J . Phys. Chem. 1990, 94, 4404-4412
4404
FEATURE ARTICLE Mechanhs of Some Chemlcal Oscillatorst Richard M. Noyes Department of Chemistry, University of Oregon, Eugene, Oregon 97403 (Received: January 1 7 , 1990)
During the past 20 years, our Eugene laboratory has made major efforts to elucidate the mechanisms of five different oscillating reactions. The Belousov-Zhabotinsky, Bray-Liebhafsky, and Briggs-Rauscher reactions are all homogeneous redox relaxation oscillators. As the chemists at du Pont who discovered these oscillations have pointed out, during the catalyzed autoxidation of benzaldehyde the transport of oxygen between gas and solution competes with autocatalytic oxidation of the substrate by free radicals. Gas-evolution oscillators involve nucleation, growth, and escape of bubbles from a solution made supersaturated by a chemical reaction. Workers elsewhere have studied many other oscillatory reactions. We are learning ever more about the types of processes necessary to cause oscillatory behavior, but we cannot yet establish exact criteria.
I. Introduction Twenty-one years ago, most chemists including myself either had never heard of oscillatory reactions or else believed that the few alleged examples were artifactual. That situation is entirely different today! Our laboratory in Eugene has been fortunate to participate in elucidating the mechanisms of several different oscillators. This review has selected five reactions to which we have devoted major effort ourselves and in sections 11-VI will present a brief summary of what is known about each. Section VI1 will list briefly a few of the many other oscillatory systems that have been studied elsewhere and will develop a classification of the various oscillatory systems that are now known. Section VI11 will then attempt to list the various criteria that seem to be required in order to generate oscillations. This review is almost completely restricted to systems in which concentrations of some intermediates oscillate with periods of the order of a minute and with amplitudes that change only slowly with time. We shall virtually ignore other phenomena such as complex bursting patterns with trains of oscillations with different amplitudes and frequencies, chaotic nonrepetitive combinations of such bursting patterns, propagation of waves at velocities dependent only on local composition of the medium, propagation of waves associated with macroscopic gradients of composition, and many other exotic phenomena. This brief list of topics to be ignored illustrates the diversity of a field which started only two decades ago! It is useful to note at the outset the distinction between harmonic and relaxation oscillations. All scientists are familiar with harmonic oscillations where the behavior of some property of a system can be described by a sine function. Many chemists are unfamiliar with so-called relaxation oscillators which exhibit almost discontinuous transitions between distinguishable states each of which persists for a significant time between transitions. Such an oscillator is illustrated in Figure 1 which shows the behavior in a typical Belousov-Zhabotinsky system of an electrode specific to bromide ion. Between points E and F, the concentration of bromide is relatively high and the system is in a reduced state. At point F, the concentration of bromide undergoes an almost discontinuous transition of about an order of magnitude to an oxidized state which persists between points G and H. These very rapid transitions between clearly defined oxidized and reduced states are characteristic of several homogeneous solution oscillators, although selected compositions which differ little from those of a stable steady state may exhibit oscillations of much smaller 'This paper is No. 86 in the series "Chemical Oscillations and Instabilities". No. 85 is: Ruoff. P.: Noyes, R. M. J. Phys. Chem. 1989. 93, 7394-7398.
