J. Phys. Chem. 1995, 99, 1228-1234
1228
Macrodynamics of Elementary Bimolecular Reactions: A Monte Carlo Bridge between Experiment and Theory Yong-Eun Lee Koo,t3S Kook Joe Shin,s and Raoul Kopelman**$ Department of Chemistry, Chung-Ang University, Seoul, Korea; Department of Chemistry, The University of Michigan, Ann Arbor, Michigan 481 09-1055; and Department of Chemistry, Seoul National University, Seoul, Korea Received: August 22, 1994@
To bridge laboratory experiments and analytical theory, we studied by Monte Carlo simulations the kinetic properties of the reaction front formed under initially separated conditions in the A B C reactiondiffusion system, for low reaction rates. We observed crossovers in (1) global reaction rates, (2) the position of the reaction front center, and (3) the width of the reaction front. The motion of the reaction front showed a U-tum in some cases but not in all the cases predicted by theory. Quantitative results, including time
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exponents, are presented and discussed. The very rich phenomenology of this simple elementary reaction is demonstrated and interpreted.
TABLE 1: Time Dependence of the Properties of the Reaction Front: Reaction-Limited vs Diffusion-Limited reaction-limited” diffusion-limitedb (early-timebehavior) (long-time behavior) tl12 t112 c center of reaction front t116 p 2 width of reaction front $12 global reaction rate tu2
I. Introduction
In this work Monte Carlo simulations are performed with the aim of bridging the apparent gaps between laboratory experiments and analytical theories. A reaction-diffusion system of the type A B C under initially separated conditions has drawn much recent interest.’-13 There are many good reasons to study such nonconvective systems. The initial These results are from refs 7 and 8. These results are from ref 1 . separation of reactants leads to a moving reaction front which See Table 2 for more detailed time-dependent behavior. is important in a variety of biological, chemical, and physical phenomena where pattern formation occur^.^^-^^ The A B TABLE 2: A Summary of the Various Time Dependences of the Center of the Reaction Front, as a Function of the C type elementary reactions have been very interesting in Parameters D and r (This Table Is from Ref 8) diffusion-limited kinetics since they show a persistent macroscopic segregation of reactants in low d i m e n s i o n ~ . l ~ We -~~ short intermediate long case ID, T I a times times times expect that such segregation will be exemplified by a reaction front that persists if the actual reaction rate (reaction probability) 0 a (D=l,r=l) 0 0 tl12 is finite. There is also a practical reason: in a real experiment b {D=l,r*l} 0 t312 c {D*I,r=l} p t112 p 2 for an initially homogenized reaction-diffusion system, the d ( 2 + f i > D > 1, < 1) or t”’ t3” with t”2 reactions have to be performed in a nonconvective media in extremum (1 l , r > 1)or homogenize a system under nonconvective conditions. It is { D < 1 , r < 1) easier to prepare two uniform reactants separately, with an a D = (DAIDB)”~ and r (a~/bo)I/~ interface, than to mix them uniformly without convection. Therefore, this is a better system for carrying out simple determining these properties. Their theoretical results predict elementary reaction experiments than an initially homogeneously various nontrivial crossovers and different universality classes mixed system. for the dynamics of the reaction center. (See Tables 1 and 2 The study of such systems was initiated by the theoretical for detailed time-dependent behaviors.) They tested such work of GAlfi and Rkcz.’ They found an anomalous long-time predictions using exact enumeration methods for the special case behavior of the kinetic properties of the reaction front using where two reagents have the same diffusion coefficients on a asymptotic scaling arguments. Specifically, the global reaction one-dimensional lattice. An experimental s t ~ d y ~adopting ,*~ a rate decreases with the front moves with (time)’”, and very slow chemical reaction has visualized some interesting the front width increases with Both e ~ p e r i m e n t a l ~ * ~ *behaviors ~ such as the change in the direction of motion of the and numerical s t u d i e ~ ~ were . ~ performed and verified their front, Le., the existence of an extremum, which is predicted predictions. There are, however, arguments6 that their predictheoretically under appropriate choices of the diffusion coeftions may be valid only above the upper critical dimension ficients and initial densities (Table 2). The experiments reveal which is dup= 2. Later, Taitelbaum et aL7s8have studied the some other features that the theory did not predict. We expected early-time behavior of the reaction front properties by using a that a numerical study would help to better understand this perturbation theory approach. They found that, unlike for the system and also bridge the results of theory and experiment. long-time limit, the system parameters play a crucial role in In this paper we present results from Monte Carlo simulations on a lattice model for a diffusion-reaction process of the type Chung-Ang University. A B C with initially separated components and with a The University of Michigan. slow reaction rate (low microscopic reaction probability). The 5 Seoul National University. Abstract published in Advance ACS Absrracfs, January 1, 1995. simulations were done only for the last two classes (d, e) of
