8898
J. Phys. Chem. 1992, 96, 8898-8904
Origin of Spontaneous Wave Generation in an Oscillatory Chemical System Yi-Xue Zhang, Petra Foerster, and John Ross* Department of Chemistry, Stanford University, Stanford, California 94305 (Received: April 24, 1992; In Final Form: July 23, 1992)
The origin of spontaneously generated chemical waves in an oscillatory Belousov-Zhabotinskii reaction has k e n investigated by numerical calculations of the deterministic reaction-diffusion equations of a modified Oregonator model and by equilibrium stochastic calculations. From numerical calculations, we obtain threshold perturbations in the phase of oscillations and in the concentrationsof HBr02 and Br- within areas of space with varying radii necessary to initiate trigger waves. Inward propagating trigger waves initiated by a phase shift in the perturbed region with respect to the bulk solution have been observed in the calculations for the first time. Perturbations smaller than the threshold perturbationsor in regions with smaller radii lead to phase-diffusion waves. Our equilibrium stochastic calculations show that the recurrence time for a thermal fluctuation to induce a change in the HBr02 concentration of sufficient magnitude within a sufficient volume for a trigger wave to propagate is many orders of magnitude larger than the observation time of traveling wave experiments. We concluded that an internal thermal fluctuation is highly unlikely to generate a trigger wave in an oscillatory chemical solution.
I. Introduction In a prior article,' we investigated the origin of spontaneous chemical wave generation in an excitable Belousov-Zhabotinskii (BZ) system by solving reaction-diffusion equations of an Oregonator model with an initial profile containing excitation of varying concentrations of either HBrO, or Br- within varying radii. Threshold excitations and the radii necessary for a wave to propagate were obtained, which compared qualitatively with experiments. On the basis of an equilibrium stochastic calculation, we concluded that an internal thermal fluctuation is highly unlikely to generate chemical waves in an excitable chemical system. In this paper, we apply a similar approach to an oscillatory BZ reaction system to investigate the possibility of chemical wave generation in such systems through thermal fluctuations. Since chemical waves can appear in various forms in an oscillatory chemical system?3 it is worthwhile to review briefly the types of chemical waves, their nomenclature, and their properties. -tic Waves. Kinematic waves, also called pseudowaves, originate in an oscillatory system when either the phase or the frequency is space dependent. A wave propagation appears in the reaction medium as an optical illusion; there is no mass transfer. This feature of pseudowaves has been demonstrated e ~ p e r i m e n t a l l y .Pseudowaves ~~ are observed in a continuous medium provided that the time scales of diffusion and apparent propagation are sufficiently different so that diffusion remains The main features of a kinematic wave are (1) a high propagation velocity (with respect to that of diffusion) without an upper limit and (2) the ability to cross impermeable barriers because diffusion plays no role in the phase propagation. Trigger Waves. Contrary to kinematic waves, trigger waves are the result of reaction and diffusion (mass transfer).' Trigger waves are characterized by relatively sharp fronts and slow velocities; they are observed in either an excitable or an oscillatory medium. Trigger waves do not interpenetrate each other but annihilate each other on contact; unlike kinematic waves, they are blocked by walls. In order for a trigger wave to appear, two requirements must be met. The first is that the chemical system must be excitable in the sense that small but finite changes in the concentrations of certain species can trigger amplified changes in themselves or other species; the second requirement is the presence of very steep spatial concentration gradients in those species able to trigger amplifications. In the BZ reaction, these gradients develop when a pulse of oxidation of F e ( ~ h e n ) ~to~ + Fe(phen)?+ occurs at a point in the reaction solution. Such pulses of oxidation can be initiated by artificial pacemakers (such as a silver wire) or they may be generated spontaneously from heterogeneous catalysts, for example, gas bubbles, dust particles, etc. The nature of the pacemaker centers of spontaneously generated trigger waves leads to the controversial issue of heterogeneouslo vs homogeneous" origin of waves. It is the purpose of this paper
to solve this problem by examining the probability of trigger wave generations through thermal fluctuations. PbMfiPsion Waves. Consider an oscillatory BZ reaction Occurring homogeneously in a thin layer of solution in a petri dish. By means of perturbation, such as the immersion of a catalyst at one point, we accelerate the overall rate of the reaction at that point; due to this difference in reaction rates, there are established differences in the concentrations or the differences in the phase of oscillation. These concentration differences bring about diffusion, and then phase diffusion occurs.12 If the perturbation is suffciently large, a trigger wave may be generated in the relaxation oscillation systems. The main features of a phase diffusion wave are (1) phase-diffusion waves do not have a sharp front and (2) the velocity of a phase-diffusion wave usually is much larger than that of a trigger wave; it increases with decreasing phase gradient and does not have an upper limit. All of the above three types of chemical waves have been observed in oscillatory BZ reaction^,^*^-^ while in an excitable medium only trigger waves are possible. Vidal and Pagola13imaged