Effect of Noise Correlation on Noise-Induced Oscillation Frequency in

Article pubs.acs.org/JPCA

Effect of Noise Correlation on Noise-Induced Oscillation Frequency in the Photosensitive Belousov−Zhabotinsky Reaction in a Continuous Stirred Tank Reactor David S. A. Simakov† and Juan Pérez-Mercader*,†,‡ †

Department of Earth and Planetary Sciences, Harvard University, Cambridge, Massachusetts 02138, United States Santa Fe Institute, Santa Fe, New Mexico 87501, United States

‡

ABSTRACT: We report on the experimental study of noise-induced oscillations in the photosensitive Ru(bpy)32+-catalyzed Belousov− Zhabotinsky reaction in a continuous stirred tank reactor (CSTR). In the absence of deterministic oscillations and any external periodic forcing, oscillations appear when the system is perturbed by stochastic fluctuations in light irradiation with sufficiently high amplitude in the vicinity of the bifurcation point. The frequency distribution of the noise-induced oscillations is strongly affected by noise correlation. There is a shift of the noise-induced oscillation frequency toward higher frequencies for an intermediate range of the noise correlation exponent, indicating the occurrence of coherence resonance. Our findings indicate that, in principle, noise correlation can be used to direct chemical reactions toward certain behavior. system without any external periodic force.18 Coherence resonance has been observed experimentally in various configurations of the BZ system, including gel reactor,9−11 cation-exchange beads,12,13 and CSTR19 and studied theoretically using the Oregonator model.20,21 The resonant behavior is typically recognized as the presence of a maximum in a plot of signal-to-noise ratio versus noise amplitude.11,18,19 Stochastic fluctuations are added to a control parameter, which can be feed flow rate of a CSTR,19 electric field,12,13 or light intensity.9,11 White (uncorrelated) noise is commonly used in this type of study. But white noise is an idealization describing uncorrelated fluctuations. In the real world, stochastic fluctuations are expected to have at least some degree of correlation.22,23 In fact, general arguments based on the dynamic renormalization group indicate that stochastic perturbations applied to a nonlinear chemical reaction can modify reaction constants in a way that depends on noise correlation.24,25 Because reaction constants are dimensional parameters, one can expect that due to the scale-dependence of colored noise correlations, subjecting a sensitive chemical system to this type of noise will result in effects that depend on the noise correlation. However, colored (correlated) noise is rarely considered in studies of noise-induced behavior in general and more specifically in studies of the BZ reaction. Two relatively recent studies should be mentioned here. Noise-induced oscillations and coherence resonance driven by colored noise have been

1. INTRODUCTION The Belousov−Zhabotinsky (BZ) reaction1,2 is undoubtedly the most studied homogeneous chemical oscillator. The basic mechanism is oxidation of organic substrate (commonly malonic acid) by bromate ions in a strongly acidic medium, in the presence of a redox catalyst (typically cerium ion, ferroin, or the ruthenium complex Ru(bpy)32+). The ruthenium complex-catalyzed BZ reaction is strongly photosensitive,3−6 a fact which can be used to control the system. Depending on operation conditions, the BZ system can exhibit a variety of nonlinear dynamic phenomena. Among them, relaxation oscillations in a batch reactor7 or continuous stirred tank reactor (CSTR)8 and waves in spatially extended systems8 are the most studied phenomena. A remarkable aspect of the dynamic behavior of nonlinear chemical systems, such as the BZ reaction, is their response to stochastic fluctuations in a control parameter (e.g., feed flow rate in a CSTR or light irradiation intensity). There are many examples demonstrating that, counterintuitively, noise can induce quasi-periodic behavior and lead to more regularity. Examples include noise-driven waves in a layer of gel,9−11 noise-induced oscillations in the BZ system immobilized in the cation exchange microbeads,12,13 noise-induced oscillations in a CSTR14 and stochastic resonance.14−16 In its classical formulation, stochastic resonance is recognized as the emergence of periodic behavior in a dynamical nonlinear system subject to a subthreshold periodic forcing and random perturbations.17 Another remarkable noise-induced phenomenon is coherence resonance, a phenomenon of the emergence of quasi-periodic behavior driven by noise and an intrinsic dynamics of the © 2013 American Chemical Society

