17446
J. Phys. Chem. 1996, 100, 17446-17451
Imperfect Annihilation Reaction in a One-Dimensional Gas Wen-Shyan Sheu Department of Chemistry, Fu-Jen Catholic UniVersity, Taipei, Taiwan 242, R.O.C. ReceiVed: July 12, 1996X
An imperfect annihilation reaction in a one-dimensional gas is studied to assess the effects of a reaction’s imperfectness on its kinetics. The reaction is initially mapped onto a random walk process with memory. After a master equation for the random walk is obtained, the system is reformulated with the first passage time approach developed in stochastic processes. Consequently, the time dependence of the reactant’s survival probability can be perturbatively found in both nearly perfect and weak reaction limits. Hence, the influence of imperfectness effects on reaction kinetics can be addressed. Although the anomalous kinetics for its diffusive counterpart in a low dimension is well-known, the conditions under which mean-field kinetics might be applicable to the current system are discussed.
I. Introduction Although it is commonly believed that a reaction’s rate depends on the order of its reactants, this statement is somehow oversimplified. The statement is based on a mean-field assumption and implies a uniform or well-stirred reaction system. However, for reactions occurring in a sufficiently low dimensional space, a well-stirred system is no longer available. Previous works indicate that a reaction’s rate slows down1,2 in the process due to the development of mesoscopic or macroscopic spatial structures. This phenomenon occurs since the reactants cannot be mixed efficiently in a low dimension. These kinds of kinetics are often referred to as “anomalous” because they differ from the classical rate laws and have been observed in several laboratory experiments3-6 as well. A great deal of research activity has been undertaken to study these anomalous kinetics, particularly for irreversible diffusioncontrolled reactions.1,2 They include one-species and twospecies annihilation reactions,1 trapping reactions,7 and coagulation reactions8 among others. For instance, in a single-species annihilation reaction A + A f L,9 the asymptotic behavior of density over a long time decays as t-d/2 for spatial dimension d < 2; meanwhile, for d g 2, the density decays according to the mean-field prediction of t-1. The differences between classical and anomalous behavior reflect the deviation of the distribution of the nearest neighboring distances from the Poissonian form that underlies the classical rate law. No mixing is found in this reaction. Nevertheless, the contrasting situation in which the reactants move ballistically has received much less attention. Consequently, only a few cases have been reported in the literature.10-12 Among them is the single-species annihilation reaction introduced by Elskens and Frisch.10 Here, a dichotomic distribution of velocities c and -c is used to model particle motion. Although the motion is different from its counterpart in the diffusion-limited reaction addressed above, the asymptotic density for the ballistic reactants is still t-1/2 for initially randomly distributed reactants in one dimension. Again, the major reason for this slowdown is due to the deviation from the Poissonian distribution as the reaction proceeds. Aggregation of particles with the same velocities emerges in the kinetics. This is attributed to the particle imbalance in any given interval length for an initially random spatial distribution of particles.10 X
Abstract published in AdVance ACS Abstracts, October 1, 1996.
