Ding Hongbo^{1,2 }Pan Zhongxiao^{1} and Renato Seeber^{2}

^{1 }Department of Applied Chemistry. University of Science and
Technology of China,Hefei,230026,China P.R. Email:

^{2} Department of Chemistry, University of Modena and Reggio
Emilia,Modena,41100,Italy

The bistable model for the process of active-passive transition of iron in sulfuric acid was given. With this model, stochastic resonance phenomenon was simulated.

**Keywords **active-passive transition, stochastic resonance,
bistability

Passivation of iron in sulfuric acid was first observed by Flade [1]. Since the seventies, it has been shown that polarization curves for iron rotating disk electrodes, obtained with a potentiostat, in 1M sulfuric acid have displayed a hysteresis loop [2~4]. From the nonlinear dynamics point of view, this can be treated as possessing bistable characteristics.

The concept of stochastic resonance (SR) was first put forward in the seminal paper by Benzi and collaborators[5] wherein they address the problem of the periodically recurrent ice ages. The basic ideas under the concept [5,6] are: for a given nonlinear system which possess the characteristic of bistability (or more generally, a form of threshold), when there’s some coherence among the nonlinear condition of the system, the input signal and the input noise, an extra dose of noise can in fact help rather that hinder the performance of the system, a kind of phenomenon of the coherent effect between the stochastic force and the signal. There’s now even a term called signal-to-noise ratio (SNR) to quantify the effect. The concept of stochastic resonance has not only theoretical importance but potential application.

In the research area of electrochemical reactions, much attention has been attracted to nonlinear dynamics, such as oscillation and surface pattern[7]. Among them, there are also some reports on electrochemical bistable systems [4,8,9]. In this paper, those experimental results from Epelboin et al[2] was analyzed, and the bistable model for the process of active-passive transition for iron in sulfuric acid was described. With this model, stochastic resonance phenomenon was simulated. Through this way, the new nonlinear dynamics concept was introduced to electrochemists and this work might help researchers to find SR in this system experimentally.

Figure 1, which was taken from Epelboin et al[2], shows a hysteresis loop in the current-voltage curve. For a positive voltage sweep, the curve a-b-d-e-h will be obtained, while for a negative sweep, the curve h-e-f-b-a is obtained.

**Fig.1 Current-voltage curve for a 5mm diam. disk in 1M sulfuric acid,
rotating at 750rpm, taken from Epelboin et al**

In the figure, the vertical line "bf" corresponds to Flade potential[2]. From the theoratical point of view, both the positive and negative polarization curve should be the same, that is: a-b-f-h or h-f-b-a. Due to the effect of ohmic potential drop, the hysteresis loop was introduced. Therefore, for a certain cross-sectional rotating disk electrode, when the controlling parameter of V(polarization potential) are in the range of "f" and "e" , the polarization curve will be in two possible stable steady states "bd" and "fe"(This corresponds to the active and the passive state of the electrode system respectively). Therefore, from the nonlinear dynamics point of view, it can be viewed as a typical bistable state[10,11]. In certain time, the system will be in one certain state which is determined by the initial condition. With the experimental data, the differential equation describing the behavior of the system can be modeled as follows:

This differential equation possess two stable steady solution [10,11]: I=2 and I=0 (corresponds to active state and passive state respectively). In addition, it has also one unstable steady solution: I=1. However, due to the effect of the inherent stochastic process of electrode reaction [11,12], it’s impossible for the system to be in this state. Therefore, the system will only be in either the active state or passive state.

The mechanism of SR is simple to explain. Consider a heavily
damped particle mass *m* and viscous friction ,
moving in a symmetric double-well potential V(x). The particle is subject to
fluctuational forces that are, for example, induced by coupling to a heat bath.
Such a model is archetypal for investigations in reaction-rate theory[13].The
fluctuational forces cause transitions between the neighboring potential wells
with a rate given by the famous Kramers rate [14] i.e.,

(2)

with being the
squared angular frequency of the potential minima at ,
and the squared angular frequency at the top of the
barrier, located at x_{b}, V is the height of the potential barrier
separating the two minima. The noise strength D=k_{B}/T is related to
the temperature T.

If we apply a weak periodic forcing to the particle, the
double-well potential is tilted asymmetrically up and down, periodically raising
and lowering the potential barrier. Although the periodic forcing is too weak to
let the particle roll periodically from one potential well into the other one,
noise induced hopping between the potential wells can become synchronized with
the weak periodic forcing. This statistical synchronization takes place when the
average waiting time T_{K}(D)=1/r_{K} between two noise-induced
inter-well transitions is comparable with *half* the period T_{O}
of the periodic forcing. This yields the time-scale matching condition for
stochastic resonance i.e.,

2 T_{K}(D)= T_{O}(3)

In short, stochastic resonance in a symmetric double-well
potential manifests itself by a synchronization of activated hopping events
between the potential minima with the weak periodic forcing. For a given period
of the forcing T_{O}, the time scale matching can be fulfilled by tuning
the noise level D_{max} to the value determined by Eq.(3).

In summary, the effect requires three basic ingredients, (i) an energetic activation barrier or, more generally, a form of threshold; (ii) a weak coherent input; (iii) a source of noise that is inherent in the system, or that ads to the input. Given these these features, the response of the system undergoes resonance-like behavior as a function of the noise level; hence the name stochastic resonance. The underlying mechanism is fairly simple and robust. As a consequence, SR has been observed in a large variety of systems, including chemical reactions [15].

SR can be envisioned as a particular problem of signal extraction from background noise. It’s quite natural that a number of authors tried to characterize SR within the formalism of data analysis, most notably by introducing the notion of signal-to noise ratio (SNR). When noise amplitude fulfilled the coherent condition of equation (3), SNR will achieve its maximum value.

According to the theoretical discussion above, the adopted stochastic differential equation with this simulation work is:

Here, A refers to the amplitude of the input sinusoid current, w refers to angular frequency. H(t) is input noise current. In this simulation, after some modification of the parameters, the amplitude of sinusoid current and the angular frequency were chosen as A=0.38 and w=0.002, then there’s only one variable: the amplitude of the noise. Here, the angular frequency of w was chosen with a very low value based on the fact that the relaxation time for the electrode system are relatively very long.[16]

**Fig.2 The optimum output of the system after modulating the
noise amplitude to an optimum value**

According to figure 2, when the noise amplitude was modified to an optimum value, the state of the system undergone periodic hopping with respect to every periodic signal. In this circumstance, the signal gives the system an optimum modulation.

In order to further discuss the relationship between the output of the system and the noise amplitude, the notion of signal-to-noise ratio (SNR) was employed. With the help of FFT technique, the time series output signal was translated into the frequency domain. For simplicity, here, the definition of SNR=S/N was taken. Figure 3 is the result of SNR as a function of the noise variance (analogous to noise amplitude).

**Fig.3 SNR-H curve for the output of the system**

According to figure 3, it can be seen that there’s a peak at the position of the optimum value of noise. On the left hand of the peak, with the increment of noise amplitude, SNR increases; while on the right hand of the peak, with the further increment of noise amplitude, there’s over output of noise, SNR decreases.

The appearances of hysteresis loop on the current-voltage curves for the process of active-passive transition of iron rotating disk electrode in 1M sulfuric acid can be described as a symmetric bistable system. Under this bistable model, the SR phenomenon can be simulated. This work might be help for further experimental verifications.

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