chemical waves in an oscillatory ferroin-catalyzed BZ system; spontaneous generation of trigger waves was observed. The core of the waves was examined with a video camera which had a resolution of up to 5.57 pm/pixel, and the center of the wave was termed either "heterogenmus" or "homogeneous". In the heterogeneous core, a particle was observed with a radius of the order of 75 pm. In the homogeneous core, no particle was observed within the resolution of 5.57 pm/pixel. Hence, there exists the possibility that a local fluctuation of concentration may generate a trigger wave. Walgraef et a1.I1 considered the probability of a thermal fluctuation inducing a chemical wave in the Brusselator modelI4under oscillatory conditions. They suggested that a local phase shift of the oscillation in the system caused by fluctuations may be the mechanism of wave generation. However, they did not consider a critical volume within which the fluctuation must occur in order for a wave to propagate. The necessity of a critical volume has been shown experimentally by Showalter et al.15 as well as by Foester et a1.I6 for wave (generation in excitable systems. Vidal et al.I7 have made careful measurements of spontaneous target patterns in oscillatory BZ reagents. They conclude (1) the temporal period of target patterns is highly variable, (2) the wave speed, on the other hand, is nearly constant from pattern to pattern, (3) the centers of the target patterns are distributed uniformly randomly in space, and (4) the wave speed is independent of frequency. Vidal et a l . I 7 pointed out that these properties of target patterns in the BZ reaction are inconsistent with the predictions of the fluctuation-nucleation theory." In this theory, target patterns arise as bifurcations from the spatially homogeneous limit cycle oscillation, and they inherit the characteristic period of that solution. Therefore, in theory, all target patterns should have
0022-3654/92/2096-8898%03.00/00 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No. 22, 1992 8899
Origin of Spontaneous Wave Generation nearly the same period. Furthermore, the fluctuation-nucleation theory predicts that the number of centers should increase exponentially with the period of the target pattern; this dependence is not supported by experiment^.^' However, Tyson'O has shown that the above properties of the target patterns deduced from experiments can be understood qualitatively in terms of singular perturbation theory of traveling waves, which is based on the hypothesis of wave generation by a catalytic particle.
II. Model and Method of Calculations The modified Oregonator model used in this paper is the same as in the previous work' for excitable systems, as well as for simulation of the pmfdes of chemical waves in the fenioncatalyzed Belousov-Zhabotinskii reaction.I8 The chemical reactions from ref 18 comprising the model are HBrOz + Br03- + H+ F! 2Br0,' Fe(phen)$+
+H20 + H+ F? F e ( ~ h e n ) ~+~HBrO, +
+ BrO,'
2HBr0,
-
HOBr
HBr02 + Br-
+ BrOy + H+
+ H+
-
-
2HOBr
-
(R3) (R4)
(R7) (R8) (R9)
A detailed discussion of reactions 8 and 9 is in ref 18. The set of three variable kinetic equations can be obtained after converting the concentrations to dimensionless variables and making steady-state approximations for the concentrations of BrOZ' and BrMA' + MA' c
dx/dr
U(C
- z) - KGZ - x + K - ~ -u x2 ~ - xy + qy
dz/dr = U(C
(1)
hz p(c - z)
+ 1 - qY - XY - z) - K+XZ p(c-z) + 1
t u dy/dr =
Equations 1-3 can be further simplified by making a steady-state approximation for y, that is, the Br- concentration. The equations for the resulting model with diffusion terms are
(Rl)
(R5)
+ Br- + 2H+ HBr02 + HOBr Fe(~hen)~'+ + BrMA + MA F! Fe(phen),2+ + H+ + BrMA' + MA' BrMA' + MA' + H 2 0 hBr- + MA + products Br0,-
TABLE I: Values of Parameters Used in the Calculations for the Oscillatory Reaction System in This Work parameters in eqs 1-5 rate const of (Rl), (R3)-(R5), (R7)-(R9) h = 0.75 k, = 40 M-2 s-I c = 1.65 X k-, = 4 X lo7 M-' s-I c = 7.31 X k3 = 1 X lo7 M-2 s-I q = 2.00 x 10-4 k-3 = 40 M-I s-I k, = 2 X 10' M-I s-I = 5.32 x 103 K-5 = 3.92 x lod k5 = 1 X lo6 M-2 s-I K+ 6.04 X lo-' k, = 2 M-3 s-I u = 1.20 x 10-2 k8 0.4 M-l S-I Kg = k8k9f k-8 = 3 X lod M-'S-I
where k3H kaB
u = -[Br02']
2k4 y = -[Br-] kl HA
(3)
where s = (ksB/D)l/2 S (unit of coordinate space). The values of the parameters used for all calculations in this work are given in Table I. The initial concentrations of the reactants are [H+], = 0.333 M, [Br03-], = 0.1999 M, [MA], = 0.043 M, and [BrMAIo = 0.0662 M; the diffusion coefficient cm2/s. D of all species is taken to be 1.5 X Method of calcuiatlolra The ondimensional reaction4ifTusion equations (4) and (5) with periodic boundary conditions are discretized with the method of lines, and the resulting ordinary differential equations are integrated by the routine DO2EBF in the NAG Fortran LibraryIgfor stiff coupled ordinary differential equations. In most calculations, the error tolerance control parameter TOL in the program was set to be l@. Further dof TOL have no effect on the solution. The number of grid points used in the calculations is varied depending on the size of the perturbation applied to the system. For a perturbation radius larger than 50 pm, the number of space grids is normally 100, whereas for smaller perturbations 200 or 400 space grids were used. In order to have sufficient space grid points covering the perturbation area, an embedding technique is employed in the calculations. With a fixed number of grid points, we start with a very small grid size and integrate the reaction diffusion equations until the perturbation area expands (through phase diffusion or trigger wave propagation) to about 60% of the total space length. Then we discontinue the integration and embed the total space length by a factor of 2 and start the integration again. The embedding is done as follows: suppose the number of total space points is 100; we take every other space point of the profile obtained by the first calculation and specify them as the middle 50 points of the initial profile of the second calculation. The remaining 50 points, 25 evenly spaced points on each of the two outskirts, are set at the values in phase with the bulk part of the profile of the first calculation. In this way, the distance between two adjacent grid points in the new profile is 2 times larger and so is the total space length. The above process is repeated several times in the calculations until a decision can be made whether a trigger wave or a phase-diffusion wave has been generated. 111. Results of Calculations
