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The A + A → 0 reaction on a critical percolation system
Volume 173, number 5,6
CHEMICAL PHYSICS LETTERS
19 October 1990
The A+ A-+0 reaction on a critical percolation system A.G. Vitukhnovsky, N.V. Kiriakova and I.M. Sokolov P.N. Lebedev Physical Institute, USSR Academy of Sciences, Leninsky Pr. 53, 1 I7924
Moscow. USSR
Received 25 April 1990; in final form 7 August 1990
The diffusion-controlled reaction A+A+O on a percolation system with finite clusters is investigated by Monte Carlo simulation. The asymptotic law of reaction kinetics n( 1) and the dependence of n(w) on n( 0) were found.
Recently, great attention has been paid to the investigation of diffusion-controlled reactions on fractal systems (refs. [ l-31, and references therein), in particular on percolation systems. The typical realization of such situations is the triplet-triplet exciton annihilation on mixed moIecular crystals or on pore-transparent matrices doped by dyes. The existence of finite clusters or isolated pores results in some deviations of the time behavior of the particle concentration n(r) from the case obtained for the AS A-0 reaction on a connected fractal system n(l)-t9’2, where D, is the spectral dimension of the system. This reaction kinetics on a percolation system with finite clusters is the object of the study in the present paper. In connected systems, the random walks of the particles are ergodic. In this case, n(t) is governed by the equation
Wt)
-=-c(t)
dt
n’(t) ,
where c(r) =dS( t)/dt and S(t) is the average number of distinct sites visited by a random walker during time t. At the furthermost stage of evolution, we have the law n(t) -S(I) -‘. If a system is disconnected (has finite clusters) and one substitutes instead of S( t) the number of distinct visited sites averaged by the position of the initial point of a walk S(t) -I, one will obtain the result n(l) -S(t) - ’ - t -DL’2 with the renormalized value of the spectral dimension 0: =D,( 2-&D), where D is the incipElsevier Science Publishers B.V. (North-Holland)
ient infinite-cluster fractal dimension and d the embedding space dimension [ 41. The S(t) time dependence itself was studied by numerous computer simulations but we shall try to demonstrate that the relation n(t) -S(t) -’ is valid only for a small reaction range and is different from the large-time asymptotic reaction behavior. This asymptotic behavior was considered in ref. [ 51, The main ideas of this approach are as follows: The reaction in each cluster must be considered separately. The reaction behavior in a finite system depends on the relation between two characteristic lengths: the diffusion length 1(t) -a( t/T)“(2ceJ and the cluster size L-&4? Here a and T are the lattice spacing and the hopping time (in what follows, we shall use dimensionless units and put a= r= 1), 8 is an anomalous diffusion exponent and M the number of sites in a cluster. Ifn(0)-2~D~L*+’ ( Psi2 > M) , the number of particles in the cluster approaches exponentially the value N, =O or 1, depending on whether the initial number of particles was even or odd, respectively. In order to obtain a scaling estimate for the number of the particles, one can use the approximation N(t,M)-N(co,M)
-IW-~~/~ for thi2-eM, -0
for t”/‘>M.
(2)
Then it is possible to determine the overall particle 521
Volume173,number 5,6
CHEMICAL PHYSICS LETTERS
concentration by averaging this expression over the cluster-size distribution, n(t)--n(m)=
;
[NhW-NwM)lIT(M), (3)
with L!(M) -JM-‘-~‘~ (the average number of clusters of M sites per n sites). Evaluation of this sum gives n(t)_n(co)_t-‘“/2’dlD=t_d/(2+e) -
(4)
The exponent (Y= d/ (2 + 0) for d= 3 is significantly different from OS/2 or D:/2: a=O.80, D,/2~0.61, 4/2~0.53 (in the three-dimensional case, Dx2.5, D,z 1.33). The characteristic time of establishing such asymptotics is of the order of town;2/“. The results of our investigations of such reactions by direct computer simulation are as follows: The reaction was simulated on a percolation system (site problem) for a cubic lattice of 40 x 40 x 40 sites. The concentration of “good” sites p=pC= 0.3 1. The good sites were occupied randomly by particles (double occupation for one site is forbidden) according to the initial concentration of particles. After the distribution of the particles, the value n(a) was computed: n(m)=
522
100~t-~1000,n(0)=0.1,0.2 ,..., l.O.Foreachvalue of n(O), the n(t)-n(m) dependence was averaged over ten realizations of the reaction. Three realizations of percolation systems were used. The time dependence of [n(r)-n(oo)]t0.8 for n(O)=O.l is shown in fig. 1. Fig. 2 presents the time dependence of n(t) -n(a) in a double-logarithmic scale (n(O)=O.l, 0.2, 0.5, 1.0 from top to bottom). The units along the Y axis are arbitrary. The curves are shifted in such a way that they have a common point at t= 1000. One can see that all the curves have the same asymptotic slope and that the establishing of this asymptotic behavior becomes slower for smaller initiai concentrations. The dependence of n (cc ) versus n(0 ) is shown in fig. 3. The ratio of the initial and final concentrations changes from a factor 2 (for small n( 0) ) to 6 0.8 ,
nW
I
d-
/N,,
where N, is the total number of sites, NCthe total number of clusters and P=O or 1 depending on the parity of the number of particles in the cluster. During one time step, each particle makes one jump to a neighbouring site of the same cluster. The jump direction is chosen by a random-number generator. Annihilation takes place for a pair of particles occurring on the same site (in the case of a multiparticle collision, only two particles annihilate). It was found that the concentration kinetics of the particles for large times t>t, follows the law n(t)-n(m)-tAa with cr=O.80?0.04. The different initial concentrations give the following results: ( 1) For n( 0) = 1, the dependence is established by a few steps. (2) For n(0) ~0.1, the dependence is established after t 22100 time steps. The presented value of cx was obtained for
19 October 1990
/A----------
5QQ
t
Fig. I. The dependencen(t) _ t-‘.* at the initial concentration n(O)=O.l.
