Y Xia

S Michelin

O Doaré

Inductive effects on energy-harvesting piezoelectric flag

L'interaction d'un drapeau flexible avec un écoulement donne lieu à une instabilité fluide-structure classique qui conduit à une vibration auto-entretenue, dont l'énergie mécanique peut être convertie en énergie électrique par le biais de matériaux piézoélectriques qui couvrent le drapeau et se déforment avec celui-ci. On étudie la possibilité de récupérer cette énergie, et en particulier l'effet d'un circuit inductif sur le processus de récupération. Une déstabilisation du système est observée par l'ajout d'une inductance. En régime non-linéaire, l'efficacité de récupération augmente significativement lors de l'accrochage entre les fréquences de battement du drapeau et du circuit électrique.

Mots clefs : piezo-electricity ; energy harvesting ; lock-in 1 Introduction

The limited availability of exploitable fossil energy resources urges increasing effort to be invested on the development of renewable energy. The omnipresence and low environmental impact make geophysical flows (e.g. tidal currents, winds, etc.) a promising energy source. Classical phenomena in fluid-solid interaction, such as vortex-induced vibrations and flag flutter, are interesting candidates of flow energy harvesting mechanisms as these interactions may result in self-sustained vibrations of the solid in an uniform flow, thus permanent energy transfer between the fluid flow and the solid.

Piezoelectric materials are known for their capability of generating electrical charge when being deformed. The periodic deformation of the flag therefore leads to a periodic charge transfer between the electrodes of piezoelectric patches positioned on the flag's surface. The feasibility and performance of harvesting the electrical charge by a resistive circuit was recently studied [START_REF] Doaré | Piezoelectric coupling in energy-harvesting fluttering flexible plates : linear stability analysis and conversion efficiency[END_REF][START_REF] Michelin | Energy harvesting efficiency of piezoelectric flags in axial flows[END_REF], and it was shown that both the stability of the system and the fluttering dynamics are influenced by electrical coupling.

Based on these works, we are interested in a slightly more complicated electric coupling : a resistiveinductive (RL) output circuit. Adding inductive elements to a purely passive circuit allows the latter to have its own dynamical properties, in particular a natural frequency, and therefore the possibility of resonance with the flapping flag. Both linear stability and nonlinear dynamics of the coupled fluid-solid-electric system are studied. In this article, some results are presented in attempt to attract readers' attention to several significant influences brought to the coupled system by adding an inductance to the output circuit.

2 Modeling of the coupled system The coupled fluid-structure system considered here is a cantilevered flag placed in an axial flow, which is illustrated in figure 1. Assuming the flag is inextensible with 2-dimensional movement in the (x, y) plane, its dynamics can be described using an Euler-Bernoulli beam model. Lighthill's Large Amplitude Elongated Body Theory (LAEBT) [START_REF] Lighthill | Large-amplitude elongated-body theory of fish locomotion[END_REF] is used to calculate the force resulting from the fluid added mass [START_REF] Eloy | The origin of hysteresis in the flag instability[END_REF] [START_REF] Michelin | Energy harvesting efficiency of piezoelectric flags in axial flows[END_REF]. The effect of crossflow separation is modeled by taking into account a drag in the fluid forcing expression, as proposed in [START_REF] Eloy | The origin of hysteresis in the flag instability[END_REF]. As shown in figure 2, the surface of the flag is entirely covered by pairs of piezoelectric patches of length d in the flow direction. The resulting three-layer sandwich plate is of lineic mass µ and bending rigidity B. Each pair is shunted with a conductance G and an inductance L, which are connected in parallel. Q and V are respectively electrical charge and voltage resulting from the piezoelectric effects.

Following [START_REF] Doaré | Piezoelectric coupling in energy-harvesting fluttering flexible plates : linear stability analysis and conversion efficiency[END_REF][7], we assume that piezoelectric pairs are continuously distributed (s i + = s i+1 -) and that they are small in front of the plate's length (d L). Following these assumptions, it is convenient to introduce lineic quantities defined by : g = G/d, l = Ld, q = Q/d, and v = V . By using lineic quantities, the electrical charge displacement across a piezoelectric pair, and the added bending moment resulting from the inverse piezoelectric effect, are given as

q = χθ + cv, (1a)
M piezo = -χv, (1b)
where χ is a mechanical/piezoelectric conversion coefficient and c is the lineic intrinsic capacity of a piezoelectric pair. The piezoelectric pair can thereby be considered as a capacitor and a current generator connected in parallel (see [START_REF] Bisegna | Optimized electric networks for vibration damping of piezoactuated beams[END_REF] for more details about the modeling of piezoelectric coupling). Thus together with the external RL circuit, the whole system is equivalent to a parallel RLC circuit connected to a current generator, and the charge conservation law of such a circuit is given by :

v + gl v + lq = 0. (2)
In equations (1a, 1b, 2), ˙and denote derivatives with respect to s and t, respectively.

In the following, the problem is nondimensionalized using c s = B/L 2 µ, the elastic wave velocity, as characteristic velocity. L, L/c s , ρHL 2 , c s µ/c, c s √ µc are respectively used as characteristic length, time, mass, voltage and lineic charge. As a result, 6 non-dimensional parameters characterize the coupled system :

M * = ρ f HL µ , U * = U ∞ c s , H * = H L , α = χ √ Bc , β = cc s gL , ω 0 = L c s √ lc .
M * is a relative measure of the fluid and solid inertia. U * is the reduced velocity representing the flow velocity, H * the aspect ratio, α the piezoelectric coupling coefficient, and β and ω 0 represent respectively the conductance and the inductance. All variables are consequently taken non-dimensional hereafter, albeit denoted with the same letters as the dimensional ones.

