^{1}, Vladimir K. Vanag

^{1}and Irving R. Epstein

^{1}

### Abstract

The Belousov–Zhabotinsky (BZ) reaction dispersed in water-in-oil aerosol OT (AOT) microemulsion has been studied at small radius of water nanodroplets . Stationary spotlike and labyrinthine Turing patterns are found close to the fully oxidized state. These patterns, islands of high concentration of the reduced state of the catalyst, can coexist either with “black” reduction waves or, under other conditions, with the “white” oxidation waves usually observed in the BZ reaction. The experimental observations are analyzed with the aid of a new Oregonator-like model and qualitatively reproduced in computer simulations.

This work was supported by Grant No. CHE-0306262 from the National Science Foundation.

I. INTRODUCTION

II. MATERIALS AND METHODS

III. EXPERIMENTAL RESULTS

A. Turing patterns

B. Turing patterns in combination with waves and oscillations

IV. THEORETICAL CONSIDERATIONS

V. DISCUSSION AND CONCLUSION

## Figures

Turing patterns in the BZ-AOT system at different concentrations of . Stationary patterns are in the middle column. Left column shows transient patterns that first emerge in the reactor. Parameters: , , , , , 0.18, (c, d) 0.24, (e, f) 0.3. All concentrations are for the aqueous pseudophase (not the total volume of microemulsion). Frame . Two-dimensional FFTs of the snapshots in the middle column are shown in the right column. Two images of the FFT correspond to two different scaling factors (1 and 2) of the FFT and allow one to see the FFT at different contrast levels. White dots correspond to FFT peaks in the allowable contrast range, 1 (black)–256 (white).

Turing patterns in the BZ-AOT system at different concentrations of . Stationary patterns are in the middle column. Left column shows transient patterns that first emerge in the reactor. Parameters: , , , , , 0.18, (c, d) 0.24, (e, f) 0.3. All concentrations are for the aqueous pseudophase (not the total volume of microemulsion). Frame . Two-dimensional FFTs of the snapshots in the middle column are shown in the right column. Two images of the FFT correspond to two different scaling factors (1 and 2) of the FFT and allow one to see the FFT at different contrast levels. White dots correspond to FFT peaks in the allowable contrast range, 1 (black)–256 (white).

(a) Conductivity of an AOT microemulsion loaded with the reactants of the BZ reaction as a function of droplet fraction at two different . Concentrations: , , . (b, c) Time series of bulk oscillations in a CSTR recorded as absorption at (arbitrary wavelength in the absorption band of the oxidized catalyst) in a square cell. Compositions of the BZ-AOT system as in Fig. 1; panels (b) and (c) correspond to (a, b) and (e, f) in Fig. 1, respectively. Period of oscillations . (d) Delay time [see (c)] vs . is dependent on stirring rate and thus can vary slightly from experiment to experiment.

(a) Conductivity of an AOT microemulsion loaded with the reactants of the BZ reaction as a function of droplet fraction at two different . Concentrations: , , . (b, c) Time series of bulk oscillations in a CSTR recorded as absorption at (arbitrary wavelength in the absorption band of the oxidized catalyst) in a square cell. Compositions of the BZ-AOT system as in Fig. 1; panels (b) and (c) correspond to (a, b) and (e, f) in Fig. 1, respectively. Period of oscillations . (d) Delay time [see (c)] vs . is dependent on stirring rate and thus can vary slightly from experiment to experiment.

Dependence of wavelength of Turing patterns on (a) [MA] at and (b) at . Other parameters: , , , , (b) (open triangles) and (closed triangles).

Dependence of wavelength of Turing patterns on (a) [MA] at and (b) at . Other parameters: , , , , (b) (open triangles) and (closed triangles).

Evolution of a Turing pattern in the BZ-AOT system. (a) [corresponds to in (e)], (b) , (c) after emergence of the first pattern. Frame . (d) Time dependence of the wavelength of Turing patterns. (e) Time series of bulk oscillations in a CSTR recorded as absorption at . Parameters: , , , , , .

Evolution of a Turing pattern in the BZ-AOT system. (a) [corresponds to in (e)], (b) , (c) after emergence of the first pattern. Frame . (d) Time dependence of the wavelength of Turing patterns. (e) Time series of bulk oscillations in a CSTR recorded as absorption at . Parameters: , , , , , .

Black spots in the BZ-AOT system at large for (a, c) specially washed glass windows and filtered microemulsion; (b, d) microemulsion without filtration and cleaning of window surfaces. Parameters: , , , , , . Frame sizes are . Time lapse between snapshots (a) and (c), as well as between (b) and (d), is .

Black spots in the BZ-AOT system at large for (a, c) specially washed glass windows and filtered microemulsion; (b, d) microemulsion without filtration and cleaning of window surfaces. Parameters: , , , , , . Frame sizes are . Time lapse between snapshots (a) and (c), as well as between (b) and (d), is .