S0022-3654(96)02082-5 CCC: $12.00
The goal of the present research is to investigate the kinetics of an imperfect annihilation reaction in a low-dimensional gas. In practice, no reaction is perfect due to energy barriers, reaction orientation, and/or many other factors. One of the major features of an imperfect reaction in a low dimension, which is distinguished from a perfect one, is the introduction of mixing into the reaction via the position or velocity exchange between nonreactive encounter partners. Due to the conservation of linear momenta, a nonreactively encountering particle pair will exchange their velocities for similar particles. This effect will introduce particles of one velocity into the above-mentioned aggregates with the other velocity and hence shrink their aggregation sizes. This mixing effect will obviously affect the distribution of reactants and, hence, a reaction’s kinetic behavior. It is noted that there is another kind of “mixing effect” discussed in the context of anomalous diffusive behaviors such as Levy’s walk.13 To avoid confusion in the terminology used, we will simply use the term “imperfectness effects” or something similar throughout the text to include the mixing effects in this context. In a recent study,14 Lindenberg et al. discussed diffusionlimited annihilation reactions in various dimensions. Besides demonstrating that the system shows the expected anomalous kinetics behavior in one dimension, they further assure that this anomaly is independent of the reactive probability upon encounter. However, this result is somewhat counterintuitive since as the weak reaction limit is approached, mean-field results should be expected to be recovered, as implied in the argument above on imperfectness effects. Therefore, it will be interesting to examine under what conditions the mean-field theory will be applicable to reaction kinetics of an imperfect reaction in a low dimension. With these considerations, we will study an imperfect annihilation reaction in a one-dimensional gas, which is accessible to exact analytic methods, to assess imperfectness effects on anomalous kinetics in low dimensions. In addition, most existing theories of anomalous kinetics adopted approximate approaches and focused only on their asymptotic behaviors at long times due to the complexities of the problems involved. Nevertheless, since asymptotic times are so long (several orders of magnitude15) for practical purposes, there has been a lot of interest in the behavior of these systems over the entire time history. This study represents one of few existing exact analytic approaches to the anomalous kinetics, making the theoretical study of the entire time history possible. © 1996 American Chemical Society
Imperfect Annihilation Reaction in a One-Dimensional Gas
J. Phys. Chem., Vol. 100, No. 44, 1996 17447
This paper is organized as follows. The model is elaborated in section II. In section III, we map the reaction onto a random process with memory and write the expression for the survival probability of particles. For the convenience of solving the problems, the survival probability is rewritten in the formalism of the first passage time approach in section IV. The resulting equations are then solved in both nearly perfect and weak reaction limits in section V. Discussions of the imperfectness effects on anomalous kinetics are also given in this section. Conclusions are stated in the final section. II. Model System Our system is an extension of that of Elskens and Frisch.10 Initially, particles of type A are randomly distributed in an infinite one-dimensional space. The particle density is σ. Each particle moves in either a positive or negative direction with probability p and q, respectively. Without loss of generality, p e q is assumed. The magnitude of velocity is c. Two particles, once meeting, annihilate each other with reaction probability R. Therefore, the nonreactive probability is β ) 1 - R, and these nonreactive particles should rebounce with their velocity interchanged to conserve momenta. This is physically equivalent to the process in which nonreactive pairs continue to move with their initial velocities, crossing each other’s trajectories. Schematically, the reaction is denoted as
A+AfL
(1)
As addressed in the Introduction, the imperfectness of the reaction will change the aggregation sizes of particles with the same velocity. The larger the β value is, the smaller the aggregation sizes will become. This will influence the particle distribution, which subsequently changes the reaction kinetics. For comparison with the microscopic theory in later sections, we first write down the survival probability obtained by the classical rate law generally assumed in physical chemistry textbooks. For convenience, particles with c and -c are denoted by X and Y, respectively. The reaction is hence transformed into
X+YfL
(2)
The densities of particle X and Y at time t are expressed as FX(t) and FY(t), respectively. Their initial values are FX(0) ) pσ and FY(0) ) qσ, and they are related by pσ - FX(t) ) qσ - FY(t). Therefore, the classical rate law is
dSX(t) dFX(t) ) ) -2cRFX(t) FY(t) dt dt
(3)
pσ
where SX(t) ) FX(t)/pσ is the survival probability of X particles at time t. Notably, the decay rate is linearly scaled down by the reaction probability R in the classical rate law. This statement is demonstrated in this study to be incorrect, as a wellstirred system is, in general, unavailable. The solution is then easily found as
FX(t) )
(q - p)σ
(4)
p R(q-p)kt -1 e q
where k ≡ 2cσ. The asymptotic behavior at long times is
q - p -R(q-p)kt for p * q e q
(5a)
SX(t) ) (Rkt/2)-1 for p ) q ) 1/2
(5b)
SX(t) )
Figure 1. Neighbors of an X mapped onto the distances of a discretetime random walk. Note that there are three options applying to Y particles in the mapping (see text for explanation).