Linear stability and threshold velocity

The flag in axial flow becomes unstable and starts to flutter when the reduced velocity exceeds a critical value U * crit . The dependence of this value on the mass ratio M * and added conductance β, in case of resistive coupling, is observed in previous studies [START_REF] Eloy | Flutter of a rectangular plate[END_REF][START_REF] Doaré | Piezoelectric coupling in energy-harvesting fluttering flexible plates : linear stability analysis and conversion efficiency[END_REF][START_REF] Michelin | Energy harvesting efficiency of piezoelectric flags in axial flows[END_REF]. In this study, we are interested in the possible dependence of the U * crit on inductance, measured by ω 0 . In order to study the influence of inductance, we consider linear configuration with a purely LC circuit by removing the second term in equation [START_REF] Bisegna | Optimized electric networks for vibration damping of piezoactuated beams[END_REF] By considering small vertical displacement, i.e y 1, we obtain the following linear equations for the coupled system :

1 + π 4 M * H * ÿ + π 2 M * H * U * ẏ + π 4 M * H * U * 2 y + y -αv = 0, (3a) v
+ 1 β v + ω 2 0 v + αÿ = 0 (3b)
with the linearized boundary conditions given by at x = 0 :

y = ẏ = 0, (4a) at x = 1 : y -αv = y -αv = 0 (4b)
We use a classical Galerkin decomposition method to study equations (3a, 3b) by assuming that the vertical displacement y is a linear combination of free vibration eigenmodes φ i (x) of a cantilevered beam without coupling, while the voltage v is decomposed as a linear combination of φ i (x), since the latter verify the boundary conditions prescribed by (4a, 4b). The resulting decomposed equations are then projected onto φ i (x) and φ i (x). Equations (3a-4b) are then recast as an eigenvalue problem and the coupled system is unstable if one of its eigenfrequencies has a positive imaginary part.

Threshold velocity U * crit is computed as a function of ω 0 for different values of M * and shown in figure 3. We observe decreasing U * crit with increasing M * . On the other hand, the influence of inductance is observed when a given M * is considered. In the limit ω 0 = 0, the inductance is infinitely large so that the output circuit is equivalent to an open loop. Consequently, no charge transfer is possible within the circuit, and voltage in piezoelectric pairs is sufficiently high that an additional rigidity is induced to raise U * crit for all M * . In the limit ω 0 → ∞, which corresponds to a very small inductance, each piezoelectric patch is directly shunted with the other one in the pair. Therefore, as explained in [START_REF] Doaré | Piezoelectric coupling in energy-harvesting fluttering flexible plates : linear stability analysis and conversion efficiency[END_REF], voltage of each pair is nearly zero as a result of instantaneous charge transfer within the circuit. This case is equivalent to α = 0, as no additional rigidity is induced. For intermediate ω 0 , an important decrease of U * crit is observed for all M * . The lowest U * crit is obtained with ω 0 ∼ 0.1 for all considered M * . These results suggest some potential benefits of an output circuit with inductance, as it decreases the instability threshold, thus allowing energy harvesting by such mechanism with slow fluid flow or lighter fluid (e.g. using air instead of water)

Nonlinear dynamics

As predicted by linear stability theory, an unstable coupled system experiences, at initial stage, an exponential growth in its amplitude, which eventually saturates because of nonlinear effects. A complete study of the system's nonlinear dynamics is therefore necessary to evaluate the harvesting efficiency η, which is defined as the ratio between the power dissipated by the conductance g, and the kinetic energy flux of the fluid passing through the rectangular outlined by the width and peak-to-peak amplitude of the flag's trailing edge.

Non-dimensional equations describing the system's nonlinear dynamics are written as

ẍ = M * (T τ ) -(θ n) + α v n + f fluid , ( 5a
) βv + v + βω 2 0 v + αβ θ = 0, ( 5b
)
f fluid = -m a M * H * un -(u n u τ ) + 1 2 u 2 n θ n - 1 2 M * C d |u n |u n n, (5c)
with the following boundary conditions

at x = 0 : x = ẋ = 0, (6a) at x = 1 : T = θ -αv = θ -αv = 0. ( 6b
)
In equations (5c), the drag coefficient C d is 1.8 for a rectangular plate in a transverse flow. u n and u τ are respectively the normal and tangential components of the non-dimensional relative velocity of the flag to the flow. They are given, with non-dimensional variables, by :

u n = -ẋy + ẏx + U * y , (7a)
u τ = ẋx + ẏy -U * x . (7b)
Equations (5a-5c) are solved using a method based on an explicit description of the fluid-solid coupling, introduced in [START_REF] Alben | Simulating the dynamics of flexible bodies and vortex sheets[END_REF], the only differences being that the fluid model is replaced in the present work by LAEBT, and that electric coupling is included. Frequency deviation is observed while studying nonlinear dynamics, as shown in figures 4a and 4b. It can be noticed that for both M * = 1 and M * = 10, when inductance is added (α = 0), the flapping frequency no longer follows the frequency-velocity relation obtained without electric coupling (α = 0). The flapping movement is actually dictated by the output circuit, as the flapping frequency is very close to the natural frequency of the circuit, ω 0 . A frequency lock-in phenomenon, as reported in some works on VIV (Vortex-Induced-Vibration) [START_REF] Williamson | Vortex-induced vibrations[END_REF] is observed here, in which the frequency of one oscillator (the flag) is dictated by the frequency of the other (the circuit). For M * = 1, lock-in with ω 0 = 17 occurs with a certain range of U * and disappears at higher U * . For M * = 10, lock-in with ω 0 = 60 is observed for all U * between 9.5 and 20. Moreover, lock-in seems not to be influenced by typical mode-switching events, which are indicated, for the uncoupled case, in figure 4b by transitions of flapping frequency between different linear modes.

With figures 4c and 4d, it is evident that the harvesting efficiency is strongly related to frequency lockin. With both M * = 1 and M * = 10, harvesting efficiency increases significantly while the flapping frequency is deviated to ω 0 . In particular, the maximal efficiency is around 0.02 with M * = 1, and reaches almost 0.4 with M * = 10. These results suggest a much higher energy harvesting performance compared with the case of a purely resistive output circuit [START_REF] Michelin | Energy harvesting efficiency of piezoelectric flags in axial flows[END_REF].

Conclusion and perspectives

The present work studies the influence of inductive elements on the piezoelectric energy-harvesting flag. It was shown that the inductance can significantly destabilize the system, leading to an increase in the operation range of the device. The results on nonlinear dynamics showed the important role played by the RL circuit. We observed frequency lock-in occurred between the flapping flag and the circuit within a certain range of fluid velocity. The energy-harvesting efficiency is significantly increased while frequency lock-in occurs.

The above results indicate some potential benefits of the coupling of the energy-harvesting flag with complex electric systems. The critical role of the natural frequency of a circuit was identified. In the first place, a circuit possessing its own natural frequency introduces the possibility of resonance and frequency lock-in with the piezoelectric flag. As a consequence, energy-harvesting performance would be improved compared with the coupling with a purely resistive circuit. In the second place, the results of nonlinear study also suggested that, by carefully choosing a circuit's natural frequency, one might be able to control the flapping frequency regardless of the varying flow conditions. Although the simplest resonant circuit is used here, the results imply that by using more complex circuits with similar types of properties, substantial improvement in harvesting efficiency and controllability of the device can be expected.

