^{1,a)}, Karin John

^{2}and Markus Bär

^{1}

### Abstract

An effective medium theory is employed to derive a simple qualitative model of a pattern formingchemical reaction in a microemulsion. This spatially heterogeneous system is composed of water nanodroplets randomly distributed in oil. While some steps of the reaction are performed only inside the droplets, the transport through the extended medium occurs by diffusion of intermediate chemical reactants as well as by collisions of the droplets. We start to model the system with heterogeneous reaction–diffusion equations and then derive an equivalent effective spatially homogeneous reaction–diffusion model by using earlier results on homogenization in heterogeneous reaction–diffusion systems [S.Alonso, M.Bär, and R.Kapral, J. Chem. Phys.134, 214102 (2009)]. We study the linear stability of the spatially homogeneous state in the resulting effective model and obtain a phase diagram of pattern formation, that is qualitatively similar to earlier experimental results for the Belousov–Zhabotinsky reaction in an aerosol OT (AOT)-water-in-oil microemulsion [V. K.Vanag and I. R.Epstein, Phys. Rev. Lett.87, 228301 (2001)]. Moreover, we reproduce many patterns that have been observed in experiments with the Belousov–Zhabotinsky reaction in an AOT oil-in-water microemulsion by direct numerical simulations.

Stimulating discussions with R. Kapral are gratefully acknowledged. We acknowledge financial support from the German Science Foundation Deutsche Forschungsgemeinschaft (DFG) within the framework of SFB 555 “Complex Nonlinear Processes” and SFB 910 “Control of Self-Organizing Nonlinear Systems.”

I. INTRODUCTION

II. MODEL

III. LINEAR STABILITY ANALYSIS

IV. NUMERICAL SIMULATIONS

V. DISCUSSION

VI. CONCLUSIONS

### Key Topics

- Fluid drops
- 36.0
- Diffusion
- 25.0
- Microemulsions
- 13.0
- Chemical reactions
- 12.0
- Percolation
- 12.0

## Figures

(a) Dependence on the dispersed phase fraction of the effective diffusion coefficient for a heterogeneous medium composed of two phases with *D* _{ d } = 0.01 and *D* _{ i } = 2 for different spatial dimensionality: *d* = 1 (black), *d* = 2 (red), *d* = 3 (green), and *d* ≈ ∞ (blue). (b) Dependence on the dispersed phase fraction of the effective reaction rate for a heterogeneous medium composed of two phases with *R* _{ i } = 1 and *R* _{oil} = 0.

(a) Dependence on the dispersed phase fraction of the effective diffusion coefficient for a heterogeneous medium composed of two phases with *D* _{ d } = 0.01 and *D* _{ i } = 2 for different spatial dimensionality: *d* = 1 (black), *d* = 2 (red), *d* = 3 (green), and *d* ≈ ∞ (blue). (b) Dependence on the dispersed phase fraction of the effective reaction rate for a heterogeneous medium composed of two phases with *R* _{ i } = 1 and *R* _{oil} = 0.

(a) Solution of Eq. (6) in a zero-dimensional system. Different regimes are observed: Steady state, oscillations (maximum and minimum are shown), and bistability. Three examples of temporal evolution are shown: (b) Oscillations corresponding to *a* = 4, (c) irregular oscillations corresponding to *a* = 8.3, and (d) bistability corresponding to *a* = 10, two different initial conditions are shown. Values of the parameters: ϕ = 0.8, *b* = 3, *c* = 3.5, ε_{1} = 1, ε_{2} = 4, α = 4/3, *D* _{ w } = 1, *D* _{ i } = 2, and *D* _{ d } = 0.01.

(a) Solution of Eq. (6) in a zero-dimensional system. Different regimes are observed: Steady state, oscillations (maximum and minimum are shown), and bistability. Three examples of temporal evolution are shown: (b) Oscillations corresponding to *a* = 4, (c) irregular oscillations corresponding to *a* = 8.3, and (d) bistability corresponding to *a* = 10, two different initial conditions are shown. Values of the parameters: ϕ = 0.8, *b* = 3, *c* = 3.5, ε_{1} = 1, ε_{2} = 4, α = 4/3, *D* _{ w } = 1, *D* _{ i } = 2, and *D* _{ d } = 0.01.

Dispersion curves obtained by linear stability analysis. Solid and dashed lines correspond to real and imaginary eigenvalues, respectively. The parameter values employed are: *a* = 2.7 and ϕ = 0.2 (a), *a* = 4 and ϕ = 0.6 (b), *a* = 3.2 and ϕ = 0.4 (c), *a* = 4.5 and ϕ = 0.5 (d), *a* = 2.7 and ϕ = 0.15 (e), and *a* = 4 and ϕ = 0.2 (f). Values of the rest of parameters: *b* = 3, *c* = 3.5, ε_{1} = 1, ε_{2} = 4, α = 4/3, *D* _{ w } = 1, *D* _{ i } = 2, and *D* _{ d } = 0.01.

