Mechanism for the color transition of the Belousov-Zhabotinsky reaction catalyzed by cerium ions and ferroin

The oscillation property of the Belousov-Zhabotinsky reaction and the color transition of its solution depend on the catalytic action of the metal ions. The solution of the reaction system catalyzed by both cerium ions and ferroin has a more complicated effect on the color than either the cerium-catalyzed case or the ferroin-catalyzed case. To theoretically elucidate the color transition of the case catalyzed by these two ions, a reduced model consisting of three differential equations is proposed, incorporating both the Rovinsky-Zhabotinsky scheme and the Field-Koros-Noyes scheme simplified by Tyson [Ann. N.Y. Acad. Sci., 316 (1979), pp.279-295]. The presented model can have a limit cycle under reasonable conditions through a Hopf bifurcation, and its existence theorem is proven by employing the bifurcation criterion established by Liu [J. Math. Anal. Appl., 182 (1994), pp.250-256].


1.
Introduction. Since its discovery in 1958, the Belousov-Zhabotinsky (BZ) reaction, a typical chemical oscillation, has been variously studied in the areas of physics, chemistry, and mathematics. Quantitative data of the rhythmic phenomenon of the BZ reaction can be gained by continuously monitoring the redox potential (oxidation-reduction potential: ORP) of the adequately stirred solution. On the other hand, by leaving the thinly stretched solution of the BZ reaction in a petri dish, it is possible to observe the time evolution of target and spiral patterns. This study deals with the rhythmic phenomenon.
Field, Körös, and Noyes (FKN) elucidated the essential mechanism of the BZ reaction in their celebrated work [3]. Since then, many articles have been published, proposing the detailed reaction mechanism from experimental and thermodynamic points of view. In particular, we refer readers to studies by Field and Försterling [2], Försterling et al. [5], Györgyi and Field [6,7], Györgyi et al. [8], and Kshirsagar and Field [10] as the key reports directly concerning the topic of the present paper. The Oregonator established by Field and Noyes [4] is a mathematical model of three ordinary differential equations that describes the FKN mechanism on the basis of the chemical kinetics. The Oregonator is a well-known limit cycle oscillator.
for the scheme of the BZ reaction. In some studies [2,12,14,18], it was pointed out that the reaction pathway of a ferroin-catalyzed system differs from that of a cerium-catalyzed system because of differences in standard redox potentials of the Ce 4+ /Ce 3+ and Fe(phen) 3+ 3 /Fe(phen) 2+ 3 couples. The RZ scheme [12] is therefore a modified FKN scheme for a ferroin-catalyzed system.
The scheme of a cerium-ferroin-catalyzed system consists of the following eleven steps selected from the combination of the RZ scheme and the FKN scheme: Briefly, to characterize the difference between the ferroin-catalyzed system and the cerium-catalyzed system, step (R9) is essentially reversible for the ferriin/ferroin couple, whereas the single-step (R11) is assumed for the ceric/cerous ion couple. We use the following notations for the reactant concentrations: We also use k n and k −n to denote the rate constants of the forward and reverse reactions of (Rn), respectively. Letting X, Y , Z, P , Q, and R represent intermediate products, and employing the following assumptions adopted by Field and Noyes [4], Györgyi et al. [8], Rovinsky and Zhabotinsky [12], and Zhabotinsky et al. [18]: • The equilibrium in (R2) is shifted almost completely to the right, that is, [HBrO + 2 ] substitutes for [BrO 2 ·]. • HOBr and Br 2 are rapidly removed through (R5), (R6), (R7), and (R8).
• For (R9) and (R10), the rate of Br − production caused by the ferriin reduction is accounted for by the stoichiometric factor q. In a similar manner, the rate of Br − production caused by the reduction of ceric ions through (R11) is accounted for by the stoichiometric factor r. See also [14] and [16].

CHIKAHIRO EGAMI
Under these assumptions, we develop the system of rate equations as follows: Taking into account the following chemical facts pointed out in [12]: • k 5 k −5 and k −9 k 10 k 9 , • The concentration of HBrO + 2 remains sufficiently small, namely, k −1 Q 2 0, and applying the quasi-steady-state approximation to P , Q, and R, we obtain a three-variable model as follows: All the parameters are positive. In particular, the kinetic reasons allow for µ < 1.