Figure 1 -

 1 Figure 1 -(a)Flapping piezoelectric flag in a uniform axial flow and (b) its 2 dimensional view. Piezoelectric patches are represented by gray strips in (a) and segments in (b)

Figure 2 -

 2 Figure 2 -(a) Flag shunted with piezoelectric pairs. The junction between two adjacent pairs are detailed in (b)

Figure 3 -

 3 Figure 3 -Evolution of threshold velocity with ω 0 at α = 0.5 and H * = 0.5, with LC coupling

  Evolution of flapping frequency at M * = 1 and ω 0 = 17, for α = 0 (black unfilled circles), α = 0.2 and β = 1 (gray filled circles) Evolution of flapping frequency at M * = 10 and ω 0 = 60, for α = 0 (black unfilled circles), α = 0.2 and β = 1 (gray filled circles) Evolution of harvesting efficiency at M * = 1, α = 0.2 and β = 1 Evolution of harvesting efficiency at M * = 10, α = 0.2 and β = 1

Figure 4 -

 4 Figure 4 -Evolution of flapping frequency (a)(b) and harvesting efficiency (c)(d) with U * . In both (a) and (b), ω 0 is represented by black solid lines. Dashed curves show linear modes frequencies evolution with U *


Miguel Piñeirua

Olivier Doaré

Sébastien Michelin

Weakly non-linear model of a piezoelectric plate in an axial flow

Keywords: Flutter instability, Energy harvesting, Piezoelectricity

We present a weakly non-linear model of a piezoelctric flag immersed in an axial flow. Numerical simulations based on Galerkin projections are carried out in order to evaluate the coupling between the flapping piezo-electric flag and a purely resistive circuit intended to simulate the electric energy harvesting circuit. The numerical simulations allow to point out the relevant physical parameters governing the apparition of the flapping instability, power conversion and system's harvesting efficiency.

Résumé :

Nous présentons un modèle faiblement non-linéaire d'une plaque piézoélectrique dans un écoulement axial. À l'aide d'une simulation numérique basée sur des projections de Galerkin, nous évaluons le comportement du système en flottement couplé à un circuit résistif simple. À partir de la simulation nous mettons en évidence les paramètres importants dans l'apparition de l'instabilité de flottement, la puissance récupérée ainsi que l'efficacité de récupération d'énergie du système.

Introduction

Recent efforts in the developement of new energy harvesting methods have been focused in study of fluid-solid coupled systems that allow the conversion of the kinetic energy of flows into electrical energy [START_REF] Westwood | Ocean power wave and tidal energy review[END_REF]. Appart from the classical technologies such as wind and water turbines, other techniques involving the deformation of elastic structures under the action of fluid forces have attracted the attention of several research groups. The present study is focused on the energy harvesting from the fluttering of a piezoelectric flag in an axial flow [START_REF] Doaré | Piezoelectric coupling in energy-harvesting fluttering flexible plates : linear stability analysis and conversion efficiency[END_REF].

A flexible plate in an axial flow can become unstable and develop self sustained flapping if the flow velocity exceeds a critical value. If the flapping plate is made of a piezoeletric material, the plate deformation can be transformed into electrical energy. Previous works [START_REF] Doaré | Piezoelectric coupling in energy-harvesting fluttering flexible plates : linear stability analysis and conversion efficiency[END_REF][START_REF] Michelin | Energy harvesting efficiency of piezelectric flags in axial flows[END_REF] have studied the fully coupled dynamics of a flexible plate covered by an infinite number of piezoelectric patches conected to simple resistive circuits.

In this work, the case of a finite number of piezoeletric patches is adressed and the dynamics of the system is modeled through a weakly non-linear model [START_REF] Eloy | The origin of hysteresis in the flag instability[END_REF]. The final objective of this work is to model the system as a nonlinear electrical impedance that can be implemented in electrical simulation software, in order to test energy harvesting circuits in a more efficient manner.

Physical Model

The studied system consists of an elastic flexible plate of length L and width H immersed in an axial flow of both uniform density ρ and velocity U ∞ . The plate is inextensible and clamped at its leading edge. The local orientation of the plate is noted θ(S, T ), where S is the curvilinear coordinate along thickness h p are fixed on both sides of the plate so that l p = L/N p . The negative electrodes of the piezo-eletric patches are shunted across the plate while their positive electrodes are connected to an external circuit. The surface density of the charge in each piezo pair is given by [START_REF] Doaré | Piezoelectric coupling in energy-harvesting fluttering flexible plates : linear stability analysis and conversion efficiency[END_REF] :

Q i = χ l p [θ] x + i x - i + CV i , (1)
where V i is the voltage between the positive electrodes of the i th piezo pair whose left and right edges are positioned at x - i and x + i respectively. C is the equivalent capacity the piezo pair and χ is the mechanical/piezoelectrical conversion factor of the piezoelectric material. Considering that the electrodes are connected to an external resistive circuit, Ohm's law leads to :

∂Q i /∂T + GV i = 0, ( 2
)
where G is the lineic conductivity of the circuit.

The voltage V i between both electrodes also generates an internal torque in the piezoeletric patches and thus on the flexible plate. The total internal torque of the bent plate will then be given by M = B∂θ/∂S -χ i V i F i , where B is the flexural rigidity of the plate and piezoeletric patches assembly (see [START_REF] Doaré | Piezoelectric coupling in energy-harvesting fluttering flexible plates : linear stability analysis and conversion efficiency[END_REF]) and F i is the polarization function of the i th patch. In the present approach,

F i = H s (x -x - i ) -H s (x -x + i
), with H s the Heaviside step function. The use of the conservation of momentum then leads to [START_REF] Michelin | Energy harvesting efficiency of piezelectric flags in axial flows[END_REF] :

µ ∂ 2 X ∂T 2 = ∂ ∂S F t τ - ∂ ∂S B ∂θ ∂S -χ i V i F i n -P n, (3)
where X(S, T ) µ is the local plate position and µ is the masss per unit area. F t is the internal tension in the plate that enforces inextensibility and P is the pressure exerted by the surrounding fluid over the plate. Following previous works on flag flapping modeling [START_REF] Eloy | The origin of hysteresis in the flag instability[END_REF], the pressure P is decomposed into a resistive part P resis that models the drag generated by the plate as it moves inside the flow, and a reactive part P reac which is due to potential flow effects,

P = P reac + P resis = m a ρH ẇ -(wu) + 1 2 w 2 κ + 1 2 ρC d |w|w, (4)
where m a is an added mass coefficient of the transverse section that equals π/4 for a rectangular plate. w and u are respectively the normal and longitudinal components of the plate's velocity relative to the fluid flow, such that Ẋ -U ∞ = uτ + wn.

