6174
J. Phys. Chem. 1984,88, 6174-6177
species formed in the decomposition of “ C H 3 0 ” CH,O(a) = CHO(a)
+ H2
is directly oxidized by adsorbed oxygen. Comparison of the Behavior of Adsorbed Methanol on Pt Metals. It seems very useful to compare the adsorption and surface decomposition of C H 3 0 H with those observed on other Pt metals (Table I). Formation of the C H 3 0 species has been assumed to occur on clean Pd(100),19but there was no indication of this species on clean Pt(l1 1)l0 and Rq( 1 surfaces. The bulk of the CH30 surface intermediate decomposes below 250 K on both Pd(100) and Rh(111) surfaces, although on Pd( 100) Christmann and DemuthI9 assumed the existence of more stable C H 3 0 in very low concentration, which decomposes only at 220-330 and 220-530 K. The existence of these latter stable surface species was not confirmed by Gates and Kesmodel.” The extent of decomposition of chemisorbed C H 3 0 H to C O and H2on clean Pt, Pd, and Rh
surfaces is nearly 10% of the monolayer. Similar results were found by Sexton et aLzl for the decomposition of C1-C4 alcohols adsorbed on a Pt( 111) surface. The presence of adsorbed atomic oxygen promoted the formation of C H 3 0 on all three metals investigated so far and, similarly as in the case of Ag and Cu surfaces,6v7enhanced the stability of C H 3 0 species. The present study demonstrates that, besides the above-mentioned effects, the adsorbed oxygen changes the final reaction products of C H 3 0 H decomposition by reacting with C H 3 0species (minor step) and/or with chemisorbed CO. The extent of this oxidation reaction amounts to even 17% of the CH30 species decomposing on the Rh( 111). In the case of Pt( 111) the participation of adsorbed oxygen in the surface oxidation reaction was not observed, and no C 0 2 formation was reported.10,20For the Pd( 111) surface the Occurrence of this reaction was assumed, but no thermal desorption measurements were performed to confirm it or to establish its extent. Registry No. Cfi,OH, 67-56-1; Rh, 7440-16-6; oxygen, 7782-44-7.
~~
(19) Christmann, K.; Demuth, J. E. J. Chem. Phys. 1982,76,6308,6312. (20) Yates, J. T.; Gqodman, D. W.; Madey, T. Proc. Int. Vac. Congr., 7th 1977, 1133.
(21) Sexton, B. A,; Rendulic, K. D.; Hughes, A. E. Surf.Sci. 1982, 121, 181. Rendulic, K. D.; Sexton, B. A. J . Catal. 1982, 78, 126.
The Dynamics of Coupled Chemical Oscillators K. Bar-Eli Department of Chemistry, “el-Aviv University, Ramat Aviv, 69978 Tel- Aviv, Israel (Received: June 29, 1984)
The Noyes-Field-Thompson model is used to calculate the behavior of two coupled CSTRs containing bromate, bromide, and cerous ions in sulfuric acid solution and operating in the oscillating region. The dynamics of the coupled system is analyzed according to the model. As the coupling rate increases one finds regions with two different types of oscillations, stability and hysteresis between oscillations and stability and entrained oscillations. A quantitative measure as to the “good((qixing Qf a CSTR i s given. The experimental difficulties involved in the verification of the above predictions are discussed.