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0022-365419512099-1228$09.0010
0 1995 American Chemical Society
Macrodynamics of Elementary Bimolecular Reactions
TABLE 3: Crossover Points of Reaction Rate I I1 I11 IV V VI VI1 VI11 IX X XI XI1 XI11 XIV XV
Pa
a:
boc
0.5 0.2 0.01 0.01 0.01 0.01 0.01 0.01 0.02 0.05 0.05 0.01 0.01 0.02 0.05
0.6 0.6 0.6 0.3 0.3 0.3 0.4 0.4 0.4 0.4 0.2 0.2 0.1 0.1 0.1
0.4 0.4 0.4 0.2 0.2 0.2 0.1 0.1 0.1 0.1 0.05 0.025 0.4 0.4 0.4
D A ~ DBe 0.66 0.66 0.66 0.3 1 1 1 2 1 1 1 1 1 1 1
1 1 1 1 3 2 2 4 3 3 3 3 3 3 3
t&
lattice
3.5 7.5 143 370 200 200 220 220 110 40 75 360 270 120 55
50 x 1000 50 x 1000 50 x 1000 100 x 200 100 x 400 100 x 400 100 x 400 100 x 400 100 x 400 100 x 400 100 x 4C;O 100 x 400 100 x 400 100 x 4 0 0 100 x 400
Reaction probability. Concentration of A at time 0. Concentration of B at time 0. Diffusion coefficient (or mobility) of A. e Diffusion coefficient (or mobility) of B. f Time step at which the crossover occurs.
Table 2, Le., for nonunity D and r. The effects of the diffusion coefficients and the initial concentrations of the two reagents on the kinetic properties were investigated. We did find the crossovers in the global reaction rates and in the position and width of the reaction front, including a U-turn of the front center. We have measured the time exponents of the properties and compare them with the theoretical results. We also give a physicochemical interpretation of the dramatically rich and antiintuitive phenomenology of this simple elementary chemical reaction. 11. Method of Computation Our Monte Carlo simulations were carried out on a twodimensional lattice. The lattice size used is typically 100 x 400 although a few simulations were done on a 50 x 1000 and 100 x 200 lattice. A 50 x 1000 lattice was not used much since it has unnecessarily many columns for the simulation conditions, and a 100 x 200 lattice was not big enough for long-time steps. However, all the lattices were good enough to observe crossovers for each condition of the run. Initially, the particles A and B are created randomly on the left and right halves of the lattice, respectively, with initial concentrations a0 and bo. No two particles of any type can occupy the same site simultaneously at this stage. At time step t = 0, particles start to diffuse (move) and react. The diffusion process is simulated as random walks by all particles. In early runs (I-IV of Table 3), particles either move only one lattice site or do not move. It is the probability of remaining in place, i.e., not moving, that enables the two particles to have different diffusion coefficients. In all later runs, the two particles are given different diffusion coefficients by letting particles move more than one lattice site, unidirectionally, but with no possibility of remaining in place. Each walker is forced to jump in a random direction to one of the available sites, the jump distance depending upon its diffusion coefficient (skipping over particles). When two particles of opposite type occupy the same site, a reaction may or may not occur since the preset reaction probability P used here is less than 100%. A random number is picked (modulo 100) and compared to the preselected P value. If the random number is less than P , a reaction occurs, and then both particles are removed from the system. If a reaction does not occur, the two reactant particles return to their original positions. The same principle applies if two particles of the same type happen to occupy the same site since we do not allow two particles of the same type (two A or two B) to react or collide. After all walkers in the system move once, the first