0022-3654/90/2094-4404$02.50/0
amplitude which can be approximated by a sine function. 11. The Belousov-Zhabotinsky Reaction A. Chemical Mechanism. By far the most studied and best understood example of a chemical relaxation oscillator was discovered by Belousovl in the early 1950s and brought to the attention of western chemists by Zhabotinsky.2 The history of the early difficulties in publication has been recounted by Winfree.j The overall reaction consists of the oxidation of an organic substrate such as malonic acid by acidic bromate often but not always catalyzed by a 1 equiv redox couple such as Ce(III)/ Ce(1V). The main features of the mechanism were elucidated by Field, Koros, and no ye^,^ and the skeleton version presented here is derived from a treatment by Ruoff and no ye^.^ The overall chemistry can be described by process T1 which
Br03- + 3RH
+ H+
-.+
2ROH
+ RBr + H 2 0
(TI)
is concerned only with the number of equivalents by which organic matter is oxidized without regard for the detailed organic chemistry. Although process TI describes the overall stoichiometry in the system, that process does not take place in a single step. Organic matter does not react directly with bromate at a significant rate, but Br03- can be reduced to HOBr either by Br- or by Mn+ according to stoichiometric process A or B, respectively. Br0,-
+ 2Br- + 3H+
Br03- + 4Mn+ + 5H+ -.+ HOBr
-
3HOBr
(A)
+ 4M("+')+ + 2 H 2 0
(B)
RBr is formed from HOBr by process C, and the rest of the overall stoichiometry can be attained with process D or E,
-
+ H20 ROH + Br- + H+ HOBr + R H 2M("+')+ + R H + H 2 0 2Mn+ + ROH + 2H+ HOBr
+ RH
-
-
RBr
(C)
(D) (E)
( I ) Belousov, B. P. Sb. Ref. Radials. Med. za 1958, Medgiz. Moscow 1959, I , 145-147. (2) Zaikin, A. N.; Zhabotinsky, A. M. Nature (London) 1970, 225, 535-537. . -.- - . . (3) Winfree, A. T. J. Chem. Educ. 1984.61, 661-663. (4) Field, R. J.; Koros, E.; Noyes, R. M. J. Am. Chem. SOC.1972, 94, 8649-8664. ( 5 ) Ruoff, P.; Noyes. R. M. J. Chem. Phys. 1986, 84, 1413-1423.
0 1990 American Chemical Society
The Journal of Physical Chemistry, Vol. 94, No. 1I, 1990 4405
Feature Article
TABLE I: Parameters for an Oscillatory Oregonator kw = 3 X IO3 kol = 1.3 ko2 = 2.4 X IO6 kos = 0.02 ko, = 34 f = l A = 0.02
b
G I50
300
&
450
G 750 960 SECONDS-
650
lk0
650
kJ0
-3.0
Figure 1. Potentiometric trace of log [Br-] in a Belousov-Zhabotinsky system at room temperature. The state is reduced between points E and F and is oxidized between G and H. Initial concentrations were [CH2(CO,H),] = 0.032 M, [KBrOJ = 0.063 M, [KBr] = 1.5 X 10" M, [Ce(NH,)2(N03)5]= 0.001 M, [H,SO,] = 0.8 M. (From ref 4.)
whichever is appropriate to counter the effect of A or of B, respectively. The stoichiometry of (TI) can then be generated by either of the sequences in eqs 1 and 2. (Tl)
= (A) + (C) 5
(B)
+ 2(D)
(1)
+ (C) + 2(E)
(2)
The same overall process can thus be accomplished by either of two independent sequences. Such a situation appears to be a necessary but not a sufficient condition if the overall mechanism is to generate oscillations. The processes as presented here involve other kinetic complications such as that process C is catalyzed by bromide ion while process D involving the same reactants as C is induced by the occurrence of process E. However, these complications are also insufficient to ensure oscillatory behavior. The oscillations arise because of the detailed mechanisms of processes A and B, which occur by the pseudoelementary processes P1-P4; the numbers are chosen so as to facilitate comparison with the Oregonator model discussed later.
-
+ Br- + 2H+ HBrO, + HOBr HBr02 + Br- + H+ 2HOBr Br03- + HBr02 + H+ s 2Br02' + HzO BrO,' + Mn+ + H+ HBr02 + M("+I)+ 2HBr0, Br03- + HOBr + H+ Br0,-
-
-
-
(PI) (P2) (P3a) (P3b) (P4)
The stoichiometries of processes A and B are then attained by the sequences of eqs 3 and 4, respectively.
+ (P2) (B) = 2(P3a) + 4(P3b) + (P4) (A)
= (Pl)
(3) (4)
When a system contains significant Br- and almost no M("+l)+, process A takes place followed by C, and Br- is depleted. Such a system is in a reduced state with relatively large [Br-] and very low concentrations of HBr0, and of M("+*)+. As Br- is consumed by a system in a reduced state, the sequence (P3a) 2(P3b) which produces HBr0, becomes ever more important compared to step (P2) which consumes it. When the rate of (P3a) + 2(P3b) becomes equal to that of (P2), [HBrO2] increases autocatalytically and process B becomes dominant followed by C, D, and E. Such a system is in an oxidized state with relatively large concentrations of HBr02 and of M(*l)+ and very low [Br-1. However, the rate of (P3a) + 2(P3b) reaches a maximum equal to twice that of second-order process P4 while processes D and E produce Br- which accelerates (P2) until it attains the new maximum rate of (P3a) + 2(P3b). Then HBrOz is being consumed more rapidly than it is being produced, and the system switches back almost discontinuously to a reduced state. Figure 1 illustrates the typical relaxation oscillations in such a system. B. The Oregonaror Model. The above qualitative discussion of the chemistry is complicated for somebody unfamiliar with the oscillating system. The argument can perhaps be appreciated better by examining the simpler Oregonator model developed by
+
-4.0 -5.0
- 6.0
s
3 -7.0
E
.u
C 0, -8.0
i
0
141
203
566
424
767
849
960
Time, s Figure 2. Logarithmic plots of the variables of eqs 7-9 with parameter values taken from Table I.