which is an exponential decay or power law depending on the relation between p and q. On the other hand, the short time behavior is
SX(t) ) 1 - qRkt + (q/2)(Rkt)2 - (q/6)(1 + 2pq)(Rkt)3 + ... (6) which is a power expansion series in t. In the following sections, a microscopic theory will be developed for the proposed model. III. Master Equation Since the velocity of each particle is independent of its position, the joint distribution of velocity and position can be written as the product of the position and velocity distribution. From the system setup, it is clear that a particle will not maintain its velocity until it reacts with an oppositely moving particle. The time it takes for a particle to react is determined by two factors: how fast it moves and how far it initially is from its reaction partner. A particle’s speed affects its survival time by scaling, which is easy to calculate. Hence, we will focus on finding a particle’s reaction partner first. In a perfect annihilation reaction, Elskens and Frisch10 used a recursive relation to determine a particle’s reaction partner. However, their method is very awkward when applied to the imperfect annihilation reaction considered in this study since all particles between a reaction pair do not need to completely annihilate each other in the imperfect annihilation reaction. Therefore, a different approach is adopted here. The reaction will be mapped onto a random walk process, which was previously applied to a coagulation reaction in a one-dimension gas.11 To begin with, we select any particle from the system and attempt to find its reaction partner. Assume the particle chosen is moving to the right, i.e., an X particle. By symmetry, a particle moving to the left can be treated similarly. Since the velocity in the system is dichotomic, this X particle can only react with a particle moving to the left. In addition, since it is impossible for the X particle to react with particles on its left, the particles on this side can be neglected. The reaction partner of the X particle can be located by making the following mapping onto a discrete-time random walk with memory (cf. Figure 1). If we take the x and y axis as the “time” and “position” axis, respectively, the particle is viewed as a random walker starting at a distance m ) 1 from the x axis at “time” n
17448 J. Phys. Chem., Vol. 100, No. 44, 1996
Sheu
) 0. The “time” variable n will correspond to the subsequent neighbors to the right of the chosen X particle. On the other hand, the “position” variable m is defined by the mapping that there are still m - 1 unpaired X particles in front of the chosen X particle at a certain “time”. An unpaired particle is a particle that is still seeking its reactive partner. The physical meaning of the description above to the system will become clearer later on. If the walker is at “position” m at “time” n - 1, it will take the following moves for the next step. If its nth neighbor is an X particle, occurring with probability p, the walker will take a step of length 1 away from the x axis. This is because the neighbor will protect the selected particle from annihilation. On the other hand, if its nth neighbor is a Y particle, happening with probability q, the walker has several options to take for its next step. It may move one step toward the x axis, remain at the same position, or return to the origin. The last option implies that the Y particle is the chosen X particle’s reaction partner. The probability of each option is described as follows. Since the walker is currently at “position” m, there are m - 1 X unpaired particles between the selected X particle and the Y particle. If the Y particle annihilates with any of the m - 1 X particles, the walker should take a step toward the x axis. This can occur if the Y particle reacts with the first X particle or the second X particle, ..., to its left. That is, the probability for this to take place is
