5388
J. Phys. Chem. 1996, 100, 5388-5392
Stochastic Resonance in Chemistry. 3. The Minimal-Bromate Reaction W. Hohmann, J. Mu1 ller, and F. W. Schneider* Institute of Physical Chemistry, UniVersity of Wu¨ rzburg, Marcusstrasse 9-11, D-97070 Wu¨ rzburg, Germany ReceiVed: NoVember 6, 1995; In Final Form: January 10, 1996X
In the present work we report the measurement of stochastic resonance in the minimal-bromate (MB) reaction. The chemical reaction is placed in an excitable focal steady state in a CSTR. When a weak periodic signal together with noise is superimposed on the focus, the Hopf bifurcation is crossed and the system responds with irregular spikes. The periodic signal alone is not sufficient to cross the threshold. An optimal noise level exists at which the periodic signal is optimally detected via the spiking behavior of the reaction. This is done through the signal-to-noise ratio or the interspike histogram that pass through a maximum at stochastic resonance. For numerical simulations we use the Noyes-Field-Thomson (NFT) model which is known to agree with the kinetics of the MB reaction.
Introduction Stochastic resonance (SR) is a nonlinear effect which describes the optimal detection of a weak periodic signal by the action of external noise in systems with a threshold. When the sum of the periodic component and the imposed noise cross the threshold, the system responds with the generation of spikes whose distribution is analyzed.1-4 The most common way for quantifying SR is to examine the signal-to-noise ratio (SNR) from the respective power spectra of the time series. An alternative is the analysis of the time series in the form of an interspike distribution at the frequency of the periodic signal. In both methods an optimal noise level for the detection of the weak periodic signal is obtained. SR occurs in several physical,5-11 biological,12-14 and technological systems.15 In concurrent work SR has been demonstrated in the BelousovZhabotinsky (BZ) reaction (part 1)16 and in the enzymatic Peroxidase-Oxidase (PO) reaction (part 2).17 In the present study (part 3) we investigate the minimal-bromate (MB) reaction in a focal steady state near a subcritical Hopf bifurcation. In the experiments a subthreshold sinusoidal signal together with noise is imposed on the flow rate of reactants into a continuousflow stirred-tank reactor (CSTR). Subsequently the flow rate noise is increased until the Hopf bifurcation is crossed such that single spikes (P1 oscillations) appear. From the analysis of the spikes by Fourier transformation and/or by the interspike histogram the phenomenon of stochastic resonance is shown to occur. SR can be modeled quantitatively by the Noyes, Field, and Thompson (NFT) mechanism (Table 1).18 Experimental Part Reactor. The CSTR (continuous flow stirred tank reactor, Figure 1) is a shortened teflon stoppered spectrophotometric cell of 1.5 mL volume. A step-motor syringe pump which is controlled by a computer via a DA/AD converter delivers three reactant feed streams. The outflow occurs through the middle of the stopper which contains a small dome to allow the passage of any accumulated air bubbles. The experiments are carried out at a temperature of 25.0 °C and a stirring rate of 1300 rpm using an unsymmetric magnetic stirrer. The time series of Ce(IV) absorption (detection wavelength at 350 nm) are measured with a UV/vis spectrophotometer (Beckman UV 5260) * To whom correspondence should be addressed. X Abstract published in AdVance ACS Abstracts, March 1, 1996.
0022-3654/96/20100-5388$12.00/0
Figure 1. Experimental CSTR. The flow rate of the three reactant feed streams is varied by a sinusoidal modulation and by stochastic noise. The light beam for spectrophotometric detection of Ce4+ at 350 nm passes unhindered between inlet tubes and stirrer.