The domain for (CF) is We remark that system (CF) is a hybrid model of the Tyson model and the RZ model. Provided F → 0 for the cerium-catalyzed case, then ξ 2 → 0, η → ∞, and β 2 → 0, so that (CF) corresponds to the Tyson model of the form On the other hand, if C → 0, then ξ 1 → 0, η → 0, α 1 → 0, and β 1 → 0, in which case it follows that (CF) corresponds to the RZ model of the form 3. Experimental section. In terms of chemical phenomena, this section illustrates the difference in oscillation properties between the cerium-, ferroin-, and cerium-ferroin-catalyzed BZ reactions. We also show the color transitions of the cerium-ferroin-catalyzed system with photographic images. The solutions were prepared for this study as shown in Table 1, and then Exp. I, II, and III were carried out by means of sufficient stirring of the mixtures shown in Table 2 in a 100 mL beaker on a magnetic stirrer hot-plate. The reaction temperature was kept constant at 30 • C. The redox potential of the mixture was measured by using a pH/ORP meter (HORIBA, D-52) with a combination electrode (HORIBA, 9300-10D) composed of a Pt wire and Ag/AgCl electrode immersed in a saturated KCl aqueous solution.  The measured potential, E m [mV], is converted to an ORP potential, E ORP , relative to a standard hydrogen electrode (SHE) with the aid of the relation where the temperature T = 30 [ • C]. Figure 1 shows the time sequences of the redox potentials measured five minutes after mixing the solutions to the reading of the steady waveforms. The amplitudes of Exp. I, II, and III were respectively 130, 50, and 245 mV. The period of Exp.
II was approximately 15 s. The periods of both Exp. I and III seemed to be approximately 55 s in the figure, whereas the period of Exp. III was found to be a few percent smaller than that of Exp. I according to a longer measurement time. The important point is that the reaction of Exp. III corresponds to that of Exp. I with a low-dose of ferroin; nevertheless, the oscillation properties clearly differ between Exp. I and III. Figure 2 shows the cycle of the color transition of Exp. III. Judging visually, the color gradually changes in six phases of four colors: red -purple -blue -greenblue -purple. The solution takes on a deep red at the lowest point on the redox potential curve. The sequence of purple -blue -green is instantaneous, whereas that of blue -purple -red is relatively gentle.  4. Numerical simulation. This section is devoted to the analysis of the reason for the color transition of the cerium-ferroin-catalyzed BZ reaction. We also discuss the adequacy of the (CF) model with the aid of numerical simulation. For the simulation, we set the parameters as shown in Table 3, the values of which are based on the experimental conditions and ref. [12] except as noted here. For 1M H 2 SO 4 , H = 1.29 was adopted in [8]. The values of k 4 and k −4 were ensured at 20 • C in [2]. The value of k 11 was estimated to be 0.4 in [15], 0.4 at 20 • C and 0.8 at 40 • C in [18], and in the range of 0.5 to 1.0 in [6]. Table 3. Fixed concentrations and rate constants for (T), (RZ), and (CF).
The redox potential, E ORP , for the solution including a single redox couple, M ox /M red , with the electron transfer reaction M ox + ne − −→ ←− M red is given by Nernst's equation of the form where R is the universal gas constant, T the temperature in Kelvin, and F the Faraday constant. For the case of n = 1 at the experimental temperature 30 • C given in Section 3, RT /(nF ) = 0.0261. In addition, E • is the standard redox potential, which depends on the redox couple, the temperature, and the measuring electrode.
this fact makes a difference in the reaction pathway between the cerium-catalyzed and the ferroin-catalyzed systems, as mentioned in Section 2. However, a theoretical determination of the standard redox potential for a general mixed solution is not only difficult but also involves indistinguishable chemical elements because of the coexistence of a number of ion equilibria. At the same time, because the color of the solution changes with changing concentration ratios of catalytic ions, the fluctuation of E ORP − E • is more significant than the level of E ORP . Systems (T), (RZ), and (CF) represent the cerium-, ferroin-, and cerium-ferroincatalyzed BZ reactions, respectively. As described in the following sections, each trajectory approaches a closed orbit in the phase space (x, y, z) under an appropriate condition. Figure 3 shows the time sequences of E ORP − E • corresponding to each closed orbit, that is, for (RZ), and where E • T , E • RZ , and E • CF are the standard redox potentials for each system. Although the detailed values are unavailable, E • RZ < E • CF < E • T is established from a general argument on electrochemistry. As can be seen in Figure 3, the amplitudes for (T), (RZ), and (CF) are respectively 180, 125, and 275 mV, and the periods 74, 37, and 69 s, respectively. The magnitude relation of the oscillation properties agrees with Figure 1. In particular, the waveform of (CF) has the same characteristics as Exp. III. However, this result does not exactly fit the experimental data with high quantitative accuracy. There are some possible reasons for the discrepancy. One is the difference in rate constant caused by the difference in temperature. Most of the rate constants of Table 3 were estimated at 25 • C, whereas the current experiments were carried out at 30 • C. The second possible reason is that the values of A and B depend on the type of catalyst and the concentration of the H 2 SO 4 medium. Obviously, system (1) is more suitable for the simulation than (CF) without the quasi-steady-state approximation if our aim is to reproduce an actual redox curve by means of modeling. There is no need look for quantitative accuracy as far as understanding the mechanism of the color transition.   Graph (c) shows the time sequences of Y (t)/C and Z(t)/F corresponding to one period of the limit cycle for (CF). It should be noted that Y (t)/C increases from the minimum to near 1.0 at a slightly earlier moment and at a more rapid rate than Z(t)/F . We assume that at approximately half of these values, the color phases of the cerium ions and the ferroin switch, as shown in gradation (d). At t ∈ (0, 26), the cerium ions become colorless in the near-entire trivalent state, whereas the ferroin turns red. As a result, the solution appears red. In the next instant, it turns blue after momentarily turning purple during the oxidation process of ferroin to ferriin. In the subsequent oxidation process of cerous to ceric ions, the solution turns green at t = 30, with the yellow supplied by Ce 4+ . The reductions of ceric ions and ferriin begin at t = 33. The solution turns blue, purple, and red, in that order, with decreasing concentrations of Ce 4+ and Fe(phen) 3+ 3 . In this way, color sequence (a) is completed. This cycle is repeated with every period of the redox potential curve.