J. Phys. Chem., Vol. 99, No. 4, 1995 1229 time step is over and the time step is incremented by one. The simulations were usually done up to 2000 time steps. A simulation run up to 5000 steps (case V of Table 3) did not give any further information on the properties of the reaction front. Periodic boundary conditions are applied in the y direction, but reflecting boundary conditions are applied to the x direction (left to right). A given run was repeated at least 8000 times and averaged. The global reaction rate is defined as the number of reactions that occurred on the whole lattice during a given time step. The center of reaction front is theoretically defined as the x position at which the local production rate is maximum. The original boundary is defined as x = 0. Since we use a two-dimensional lattice, the position of the center of the reaction front xf,at a given step, corresponds to the difference between the column with the maximum number of products and the column at the center, Le., the original boundary. We also calculated the average center location of the reaction front, (xf), using the following f o r m ~ l a . ~
where R(x,t) is defined as the number of products on the xth column during time step t, and xd0) is the position of the column at the original boundary. (We take x40) as 200 for the 100 x 400 lattice.) However, this averaging procedure was inferior to taking the maximum. We note that the theoretical appro ache^'^^^* used the peak method, in contrast to some previous sim~lations.~ 111. Results and Discussion
Global Reaction Rate. The crossover phenomena in the global reaction rate were observed in runs with reaction probability less than 1. Figure la,b clearly shows such crossover phenomena, Le., an initial increase and later decrease in the rate with time, both for the fast minority case (the diffusion coefficient of the particle type with lower concentration is larger than that with higher concentration) and for the slow minority case (diffusion coefficient of the particle type with lower concentration is smaller than that with higher concentration), respectively. The crossover time steps for runs with various different conditions are summarized in Table 3. We investigated the effects on the crossover time steps of the reaction probability and of the initial concentrations so as to test the theoretical prediction by Taitelbaum et al.’ for the crossover time
where k is the microscopic reaction constant and a0 and bo are the initial concentrations of A and B. We note that in our simulations we use the reaction probability P which is linearly proportional to k (but normalized differently). The crossover times, with various initial concentrations and diffusion coefficients for the two particle types and with a fixed reaction probability of 0.01 for the fast minority case, are shown in Table 4 and Figure 2. We can see that the crossover times are linearly proportional to the inverse of the square root of the product of the initial concentrations of the two reagents (but see below). Also in agreement with Taitelbaum et al.,’ they are not affected by the diffusion coefficients. However, the data are actually fitted to a modified eq 2, which also accounts for
1230 J. Phys. Chem., Vol. 99, No. 4, 1995
Koo et al. 400
y = 15.750~+ 138.201
0
? = 0.999
m
0
1
2
3
4
5
6
7
8
In (time step)
8000
2 c .-0
-2
-!m! n -0
0)
7000
150
2.5
6000
5
12.5
10
7.5
i
(a,bJ1/2
5000
Figure 2. Crossover time vs l/&
for fast minority with P = 0.01.
4000 3000
2000 1000 ~ ~ " ' 1 " " " " " " " " " " ' " ' " ' " 1 ' ' ' ~ ~ 0 1 2 3 4 5 6 7
8
In (time step) Figure 1. A plot of the global reaction rate vs ln(time step). The units of the global reaction rate are the total number of accumulated product particles (C) per unit time step. (a) The results after 28 000 runs using the parameters a0 = 0.2, bo = 0.05, DA= 1, DB = 3, and P = 0.05. (b) The results after 8000 runs using the parameters a0 = 0.1, bo = 0.4, DA= 1, DB= 3, and P = 0.05. TABLE 4: tco vs (.Ja,b,)-'for Fast Minority Case with Fixed Reaction Probability 0.01 4.1" 4.1b 5b a
5' 14.1"
200 200 220
220
360
DAIDB = 113. DAIDB= 112. DAIDB= 214 5
the discrete simulation time steps
1/P PJ
where to and U are constants. A plot of the crossover time vs the inverse of the reaction probability, for both the fast and slow minority cases, with fixed diffusion coefficients (B moves 3 times faster than A) and the fixed initial concentrations of the two reagents, is given in Figure 3. It shows that the crossover times are inversely proportional to the reaction probability P. (Note that here to is on the order of unity.) Again this result is in accord with eq 2'. We also investigated the time exponent of the global reaction rate R
R
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t-6
(3)
by plotting ln(g1obal rate) vs ln(time step) before and after the crossover time (see Figure 4a,b). The resultant time exponents 6 are in Table 5. The average values of the time exponent in each case are -0.48 and 0.51 before and after the crossover,
fast minority
-
y = 2 . 2 4 1 ~3.743 r2 = 1,000
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0 slow minority y = 2.7l9x 5.765 ? = 0.993 Figure 3. Crossover time vs UP. For fast minority, a0 = 0.4, bo = 0.1, DA= 1, and DB = 3. For slow minority, a0 = 0.1, bo = 0.4, DA = 1, and DB= 3.
