the models studied in this work the sites of the lattice may be V, A, B and AB when they are vacant, or occupied by A, B and AB-species, respectively. So, all contributions to the Hamming distance given by V-A, V-B, V-AB, A-B, A-AB and B-AB are taken to be equal to unity in the evaluation of Eq. (2). According to the discussion in the introduction, the dynamic behavior of the system is called "chaotic" if D(t) takes a finite value for long times if D(0) -~ 0 [40]. On the contrary, the damage is healed in the "frozen" phase, (D(z) = 0 for sufficiently large r), this means that during their respective time evolutions both configurations become identical. Since it is necessary to work with finite, but small, D(0)-values it is necessary to take the limit D(0) --, 0 to obtain reliable results. However, this tedious work can be avoided if three configurations {o'A}, {0"B} and {ac}, such a s DAB(0 ) = DBc(0)= ½DAc(0) = s, are considered [29, 40]. Then
D(t) = DAB(t) + Dac(t) -- OAc(t)
(3)
is a very good approximation to D(0) ~ 0. In all cases considered in the present work the initial damage was created by changing a fraction s = 0.1 of the occupied sites at random. So, we are interested in studying the spreading of a global initial damage distributed throughout the system, in contrast with another approach which only considers the spreading of a local initial damage [40].
3. Results and discussion
3.1. Results from model M1 Fig. 1 shows the phase diagram of model M 1. As follows from Fig. 1, model M 1 has a single critical point at Yc = 2/3, such that for Y < Yc the surface becomes poisoned by a binary mixture of {A(a)+ AB(a)}, while for Y > Yc the surface becomes saturated by B(a). Exactly at Y -- Y¢, sustained reaction with B2A-production was found in the simulations. However, it is expected that in the t --* ~ limit the reactive state will vanish and the surface will ultimately become poisoned with either a binary mixture of {A(a)+ AB(a)} or B(a) [23-27]. The poisoned state with the binary mixture of {A(a) + AB(a)} is jammed, that is the surface cannot be fully covered due to the blocking effect of the previously adsorbed species towards dimer adsorption.
E.V. Albano/Physica A 215 (1995) 451 460 1.00
I
f
455
I
0.75
0
0.50
N
7
v"
v"
"~
0.25
x~ 0.00 0.00
--" 0.50
"0.25"
0.75
1.00
Y Fig. 1. Plot of the surface c o v e r a g e 0 of the a d s o r b e d species A, AB and B, versus Y for m o d e l M 1.
0.8
0.6
0.4
0.2
0.0 0
I
i
f
J
1000
2000
3000
4000
5000
t Fig. 2. Plot of D(t) versus t t a k e n at Yc = z for model M1.
Within this phase the total coverage with {A(a)+ AB(a)} is close to 0.907 (the jamming coverage for the random dimer filling problem in two dimensions [41,42]). In the case of model M1 we have investigated the spreading of damage just at criticality. Fig. 2 shows a plot of D(t) versus t. It was found that the damage spreads
E, V. Albano / Physica A 215 (1995) 451-460
456
very fast. This finding is in contrast to results of the monomer-monomer model where, just at the single critical point Y¢ = ½, the damage grows according to a single power law O(t) oz t ~ (6 ~ 0.28) [29]. The growth of the damage is consistent with the fact that the final state of the surface will be a poisoned one. In order to explain this result, three possible scenarios must be taken into account: (i) if the different configurations used to evaluate the damage evolve towards the poisoned state with B D(ov)= 0, (ii) if the configurations evolve towards differently poisoned states D ( ~ ) = 1, and (iii) if the final poisoned states correspond to the binary mixture D ( ~ ) > 0 since the states are jammed.