Coexistence of black spots with (a) black reduction waves and (b) white oxidation waves at different times. (c) Space-time dependence at the gray line in snapshots (a) and (b). Arrows in plot (c) indicate times at which snapshots (a) and (b) were taken. Time lapse between snapshots (a) and (b) is . Parameters: , , , , , . Frame sizes: (a), (b) , (c) . Wave velocity for black waves [first waves in (c)], for white waves in (b), and for last set of waves in (c). Arrows in (a) and (b) indicate direction of wave propagation.

Coexistence of black spots with (a) black reduction waves and (b) white oxidation waves at different times. (c) Space-time dependence at the gray line in snapshots (a) and (b). Arrows in plot (c) indicate times at which snapshots (a) and (b) were taken. Time lapse between snapshots (a) and (b) is . Parameters: , , , , , . Frame sizes: (a), (b) , (c) . Wave velocity for black waves [first waves in (c)], for white waves in (b), and for last set of waves in (c). Arrows in (a) and (b) indicate direction of wave propagation.

Stationary (a), (b) and oscillatory (c), (d) Turing patterns in the BZ-AOT system at (a) , (b) , (c) , (d) after the emergence of the pattern. (e) Space-time dependence at the gray line in snapshot (b). Arrows in plot (e) correspond to snapshots (a)–(d). Parameters: , , , , , . Frame sizes: (a)–(d) , (e) .

Stationary (a), (b) and oscillatory (c), (d) Turing patterns in the BZ-AOT system at (a) , (b) , (c) , (d) after the emergence of the pattern. (e) Space-time dependence at the gray line in snapshot (b). Arrows in plot (e) correspond to snapshots (a)–(d). Parameters: , , , , , . Frame sizes: (a)–(d) , (e) .

(a) Oscillations in 0D system (10) and (11) with , , , , and (curve 1), 0.0097 (curve 2); . The same shape of oscillations as in curve (2) is obtained at and . (b) Curves 1 and 2 are nullclines for Eqs. (10) and (11), respectively, with parameters: , , , , and (curve 3, trajectory of large amplitude oscillations), 0.01 (curve 4, trajectory of small amplitude oscillations). Steady state (cross point of curves 1 and 2) is inside the loops 3 and 4. (c) Phase diagram of system (10) and (11) in logarithmic coordinates. Parameters , , , . SS is steady state with fully oxidized catalyst, means instabilities. Curve (1) (white squares) corresponds to the onset of Turing instability, curve (2) to the onset of Hopf instability. Points of curve (2) are fitted by a linear trend line with slope . (d) Dispersion curves for system (10) and (11) with , , , , , . Curves (1) correspond to Fig. 10(a), and curves (2) to Fig. 10(b). Curves and are real part of eigenvalue, curves and are corresponding imaginary parts. Two horizontal dashes indicate half of the imaginary part at . Two vertical dashes indicate half of [ has a maximum at ].

(a) Oscillations in 0D system (10) and (11) with , , , , and (curve 1), 0.0097 (curve 2); . The same shape of oscillations as in curve (2) is obtained at and . (b) Curves 1 and 2 are nullclines for Eqs. (10) and (11), respectively, with parameters: , , , , and (curve 3, trajectory of large amplitude oscillations), 0.01 (curve 4, trajectory of small amplitude oscillations). Steady state (cross point of curves 1 and 2) is inside the loops 3 and 4. (c) Phase diagram of system (10) and (11) in logarithmic coordinates. Parameters , , , . SS is steady state with fully oxidized catalyst, means instabilities. Curve (1) (white squares) corresponds to the onset of Turing instability, curve (2) to the onset of Hopf instability. Points of curve (2) are fitted by a linear trend line with slope . (d) Dispersion curves for system (10) and (11) with , , , , , . Curves (1) correspond to Fig. 10(a), and curves (2) to Fig. 10(b). Curves and are real part of eigenvalue, curves and are corresponding imaginary parts. Two horizontal dashes indicate half of the imaginary part at . Two vertical dashes indicate half of [ has a maximum at ].

Turing patterns in model (10) and (11). (a) Parameters: , , , , , , , , . At these parameters, and , thus the system exhibits pure Turing instability; . (b) An increase in by increasing to 100, transforms black spots in (a) into stripes. (c) Localized Turing spot. Parameters: , , , , , , ; , . , . All eigenvalues have negative real part at all wave numbers . Stationary single spot emerges only after a large local perturbation of the steady state, at the center of the square area, where is the steady state value of . (d) Increasing to by decreasing to 22 leads to the slow spreading of Turing spots, which finally occupy the entire area; all other parameters as in (c).

