J . Phys. Chem. 1985,89, 2022-2026
2022
Velocity of Trigger Waves and Temperature Dependence of Autowave Processes in the Belousov-Zhabotinsky Reaction L. Kuhnert,* H.-J. Krug, and L. Pohlmann Zentralinstitut fiir physikalische Chemie, Akademie der Wissenschaften der DDR, DDR- 1 1 99 Berlin, Rudower Chaussee 5. GDR (Received: February 15, 1983; In Final Form: July 20, 1984)
Reaction-diffusion structures are models for generating biological structures (morphogenesis). The Belousov-Zhabotinsky reaction (BZR; bromate, malonic acid, ferroin) is the favored object for experimental investigations in this field. Trigger waves propagate with a constant velocity according to the formula u = 2(Dqlho[HBr03])1/2.The kinetic constant q1 and its temperature dependencewere determined from experimental investigations of the autocatalytic step. In this way we obtained a good concordance between experimental and theoretical values for velocity of trigger waves and their activation parameters.
Introduction
In reaction-diffusion systems involving autocatalytic steps, so-called symmetry-breaking instabilities can be observed. In 1905 Luther' first described impulse migration of autocatalytic reactions in liquid medium, for example oxidation of oxalic acid by permanganate. Already Luther has treated the appropriate differential equation describing linear autocatalysis coupled with diffusion of the autocatalyzer and derived an expression for velocity of the reaction front u
-
(kD)'/2
(1)
where k is the velocity constant of the rate-determining step of the autocatalytic reaction and D is the diffusion coefficient of the autocatalytic species. similar phenomena also occur in population dynamics and combustion. A review of the mathematical treatment of that class of wave phenomena was given by Fife.2 Recently a great multitude of spatial structures and traveling waves was theoretically proposed by considering further feedback steps in reaction systems and diffusion of more than one comp ~ n e n t . Motivation ~ for these investigations was the importance of these reaction-diffusion structures as models for generation of biological structures (morph~genesis).~The Belousov-Zhabotinsky reaction is the favored object for experimental investigations in this field,5 though in other reaction systems spatial structures and traveling waves also were identified recently.6 For such investigations the preferred reaction mixture contains malonic acid, bromate, and ferroin in diluted sulfuric acid. In this system a sharp color change from red (Fe(phen)?+) to blue (Fe(phen)?+) occurs. Therefore, it is easy to detect the chemical structure visually. Our present paper deals with mechanisms of autowave phenomena. In the BZ system three types (a+) of wave phenomena have been observed: (a) Phase waves or kinematic waves. These kind of waves are generated by phase gradients in bulk oscillations of the reagent. They are not due to transport of impulse or matter (diffusion) in analogy to physical phase waves. Velocity is inversely proportional to the phase gradient that can be established as a gradient in temperature or as a concentration gradient of a chemical component.' (b) Trigger waves. Trigger waves can be observed in excitable but not necessarily oscillatory media. This wave type is the (1) R. Luther, 2. Elekfrochem., 12, 596 (1906). (2) P. C. Fife, Lecf. Notes Biomarh., 28 (1979). (3) V. A. Vasilev, J. M. Romanovsky, and V. G. Yachno, U s p f i z . Nauk, 128, 685 (1979). (4) A. M. Turing, Philos. Trans. R. SOC.London, Ser. B, 237,37(1952). ( 5 ) A. M. Zhabotinsky, "Koncentracionnije Avtokolebanija", Nauka, Moscow, 1974. (6) K. W. Pehl, L. Kuhnert, and H. Linde, Nafure (London), 282, 198 (1979). ( 7 ) D. Thoenes, Nafure (London) Phys. Sci., 243, 18 (1973).
