2954
J. Phys. Chem. 1982, 86, 2954-2958
Trigger Wave Fronts of Rapid Consumption in a Belousov-Zhabotinskii System John Rlnzel' and G. Bard Ermentrout h4athematical Research Branch, National Institote of Arthrhis, Diabetes and Digestive and Kklney Diseases, National Institutes of Health, Bethesda, Wryland 20205 (Received:January 5, 1982; I n Final Form: March 23, 1982)
Showalter (J.Phys. Chem. 1981,85,440) observed front-type trigger waves in the acidic bromate oxidation of ferroin, a BZ system without malonic acid. To model these waves of autocatalytic growth of X and consumption of Y , we consider an Oregonator model with two species, X = [HBrOJ, Y = [Br-1, in which the consumption rate of Y rapidly increases as Y falls below Y,. Such waves and models have also been used to describe the leading edge of pulse-like trigger waves in the full BZ system. For an appropriate parameter range (notably, when [H+]is small), we exploit a pseudo-steady-state assumption on X to reduce the two-variable model to a single nonlinear reaction-diffusion equation for Y. Our approximate one-variable model admits a traveling front solution which corresponds to the total consumption of Br- from Y,, [Br-] ahead of the wave, down to Y = 0. Propagation speed as a function of Y , is easily computed for this model. We also show analytically that for large Y , speed is inversely proportional to Y,. We have considered two other simplified consumption models which are analytically solvable. The speed dependence on Y , is compared for all the models (Figure 3). Our theoretical results are also compared to the experimental dependence of speed on initial reactant concentrations. Our study has considered parameter ranges which complement those of previous theoretical treatments.
1. Introduction To understand propagation of chemical waves in Belousov-Zhabotinskii (BZ)-like it is insightful to consider the Oregonatof model with diffusion. This model has three dynamic species X = [HBr02], Y = [Br-I, and 2 = [CeI"]. Since 2 changes on a much slower time scale than X and Y it is convenient (e.g., ref 5-8) and not unreasonable to neglect the 2 dynamics in describing the wave's rapid upstroke of the autocatalytic X . This upstroke is accompanied by rapid consumption of Y. Unless the slower dynamics of 2 are subsequently included, the wave profile is a traveling front. A simplified chemical modelg applicable to this subsystem is
A+Y-X+P X+Y-.2P A+X-2X 2X--+A+P (1.1) where P = [HOBr] and A = [BrOy] which are assumed constant. Beyond its applicability to the leading edge of BZ pulselike waves, this system bears direct relevance to the experiments of Showalterloon the acidic bromate oxidation of ferroin, a BZ system without malonic acid. For a spatially homogeneous batch system, initially in the reduced state (low X),there is an induction period of slow Brdepletion. When a critical value Y, of [Br-] is reached, after perhaps 10-15 min, autocatalytic growth of X and rapid, apparently total, consumption of Y occurs and the mixture is left permanently in the oxidized state. If the solution is spread thinly in a Petri dish one can locally deplete Br- sufficiently, by using a Ag electrode, to prematurely initiate the rapid transition to the oxidized state which then propagates outward as a wave front. Showalter (1) Belousov, B. P. Ref. Radiat. Med. 1959, 1958, 145. (2) Zhabotinskii, A. M. Dokl. Acad. Sci. Nauk SSSR 1964,157,392. (3) Zhabotinskii, A. M. Biophysics 1964, 9, 329. (4) Field, R. J.; Noyes, R. M. J. Chem. Phys. 1974, 60,1877. (5) Field, R. J.; Noyes, R. M. J. Am. Chem. SOC.1974, 96, 2001. (6) Murray, J. D. J. Theor. Biol. 1976, 56, 329. (7) Schmidt, S.; Ortoleva, P. J. Chem. Phys. 1980, 72, 2733. (8) Tyson, J. J.; Fife, P. C. J. Chem. Phys. 1980, 73, 2224. (9) Noyea, R. M. J. Chem. Phys. 1980, 72, 3454. (10) Showalter, K. J. Phys. Chem. 1981,85, 440.
