The Journal of
Physical Chemistry
0 Copyright, 1986, by the American Chemical Society
VOLUME 90, NUMBER 2 JANUARY 16,1986
LETTERS Spiral Waves in a Model of the Ferroln Catalyzed Belousov-Zhabotinsky Reaction A. B. Rovinsky Institute of Biological Physics of the Academy of Sciences, Pushchino, Moscow Region, 142292, USSR (Received: June 3, 1985; In Final Form: September 20, 1985)
A two-dimensional Belousov-Zhabotinsky chemical medium is simulated (the bromate-ferroin-bromomalonic acid-sulfuric acid system). Solutions in the form of propagating spiral waves with one, two, and four arms are obtained. The dependences of the period of the waves on the concentrations of bromate and sulfuric acid are falling branches of a U-shaped curve. The period rises slightly with rising concentrations of bromomalonic acid and ferroin and does not depend on the number of the arms of a vortex. A complex behavior of the core of the vortices is observed, just as found earlier experimentally by Agladze and Krinsky.
Introduction Formation of spatio-temporal structure is one of the general ways of self-organizing nonlinear active media. It is rather common in biological, chemical, and physical systems.I4 Some of the most fascinating examples of this process are spiral waves. Spiral waves were theoretically predicted by Wiener and Rosenblueth5 (spiral wave rotating around a hole) and Balakhovsky6 (spiral in a homogeneous medium). They were experimentally discovered by Zhabotinsky and Zaikin7 and by Winfree.8s9 Agladze and Krinsky'O obtained two, three- and four-armed (1) v o l i s , G.; Prigogin, I. "Self Organization in Non-Equilibrium Systems ; Wiley: New York, 1977. (2) Haken, G. 'Synergetics"; Springer-Verlag: West Berlin, 1983. (3) 'Oscillations and Travelling Waves in Chemical Systems"; Field, R. J., Burger, M., Eds.; Wiley: New York, 1985. (4) "Self-Organization. Auto-waves and structures for from equilibrium"; Krinsky, V. I., Ed.; Springer-Verlag: West Berlin, 1984. (5) Wiener, N.; Rosenblueth, A. Arch. Inst. Cardiol. Mex. 1946, 16, 205. (6) Balakhovsky, I. S. Biofizika 1965, 10, 1063. (7) Zhabotinsky, A. M.; Zaikin, A. M. "Oscillatory Process in Biological and Chemical Systems"; Nauka: Pushchino, 1971; Vol. 2. (8) Winfree, A. T. Science 1972, 175, 634. (9) Winfree, A. T. Science 1973, 185, 937.
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vortices in the same medium. They observed an unexpectedly complex behavior in the core of the vortices: the ends of the arms periodically joined together and separated and did not rotate stationarily . While the principal chemical mechanism underlying the oscillatory process in the BZ reaction seems understood,",l2 it has not been clear whether the appearance of the spiral waves could be described within the same framework. That question is studied here. The active chemical medium is simulated on the basis of a model of the bromate-ferroin-bromomalonic acid reaction.'* Spiral wave solutions are obtained. The dependence of the period on the parameters is found. The vortices behave just as observed experimental1y.l0
Model and Calculations It has been shown12 that the bromate-ferroin-bromomalonic acid reaction can be adequately described in the framework of (10) Agladze, K. I.; Krinsky, V. I. Nature 1982, 296, 424. (11) Field, R. J.; Koros, E. E.; Noyes, R. M. J. Am. Chem. Soc. 1972, 94, 8649. (12) Rovinsky, A. B.; Zhabotinsky, A. M. J . Phys. Chem. 1984,88,6081.