Our most important result is the following theorem for the case in which we take ε as the bifurcation parameter. .
Then system (CF) has a unique simple Hopf bifurcation point.
Theorem 5.4. Suppose that condition (10) holds. The number of simple Hopf bifurcation points for (CF) and the local stability ofĒ are sorted by the signs of q 2 , q 10 , and q 20 into ten types, as shown in Table 4.
As seen from the above and noting that −13+5 √ 7 3 0.0762522, condition (10) holds. The first example shown in Figure 5 was performed by varying σ (= ε −1 ) instead of varying ε. Now, p 2 = −29.5534, which corresponds to case (ii) in the proof of Theorem 5.3. In fact, σ 1 = 43.5155, σ 2 = 2.77303, and the Hopf bifurcation point σ * = 2.76166. Because an easy simulation near the bifurcation point indicates the existence of a limit cycle with a small amplitude for some σ < σ * , e.g. σ = 2.740, the Hopf bifurcation caused by this parameter set is expected to be the subcritical type, however, its theoretical proof is beyond the aim of the current paper. Figure 6 illustrates an example of varying κ (= ξ −1 1 ). In this case, we have q 1 = 0.358995, q 10 = −83.1873, q 2 = −0.0948147, q 20 = −3812.08, and q 3 = 94.3284, and therefore system (CF) shows the dynamics of the T0-type stated in Theorem 5.4. Figure 7 gives an example of varying κ (= ξ −1 2 ). In this situation, q 1 = 0.122226, q 10 = −118.516, q 2 = −12.1229, q 20 = −82.5546, and q 3 = 1.94646, and as a result, the dynamics come to the same type as that of the last example. In both figures, each ω-limit set forms a closed orbit in Ω 0 , but a bifurcation phenomenon ofĒ cannot be observed. The parameter set of Table 3 does not generate a Hopf bifurcation with respect to ξ 1 and ξ 2 . Considering from a different perspective, this means that a bifurcation point does not exist for any ξ 1 or any ξ 2 . These results also coincide with the fact that the redox potential oscillates even in the absence of either ferroin or cerium ions. It is apparent that ε, ξ 1 , and ξ 2 each have a role in the time constant for (CF), so that the bifurcation parameter of the present analysis has a direct effect on the frequency or period of oscillation through numerical simulation. Generally speaking, smaller time constants, namely, larger values of σ and κ, provide more rapid responses to the system. 7. Concluding remarks. The color transition of the BZ reaction system catalyzed by cerium ions and ferroin is a well-recognized phenomenon from the experimental aspect. This paper presents a three-variable model (CF) by interlacing the RZ model and the Tyson model to theoretically describe the six-phase color transition of four colors. In particular, we not only established a Hopf bifurcation theorem with a simple condition for the value of µ by using Liu's method, but also gave the bifurcation point σ * with an exact formula.
Real oscillations of the BZ reaction systems catalyzed by multiple ion types sometimes exhibit chaotic behavior. However, the model (CF) has no chaotic orbits because the system itself is composed of a homotopy between the RZ model and the Tyson model, both of which never show chaotic behavior. This indicates the limitation of (CF) as a model. It should also be emphasized that widely varying   Figure 7. ω-limit sets obtained by varying ξ 2 : (a) The phasespace plots for κ (= ξ −1 2 ) between 0.1 and 50.1 every 2.5; the equi-libriumĒ is always unstable. (b) The time-sequence y(τ ), z(τ ) of the solution for the initial condition (x 0 , y 0 , z 0 ) = (0.5, 0.1, 0.3) at κ = 0.1, 3.0, 10.0, 50.0; the period of limit cycle is almost proportional to κ as long as κ is small. the specified bifurcation parameter is impractical for real reaction systems because of the loss of hypothesis at fixed concentrations and rate constants.
One of the remaining problems conceived immediately is to analyze the cases of general r and q. When these are answered, (CF) will extend the describable oscillation property. As a result, a mathematical model in better agreement with the experimental data more precisely will be realized.
Appendix. This section introduces a Hopf bifurcation theorem established by Liu [11] without the proof. In [1], the author developed the techniques to apply Liu's theorem to a higher-order system with a rigorous computation through the bifurcation analysis for CTL dynamics.
Consider a system of differential equations with a single parameteṙ where f µ is smooth for every µ. Assume that (19) has an equilibrium (x 0 , µ 0 ), that is, f µ0 (x 0 ) = 0.
where a i (µ) ≡ 0 for i > n. Let us denote by ∆ i (µ) the ith leading principal minor of H p (µ), which is called the Hurwitz determinant of order i.