measured the dependence of conduction velocity and wave initiation time on various initial reactant concentrations. For example, speed decreases as a function of Y,, i.e., [Br-] ahead of the wave. Showalter interprets the waves by heuristically modifying system (1.1)so that it exhibits bistability. With the above applications in mind we are motivated to study the front propagation characteristics of system (1.1). Our treatment differs from other modeling studiess7 in that we consider a parameter range for which X may be assumed to be at rapid equilibrium. For the FKN mechanism4 of bromate chemistry this parameter range corresponds to relatively low [H+]. With the pseudosteady-state assumption, the model for (1.1)reduces to a single nonlinear reaction-diffusion equation for Y. By using this simplified one-variable model we expose, and study in detail, the mechanism for front propagation: the nonlinear consumption rate of Y which rapidly increases as Y falls below Y,. By appropriate scaling we stretch out the sharp decay of Y and obtain an approximate model in which Y , is constant. With this singular perturbation approximation we easily compute speed as a function of Y , and other parameters. Numerical results for this simplification compare well with those for the full X-Y system over a sizable parameter range. We have further considered other nonlinear consumption laws for which speed vs. Y, is obtained analytically. For each consumption model, we find that speed is proportional to 1/ Y , for large Y , but this simple dependence does not apply as Y , decreases toward Y,. 2. T h e Model Equations and Scaling For simplicity we consider a one-dimensional medium. Let T and R denote physical time and distance. Then if X = [HBrOJ and Y = [Br-] we may nondimensionalize variables in a manner similar to Field and Noyes4 and obtain the reaction-diffusion model for (1.1): 1
at
s
a7
- - = 1)(1 - 5)
1 a2t + ((1- Qt) + s2
This article not subject to U S . Copyright. Published 1982 by the American Chemical Society
(2.1)
ap2
(2.2)
The Joumal of phvslcal Chemlstty, Vol. 86, No. 15, 1982 2955
Trigger Wave Fronts In a BZ System
where
Y 1.5
The diffusion constant D = 1.8 X cm2 s-’ and we set A = [BrO,-] = 0.06 M. We note the [H+] dependence (taken from ref 11) of the rate constants:
I
0 t - J
1
I 10
0
20
kl = 2.1[H+J2M-ls-l kz = 2
X
10g[H+]M-ls-’
k3 = 1
X
104[H+]M-’ s-’ (2.4)
To motivate a useful parameter restriction we briefly consider the bulk dynamics:
- 5) + 5(1 - q5)
sq = -q(1
(2.5)
+ 5)
(2.6)
For any positive initial concentrations, ( 5 , ~ tends ) to (1/ q,O), the unique steady state of high oxidation. Because s is large it seems reasonable to expect that 5 will tend to rapid equilibrium in which case 7 = 5(1 - q5)/(5
- 1)
(2.7)
-
Under this supposition, an initial rapid transient with Ii/Sl 0 (s) takes 5 to the value dictated by (2.7) with q = q(O), its initial value. Then (2.7) and (2.6) govern the eventual total consumption of q. Since q is small, (2.7) implies that 5 = (1 - q)/q for q < 1 and 5 = 1 + l / ( q - 1) for q > 1. Hence when 9 is large, q decays slowly like e-2r/sbut for q small the rate of consumption grows rapidly: l*/ql O(l/qs). Because of this significantly increased rate in the q dynamics, the pseudo-steady-state assumption for 5 is no longer valid when q is small unless 5 is still much faster than q, i.e., (2.7) is valid, even for small q , if s >> l/qs. Next we introduce a different scaling for (2.1H2.2) to exploit the preceding observation. At the same time we stretch 7 and p to examine in detail the frontlike traveling profiles; with the scaling (2.3), the fronts are very steep and they travel at speeds APIA? O(l/q1/2). To these ends, we let
-
-
t = 7/95 =
x = 95
T Ak2k3 2k4
r =
P
q 1 / 2 = R(
b = l/qs2 = k2/2k4
=)
b-
at
= y(q - X ) aY = -y(q at
y =q
a2X
+ ~ ( 1 X- ) + b-a$ a2Y + x) + ar2
Figure 1. Concentration profiles vs. distance, r , for two different limes with At = I O . Solid curves represent solutlons to (2.9H2.10) with q = 8.375 X lo4, b = 0.25. Dashed curves satlsfy (3.1), with q = b = 0. For the case b = 0.25,concentrations at t = -8 were specified as x(r,-8) = 0 and y(r,-8) = 0.5 for r < 10, y(r,-8) = y m for r > 10 where y.. was chosen slightly greater than 1.5 so that at t = 0, y m = 1.5. For b = 0, the y(r,-8) profile is as above with y - = 1.5.
