# American Institute of Mathematical Sciences

August 2018, 23(6): 2527-2544. doi: 10.3934/dcdsb.2018061

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

 Laboratory of Mathematics, Faculty of Regional Environment Science, Tokyo University of Agriculture, 1-1-1 Sakuragaoka, Setagaya-ku, Tokyo 156-8502, Japan

Received  May 2017 Revised  September 2017 Published  February 2018

Fund Project: This work was supported by JSPS KAKENHI (Grant Number 22740114,26400183)

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-Körös-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].

Citation: Chikahiro Egami. Mechanism for the color transition of the Belousov-Zhabotinsky reaction catalyzed by cerium ions and ferroin. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2527-2544. doi: 10.3934/dcdsb.2018061
##### References:
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##### References:
 [1] C. Egami, Bifurcation analysis of the Nowak-Bangham model in CTL dynamics, Math. Biosci., 221 (2009), 33-42. doi: 10.1016/j.mbs.2009.06.005. [2] R. J. Field and H.-D. Försterling, On the oxybromine chemistry rate constants with cerium ions in the Field-Körös-Noyes mechanism of the Belousov-Zhabotinskii reaction: The equilibrium $\rm HBrO_2 + BrO_3^{-} + H^{+} \rightleftharpoons 2BrO_2{·} + H_2O$, J. Phys. Chem., 90 (1986), 5400-5407. [3] R. J. Field, E. Körös and R. M. Noyes, Oscillations in chemical systems. Ⅱ. Thorough analysis of temporal oscillation in the Bromate-Cerium-Malonic acid system, J. Am. Chem. Soc., 94 (1972), 8649-8664. doi: 10.1021/ja00780a001. [4] R. J. Field and R. M. Noyes, Oscillation in chemical systems. Ⅳ. Limit cycle behavior in a model of a real chemical reaction, J. Chem. Phys., 60 (1974), 1877-1884. doi: 10.1063/1.1681288. [5] H.-D. Försterling, L. Stuk, A. Barr and W. D. McCormick, Stoichiometry of bromide production from ceric oxidation of bromomalonic acid in the Belousov-Zhabotinskii reaction, J. Phys. Chem., 97 (1993), 7578-7584. [6] L. Györgyi and R. J. Field, Aperiodicity resulting from two-cycle coupling in the Belousov-Zhabotinskii reaction. ⅲ. Analysis of a model of the effect of spatial inhomogeneities at the input ports of a continuous-flow, stirred tank reactor, J. Chem. Phys., 91 (1989), 6131-6141. doi: 10.1063/1.457432. [7] L. Györgyi and R. J. Field, Simple models of deterministic chaos in the Belousov-Zhabotinskii reaction, J. Phys. Chem., 95 (1991), 6594-6602. [8] L. Györgyi, T. Turányi and R. J. Field, Mechanistic details of the oscillatory Belousov-Zhabotinskii reaction, J. Phys. Chem., 94 (1990), 7162-7170. [9] J. P. Keener and J. J. Tyson, Target patterns in a realistic model of the Belousov-Zhabotinskii reaction, Phys. D, 21 (1986), 307-324. doi: 10.1016/0167-2789(86)90007-2. [10] G. Kshirsagar and R. J. Field, A kinetic and thermodynamic study of component processes in the equilibrium $\rm 5HOBr \rightleftharpoons 2BR_2+BrO_3^{-}+2H_2O+H^{+}$, J. Phys. Chem., 92 (1988), 7074-7079. [11] W. M. Liu, Criterion of Hopf bifurcations without using eigenvalues, J. Math. Anal. Appl., 182 (1994), 250-256. doi: 10.1006/jmaa.1994.1079. [12] A. B. Rovinsky and A. M. Zhabotinsky, Mechanism and mathematical model of the oscillating Bromate-Ferroin-Bromomalonic acid reaction, J. Phys. Chem.