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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