Received: September 10, 2013 Revised: November 25, 2013 Published: November 25, 2013 13999

dx.doi.org/10.1021/jp409033j | J. Phys. Chem. A 2013, 117, 13999−14005

The Journal of Physical Chemistry A

Article

Figure 1. Experimental setup: a diagram representing the experimental system infrastructure (a) and a schematic of the CSTR (b). S-1 and S-2 denote two different feed solutions (as explained in the Experimental Section). TC-1 and TC-2 stand for two thermocouples used to monitor the reaction mixture temperature (the difference between the reading of TC-1 and TC-2 was less than 0.05 °C).

the PC using a PID control program (LabVIEW, National Instruments) and a digital-to-analog converter (NI USB-6008, National Instruments). The CSTR temperature was continuously monitored with two PFA-coated thermocouples (OMEGA Engineering Inc.) using an analog input module (NI 9211, National Instruments), providing the required feedback for the PID controller (LabVIEW). Peristaltic pumps with external control (BT100−2J, Langer Instruments Inc.; NI USB-6008, LabVIEW, National Instruments) were used to supply feed solutions and to remove the excess reaction mixture smoothly (Figure 1) to ensure that there were no significant fluctuations in the reaction mixture volume. The first solution (S-1) contained NaBrO3 and H2SO4 and the other solution (S-2) contained KBr, CH2(COOH)2, and Ru(bpy)3Cl2. The reagents were analytical grade (SigmaAldrich) and deionized water was used for preparation of solutions. The total feed concentrations (calculated as after mixing of S-1 and S-2) were 0.60 M H2SO4, 0.24 M NaBrO3, 0.06 M KBr, 0.18 M CH 2 (COOH) 2 , and 0.48 mM Ru(bpy)3Cl2. The residence time in the CSTR was 30 min in all experiments. The optical setup comprised a collimated LED light source with a nominal wavelength of 455 nm and a fwhm of 18 nm (Thorlabs), which is in the range of the Ru(bpy) 3 Cl 2 absorption.6 To eliminate any light irradiation disturbances, the CSTR was insulated by several layers of aluminum foil and the irradiation window (1 cm2) was protected by a blue filter (center wavelength 450 nm, 80 nm fwhm, Edmund Optics) (Figure 1b). The light intensity was controlled using the LED driver with an external 0−5 V control (Thorlabs) and a digitalto-analog converter (NI USB-6008, LabVIEW, National Instruments). The light transmitted through the reaction mixture was recorded with a silicon photodetector and a power meter (918D-UV-OD3R, 841-P-USB, Newport); this setup was also used for system calibration without reaction mixture. The power meter was connected to the PC using the analog input module (NI USB-6008, LabVIEW, National Instruments) for continuous monitoring. Redox potential of the reaction mixture was measured by the ORP microelectrode (Microelectrodes Inc., US) and continuously recorded using a

demonstrated numerically using the modified (photosensitive) Oregonator model.20 Coherence resonance in the wave initiation induced by colored noise also has been also observed experimentally in an open gel reactor using a photosensitive version of the BZ reaction.11 In this paper, we aimed to report the relation between noise correlation and nonlinear reaction kinetics which is not affected by any transport process (such as diffusion in a gel reactor). With this purpose, we studied experimentally the noise-induced oscillations driven by colored noise in the photosensitive Belousov−Zhabotinsky reaction in a continuous stirred tank reactor. We show that the frequency distribution of the resulting stochastically driven chemical oscillations is strongly affected by noise correlation. For an intermediate range of the noise correlation exponent, there is a significant shift of the noise-induced oscillation frequency toward higher frequency, indicating the occurrence of coherence resonance. Our results provide new experimental evidence of the importance of noise correlation, which can affect the reaction kinetics in a nontrivial way, for noise-induced phenomena.