Introduction The coupling of chemical oscillators shows strange and rather unexpected phenomena. The dynamics of coupled CSTRs (continuous stirred tank reactor) may be totally different than that of the single CSTR. stability domains, chaos, entrainment of oscillators, different periods, and different amplitudes were all found. Lefever and Prigoginel showed that two coupled Brusselators may have an inhomogeneous stable steady state. Bar-Elia has shown that stable steady states could be achieved when many other oscillators, in addition to the Brusselator, are coupled. Such oscillators include the Noyes-Field-Thompson ( N I T ) mode1,j the Field-Koros-Noyes4 (FKN) model of the Belousov-Zhabotinskii r e a ~ t i o nthe , ~ Oregonator model6 devised as a simple mechanism for the same reaction, the Kumar7 model for gaseous reaction, (1) Lefever, R.; Prigogine, I. J . Chem. Phys. 1968, 48, 1695. (2) Bar-Eli, K. J . Phys. Chem. 1984, 88, 3636. (3) Noyes, R. M.; Field, R. J.; Thompson, R. C. J. Am. Chem. SOC.1971, 93, 7315. (4) Field, R. J.; Koros, E.; Noyes, R. M. J . Am. Chem. SOC.1972, 94, 8649. (5) (a) Belousov, B. P. Sb. Ref. Radiats. Med. 1958, 145. (b) Zhabotinskii, A. M. Dokl. Akad Nauk. SSSR 1969, 157, 392. (c) Winfree, A. T. Science 1973, 181, 937. ( 6 ) Field, R. J.; Noyes, R. M. J . Chem. Phys. 1974, 60, 1877. (7) Kumar, V. R.; Jayaraman, V. K.; Kulkami, B. D.; Doraiswamy, L. K. Chem. Eng. Sci. 1983, 38, 613.
0022-3654 /84/2088-6 174$01.50/0 , I
,
and the Lotka-Volterras prey-predator model. These oscillators differ quite appreciably from each other in the number of species, the nature of the nonlinearity, and the nature of the oscillations whether conservative or of the limit cycle type. Tyson’O has dealt with the problem of Brusselators coupled in series, where the output of one oscillator serves as the input for the second. In the other work as well as the present one, the coupling is in parallel in a diffusionlike manner, Le., the rate of transfer is proportjonal to the concentration difference in the cells. In this work the dynamics of coupled N F T oscillators3 is investigated. This model was shown theoretically11J2and verified e~perimentally’~-’~ to oscillate in addition to its well-known b i ~ t a b i l i t y . ’ ~ .This ~ ~ oscillator is the only one for which the (8) (a) Lotka, A. J . Phys. Chem. 1910, 14, 271. (b) Lotka, A. J . Am. Chem. SOC.1920, 42, 1595. (c) Volterra, V. “Lecons sur la Theorie Mathematique de la Lutte pour la Vie”; Gauthier-Villars: Paris, 193 1 . (9) (a) Schreiber, I.; Marek, M. Physica D (Amsterdam) 1982,’SD,258. (b) Schreiber, 1.; Marek, M. Phys. Lett. A 1982, 91A, 263. (10) Tyson, J. J. J . Chem. Phys. 1973, 58, 3919. (1 1) Bar-Eli, K. In “Nonlinear Phenomena in Chemical Dynamics”; Vidal, C., Pacault, A., Eds.; Springer-Verlag: West Berlin, 198 1; Spriiger Series in Synergetics, Vol. 12, pp 228-239. (12) Bar-Eli, K.; Geiseler, W. J . Phys. Chem. 1983, 87, 3769. (13) Orban, M,; DeKepper, P.; Epstein, I. R. J . Am. Chem. Soc. 1982, 104, 2651. (14) (a) Geiseler, W. Ber. Bunsenges. Phys. Chem. 1982, 86, 721. (b) Geiseler, W. J . Phys. Chem. 1982, 86, 4394. (15) Geiseler, W.; FOIlner, H. Biophys. Chem. 1977, 6, 107.
