J . Phys. Chem. 1984, 88, 3616-3622
3616
Coupling of Chemical Oscillators K. Bar-Eli Department of Chemistry, Tel Aviv University, Ramat Aviv, 69 978 Tel Aviv, Israel (Received: September 20, 1983)
Two CSTR’s (continuous stirred tank reactors) are coupled by transferring material from one CSTR to its neighbor and vice versa. The CSTR’s when uncoupled are operating in the oscillating mode; however, each is operating under different conditions. At low coupling rates the two CSTR’s oscillate at their own frequency and amplitude. At very high coupling rates the oscillation will resemble that of a CSTR operating under average conditions. In some medium coupling rates, a region is found where, rather unexpectedly, the two oscillators will arrive at a stable steady state and will stop oscillating. Several well-known chemical oscillators were examined and all were found to follow this behavior. The models used were the following: (1) the Noyes-Field-Thompson model for the oxidation of cerous ions by bromate ions; (2) the oregonator model and (3) the Field-Koros-Noyes model, both modeling the Belousov-Zhabotinski reaction; (4) the Brusselator due to Lefevre and Prigogine; ( 5 ) an autocatalytic first-order decomposition due to Kumar; and (6) the Lotka-Volterra model. Because of its generality, we expect that experimental verification of the findings will soon follow.
Introduction In a previous paper1 we have tried to follow ideas of SmaleZ and Turing3 and to introduce oscillations in coupled CSTR’s (continuous stirred tank reactors) each one in a different steady state (SS). The particular chemical system chosen was bromatecerium-bromide in a sulfuric acid sol~tion.”~This system behavior is described adequately by the Noyes-Field-Thompson (NFT) model8 as far as dependence of the SS on the various constraints is c ~ n c e r n e d . ~Recently it was predicted’O and confirmed experimentally’ 1-14 that the system can also oscillate in a fairly narrow range of constraints in the vicinity of the critical point. In this paper we investigate the problem of coupling two CSTR’s each operating under different conditions. The constraints in both CSTR’s are such that the systems will oscillate. The two oscillators are coupled by a diffusion-like process; namely, material is transferred from one CSTR to its neighbor at a rate which is proportional to the difference in the concentrations of the two cells. Figure 1 shows a schematic description of the system. We have investigated the coupling of a few well-known chemical oscillators. These are the following: ( 1 ) the NFT model, ( 2 ) the simple oregonator model (SOM),15(3) the Field-Koros-Noyes (FKN) model,16 (4)the “Bru~selator”,’~ (5) an autocatalytic decomposition model by Kumar,18and (6) the L o t k a - V ~ l t e r r a ~ z ~ ~ model. In all six oscillators there is a range of coupling rates in which, unexpectedly, the whole system becomes stable and the oscillations stop. (1) K. Bar-Eli and W. Geiseler, J . Phys. Chem., 85, 3461 (1981). (2) S. Smale in “Lectures in Mathematics in Life Sciences”, Vol. 6, J. D. Cowan, Ed., American Mathematical Society, Providence, RI, 1974, p 17. (3) A. Turing, Philos. Trans. R. SOC.London, Ser. B., 237, 37 (1952). (4) W. Geiseler and H. Follner, Biophys. Chem., 6, 107 (1977). (5) K. Bar-Eli and R. M. Noyes, J. Phys. Chem., 81, 1988 (1977). (6) K. Bar-Eli and R. M. Noyes, J . Phys. Chem., 82, 1352 (1978). (7) S. Barkin, M. Bixon, R. M. Noyes, and K. Bar-Eli, Int. J. Chem. Kiner., 11, 841 (1977). (8) R. M. Noyes, R. J. Field, and R. C. Thompson, J . Am. Chem. SOC., 93,7315 (1971). (9) W. Geiseler and K. Bar-Eli, J. Phys. Chem., 85, 908 (1981). (10) K. Bar-Eli in “Nonlinear Phenomena in Chemical Dynamics”, Vol. 12, C. Vidal and A. Pacault, Eds., Springer Series in Synergetics, SpringerVerlag, West Berlin, 1981, pp 228-39. (11) M. Orban, P. DeKepper, and I. R. Epstein, J . Am. Chem. SOC.,104, 2657 (1982). (12) W. Geiseler, Ber. Bunsenges. Phys. Chem., 86, 721 (1982). (13) W. Geiseler, J . Phys. Chem., 86, 4394 (1982). (14) K. Bar-Eli and W. Geiseler, J. Phys. Chem., 87, 3769 (1983). (15) R. J. Field and R. M. Noyes, J . Chem. Phys., 60, 1877 (1974). (161 R. J. Field. E. Koras. and R. M. Noyes, J . Am. Chem. SOC., 94, 8649 (1972 j. (17) R. Lefevre and I. Prigogine, J. Chem. Phys., 48, 1695 (1968). (18) V. R. Kumar, V. K. Jayaraman, B. D. Kulkarni, and L. K. Doraiswamy, Chem. Eng. Sci., 38, 673 (1983).
