State-of-the-Art Symposium: Self-Organization in Chemistry /
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Some Models of ~hemical~scillators Rlchard M. Noyes University of Oregon, Eugene, OR 97403 This review will concentrate on models of chemical oscillations, which constitute the self-organization of a system in time without any accompanying organization in space. Manvfeatures of the threemodels that have been chosen can beappreciated by undergraduatestudents and by professors who have no background in the field. The Lotka Model In 1920, A. J. Lotka wasprobably the first to suggest that a chemical system might undergo sustained oscillations. Consider the three steps L1 to L3.
Y-P
(L3)
Although Lotka was trying to model a chemical system, the examples of this sort of behavior are found in ecology. Let A represent a large meadow receiving plenty of rain and sunshine. Let X be members of a family of rabbits who take advantage of these favorable conditions to procreate by step L1. Let Ybe members of a family of foxes who can reproduce only if they find rabbits to eat by step L2, and let step L3 be the death of foxes. The net effect of the three steps is A -P. For any arbitrary initial values of X a n d Y, this system will move on a trajectory that repeatedly passes around the steady state on a repetitive path that depends upon those initial values. Different initial states generate different trajectories. The Oregonator Model The Oregonator model was developed by R.J. Field and R. M. Noyes at the University of Oregon in 1974 to illustrate the mechanism of the Belousov-Zhah~tinlikv oscillatorv reaction. I t has three independent compositidn variablesand the five irreversible steps 01to 05.
catalytic production of X much like steps L1 and L2 in the Lotka model. All of the presently understood examples of chemical oscillators contain such processes. Still another important feature is that X and Y are in competition with each other. A critical composition may be defined as one in which the rate of formation of X by steps 01 and 0 3 is equal to the rate of its destruction by steps 0 2 and 04. If relative change in Z is much slower than in X and Y, a critical composition is unstable to perturbation in X. Any small increase in X will cause a further increase by step 0 3 while Y is decreased by step 02. A small decrease in X will have opposite effects. The value of Y,,it is almost independent of X and Z. Whenever Y is increasing or decreasing while Z is changing more slowly, the system will almost discontinuously switch dominance by a to dominance by b or vice versa whenever Y passes through Yc,it. The behavior in the preceding paragraph resembles the initiation of any chemical explosion a t a threshold. I t is step 0 5 that causes the Oregonator to model an oscillator. If process a is dominant and Z is small, Y will decrease until i t falls to Yc,itand process b bemmes dominant instead. Process b produces Z, which then causes Y to increase again. When Yrises to Y,it, the system will suddenly switch hack t o process a. Step 0 5 is an excellent example of feedback. A switch between dominance by process a or b causes Z t o change in a direction that drives Y hack toward Y,,it, but that change is delayed in its action. If ks were sufficiently large, the system would not oscillate but would approach a stable steady state. The Bubbelator Model The Bubbelator model describes a gas-evolution oscillator where a reaction that produces a gas will cause the syscem to foam up and calm down repeatedly. 'The ideas which led to this model were formulated hy K. W. Smithand R. M. Noyes about 1982. A-M M eG
Steps 01 + 0 2 generate process a. Twice step 0 3 plus step 0 4 generate process h.
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If the stoichiometric factor f is unity, a and b add to twice 0 5 to generate the overall reaction 2A 4P. If only process a took place, X. = klAlkz would be the steady-state concentration of X. If only process b took place, Xb = kaAI2k4. An important feature of the model is that for the rate constants appropriate to the chemistry Xb is about 100,000 times X,. Another important feature is that step 0 3 involves auto190
Journal of Chemical Education
Reactant A produces molecules of dissolved gas, M, by step B1, and those molecules escape from the surface of the supersaturated solution by step B2. If no other processes took place, such a system would evolve to an uninteresting stable steady state. If the supersaturation becomes sufficiently great, there is a critical threshold a t which bubble nuclei, B., form spontaneously. Any bubbles larger than these nuclei can either grow or shrink by reversible reaction B4j, and (in a gravitational field) they can escape to the gas phase by irreversible reaction B5j. An important feature of the model is that any population of bubbles is unstable even if all bubbles are initially of identical size. Let a he the surface tension and let Pj be
pressure inside a huhble of radius rj. P, can he calculated from eq 1, where P, is the pressure outside the huhble. If a solution contains dissolved gas that would be in equilibrium with bulk gas a t pressure P,, then a bubble of radius rj would neither grow nor shrink in such a solution. However, molecules from this solution will evanorate into anv bubble larger than rj,.and any bubble smaller than rj will &inkand disappear whlle the gas in i t dissolves in the solution. Another feature of the model is that bubbles little larger than nuclei have such small surface area that they exchange gas molecules only slowly with solution while large bubbles grow much more rapidly. Similarly, bubbles rise toward the surface a t a rate proportional tori; larger bubbles transport gas out of solution much more rapidly than do very small bubbles. Both of these effects generate adelayedfeedback so that there is a significant time between the formation of nuclei by stepB3 and the removalof the resulting bubbles by one of the steps B5j. General Comments This review has considered why three different models can exhibit oscillatory behavior. The Lorka model can generate an indefinitely large number of trajectories corresponding to
different initial conditions; the Oregonator and the Bubbelator each generate a limit-cycle trajectory that is independent of initial conditions. Both the Oregonator and the Bubbelator exhibit what are j u m- ~almost discontinucalled relaxation oscillations that . ously between two different types of reaction whenever a control variable passes through Y,,it or through the critical concentration for nucleation of bubbles, respectively. Furthermore, whenever the control variable passes through the critical composition, the system is unstable t o perturbation in either direction. Finally, each of these two models oscillates because there is adelay in the feedback that returns the control variable to its critical value. These ideas cannot be developed in detail here, but they may help to suggest criteria to look for when models for chemical oscillators are developed. Other models are already well established for known systems. A discussion of the Oregonator model and the experimental justification for it will be found in R. J. Field and R. M. Noyes, Acc. Chem. Res. 1977,10,214-221,273-280. A demonstration of the Belousov-Zhabotinsky reaction is described in R. J. Field. J. Chem. Educ. 1972.49.308-309. A demonstration of agas-evolution oscillator is described in S. M. Kaushik, 2. Yuan. and R. M. Noves. J. Chem. Educ.
Volume 66 Number 3 March 1969
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