R + Rβ + Rβ + ... + Rβ 2
m-2
)1-β
m-1
(7)
If the Y particle does not react with any of the m - 1 and chosen X particles, the walker should remain at the same “position”. By an argument similar to that above, the probability is βm. The last probability is that the Y particle might nonreactively encounter all of the m - 1 X particles in between before reacting with the chosen X particle. Obviously, the probability is Rβm-1. As a result of this mapping procedure, we can write the master equation for the probability distribution to reach “position” m in n “time” steps, Pm(n). According to the description in the previous paragraph, the equation for Pm(n) is
Pm(n + 1) ) q(1 - βm)Pm+1(n) + qβmPm(n) + pPm-1(n) (8) The first term on the right-hand side of the equal sign arises when the (n + 1)th neighbor of the chosen X particle is a Y particle, occurring with probability q, and it will react with any of the m X unpaired particles between itself and the selected X particle left at “time” n. The second term is produced by the nonreactive encounters of the X particles of interest and the Y particle. The last term is obviously due to the fact that the (n + 1)th neighbor is an X particle. Due to the chosen X particle, this master equation is to be solved with the initial condition that Pm(0) ) δm,1, where δm,1 is the Kronecker delta. The reaction partner is identified with the first “time” when the walker reaches the “position” of origin. Therefore, the absorption boundary condition P0(n) ) 0 is imposed upon the master equation. Upon solving the above master equation, the reaction partner can be identified. The “survival” probability, defined as the probability that the reaction partner of the chosen X particle at ∞ Pm(n). Hence, the prob“time” n is still not found, is ∑m)1 ability that the reaction partner is the nth neighbor, G1(n), is the difference between the survival probability at “time” n - 1 and n, or
∞
G1(n) )
∑ {Pm(n - 1) - Pm(n)}
(9)
m)1
In terms of random walk language, G1(n) is the first passage “time” that the walker starting from position “1” reaches the origin at “time” n. However, this “time” is not a real time. The velocity and initial position distribution of particles must be considered in order to obtain the real time survival probability SX(t) of the chosen X particle.10,11 SX(t) is determined from the probability F(∆xe2ct) that the initial distance ∆x between the chosen X particle and its reaction partner is less than 2ct, which is the greatest distance traveled as the particles move toward each other in the time interval t. This survival probability is equal to 1 minus all probabilities that the X particle reacts within time t or, namely, ∞
SX(t) ) 1 - ∑G1(n) F(xn-x0e2ct)
(10)
n)1
where xn is the initial position of the nth neighbor. The summation is used to include all of the possible reaction neighbors. If the initial distances yn ) xn - xn-1 are identically distributed independent random variables with probability density Ψ(yn), F(xn-x0e2ct) can be expressed in terms of Ψ as
F(xn-x0e2ct) ) n
∫0
2ct
Ψ(yn) ∫0
2ct-yn
Ψ(yn-1) ...∫
2ct-∑yk k)2
Ψ(y1) dy1...dyn-1 dyn
0
(11) The Laplace transform of SX(t) is therefore equal to
1 1∞ ˆ (s)]n Sˆ X(s) ) - ∑G1(n)[Ψ s s n)0 1 1 ) - G ˆ (s)) ˆ (Ψ s s 1
(12)
where G ˆ 1(z) is the generating function of G1(n), ∞
G ˆ 1(z) ) ∑G1(n)zn
(13)
n)0
and Ψ ˆ (s) is the Laplace transform of Ψ(x). For an initially random distribution of distances between adjacent particles, the distance distribution is Poissonian:
Ψ(x) ) σe-σx Ψ ˆ (s) ) k/(s + k)
(14)