Non-dimensional equations

Using L, L/U ∞ , U ∞ µ/c et U ∞ √ µc as characteristic length, time, voltage and charge respectively, equations ( 1), ( 2) and ( 3) take the non-dimensional form :

∂x ∂t = ∂f t ∂s τ - 1 U * 2 ∂ 3 θ ∂s 3 n + α U * i v i δ (x - i ) -δ (x + i ) n -M f p res -M f H * m a p reac , (5)
q i = v i + α U * γ [θ] x + i x - i , (6)
β ∂q i ∂t + v i = 0, (7)
with the following non-dimmensional parameters :

U * = LU ∞ µ/B, α = χ √ BC , M f = ρL µ , H * = H L , γ = l p L , β = U ∞ C
LG .

The associated clamped-free boundary conditions are :

at s = 0, y = 0, θ = 0 (8) at s = 1, f t = ∂θ ∂s -αU * v = ∂ 2 θ ∂s 2 -αU * i v i (δ(x - i ) -δ(x + i )) = 0. ( 9
)
Following [START_REF] Eloy | The origin of hysteresis in the flag instability[END_REF], equation ( 5) is projected onto x and y in order to obtain two equations for x(s, t) and y(s, t) respectively. The x projection is to eliminate the tension term F t from the y projection. Finally, x and its derivatives are eliminated using the inextensibility condition. Keeping terms up to O(y 3 ) one obtains a weakly non-linear equation for y(s, t) :

L(y) + f m (y) + 1 U * 2 f B (y) - α U * f χ (y, v) + M f f res (y) + M f H * m a f reac (y) = 0, (10)
with

L(y) = ÿ + 1 U * 2 y (4) - α U * i v i (δ (x - i ) -δ (x + i )) + M f H * (y + 2 ẏ + ÿ), f m (y) = y s 0 ( ẏ 2 + y ÿ )ds -y 1 s s 0 ( ẏ 2 + y ÿ )dsds , f B (y) = 4y y y + y 2 y (4) + y 3 , f χ (y, q) = 1 2 y 2 i v i (δ (x - i ) -δ (x + i )) + y y i v i (δ(x - i ) -δ(x + i )), f res = 1 2 C D |y + ẏ|(y + ẏ), f reac = - 1 2 y y 2 + ẏ y 2 -3y y ẏ -2 ẏ y ẏ - 1 2 y ẏ2 + y s 0 ( ẏ 2 + y ÿ )ds + 2(y + ẏ ) s 0 ẏ y ds -y 1 s y (y + 2 ẏ + ÿ)ds.
A classical Galerkin decomposition is then employed : the vertical displacement y is expanded on clamped-free beam eigenmodes φ p (s),

y(s, t) = ∞ p=1 X p (t)φ p (s). (11)
Next, equation ( 10) is projected on the same set of eigenmodes. After truncation to N linear modes, the resulting coupled system of equations is integrated numerically using an explicit step-adaptive fourth order Runge-Kutta method.

Results

In this section results of the present model are first compared to that of reference [START_REF] Michelin | Energy harvesting efficiency of piezelectric flags in axial flows[END_REF]. Both models differ in two aspects : (1) a discrete number of finite length piezoelectric patches is considered here whereas a continuous distribution of infinitesimal electrodes is used in reference [START_REF] Michelin | Energy harvesting efficiency of piezelectric flags in axial flows[END_REF], (2) a third order weakly nonlinear approximation is used in the present paper when a fully nonlinear model is developed in reference [START_REF] Michelin | Energy harvesting efficiency of piezelectric flags in axial flows[END_REF]. Both models should give similar results when a large number of discrete piezoelectric patches are used and when the flapping amplitude is small.

Next, we present the impact of the number of piezoelectric patches on the linear stability threshold, the limit-cycle flapping amplitude and the harvested energy.

Stability Analysis

In figure 2-a we present the evolution of the stability threshold as a function of M f for different values of α when a large number of piezoelectric patches is used (N p = 100). For small values of M f , the damping induced by the electro-mechanical coupling increases the critical velocity U * s . This effect is more important as the coupling factor α is increased. However, for high values of M f , the electromechanical coupling has a destabilizing effect, which was interpreted in [START_REF] Doaré | Piezoelectric coupling in energy-harvesting fluttering flexible plates : linear stability analysis and conversion efficiency[END_REF] as a consequence of the existence of negative energy waves in the non-dissipative flag [START_REF] Benjamin | The threefold classification of unstable disturbances in flexible surfaces bounding inviscide flows[END_REF]. Our discrete model also allows to study the influence of the number of piezoelectric patches. As seen on figure 2-b, the stability threshold tends to a constant value when N p > 10. For smaller values of M f the threshold differs.

Flapping regime non-linear dynamics

If the critical velocity of the stability threshold is over passed, a small perturbation of the system is enough to jump from the rest state to a flapping regime. From the linear analysis of the system the flapping amplitude should grow exponentially, however, nonlinear effects cause the saturation of the flapping amplitude leading to a permanent regime. In figure 3 we present the evolution of the flapping amplitude as a function of the non dimensional velocity U * for a flag with 100 piezoelectric patches and fixed values M f = 0.6 and α = 0.5. In the graph our results are compared to a continuous fully non linear model [START_REF] Michelin | Energy harvesting efficiency of piezelectric flags in axial flows[END_REF]. Near threshold both models show good agreement but for higher values of U * the weakly nonlinear model estimates flapping amplitudes that are significantly higher than those obtained by fully nonlinear model. The same differences between the two models can be observed for the energy harvesting efficiency (figure 3-b). The piezoelectric coupling factor also plays an important role in determining the flapping amplitude in the permanent regime. Because the flag tends to become stiffer as α is increased and because more and more energy is dissipated in the resistance, one should expect a reduction of the flapping amplitude as a function of α. This is confirmed by the results shown in figure 4-a. For flags with more than three piezoelectric patches and for fixed values of U * = 15 and M f = 0.5 we observe that the flapping amplitude continuously decreases as the parametes α is increased untill it eventually reaches a critical value for which the flag stops. This behavior is not observed in the cases N p = 1 and 2, for which the amplitude increases with α. For flags with more than three patches the harvesting efficiency exhibits a maximum in the range 0 < α < 1 (figure 4-b).