2. EXPERIMENTAL SECTION The experimental system (Figure 1) comprised a continuous stirred tank reactor (CSTR), a cooling system for temperature control, a flow system for continuous supply of reagents, an optical setup for the purpose of controlling the photosensitive BZ reaction, and a monitoring system that included thermocouples, an oxidation reduction potential (ORP) electrode, and a photodiode. The entire system was controlled and monitored from a personal computer (PC). A jacketed beaker (Sigma-Aldrich) was used for the CSTR compartment which was thermally insulated (Figure 1b). The volume of the reaction mixture was 12 mL and the CSTR was stirred using a magnetic stirrer and a 1.5 cm long Teflon stirring bar at 300 rpm The CSTR temperature was kept at 22 ± 0.1 °C by circulating cooled water (kept at 10 °C in a refrigerator) through the CSTR jacket (Figure 1b) using a peristaltic pump (model BT100−2J, Langer Instruments Inc.). To achieve high precision of the temperature control, the peristaltic pump was equipped with an external 0−5 V control module (Langer Instruments Inc.) and the coolant flow rate was regulated from 14000



Article

Figure 2. Deterministic oscillations: time series (oxidation−reduction potential vs time) of the oscillations observed without light irradiation (a), probability distribution of the deterministic frequency (b), and a plot of the oscillation frequency as a function of the (normalized) irradiation intensity (c). Note the existence of the region of bistability between the oscillatory and nonoscillatory states for 0.36 < I/Imax < 0.46. The maximum power density was Imax = 90 μW/cm3.

accumulated after ∼200 runs (each one for one residence time) repeatedly carried out during the course of experiments was as follows (in mHz): mean = 24.52 ± 0.55, median = 24.31 ± 0.12, mode = 24.28 ± 0.16, STD = 3.31 ± 0.72, MAD = 0.36 ± 0.21. Because virtually identical values were obtained for mean, median, and mode, it can be safely concluded that the deviations from the main frequency were normally distributed. Together with relatively small values of the standard deviation (STD) and median absolute deviation (MAD), this indicates that the system was well-controlled and stable during the experiments. The effect of light irradiation on the deterministic oscillations is shown in Figure 2c (the maximum power density was Imax = 90 μW cm−3); error bars show standard deviation accumulated after 3−5 runs. The upper curve was obtained by a gradual increase in the intensity of light irradiation, while the lower curve was recorded when the light irradiation power was gradually decreased, starting from the maximum value. For I/Imax < 0.35 the oscillation frequency is insensitive to light irradiation. However, for higher irradiation intensity a sharp transition to the lower oscillation frequency occurs, and it is followed by a complete inhibition of the oscillations. The inhibitory effect of light irradiation in the Ru(bpy)32+-catalyzed BZ reaction is well-documented and quite well-understood.4−6 Note that there is a region where oscillatory and nonoscillatory states coexist (Figure 2c). 3.2. Noise-Induced Oscillations. Stochastic fluctuations in the light irradiation power were applied at I/Imax = 0.5 in the vicinity of the bifurcation point (Figure 2c). Random numerical sequences were first generated in a computer and then translated into the LED irradiation intensity using the experimental setup (explained in detail in the Experimental Section). Figure 3 shows typical results obtained with white

benchtop meter (Sper Scientific, US) and the analog input module (NI USB-6008, LabVIEW, National Instruments). To generate numerical sequences of temporally correlated noise, we used an algorithm based on the modified Fourier filtering method.26 Briefly, a sequence of random numbers with power-law correlation is generated by filtering the Fourier components of an uncorrelated sequence of random Gaussiandistributed numbers. The resulting (still Gaussian-distributed) sequence is characterized with a correlation exponent according to the following formula (C(t) is correlation function; ηi are correlated random numbers, t is time, and γ stands for the correlation exponent): C(t ) ≡ ⟨ηη ⟩ ∝ (1 + t 2)−γ /2 i i+t

(1)