Field6 which reproduces many but not all features of the full chemistry. Consider the five irreversible steps 01 to 05, where A+Y+X+P (01) X+Y-2P (02) (03) A X j 2X + Z 2X-A+P (04)
+
z -fy (05) A = BrO), X = HBr02, Y Br-, Z 2M("+I)+,and P HOBr, whilefis a stoichiometric factor associated with the relative rates of processes C, D, and E in the full chemical system. If processes a and /3 in the Oregonator model simulate processes A and B in the chemical system, we obtain eqs 5 and 6 . Iff = 1, the overall stoichiometry 2A 4P is generated by the sequence (a)+ (0) + 2(05). A + 2Y 3P (a)
-
-+
+
(e)= (01) (02)
(5)
A-P+2Z (6) (p) = 2(03) + ( 0 4 ) (6) The qualitative argument for the chemical system above suggests that process a will be dominant for a reduced state with large Y and small X and Z while process p will be dominant for an oxidized state with small Y and large X and Z. The following semiquantitative argument will support those suggestions and show why the oscillating system repeatedly changes between oxidized and reduced states. The dynamic behavior of the Oregonator model is described by differential eqs 7-9. dX/dt = ko1AY.- ko2XY + ko3AX- 2kWXZ (7) dY/dt =fkosZ - (kolA + ko2X)Y dZ/dt = ko3AX - kosZ
(8) (9)
(6) Field, R. J.: Noyes, R. M. J . Chem. Phys. 1974, 60, 1877-1884.
Noyes
4406 The Journal of Physical Chemistry, Vol. 94, No. 11, 1990
z, = 3.85 x
Finally, eq 8 shows that Y will tend to decrease if it is greater thanflo5Z/(koiA + ko2X). In such a reduced stae, process a is dominant and Y will decrease as long as it is greater than fkosZ,/(kolA kozXa) = 1.42 X therefore, it will pass through Ycrlt and cause X to switch to X,. Furthermore, the transition from X , to X, is very much faster than that from Z, to Z,, and Y will overshoot YCrltand continue to decrease toward a minimum value given by eq 17. Figures 2 and 3 show that increasing Z halts the decrease in Y long before Y,, has been attained.
-
-3.0 -4.0
+
-
c7 0 -5.0-
-6.0 -7.0 4 -9.0
-a0
-7.0
-6.0 -5.0 LOG Y
-4.0
Y,,,
4
-so
Figure 3. Behavior of log Y and log 2 during one period of the limit cycle shown in Figure 2.
Table 1 presents a set of rate constants and composition parameters developed by Field and Forsterling7 as appropriate to a real oscillatory system. Concentrations are expressed in moles per liter and time is in seconds. Figure 2 shows the variation with time of the variables in eq 7-9 solved for the parameter values in Table I. The plot of log Y exhibits relaxation oscillations very similar to the experimental behavior of log [Br-] in Figure I . Figure 3 shows the behaviors of log Y and log Z from Figure 2 during one cycle of 422 s. The way eqs 7-9 generate the curve in Figure 3 is as follows: Except perhaps briefly in strongly oxidized states, it will be a good approximation to set eq 7 equal to zero to estimate X,, the steady-state concentration of X when the absolute magnitude of dX/dt is very small compared to the rates of change of the other variables. ko3A - ko2Y
Xs, =
(16)
10-3
+ [(ko,A - kozv)' + 8kolko4AYjl12
(10)
4ko4 X , is a function of Y only, and representative solutions are given by eqs 11-13.
if Y >> ko3A/koz: X , = kolA/ko2 = 1.08 X IO-* i f Y = ko3A/ko2: X,, =
[koiko3A2 2k02k04]
= 1 . 1 1 x 10-6
if Y