0 1986 American Chemical Society
Letters
218 The Journal of Physical Chemistry, Vol. 90, No. 2, 1986
the modified FKN" mechanism:
H*
+ HBrO3 + HBrO, * HBrOZ' + BrOy + H,O BrOy + H+ * HBr02+ + HBrOz++ Fe(phen),'* + HBr0, 2HBrOz+ HOBr + HBrO, H++ Br- + HBrO, * 2HOBr
Fe(phen),,*
+ Br- + HOBr e Br, + H,O H+ + Br- + HBr03 * HBrO, + HOBr H+
Fe(phen)?+
(2) (3) (4) (5)
(6) L9[HVOJ
(7)
+ .CBr(COOH)2 (8)
+ .CBr(COOH), * H* + Br- + COH(COOH), HOBr + CHBr(COOH), * CBr,(COOH), + H 2 0 H,O
+ CHBr(COOH),
t
Figare 1. Null clines for the local part of model 14. The cross point
+ CHBr(COOH), * Fe(phen)P
Br,
(1)
CBr,(COOH)2 + H* + B i
corresponds to the steady state.
(9)
(10) (1 1)
Directly from the scheme 1-11 a model was obtained: dx
c-
dr
= x(l
x-a - x ) - 2qa-1 7.- 7 . x+a
dz -= dr
X-m-
(12)
7.
1-7.
where [Fe(phenh3+]= Cz, [HBrOJ = (k,A/2k4)x,e = klA/k4C, 01 = k&B/k12A2h02, a = 2k4k,/k,ks, f = k4C/kl2A2hQ2, KS= k8k9/k.8,C = [Fe(phen),f+] [Fe(~hen),~+]; A = [HBrO,], B = [CHBr(COOH),J, ho IS the acidity function, q is the stoichiometric factor, k* are rate constants. Estimates for the rate constants are k , = 100, k4 = 1.7 x 104, k , = lo', k, = 15, and ks = 2 X In ref 13 some evidence was supplied for the fact that in this system a slow hydrolysis of hramomalonic acid takes place
+
CHBr(COOH),
+ H,O
-
CHOH(COOH),
+ H+ + B i
Figure 2. Wave evolution in the core of single-, double- and four-armed spirals. Time intervals 1.45 s. A = 0.03,B = 0.3,C = 0.001. h, = 2.
(13)
Although slow, it can play a role at-large concentrations ofbromomalonic acid and should be taken into consideration. Thus, the reaction-diffusion system considered is
az 7. - = x - aa? 1-2
+ dzAz
where
The following set of constants was ultimately fixed: k , = 100, k4 = 1.7 X lo4, kS = lo', k7 = 15, ks = 2 X k I 3= lod, 0, = 0, = lWS,and q = 0.5. Model 14 looks very like the model of Tyson and Fife14 which was used for studies of target pattern formation. It should he noted, however, that those model are not the same. Their distinctions were discussed earlier." Systems of differential equations like (12) and (14) are often stiff. However, in the case considered here the fastest motions of the variables had to be followed. Therefore, the model 14 was not actually stiff. For this reason and since there was no need (13) Rwinsky, A. 6.; Zhabatinsky, A. M. ' P r d i n g s of the 4th In&national Symposium on Homogeneous Catalysis": Gordon and Breach: New York, 1985. (14) Tyson, I. J.; Fife, P. C. J. P h p . C h e k 1980, 73, 2224.
Figure 3. Concentration distribution and maps of the levels in the cores of the vortices.
for high accuracy the simplest Euler technique turned out the best choice for the computations. The model medium was a square of 40 X 40 elements. The error caused by space discretization did not exceed 5% for the period of waves and was even less for other values. The parameters of the model were chosen such that the local system was in an excitable (nonoscillatory) steady state c o n e s p d i n g to the reduced form of the catalyst (Figure 1). The initial distribution of the concentrations in the medium was created as follows, First the calculations were carried out in one half of the square (20 X 40
J . Phys. Chem. 1986,90, 219-222
a
Figure 4. Dependence of the wave period of the single-armed spiral. The common point for all the plots is A = 0.03, B = 0.3,C = 0.001, ho = 2.