From eq 12, it can be shown that the physical meaning of ˆ (s)) is the Laplace-transformed decay rate of the survival G ˆ 1(Ψ probability. Hence, we have obtained the expression for the survival probability in real time. The kinetics of the annihilation reaction is ready for study once the behavior of Pm(n) or G1(n) is known. The final task involves solving the above master equation. Nevertheless, the master equation is noted to be a difference equation with variable coefficients, which corresponds to a random walk with memory and reduces to the master equation for an ordinary random walk at β ) 0. Just as in differential equations with variable coefficients, obtaining the exact solution of this master equation is difficult. In addition, even if Pm(n) can be obtained, the summation in eq 9 may be difficult to
Imperfect Annihilation Reaction in a One-Dimensional Gas
J. Phys. Chem., Vol. 100, No. 44, 1996 17449
perform. Therefore, we will choose to work directly with G1(n), which we will describe in the next section. IV. First Passage Time Approach It is more convenient to rewrite the master equation in terms of first passage time formalism. Define Gm(n) as the probability for the walker to reach the origin for the first time in n “time” steps after starting from position m. Following similar arguments used to obtain eq 8, the walker at position m can take its first step to position m + 1, m - 1, m, or “0”. Therefore, the equations for Gm(n) are
Gm(n) ) q(1-βm-1) Gm-1(n - 1) + qβmGm(n - 1) + pGm+1(n - 1) + qRβ
m-1
δn,1 for m g 2 (15a)
G1(n) ) qβG1(n-1) + pG2(n - 1) + qRδn,1 for m ) 1 (15b) where n g 0. The initial and boundary condition are Gm(0) ) 0 and G0(n) ) 0, respectively. The quantity of interest is G1(n), which is the first passage time distribution to reach position “0” in n steps starting from position m ) 1. Multiplying eqs 15 by zn and summing the results from n ) 1 up to n ) ∞, the equations for the generating function of Gm(n) can be obtained:
pzG ˆ m+1(z) + qRβm-1z for m g 2 (16a)
V. Asymptotic Solutions Here, the asymptotic solutions of G ˆ 1(z), SX(t), and its Laplace transforms will be developed for both limiting cases of β f 0 and β ) 1 - R f 1. A. β f 0. The nearly perfect reaction limit is first discussed. ˆ m(0)(z) + βG ˆ m(1)(z) + β2G ˆ m(2)(z) + .... Then, Express G ˆ m(z) ) G by combining it with eqs 16 and collecting the terms with the same order in β, the equations for each order of β are obtained, which are discussed further. ˆ m(0)(z). The equations for the zeroth (a) β0 Order Term G order of β are
ˆ m-1 (z) + pzG ˆ m+1 (z) for m g 2 G ˆ m (z) ) qzG (0)
(0)
ˆ 2(0)(z) + qz for m ) 1 G ˆ 1(0)(z) ) pzG
(17a) (17b)
These are the equations for the perfect reaction. With standard methods and the condition that G ˆ m(z) should be finite for all m, it is easy to show that
G ˆ m(0)(z) ) γ-m where
(19)
Therefore, G ˆ 1(0)(z) ) γ-, the survival probability of the prefect reaction, is recovered.10 (b) β1 Order Term G ˆ m(1)(z). The equations for the first-order terms in β are more complicated. They are
ˆ m-1(1)(z) + pzG ˆ m+1(1)(z) for m g 3 G ˆ m(1)(z) ) qzG
(20a)
ˆ 1(1)(z) + pzG ˆ 3(1)(z) - qzG ˆ 1(0)(z) + qz (20b) G ˆ 2(1)(z) ) qzG ˆ 1(0)(z) + pzG ˆ 2(1)(z) - qz G ˆ 1(1)(z) ) qzG
(20c)
As in the case of the zeroth-order eqs 17, the solution of eq 20a with the boundary conditions at infinity is
G ˆ m(1)(z) ) dγ-m for m g 2
(21)
where d is a constant to be determined from eqs 20b and 20c. Substituting this result and those from the zeroth-order calculation into these two equations leads to
ˆ 1(1)(z) + pzdγ-3 - qzγ- + qz dγ-2 ) qzG
Therefore, G ˆ m(1)(z) can be easily found to be
1-z 2 G ˆ 1(1)(z) ) γ qz -
(16b)
Notably, the coefficients of G ˆ m(z) still depend on m. However, these equations are easier to work with than the equation for Pm(n). To proceed further, an attempt is made to find its asymptotic solutions by the perturbation method in two limits: β f 0 and β ) 1 - R f 1, which correspond to the nearly perfect and weak reaction limits, respectively. On the basis of these results, the imperfectness effects in this reaction will be discussed.