0022-3654/85/2089-2022$01.50/0
isothermal analogue to flame fronts that also propagate with constant velocity. Trigger waves are generated due to a feedback reaction (autocatalysis) in connection with diffusion of the reactants relative to each other. That reaction-diffusion mechanism was first proposed by Luther.' (c) Self-accelerating waves or metawaves. These wave fronts propagate with growing velocity in analogy to explosive phenomena. They are assumed to be caused by interference of autocatalysis with thermokinetic or other kinds of additional feedback. The generation of various types of waves depends on intrinsic conditions of the chemical system (temperature, concentration, etc.). In some cases we observe coexistence between various regimes or a transition from one type to another, e.g. transition of phase to trigger waves or of trigger waves to self-accelerating waves.8 Experimental Section
NaBr03 was three-times recrystallized. The other chemicals, malonic acid, sulfuric acid, 1,IO-phenanthroline, Fe(NH,),(S04)2.6H20,NH4Br, KBr03, and KBr, were of commercial analytical reagent grade and were used without further purification. All solutions were prepared with triply distilled water. The velocity of wave propagation was measured in a thin layer of liquid in a thermostated Petri dish. The solution was photographed at definite times, and the distance of bands from a fixed reference point was measured in each photograph. Kinetics measurements of the autocatalytic oxidation of ferroin by bromate were carried out by following the light absorption at X = 505 nm (qerroin = 11 000 M-' cm-', eferriin= 350 M-' cm-') using a Specord UV-vis spectrometer (Carl Zeiss, Jena). The reaction was simultaneously followed by recording the potentials of a redox-sensitive platinum electrode and a bromide-sensitive electrode (Keramische Werke, Hermsdorf). All potentials were measured against a calomel reference electrode (Forschungsinstitut, Meinsberg) in a saturated potassium chloride solution via a salt bridge containing potassium nitrate with agar-agar gel. The well-stirred reaction solution was contained in a thermostated reaction vessel of 60 mL connected with a quartz cuvette for colorimetric measurements. Velocity of Trigger Waves
Trigger waves propagate with constant velocity according to a reaction-diffusion mechanism. Their velocity is in first approximation determined by the velocity of autocatalytic reaction and the diffusion coefficient of the autocatalyzer according to Luther's formula (1). Field and Noyes9 investigated trigger wave movement in the bromate-malonic acid-ferroin (BZ) system. With the auto(8) L. Kuhnert and L. Pohlmann, in preparation. (9) R. J. Field and R. M. Noyes, J . Am. Chem. Soc., 96, 20 I1 (1974).
0 1985 American Chemical Society
The Journal of Physical Chemistry, Vol. 89, No. 10, 1985 2023
Belousov-Zhabotinsky Reaction catalytic reaction in BZ systems 3Ht
-
+ BrO; + 2 F e ( ~ h e n ) , ~++ HBr02 2HBr02 + 2Fe(phen):+ + H20 kni
and neglecting disproportionation of HBrOz and inhibition by bromide ion, they obtained a reaction-diffusion equation of the form
where X = [HBr02], r is a space variable, kR1is the kinetic constant for the rate-determining step of the autocatalytic reproduction of HBr02, and D, is the diffusion coefficient of HBr02. From (2) they derived an expression for trigger wave velocity utr similar to (1) ut, = (4kR1D,)1/2([H+][Br03-])'/2 = m,([H+] [Br03-])1/2 (3)
(b) The dimer form Br204of bromine(1V) oxide radical must be taken into account. For example, LamberzI6 pointed out that the equilibrium constant for (R2) given by Buxton and Daintonl' (KR2= 1/19 M) was not derived correctly. By correct evaluation of the results in ref 17 one will obtain K R= ~ 5.3
+ 27.87([H+] [Br03-])1/2
(mm m i d ) (4)
This result was underlined by Showalter,Io who found a similar dependence of utr in the bromate-4-cyclohexene-1 ,Zdicarboxylic acid-ferroin system: ut, = -0.999
+ 27.33([H+][Br03-])'/2
(mm min-')
(5)
Recently, SevEikova and Marek" obtained utr also to be a linear function of ([H+] [Br0
4 200
a"
IU I 50
10
100
I
150
200
t [secl Figure 1. Plots of ferroine concentration, potential of bromide-sensitive
electrode, and redox-sensitive platinum electrode for the ferroin-bromate reaction. Initial concentrations in the sequence of adding are [HZSO4I0 = 0.613 M (h, = 0.83 M), [KBr], = 1.5 X lV5M, [Fe(II)], = 1 X lo4 M, and [NaBrO,], = 1 X lo-* M (T = 25 f 0.5 "C). If disproportionation R5 can be neglected, [HBrO,] grows exponentially until Fe(I1) is consumed completely [HBrO,] = [HBrOrloeUf
(14)
[Fe(III)] = 2[HBr02]0euf+ C'
(15)
C' = -2[HBrOzlo if [Fe(III)lo = 0
(16)
as does [Fe(III)] where and a = qlho[HBr03] (17) ho is the acidity function of HzS0418used instead of [H+]. Bromate is provided to be given in excess; thus, [HBr03] can be regarded as constant. At the end of reaction there is
[Fe(WlIi, = [Fe(II)lo
(18)
and [HBr021,*,
!MFe(II)lo
(19)
Disproportionation R5 can only be neglected when qlho[HBr03] >> 2kR5[HBro2] or with (19) and (20)
(20)
[Fe(II)IO