points). On one of the short walls, the concentration of the HBr02 was 5 X lo2 times as great as the steady-state concentration of this variable. That wall became a plane wave generator. The boundary conditions corresponded to zero flux at the other walls. As the wave reached the middle of the container all the walls were made impenetrable (zero flux) and the other half of the square was filled with the steady-state concentrations. Thus a broken
219
wave was created. The calculations were then continued in the whole square. The broken wave twisted into a single-armed spiral. A double-armed spiral was created similarly, but in the second half of the container the first one was reproduced in the opposite direction. To give rise to a four-armed vortex, the quarter of the square contiguous to the wave generator wall was reproduced in the other three with consequent rotation by 90'. Wave evolution in the single-, two- and four-armed vortices is shown in Figure 2. This figure represents the maps of the levels of ferroin concentration. Figure 3 shows how these maps are related to the concentration distribution of HBr02 and Fehen)^^'. The dependence of the period of the single-armed spiral wave on various parameters is shown in Figure 4. It should be noted that the period depends rather slightly on the bromomalonic acid and ferroin concentration. The period of the waves sent into the medium by multiarmed spirals does not depend on the number of arms. The calculations were carried out up to 10-12 periods. No evidence for instability of the vortices was found. Alternate junctions and disjunctions of the arms in the core can be seen just as it was observed in the experiment.I0 The rate constants in the model 14 fit the experimental data obtained at 40 OC while most of the experiments with spiral waves in the Belousov-Zhabotinsky system were carried out at 20-25 OC. Taking this remark into account one can see that the calculations presented agree quite well with the experiments.'-I0 Thus, the mechanisms 1-1 1 and 13 and the model 14 describe quite adequately most of the complicated spatio-temporal behavior of the Belousov-Zhabotinsky system.
Acknowledgment. I am grateful to M. F. F. Highrullina for her help in calculations and to Dr. Tov. V. Samoilov for his competent consultations on software problems.
Hot-Band Study of Dabco Using Resonant Multiphoton Optogalvanic Spectroscopy Mark A. Quesada, Zheng-Woo Wang, and David H. Parker* Chemistry Board of Studies, University of California, Santa Cruz, Santa Cruz, California 95064 (Received: October 1.5, 1985)
The application of MPI-optogalvanic spectroscopyto the molecule 1,4-diazabicyclo[2.2.2]octaneat various levels in a methane/air flame environment is described. The method employs a burner design that permits access to preheated and primary reaction zones of the flame for laser probing. Hot bands arising from two-photon resonant (]Al' 'XI') transitions are measured and the intramolecular vibrational potentials for the ground and first excited electronic state are determined.
-
Introduction A wide variety of laser-based optical methods have shown great promise in the detection of reactive species in flame environments.'s2 Nonlinear Raman techniques such as coherent antiStokes Raman spectroscopy (CARS), laser-induced fluorescence (LIF) methods, and resonance enhanced multiphoton ionization (REMPI) spectroscopies have all allowed access to the behavior of electronically excited molecules in thermally excited media. In the present paper, we report the two-photon resonant multiphoton ionization (2 + 2) spectra of the SI So transition3" of
1,4-diazabicycl0[2.2.2]octane (Dabco) at various regions in a seeded atmospheric pressure flame. The electronic spectra of this rigid caged amine exhibits a well-resolved vibrational structure rich in sequence (Au = 0, f2) bands involving the low-frequency torsional vibration of the two N pyramids about the C3symmetry
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(1) D. R. Crosley, Ed.,"Laser Probes for Combustion Diagnostics", American Chemical Society, Washington, DC, 1980, ACS Symp. Ser., No. 134. (2) Opt. Eng., 20, 493-545 (1981). (3) D. H. Parker and Ph. Avouris, J. Chem. Phys., 71, 1241 (1981). (4) N. Gonohe, N. Yatsuda, N. Mikami, and M. Ito, Bull. Chem. SOC. Jpn., 55, 2796-2802 (1982). (5) G. J. Fisanick, T. S . Eichelberger IV, M. B. Robin, and N. A. Kuebler, J . Phys. Chem., 87, 2240-2246 (1983).
0022-3654/86/2090-0219$01 SO/O
DABCO axis (1). The ability to probe the preheat zone of the methane/air (6) D. H. Parker and Ph. Avouris, Chem. Phys. Lett., 53, 515 (1978).
0 1986 American Chemical Society