For this particular set of parameters the optimal value of α is α ∼ 0.55. However, in practice α is restricted by the physical properties of the piezoeletric material so it is possible that this optimal value could not be reached. 1 . For this reason we must also treat prudently the efficiency of flags with one and two patches. Even if they can go have efficiencies three times greater than those of the flags with several patches, their optimal α value will probably be impossible to reach in real applications.

The evolution of the trailing edge amplitude and of the power and energy harvesting efficiency as a function of the non-dimensional resistance β are shown in figure 5. For the case β 1 the output electric circuit behaves like a short circuit thus there is no effect of the electric part on the flag dynamics (an equivalent case is seen when α = 0). In the opposite case, β 1, the output circuit behaves like an open circuit, so with no current flowing through the resistance no power is dissipated. Between these two extreme cases we observe a maximum in the harvesting efficiency and dissipated power for β ∼ 0.4. Contrary to what is observed when α is varied, the dependence of the harvesting efficiency on β has small variations when using 100 or 2 piezoelectric patches. Results obtained with 100 patches are in agreement with those presented in reference [START_REF] Michelin | Energy harvesting efficiency of piezelectric flags in axial flows[END_REF], where a continuous distribution of piezoelectric patches is modeled with a fully nonlinear model.

Conclusions

We have developed a weakly non linear model that describes the mechanical-electrical interactions between a fluttering flag with a discrete distribution of piezoeletric patches and a resistive circuit.

Results obtained for the fluttering amplitude and harvested energy efficiency as a function of the various system's parameters are compared to previous results obtained with a fully nonlinear model that considers a continuous distribution of piezoelectric patches. For a sufficiently large number of piezo electric patches our results for the flag dynamics near the instability threshold are in good agreement with the continuous fully non linear model. However, far from threshold discrepancies between the two models are observed, mainly due to the differences of the nonlinearity order considered in each model.

In the case when a small number of piezoelectric pacthes is considered, the system has a very different behavior. This effect is still to be understood. Further work shall be focused on the integration of our model as a nonlinear impedance in the developpement of electric circuit simulation software.

Figure 1 -

 1 Figure 1 -(a) Two dimensional flapping of a flexible plate covered by pires of piezoeletric patches. (b) Flexible plate flapping in uniform axial flow.

Figure 2 -

 2 Figure 2 -(Left) Evolution of the stability threshold as a function of M f for different values of the coupling factor α. (Right) Evolution of the stability threshold for three different values of M f and fixed α = 0.5 as a function of the number of piezoelectric patches. U sref stands for the stability threshold speed with N p = 100. For both graphs r = cst = 1.

3 Figure 3 -

 33 Figure 3 -Comparison of the the discrete weakly non-linear model with N p = 100 to the continuous fully non-linear model developed by Michelin et al.. (Left) : non-dimensional fluttering amplitude of the trailing edge A e as a function of U * . (Right) : efficiency of the energy harvesting as a function of U * . Efficency is defined η = P/(ρU 3 ∞ AH), where P = Np i GV 2 i .

3 Figure 4 -

 34 Figure 4 -Evolution of (Left), the trailing edge amplitude and (Right), the energy harvesting efficiency as function of the coupling coefficient α, for U * = 15, r = 1 and M f = 0.5.

3 Figure 5 -

 35 Figure 5 -Evolution of (left), the trailing edge amplitude and the power and (right), the energy harvesting efficiency as a function of the non-dimmensional parameter β, for U * = 15, α = 0.5 and M f = 1.

In a typical application involving piezoeletric materials such as PZT, α has been estimated in reference[START_REF] Doaré | Piezoelectric coupling in energy-harvesting fluttering flexible plates : linear stability analysis and conversion efficiency[END_REF] to be around 0.3.

Acknowledgments

This work was funded by the Laboratoire d'Excellence LaSIPS ("PIEZOFLAG" project).

Références


A Fakhry

H Cachet

C Debiemme-Chouvy
email: catherine.debiemme-chouvy@upmc.fr

Electrochemical Characterisations of Ultra Thin Overoxidized Polypyrrole Films Obtained by One-step Electrosynthesis

Keywords: overoxidized polypyrrole, EIS, one-step electrochemical synthesis, anion rejection

Ultra thin (10-20 nm) and pin-hole free overoxidized polypyrrole films were directly electrosynthezised under potentiostatic conditions in the presence of weak-acid anions, specifically monohydrogenophosphate. Various overoxidized polypyrrole membranes were fabricated in these conditions at different potentials in the range of 0.70 to 0.90 V/SCE. Their non-conductive property and blocking character towards anions were verified by cyclic voltammetry and electrochemical impedance spectroscopy in the presence of the ferricyanide/ferrocyanide redox couple. The results show that these films are definitively ultra thin pine-hole free and non-conductive membrane. For comparison purposes, classical polypyrrole films were also prepared in the presence of perchlorate ions, overoxidized in KOH solution, and finally electrochemically characterized.

INTRODUCTION

Polypyrrole (PPy) is one of the most widely investigated conducting organic polymer notably due to its high conductivity under its doped oxidized form. However, some analytical applications concern overoxidized polypyrrole (OPPy) films. [1,2] The main characteristics of overoxidized polypyrrole are the following: (i) the polymer is de-doped i.e. the anions which dope the polymer under its oxidized conducting form are expelled from the polymer, (ii) some carbonyl groups are present in β-position of some pyrrole rings, therefore the conjugation is disrupted and the polymer film is no more conductive. [1,3,4,5,6] It has been shown that these OPPy films could be used as a rejection membrane or as a permselective coating against anions. [3,4,7,8,9,10,11] Actually, they are as effective as Nafion membrane for the rejection of the anions. [12,13,14] Generally the fabrication of overoxidized polypyrrole membrane is performed in two steps, the first one being the electrogeneration of a PPy film and the second one is the overoxidation of the film in an indifferent electrolyte, often sodium hydroxide. [6,8,15,16,17,18] The overoxidation of a conductive PPy film (doped film) can take place when it is anodically polarized in an indifferent aqueous electrolyte i.e. when water oxidation occurs leading to the generation of hydroxyl radicals. [19] Sometimes this second step can last several hours, until the anodic current is very low and constant. Recently, Debiemme-Chouvy has shown that the presence of solely anions of a weak acid (weak-acid anions) in the pyrrole solution, such as monohydrogenophosphate or carbonate, allows one to rapidly and directly deposit an ultra thin film of OPPy on the metal electrode. [20] This kind of films could be obtained under potentiodynamic or potentiostatic conditions. Actually, this process has been explained considering the proton release during the pyrrole oxidation: (1) 3 γ stands for the doping level of the polymer. Note that this equation shows that the presence of anions (A -) in the pyrrole (Py) solution is essential for polymer generation.