TABLE 1: NFT Model BrO3- + Br- + 2H+ f HBrO2 + HOBr HBrO2 + Br- + H+ f 2HOBr HOBr + Br- + H+ h Br2 + H2O BrO3- + HBrO2 + H+ h 2BrO2• + H2O Ce3+ + BrO2• + H+ h Ce4+ + HBrO2 k1 ) 2.1 s-1 M-3 k2 ) 2.0 × 109 s-1 M-2 k3 ) 8.0 × 109 s-1 M-2 k4 ) 1.0 × 104 s-1 M-2 k5 ) 6.5 × 105 s-1 M-2
k-3 ) 1.1 × 102 s-1 M-1 k-4 ) 2.0 × 107 s-1 M-1 k-5 ) 2.4 × 107 s-1 M-1
and the data are computer-collected at a rate of 0.5 Hz. The experimental time series have a length of 12800 s. Materials. Sulfuric acid, cerous(III) sulfate (Riedel-deHae¨n), potassium bromate, and potassium bromide (Merck) were of analytical grade and used without further purification. The reactor concentrations of the inflow species are given in Table 2. Modulation of the Flow Rate. At the given reactor concentrations the experimental MB system shows P1 oscillations at flow rates between 0.0025 and 0.0035 s-1. The amplitude and period of the oscillations increase with increasing flow rate. The transition from the oscillations to the focus © 1996 American Chemical Society
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TABLE 2: Concentrations (M) of Inflow Species [H+] [BrO3-] [Br-] [Ce3+]
model
experiment
1.5 6.0 × 10-2 3.0 × 10-4 1.5 × 10-4
1.5 1.0 × 10-1 4.0 × 10-4 3.0 × 10-4
occurs at a subcritical Hopf bifurcation (HB2 in the model, Figure 2).19 For the perturbation experiments a weak sinusoidal signal and equally distributed noise are superimposed on the flow rate according to eq 1
kf ) kf0(1 + R sin(ωt) + βR(δ))
(1)
where kf0 (3.81 × 10-3 s-1) is the experimental flow rate at a focal steady state located 8.9% above the Hopf bifurcation, R is the amplitude of the sinusoidal signal of frequency ω, β is the noise amplitude, while R(δ) are equally distributed random numbers between -1 and 1 whose average is zero. δ stands for the pulse length of a single noise event. The signal amplitude has been kept constant at R ) 0.0513. The signal frequency ω ) 0.007 85 rad s-1 (period T ) 800 s) is chosen in such a fashion that the refractory time of the system is about half of the period of the sinusoidal signal. The pulse length of the noise is set to δ ) 10 s for an optimal SR effect. The noise amplitude β is varied between 0 and 0.179 while all other parameters are kept constant. Results. Figure 3a shows the subthreshold sinusoidal variation of the flow rate without added noise (β ) 0, lower part). The MB system responds to this subthreshold signal with oscillations of low amplitude (upper part). The first spikes appear for a noise amplitude of β ) 0.05, where β/R ≈ 1 (not shown). Increasing the noise level to β ) 0.0769 (β/R ) 1.5, Figure 3b, lower part) more spikes appear (Figure 3b, upper part). For β ) 0.103 (β/R ) 2) the spiking behavior becomes quite regular (Figure 3c, upper part). Here the noise level (Figure 3c, lower part) is optimal for stochastic resonance. A further increase of the noise amplitude produces even more than one spike per period, since the flow rate crosses the threshold independently of the weak sinusoidal periodic signal. At a high noise level of β ) 0.179 (β/R ) 3.5, Figure 3d, lower part) the interval times between two spikes are about 400 s (Figure 3d, upper part), which approximately represents the refractory time of the present MB system. When the number of spikes in the time interval between 720 and 880 s is plotted versus the relative noise amplitude β/R (Figure 4) a maximum is obtained at β/R ) 2.0. This maximum denotes the optimal noise level (β ) 0.103) for the detection of the weak sinusoidal flow rate signal of T ) 800 s and R ) 0.0513. The plot of the signal to noise ratio from the Fourier spectra versus the relative noise level β/R (Figure 5) shows a maximum at the same relative noise level. Model Calculations NFT Model. The MB reaction can be modeled with a mechanism suggested by Noyes, Field, and Thompson (NFT).18 The agreement between experiments and simulations are almost quantitative.20-23 The original 10-variable model may be reduced according to Bar-Eli24 (stage e) to a seven-variable model (Table 1). Here we use the reduced version of the NFT model which shows the same dynamic features as the original model and it reproduces the experimental behavior quite well. The reactor concentrations of the inflow species are given in Table 2.
Figure 2. Bifurcation diagram of the NFT model with the parameters given in Tables 1 and 2. The region of period 1 oscillations (P1) is flanked by focal steady states (SS), where the lower and the upper Hopf bifurcations (HB1 and HB2) are supercritical and subcritical, respectively. The width of the subcritical region at HB2 is extremely narrow. This is indicated by the dotted line.