3. RESULTS AND DISCUSSION 3.1. Deterministic Oscillations. A representative example of oscillations in the absence of light irradiation is shown in Figure 2a. Figure 2b shows a discrete probability distribution for the oscillation frequency (which we call below “frequency distribution”) accumulated after 36 runs. Each run was carried out for one residence time, which was 30 min in all experiments. We calculated the frequency distribution as the reciprocal of the distribution of the interspike intervals (intervals between maxima) calculated from the recorded time series (such as shown on Figure 2a). In all our experiments, we counted only spikes with an amplitude above the threshold of 100 mV, ignoring the small amplitude peaks ( 20 mHz for γ = 0.65 is larger than that for γ = 2.00. The quasiperiodic appearance of the noise-induced periodic orbit with frequencies close to the deterministic ones can occur when the characteristic frequency of the colored noise (noise correlation

Gaussian-distributed noise, whose amplitude is characterized by a standard deviation (σ) from the mean value. A modulation frequency of the noise amplitude was 1 s, i.e., the LED irradiation intensity value was updated every second. This amplitude modulation frequency was used in all experiments. It was confirmed in a set of separate calibration experiments, which were carried out without reaction mixture, that the fluctuating light intensity recorded by the photodetector (see Experimental Section) indeed tracks the numerical sequence. For weak noise, i.e., with a small amplitude of fluctuations around the mean value of I/Imax = 0.5 (σ = 0.01, Figure 3a), the system does not respond (typical time series of the recorded redox potential are shown on the left panel, and the right panel shows the normalized frequency distribution accumulated after 5 runs). Noise-induced oscillations appear for a sufficiently high amplitude (σ = 0.11, Figure 3b). This observation is not surprising and a similar phenomenon has been reported, although using the CSTR feed flow rate as a control parameter.19 The Belousov−Zhabotinsky reaction is a relaxation limit cycle oscillator. The onset of a stochastically driven periodic orbit can occur when the chemical oscillator is perturbed in the vicinity of the Hopf bifurcation. 14002



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Figure 4. Colored-noise-induced oscillations: time series recorded with colored noise applied in the vicinity of the bifurcation point, i.e., around I/ Imax = 0.5, with different correlation exponents (Gaussian-distributed noise was used in both cases). Left panels show time series of the noise (I/Imax vs t) and of the noise-induced oscillations (E vs t) for γ = 0.65 (a) and γ = 2.00 (b). Right panels show discrete probability distributions of the noiseinduced frequency for γ = 0.65 (a) and γ = 2.00 (b), accumulated after 5 runs (for each γ). Each run was carried out for two residence times, i.e., for 1 h.

Figure 5. Coherence resonance: fraction (ϕ) of the high-frequency oscillations ( f > 20mHz) as a function of the correlation exponent (γ) for different noise amplitudes (σ = 0.11 and 0.25). The polynomial fits (dashed lines) are shown only to guide the eye. The fraction calculated for the oscillations induced by white noise is also shown on the same plot (not related to the γ axis) for comparison (gray circles at the right side of the plots). Note that for a wide range of correlation exponents colored noise can be much more effective than white noise in inducing high-frequency oscillations.

refers to long time dynamics of fluctuations) is close to the frequencies for which the deterministic limit cycle appears. A

similar phenomenon has been studied numerically using the modified Oregonator model.20 14003



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It has been suggested that the appearance of coherence resonance in the excitable system driven by noise (in the absence of any periodic forcing) as a function of noise amplitude could be attributed to an optimal balance between the relaxation oscillator excitation and excursion times, which are affected by noise in a different way.18 Beyond the above, we provide experimental evidence for the existence of coherence resonance driven by noise in a way that depends on the noise correlation. Our findings are consistent with the previously reported numerical work11,20 and the experimental observations of colored-noise-induced waves11 in the photosensitive BZ reaction. The mechanism of coherence resonance in nonlinear chemical systems, perturbed in the vicinity of the bifurcation point of transition between the oscillatory and nonoscillatory state, could be generic and can be related to the overlap between the enhanced frequencies of the colored noise power density spectrum and the natural frequency of the oscillator.27