0 1984 American Chemical Society
The Journal of Physical Chemistry, Vol. 88, No. 25, 1984 6175
Dynamics of Coupled Chemical Oscillators mechanism is known completely, all the chemicals are well defined, and it does not contain any undefined stoichiometric factors. The rate constants for the model give a fairly good agreement with all the experimental data.16J7 Tyson18 criticized some of the rate constants, but Bar-Eli and RonkinI7 showed that the previously suggested rates by Noyes3 et al. give a better agreement with experiments.
r
0.48rm a
b
0.3 (D
0
‘ii 0.1
F
L
Calculations The N F T mechanism is made up of the following seven chemical reactions, the rate constants of which are shown. These rate constants are partly measured: partly estimated,lg and slightly corrected to agree better with kinetic data.I7p2O Br03- + Br- + 2H+ F! H B r 0 2 + HOBr (1) kl = 2.1 M-3 s-l HBr02
k-l = 1 X lo4 M-I
+
+
HOBr Br- H+ k3 = 8 X lo9 M-2 s-l
+
(2)
+
Br2 H 2 0 k-3 = 110 s - ~
+
(3) I C , ’ ,
+
Br03- H B r 0 2 H+ 9 2Br02. H 2 0 k4 = 1 X lo4 M-2 s-l k-4 = 2 X lo7 M-’ s-l
+
+
+
Ce3+ Br02. H+ Ce4+ H B r 0 2 k, = 6.5 X lo5 M-2 s-I k-, = 2.4 X lo7 M-’
(4)
0
1
I
I
,
,
I , , ,
1000 2000
j
s-l
(6)
+ HOBr + H+
(7) M-2 s-l k7 = 4 X lo7 M-’ s-I k-7 = 2.1 X The rate equations for the coupled cells are thus = R(C(1)) + ko(C0(1)-C(1)) + kx(C(2)-C(l)) = R(C(2)) + ko(Co(2)-C(2)) + kx(C(l)-W)) where C(1) and C(2) are the concentrations of the various species in cell 1 and 2, respectively. R(C(1)) and R(C(2))are the mass action rates formed from the above reactions; ko is the flow rate divided the cell volume, and k, is the coupling rate. Co are the concentrations of the species in the inflow taken to be [BrO,-], = 60 X M, [Ce3+Io= 1.5 X lo4 M, [H+], = 1.5 M for both cells, and [Br-],(l) = 320 X 10” M, [Br-lO(2)= 280 X M for cell 1 and 2, repectively. The inflow ko is taken to be 4 X s-l for both cells. According to previous work1’J2 the two cells are well within the oscillating region as is seen in Figure 1 when no coupling is operating. Details of the bistability and oscillating domains as a function of various parameters and comparison to experimental data can be found elsewhere.”-14 The steady states were found by Newton’s method; the eigenvalues and thus the stability of the steady states were calculated by a standard computer package (EISPACK). The Hopf bifurcations, whether supercritical or subcritical, were calculated by Hassard’s22routine BIFOR2. The dynamics of the coupled system was found by solving the resulting differential equations by the Gear23method. (16) Geiseler, W.; Bar-Eli, K. J . Phys. Chem. 1981, 85, 908. (17) Bar-Eli, K.; Ronkin, J. J . Phys. Chem. 1984, 88, 2844. (18) Tyson, J. J. In ’Oscillation and Travelling Waves in Chemical Systems”; Field, R. J., Burger, M., Eds.; Wiley: New York, in press. (19) Latimer, W. M.; “Oxidation Potentials”, 2nd ed.;Prentice-Hall: New York, 1952. (20) Barkin, S.;Bixon, M.; Noyes, R. M.; Bar-Eli, K. Int. J. Chem. Kinet. 1977, 1 1 , 841. (21) (a) Ioos, G.; Joseph, D. D. “Elementary Stability and Bifurcation Theory”; Springer-Verlag: New York, 1980. (b) Marsden, J. E.; McCraken, M. “The Hopf Bifurcation and Its Applications”;Springer-Verlag: New York, 1976. (22) Hassard, B. D.; Kazarinoff, N . D.; Wan, Y.-H. “Theory and Applications of Hopf Bifurcation”; Cambridge University Press: Cambridge, England, London Mathematical Society Lecture Notes Series 41.