We will study (2.9H2.10) in the parameter range 0 < b > l / q . Since from (2.41, b = 25[H+], our assumption that b is small means [H+] must be small. The scaling (2.8) is quite similar to that of Murray;6 he studied this system with q = 0. Schmidt and Ortoleva’ considered wave front solutions for q = 0 and b large.
3. Traveling Wave Fronts Recall that Showalter’s’oexperimental setup consists of starting the medium with x small and y = y m > 1 and forcing y < 1 in a small spatial region. This results in a propagating “wave front” which joins y = 0 to y = y-. The corresponding solution to (2.9H2.10) is not a true traveling wave front since y cannot remain at y mas it is not a true steady state. Rather, y ahead of the front slowly decays as e-29’. For q 1, w = 0 are singular points. It is this degeneracy which allows the proof of existence and stability12J3and computati~n~~' of true traveling waves of the system (2.9H2.10) with q = 0. We therefore emphasize that when q # 0 the waves are only transients since y eventually must decay to 0. Nevertheless Figure 1 gives convincing evidence that the q = 0 approximation is a valid and useful one for times less than l / q . We turn our attention to solving (3.3) when q = 0. We seek a trajectory such as the one shown in Figure 2, which comes out of (u,w)= (0,O) and ends on (u,w) = (y,,O). Our goal is to find a relationship between y- and c, the velocity of the front. If we divide (3.3b) by (3.3a) we obtain the following problem: (3.4a) wI,=o = 0 wlu=ym= 0
I
3
Figuro 2. Phase plane for (3.3). Soli curve represents w' = 0 null cline when Q = 0;dashed curve is w' = 0 null cline when q = 0.005. Traiectoty with c = 0.25 and y , = 2.70888 for Q = 0 is indited with arrows.
dw/d{ = cw
1
(3.4b) (3.4c)
For u I 1, w = -cu + A,,, where (3.4~)tells us that A, = cy,. If this solution is matched at u = 1with the solution for u I1 we get c = wl,=,/(Ym - 1) (3.5) It is straightforward to solve (3.4a) by numerical shooting; we choose c and follow the outgoing saddle trajectory until (12)Troy, W. C. J.Diff.Eqs. 1980, 36,89. (13) Klaasen, G . A.; Troy, W. C. SIAM J. Appl. Math. 1981,41,145.
.5
0
1
1.5
2
2.5
LQ, v
Flgm 3. Propagation speed vs. [Br] ahead of wave in dimensionless variables; log c vs. log y,. Curves are for different approxiinate Consumptlon models: solid is for (3.2) and (3.4) with long dashed its asymptotic approximation (3.9) for y, >> 1; short dashed from (4.5) for (4.la,b) and (4.3) with M = 3; mixed dashed is from (4.9) for (4.la,c) and (4.3). The points marked X are the velocities computed from solutions to (2.9)-(2.10) with Q = 8.375 X lo-', b = 0.25.