Fig 2. The dependence In [ n ( t) ] on In (t ) at different initial concentrations: (1) 1.0; (2) 0.5; (3) 0.2; (4) 0.1.
Volume 173, number 5,6
CHEMICAL PHYSICS LETTERS
I9 October 1990
lowance or prohibition of double occupation of sites). Thus, we have proved that the relaxation process of excitation concentration annihilating via the A+A-0 bimolecular reaction in a percolation system with finite clusters obeys the asymptotic law proposed in ref. [ 5 1. .The n(t)-n(c+t-d”2+e’ dependence of n(m) on n (0) is also investigated. References Fig. 3. The finite concentration dependence n(m) on the initial concentration n(O).
(for n(0) SE1). We note, however, that this dependence must be different for the different lattices and different initial distributions of the particles (e.g. al-
[ I ] J. Klafter, A. Blumen and J. Zumofen, J. Luminescence 31/ 32 (1984) 627. [ 21 K. Kang and S. Render, Phys. Rev. A 32 ( 1985) 435. [ 3 ] D.F. Calef and J.M. Deutch, Ann. Rev. Phys. Chem. 34 (1983) 493. [4] 1. Webman, Phys. Rev. Letters 52 ( 1984) 220. [S] I.M. Sokolov,Fiz. Tverd. Tela, 31 (1989) 57.
523
CHEMICAL PHYSICS LETTERS
19 October 1990
The A+ A-+0 reaction on a critical percolation system A.G. Vitukhnovsky, N.V. Kiriakova and I.M. Sokolov P.N. Lebedev Physical Institute, USSR Academy of Sciences, Leninsky Pr. 53, 1 I7924
Moscow. USSR
Received 25 April 1990; in final form 7 August 1990
The diffusion-controlled reaction A+A+O on a percolation system with finite clusters is investigated by Monte Carlo simulation. The asymptotic law of reaction kinetics n( 1) and the dependence of n(w) on n( 0) were found.
Recently, great attention has been paid to the investigation of diffusion-controlled reactions on fractal systems (refs. [ l-31, and references therein), in particular on percolation systems. The typical realization of such situations is the triplet-triplet exciton annihilation on mixed moIecular crystals or on pore-transparent matrices doped by dyes. The existence of finite clusters or isolated pores results in some deviations of the time behavior of the particle concentration n(r) from the case obtained for the AS A-0 reaction on a connected fractal system n(l)-t9’2, where D, is the spectral dimension of the system. This reaction kinetics on a percolation system with finite clusters is the object of the study in the present paper. In connected systems, the random walks of the particles are ergodic. In this case, n(t) is governed by the equation
Wt)
-=-c(t)
dt
n’(t) ,
where c(r) =dS( t)/dt and S(t) is the average number of distinct sites visited by a random walker during time t. At the furthermost stage of evolution, we have the law n(t) -S(I) -‘. If a system is disconnected (has finite clusters) and one substitutes instead of S( t) the number of distinct visited sites averaged by the position of the initial point of a walk S(t) -I, one will obtain the result n(l) -S(t) - ’ - t -DL’2 with the renormalized value of the spectral dimension 0: =D,( 2-&D), where D is the incipElsevier Science Publishers B.V. (North-Holland)
ient infinite-cluster fractal dimension and d the embedding space dimension [ 41. The S(t) time dependence itself was studied by numerous computer simulations but we shall try to demonstrate that the relation n(t) -S(t) -’ is valid only for a small reaction range and is different from the large-time asymptotic reaction behavior. This asymptotic behavior was considered in ref. [ 51, The main ideas of this approach are as follows: The reaction in each cluster must be considered separately. The reaction behavior in a finite system depends on the relation between two characteristic lengths: the diffusion length 1(t) -a( t/T)“(2ceJ and the cluster size L-&4? Here a and T are the lattice spacing and the hopping time (in what follows, we shall use dimensionless units and put a= r= 1), 8 is an anomalous diffusion exponent and M the number of sites in a cluster. Ifn(0)-2~D~
-IW-~~/~ for thi2-eM, -0
for t”/‘>M.