(0)
1 - x1 - 4pqz2 2pz
G ˆ 1(1)(z) ) pzdγ-2 - qz + qzγ-
ˆ m-1(z) + qβmzG ˆ m(z) + G ˆ m(z) ) q(1 - βm-1)zG
ˆ 1(z) + pzG ˆ ∞(z) + qRz for m ) 1 G ˆ 1(z) ) qβzG
γ- )
(22)
of which the definition of γ- has been used in the derivation. To sum up, the solution of G ˆ 1(z) up to first order is
G ˆ 1(z) ) γ- -
1-z 2 γ β qz -
(23)
As previously mentioned, G ˆ 1(z) is the decay rate of the survival probability in the Laplace domain. Notably, the second term of eq 23 on the right-hand side is negative, implying that the rate is slowing down compared with the case of β ) 0. Although this is expected due to the imperfectness of the reaction, the slowdown is not as much as expected based on the linear rate with respect to R as predicted by the mean-field theory (cf. eq 3). In addition, the rate is not scaled linearly in R as usually assumed in mean-field theory. To see these points more closely, we rewrite eq 23 as
(
G ˆ 1(z) ) Rγ- + βγ- 1 -
1-z γ qz -
)
(24)
From eqs 14 and 19, it is a simple matter to show that the second term is positive at a long time limit. Therefore, the rate is βγ-(1 - [(1 - z)/qz]γ-) larger than the linear scaled rate Rγ- in the mean-field theory at a long time limit. Obviously, this speedup is due to the shrinkage of aggregation sizes previously discussed in the Introduction, which affects particle distribution and increases the number of reacting pairs in the imperfect reaction. To examine the survival probability in the time domain, eq 23 is inserted into eq 12 and the inverse Laplace transform of Sˆ X(s) is performed to find
(18) SX(t) ) 1 -
x
-kτ
q te ∫ p 0
-kt I1(bτ) 2β e I2(bt) dτ + (25) τ p kt
17450 J. Phys. Chem., Vol. 100, No. 44, 1996
Sheu
where b ≡ 2xpqk and In(x) is a modified Bessel function. Hence, the long time behavior of SX(t) is
SX(t) )
q(kt)-3/2e-(1-2xpq)kt (pq)3/4(4π)1/2
[
SX(t) ) (πkt/2)-1/2 1 +
[
]
1
2β + + ... (1 - 2xpq) xpq for p < q (26a)
1 1 β - + ... 2kt 4
) ]
(
for p ) q ) 1/2 (26b)
which is an exponential decay or power law, depending on the relation between p and q. However, in either case, the exponent or power is different from the results of the mean-field theory (cf. eqs 5). In addition, when xpq is very small, the second term of eq 26a may become so large that the expansion breaks down. This occurrence implies that when xpq f 0, the asymptotic behavior of SX(t) for an imperfect reaction may be different from that of the perfect one, no matter how small the nonreactive probability is. This phenomenon indicates the possible existence of unstable kinetics for the small pq case and is probably due to the rare events dominated by the tail of the particle distribution. This point will be pursued further in future studies. On the other hand, the short time behavior is
(kt)2 2
SX(t) ) 1 - Rqkt + (1 - 2β)q
3
[(1 + pq) - β(3 + 2pq)]q
(kt) + ... (27) 6
Inspecting the above equation, the expected deviation of the kinetics from the mean-field results can be found. However, surprisingly, the deviation arises only at a much later time. To the first order in β, the deviation initiates from the third-order terms and is proportional to pq. B. β f 1. This case corresponds to the weak reaction limit. Since the better expansion parameter is R, we will write G ˆ m(z) ˆ m(1)(z) + R2G ˆ m(2)(z) + .... As in the case of a )G ˆ m(0)(z) + RG nearly perfect reaction limit, we combine it with eqs 16 and collect terms of the same order in R to obtain their respective equations. ˆ m(0)(z). The equation for the zeroth order (a) R0 Order Term G of R is
ˆ m(0)(z) + pzG ˆ m+1(0)(z) G ˆ m(0)(z) ) qzG
(28)
This is the equation with zero reaction probability. The solution is easily found as
G ˆ m(0)(z) )
(1 -pzqz)
m-1
G ˆ 1(0)(z) ) 0
(29)
The last equality is obtained by imposing the condition that ˆ m(0)(z) is 0 for all G ˆ m(z) should be finite for all m. Therefore, G m. This result is anticipated since no reaction should occur with zero reactive probability. ˆ m(1)(z). The equation of the first order (b) R1 Order Term G is