1
,
8
’
8
1
3000 4000 0
TIME
(5)
Ce4+ + Br02- + H 2 0 F? Ce3+ + Br03- + 2H+ M-3 S-l k6 = 9.6 M-’ S-l k-6 = 1.3 X 2HBr02 F? Br03-
r1 L
M-l s-l
k-2 = 5 X
:
L d
s-l
+ Br- + H+ e 2HOBr
k2 = 2 X lo9 M-2 s-I
I ’
1000
2000 30 0
4000
TIME
Figure 1. Plots of time vs. bromide ion concentration in the two cells in the following coupling rates: (a) k, = 0; (b) k , = 3 X (c) k, = 4 X lo4; (d) k, = 5 X lo4 s-I. Constant parameters are given in the text. Arrow shows the steady state.
Results Small Coupling Rates. There are ten chemical species in each cell but because of conservation laws, only five are independent. Thus the phase space of the whole system is ten-dimensional. All the results are shown in terms of [Br-](l) and [Br-](2), although any other species could have been chosen. All phase space plots are shown as these two-dimensional projections of the ten-dimensional phase space. In Figure 1 bromide ion concentrations in the two cells are shown as a function of time. When k, = 0, Le., no coupling, the two cells are independent, and each is operating without reference to the other. Cell 1 is seen to be oscillating with period of 506 s while cell 2 is oscillating with period of 160 s. The periods are not commensurable and thus the phase space plots (not shown in this case) will slowly cover up the whole space between the minimum and maximum values of the species. The character of the two oscillations seems to be quite different. The slower oscillations of the first cell are of the relaxation type. Most of the time is spent near the steady state (in fact slightly above it) with fast and short excursions to the low and high concentrations range. On the other hand, the faster oscillations of the second cell are of smaller amplitude and resemble more a sinusoidal oscillations around the steady state. The amplitude ratios of the two cells, Le., [Br-],,x/[Br-],in, are 8.9 and 2.6, respectively. The number of eigenvalues with positive real values is four. Since there is no coupling, the steady states and the corresponding eigenvalues are the same as those of the separate cells, which, being in the oscillatory state, have each two eigenvalues with positive real part. As the coupling rate increases, the form of the oscillations changes with it. Both cells still oscillate with roughly the same amplitude ratio. The complete period of the first cell increases s-’ for instance. This to 11 15 s, for a coupling rate of 3 X period is made, however, from two subperiods of 542 and 578 s. The second cell oscillates in the same period as before, but the oscillations can easily be grouped to two groups each with three and four oscillations. These changes make the coupled oscillations (23) (a) Gear, C. W. “Numerical Initial Value Problems in Ordinary Differential Equations”; Prentice-Hall: Englewood Cliffs, NJ, 1971; pp 209-229. (b) Hindmarsh, A. C. “Gear: Ordinary Differential Equations Systems Solver”; VCID 2001, Ref. 3, Dec 1974.
6176 The Journal of Physical Chemistry, Vol. 88, No. 25, 1984 I
I
I
I
Bar-Eli I
I
I
I
I
I
I
I
1
I
I
2
3
4
5
6
[Br-](l )IO7
Figure 2. Phase space projection of [Br-](l) vs. [Br-](2) for k, = 3 X
d. Point shows the unstable steady state.
Figure 4. Phase portrait for k, = 110 X 0567
&
3
I
1
/
s-I.