(25), 88 (), 6081-6084,1984. doi: 10.1021/j150669a001. [13] C. E. Sjogren, S. Kolboe and P. Ruoff, Transitions between two oscillatory states in a closed malonic acid Belousov-Zhabotinsky reaction simultaneously catalyzed by ferroin and cerium ions, Chem. Phys. Lett., 130 (1986), 72-75. [14] A. F. Taylor, V. Gáspár, B. R. Johnson and S. K. Scott, Analysis of reaction-diffusion waves in the ferroin-catalysed Belousov-Zhabotinsky reaction, Phys. Chem. Chem. Phys., 1 (1999), 4595-4599. doi: 10.1039/a904994k. [15] J. J. Tyson, Oscillations, bistability, and echo waves in models of the Belusov-Zhabotinskii reaction, Ann. N.Y. Acad. Sci., 316 (1979), 279-295. [16] J. J. Tyson, Relaxation oscillations in the revised Oregonator, J. Chem. Phys., 80 (1984), 6079-6082. doi: 10.1063/1.446690. [17] J. J. Tyson and P. C. Fife, Target patterns in a realistic model of the Belousov-Zhabotinskii reaction, J. Chem. Phys., 73 (1980), 2224-2237. doi: 10.1063/1.440418. [18] A. M. Zhabotinsky, F. Buchholtz, A. B. Kiyatkin and I. R. Epstein, Oscillations and waves in metal-ion-catalyzed Bromate oscillating reactions in highly oxidized states, J. Phys. Chem., 97 (1993), 7578-7584. doi: 10.1021/j100131a030.
Periodic color transition of the BZ reaction solution (Exp. Ⅲ) catalyzed by cerium ions and ferroin. The value of $E_{\rm ORP}$ shows the redox potential of solution at the moment the image was captured
Redox potential curves recorded by the three catalytic types of BZ reactions
Redox potential curves represented by the solutions of (T), (RZ), and (CF)
The rule of color transition for the BZ reaction catalyzed by cerium ions and ferroin: (a) the color sequence of the reaction solution, (b) the redox potential curve generated by (CF), (c) the time sequences of the ratios of $\rm Ce^{4+}$ and $\rm Fe(phen)_3^{3+}$ to the total cerium ion and ferroin concentrations, (d) the color phases exhibited by the cerium ions and ferroin. The solution color changes in six phases of four colors because of the simultaneous continuous fluctuation of the ratios between the oxidant and reductant of each ion
Bifurcation diagram obtained by varying $\varepsilon$: {(a)} The state-space plot of the $\omega$-limit sets by setting $\sigma$ ($= \varepsilon^{-1}$) between 1.0 and 20.0 every 1.0; the orbits of (CF) approach the equilibrium $\bar{E}$ for every $\sigma<\sigma^*$, whereas a limit cycle appears around $\bar{E}$ for each $\sigma>\sigma^*$. {(b)} The time sequence $y(\tau)$, $z(\tau)$ of the solution for the initial condition $(x_0, y_0, z_0) = (0.5, 0.1, 0.3)$ at $\sigma = 2.7, 2.8, 5.0, 10.0$. An increase in $\sigma$ enlarges the frequency and the amplitude of oscillation
$\omega$-limit sets obtained by varying $\xi_1$: (a) The phase-space plots for $\kappa$ ($= \xi_1^{-1}$) between 0.1 and 10.1 every 0.5; the equilibrium $\bar{E}$ is always unstable. (b) The time-sequence $y(\tau)$, $z(\tau)$ of the solution for the initial condition $(x_0, y_0, z_0) = (0.5, 0.1, 0.3)$ at $\kappa = 0.1, 0.3, 1.0, 10.0$; the period of the limit cycle is almost proportional to $\kappa$ as long as $\kappa$ is small
