18992
J. Phys. Chem. 1996, 100, 18992-18996
Light-Induced Frequency Shift in Chemical Spirals Valery Petrov, Qi Ouyang, Ge Li, and Harry L. Swinney* Center for Nonlinear Dynamics and Department of Physics, UniVersity of Texas, Austin, Texas 78712 ReceiVed: May 13, 1996; In Final Form: July 8, 1996X
Illumination of ruthenium-catalyzed Belousov-Zhabotinsky reaction decreases the rotational frequency of spirals at low bromate concentrations but increases the frequency at high bromate concentrations. The effective diffusion coefficient D deduced from the Keener-Tyson relation for the spirals, D ≈ ω/3k2, is independent of light intensity (D ) 2.5 × 10-6 cm2/s).
Introduction The light-sensitive Belousov-Zhabotinsky (BZ) reaction with a ruthenium-based catalyst is a convenient system to study the effect of external perturbations on dynamical behavior of chemical reactions. The Ru(bpy)3Cl2 complex was first suggested as a luminescent indicator for the BZ reaction by Demas and Diemente.1 Later the reaction was found to be sensitive to visible light by Gaspar et al.,2 and several studies have elucidated the mechanism of the photosensitivity.3,4 The light-sensitive BZ reaction can also be used in spatially extended reactors where light can alter the spatiotemporal behavior of the system. Kuhnert et al. suggested that the light-sensitive BZ reaction could be used for image processing.5 Steinbock et al. used periodic light perturbations to stimulate meandering motion of the spiral core.6 These studies were, however, conducted in a limited range of concentrations where the reaction is inhibited by the light. We have examined the behavior of spirals in an open reactor for a wide range of bromate concentrations. Examples of system behavior for different bromate concentrations are shown in Figure 1. Projection of blue light with intensity 20 mW/cm2 on the region inside the dashed circle results in the suppression of the wave propagation (Figure 1a), the decrease of the wave speed (Figure 1b), the increase of the wave speed (Figure 1c), and the formation of a target pattern (Figure 1d). The BZ reaction is famous for its ability to sustain spiral waves.7 We present here a study of the primary bifurcations of BZ spirals as a function of malonic acid and bromate concentrations and the illumination intensity. The important characteristics of the spirals are their frequency and wavelength. These properties are governed by the spiral core, which is a self-sustained wave generator. In the ruthenium-catalyzed BZ reaction, light with the wavelength around 450 nm changes the underlying chemical reaction, resulting in a change in the frequency of the oscillations. We measure the spiral frequencies and wavelengths to obtain the dispersion relation as a function of bromate concentration. Experimental Section The experiments were carried out in an open spatial reactor similar to the one used by Ouyang and Swinney.9 The reactor architecture allows constant concentrations of the reactants to be maintained indefinitely, allowing a precise characterization of the bifurcation sequence of the spatiotemporal dynamics. Two 10 mL compartments with continuously refreshed chemicals are separated by a Vycor glass membrane that is 0.4 mm thick and X
Abstract published in AdVance ACS Abstracts, November 1, 1996.
