^{1,a)}and Irving R. Epstein

^{1,b)}

### Abstract

We develop a four-variable model, based on the classic Field–Kőrös–Noyes mechanism for the oscillatory Belousov-Zhabotinsky (BZ) reaction, that describes recently discovered jumping waves and bubble waves in the BZ reaction in aerosol OT microemulsion and provides insight into their origins. Contrary to suggestions based on previous models, it appears that these phenomena can arise from interaction between a Turing instability and either excitability or a Hopf instability of the steady state, rather than requiring a wave instability. The model should be useful for studying other patterns in BZ microemulsions as well as the behavior of systems of BZ microdroplets coupled through bromine diffusion.

This work was supported by the National Science Foundation through Grant No. CHE-0615507.

I. INTRODUCTION

II. MODELING THE BZ REACTION

A. Steady state

B. Linear stability analysis of model (A)

III. MODELING THE BZ-AOT SYSTEM

A. Linear stability analysis of model (B)

B. Computer simulation of model (B)

IV. DISCUSSION

### Key Topics

- Diffusion
- 14.0
- Fluid drops
- 10.0
- Numerical modeling
- 9.0
- Surfactants
- 7.0
- Microemulsions
- 6.0

## Figures

Comparison of the Oregonator model (curve 1) and model (29′)–(32′) (curves 2–4). Parameters: , , , and , , (4) 0.879, , (3) 0.0857, , (3) 0.0404.

Comparison of the Oregonator model (curve 1) and model (29′)–(32′) (curves 2–4). Parameters: , , , and , , (4) 0.879, , (3) 0.0857, , (3) 0.0404.

Phase diagram in the plane for models (29)–(32). All curves correspond to the onset of Hopf instability. Parameters: , (2, 6) 0.2, (4 and 5) 0.4, , (2, 3) 0.2, (4) 0.06, (5, 6) 0.1. “Osc” denotes the oscillatory region. Curves 1–3 are well fitted by lines , where , (2) 3, (3) 1.92, while curves 4 and 5 are well fitted by lines , where and , (5) 0.064, and (6) 0.0207. For curves 1–3, the SS (reduced) is above and to the left of the curves, while for curves 4–6, the SS (oxidized) is below and to the right.

Phase diagram in the plane for models (29)–(32). All curves correspond to the onset of Hopf instability. Parameters: , (2, 6) 0.2, (4 and 5) 0.4, , (2, 3) 0.2, (4) 0.06, (5, 6) 0.1. “Osc” denotes the oscillatory region. Curves 1–3 are well fitted by lines , where , (2) 3, (3) 1.92, while curves 4 and 5 are well fitted by lines , where and , (5) 0.064, and (6) 0.0207. For curves 1–3, the SS (reduced) is above and to the left of the curves, while for curves 4–6, the SS (oxidized) is below and to the right.

Dispersion curves for model (B). (a) Turing instability. (b) Wave instability. Curves 1 and 2 are, respectively, and , where is the eigenvalue of the linearized model (B) with the largest real part. Parameters: (a) , , , , , , , , , , , is calculated from Eq. (28) at and , , . (b) , , , , , , , , , , , , , .

Dispersion curves for model (B). (a) Turing instability. (b) Wave instability. Curves 1 and 2 are, respectively, and , where is the eigenvalue of the linearized model (B) with the largest real part. Parameters: (a) , , , , , , , , , , , is calculated from Eq. (28) at and , , . (b) , , , , , , , , , , , , , .

(a) Space-time plot for JWs found in model (B) in 1D with zero-flux boundary conditions and a small perturbation of the SS at the left end. Total length is 280 a.u. for dimensionless diffusion coefficients , or 5.6 mm for dimensional coefficients and . Total time . White short segments correspond to high concentration of the oxidized catalyst . Dotted line is drawn through the first wave. (b) Four concentration profiles of the catalyst and inhibitor . Space is in a.u.; concentrations are in milimolar. Parameters of model (B) are the same as in Fig. 3(a).

(a) Space-time plot for JWs found in model (B) in 1D with zero-flux boundary conditions and a small perturbation of the SS at the left end. Total length is 280 a.u. for dimensionless diffusion coefficients , or 5.6 mm for dimensional coefficients and . Total time . White short segments correspond to high concentration of the oxidized catalyst . Dotted line is drawn through the first wave. (b) Four concentration profiles of the catalyst and inhibitor . Space is in a.u.; concentrations are in milimolar. Parameters of model (B) are the same as in Fig. 3(a).

Space-time plot for JWs in model (B) for 1D with zero-flux boundary conditions, small perturbation of the SS at the left end, and bulk oscillations (Hopf bifurcation). Parameters as in Fig. 4 except . .

Space-time plot for JWs in model (B) for 1D with zero-flux boundary conditions, small perturbation of the SS at the left end, and bulk oscillations (Hopf bifurcation). Parameters as in Fig. 4 except . .

(a) Turing pattern found in model (B) in 1D with zero-flux boundary conditions and small local perturbation of the SS at the center. Total Concentrations of and are in M. Parameters are the same as in Fig. 3(a) except . (b) Space-time plot of a simple (continuous) traveling pulse in model (B) in 1D with zero-flux boundary conditions and small local perturbation of the SS at the left end of the segment. Parameters: , , , , , , , , other parameters as in Fig. 3(a).

(a) Turing pattern found in model (B) in 1D with zero-flux boundary conditions and small local perturbation of the SS at the center. Total Concentrations of and are in M. Parameters are the same as in Fig. 3(a) except . (b) Space-time plot of a simple (continuous) traveling pulse in model (B) in 1D with zero-flux boundary conditions and small local perturbation of the SS at the left end of the segment. Parameters: , , , , , , , , other parameters as in Fig. 3(a).

Snapshots of JWs found in model (B) in 2D with a small perturbation at the center of a circle of radius 80; zero-flux boundary conditions, -variable. Darker color indicates higher concentration of . All parameters are the same as in Fig. 3(a).

Snapshots of JWs found in model (B) in 2D with a small perturbation at the center of a circle of radius 80; zero-flux boundary conditions, -variable. Darker color indicates higher concentration of . All parameters are the same as in Fig. 3(a).

Snapshots of bubble waves in model (B) in 2D with a small perturbation in the center of a circle of radius 30; zero-flux boundary condition, -variable. Darker color indicates higher concentration of . Parameters: , , , , , , , , all other parameters as in Fig. 3(a).

Snapshots of bubble waves in model (B) in 2D with a small perturbation in the center of a circle of radius 30; zero-flux boundary condition, -variable. Darker color indicates higher concentration of . Parameters: , , , , , , , , all other parameters as in Fig. 3(a).

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