The boundary conditions of a CSTR are modeled by adding flow terms to the differential rate equations:
dyi/dt ) f(y) - kf(yi - yi0);
i ) 1, 2, ..., 7
(2)
The vector y stands for the actual concentrations yi of all components and the function f(y) represents the rate law for the chemical reaction. For the reactor concentrations yi0 (given in Table 2) the model shows oscillations between kf ) 3.542 × 10-3 s-1 and kf ) 4.337 × 10-3 s-1. The corresponding bifurcation diagram is shown in Figure 2. As in the experiment the amplitudes and periods of the oscillations increase with increasing flow rate. At flow rates higher than the Hopf bifurcation (kfHB2 ) 4.337 × 10-3 s-1) the NFT model shows a focal steady state. In our model calculations we chose a flow rate of kf0 ) 4.725 × 10-3 s-1 for the focus which is located 8.9% above the Hopf bifurcation. Modulation of the Flow Rate. To simulate the sinusoidal signal and the equally distributed noise the expression for kf in eq 1 is used, where kf0 is the flow rate of the focal steady state at 4.725 × 10-3 s-1. Similarly to the experiment R is the amplitude, ω the frequency of the sinusoidal signal, β is the noise amplitude, and R(δ) are equally distributed random numbers between -1 and 1 with the pulse length δ as in the experiment (10 s). The amplitude was kept constant at R ) 0.0513 as in the experiment. In analogy to the experiment the frequency is chosen at ω ) 0.007 85 rad s-1 (T ) 800 s). The maximal noise amplitude β is varied from 0 to 0.2565. For the Fourier transformation we used 60 000 points at intervals of 1 s. Results. When the sinusoidal flow rate signal (ω ) 0.007 85 rad s-1, R ) 0.0513, Figure 6a, lower part) is imposed on the focus without noise (β ) 0, β/R ) 0), the system responds with subthreshold oscillations of identical period (T ) 800 s, Figure 6a, upper part). At a noise amplitude of β ) 0.0513 (β/R ) 1) the threshold is crossed and the first irregular spikes appear. Increasing the noise level to β ) 0.0718 (β/R ) 1.4, Figure 6b, lower part) more spikes occur (Figure 6b, upper part). For β ) 0.123 (β/R ) 2.4, Figure 6c, lower part) spiking becomes quite regular (Figure 6c, upper part). Here the maximum of the stochastic resonance effect is obtained in the simulation. A further increase of the noise amplitude to β ) 0.205 (β/R ) 4.0, Figure 6d, lower part) generates more spikes per period. However, the number of spikes in the interval between 720 and 880 s has decreased (Figure 6d, upper part). A maximum in the number of spikes in the time interval between 720 and 880 s is obtained at a noise level of β ) 0.123 (β/R ) 2.4, Figure 7). Likewise, the maximum in the signal-
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Hohmann et al.
Figure 3. (a) Ce(IV) absorption (350 nm, upper part) and flow rate (lower part) versus time at a noise amplitude β ) 0. (b) Ce(IV) absorption (350 nm, upper part) and flow rate (lower part) versus time at β ) 0.0770. (c) Ce(IV) absorption (350 nm, upper part) and flow rate (lower part) versus time at β ) 0.103, which represents the optimal noise level at SR. (d) Ce(IV) absorption (350 nm, upper part) and flow rate (lower part) versus time at a high noise amplitude β ) 0.180.
Figure 4. Normalized number of spikes in the experiments occurring in the interval from 720 to 880 s versus the relative noise amplitude β/R. The maximum is located at a relative noise amplitude of β/R ) 2.0, corresponding to β ) 0.103.
to-noise ratio taken from the individual Fourier spectra at the signal frequency occurs at the same noise level (Figure 8) as expected for SR. Discussion In this paper we demonstrate the phenomenon of stochastic resonance in a further nonlinear chemical reaction, namely the
Figure 5. Experimental signal-to-noise ratio from the Fourier spectra at ω ) 0.0078 rad s-1 (T ) 800 s) versus the relative noise amplitude. The maximum (β/R ) 2.0) is identical with that of Figure 4.
MB system, where the experiments are in almost quantitative agreement with the NFT model. The calculations and the experiments show the existence of an optimal noise level for the detection of a subthreshold periodic signal in the flow rate. The detection of the periodic signal via the spikes has been greatly enhanced by noise. The optimal noise level is shifted to higher values when the noise frequency and the signal frequency increase and when the signal amplitude decreases. A larger distance of the focus from the threshold requires a
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J. Phys. Chem., Vol. 100, No. 13, 1996 5391
Figure 6. (a) Calculated Ce(IV) concentration (upper part) and flow rate (lower part) versus time at a noise amplitude β ) 0. (b) Calculated Ce(IV) concentration (upper part) and flow rate (lower part) versus time at β ) 0.718. (c) Calculated Ce(IV) concentration (upper part) and flow rate (lower part) versus time at β ) 0.123, which represents the optimal noise level at SR. (d) Calculated Ce(IV) concentration (upper part) and flow rate (lower part) versus time at a high noise amplitude β ) 0.205.
Figure 7. Normalized number of spikes in the model calculations occuring in the interval from 720 to 880 s versus the relative noise amplitude. The maximum is located at a relative noise amplitude β ) 0.123.