In the presence of weak-acid anions, the released protons are captured by these anions implying two phenomena: a limitation of the decrease of the interfacial pH and a huge decrease of the interfacial concentration of anions. Thus, since none or only a few anions are present at the electrode/solution interface, reaction (1) cannot take place and as the interfacial pH is high water oxidation can occur (E = 1.0 -0.06 pH (V vs. SCE), at 25°C). Water oxidation leads to the production of hydroxyl radicals which overoxidize the PPy film formed at the very beginning of the electrochemical process.

.

.. A n PPy O OH x ] A n , [(Py) - - ) n ( n + γ + → + γ • + γ (2)
In the present work, the electrical characteristics of OPPy films synthesized in the presence of only monohydrogenophosphate (HPO 4 2-) as anions are described. They have been determined by cyclic voltammetry (CV) and by electrochemical impedance spectroscopy (EIS) in the presence of a monoelectronic rapid redox couple (Fe(CN) 6

3-/4-

). Moreover, 'classical' PPy films were also electrogenerated and then overoxidized in KOH solution. The performances of very thin OPPy films formed by a one step process (using weak-acid anions) are compared with those of a film formed by a classical two-step process.

EXPERIMENTAL

The polypyrrole films were synthesized using pyrrole solution preliminarily distilled. [21] All of the solutions were prepared with bi-distilled water. Py solution, K 2 HPO 4 , LiClO 4, K 3 Fe(CN) 6 , K 4 Fe(CN) 6 , KOH and KCl salts were purchased from Aldrich.

The electrochemical experiments were performed in a classical three-electrode electrochemical cell. A platinum foil was used as counter electrode and a saturated calomel electrode (SCE) as the reference electrode. A double junction was used in order to avoid chloride diffusion into the Py solution. The working electrode was a Pt electrode (S=0.071 cm²) for electrochemical studies and Au/mica or a-CNx [22,23] substrates for SEM observations. The cyclic voltammetry and electrochemical impedance spectroscopy experiments were performed using an Autolab PGSTAT30 (Ecochemie) potentiostat/galvanostat controlled with the GPES or FRA softwares. The EIS experiments

were performed at the formal potential of the redox system, from 10 mHz to 10 kHz (9 points per decade) and a 10 mV (r.m.s.) ac perturbation for a 0.01 mol L -1 K 4 Fe(CN) 6 / K 3 Fe(CN) 6 in 0.5 mol L -1 KCl solution.

EIS spectrum simulations were done on the basis of electrical equivalent circuits. All the elements contained in a given equivalent circuit were determined by fitting its frequency dependence to the experimental impedance with a home-made simplex program using Nelder and Mead down-hill strategy. [24] The scanning electron microscopy (SEM) images were obtained from a SEM-field-emission gun (FEG) microscope, Ultra 55 Zeiss operating at 3 kV.

RESULTS AND DISCUSSION

One-step electrosynthesis of OPPy film

The OPPy films were prepared from a pyrrole aqueous solution containing only monohydrogenophosphate species as anions. The pH of this solution was 8.6. The film synthesis was performed under potentiostatic conditions and various potentials were tested.

From Figure 1A, it is clear that whatever the applied potential (from 0.70 to 0.90 V/SCE) the shape of the chronoamperometric curves is the same. First, the anodic current is high, between 160 and 400 µA (see Table 1), then it decreases very rapidly until a very low value, between 0.4 and 0.7 µA. For each potential tested, the initial (I 0 ) and final (I f ) currents are reported in Table 1 as well as the anodic charge that passed through the electrode for 600 seconds, which is the duration of the film synthesis. The ratio between I f and I 0 is in the range of 0.0014 to 0.0023. All these results indicate that the Py oxidation rate is not constant and rapidly decreases with time. For a diffusion limited process, the Cotterell equation is I=kt -1/2 where I stands for the current, k the kinetic constant and t the time. Thus, if I 0 is determined at t=0.1s and I f after 600 s of polarization, then I f /I 0 = (0.1/600) 1/2 = 0.013. This value is about ten times higher than the values determined in our experimental conditions (see Table 1). Therefore the pyrrole polymerization process is not limited by the diffusion, actually it is progressively blocked due to the overoxidation of the polymer already deposited on the electrode. This finding was confirmed by the fact that the relationship between I and t -1/2 is not linear.

Besides, it has been observed that for all the potentials tested, the films electrogenerated on the Pt electrode were not visible to the naked eye the electrode surface appeared yellow.

These results are in good agreement with the very low anodic charges passed through the electrode during its polarization (see Table 1), about five hundred times lower than the anodic charge in the presence of perchlorate (see Figure 1B).

To confirm this finding another test was V/SCE, the addition of LiClO 4 potential was increased up to 0.78 V/SCE. In this case, the electrochemical response of the ferri-ferrocyanide redox system is detected in good agreement with the fact that the polymer is under its oxidized state (doped and conductive form).

EIS in the presence of redox species

In order to specify the results obtained by CV, EIS experiments were carried out. Figure 4 shows Such a situation has been formally addressed by Young in the case of blocking oxide layers, assuming that the conductivity σ(x) within the films varies exponentially as σ(x) = σ(0) exp(x/δ) where δ is a characteristic length, shorter than the film thickness d. [25,[START_REF] Tribollet | Electrochemical Impedance Spectroscopy[END_REF][START_REF] Schiller | [END_REF] The relative dielectric constant ε inside the film is assumed to be constant. Within these assumptions, the impedance Z Young can be expressed as:

        ωτ + δ ωτ + ⋅ ω δ = j 1 ) / d exp( j 1 ln j d / f Young C Z ( 3
)
where 2. The reduced standard deviation calculated over the real and imaginary components assuming a 1% error on impedance modulus are also reported. 3), ( 3) two-step OPPy.