3.2. Results from model M3
Fig. 3 shows the phase diagram of model M3 which exhibits a continuous irreversible transition between a poisoned (jammed) state with {A(a) + AB(a)} and a stationary regime with BzA-production at the critical point Y~ -~ 0.7014 [23-25]. For Y > Y¢ the damage spreads and after some time, usually 20 < t < 50, it reaches a stationary value as it is shown in Fig. 4. For Y < Yc, D > 0 since the poisoned state is jammed and the damage created in the poisoned state can not be healed out completely. Fig. 5 shows a plot of the stationary value of the damage {D(t ~ ~)} versus Y. From the figure it follows that D has a discontinuity just at the critical point Y¢. This result strongly suggests that the damage undergoes a first order transition
0.6
A
R,O 0.4
...............".......
~
~
BaA
""V 0.0 0.50
=
=
Cl
0.75
1.00
Y Fig. 3. Plot of the surface coverage 0 of the a d s o r b e d species A, AB a n d B a n d the rate of B2A-production (R), versus Y for m o d e l M3.
E.V. Albano/'Physica A 215 (1995) 451 460
45?
when the parameter Y is finely tuned in order to irreversibly change the state of the system from the reactive regime (Y > Yc) to the poisoned phase (Y ~< Yc). The observed discontinuous transition of D just at Yc is in dramatic contrast with the continuous irreversible phase transition from the poisoned to the reactive state, which
0,2
--
i
I
I
I
i 4O
i 60
i 8O
0.1
V.V"
0.0
v
i 20
100
t Fig. 4. P l o t s o f D(t) v e r s u s t for m o d e l M 3 for t w o v a l u e s o f Y. • : Y = 0.85 a n d V : Y = 0.95.
1
l
T
0.6
0.4
\ \
0.2
\
\
0.0 0.7
0.8
0.9
Y Fig. 5. P l o t o f D(I ---, ~ ) v e r s u s Y for m o d e l M 3 .
E.K Albano/Physica A 215 (1995) 451 460
458
I
t
10 -1
10 - 2
10 -3
10 -3
I
I
10-2
10 -1
AY Fig. 6. Plot of D(t ~ ~ ) versus AY = YH - Y for m o d e l M3. The s t r a i g h t line with slope fl = 1.075 was o b t a i n e d by m e a n s of a least s q u a r e fit of the data.
the system undergoes at the same critical point. This novel behavior, which is in contrast to results obtained for the ZGB model and the m o n o m e r - m o n o m e r model where the damage undergoes smooth transitions [29], also indicates that damage spreading introduces a new kind of dynamic critical phenomena in the variant M3 of the dimer-dimer model. As shown in Fig. 5 the damage becomes healed just in the limit Yn = 1. For this Y-value model M3 corresponds to a simple adsorption-desorption process of B2 which obviously belongs to the "frozen" phase. Since the damage itself is the natural order parameter, it may be anticipated that the following behavior should hold
D(t
~
oo)~(Y
H -
Y)a,
(4)
close to the chaotic-frozen transition, where fl is the order parameter critical exponent. Fig. 6 shows that a log-log plot of D versus A Y = YH -- Y exhibits a linear behavior which allow us to determine fl by means of a least square regression, e.g. fl - 1.075 _+ 0.019, where the error bars merely reflect the statistical error.
4. Conclusions The spreading of damage was studied in two versions of a dimer-dimer reaction model in two dimensions. In the case of model M 1 the damage spreads very fast just at criticality. Model M3 exhibits a richer behavior since a discontinuity of the damage is
E.V. Albano/Phvsica A 215 (1995) 451 460
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observed just at the critical point corresponding to the (standard) continuous irreversible phase transition from the poisoned state to the reactive regime. This behavior is consistent with the fact that the poisoned state is jammed, however the understanding of the discontinuity deserves further studies. In the limit Y = 1 damage healing was observed and the corresponding order parameter critical exponent of the chaoticfrozen transition was found to be/3 =~ 1.075.
Acknowledgements This work was supported by the Consejo Nacional de Investigaciones Cientificas y T6cnicas (CONICET) de la Republica Argentina. The Alexander von Humboldt Foundation (Germany), the Volkswagen Foundation (Germany) and Fundaci6n Antorchas (Argentina) are greatly acknowledged for the provision of valuable equipment.
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