$\omega$-limit sets obtained by varying $\xi_2$: (a) The phase-space plots for $\kappa$ ($= \xi_2^{-1}$) between 0.1 and 50.1 every 2.5; the equilibrium $\bar{E}$ is always unstable. (b) The time-sequence $y(\tau)$, $z(\tau)$ of the solution for the initial condition $(x_0, y_0, z_0) = (0.5, 0.1, 0.3)$ at $\kappa = 0.1, 3.0, 10.0, 50.0$; the period of limit cycle is almost proportional to $\kappa$ as long as $\kappa$ is small
Solution preparation. (M = mol/L)
 No. Contents and concentrations (S1) $\rm KBrO_3$ 0.23M (S2) $\rm CH_2(COOH)_2$ 0.31M, $\rm KBr$ 0.059M (S3) $\rm H_2SO_4$ 3M, $\rm Ce(NH_4)_2(NO_3)_6$ 0.019M (S4) $\rm H_2SO_4$ 4M (S5) $\rm FeSO_4\!\cdot\!7H_2O$ 7.6mM, $\rm C_{12}H_8N_2$ 22.8mM
 No. Contents and concentrations (S1) $\rm KBrO_3$ 0.23M (S2) $\rm CH_2(COOH)_2$ 0.31M, $\rm KBr$ 0.059M (S3) $\rm H_2SO_4$ 3M, $\rm Ce(NH_4)_2(NO_3)_6$ 0.019M (S4) $\rm H_2SO_4$ 4M (S5) $\rm FeSO_4\!\cdot\!7H_2O$ 7.6mM, $\rm C_{12}H_8N_2$ 22.8mM
Mixture quantities for the three kinds of BZ reactions
 Exp. Solution number and volume [mL] Type of catalyst Ⅰ (S1) 20, (S2) 20, (S3) 20 cerium-catalyzed Ⅱ (S1) 20, (S2) 20, (S4) 15, (S5) 5.0 ferroin-catalyzed Ⅲ (S1) 20, (S2) 20, (S3) 20, (S5) 1.0 cerium-ferroin-catalyzed
 Exp. Solution number and volume [mL] Type of catalyst Ⅰ (S1) 20, (S2) 20, (S3) 20 cerium-catalyzed Ⅱ (S1) 20, (S2) 20, (S4) 15, (S5) 5.0 ferroin-catalyzed Ⅲ (S1) 20, (S2) 20, (S3) 20, (S5) 1.0 cerium-ferroin-catalyzed
Fixed concentrations and rate constants for (T), (RZ), and (CF)
 Concentration Rate constant $A$ 0.04 M $k_1$ $100$ $\rm {M^{-2}s^{-1}}$ $k_6$ $10^7$ $\rm {M^{-2}s^{-1}}$ $B$ 0.10 M $k_3$ $10^7$ $\rm {M^{-1}s^{-1}}$ $k_8$ $15$ $\rm {M^{-2}s^{-1}}$ $C$ (Exp. Ⅰ, Ⅲ) 6.3 mM $k_{-3}$ $10^3$ $\rm {M^{-1}s^{-1}}$ $k_9$ $2$ $\rm {M^{-1}s^{-1}}$ $F$ (Exp. Ⅲ) 0.13 mM $k_4$ $8.0\times 10^4$ $\rm {M^{-2}s^{-1}}$ $k_{-9}$ $2.0\times 10^8$ $\rm {M^{-2}s^{-1}}$ $F$ (Exp. Ⅱ) 0.65 mM $k_{-4}$ $8.9\times 10^3$ $\rm {M^{-1}s^{-1}}$ $k_{10}$ $2.0 \times 10^3$ $\rm s^{-1}$ $H$ 1.29 M $k_5$ $1.7\times 10^4$ $\rm {M^{-1}s^{-1}}$ $k_{11}$ $0.8$ $\rm {M^{-1}s^{-1}}$
 Concentration Rate constant $A$ 0.04 M $k_1$ $100$ $\rm {M^{-2}s^{-1}}$ $k_6$ $10^7$ $\rm {M^{-2}s^{-1}}$ $B$ 0.10 M $k_3$ $10^7$ $\rm {M^{-1}s^{-1}}$ $k_8$ $15$ $\rm {M^{-2}s^{-1}}$ $C$ (Exp. Ⅰ, Ⅲ) 6.3 mM $k_{-3}$ $10^3$ $\rm {M^{-1}s^{-1}}$ $k_9$ $2$ $\rm {M^{-1}s^{-1}}$ $F$ (Exp. Ⅲ) 0.13 mM $k_4$ $8.0\times 10^4$ $\rm {M^{-2}s^{-1}}$ $k_{-9}$ $2.0\times 10^8$ $\rm {M^{-2}s^{-1}}$ $F$ (Exp. Ⅱ) 0.65 mM $k_{-4}$ $8.9\times 10^3$ $\rm {M^{-1}s^{-1}}$ $k_{10}$ $2.0 \times 10^3$ $\rm s^{-1}$ $H$ 1.29 M $k_5$ $1.7\times 10^4$ $\rm {M^{-1}s^{-1}}$ $k_{11}$ $0.8$ $\rm {M^{-1}s^{-1}}$
Relation between signs of $q'_2$, $q_{10}$, $q_{20}$, and bifurcation type
 $q'_2$ $q_{10}$ $q_{20}$ type + + + T1 + + 0 T5 + + - T3 + 0 + T6 + 0 0 T3 + 0 - T3 + - + T3 + - 0 T3 + - - T3 0 + + T7 0 + 0 T0 0 + - T0 0 0 + T9 0 0 0 T0 0 0 - T0 0 - + T8 0 - 0 T0 0 - - T0 - + + T4 - + 0 T0 - + - T0 - 0 + T4 - 0 0 T0 - 0 - T0 - - + T2 - - 0 T0 - - - T0
 $q'_2$ $q_{10}$ $q_{20}$ type + + + T1 + + 0 T5 + + - T3 + 0 + T6 + 0 0 T3 + 0 - T3 + - + T3 + - 0 T3 + - - T3 0 + + T7 0 + 0 T0 0 + - T0 0 0 + T9 0 0 0 T0 0 0 - T0 0 - + T8 0 - 0 T0 0 - - T0 - + + T4 - + 0 T0 - + - T0 - 0 + T4 - 0 0 T0 - 0 - T0 - - + T2 - - 0 T0 - - - T0
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