C f = ε o εS/d
The first information obtained from the data reported in Table 2 is the relative dielectric constant ε of the polymeric film. This parameter was determined by using ten film thicknesses d measured for four different samples. Therefore, ten values of ε were obtained assuming the film capacity C f to be described by that of a plane condenser with the surface S = 0.071 cm 2 , according to:

S C d f ⋅ ⋅ = 0 ε ε (5)
It reads a mean value of ε = 12 ± 2. This value seems reasonable for a non conducting polypyrrole material, but it is hardly comparable to literature data which are scarce. A value of 13.6 was found for chemically synthesized PPy films. [28,29] A number of investigations was performed to measure the dielectric permittivity ε of PPy films at high frequencies in the GHz range. [30] At a frequency of 2.45 GHz, ε was found around 10 for lightly doped PPy films.

Table 2: EIS analyses. Fitted parameters of OPPy films directly synthesized at different

potentials E ox according to the electrical model depicted in Fig. 5(1) (electrode surface: 0.071 cm 2 ). The experimental spectra are shown in Fig. 4. Calculated parameters: σ(0), σ s , θ (see text).

Accordingly the largest film conductivity σ(0) at the OPPy/solution interface is given by:

τ εε σ 0 ) 0 ( = (6)
The calculated values for each applied potential tested are given in

) exp( ) 0 ( δ σ σ d s - ⋅ = (7)
The attenuation factor (exp(-d/δ)) is of the order of 10 4 -10 8 yielding the σ s values in the range 10 -12 S/cm (Table 2) whatever the potential used for the film synthesis with an optimum value of 0.39 10 -12 S/cm for E ox = 0.72 V/SCE. These findings validate the synthesis route tested herein to produce in one step a highly blocking ultra thin polymeric film. In the presence of redox species, a residual current can be observed involving a very small fraction θ of the electrode surface. This fraction can be evaluated by comparing the EIS response of a very thin PPy conductive (doped) film generated with an anodic charge of 5 mC, investigated in the same electrochemical conditions (see Fig. 7). The frequency dependence of the impedance is well accounted for by a Randles circuit involving a Warburg diffusion impedance in series with the transfer resistance.

One-step OPPy film

PPy film as prepared PPy film after overoxidation

Diff t t Z R Z + = ( ) α ω + = dl t t t j 1 C R R Z β ω + + ω = ) j /( 1 j 1 w t dl t R R C Z t Young 1 1 Z Z Y + = w f f R ) j ( C j Z 1 β ω + ω = f f f 1 j 1 R C Z + ω = Y R 1 Z s + = f t s Z Z R Z + + = f t s Z Z R Z + + =
However better simulations of the experimental data were obtained by considering two capacitive contributions in series according to the equivalent circuit reported in Figure 5(2) and Table 3 column 2, and discussed in the following. The Warburg coefficient was R W = 312 Ω s -0.5 , similar to the value found for a bare platinum electrode. R W is given by the following expression written for equimolar oxidized and reduced species at concentration C 0 :

D S C n RT 2 W 0 2 2 F R W = (8)
where F stands for the Faraday constant (RT/F = 0.0258 V), n the number of electrons exchanged (n=1 herein), S W the surface concerned by the redox reaction and D the mean diffusion coefficient for both redox species taken as D = 7.10 -6 cm 2 s -1 . [31] The measured value of R W corresponds to an electrode surface of 0.065 cm 2 , in good agreement with the geometrical electrode surface (0.071 cm 2 ). The apparent transfer rate constant, k 0, can be calculated according to the expression:

t W R S C nF RT k 0 2 0 = (9)
k 0 = 0.15 ± 0.02 cm s -1 is deduced from the transfer resistance R t (PPy) = 3.7 ± 0.5

Ω (0.26 Ω cm 2 ).
Therefore, the conductive surface fraction θ of the overoxidized OPPy films can be evaluated by comparing the values of the charge transfer resistance of the OPPy films (R t , Table 2) to that of the conductive PPy film. θ can be calculated as shown in Equation 10.

) OPPy ( ) PPy ( 100 (%) t t R R = θ (10)
θ values determined using this equation are listed in Table 2. It can be seen that θ represents less than 0.001% of the electrode surface.

OPPy film obtained by a two-step electrosynthesis process

As mentioned in the introduction, classically a blocking behaviour of a conducting PPy film is achieved via an additional oxidation step in KOH solution. Figures 3B and7 show the CV and EIS responses, respectively, of a doped PPy film in contact with the Fe(CN) 6 4-/3-solution. This film was synthesized in a LiClO 4 solution at +0.6 V/SCE and was then further overoxidized at 0.75 V/SCE in 0.1M KOH solution. The anodic charge passed during the PPy film deposition was 5 mC and its overoxidation treatment was performed for 20 or 40 minutes.

For the doped PPy film, as already mentioned, the EIS impedance response can be modeled by assuming two contributions in series with the electrolyte resistance, the charge transfer impedance Z t and that of redox species transport inside the film Z w (see Figure 5(2) and Table 3). The charge transfer is represented by a parallel (R t , C dl ) circuit assuming a Cole-Cole type distribution of time constants. The charge transport inside the film is represented by a generalized Warburg impedance R w /(jω) β in parallel with the PPy film pseudo capacitance C f .

After overoxidation in KOH solution, the EIS scheme is modified: the Warburg impedance becomes in series with the transfer resistance R t , both in parallel with the double layer capacitance C dl ; the film capacitance C f lies in parallel with a resistance R f . Both models are detailed in Table 3 and the fitted parameters are gathered in Table 4. Taking into account the values of these parameters, two different pictures can be proposed for PPy films one before and one after overoxidation. For the as-prepared film, charge transfer takes place inside the porous film in the vicinity of the platinum substrate. Transport of redox species is represented by a generalized Warburg impedance in parallel with the large chemical film capacitance (around 400 µF). On the other hand, after an overoxidation step, charge transfer occurs close to the film/electrolyte interface: the transfer resistance, only increased by a factor 5-6, is in parallel with the double layer capacitance having a "standard" value (40-50 µFcm -2 ). But this external electroactive part of the film is connected to the platinum rear contact through a highly resistive material as pointed out by the very large value of the R f resistance.

It has been noticed that after overoxidation, the color of the films slightly changes from grey to yellow, indicating that during the overoxidation treatment the polymeric film is partially dissolved. Finally, by comparing graphs A and C (same scale) in Figure 3 and the data in Tables 2 and4, it can be concluded that the lowest electrochemical reactivity of the OPPy films is achieved by the one-step strategy. The OPPy films synthesized by this way are ultra thin and pine-hole free. Indeed the transfer resistance with respect to the ferri-ferrocyanide redox system is about ten thousand times higher using the one-step synthesis than using the two-step one, allowing to realize compact permselective ultra thin membranes.