2k4t c=c klHA
A = [BrO,-]
B = [MA]
k8
p=-c
KSC
+ [BrMAl
+
H = [H+l
C = [ F e ( ~ h e n ) ~ ~ +[]F e ( ~ h e n ) ~ ~ + ]
In this section, we present the results of numerical calculations obtained from solving the one-dimensional reaction-diffusion equations (4)and (5) with initial profiles containing excitations in either the phase of the adlation or in the HBr0, concentration, within varying radii. From these calculations, we determine the critical perturbations and radii necessary for propagation of a trigger wave. We also discuss the phase-diffusion waves observed in the calculations. We begin by discussing the differences and similarities between an excitable and an oscillatory system re-
Zhang et al.
8900 The Journal of Physical Chemistry, Vol. 96, No. 22, 1992
TABLE 11: Results of the Reaction-Diffusion Equatioss (4) and (5) in Predicting Propagation of a Trigger Wave or a Pbrse-Diffusion Wave"
radius, ym A@, deg D
I\ I
25
50
P P P
P P P
P T P P
100
500
250
O 0 0 T T T 250 T T T 161 P T T 101 P T T 80.6 P T T 55.5 P P T 30 P P P A4 is the phase difference between the bulk and the perturbed region in the initial profile. The phase of the bulk part in the initial profiles is 4 = Oo. The phase shift A 4 is indicated by the first column, and radii of the perturbed region in the initial profile are indicated by the top row. If a trigger wave is generated for the given perturbation, a T is marked; if a phase-diffusion wave is generated, a P is marked. Inward propagating trigger wave is indicated by 0. The unmarked spaces in the table represent phase-diffusion waves, but no calculations were done in these cases. 334.9 331
J
10
TABLE 111: Results of Similar Calculations to Those in Table 11"
radius, km - 4600
- 04 -40
-30
-20
-1
0
00
00
10
20
30
40
50
60
Log x Dim.nsion1.m
Tim.
Figure 1. (a, b) Excitable system. (a) The nullclines of the concentra-
tions of x and z of eqs 4 and 5 without diffusion terms, wheref= dx/dT, parameter values used are given in Table I, except h = 1.75. The threshold values of excitation in both x and z are indicated. The dashed line represents the trajectory of the system started at X * m b l d and I, where I , is the value of z at the steady state. The trajectory coincides with the curve off = 0 from A to the steady state. (b) The excitation phenomena correspnding to the dashed line in a. The dashed line is the steady state of x , and the solid line is the excursion curve which results by changing x at a from x, to Xth&,,,ld. (c, d) Oscillatory system. (c) The nullclines of the concentrationsof x and z of eqs 4 and 5 without diffusion terms. All parameter values are given in Table I. The dashed line and the solid line from A to B represent the limit cycle of the system. Numbers along the limit cycle are the phase of the oscillation. The part from A to B is excitable, and the horizontal lines represent the magnitude of the threshold values of excitation at a given point on this part of the limit cycle. (d) Oscillations in log x . A to B is the excitable branch. Numbers along the curve indicate the phase of the oscillation. g = dz/d7. All
garding trigger wave formations. Reusser and Fieldz0pointed out that in order for trigger waves to appear the system needs to be excitable; an oscillatory BZ system is also excitable and therefore can support trigger waves. However, we point out some differences in the excitability of an excitable and an oscillatory system. Figure 1 contains the phase planes and the concentration vs time curves of an excitable and an oscillatory BZ system. The excitable system has a single stable steady state. When small disturbances of the concentrations of either x or z occur beyond the corresponding threshold values, the system goes off on a long excursion in the concentration space before returning to the steady state. In a spatially distributed unstirred system (such as a thin layer of solution in a petri dish), a trigger wave can be initiated by perturbing a small area of the solution in the spccies concentration^'^ or in the rate of reactions by applying a laser beamz1on a small area. Next we examine an oscillatory system, which is also excitable: if the system happens to be on the lower branch (A to B) of the slow drift part of the log x vs time curve as indicated in Figure Id or the portion A to B on the limit cycle as in Figure IC, small perturbations in the concentration of x may be amplified to the maximum value of x, and then x relaxes back to the branch from A to B again. Perturbations can be applied in any part of the stable l i i t cycle for an oscillatory system although not everywhere in the l i i t cycle is the system excitable. The excitability property of an oscillatory BZ system can only be seen on the lower branch of the x variable (we call it the excitable branch) in the log x vs
Ab. dea
50
100
200
T T T T P P "The phase of the bulk part in the initial profiles is 4 = Symbols have the same meanings as in Table 11. 19.88 14.648 9.416
P
290.93".