Point shows the stable
steady state. n
r
h
I I
m
Y
Figure 5. Time concentration plots for (a) k, = 110 X s-I and (b) k, = 1000 X Arrow shows the steady state, stable for (a),
unstable for (b). I1
4 90
I
I
4 95
5 00
5 05
[ Br-](l)xIO' Figure 3. Phase portrait for k, = 5 X lo4 s-l. Point shows the unstable
steady state. a true limit cycle and not chaotic. As can be seen from Figure 2, the phase space plot, although rather complicated, is a closed curve which repeats itself each 1115 s unlike the uncoupled case. As the differences in the coupling rates among plots b, c, and d of Figure 1 are rather small, one should take extreme care when testing this phenomena experimentally. As the coupling rate changes only slightly, large differences in the pattern of oscillations may arise and experiments may show a chaotic behavior although the computations give oscillations which are complex but not chaotic. As the coupling rate increases, the period increases from 1 115 s at k, = 3 X 10" to 1589 s at k, = 4 X The humps in the middle of the oscillations in first cell become more pronounced and the subgroups of oscillations in the second cell contain now four and six periods of 160 s. This means that the subperiods become longer and the system spends more time around the steady state. The phase space portrait looks qualitatively similar to that of Figure 2. The many periods in the second cell are easily seen in the nearly closed cycles a t the right-hand side of the figure, with the large and sparse excursions in [Br-](l). At k, = 4.247 X 10" both cells go over to small sinusoidal oscillations around the steady state with simple phase space diagram as seen in the time plots of Figure s-'. 1 and the phase portrait in Figure 3 for k, = 5 X In fact, the oscillations in the second cell did not change much, while those of the first cell became suddenly small. The phase portrait of Figure 3 is in fact very thin (note the scale) and looks
very much like one of the nearly closed thin cycles on the right-hand side of Figure 2. We note that a t k, = 3.5185 X s-l the number of eigenvalues with positive real part changes from four to two with no noticeable qualitative change in the pattern shown and described. The limit cycle shown in Figure 3 continues to shrink and falls to zero at the supercritical Hopf bifurcation which occurs at k, = 10.9769 X lo4 s-l where all the real parts of all the eigenvalues become negative. If the system starts at the small simple limit cycle, shown in Figure 3, and the coupling is decreased, the limit cycle will remain simple and the oscillations sinusoidal until k, = 3.800 X lo4 s-l. < k, < 4.247 X In the small region 3.800 X both limit cycles, the large and complicated, Figure 2, and small simple one, Figure 3, coexist. The system will choose one or the other according to the initial conditions. Medium Coupling Rates. In addition to the Hopf bifurcation mentioned above, another one occurs at higher coupling rates where two eigenvalues become positive real valued. Thus in the range 10.9769 X lo4 C k, < 285.362 X lo4 s-' all the eigenvalues are negative and the system is stable. The upper Hopf bifurcation is subcritical, so that there is a range below this point where oscillations coexist with the stable steady state. Thus in the range 10.9769 X 10" < k, C 102.1 X lo4 s-' only the stable steady state exists and the system will be stabilized regardless of the initial conditions. In the coupling range 102.1 X < k, < 285.362 X lo4 s-l both stable steady states and the limit cycle shown in Figure 4 and time plots in Figure 5 are feasible. The steady state marked in both figures is stable, meaning that an unstable limit cycle must also exist between the steady state and the limit cycle. This unstable limit cycle will fall onto the
The Journal of Physical Chemistry, Vol. 88, No. 25, 1984 6177
Dynamics of Coupled Chemical Oscillators TABLE I: Periods and Amplitude Ratio in Medium and High Coupling Rates
k,
X IO4, s-l
102.1 105 110 150 250 500
1000 2500 a”
period, s 196 193 190 177 172 178 192 202 203
amplitude ratio cel! 2 cell 1 2.91 3.13 3.3 3.75 4.09 4.33 4.77 4.84 4.89
2.89 3.03 3.06 3.28 3.65 4.04 4.51 4.53 4.89
“The infinite value was taken to be the same as the average cell, Le., with [Br-](l) = [Br-](2) = 300 X 10” M.
IC
--0 X
N
0
[Br-](l)x107
Figure 6. Phase portrait for k, = 1000 X lo4
s-l.