respectively, which also matches well the theoretically predicted values of -0.5 and 0.5, respectively. Center of the Reaction Front. The movement of the center of the reaction front was barely observed in the runs with nonforced moves. The fronts just move back and forth around the initial position, although the fluctuations become larger with time. However, the runs where the particles are forced to move more than one lattice unit show very interesting results. Most of the runs for the fast minority where the ratio of the concentrations of the two reagents, a&, is higher than 4 (cases VII-XI of Table 3) show a U-turn in the reaction front. One exception is the run with the longest crossover time (case XI1 of Table 3); however, it might be "on its way" to a U-turn in
J. Phys. Chem., Vol. 99, No. 4, 1995 1231
Macrodynamics of Elementary Bimolecular Reactions
TABLE 5: Time Exponents & of the Global Reaction Rates R in Two Different Regions (before and after Crossover Step) for Various Simulation Runs measured 6 average 6 predicted 6
a 9.2
before -0.48, -0.43, -0.46, -0.46, crossover -0.47, -0.49, -0.49, -0.51, -0.50, -0.50, -0.50, -0.47 after 0.51.0.49, 0.50,0.50,0.50, crossover 0.53,0.53
h
2
E
-0.5
-0.48
0.5
0.5 1
6 is the time exponent of the global reaction rate, Le., R
-
t-6.
CI
2
,o M
U
In (time step) 2
0
8
z
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1
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In (time step) 4
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8
6
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-8 2
0
6
4
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In (time step) Figure 5. (a) Time dependence of the center of reaction front and (b) the average center of reaction front, for the fast minority case which shows a U-turn. The parameters for the simulations are a0 = 0.4, bo = 0.1, DA = 1, DB = 3, and P = 0.05. I
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I
5.oooO 5.5000 6.oooO 6.5000 7.oooO 7.5000 8.oooO
In (time step) Figure 4. A plot of ln(globa1 rate) vs ln(time step). The units of the global reaction rate are the total number of accumulated product particles (C) per unit time step. (a) Before the crossover, steps 2-6, and (b) after the crossover, steps 250-2000, for the runs with the parameters a0 = 0.4, bo = 0.1, DA = 1, DB= 3, and P = 0.05, where crossover occurs at time step 40.
its moving direction since it shows large fluctuations toward the end of the run. Still, the other runs for the fast minority (cases V and VI of Table 3) do not show a U-tum of the reaction front. Theoretically, these cases are expected to show a U-tum, since they belong to case d of Table 2, which should show an extremum with time in the front position. The question arises whether the simulation runs are long enough so as to see the U-turn. To test such a possibility, the simulation was run up to 5000 steps for case V of Table 3. (Typically, the simulations run up to only 2000 steps.) The runs of the slow minority again show no U-tum, in agreement with theory. The motion of the
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0
1
2
3
4
5
6
7
In (time step) Figure 6. Time dependence of the center of reaction front for the fast minority case which does not show a U-turn. The parameters for the simulations are a0 = 0.3, bo = 0.2, DA = 1, DB = 3, and P = 0.05.
reaction front for the above three cases is shown in Figures 5-7, as described below. Figure 5a is a plot of the position of the center of reaction
Koo et al.
1232 J. Phys. Chem., Vol. 99, No. 4, 1995 5.000
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y = 0.744~ 0.990
? = 0.970
/
4.000
V
3.000 -20 0
1
2
3
4
5
6
7
8
In (time step) Figure 7. Time dependence of the center of the reaciton front for the slow minority case. The parameters for the simulations are a0 = 0.1, bo = 0.4, D A = 1, D B = 3, and P = 0.05.
