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

Chikahiro Egami

doi:10.3934/dcdsb.2018061

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2018, Volume 23, Issue 6: 2527-2544. Doi: 10.3934/dcdsb.2018061

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

Chikahiro Egami

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

Received: April 30, 2017

Revised: August 31, 2017

Early access: February 2018

Published: June 2018

This work was supported by JSPS KAKENHI (Grant Number 22740114,26400183).

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Abstract

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].

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Mathematics Subject Classification: Primary: 34C26; Secondary: 92E20.

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Figure 2. 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

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Figure 1. Redox potential curves recorded by the three catalytic types of BZ reactions

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Figure 3. Redox potential curves represented by the solutions of (T), (RZ), and (CF)

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Figure 4. 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

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Figure 5. 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

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Figure 6. $\omega$-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

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Figure 7. $\omega$-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

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Table 1. 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

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Table 2. 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

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Table 3. 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	$k_5$	$1.7\times 10^4$ $\rm {M^{-1}s^{-1}}$	$k_{11}$	$0.8$ $\rm {M^{-1}s^{-1}}$

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Table 4. 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

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