Point shows unstable
steady state. steady state at the upper bifurcation point, Le., at k, = 285.362 x 10-4 s-1. The initial conditions of the system will1 determine whether the stable limit cycle or the stable steady state will be achieved. The oscillations look fairly simple although they are not symmetrical, concentration wise, around the steady state. Each cell spends about half its time above and half below the steady state. High Coupling Rates. From the upper Hopf bifurction point to infinite rates the oscillations continue to be small and Table I summarizes the periods and amplitude ratios in the medium and high coupling ranges. A typical phase portrait in this range is shown in Figure 6. Both cells are oscillating with the same period; however, their phase and amplitude are different. As the coupling rate increases, the oscillations become more symmetrical around the steady state, Figure 5b. The phase portrait becomes thinner and approaches the line bisecting the coordinates, since the amplitude and the phase of the two cells approach each other. However, even at very high coupling rates, about 25 times faster than the flow rate, there is still a small but finite phase difference, as manifested by the width of the limit cycle in Figure 6 between the two cells. This width, Le., the phase difference between the two cells, can still be noticed even at coupling rates s-l. as high as k , = 5000 X
The amplitudes, given in Table I, should be the same at very high coupling rates. The data in the table, as well as the deviation of the phase plot from the bisecting line, show that these high coupling rates are not enough to be considered “infinite”. This means that the mixing of a single CSTR should be very fast in order to ensure that the CSTR really behaves as one unit. Experimentally, SorensenZ4has noticed, sometime ago, that the rate of mixing has a profound effect on the oscillations of the Belousov-Zhabotinskii and similar systems. From Table I we see that the period goes through a minimum, near the bifurcation point, as the coupling rate increases from 102.1 X lo4 s-l to infinity, where the period approaches the value of the average cell. At the same time, the amplitude ratio of both cells rise monotonically to approach the average cell value. The ratio of the amplitudes, Le., the ratio of the last two columns of Table I, goes through a small maximum, again, near the bifurcation point.
Discussion The results presented above depend, of course, on the rate constants used for reactions 1-7. If the rate constants used are not exact, the particular numbers given in the text should be changed. The main results and conclusions will remain, however, unaltered. Our computations reveal three interesting coupling regions. The first is a range of the coexistence of two types of limit cycles as seen typically in Figures 2 and 3. Hysteresis phenomena between these two types of oscillations or limit cycles will therefore occur. As one increases the coupling rate, the large amplitude oscillations of cell 1 will suddenly become small at k, = 4.247 X lo4 s-I. Starting, however, from the small oscillations and decreasing the coupling rate, the amplitude will s-l. suddenly increase at k, = 3.800 X The second is a large domain of stability in which the oscillating CSTRs should stabilize each other, regardless of the initial conditions. The third region of coupling rates should show a hysteresis between a stable state and an oscillating limit cycle. The last conclusion resulting from the above calculations is the tremendously fast mixing necessary to obtain a veritable single CSTR. This point should be kept in mind in all experiments of this type, with this as well as in other systems. A rough estimate for the necessary fast mixing should be taken to be a hundred times faster than the rate of flow to and from the CSTR. Insufficient mixing may also manifest itself experimentally by a seemingly chaotic behavior where computations do not show it (cf. above). The experimental verification of this phenomenon is very difficult since it calls for extreme care in the control of the coupling rates. Obviously, the peristallic pumps with which most of the work has been usually done should be replaced by a continuously working pumps,25both for the inflow and for the coupling. The parameter range of obtaining the oscillations is fairly narrow.’ 1-14 The experimental verification of the above results becomes thus even more challenging. Registry No. BrO,-, 15541-45-4; Br-, 24959-67-9; Ce, 7440-45-1. (24) (a) Sorensen, P. G. Ber. Bunsenges. Phys. Chew. 1980,84,408. (b) Sorensen, P. G. In “Kinetics of Physicochemical Oscillations”; Discussion Meeting Deutsche Bunsengesellschaft fur Physikalische Chemie: Aachen, West Germany, Sept 1979; p 41. (25) Lachman, H., private communication.
