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Local electrochemical impedance measurement: scanning vibrating electrode technique in ac mode
Electrochimica Acta 44 (1999) 4117±4127
Local electrochemical impedance measurement: scanning vibrating electrode technique in ac mode E. Bayet a, F. Huet b, M. Keddam b,*, K. Ogle a, H. Takenouti b a IRSID, BP 30320, 57283 MaizieÁres-leÁs-Metz Cedex, France CNRS-UPR15, ``Physique des Liquides et Electrochimie'', Universite P. et M. Curie, CP 133, 4 Place Jussieu, 75252 Paris Cedex 05, France
b
Received 7 August 1998; received in revised form 28 January 1999; accepted 28 January 1999
Abstract The paper gives a new insight into the meaning of local impedances and the related measurement techniques. By numerical simulation it is shown that the only fully relevant quantity is obtained through local values of both the current density and potential. A recently developed equipment based on a commercial SVET device modi®ed for ac polarization is applied to corrosion situations. Under oxygen reduction control of the corrosion rate, serious artifacts are found on the dc current density mapping and local impedance. The drawback is explained by the contribution of a chemical term (redox potential) in the probe potential. A model is elaborated accounting for the resulting side eects in both dc and ac measurements. # 1999 Published by Elsevier Science Ltd. All rights reserved. Keywords: Potential mapping; Laplace's equation; Potential probe; SVET; Corrosion
1. Introduction EIS and alike frequency resolved techniques have nowadays attained a great degree of sophistication owing to both advanced equipments and computer modelling. However their quantitative application may be dramatically obliterated by their lack of space resolution. Classically the electrode impedance is exclusively de®ned and measured as the complex ratio of two global (surface averaged) ac quantities: the potential dierence between the reference electrode (RE) and the working electrode (WE) and the overall current ¯owing across the interface. Practically any real life solid electrode exhibits for structural and/or geometrical
* Corresponding author. Tel.: +33-1-4427-4148; fax: +331-4427-4074. E-mail address: [email protected] (M. Keddam)
reasons heterogeneities in surface properties and therefore in impedance. The characteristic dimension may lie between the nanometric scale and the macroscopic size of the electrode. Local impedance can be achieved by measuring the local values of both potential and current density. Local potential has been long measured by means of local probes for canceling ohmic drop polarization (Piontelli and Haber±Luggin capillary electrodes). They can be extended to the ac regime. In contrast, measurement of the local c.d. in ac regime requires more sophisticated equipment and signal processing. All the devices presently known are based on the measurement of the local ohmic drop across a short range of current path in the solution close to the surface. Most of the works were devoted to corrosion and used twin-electrode probes [1,2]. The so-called local impedance is considered as the ratio of the global potential at a remote distance by the local
0013-4686/99/$ - see front matter # 1999 Published by Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 4 6 8 6 ( 9 9 ) 0 0 1 2 6 - 7
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Fig. 1. Model cell used for the numerical simulation of global, semi-local and local impedances. left: top view, right: axial section. r1 and r2: radii of border lines for concentric areas 1 and 2 with dierent surface impedances.
current density. Two weak points of this approach must be mentioned: . It is dicult to make twin-electrodes separated by less than 0.1 mm and to keep a low inter-electrode capacity. Twin-electrodes are restricted to ac mode since on the one hand the signal/noise ratio cannot be improved in the absence of modulation and on the other hand any residual dierences between the open-circuit potentials of the two electrodes will be interpreted as a dc current density. Therefore the gap between dc and ac cannot be ®lled in a reliable way. . The global potential/local current density impedance is likely to be seriously biased since the distant potential is re¯ecting in an unpredictable manner a frequency dependent distribution in the cell. With respect to the ®rst point, a new measurement technique based on the adaptation of the SVET (scanning vibrating electrode technique) to ac polarization was designed and its performances on model systems were illustrated [3,4]. It will not be dealt with. This paper is emphasizing two aspects: . in order to better de®ne the nature of the local impedance, the in¯uence of the position of the potential and current probes on the impedance was investigated numerically on a model interface likely to generate a huge frequency dependence of the current distribution. . investigation of ac current density distribution on cell of well-known geometry revealed serious distortions pointing to a totally discarded contribution of chemical potential to the probe signal. This side eect is likely to generate wrong results even in dc conditions, therefore it was analysed in deeper details and a model accounting for it is proposed.