(2)
Then it is possible to determine the overall particle 521
Volume173,number 5,6
CHEMICAL PHYSICS LETTERS
concentration by averaging this expression over the cluster-size distribution, n(t)--n(m)=
;
[NhW-NwM)lIT(M), (3)
with L!(M) -JM-‘-~‘~ (the average number of clusters of M sites per n sites). Evaluation of this sum gives n(t)_n(co)_t-‘“/2’dlD=t_d/(2+e) -
(4)
The exponent (Y= d/ (2 + 0) for d= 3 is significantly different from OS/2 or D:/2: a=O.80, D,/2~0.61, 4/2~0.53 (in the three-dimensional case, Dx2.5, D,z 1.33). The characteristic time of establishing such asymptotics is of the order of town;2/“. The results of our investigations of such reactions by direct computer simulation are as follows: The reaction was simulated on a percolation system (site problem) for a cubic lattice of 40 x 40 x 40 sites. The concentration of “good” sites p=pC= 0.3 1. The good sites were occupied randomly by particles (double occupation for one site is forbidden) according to the initial concentration of particles. After the distribution of the particles, the value n(a) was computed: n(m)=
522
100~t-~1000,n(0)=0.1,0.2 ,..., l.O.Foreachvalue of n(O), the n(t)-n(m) dependence was averaged over ten realizations of the reaction. Three realizations of percolation systems were used. The time dependence of [n(r)-n(oo)]t0.8 for n(O)=O.l is shown in fig. 1. Fig. 2 presents the time dependence of n(t) -n(a) in a double-logarithmic scale (n(O)=O.l, 0.2, 0.5, 1.0 from top to bottom). The units along the Y axis are arbitrary. The curves are shifted in such a way that they have a common point at t= 1000. One can see that all the curves have the same asymptotic slope and that the establishing of this asymptotic behavior becomes slower for smaller initiai concentrations. The dependence of n (cc ) versus n(0 ) is shown in fig. 3. The ratio of the initial and final concentrations changes from a factor 2 (for small n( 0) ) to 6 0.8 ,
nW
I
d-
/N,,
where N, is the total number of sites, NCthe total number of clusters and P=O or 1 depending on the parity of the number of particles in the cluster. During one time step, each particle makes one jump to a neighbouring site of the same cluster. The jump direction is chosen by a random-number generator. Annihilation takes place for a pair of particles occurring on the same site (in the case of a multiparticle collision, only two particles annihilate). It was found that the concentration kinetics of the particles for large times t>t, follows the law n(t)-n(m)-tAa with cr=O.80?0.04. The different initial concentrations give the following results: ( 1) For n( 0) = 1, the dependence is established by a few steps. (2) For n(0) ~0.1, the dependence is established after t 22100 time steps. The presented value of cx was obtained for
19 October 1990
/A----------
5QQ
t
Fig. I. The dependencen(t) _ t-‘.* at the initial concentration n(O)=O.l.
Fig 2. The dependence In [ n ( t) ] on In (t ) at different initial concentrations: (1) 1.0; (2) 0.5; (3) 0.2; (4) 0.1.
Volume 173, number 5,6
CHEMICAL PHYSICS LETTERS
I9 October 1990
lowance or prohibition of double occupation of sites). Thus, we have proved that the relaxation process of excitation concentration annihilating via the A+A-0 bimolecular reaction in a percolation system with finite clusters obeys the asymptotic law proposed in ref. [ 5 1. .The n(t)-n(c+t-d”2+e’ dependence of n(m) on n (0) is also investigated. References Fig. 3. The finite concentration dependence n(m) on the initial concentration n(O).
(for n(0) SE1). We note, however, that this dependence must be different for the different lattices and different initial distributions of the particles (e.g. al-
[ I ] J. Klafter, A. Blumen and J. Zumofen, J. Luminescence 31/ 32 (1984) 627. [ 21 K. Kang and S. Render, Phys. Rev. A 32 ( 1985) 435. [ 3 ] D.F. Calef and J.M. Deutch, Ann. Rev. Phys. Chem. 34 (1983) 493. [4] 1. Webman, Phys. Rev. Letters 52 ( 1984) 220. [S] I.M. Sokolov,Fiz. Tverd. Tela, 31 (1989) 57.
523