Ding Hongbo^{1}, Pan Zhongxiao^{1}, Lin
Changjian^{2} and Renato Seeber^{3}

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

^{2}Dept. of Materials Sci., Ins. Phys. Chem., State Key Lab of Phy.
Chem. of the solid Surf., Xiamen Univ., Xiamen 361005, P.R. China

^{3}Dipartimento di Chimica, Universita di Modena, 41100, Modena,
Italy.

Author's e-mail:

Stochastic resonance (SR) is a generic term to describe the phenomenon which is manifest in nonlinear systems whereby a – generally feeble – input information (such as a weak signal) can be amplified, and optimized, by the assistance of noise. In this paper, some basic ideas concerning a non-coated passive metal corrosion system are discussed, and the theory of SR is used to interpret the mechanism of the passivity.

Keywords

Stochastic resonance, corrosion, passivity, noise, nonlinear system

The concept of stochastic resonance was originally put
forward in seminal papers by Benzi and collaborators[1~3] wherein they address
the problem of the periodically recurrent ice ages. A statistical analysis of
continental ice volume variations over the last 10^{6} yr shows that
the glaciation sequence has an average periodicity of about 10^{5} yr.
This conclusion is intriguing because the only comparable astronomical time
scale in earth dynamics known so far is the modulation period of its orbital
eccentricity caused by planetary gravitational perturbations. The ensuing
variations of the solar energy influx (or solar constant) on the earth surface
are exceedingly small, about 0.1%. The question climatologists (still) debate
is whether a geodynamical model can be devised, capable of enhancing the
climate sensitivity to such a small external periodic forcing. Stochastic
resonance provides a simple, although not conclusive answer to this question.
In the model of Benzi et al, the global climate is represented by a
double-well potential, where one minimum represents a small temperature
corresponding to a largely ice-covered earth. The small modulation of the
earth’s orbital eccentricity is represented by a weak periodic forcing.
Short-term climate fluctuations, such as the annual fluctuations in solar
radiation, are modelled by Gaussion white noise. If the noise is tuned
suitably, synchronised hopping between the cold and warm climate could
significantly enhance the response of the earth’s climate to the weak
perturbations caused by the earth’s orbital eccentricity, according to
arguments by Benzi et al.

The processes of corrosion are most complicated, showing affluent nonlinear phenomena. One example of this is metal passivity. Because of the appearing of passivity, the corrosion processes of metals are inhibited. So, it’s worth of the attention for corrosionists to the phenomenon of passivity.

The most wide-spread and important theories of passivity are the thin-film theory and the absorption theory. In the thin-film theory, it is assumed that the cause of passivity is the formation of a thin film on the surface of metals because of the coherent reaction between metals and its environment. Lots of experimental evidence has confirmed this kind of assumption. But, it is also argued that, to promote the corrosion-resistance of metals, the film must be thorough, completely covering the surface of metals. On the other hand, some steady metals without film on their surfaces, e.g. Pt, have also the property of passivity. Then, how to explain this? The absorption theory, therefore, offers another explanation for passivity phenomenon. For example, when 6% of the surface area of Pt in HCl solution is covered with oxygen, its potential moves upwards 0.12V, and its corrosion rate is reduced to 10% of its original rate[4].

Unfortunately, there are still questionable points inside the absorption theory. Scientists failed to give a satisfactory explanation concerning why so little absorption can lead to complete passivity of a metal. Modern nonlinear science theory have undergone an extraordinary growth in last few decades, and this has, in turn, been changing the general views of human being about the world. Hence, human being has now got deeper and deeper understanding about the nature of the world. The complexity of corrosion process requires corrosionists to use modern nonlinear science theory to explore the inner rules of corrosion process.

In this paper, after the simple description of some basic ideas of SR theory, the characteristics of corrosion process for a passive system is discussed, and SR theory is tried to the interpretation of the mechanism of the passivity along the absorption 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[6]. The fluctuational forces cause transitions between the neighboring
potential wells with a rate given by the famous Kramers rate[7]i.e.,

(1)

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},
ΔV 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} (2)

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[8]. 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. (2).

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 adds to the input. Given 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[9].

**1. The two stable steady-state characteristic of a passive system**

For a passive metal corrosion system, the typical anodic polarization curve can be shown as figure 1.

Fig.1 The typical anodic polarization curve for a passive metal corrosion system

In figure 1, the Region OA is active dissolution area, the polarization current increase with the improvement of polarization potential; Region AB is temporary area, the polarization current decreases rapidly with the improvement of polarization potential; Region BC is in passive state, the polarization current shows little increase with the improvement of polarization potential, the metal shows little corrosion, almost none; with the further increase of polarization potential, there will possibly again be some increase of polarization current, so, the Region of CD can be called as trans-passive area.

In certain circumstance, the determining factor for the state of a passive metal corrosion system is the cross-point between anodic polarization curve and cathodic polarization curve. The actual corrosion system is an typical nonlinear dynamics system. This can be shown as figure 2[10]

Fig.2 The schematic representation for the corrosion state of a passive metal corrosion system <Comment from RAC - it is rather messy using three separate images for these three diagrams - could you combine them onto one?>

In figure 2 (b), it is shown that, there are three cross-points between anodic polarization curve and cathodic polarization curve. This points out that this nonlinear dynamics system has three state, among them, A and C are steady state, B is unsteady. Because of the nonlinear characteristic of the system and the inner stochastic force (electrochemical noise or, ECN)[11~13], the system will not possibly be in B, so this system is a kind of two stable steady-state nonlinear system, it has just two possible state: when at A, active dissolving (showing as figure2a); when at C, passive state (shown as figure 2c).

**2. The source of noise[11~13]**

From current knowledge, there are possibly three kind of sources which generated noise: the noise generated by the thermal vibrations of the equivalent circuit of underlying electrochemical processes of corrosion; the noise generated by the electrolyte solution; and most importantly, the noise generated by the process of absorption-desorption which will generate noise with large enough energy for SR to take place.

**3. The source of "signal"**

From the discussions above, it is seen that whether the corrosion system is in the state of passivity or not lies in the shape of cathodic polarization curve. While, the process of the reduction of absorbents was controlled by its absorption degree on the surface of metals. When surface absorption increases, polarization current develops quickly with polarization potential moving toward cathodic direction. This stands for a perturbation signal to the two stable steady-state nonlinear system.

Now, we can make some tentative explanations. Although the absorption degree of absorbents on the surface of metal was very small, this weak "signal" can be modulated by the two stable steady-state nonlinear system and the noise, there will be a stochastic resonance response, a kind of mechanism that noise energy moves on to signal energy, to take place. Therefore, the corrosion system changes its state from active to passive.

Due to the nonlinear characteristic of corrosion system, and the inherent stochastic force (ECN) of corrosion process, it is promising to discover more and understand deeply about the inner rules of corrosion process with the further study of the effect of stochastic force to the nonlinear system.

Passive metal corrosion system can be treated as a two stable steady-state nonlinear dynamics system. Due to the process of absorption-desorption of the absorbents, noise with large enough energy can be produced, and the absorption of reduction agents can also change the shape of cathodic polarization curve, this equals to a perturbation signal to the system. Because of the coherent effect among the system, signal and noise, stochastic resonance may take place. Therefore, the formation of the passive state of metal corrosion system may be due to this kind of effect. Experimental verifications are now under considering.

The complexity of the process of metal corrosion tells us that, there are plenty of nolinear process inside the system, and the process can produce much stochastic forces (ECN). It is certain that the study of the effect of stochastic force to the nonlinear system can be a help in the research of metal corrosion.

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