ˆ m-1(0)(z) + qz[-mG ˆ m(0)(z) + G ˆ m(1)(z) ) qz(m - 1)G G ˆ m(1)(z)] + pzG ˆ m+1(1)(z) + qz (30) Letting G ˆ m(1)(z) be independent of m and substituting it into the equation, we readily find
G ˆ m(1)(z) ) qz/(1 - z)
(31)
The constant is a reflection of each particle experiencing the same reaction condition, independent of its neighboring particle distribution at the lowest order of the low reactive probability. It is because, in most cases, particles will only pass each other without reacting. Therefore, the reaction probability does not depend on the number of X particles between the chosen X particle and its reactive partner. (c) R2 Order Term G ˆ m(2)(z). With the assistance of G ˆ m(0)(z) ) 0 and eq 31 and simple manipulation, the second-order equation is as follows:
ˆ m(2)(z) + pzG ˆ m+1(2)(z) - qzG ˆ m(1)(z) G ˆ m(2)(z) ) qzG qz(m - 1) (32) By allowing G ˆ m(2)(z) ) am + b and inserting it into this equation, the solution is
G ˆ m(2)(z) ) -
qz(1 - 2z) qz m+ 1-z (1 - z)2
(33)
To sum up, the solution of G ˆ 1(z) up to the second order is
qz qz2 - R2 1-z (1 - z)2
(
G ˆ 1(z) ) R
)
(34)
The physical meaning of the equation will be transparent after inserting it into eq 12 for Sˆ X(s) and performing the inverse Laplace transform on the results. The final equation is
SX(t) ) 1 - Rqkt + (q/2)(Rkt)2 + ...
(35)
Unlike the expansion of the survival probability in the nearly perfect reaction limit, eq 25, the expansion in the weak reaction limit is valid only at short times. This has a simple explanation. The first-order term arises from the chosen X particle reacting with any Y particle to its right, while nonreactively passing any particles between this reactive pair. In addition, no reaction should occur among the in-between particles in order to contribute to the survival probability up to first order in R. Owing to this reason, the reactive pair cannot be too far apart; otherwise, reactions of other pairs may first occur, which will give higher order terms in R instead. A direct comparison of eq 35 with classical results at short times, eq 6, reveals that they are identical to the second order in R, as in the case of the nearly perfect reaction limit. Calculating further to the third order in R and comparing the results with the mean-field theory, we still find an agreement. This result suggests the difficulties in forming particle aggregates of the same velocities in the weak reaction system. It should also be compared with the deviation starting at this order in the nearly perfect reaction case. Therefore, we can conclude that, in the case of the weak reaction limit, the kinetics is of meanfield type at short to medium times, independent of p or q. Since the imperfectness effects in the weak reaction limit are the strongest, particle aggregation may not have a chance to form. Hence, the kinetics is very likely of mean-field type even for a long period. This point will be further investigated in future studies. In addition, the results should be compared with its diffusive counterpart,14 of which the kinetics shows anomalous behaviors regardless of the reaction probability upon contact. The difference, if any, of these kinetic behaviors may be due to the encounter number between any particle pairs. For ballistic particles, there is at most one encounter between any set of pairs. However, for diffusive particles, any encounter pair can re-