CONCLUSIONS

In the presence of weak-acid anions in the pyrrole solution, strongly isolating overoxidized polypyrrole films were obtained and their blocking character is assessed by CV and EIS. The films formed at potentials between 0.7 and 0.9 V/SCE in the presence of HPO 4 2-anions in a one-step process behave as ultra thin (15 ±5 nm) and isolating membranes. Their impedance response is well accounted by using the Young model assuming uniform dielectric properties and a conductivity gradient perpendicular to the surface. By checking the electrochemical response to the ferri-ferrocyanide redox system, the remaining conductive part of the membrane was shown to be less than 0.001% of the geometric surface.

For comparison, conducting PPy films have been synthesized in the presence of perchlorate anions and subsequently electro-overoxidized in a KOH medium. For the as-prepared doped films, charge transfer is found to take place inside the porous film. On the other hand, after the overoxidation step, charge transfer occurs in the vicinity of the film/electrolyte interface, the electroactive interface being connected to the metal substrate via a highly resistive material. From the present studies, it can be concluded that for preparing ultra thin strongly isolating membranes the one-step strategy is more efficient in terms of a much lower electrochemical reactivity than that of requiring two steps. Thus, OPPy ultra thin membranes would constitute excellent candidates for electrochemical biosensors or for anticorrosion coatings.

Figure 2 : 7 oFigure 3 :

 273 Figure 2: OPPy membranes. SEM image

Figure 4 :

 4 Figure 4: EIS response of OPPy-coated electrode. Nyquist EIS diagrams recorded at 0.21 V/SCE for OPPy-coated Pt electrodes in contact with a 10 mM Fe(CN) 6 4-/3-+ 0.5 M KCl

Figure 5 ( 1 )

 51 Figure 5(1) shows the equivalent electrical circuit used to analyze the frequency dependence

Figure 5 :

 5 Figure 5: Equivalent electrical circuits. Electrical models used to fit the impedance

Figure 6 :

 6 Figure 6: Conductivity of OPPy films. Variation of the highest conductivity (σ(0)) of the OPPy

Table 3 :

 3 Impedance models used for modelling the EIS responses of the OPPy films synthesized in one-step and the PPy films, as prepared and subsequently overoxidized in KOH solution. The equivalent electrical circuits are shown in Fig.5in which Z W is defined as R w /(jω) β .

Figure 7 :

 7 Figure 7: EIS response of a PPy-coated electrode. Nyquist EIS diagrams of a PPy-coated Pt electrode in contact with a 10 mM Fe(CN) 6 4-/3-+ 0.5 M KCl solution recorded at 0.21 V/SCE.

Figure 8

 8 Figure8shows the dependence of the dielectric film capacity C f upon the anodic charge passed during the film synthesis at different applied potentials. The film capacity is found to

Figure 8 :

 8 Figure 8: Capacity of OPPy films. Variation of the capacity of the OPPy films as a function of

Table 1 :

 1 One-step electrosynthesis of OPPy films. Applied potentials, initial and final

	currents and anodic charges passed through the Pt electrode polarized for 600 s in 0.15 M Py
	+ 0.2 M K 2 HPO 4 solution.
	E (V/SCE)	I 0 (µA)	I f (µA)	Q anodic (mC)	I f / I 0
	0.70	161	0.364	0.63	0.0023
	0.72	201	0.383	0.66	0.0019
	0.75	271	0.436	0.80	0.0016
	0.78	285	0.463	0.83	0.0016
	0.80	367	0.499	0.85	0.0014
	0.82	373	0.548	0.99	0.0015
	0.85	394	0.653	1.26	0.0016
	0.90	405	0.715	1.34	0.0018

Table 2

 2 ) lies in the range of 0.70 ≤ E ox ≤ 0.82 V/SCE (see Fig.6). The conductivity gradient mainly applies to about a tenth of the film thickness, regarding the δ/d values closed to 0.1. According to Young model, the film conductivity σ s at the Pt/OPPy interface can be calculated as:

	E ox (V/SCE)	0.70	0.72	0.75	0.78	0.80	0.82	0.85	0.90
	δ/d	0.1000 0.0965 0.1001 0.1000 0.1075 0.0959 0.0911 0.0936
	C f (nF)	70.2	64.0	57.1	54.6	46.0	45.8	37.2	32.6
	τ (10 -6 s)	71.8	58.5	52.2	47.9	53.7	40.9	9.5	11.4
	R t (Ω)	458710 640320 630110 696090 662940 690940 389180 368520
	R N (MΩ)	7.58	7.54	8.17	11.1	66.2	10.4	67.2	258
	τ d	161	143	145	128	10870	146	5530	37000
	(s)
	R s (Ω)	42.7	42.6	45.3	43.3	58.3	44.8	34.0	33.9
	Standard deviation	1.6398 1.7232 1.6833 2.1829 1.5029 2.0158 1.5931 2.6213
	σ(0) (µS cm -1 )	0.010	0.012	0.014	0.015	0.013	0.018	0.076	0.063
	σ s (10 -12 S cm -1 )	0.46	0.39	0.63	0.68	1.225	0.52	1.30	1.45
	θ (%)	0.00081 0.00058 0.00059 0.00053 0.00056 0.00054 0.00095 0.0010

. From these data, it is clear that the electrical isolating character of the OPPy film is the best when the applied ox

Table 4 :

 4 EIS fitted parameters for a PPy film synthesized in pyrrole solution containingLiClO 4 (Q anodic = 5 mC) before and after an overoxidation step of 20 and 40 minutes according to the impedance models given in Table3(SD: standard deviation). Dashed boxes stand for non relevant parameters for the model used.

	PPy	R s	R t	C dl		C f	R w		R f		Equiv.
	film	(Ω)	(Ω)	(µF)	α	(µF)	(Ω s -β )	β	(Ω)	SD	electrical circuit
	As
		28.3	3.7	60	0.86	362	312	0.5		1.031 Fig. 5(2)
	prepared
	Overox.
		28.5	36	3.0		306	252 0.352 131920 0.9898 Fig. 5(3)
	20 mn
	Overox.
		31.3	33	3.7		421	258 0.384 28500 0.6907 Fig. 5(3)
	40 mn

Acknowledgements

The authors would like to thank Françoise Pillier for performing the SEM observations.