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:
[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.  Google Scholar

[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.   Google Scholar

[3]

R. J. FieldE. 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.  Google Scholar

[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.  Google Scholar

[5]

H.-D. FörsterlingL. StukA. 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.   Google Scholar

[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.  Google Scholar

[7]

L. Györgyi and R. J. Field, Simple models of deterministic chaos in the Belousov-Zhabotinskii reaction, J. Phys. Chem., 95 (1991), 6594-6602.   Google Scholar

[8]

L. GyörgyiT. Turányi and R. J. Field, Mechanistic details of the oscillatory Belousov-Zhabotinskii reaction, J. Phys. Chem., 94 (1990), 7162-7170.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

C. E. SjogrenS. 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.   Google Scholar

[14]

A. F. TaylorV. GáspárB. 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.  Google Scholar

[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.   Google Scholar

[16]

J. J. Tyson, Relaxation oscillations in the revised Oregonator, J. Chem. Phys., 80 (1984), 6079-6082.  doi: 10.1063/1.446690.  Google Scholar

[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.  Google Scholar

[18]

A. M. ZhabotinskyF. BuchholtzA. 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.  Google Scholar

show all references

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.  Google Scholar

[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.   Google Scholar

[3]

R. J. FieldE. 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.  Google Scholar

[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.  Google Scholar

[5]

H.-D. FörsterlingL. StukA. 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.   Google Scholar

[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.  Google Scholar

[7]

L. Györgyi and R. J. Field, Simple models of deterministic chaos in the Belousov-Zhabotinskii reaction, J. Phys. Chem., 95 (1991), 6594-6602.   Google Scholar

[8]

L. GyörgyiT. Turányi and R. J. Field, Mechanistic details of the oscillatory Belousov-Zhabotinskii reaction, J. Phys. Chem., 94 (1990), 7162-7170.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

C. E. SjogrenS. 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.   Google Scholar

[14]

A. F. TaylorV. GáspárB. 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.  Google Scholar

[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.   Google Scholar

[16]

J. J. Tyson, Relaxation oscillations in the revised Oregonator, J. Chem. Phys., 80 (1984), 6079-6082.  doi: 10.1063/1.446690.  Google Scholar

[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.  Google Scholar

[18]

A. M. ZhabotinskyF. BuchholtzA. 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.  Google Scholar

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
Figure 1.  Redox potential curves recorded by the three catalytic types of BZ reactions
Figure 3.  Redox potential curves represented by the solutions of (T), (RZ), and (CF)
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
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
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
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
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
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
Table 2.  Mixture quantities for the three kinds of BZ reactions
Exp.Solution number and volume [mL]Type of catalyst
(S1) 20, (S2) 20, (S3) 20cerium-catalyzed
(S1) 20, (S2) 20, (S4) 15, (S5) 5.0ferroin-catalyzed
(S1) 20, (S2) 20, (S3) 20, (S5) 1.0cerium-ferroin-catalyzed
Exp.Solution number and volume [mL]Type of catalyst
(S1) 20, (S2) 20, (S3) 20cerium-catalyzed
(S1) 20, (S2) 20, (S4) 15, (S5) 5.0ferroin-catalyzed
(S1) 20, (S2) 20, (S3) 20, (S5) 1.0cerium-ferroin-catalyzed
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 $k_5$ $1.7\times 10^4$ $\rm {M^{-1}s^{-1}}$ $k_{11}$ $0.8$ $\rm {M^{-1}s^{-1}}$
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 $k_5$ $1.7\times 10^4$ $\rm {M^{-1}s^{-1}}$ $k_{11}$ $0.8$ $\rm {M^{-1}s^{-1}}$
Table 4.  Relation between signs of $q'_2$, $q_{10}$, $q_{20}$, and bifurcation type
$q'_2$ $q_{10}$ $q_{20}$type
+++T1
++0T5
++-T3
+0+T6
+00T3
+0-T3
+-+T3
+-0T3
+--T3
0++T7
0+0T0
0+-T0
00+T9
000T0
00-T0
0-+T8
0-0T0
0--T0
-++T4
-+0T0
-+-T0
-0+T4
-00T0
-0-T0
--+T2
--0T0
---T0
$q'_2$ $q_{10}$ $q_{20}$type
+++T1
++0T5
++-T3
+0+T6
+00T3
+0-T3
+-+T3
+-0T3
+--T3
0++T7
0+0T0
0+-T0
00+T9
000T0
00-T0
0-+T8
0-0T0
0--T0
-++T4
-+0T0
-+-T0
-0+T4
-00T0
-0-T0
--+T2
--0T0
---T0
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