time plot or the portion from A to B in the limit cycle as indicated in Figure IC. Furthermore, in a spatially distributed unstirred ascillatory system, there is a consideration other than a sufficiently large perturbation on the excitable branch necessary to initiate a trigger wave: we also need a sufficiently large phase difference between the bulk and the perturbed area. We return to this point later. A. Critical Radiw and Critical Perturbsttiom for Trigger Wave Generations from Numerical Solution of the Deterministic Reaction-Diffusion Equations. The threshold perturbations of x on the excitable branch are indicated by the horizontal lines shown in Figure IC. It can be seen from this phase plane diagram that the threshold perturbation in x decreases as the system moves along the excitable branch from point A to B; Le., the excitability of the oscillatory system increases as we move along the limit cycle from A to B in Figure IC. For this reason, selected points between A and B are taken as the state of the unperturbed part of the space in the initial profiles of the calculations. We present the results of the calculations in the following sections for perturbations due to a phase shift and due to a change in [HBrO,] separately. 1. Perturbation in Phase of the M i a t i o n in a Given Area of the Reacting Solution. In these calculations, the initial profiles of both [HBrOJ and [ferriin] contain a region of perturbation in the phase of the oscillation, i.e., a finite phase shift with respect to the unperturbed region. We calculate the development of the perturbed region. For sufficiently large phase shifts and perturbation areas, trigger wave propagation is observed. In cases where the nature of the wave cannot be easily judged, the velocity of the wave propagation is calculated. Under the conditions we specified in section I1 of this paper, the calculated trigger waves travel at a speed of about 9 mm/min, whereas the velocity of the phase waves is usually larger than 20 mm/min. Once a trigger wave is initiated, the concentrations of [HBrO,] and [ferriin] quickly reach their maximum values and then propagate outward. If the perturbation is not large enough or the region of perturbation is too small for trigger wave initiation, then a phase-diffusion wave propagates. We discuss the phasediffusion waves in some detail in the next section. Tables I1 and 111 summarize the results of the calculations. The phase of the unperturbed part of the initial profile is indicated
The Journal of Physical Chemistry, Vol. 96, No. 22, 1992 8901
Origin of Spontaneous Wave Generation
-50 -'Ot-----l
-I
-701)_1 ::::KI
-50
-4 0
-5 0
-2 56 -6 0
041 -7 0
00
05
10
I 5
DISTANCE cm
20
25
- 2 50 00
05
15 DISTANCE em 1 0
10
2 5
Figure 2, Initiation of a trigger wave by perturbation of the phase in a given spatial region. Calculated sequence of profiles of [HBrO,] and [ferriin]. Profiles 1-5 follow in time (seconds): (1) 0.00, (2) 70.51, (3) 85.62, (4) 112.18, (5) 169.42. In the initial profile, the phase of the bulk 0 and the phase in the perturbed region is @I = 250.1'. The part is 4 = ' radius of the perturbed region is 250 pm. The period of the bulk oscillationsis 94.5 s. Profiles 2 and 3 are propagating trigger waves in the first cycle of the oscillation, and 5 is the trigger wave occurring in the second cycle of oscillation of the bulk.