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22 mm in diameter. The membrane is transparent to visible light and chemically inert. It serves as a continuously fed unstirred reactor (CFUR) by allowing fresh reactants to diffuse from the sides but preventing convective mixing. Each compartment has two inlets and one outlet. The first compartment (I) is fed with a mixture of malonic acid (MA) and sodium bromide (NaBr) in one inlet and H2SO4 and sodium bromate (NaBrO3) in the other inlet. A second compartment (II) is fed with Ru(bpy)3Cl2 in one inlet and H2SO4 and NaBrO3 in the other inlet. Ru(bpy)32+ is immediately oxidized by the bromate to the Ru(bpy)33+, making the reactor transparent in transmitted light. The flow rate in each inlet was 20 mL/h, resulting in a residence time of 15 min in each compartment. The concentrations of the H2SO4 and NaBrO3 were maintained at the same level on both sides of the membrane resulting in a constant concentration profile of these reactants. Ru(bpy)3Cl2 and the mixture of MA and NaBr were separated in different compartments, resulting in gradients across the membrane. The crossflux of the chemicals through the membrane was much lower than their feeding streams, and conditions in one compartment have very little effect on the other one. MA and Br- on one side and Ru(bpy)33+ on the other side diffuse into the membrane, and the reaction takes place in a thin layer in the membrane. The quasi-two-dimensional spatial patterns were monitored in transmitted light from a low-power light source using a CCD camera and a band-pass filter at 450 ( 20 nm. This wavelength corresponds to the maximum in the absorption spectrum of Ru(bpy)32+. The observed images were filtered in time using a digital first-order band-pass filter with the band frequencies tuned to be transparent to the characteristic frequency of the BZ reaction (0.10 ( 0.05 Hz). Processed images obtained at 1 s intervals were stored and later analyzed in time and space domains using Fourier transforms. A Sanyo PLC-220N video projector was used to illuminate the reaction. Light from the projector was directed to the membrane by a sequence of lenses and a beam splitter. The spatial inhomogeneity of the light intensity in the projected image was originally as large as 50%. It was equalized within 5% by using the correction matrix calculated in advance based on the reflectance from a diffusive screen. The spectral power of the light in the 430-470 nm wavelength range was 20 mW/ m2. Since the spectrum of the light used for the observation and the perturbation was the same, the intensity of the observed light was 2 orders of magnitude lower than that of the light from the video projector. Also, the direction of the perturbing illumination had a slight angle with respect to the observation axis to remove interference between two light sources. © 1996 American Chemical Society
Frequency Shift in Chemical Spirals
J. Phys. Chem., Vol. 100, No. 49, 1996 18993
Figure 1. Effect of uniform illumination of the region inside the dashed circle: (a) inhibition of wave propagation ([MA]I ) 0.03 M, [BrO3-]I,II ) 0.05); (b) decrease in wave speed ([MA]I ) 0.03 M, [BrO3-]I,II ) 0.10); (c) increase in wave speed ([MA]I ) 0.05 M, [BrO3-]I,II ) 0.25); (d) generation of a target pattern ([MA]I ) 0.05 M, [BrO3-]I,II ) 0.4). Other control parameters were held fixed: [H2SO4]I,II ) 0.5 M, [Br-]I ) 10 mM, [Ru(bpy)32+]II ) 0.5 mM. Each image is 11 × 11 mm2.
Results and Discussion Figure 2 shows the two-dimensional bifurcation (phase) diagram that defines the regions of the temporal behavior in the BZ reaction for different MA and BrO3- concentrations. For low bromate concentration the system is in the reduced stationary state. Under these conditions the reactor is dark when observed in transmitted blue light. As the bromate concentration is increased above the value defined by the line with filled circles, the system becomes excitable and wave pulses can propagate through it. Several types of waves can be observed in the BZ reaction.10 Trigger waves and phase waves can be observed in excitable and oscillatory media, respectively. For low concentrations of the bromate the system is excitable, and the trigger wave can propagate only if it is initiated somewhere. A spiral wave is the only self-sustaining structure in this region; the spiral core generates the propagating pulses. The number of spiral centers can vary depending on the initial conditions, but eventually the single spiral with the highest frequency will
survive.11 The spatial dominance by the structure with the highest frequency seems to be a generic phenomenon in the spatially extended BZ reaction. Region (a) in Figure 2 defines the concentration range where the light illumination inhibits wave propagation. The waves can be suppressed by illuminating the reaction with blue light of 20mW/m2 intensity, as shown in Figure 1a where light was projected on the region inside the dashed circle. If the bromate concentration is further increased to move the system into the region (b) in Figure 2, the light only decreases the propagation velocity of the waves, as shown in Figure 1b. We did not calculate the change in the wave velocity; instead, the spiral frequency and wavelength were deduced using Fourier spectra of the single-probe time series and the reaction images, respectively. Figure 3 shows the power spectrum of an intensity time series in the center of the reactor (averaged over 3 × 3 pixel region) calculated from 1024 data points. The solid line corresponds to the power spectrum of the system in the dark.