In all our experiments, bimodal distributions similar (but not identical) to those shown in Figure 4 were observed for σ ≥ 0.06 for both colored and white noise (Figure 3b). These experiments were carried out for different values of noise amplitude (σ = 0.01, 0.06, 0.11, 0.16, 0.20, 0.25) and, for colored noise, of correlation exponent (γ = 0.30, 0.65, 0.10, 0.15, 0.20, 0.25). Noise is a stochastic process and a significant amount of data has to be collected in order to identify noiseinduced effects in a reliable way. We tested all possible combinations (σ, γ) and each combination was repeated 4−6 times, resulting in a total of ∼200 runs (each run for two residence times, i.e., for 1 h); representative examples are provided in the next section. In addition, it was verified that the natural frequency observed for I/Imax = 0 (Figure 2c) did not change through the course of experiments (for statistics, see 3.1 Deterministic Oscillations). 3.3. Coherence Resonance. Figure 5 shows representative examples of the plot of the relative fraction of the deterministic frequency (ϕ) versus the correlation exponent (γ). This fraction is calculated as the ratio of the integrated probability distribution for f > 20 mHz (Figure 2c) to the total integrated probability distribution (probability distribution examples are shown in Figure 4):

4. CONCLUSIONS We provide experimental evidence of the existence of coherence resonance in the photosensitive Ru(bpy)32+-catalyzed Belousov−Zhabotinsky (BZ) reaction in an open system subject to colored noise. In our study, we focused on investigating the effects of temporally correlated fluctuations on the intrinsic kinetics of a nonlinear chemical system in the vicinity of the bifurcation point in the absence of a deterministic limit cycle or any periodic forcing. To eliminate transport effects (such as diffusion), we used a continuous stirred tank reactor (CSTR) configuration with a high-precision system control and light irradiation as a stochastic control parameter. To the best of our knowledge, our paper is the first one to report the occurrence of coherence resonance driven by colored noise in the photosensitive BZ reaction in a CSTR. By analyzing the probability distribution of the frequency of the resulting noise-induced oscillations, we find that the dominant frequencies of the stochastic oscillator are similar to those of the corresponding deterministic oscillator. The noise-induced frequency distributions were strongly affected by noise correlation. To characterize these distributions, we defined a parameter that provides a measure of the ability of noise to induce oscillations at frequencies similar to the deterministic frequency in the absence of any forcing (i.e., natural frequency). We have found that there is an optimal range of the noise correlation exponents, where the noiseinduced frequency is shifted toward the natural frequency of the BZ oscillator. Importantly, in this range of correlation exponents colored noise is significantly more effective than white noise (which has no correlation) in inducing oscillations at natural frequencies. Our results highlight the importance of temporal correlation of random fluctuations in a control parameter affecting a nonlinear chemical system. Depending on the correlation of fluctuations, the reaction kinetics can be affected in a different way. This is expected to hold also for biological reaction systems, such as metabolic oscillators, which are naturally exposed to a variety of intrinsic and extrinsic sources of noise. Our findings also suggest that, in principle, chemical reactions can be directed toward certain behavior using noise correlation as a control parameter. This type of control is expected to be realized most efficiently when a nonlinear chemical system is in the vicinity of the point of transition between different states (e.g., oscillatory and nonoscillatory).

50

φ=

∫20 p(f ) df 50

∫0 p(f ) df

(2)

