Volume 1 Paper 16
A Study on Stochastic Resonance for the Process of Active-passive Transition of Iron in Sulfuric Acid
Ding Hongbo, Pan Zhongxiao and Renato Seeber
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JCSE Volume 1 Paper 16
Submitted 5 June 1999, published for public review 29 October 1999
A Study on Stochastic Resonance for the Process of Active-passive
Transition of Iron in Sulfuric Acid
Ding Hongbo1,2 Pan Zhongxiao1 and Renato Seeber2
1 Department of Applied Chemistry. University of Science and
Technology of China,Hefei,230026,China P.R. Email: mailto2('hbding','263.net')
2 Department of Chemistry, University of Modena and Reggio
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.
§2 Keywords active-passive transition, stochastic resonance,
Passivation of iron in sulfuric acid was first observed by
Flade . 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.
§4 The concept of stochastic resonance (SR) was first put
forward in the seminal paper by Benzi and collaborators 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.
§5 In the research area of electrochemical reactions, much
attention has been attracted to nonlinear dynamics, such as oscillation and
surface pattern. Among them, there are also some reports on electrochemical
bistable systems [4,8,9]. In this paper, those experimental results from
Epelboin et al 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.
§6 Theoretical Aspects
1. The description of the bistable model for the actual
Figure 1, which was taken from Epelboin et al, 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.
§7 Fig.1 Current-voltage curve for a 5mm diam. disk in 1M sulfuric acid,
rotating at 750rpm, taken from Epelboin et al
§8 In the figure, the vertical line "bf" corresponds
to Flade potential. 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
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.
§9 2. The simple description of SR theory
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.The
fluctuational forces cause transitions between the neighboring potential wells
with a rate given by the famous Kramers rate  i.e.,
with being the
squared angular frequency of the potential minima at ,
and the squared angular frequency at the top of the
barrier, located at xb, V is the height of the potential barrier
separating the two minima. The noise strength D=kB/T is related to
the temperature T.
§10 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 TK(D)=1/rK between two noise-induced
inter-well transitions is comparable with half the period TO
of the periodic forcing. This yields the time-scale matching condition for
stochastic resonance i.e.,
2 TK(D)= TO(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 TO, the time scale matching can be fulfilled by tuning
the noise level Dmax to the value determined by Eq.(3).
§11 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 .
§12 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
§13 Results and discussion
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.
§14 Fig.2 The optimum output of the system after modulating the
noise amplitude to an optimum value
§15 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.
§16 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).
§17 Fig.3 SNR-H curve for the output of the system
§18 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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