18994 J. Phys. Chem., Vol. 100, No. 49, 1996
Figure 2. Phase diagram of the BZ reaction with Ru(bpy)32+ catalyst. The line with the open circles defines the Hopf bifurcation for the system in the dark. The line with the open squares represents the locus of the Hopf bifurcation of the illuminated reaction. The lower limit for the propagating waves in the system is defined by the line with filled squares for the illuminated system and by the line with filled circles for the system in the dark. The dashed line shows an approximate position of the parameters where the light has no effect. Other control parameters are the same as in Figure 1. Measurements were made with increments of 0.05 M in bromate concentration for the illuminated and the dark reactor for each malonic acid concentration shown; only the bromate concentrations corresponding to a change in the spatiotemporal behavior are plotted.
Petrov et al.
Figure 4. Spatial power spectrum of a two-dimensional image of the spiral, averaged over all the angles in k-space. Solid (dashed) line represents dark (illuminated) system. Parameters are the same as in Figure 3.
Figure 5. Dependence of the angular frequency ω of the most stable spirals on bromate concentration for the dark (solid line) and the illuminated (dashed line) system. The values of ω were obtained from the first peak in Fourier spectra like that in Figure 3. The concentration of malonic acid was 0.1 M. Other control parameters are the same as in Figure 1.
Figure 3. Power spectrum of temporal oscillations in the center of membrane of the dark (solid line) and the illuminated (dashed line) system. Arrows indicate location of the fundamental frequencies. Concentrations of malonic acid and bromate were 0.1 and 0.097 M, respectively, corresponding to region (b) in Figure 2. Other control parameters are the same as in Figure 1.
The dashed line represents the power spectrum of the illuminated system. The oscillations are nonlinear, and the first maximum shows the fundamental frequency. The two-dimensional Fourier spectrum of the spiral image has characteristic wavelengths positioned at different angles but at the same distance from the origin in k-space. To improve the signal-to-noise ratio, the spatial spectrum was azimuthally averaged, and the result is shown in Figure 4. The position of the maximum in the Fourier spectrum was interpolated from the three closest points, allowing wavenumbers that are not an integer of the inverse reactor length. The maximum frequencies and wavelengths determined
for different values of the bromate concentration are shown in Figures 5 and 6. The difference between the illuminated and the dark systems decreases as the bromate concentration approaches the dashed line in Figure 2. Along this line the light seems to have no static effect. The position is only approximate since the zerofrequency change is observed at a slightly higher bromate concentration than the zero shift in the spiral wavelength (Figures 5 and 6). The light perturbation increases the wave speed in region (c) above the inflection line in Figure 2. Figure 1c shows deformation of the wave that enters the illuminated area defined by the dashed circle. If the spiral core is placed in the illuminated region, the spiral frequency and the characteristic wavenumber will also increase. Further increase of the bromate concentration will increase the frequency of the oscillations and decrease the amplitude. At high bromate concentrations, oscillations in the BZ reaction are terminated, and the whole system settles into the oxidized state. The solubility limit of NaBrO3 prohibited the exploration of the position of this bifurcation for all values of MA concentration.
Frequency Shift in Chemical Spirals
J. Phys. Chem., Vol. 100, No. 49, 1996 18995
Figure 6. Dependence of the characteristic wavenumber k ) 2π/λ of the most stable spirals as a function of the bromate concentration for the dark and the illuminated system. The values of k were obtained from the first peak in Fourier spectra like that in Figure 4. Parameters are the same as in Figure 5.
Figure 7. Dispersion relation ω vs k2 deduced from the data in Figures 5 and 6 for low bromate concentrations (0.05 < [BrO3-]I,II < 0.10 M).