0022-3654/84/2088-3616$01.50/0
This result is rather surprising but because of its generality it seems that it would not be difficult to confirm it experimentally. Computations The systems can be generally described by the following equations:
a11 = R ( C ( 1 ) ) + ko(Co(1) - C ( 1 ) ) + kx(C(2) - C ( 1 ) )
(1)
C(1) and C(2) are the vector concentrations of CSTR’s 1 and 2, respectively (see Figure 1). ko is the flow rate divided by the cell volume (reciprocal of the retention time). k, is the coupling rate (volume per second transferred from one cell to the next divided by the cell volume). Co(l) and C0(2) are the vector concentrations of the species introduced into the CSTR’s. R(C( 1)) and R ( C ( 2 ) )describe the chemical mass action reaction rates as derived from the appropriate chemical equations of each model. In some cases the flow term ko(Co- C ) is incorporated directly into the functions R(C). These cases include the “Brusselator” and Lotka-Volterra models. In these cases, the R(C) terms should be changed to R l ( C ( l ) )and R2(C(2))in eq 1 and 2, respectively. The coupling terms remain unchanged. These changes do not affect any of the results given below. The steady states (SS) were found by solving the equations C(1) = C(2) = 0 by Newton’s method. The stability of the resulting SS was found by calculating the eigenvalues by standard computer package (EISPACK). Whenever Hopf‘s b i f ~ r c a t i o n ’was ~ encountered, Hassard’s20 routine B I F O R ~was used in order to find whether the bifurcation is subcritical or supercritical. This method is not applicable when some of the other eigenvalues are positive or have positive real parts. The properties of such bifurcations were not studied in detail. Oscillations and limit cycles were calculated by solving eq 1 and 2 directly by Gear’sz1method. The various models will now be described below, and the results given in order. Noyes-Field-Thompson (NFT) Model This model was first devised by Noyes, Field, and Thompson8 in order to explain the oxidation of cerous by bromate ions. The (19) J. E. Marsden and M. McCraken, “The Hopf Bifurcation and Its Applications”, Springer-Verlag, West Berlin, 1976. (20) B. D. Hassard, N. D. Kazarinoff, and Y.-H. Wan, “Theory and Applications of Hopf Bifurcation”, London Mathematical Society Lecture Note Series 41, Cambridge University Press, London, 1981. (21) (a) C. W. Gear, “Numerical Initial Value Problems in Ordinary Differential Equations”, Prentice-Hall, Englewood Cliffs, NJ, 1971, pp 20929; (b) A. C. Hindmarsh, “Gear: Ordinary Differential Equation System Solver”, UCID 30001, Rev. 3, Lawrence Livermore Laboratory, University of California, Livermore, CA, Dec 1974.