in the captions to the tables. If a trigger wave is generated for a given phase shift within a given radius, a T is placed in the appropriate place in the table. If a phase-diffusion wave is generated instead of a trigger wave, a P is marked. The larger the phase shift, the smaller the radius of excitation sufficient for trigger wave generation. A typical trigger wave initiation and propagation process is presented in Figure 2. In this example, the spikes in [HBrO,] and [ferriin] introduced in the initial profile grow to their maximum values first and then propagate outward. As the front moves, the medium behind the front relaxes back to the reduced state. In the meantime, the bulk part of the system is also moving along the slow manifold toward the fast manifold in Figure IC and d. Once the bulk part reaches the point where the system jumps from the reduced state to the oxidized state, the trigger wave disappears. However, it reappears in the next cycle of the bulk oscillation. Therefore, the periodicity of the trigger wave generated in this way is very close to the period of oscillations of the system. Unlike in an excitable medium, where a trigger wave can propagate far away from the center as long as the reaction vessel is large enough, there is a finite extent of spread for the trigger wave in an oscillatory medium. Trigger waves in this medium disappear when the bulk part of the solution jumps from the slow manifold to the fast manifold. The extent of spread varies depending upon the choice of the phase of the unperturbed part of the system initially. There is another type of trigger wave which has been observed for the first time in the present calculations, one which propagates inward toward the center of perturbation, as shown in Figure 3. In a two-dimensional system such as a thin layer of BZ solution in a petri dish, one would see a circular trigger wave shrinking instead of expanding. For this type of trigger waves, a 0 is marked in Table 11. As we can see from the table, this phenomenon occurs in cases where a very large phase shift is applied in the region of perturbation when the unperturbed part has a phase of Q, = Oo in the initial profile. In Figure 3, the initial profile contains a region of perturbation with a radius of 250 wm; the unperturbed
-
0 000 0
025
050 075 DISTANCE cm
100
125
2 61 000
025
050
075
100
I
125
DISTANCE cm
Figure 3. Calculated sequence of [HBrO,] and [ferriin] for the development of an inward propagating trigger wave. Profiles 1-5 follow in time (seconds): (1) 0.00,(2) 1.14, (3) 5.73, (4) 68.68, (5) 91.57. In the initial profile, the phases of the bulk part and the perturbed region are ' 0 and 334.9O, respectively; the radius of the perturbed region is 250 pm.
region has a phase of Q, = Oo, and the phase in the perturbed region is Q, = 334.9O. As we can see from Figure 3, the perturbed area grows to the maximum value of [HBrOJ very quickly, but it relaxes back to the low [HBr02] branch instead of initiating a trigger wave. An inward propagating trigger wave is generated in about 68 s later. The explanation of this phenomenon is the following: once the system begins to evolve from the initial profile, both parts of the system (perturbed and unperturbed regions) are changing. The perturbed area grows to the maximum value of [HBrO,] very quickly; at the same time, the bulk part moves slowly along the reduced branch from A to B in Figure IC and d. Under the conditions in Figure 3, the time needed for the perturbed area to grow to the maximum value of [HBr02] is so short that the bulk part of the solution is only slightly different from the point Q, = Oo in Figure IC and d. From the discussion in the beginning of this section, we know that $I = Oo is the least excitable point on the excitable branch. An area perturbed even with [HBrO,],, cannot initiate a trigger wave in this medium. Therefore, the amplified perturbation area relaxes back to the reduced branch instead of propagating outward. After the perturbed region relaxes back to the reduced branch, the perturbed and the unperturbed regions switch their roles. Both of them are moving along the excitable branch A to B in Figure IC and d, but the bulk part of the solution reaches point B and jumps to the higher [HBrO,] branch first, while the perturbed region lags behind on the lower [HBrO,] branch, which is the excitable branch. Therefore, there results a trigger wave propagating inward. When the two inward propagating fronts collide, the trigger wave disappears. An inward propagating trigger wave can also be generated by setting the initial profile with the unperturbed region having q5 > Oo and the perturbed area having Q, = 0'. The inward propagating process described here is not related to the shrinkage of trigger waves in a two-dimensional space due to high positive curvature; neither is it related to the shrinkage of a curved filament in a three-dimensional system.22 The effect of diffusion at the very beginning of the evolution of the system is the reason for the existence of a critical radius for wave generation in both excitable and oscillatory BZ systems for a sufficiently large perturbation in the concentration of a chemical species or in the phase of oscillations. This can been
Zhang et al.
8902 The Journal of Physical Chemistry, Vol. 96, No. 22, 1992
TABLE IV: Results of the Reaction-Diffusion Equatiom (4) and (5) in Predicting Propagation of a Trigger Wave or a Phase-Diffusion Wavea
excitation of HBr02, pM 238
23.8 IO
00
30
10
00
03
0 6
09
I
11
T P
T T T P
radius, pm 25 50
T T T T P
2.39 1.72 1.46 1.33
DISTANCE, cm
n
10
7.16 4.77
IO
I I
1
1 5
DISTANCE. cm
Figure 4. Results of calculations for two initial profiles of [HBrO,] with the same phase settings for the bulk and the perturbed regions but different radii of perturbation. In the initial profiles of both a and b, $(bulk) = 0’ and $(perturbed region) = 250.1’. The radius of the perturbed area is 250 pm in a and 50 pm in b. Further developments of a and b lead to a trigger wave and a phasediffusion wave, respectively. The time sequence of the curves in both a and b is (-) 0.0 s, (- -) 0.2289 s, and (- -) 2.289 s.