Imperfect Annihilation Reaction in a One-Dimensional Gas encounter a great number of times due to the well-known “cage effects”, which makes an imperfect reaction become perfect in effect. VI. Conclusions This work has investigated an imperfect annihilation reaction in a one-dimensional gas to evaluate the effects of a reaction’s imperfectness on its kinetics. The reaction was first mapped onto a random walk process with memory. The first passage time approach was then adopted to find the survival probability of particles in both perfect and weak reaction limits. In addition, the results obtained by this microscopic theory and the classical mean-field theory were compared. The imperfectness of a reaction is shown to have different effects on the kinetics in different time ranges. It makes the reaction rate nonlinearly depend on the reaction probability upon contact, R. This is different from the linear rate of R expected in the mean-field theory. At R ) 1, the results of a perfect annihilation reaction first given by Elskens et al.10 (cf. eq 18) is recovered by this mapping. From the long time behavior of the survival probability of particles, we find the effects of a reaction’s imperfectness do enhance the reaction rate as compared to the mean-field results (cf. eqs 23 and 24). This is attributed to the imperfectness effects on the particle distribution, probably by erosion of the formed particle aggregation due to introduction of particles of one velocity into particle aggregates with another velocity. In addition, although anomalous kinetics are expected in a low dimension for a nearly perfect reaction, they appear much later in time. Regardless of the reaction probability upon contact, the mean-field theory is applicable up to second order in time (cf. eqs 27 and 35). This may be due to the slow development of the mesoscopic or macroscopic structures of particles. Therefore, if only a not-too-long time range is of interest, the mean-field theory might still be applicable even to a reaction following anomalous kinetics. This will tremendously simplify the analysis of a reaction. As the reaction probability becomes weaker, the mean-field theory is even better, up to third or higher order in time. This may be
J. Phys. Chem., Vol. 100, No. 44, 1996 17451 attributed to the difficulties in forming particle aggregates of the same velocities in the weak reaction system. Hence we may conjecture that the mean-field theory is correct in the weak reaction limit, which is currently under investigation in our laboratory. Acknowledgment. The author is pleased to express his gratitude to Prof. Katja Lindenberg for many stimulating discussions. In addition, the author would like to thank the National Science Council, ROC, for financial support of this work under Contract No. NSC 85-2113-M-030-005. References and Notes (1) Kuzovokov, V.; Kotomin, E. Rep. Prog. Phys. 1988, 51, 1479 and references therein. (2) Special Issue of J. Phys. Chem., dedicated to R. Kopelman (July 1994). (3) Klymko, P. W.; Kopelman, R. J. Lumin. 1981, 24/25, 457; Ibid. J. Phys. Chem. 1982, 86, 3686. (4) Evesque, P.; Duran, J. J. Chem. Phys. 1984, 80, 3016. Klymko, P. W.; Kopelman, R. J. Phys. Chem. 1983, 87, 4565. (5) Hunt, I. G.; Bloor, D.; Movaghar, B. J. Phys. C 1983, 16, L623; Ibid. 1985, 18, 3497. (6) Koo, Y.-E.; Kopelman, R.; Yen, A.; Lin, A.; In Dynamics in Small Confining Systems; Drake, J. M., Klafter, J., Kopelman, R., Awshalom, D. D., Eds.; Mater. Res. Symp. Proc. 1993, 298, 273. (7) Grassberger, P.; Procaccia, I. J. Chem. Phys. 1982, 77, 6281. (8) Doering, C. R.; ben-Avraham, D. Phys. ReV. A 1988, 38, 3035; Phys. ReV. Lett. 1989, 62, 2563. Bursvhka, M. A.; Doering, C. R.; benAvraham, D. Phys. ReV. Lett. 1989, 63, 700. (9) Torney, D. C.; McConnell, H. J. Phys. Chem. 1983, 87, 1941. (10) Elskens, Y.; Frisch, H. L. Phys. ReV. A 1985, 31, 3812. (11) Sheu, W.-S.; Van den Broeck, C.; Lindenberg, K. Phys. ReV. A 1991, 43, 4401. (12) Ben-Naim, E.; Kreapivsky, P.; Leyvraz, F.; Redner, S. J. Phys. Chem. 1994, 98, 7284 and references therein. (13) Zumofen, G.; Klafter, J. Phys. ReV. E 1994, 50, 5119. Klafter, J.; Shlesinger, M. F.; Zumofen, G. Phys. Today 1996, 49 (2), 33. (14) Lindenberg, K.; Argyrakis, P.; Kopelman, R. J. Chem. Phys. 1995, 99, 7542. (15) Lin, A.; Kopelman, R.; Argyrakis, P. Phys. ReV. E 1996, 53, 1502 and references therein.
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