0 1984 American Chemical Society
The Journal of Physical Chemistry, Vol. 88, No. 16, 1984 3617
Coupling of Chemical Oscillators
I
ko
1
Iko
L
m
Figure 1. Scheme of the system of coupled CSTR's 1 and 2 each with flow rate ko in and out, and coupling k, between them.
model is given by the following chemical reactions together with their rate constants: Br03- Br- 2H+ s H B r 0 2 HOBr (3) kl = 2.1 M-3 s-l k-l = 1 X lo4 M-I s-I
+
+
HBr02
+
+ Br- + H+ F? 2HOBr
k2 = 2 X lo9 M-2 s-l
+
+
s-l
+
Br2 H 2 0 k-3 = 110
k-4 = 2
(5)
X
lo7 M-I
(6) s-l
+ Br02. + H+ F? Ce4+ + H B r 0 2
+ Br02. + H 2 0 e Ce3+ + Br03- + 2H+
k6 = 9.6 M-'
S-'
(8)
M-3 S-I
k-6 = 1.3 x
2HBr02 F! Br03-
k7 = 4 X lo7 M-I s-l
(7)
k-, = 2.4 X lo7 M-I s-]
k5 = 6.5 X lo5 M-2 s-l Ce4+
M-'
+ H B r 0 2 + H+ e 2Br02. + H 2 0
k4 = 1 X lo4 M-2 s-l Ce3+
(4)
k-2 = 5 X
HOBr Br- H+ k3 = 8 X lo9 M-2 s-' Br03-
261
+ HOBr + H+
(9)
M-2 s-'
k-7 = 2.1 X
'ryson22 has recently criticized these rate constants. His Sullivan and criticism relies on recent work of Forsterling et Thompson,2 Noszticzius et a1.,25 and Rovinskii and Zhabotinskii.26 Each of these groups has measured several component steps of the mechanism. Also a different estimate is given for the dissociation constant of bromous acid which bears on the estimate of some rate constants.27 As a result, some rate constants differ by as much as a few orders of magnitude. Bar-Eli and RonkinZ8 have shown that the old set of rate constants, given above, fits best most of the known results in CSTR. We have used therefore this set in the present work. Typical results for a single CSTR are given in Figure 2, in which the [Br03-]o-[Br-]o subspace of constraints is shown near the critical point. It is seen that below the critical point three SS can coexist (with one SS being unstable). Transitions via hysteresis between the SS were measured and compared to the calculated ones by Geiseler and Bar-ELg Above the critical point where only one SS exists, the possibility of oscillations was predicted'O and verified e~perimentally,"-'~ in the small region enclosed by the dotted line. The dependence of the oscillation region on the various constraints is given by Bar-Eli and Gei~e1er.l~ The two C S T R s are taken to be inside the dotted region of Figure 2; namely, both of them will oscillate independently. The ~
~~~~
(22) J. J. Tyson in "Oscillations and Travelling Waves in Chemical Systems", R.J. Field and M.Burger, Eds., Wiley, New York, 1983. (23) (a) H. D. Forsterling, H. J. Lamberz, and H. Schreiber, Z.Nutuforsch. A, 35, 329 (1980); (b) H. D. Forsterling, H. Lamberz, and H. Schreiber, ibid., 35, 1354 (1980). (24) J. C. Sullivan and R. C. Thompson, Inorg. Chem., 18, 2375 (1979). (25) Z. Noszticzius, E. Noszticzius, and Z. A. Schelly, J . Phys. Chem., 87, 510 (1983). (26) A. B. Rovinskii and A. M.Zhabotinskii, Theor. Exp. Chem. (Engl. Trunsl.), 15, 17 (1979). (27) A. Massagli, A. Indeli, and F. Pergola, Inorg. Chim.A m , 4, 593 (1970). (28) K. Bar-Eli and J. Ronkin, J . Phys. Chem., in press
, 35
40
45
I
I
50 55 [BrOiIo (IO3)
I
1
60
65
I
Figure 2. Multistability and oscillations domaie of the NFT mechanism near the critical point of the [BrO