u = 1, this gives WI,,~. Equation 3.5 relates this to ym. In Figure 3 we have plotted (solid line) log c as a function of log ym. We include the values of the velocity for different ymin the full problem, (2.9)-(2.10). Our approximation is fairly good over most of the range of y, even for b = 0.25. For y.. large c l / y m ,so this suggests a perturbation with y, large. If we let a = l/(y, - l), then (3.4) becomes (3.6a) dw/du = -C + ( U / W ) ( ~ - U )
-
w(0) = 0 w(1) = c / a
(3.6b) (3.6~)
We let u = aw/c and z = 1 - u so that (3.6) becomes dv _ -- a + a2z(z - 1) v(0)= 1 u(1) = 0 (3.7) dz c2u Schmidt and Ortoleva' studied (3.7) for (a,c) small (i.e., for y, large, c small). They find a 2 / c 2 = 3 + Pla + P2a2 (3.8) where
01 = 2(3512- 6)/5
& = 0.778
or for ymlarge
-
1/[3(y, - 1)' + &(ym- 1) + P211'2 (3.9) To lowest order, c 1/(d3ym) In Figure 3, we have plotted the approximation (3.9) in order to compare it to the actual speeds. It is fairly good for y, large but in the limit as ym 1, the value of c from (3.9) tends to 1.13. For ym 1,the actual velocity is c = 2, the lowest wave front velocity to Fisher's equation. At critically (ym= 1) there is a wave front for each c 2 2. But this degeneracy could never be seen in a real experiment since at the next instant in time y would rapidly decay from y = yc = 1. We conclude that the approximation, b = 0, q = 0 is fairly good and predicts the wave shape and velocity of the full system quite well. c
-
--
4. Solvable Fast Consumption Models In the previous section, a second-order nonlinear differential equation (3.3) was derived which governs the dynamics of the chemical wave front. It was necessary to solve this numerically in order to obtain a velocity vs. ym
Trigger Wave Fronts in a BZ System
The Journal of Physical Chemistry, Vol. 86, No. 15, 7982 2957
relationship. Here, we propose two simple models which qualitatively mimic (3.1) but have the advantage of being exactly solvable. We consider the problem: (4.la) y(-m,t) = 0
y(m,t) = ym (4.lb)
B(1- Y ) ( l - Y l A ) Y < 1
f(y)=
=o
(4. IC)
Y > l
Thus, (4.lb) represents a piecewise constant consumption rate and (4.1~)a quadratic consumption rate as opposed to the linear consumption rate, (1- y) in section 3. When the variables, [ = -ct r, y(r,t) = u ( [ ) are introduced, (4.1) becomes du/d[ = w (4.2a)
+
dw/d[ = -CW
u(-=) = w(-m) = 0
+ u f(u)
= ym
~ ( m )
(4.2b)
~ ( m = )
0
(4.2~)
As in section 3, we divide (4.2b) by (4.2a) and use (3.5) to obtain the problem dw = - c + - l.4 fW O S U S l (4.3a) W du w(0) = 0 w(1) = c(ym - 1) (4.3b) For the piecewise constant model, (4.lb), the solution is -c (c2 4Mp w= U (4.4) 2 and (4.3b) implies
+ +
(4.5)
-
This relationship is depicted in Figure 3 for M = 1/3 to agree with the other velocity relationships as ym a. As y.. 1, c 00 for this model so the relationship is not accurate near y.. = 1. Qualitatively though, there is a similarity for the two models. For the cubic model, we look for a solution to (4.3a) of the form
- -
w = Eu( 1 -
z)
(4.6)
where E is an undetermined constant. Note that A depends on the particular y.. picked and B will be chosen so that the velocity relationship is correct for large ym. When (4.6) is substituted into (4.3a) and terms are matched, we find E = (BA/2)'I2 c = -B-- E 2
E
-
[
F(l-AI2)
The boundary condition implies 2Y- - 1 A=---
]112
(4.7)
-
for ym 0.
m.