--
seen in Figure 4. In both a and b of this figure, the initial profiles
have the same magnitude of phase shift and same phase setting for the bulk. The difference is the radius of the perturbation region. From both a and b, we can see that once the systems begin to evolve, diffusion tends to smooth out the initial wave form and thus distort the initial rectangular wave form of the perturbation. However, the degree of this distortion depends on the radius of the perturbed region: the smaller the radius of perturbation, the larger the distortion. Therefore, a decrease in the radius of the perturbation area is equivalent to a decrease in the magnitude of the phase shift in the initial profile. If this decrease is sufficiently large, the initial perturbation introduced to the system fails to initiate a trigger wave, as in Figure 4b, which leads to a phasediffusion wave instead. In Figure 4a, the area of perturbation is sufficiently large and, therefore, diffusion is less effective in the middle of the perturbed area. The perturbation is amplified to the maximum value of [HBr02],and a trigger wave is initiated. 2. Pertllrktioae in HBd& C ” t i o n . Since HBrOz is an autocatalytic species in the BZ reaction, perturbations of its concentration may excite the system by turning on the autocatalytic process. Calculations have been carried out by setting the HBr02 concentration in a given region to a value larger than that in the bulk, and the resulting initial profile of [HBrOJ in space then resembles a single square wave pulse. The initial profile of [ferriin] is constant in space. Several different values of concentration in HBrOz are used. For each concentration of perturbation, several different radii within which the excitation is applied are considered in the calculations. Tables IV and V summarize the results of the calculations performed for two different points on the excitable branch in Figure IC and d. An oscillatory system generally needs larger perturbations in HBr02and Br- concentrations for a given region of space, in order to initiate a trigger wave, than an excitable system.’ The minimum critical radius of perturbation is also much larger than that in the excitable system. B. CeamtiOn of Pbise-Diff\lSlon Wives. After a weak perturbation in a given area of space in an excitable medium of the BZ reaction, the system relaxes back to the single steady state by means of both diffusion and chemical reaction, in particular due to the presence of the attractor. For an oscillatory system
100
250
T T T T T
T T T T T
T T T T T
P
T T
T T
P
P
“In an initial profile the region of excitation is constituted by a change in the concentration of HBrO,. The initial region of perturbation has a concentration as indicated by the first column and a radius indicated by the top row. The phase of the bulk part of the initial profile is 4 = 150.7’. The meanings of the symbols are the same as in Table 11. TABLE V Results of Similar Calculatiollrp to Those in Table IVa excitation of HBrO,,pM 238
23.8 7.16 4.77
2.39 1.33 1.06 0.795
radius, pm
T
250 T T T
T T T T
T T T T
1
10
50
100
T T T P
T T T T
T T T T
T
T P
T T T P
T
P
P
“The phase of the bulk part of the initial profile here is 4 = 310.8’. perturbed by a phase shift in a given area, the perturbation is removed mainly by diffusion. Reactions periodically change the concentration gradients between the perturbed and unperturbed regions during each cycle of the bulk oscillations, and therefore, it may take many cycles of oscillations of the bulk before the pmfde becomes uniform again. This process has been observed by Bodet and RossZand can be explained by the phase-diffusion theory developed by Ortoleva and Ross.8 Our calculations show that a phase-diffusion wave can be observed for small perturbations introduced even in small regions of space in the initial profile. Therefore, no minimum perturbations and radii are obtained for phasediffusion waves. A typical phase-diffusion wave is presented in Figure 5. In this example, the initial profile contains a perturbation region of 100 pm in radius with a phase shift slightly smaller than the critical phase shift necessary to generate a trigger wave. As the whole system moves along the limit cycle of oscillations, a phase-diffusion wave propagates. Although a circular wave front periodically appears similar in shape to a trigger wave, it is a phase wave because of its much higher velocity (=20mm/min) than a trigger wave. The sharpness of the front, less than that of trigger waves, decreases as time goes on. Transitions from a phase-diffusion wave to a trigger wave or vice versa have been observed in the calculations; the transition occurs to the type of wave which has the larger velocity. Before the onset of a trigger wave, there is usually a phasediffusion wave. The trigger wave itself gradually evolves to a phase-diifusionwave after many cycles of bulk oscillationsdue to mass diffusion. This transition was seen by Bodet and RossZin calculations with the Brusselator model. IV. Estimate of Probability of Chemical Wave Initiation by Thermal Fluctuations We have analyzed previously’ the possibility of spontaneous wave generation through thermal fluctuations for an excitable system of the BZ reaction. In this section, we apply a similar approach to an oscillatory system to make an estimate of the probability of a thermal fluctuation of sufficient magnitude in concentration occurring within a sufficient volume of critical size.
rhe Journal of Physical Chemistry, Vol. 96, No. 22, 1992 8903
Origin of Spontaneous Wave Generation
2 7E-I
004
4
I
1
I OE-7J 00
20
40
6.0
00
100
8 0
I
20
40
DISTANCE, mm
80
60
100
DISTANCE, mm
Figure 5. Calculated sequence of profiles of [HBr02]and [ferriin] in a propagating phase-diffusionwave. Numbers in the figure are times in seconds. In the initial profile, #(bulk) = Oo, #(perturbed region) = 30.35O, and the radius of the perturbed region is 100 pm. Period of bulk oscillations is 94.5 S.