front vs ln(time step) for the fast minority case. It does show a U-turn (case X of Table 3). It shows that the center first moves toward the higher concentration side and then practically stays at a certain point for quite a while, initially with small fluctuations and later with larger fluctuations. Eventually it moves at a fast rate in the opposite direction, Le., toward the lower concentration side. The crossover time in the global reaction rate roughly corresponds to the beginning of the fluctuation of the center of reaction front. Unfortunately, the time exponent of the position of the center of the reaction front could not be measured quantitatively due to the large fluctuation. The average centers of the reaction front are calculated for the same run (case X) so as to see the U-turn of the reaction front clearly. We note that the averaging process of eq 1 reduces the fluctuations but it may lose many interesting features of local behavior. The results are plotted against ln(time step) in Figure 5b. We observe that the crossover time step in the global reaction rate does match with the turning point (U-turn) of the average center of reaction front (Figure 5a). For instance, compare the crossover time of Table 3 (row IX of Table 3) with that in Figure 5b. [We can understand this since both properties are the result of averaging over local reaction rate.] Figure 6 shows the position of the center of reaction front vs ln(time step) for one of the fast minority cases (case V of Table 3) which did not show a U-turn. In this case, the center first moves toward the higher concentration side (phase 1) and essentially stays there at a certain point for a while, initially with almost no fluctuations and later with fluctuations around this position (phase 2 ) . Finally, it again moves toward the higher concentration side, again with significant fluctuations (phase 3). Here, the crossover time in the global reaction rate happens during the third phase. In contrast, according to the theoretical predictions,8 an extremum point should have existed in this case. In the case of the slow minority (see Figure 7 ) ,the center of the reaction front does not show an extremum, this time in agreement with the theoretical prediction: it first moves slowly toward the lower concentration side and then moves faster in the same direction. Here, the position of the center of reaction front does not show much fluctuation, in contrast to the case of the fast minority. A quantitative analysis has been done for this case in order to get the time exponent a for the center of reaction front, Xf
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ta
(4)
By plotting ln(xf) vs ln(time step), as in Figure 8, we have found a to be 0.80,0.77, and 0.74 after the crossover time of the global reaction rate for three simulation runs (cases XIII-XV of Table
2.000
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I
1
4.000
5.000
6.000
7.000
8.
M
In (time step) Figure 8. The ln(center of reaction front) vs ln(time step) between step 70 and 2000 for the run with a0 = 0.1, bo = 0.4, DA = 1, DB = 3, and P = 0.05.
3). Although this result (average value is 0.77) does not match the theoretical prediction (a= 1.5 for intermediate time and a = 0.5 for both short and long times), it is definitely higher than 0.5, and we infer from Figure 7 that it may still increase if the runs were longer. Obviously, we find no sharp crossovers between slopes 0.5 and 1.5. The above results may be interpreted in the following way. Diffusion and reaction are two competing processes in the system. The relative influence of these two processes varies in time. The diffusion process mainly limits the global reaction rate at very early times, since the two reagents are initially separated. However, as the two reagents are mixed in time, the diffusion process and the reaction process compete if the reaction probability is low. Later, the concentrations of the two reagents inside the reaction front become very low, and the diffusion process again becomes the limiting factor. The global reaction rate shows the average behavior of the reaction front at each moment, but the center (peak) of the reaction front shows a local behavior distribution inside the reaction front. Therefore, we expect the center of the reaction front to show a complicated behavior depending upon the driving force and other system conditions, while the global reaction rate would just show the typical behavior of the major process. As pointed out above, the crossover times of the global rate coincide roughly with those of the average center of the reaction front. Consider a fast minority case (a0 > bo and DA < DB). At very early times where there is not much mixing of the two reagents, the reaction front moves toward the lower diffusion coefficient side (higher concentration side) since the fast particles penetrate into the slow particle domain and form more products than at the other side even though the reaction probability is very low. This is a diffusion-limited regime. At intermediate times, the two processes start to "compete" with each other as to which is the limiting factor. When the two reagents become mixed enough due to diffusion, the reaction may become the dominating limiting process. The reaction process influences the reaction front to move toward the lower concentration side, which is the opposite behavior to the early diffusion-limited process. This is due to the fact that the B particles which
J. Phys. Chem., Vol. 99, No. 4, 1995 1233