Turing patterns in model (10) and (11). (a) Parameters: , , , , , , , , . At these parameters, and , thus the system exhibits pure Turing instability; . (b) An increase in by increasing to 100, transforms black spots in (a) into stripes. (c) Localized Turing spot. Parameters: , , , , , , ; , . , . All eigenvalues have negative real part at all wave numbers . Stationary single spot emerges only after a large local perturbation of the steady state, at the center of the square area, where is the steady state value of . (d) Increasing to by decreasing to 22 leads to the slow spreading of Turing spots, which finally occupy the entire area; all other parameters as in (c).

Space-time plot for oscillatory Turing patterns in 1D. Model parameters: , , , , , , , 1.78, (b) 1.7; 3.2, (b) 2.72. White corresponds to maximum of activator and black to minimum. Total 0.6, (b) 1.08.

Space-time plot for oscillatory Turing patterns in 1D. Model parameters: , , , , , , , 1.78, (b) 1.7; 3.2, (b) 2.72. White corresponds to maximum of activator and black to minimum. Total 0.6, (b) 1.08.

FFT spectra for time series of 1D oscillatory Turing patterns shown in Fig. 10 for 3.2, (b) 2.9, (c) 2.8, (d) 2.72. Bold solid and thin dotted lines in (b) and (c) correspond to different spatial points.

FFT spectra for time series of 1D oscillatory Turing patterns shown in Fig. 10 for 3.2, (b) 2.9, (c) 2.8, (d) 2.72. Bold solid and thin dotted lines in (b) and (c) correspond to different spatial points.

Oscillatory Turing patterns in 2D for model (10) and (11). Parameters: , , , , , . For , and (between snapshots); one full period , , . For , , and ( a period), , .

Oscillatory Turing patterns in 2D for model (10) and (11). Parameters: , , , , , . For , and (between snapshots); one full period , , . For , , and ( a period), , .

(a) One-dimensional Reduction waves in model (10) and (11). Bold line is catalyst ; thin dotted line is activator . Arrow shows direction of wave propagation. At initial moment of time, left edge of the segment (with zero flux boundary conditions) was perturbed. Parameters: , , , , , , (similar waves are obtained at ). The system is close to the subcritical Hopf bifurcation. At these parameters, at , while at , at . (b) Homogeneous large-amplitude oscillations on a segment of length 80 after homogeneous large-amplitude perturbation ; , , . Oscillations are the same as in 0D case.

(a) One-dimensional Reduction waves in model (10) and (11). Bold line is catalyst ; thin dotted line is activator . Arrow shows direction of wave propagation. At initial moment of time, left edge of the segment (with zero flux boundary conditions) was perturbed. Parameters: , , , , , , (similar waves are obtained at ). The system is close to the subcritical Hopf bifurcation. At these parameters, at , while at , at . (b) Homogeneous large-amplitude oscillations on a segment of length 80 after homogeneous large-amplitude perturbation ; , , . Oscillations are the same as in 0D case.

White Turing spots at large . , , , , , , . (a, b) , , 1D case, . For (c)–(g), , 17, 20, 45, and 300, respectively, , (h) , . In all cases, vertical thin line in the center is perturbed initially.

White Turing spots at large . , , , , , , . (a, b) , , 1D case, . For (c)–(g), , 17, 20, 45, and 300, respectively, , (h) , . In all cases, vertical thin line in the center is perturbed initially.

Stationary Turing patterns in model (10) and (11) in the oxidized steady state of the system. Parameters: , , , , , (as in Fig. 8), . (a)–(c) small random initial perturbation ; , ; (a, b) (light line) and (dark line) profiles at and , respectively; (c) profiles at 75, (2) 79, (3) 90, (4) 135, (5) 375, is shifted down by (1) 0, (2) 0.01, (3) 0.02, (4) 0.1, and (5) 0.2. Small amplitude oscillations at each spatial point with a period of 2.14 gradually die and at first Turing patterns [(shown in (a)] start to emerge. (d) Large initial perturbation at one edge of the segment. Horizontal lines in (d) indicate steady states. Wavelength 8.64, (b) 12.3; (d) 14.5.

Stationary Turing patterns in model (10) and (11) in the oxidized steady state of the system. Parameters: , , , , , (as in Fig. 8), . (a)–(c) small random initial perturbation ; , ; (a, b) (light line) and (dark line) profiles at and , respectively; (c) profiles at 75, (2) 79, (3) 90, (4) 135, (5) 375, is shifted down by (1) 0, (2) 0.01, (3) 0.02, (4) 0.1, and (5) 0.2. Small amplitude oscillations at each spatial point with a period of 2.14 gradually die and at first Turing patterns [(shown in (a)] start to emerge. (d) Large initial perturbation at one edge of the segment. Horizontal lines in (d) indicate steady states. Wavelength 8.64, (b) 12.3; (d) 14.5.

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