TABLE VI: V a l w of Y, n, IY- nl, 6, and 8,"
radius, pm Y WG1initi.i
n [HBrO~lcr In YI
b
-
en, s
'
1
IO
984 6.00 X IO5 5.99 x 105 31.4 10l410939
9.84 X IO5 1.81 x 107 1.71 x 107 992 1015457381
25 1.54 x 107 1.88 X lo8 1.73 X lo8 3.92 X IO3 10129328550
50 1.23 X 7.53 x 6.30 X 1.11 x
100
IO8 108
IO8 104
10318922540
9.84 X IO8 3.68 x 109 2.70 X IO9 3.34 x 104 1093724W90
250 1.54 x 1010 5.75 x 1010 4.21 X 1O'O 1.24 x 105 ~0146l4662oOo
" Y , the average number of particles in a volume with a critical radius in Table IV;n, the number of particles must be contained in the above volume to initiate a trigger wave; IY- nl, the critical fluctuation, 6, the thermal fluctuation; and e,, the recurrence time for critical fluctuations occurring in [HBrOZ]in spheres of different radii in Table IV from the deterministic reaction-diffusion equations (4) and (5).
TABLE VII: Saw Quantities as Given in Table VI for a Critical Fluctuation Occurring in [HBr02] in Spheres of Different Radii in Table V Calculated by the Deterministic Reaction-Diffusion Equations
radius, gm 1 Y
[HBWinitin~
n [HBGlcr In
b
e,,
-4 s
1.63 X 10' 1.81 x 104 1.64 x 104 40.4 1011762
10 1.63 X IO6 6.03 X IO6 4.40 X 10" 1.28 X IO3 10'5'492'
Our estimate is based on fluctuations in an equilibrium system. Limitations of such an estimate for a nonequilibrium system are recognized, but we think it is unlikely that the estimate is incorrect by orders of magnitude. The formulation of the problem has been given in the previous pub1ication.l Consider a chemical reaction system in either an excitable or an oscillatory state which is at constant temperature and spatially homogeneous. The volume of the entire system is V,and we select a small volume, u, which is in equilibrium with the larger volume surrounding it, equilibrium being assumed with respect to concentration fluctuations. The number of small volumes comprising the entire system is N. Consider the measurement of the number, n, of particles in u made at regular time intervals, 7, and let the average number of particles in u be Y . Then
50
2.04 X lo8 3.34 x 108 1.30 X lo8 1.43 x 104 1015056558
100 1.63 X IO9 2.67 x 109 1.04 x 109 4.04 x 104
200 1.31 X 1Olo 2.14 X 1OIo 8.33 x 109 1.14 x 105 10956604820
1ol205l79OO
the frequency, W(n), with which n independent particles are observed in u is given by the Poisson distribution: W(n) =
e-"u" n!
The root-mean-square fluctuation from the average number of particles, Y, is 6=
Y1/2
(7)
Mori and Ross1have shown that if the total volume of the entire solution is 6 mL and u is a sphere with a radius of 0.5 pm, the recurrence time, On, can be approximated with a high degree of accuracy as
8904
J . Phys. Chem. 1992, 96, 8904-8909
8, is defined as the time necessary to wait before observing the state of n particles in any v after n has been observed previously in any u; 7 is the observation interval. For all calculations in this paper, 7 is taken as 1 s and the total volume, V,is 6 mL. The recurrence times for fluctuations of [HBrO,] with varying size of perturbation area are calculated with the above formula. Their magnitudes determine the likehood of observing a trigger wave which has been induced by an internal fluctuation in the time the experiment is carried out. Tables VI and VI1 summarize the results of the calculations for the perturbations in [HBrO,]. In both tables, Y is the average number of HBr02 molecules contained in a sphere of a given radius in the initial profile; n is the number of HBr0, molecules which must be contained in that volume in order for a trigger wave to propagate, as determined by the calculations of the reaction-diffusion equations presented in section 11. The size of the fluctuation is the number of HBr02 molecules (the difference between n and v) necessary for trigger wave generations, and 6 is the mean fluctuation for the given average number of HBr02 molecules within the considered volume. 8, is the recurrence time for the critical fluctuation. By comparing the magnitude of critical fluctuations and the average thermal fluctuations in HBr02,we see that thermal fluctuations are orders of magnitude too small for initiation of a trigger wave; further, calculated recurrence times are many orders of magnitude larger than the observation times of traveling wave experiments. The critical perturbations predicted by the deterministic equations necessary for the initiation of a trigger wave in an oscillatory chemical system have vanishingly small probabilities of occurring spontaneously in solution. Calculations have also been carried out by solving three variable reaction-diffusion equations from eqs 1-3 with perturbations in [Br-] in the initial profiles. The results are very similar to the results we discussed above for perturbations in HBrO,. The minimum critical radius to initiate a trigger wave is even larger than the values in Tables IV and V. V. Conclusions We have used numerical solutions of the deterministic reaction-diffusion equations and equilibrium fluctuation theory to investigate the probability of trigger wave generation in an oscillatory Belousov-Zhabotinskii system. Critical perturbations
in the phase of the oscillations, in [HBrO,] and in [Br-1, as well as critical radii of perturbation area are obtained by solving the reaction-diffusion equations in one dimension of a modified Oregonator model. Phase-diffusion waves are generated when weak perturbations are applied to the system in a given area of space. Estimates of the critical radii may change with the change of the dimension of the system but not significantly. Equilibrium stochastic calculations show that the probability of a thermal fluctuation initiating a trigger wave through internal thermal fluctuations is vanishingly small. Therefore, we conclude that the spontaneous traveling waves observed experimentally in oscillatory BZ systems are most likely initiated by a heterogeneous mechanism.