Dispersion curves obtained by linear stability analysis. Solid and dashed lines correspond to real and imaginary eigenvalues, respectively. The parameter values employed are: *a* = 2.7 and ϕ = 0.2 (a), *a* = 4 and ϕ = 0.6 (b), *a* = 3.2 and ϕ = 0.4 (c), *a* = 4.5 and ϕ = 0.5 (d), *a* = 2.7 and ϕ = 0.15 (e), and *a* = 4 and ϕ = 0.2 (f). Values of the rest of parameters: *b* = 3, *c* = 3.5, ε_{1} = 1, ε_{2} = 4, α = 4/3, *D* _{ w } = 1, *D* _{ i } = 2, and *D* _{ d } = 0.01.

Phase diagram obtained by linear stability analysis of the solutions obtained in Eqs. (7) and (8). Lines correspond to Hopf (solid), bistability (solid), turing (solid), and wave (dashed) instabilities. Right of the dotted-dashed line the two solutions *u* _{ o2, 3} become physically relevant. Crosses × and + correspond, respectively, to the parameter values employed in the numerical simulations shown in Figs. 5 and 7. Values of the parameters: *b* = 3, *c* = 3.5, ε_{1} = 1, ε_{2} = 4, α = 4/3, *D* _{ w } = 1, *D* _{ i } = 2, and *D* _{ d } = 0.01.

Phase diagram obtained by linear stability analysis of the solutions obtained in Eqs. (7) and (8). Lines correspond to Hopf (solid), bistability (solid), turing (solid), and wave (dashed) instabilities. Right of the dotted-dashed line the two solutions *u* _{ o2, 3} become physically relevant. Crosses × and + correspond, respectively, to the parameter values employed in the numerical simulations shown in Figs. 5 and 7. Values of the parameters: *b* = 3, *c* = 3.5, ε_{1} = 1, ε_{2} = 4, α = 4/3, *D* _{ w } = 1, *D* _{ i } = 2, and *D* _{ d } = 0.01.

Typical evolution of different patterns obtained in phase space of Fig. 4 corresponding to: *a* = 9 and ϕ = 0.64 (a), *a* = 5 and ϕ = 0.8 (b) and (c), *a* = 7 and ϕ = 0.55 (d), and *a* = 2.8 and ϕ = 0.3 (e). Bright and dark colors in the snapshots correspond, respectively, to high and low concentrations of the activator. Values of the parameters as in Fig. 4. Numerical parameters: Δ*x* = 0.3 and Δ*t* = 0.01 in a grid of 200 × 200 pixels.

Typical evolution of different patterns obtained in phase space of Fig. 4 corresponding to: *a* = 9 and ϕ = 0.64 (a), *a* = 5 and ϕ = 0.8 (b) and (c), *a* = 7 and ϕ = 0.55 (d), and *a* = 2.8 and ϕ = 0.3 (e). Bright and dark colors in the snapshots correspond, respectively, to high and low concentrations of the activator. Values of the parameters as in Fig. 4. Numerical parameters: Δ*x* = 0.3 and Δ*t* = 0.01 in a grid of 200 × 200 pixels.

Collection of patterns obtained at the end of the numerical simulations and arranged as a numerical phase diagram. Bright and dark colors in the snapshots correspond, respectively, to high and low concentrations of the activator. Values of the parameters as in Fig. 4. Numerical parameters: Δ*x* = 0.3 and Δ*t* = 0.01 in a grid of 200 × 200 pixels.

Collection of patterns obtained at the end of the numerical simulations and arranged as a numerical phase diagram. Bright and dark colors in the snapshots correspond, respectively, to high and low concentrations of the activator. Values of the parameters as in Fig. 4. Numerical parameters: Δ*x* = 0.3 and Δ*t* = 0.01 in a grid of 200 × 200 pixels.

Evolution of different patterns obtained in phase space of Fig. 4 corresponding to: *a* = 5 and ϕ = 0.62 (a), *a* = 9 and ϕ = 0.64 (b), *a* = 6 and ϕ = 0.62 (c), *a* = 5 and ϕ = 0.5 (d), and *a* = 4.22 and ϕ = 0.48 (e). Bright and dark colors in the snapshots correspond, respectively, to high and low concentrations of the activator. Values of the parameters as in Fig. 4. Numerical parameters: Δ*x* = 0.3 and Δ*t* = 0.01 in a grid of 200 × 200 pixels.

Evolution of different patterns obtained in phase space of Fig. 4 corresponding to: *a* = 5 and ϕ = 0.62 (a), *a* = 9 and ϕ = 0.64 (b), *a* = 6 and ϕ = 0.62 (c), *a* = 5 and ϕ = 0.5 (d), and *a* = 4.22 and ϕ = 0.48 (e). Bright and dark colors in the snapshots correspond, respectively, to high and low concentrations of the activator. Values of the parameters as in Fig. 4. Numerical parameters: Δ*x* = 0.3 and Δ*t* = 0.01 in a grid of 200 × 200 pixels.

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