Results for two noise intensities are shown (σ = 0.11 and 0.25); the polynomial fits are shown only to guide the eye. The fraction calculated for the oscillations induced by white noise is also shown in Figure 5 for comparison (gray circles at the right side of the plots). The fraction ϕ provides a measure of the ability of stochastic fluctuations to induce oscillations at frequencies similar to the deterministic one observed for I/ Imax < 0.35 (Figure 2c). The maxima in ϕ versus γ plots indicate that there is an enhanced coherence between the noise and the intrinsic dynamics of the oscillatory BZ system for a certain range of the correlation exponent. Note that for high values of γ (weak correlation) the magnitude of ϕ is similar to that of white noise (which is uncorrelated by definition). On the other hand, as γ is brought below 2 (stronger correlation) the fraction of highfrequency components increases significantly and it is followed by a sharp drop for γ < 0.5; this is a typical resonant behavior, which is characterized by the existence of an optimum. The chemical mechanism of this resonant phenomenon is expected to be related to both inhibitory regulation (photocatalytic production of bromide) and the autocatalytic process (production of bromous acid). Fluctuations in light irradiation in the vicinity of the bifurcation point (I/Imax = 0.5, Figure 2c) where the oscillations are completely suppressed should result in temporal changes in the rate of bromide production, which is related to irradiation intensity.4−6 This can allow transitory production of bromous acid (when irradiation intensity is low) with a certain probability of inducing the autocatalytic positive feedback, which can result in a burst. Interestingly, these bursts (noise-induced oscillations) are not completely random but have some characteristic probability distribution (Figure 4), though random external forcing (noise) is applied. Even more interestingly, their probability distribution is strongly affected by noise correlation showing the coherence resonance. 14004



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(16) Amemiya, T.; Ohmori, T.; Yamamoto, T.; Yamaguchi, T. Stochastic Resonance under Two-Parameter Modulation in a Chemical Model System. J. Phys. Chem. A 1999, 103, 3451−3454. (17) Benzi, R.; Sutera, A.; Vulpiani, A. The Mechanism of Stochastic Resonance. J. Phys. A: Math. Gen. 1981, 14, L453−L457. (18) Pikovsky, A. S.; Kurths, J. Coherence Resonance in a NoiseDriven Excitable System. Phys. Rev. Lett. 1997, 78, 775−778. (19) Jiang, Y.; Zhong, S.; Xin, H. Experimental Observation of Internal Signal Stochastic Resonance in the Belousov−Zhabotinsky Reaction. J. Phys. Chem. A 2000, 104, 8521−8523. (20) Zhong, S.; Xin, H. Internal Signal Stochastic Resonance in a Modified Flow Oregonator Model Driven by Colored Noise. J. Phys. Chem. A 2000, 104, 297−300. (21) Lia, Q. S.; Zhu, R. Stochastic Resonance with Explicit Internal Signal. J. Chem. Phys. 2001, 115, 6590−6595. (22) Weissman, M. B. 1/f Noise and Other Slow, Nonexponential Kinetics in Condensed Matter. Rev. Mod. Phys. 1988, 60, 537−571. (23) Hausdorff, J. M.; Peng, C.-K. Multiscaled Randomness: A Possible Source of 1/f Noise in Biology. Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 1996, 54, 2154−2157. (24) Hochberg, D.; Lesmes, F.; Morán, F.; Pérez-Mercader, J. LargeScale Emergent Properties of an Autocatalytic Reaction-Diffusion Model Subject to Noise. Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 2003, 68, 066114. (25) Lesmes, F.; Hochberg, D.; Morán, F.; Pérez-Mercader, J. NoiseControlled Self-Replicating Patterns. Phys. Rev. Lett. 2003, 91, 238301. (26) Makse, H. A.; Havlin, S.; Schwartz, M.; Stanley, H. E. Method for Generating Long-Range Correlations for Large Systems. Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 1996, 53, 5445−5449. (27) Simakov, D. S. A.; Pérez-Mercader, J. Noise induced oscillations and coherence resonance in a generic model of the nonisothermal chemical oscillator. Sci. Rep. 2013, 3, 2404.

AUTHOR INFORMATION

Corresponding Author

*Tel: +1-617-496-4778. E-mail: [email protected]. edu. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Michael Rudenko (Harvard University) for help with the implementation of the correlated noise generation procedure and Chris Stokes (The Rowland Institute at Harvard) for help with experimental setup design. This work was funded by Repsol, S.A.. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

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REFERENCES

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Effect of Noise Correlation on Noise-Induced Oscillation Frequency in

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