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Complex wave patterns in an effective reaction–diffusion model for chemical reactions in microemulsions
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10.1063/1.3559154
/content/aip/journal/jcp/134/9/10.1063/1.3559154
http://aip.metastore.ingenta.com/content/aip/journal/jcp/134/9/10.1063/1.3559154
View: Figures

Figures

Image of FIG. 1.
FIG. 1.

(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.

Image of FIG. 2.
FIG. 2.

(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, ε1 = 1, ε2 = 4, α = 4/3, D w = 1, D i = 2, and D d = 0.01.

Image of FIG. 3.
FIG. 3.

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, ε1 = 1, ε2 = 4, α = 4/3, D w = 1, D i = 2, and D d = 0.01.

Image of FIG. 4.
FIG. 4.

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, ε1 = 1, ε2 = 4, α = 4/3, D w = 1, D i = 2, and D d = 0.01.

Image of FIG. 5.
FIG. 5.

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: Δx = 0.3 and Δt = 0.01 in a grid of 200 × 200 pixels.

Image of FIG. 6.
FIG. 6.

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: Δx = 0.3 and Δt = 0.01 in a grid of 200 × 200 pixels.

Image of FIG. 7.
FIG. 7.

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: Δx = 0.3 and Δt = 0.01 in a grid of 200 × 200 pixels.

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/content/aip/journal/jcp/134/9/10.1063/1.3559154
2011-03-07
2014-04-16
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Complex wave patterns in an effective reaction–diffusion model for chemical reactions in microemulsions
http://aip.metastore.ingenta.com/content/aip/journal/jcp/134/9/10.1063/1.3559154
10.1063/1.3559154
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