related to the frequency and the wavenumber of the spiral as16
D = ω/3k2 The amplitude of oscillations is very small in this region, and it was difficult to detect reliably the exact position and nature of the bifurcation that terminates the oscillatory behavior. We assume that it is a supercritical Hopf bifurcation. The Hopf bifurcation line can be lowered by the application of the light, and if the parameters are within region (d) in Figure 2, the system can be effectively moved through the Hopf bifurcation by varying the light intensity. An interesting property of the BZ reaction under these conditions is the target pattern generation from locally illuminated areas (Figure 1d). Normally, target patterns observed in the BZ reaction appear due to the uneven distribution of chemicals or other media defects.12 Long-lived target patterns have not been observed in our experiments in the absence of illumination since the frequency of spirals is higher than the frequency of homogeneous oscillations of the BZ reaction. If, however, a part of the membrane is illuminated, the system is shifted above the Hopf bifurcation, thus creating a defect. Regions around the illuminated spot that are closer to the Hopf bifurcation have a higher oscillation frequency than the rest of the media.13 As a result, the target pattern will eventually suppress all the spirals. Kinematic theory developed by Keener and Tyson 8 provides a framework for explaining the spiral structure and relates the observed rotational frequency to the spiral wavelength. The kinematic theory assumes that propagation speed of the spiral front depends on its curvature
V ) V0 + Dκ
(1)
where V0 is the speed of the planar front, κ is the local curvature, and D is an effective diffusion coefficient. Equation 1 is known as an eikonal equation, and the effective diffusion coefficient D is, in general, a function of diffusion coefficients of the species involved in the front propagation.14 For example, D can be treated as the diffusion coefficient of the HBrO2 in the Oregonator15 model of the BZ reaction. The estimation of D from direct measurements of curvature and velocity is difficult due to the error amplification involved in the calculation of the second derivative of the front. Instead, one can evaluate D from the frequency-wavenumber measurements of the spirals. In the first-order approximation of the kinematic theory, D is
(2)
To estimate the diffusion coefficient, the slopes of the ω vs k2 dependence in Figure 7 were calculated using ω and k values for the low bromate concentrations. The corresponding values of the diffusion constants are (2.6 ( 0.2) × 10-6 and (2.5 ( 0.2) × 10-6 cm2/s for the illuminated and the dark systems, respectively. Conclusions Static homogeneous illumination of the ruthenium-catalyzed BZ reaction changes the rotational frequency, wavelength, and wave speed of spirals. The magnitude and the sign of the effect depend on the concentration values of the inflow reactants, particularly the bromate concentration. It has been suggested3,4 that light couples with the dynamics of the BZ reaction through the production of the Br- from the bromate by the following mechanism:
ν + Ru(bpy)32+ + BrO3- + 6H+ ) Ru(bpy)33+ + 3H2O + Br- (3) We did not systematically study the effect of varying bromide concentration, but the few measurements that were made indicate that an increase of Br- concentration does not lower the Hopf bifurcation line when the system is above the dashed line in Figure 2. Further investigation is necessary to clarify the mechanisms of the light illumination at high bromate and sulfuric acid concentrations. The light-induced shift can be used to generate target patterns or to induce the acceleration of wave propagation at the illuminated areas. Acknowledgment. This work was supported by the U.S. Department of Energy Office of Basic Energy Sciences and the Robert A. Welch Foundation. References and Notes (1) Demas, J. N.; Diemente, D. J. Chem. Educ. 1973, 50, 357. (2) Gaspar, V.; Bazsa, G.; Beck, M. T. Z. Phys. Chem. (Leipzig) 1983, 264, 43. (3) Kuhnert, L. Nature 1986, 319, 393. (4) Junguji, M.; Ishihara, M.; Nakazawa, T. J. Phys. Chem. 1992, 96, 4279. (5) Kuhnert, L.; Agladze, K. I.; Krinsky, V. I. Nature 1989, 337, 244. (6) Steinbock, O.; Muller, S. J. Bifurcation. Chaos 1993, 3, 437.
18996 J. Phys. Chem., Vol. 100, No. 49, 1996 (7) Winfree, A. T. Science 1972, 175, 634. (8) Keener, J. P.; Tyson, J. J. Physica D 1986, 21, 307. (9) Ouyang, Q.; Swinney, H. L. Chaos 1991, 1, 411. (10) Ross, J.; Muller, S. C.; Vidal, C. Science 1988, 240, 365. (11) Krinsky, V. I.; Agladze, K. I. Physica D 1983, 8, 50. (12) Agladze, K. I.; Krinsky, V. I. In Self-Organization. AutowaVes and Structures Far from Equlibrium; Krinsky, V. I., Ed.; Springler-Verlag: Berlin, 1984; p 147.
Petrov et al. (13) Figure 3 demonstrates that frequency of oscillations is increased when bromate concentration is increased, moving the BZ reaction closer to the Hopf bifurcation. (14) Meron, E. Phys. Rep. 1992, 218, 1. (15) Tyson, J. J.; Fife, P. C. J. Chem. Phys. 1980, 73, 2224. (16) The Keener and Tyson8 formula was originally written as (period)(speed)2 = 6πD.
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