Macrodynamics of Elementary Bimolecular Reactions invaded the A domain have eventually been eliminated. The center of the reaction front may also stay at a certain point (with fluctuations) for quite a while (crossover regime). The mobility difference and concentration difference drive the reaction front in opposite directions. Also, the reaction process usually does not limit the reaction as much as the diffusion process in cases of low concentration. At long times the diffusion is always the rate-limiting process since then the reaction front domain is depleted in both reagents. At this point the diffusion process affects the motion of the reaction front in two different ways, depending upon the initial conditions of the system, as described below. Consider first the case where the difference in the initial concentrations is large. The reaction front is depleted more in the reagent with lower initial concentration than that with higher initial concentration. This leads to the crossover time, after which a penetration of the fast minority particles into the slow majority particle domain is not effective since they have to pass the “gap”, the reaction front region. However, considerable numbers of the slow particles with the initial high concentration are already located very close to the other reactants, Le., inside the reaction front. Their movement into the minority reagent domain may make more products since they have much easier access to the other species. (They have not been completely eliminated on the way.) Therefore, the same diffusion process which drives the reaction front toward the higher concentration side at the very early time now propels the reaction front toward the lower concentration side (albeit, first the diffusion process is dominated by the fast particles but later by the slow majority). Finally, the reaction front always heads at a fast rate toward the lower concentration side. However, if the difference in the concentration is not so high, then the penetration is more important since now the reaction front is a “gap” for both kinds of reactants, and thus the diffusion process will move the reaction front toward the lower diffusion coefficient side (i.e., higher concentration side). For the slow minority case (a0 bo and DA < DB), both the diffusion and reaction processes affect the motion of the reaction front in the same direction. The reaction front moves toward the lower concentration and lower diffusion coefficient side without any fluctuation, although the movement of the reaction front in the reaction-limited regime is much slower than in the diffusion-limited regime. Width of Reaction Front. We were able to quantitatively analyze the width w of the reaction front only for the cases with early crossover time because the peaks become quickly broader for the reaction-limited regime and thus have more noisy side peaks, which makes it very hard to measure the width. We assume that the width of the reaction front has the same crossover time as the global reaction rate. The ln(widths) vs ln(time step) are plotted to find the time exponent of the width of reaction front in Figure 9a-c for the slow minority and fast minority cases. w-B
a
3.000 I
I
2’m1
y = 0.527~+ 0.793 r2 = 0.974
/
2.600
3
s
2.400
Y
I’
1.600 1500
I
I
1
I
2.m
2.500
3.000
3.500
4
ln(time step)
b
3500
y = 0.160~+ 2.417
3.300 3-4001
I
3
? = 0.918
/
3.200
Y
I
2.900 I 1 I I I I 3.500 4.000 4.500 5.000 5.5W 6.000 6.500 7
ln(time step)
c
3.400
3.200
3
3.000
4
Y
5
2.800
(5)
The time exponents p were found for the slow minority case (case XV of Table 3) to be 0.53 before the crossover time of the global reaction rate and 0.16 after the crossover of the global reaction rate, which is in good accordance with the theory (l/* and lI6, respectively). In the case of the fast minority, we could measure the time exponent only after the crossover time, due to more complicated peak shapes at early times associated with the U-turn of the reaction front. The resultant time exponent p of this case (case X of Table 3) is 0.30 between step 50 and
2.400 I I I 3 M o 4.000 4MO 5.000 5
I
I
I
s 6.000 6.500 7
In (time step) Figure 9. Time dependence of the width of the reaction front. (a) Between steps 6 and 40 for the runs with uo = 0.1, bo = 0.4,DA = 1, DB= 3, and P = 0.05, (b) between steps 70 and 500 for the same runs as in (a), and (c) between steps 50 and 700 for the runs with a0 = 0.4, = 3, and P = 0.05. bo = 0.1, D A = 1,
Koo et al.
1234 J. Phys. Chem., Vol. 99, No. 4, 1995 step 700; it might go lower to ‘16 if we could avoid the fluctuations at the longer time range.
IV. Summary We have studied numerically, by the Monte Carlo simulation method, the kinetics of the reaction front in the A B C reaction-diffusion system with initially separated components for cases where the microscopic reaction probability is low. Dramatic crossover phenomena in the kinetic properties of the reaction front were observed. The crossover times in the global reaction rate were analyzed to test their dependence on both the initial concentrations of the reagents and their reaction probabilities. These were found to be in good accord with the theoretical prediction. The motion of the center of reaction was analyzed in detail for two different classes, i.e., fast minority and slow minority. In particular, we observed a U-turn of the front in some cases of the fast minority but not in all the cases predicted by theory. The width of reaction front also shows the different time-dependent behaviors before and after the crossover, in agreement with theory.
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Acknowledgment. This work has been partially supported by the NSF Grant DMR-91-11622. Partial support is also acknowledged to Grant BSRI-93-3 11 from the Basic Science Research Program, Ministry of Education of Korea, 1993. References and Notes ( 1 ) Gilfi, L.; Ricz, Z. Phys. Rev. A 1988, 38, 3151.
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