Acknowledgment. We thank Igor Schreiber and Eugenia Mori for helpful discussions. This work was supported in part by the National Science Foundation and the Air Force Office of Scientific Research. References and Notes Mori, E.; Ross, J. J . Phys. Chem. 1992, 96, 8053. Bodet, J. M.; Ross,J. J . Chem. Phys. 1987, 86, 4418. Ross, J.; Muller, S. C.; Vidal, C. Science 1988, 240, 460. Vidal, C.; Pacault, A. In Evolution of Order and Chaos; Haken, H., Ed.; Springer-Verlag: Heidelberg, 1982. (5) Sadoun-Goupil, M.; De Kepper, P.; Pacault, A,; Vidal, C. Acta Chim. 1982, 7, 37. (6) Kopell, N.; Howard, L. N. Science 1973, 180, 1171. (7) Winfree, A. Faraday Symp. Chem. SOC.1974, No. 9, 38. (8) Ortoleva, P.; Ross, J. J . Chem. Phys. 1974, 60, 5090. (9) Beck, M. T.; Varadi, 2.B. Nature (Phys. Sci.) 1972, 235, 15. (10) Tyson, J. J. J. Chim. Phys. 1987, 84, 1359. (1 1) Walgraef, D.; Dewel, G.; Borckmans, P. J . Chem. Phys. 1983, 78, (1) (2) (3) (4)
3043. (12) Orteleva, P.; Ross, J. J. Chem. Phys. 1973, 58, 5673. (13) Vidal, C.; Pagola, A. J. Phys. Chem. 1989, 93, 271 1. (14) Nicolis, G.; Prigigine, I. Self-organization in Nonequilibrium Systems; Wiley: New York, 1977. (15) Showalter, K.; Noyes, R. M.; Turner, H. J . Am. Chem. SOC.1979, 101, 7463. (16) Foester, P.; Muller, S. C.; Hess, B. Proc. Natl. Acad. Sci. U.S.A. 1989,86, 683 1. (17) Vidal, C.; Pagola, A.; Bodet, J. M.; Hanusse, P.; Bastardie, E. J . Phys. 1986, 47, 1999. (18) Mori, E.; Schreiber, I.; Ross, J. J . Phys. Chem. 1991, 95, 9359. (19) NAG. Fortran Library, Mark 13. Vol. 2, 1988. (20) Reusser, E. J.; Field, R. J. J. Am. Chem. SOC.1979, 101, 1063. (21) Wood, P. M.; Ross, J. J. Chem. Phys. 1985, 82, 1924. (22) Tyson, J. J.; Keener, J. P. Phys. D 1988, 32, 327.
Cooperative Effects of Scavengers on the Scavenged Yield of the Hydrated Electron Simon M. Pimblott* and Jay A. LaVerne Radiation Laboratory, University of Notre Dame, Notre Dame, Indiana 46556 (Received: April 2, 1992; In Final Form: July 23, 1992)
An analytic description of the cooperative effects of scavengers on the radiation chemistry of the hydrated electron in aqueous
solution is developed by using the available experimental data and the results of deterministic diffusion-kinetic calculations. The formulation has two components: the first corresponds to the effect of the primary (electron) scavenger in isolation and the second represents the suppression of the eaq-+ OH spur reaction by the secondary (hydroxyl radical) scavenger. The two aqueous systems that have been considered in detail are nitrate/formate and nitrous oxide/Zpropanol. The agreement between the analytic treatment, the experimental data, and the kinetic modeling is good.
1. Introduction
A variety of different experimental techniques have been used to probe the factors influencing the radiation chemistry of aqueous so1utions.I One area of extensive effort has been the examination of the effect of solute concentration on the yield of an observed product.2d These scavenger experiments can give the yields of the different radiation-induced species,’,8 and in addition they can 0022-36S4/92/2096-8904$03.00/0
provide direct information that helps to elucidate the details of the radiation-hemical reaction scheme? It is frequently desirable to be able to estimate the yield in a particular scavenger experiment. To this end the concentration dependence of measured yields are usually fitted to one of several simple empirical equations so as to allow for facile e x t r a p o l a t i ~ n . ~ .Thus ~ ~ ~ far, ~ ~ l these ~ experimental formulas have only been developed to describe the 0 1992 American Chemical Society