This follows from integrating (4.3) when c =
5. Discussion
Showalterlo observed front-type trigger waves in the acidic bromate oxidation of ferroin, a BZ system without malonic acid. An Oregonator model (2.9)-(2.10) with two species, X = [HBr02], Y = [Br-] (2 = [FeI"] or [Ce'"] is not present) is considered to model these waves of autocatalytic growth of X and consumption of Y. Such waves and models have also been used to describe the leading edge of pulselike trigger waves in the full BZ system. By appropriate scaling of space and time variables, (eq 2.8) and by exploiting a pseudo-steady-state assumption on X we reduce the two-variable model to a single nonlinear reaction-diffusion equation for Y, eq 3.1. We also set the small Field-Noyes4 parameter q to zero. In this approximation y-, the value of [Br-] ahead of the wave, is constant and the one-variable model admits a traveling front solution which corresponds to the total consumption of Brfrom ymdown to y = 0. The propagation speed c as a function of ymis easily computed for this model. We have also described this relationship analytically (3.9) for large ymand find c 0: l/ym. Motivated by the utility of the scalar model we have considered two other simplified consumption models which are analytically solvable. We have compared the velocity dependence on y mfor all the models (Figure 3). The validity of our rapid equilibrium assumption for x in (2.9H2.10) hinges on 0 < b > 1 and b > 0.04) and the experimental result; to reconcile this difference they suggest that Y , [H+][BrO,-]. A similar tack for (5.1) might lead one to suggest
Y , 0: [Br03-][H+]1/2 (5.2) although we have no chemical intuition for such a proposition. Moreover, one may wonder if (5.2) contradicts the relationship between Y, and [BrOg], [H+]which is implied by Showalter's data on wave initiation time. To consider this let Yo = [Br-]T=O,let u be the proportionality factor of (5.2)' and let Y* = Y , denote the bulk [Br-] at P,the time of wave initiation. Since in the bulk medium, Y is consumed exponentially, Showalter can derive the relationship (eq 3 of ref 10)
1/P = 4.2[H+l2[BrO3-I/ln (Yo/Y*)
(5.3)
when In (Yo/Y*)is assumed to be independent of [H+]and [Br03-]. When (5.2) is used in (5.3) we obtain 4.2 [H+]2[Br03-] 1/T* = In ( Y o / a )- In [Br03-] - '/z In [H+] (5.4) Because the logarithmic dependence is rather weak, (5.4) would still basically match the experimental data (Figures 6 and 7 of ref 10) over a twofold range of variation in [BrO,] and [H']. Tilden14and Field and Noyes5have also considered theoretically the dependence of speed on parameters but their linear treatments were not for true traveling waves of the nonlinear Oregonator model. To get an order of magnitude prediction for 0 in a low [H+]system use (5.1) with [BrOg] = 0.06 and [H+] = 0.01. For Y , = lOY,, this yields (a physically detectable value of) 0.18 mm/min. The low [H+]means, however, that T* is increased substantially. We hope experiments at low [H+] are feasible so that the analytic result (5.1) could be tested directly. Furthermore, if [Br-] ahead of the wave were measured, the suggested dependence of (5.2) could be assessed. For a theoretical investigation of wave initiation and of (5.2) one may try to include the Ag electrode stimulus into the mathematical model. Results on the stimulus-response properties of this system should also shed insight on the threshold behavior15 of the full, excitable, BZ system. (14) Tilden, J. J . Chem. Phys. 1974, 60, 3349. (15) Showalter K., Noyes, R. M. Turner, H. J. Am. Chem. SOC.1979, 101, 7463.
Photolonizatlon Mass Spectrometer Studies of the Colllsionally Stabillzed Product Distribution in the Reaction of OH Radicals with Selected Alkenes at 298 K H. W. Blermann, 0. W. Harrls, and J. N. PMs, Jr.' O e p a m n t of Chemistry and Statewkle Air Pollution Research Center, University of California,Rivers&, California 9252 1 (Received:January 28, 1981; In Final Form: March 11, 1982)
The reactions of OH radicals with propene, 1-butene,and 1-pentenehave been studied in a discharge flow system coupled to a photoionization mass spectrometer. A t totalpressures near 2 torr a collisionally stabilized OH-alkene adduct was observed in each case. For 1-buteneand 1-pentenethe radicals arising from abstraction of a hydrogen atom from the alkene by OH were also observed and their absolute concentrations measured under controlled conditions. Contributions of the abstraction pathway to the overall reaction were as follows: propene,