Figure 8. Signal-to-noise ratio from the Fourier spectra at ω ) 0.0078 rad s-1 (T ) 800 s) versus the relative noise amplitude in the model calculations. The maximum (β/R ) 2.4) is identical with that of Figure 7.
higher optimal noise level. Multiple spikes are obtained when the period of the signal is too large. This reduces the optimal detection of SR. Therefore the optimal noise level depends on several parameters in a complex way. Naturally, the period of the signal must not be smaller than the refractory time of the system, which denotes the shortest possible time between two subsequent spikes. Since the fluctuations on the sinusoidal signal are imposed using random numbers, the calculated time
series differ slightly in each calculation for the same imposed noise amplitude. Therefore, the number of spikes and the signal to noise ratio (Figure 7 and 8) is the average of 10 calculations for each noise amplitude. The MB reaction has been found to be sensitive to the stirring rate between ∼600 and ∼1100 rpm.19,25 The sensitivity to the stirring rate is greatly reduced at stirring rates above 1100 rpm as used in this study, where the effective natural noise is
5392 J. Phys. Chem., Vol. 100, No. 13, 1996 estimated to be less than ∼2% expressed in terms of flow rate variations. SR has also been demonstrated in other nonlinear chemical reactions with Hopf bifurcations displaying the required threshold properties, namely in the Belousov-Zhabotinsky reaction16 (part 1) and in the enzymatic peroxidaseoxidase reactions17 (part 2). The observed SR behavior shows similar properties and dependencies in all three chemical systems studied so far. Further work is in progress. References and Notes (1) Wiesenfeld, K.; Moss, F. Nature 1995, 373, 33. (2) Benzi, R.; Sutera, A.; Vulpiani, A. J. Phys. 1981, 14A, L453. (3) Nicolis, C. Tellus 1982, 34, 1. (4) Benzi, R.; Parisi, G.; Sutera, A.; Vulpiani, A. Tellus 1982, 34, 10. (5) Fauve, S.; Heslot, F. Phys. Lett. 1983, 97A, 5. (6) McNamara, B.; Wiesenfeld, K.; Roy, R. Phys. ReV. Lett. 1988, 60, 2626. (7) Simon, A.; Libchaber, A. Phys. ReV. Lett. 1992, 68, 3375. (8) Gammaitoni, L.; Martinelli, M.; Pardi, L.; Santucci, S. Phys. ReV. Lett. 1991, 67, 1799; J. Statist. Phys. 1993, 70, 425. (9) Dykman, M, I.; Velikovich, A. L.; Golubev, G. P.; Luchinskii, D. G.; Tsuprikov, S. V. Pis‘ma Zh. Eksp. Teor. Fiz. 1991, 53, 182. Dykman, M. I.; Horita, T.; Ross, J. J. Chem. Phys. 1995, 103, 966. (10) Grohs, J.; Apanasevich, S.; Jung, P.; Issler, H.; Burak, D.; Klingshirn, C. Phys. ReV. 1994, 49a, 2199.
Hohmann et al. (11) Spano, M. L.; Wun-Fogle, M.; Ditto, W. L. Phys. ReV. 1992, 46a, 5253. (12) Douglass, J. K.; Wilkens, L.; Pantazelou, E.; Moss, F. Nature 1993, 365, 337. (13) Moss, F.; Douglass, J. K.; Wilkens, L.; Pierson, D.; Pantazelou, E. Ann. N.Y. Acad. Sci. 1993, 706, 26. (14) Chialvo, D. R.; Apkarian, A. V. J. Statist. Phys. 1993, 70, 375. (15) Hibbs, A. D.; Singsaas, A. L.; Jacobs, E. W.; Bulsara, A. R.; Bekkedahl, J. J.; Moss, F. J. Appl. Phys. 1995, 77, 2582. (16) Guderian, A.; Dechert, G.; Zeyer, K. P. W.; Schneider, F. W. part 1, in press. (17) Fo¨rster, A.; Merget, M.; Schneider, F. W. part 2, in press. (18) Noyes, R. M.; Field, R. J.; Thompson, R. C. J. Am. Chem. Soc. 1971, 93, 7315. (19) Hauser, M. J. B.; Lebender, D.; Schneider, F. W. J. Phys. Chem. 1992, 96, 9332. (20) Bar-Eli, K.; Noyes, R. M. J. Phys. Chem. 1976, 81, 1988; J Phys. Chem. 1978, 82, 1352. (21) Geiseler, W.; Bar-Eli, K. J. Phys. Chem. 1981, 85, 908. (22) Bar-Eli, K.; Geiseler, W. J. Phys. Chem. 1981, 85, 3461; J. Phys. Chem. 1983, 87, 3769. (23) Geiseler, W. J. Phys. Chem. 1982, 86, 4394; Ber. Bunsen-Ges. Phys. Chem. 1982, 86, 721. (24) Bar-Eli, K. J. Phys. Chem. 1985, 89, 2855. (25) Schneider, F. W.; Mu¨nster, A. F. J. Phys. Chem. 1991, 95, 2130.
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