2. Numerical approach of the local impedance problem 2.1. Theoretical framework Provided local values of potential and current densities are available over a wide frequency range, several impedances can be de®ned and computed numerically. Potential distribution in a cell obeys the Laplace equation r2 E 0
1
(absence of electrical space charge). The thickness of the electrical double layer, non-zero charge region, is regarded as negligibly small with respect to the cell size. Mass transport is also discarded. Electrode polarizations are depicted by mixed boundary conditions linear relationship between the potential drop across the interface and the current density of the form Z i @E i i i E ÿ Ex,y,z 2 r @n where Ei is the inner potential of the i-th electrode, Eix,y,z the surface potential of the i-th electrode on the outer side of the double layer, r the solution resistivity and Zi the interfacial impedance of the i-th electrode for a surface unit. In general no analytical solution of the Laplace equation is available except for highly symmetrical cell geometry and zero surface impedance (e.g. primary ®eld distribution on the disc electrode). A ®nite element method (FEM) was used in MacIntosh environment (MacGfem software by Numerica1 based on FEM freeware).
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Fig. 2. Equipotential lines at 3 dierent frequencies issued by numerical integration of Laplace equation for the cell shown in Fig. 1. Transition from primary to secondary ®eld distributions. Equivalent circuits for areas 1 and 2. De®nitions of impedances dealt with in the paper. R2/R1=104. C1=C2.
A cell model with uni-axial symmetry was used in order to reduce the 3D problem to a 2D one by solving the Laplace equation r2 E
@ 2E 1 @ E 1 @ 2E @ 2E 2 2 2 @ r2 r @r r @y @z
3
in an axial plane: (@E/@y )=0 The cell layout is shown in Fig. 1. The working electrode (cell bottom) is made of two concentric areas indexed 1 and 2 of radii r1 and r2 to which dierent surface impedance Z 1 and Z 2 were ascribed in order to simulate electrodes with sharply heterogeneous reactivity. They are represented by a single parallel R±C circuit (R1±C1 and R2±C2). The overall problem involved 9 dierent boundaries determined by electrodes and insulating parts. The accuracy and computing time are very dependent on the space meshing, most of the runs used a 4000-node mesh. Higher mesh density was forced in front of the working electrode where the current lines are converging. The potential and current pro®les over the central electrode were de®ned by at least 50 points. Instabilities for r 4 0 were eliminated by shifting the axial boundary to a small positive value
0.01 r1. An average computing time was 20 s on a 180 MHz MacIntosh 4400 P.C. MacGfem algorithm handles the linear mixed boundary condition between E and its normal gradient ÿ(@ E/@n ) in the following way: E:
1 @E 0, ÿ ZL @n
4
the normal potential gradient is related to the current density j and the solution resistivity r through Ohm's law: (@E/@n )=rj, therefore one gets ZL
E Z , rj r
5
where Z is the interfacial impedance per surface unit. ZL is a surface impedance `normalized' by the solution resistivity, for real-valued Z it is the length of the current path in the solution having an ohmic resistance equal to the polarization resistance of its cross-section on the electrode (a characteristic distance in the theory of cell polarization developed by Wagner and Waber). Impedance data as entry parameters and results will therefore be expressed in length unit of the cell layout
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Table 1 Parameter values for computing the data of Figs. 2 and 3. Electrode radius and resistances (reduced by the solution resistivity) are given in length unit used by the MacGfem program. Capacitance are expressed in s/length unit W.E. region
Radius
Resistance
Capacitance
Central area #1 Surrounding area #2
r1=10 r2=120
R1=10 R2=105
C1=10ÿ4 C2=10ÿ4
used by the program. Transposition to a particular real cell in terms of size (e.g. in cm), electrode surface impedance (in O cm2) and solution resistivity (in O cm) is straightforward by dimension consideration1. 2.2. Potential mapping, impedance de®nition and results Fig. 2 displays the potential mapping provided by the numerical integration of the Laplace equation at three typical angular frequencies o=1, 100 and 1000 sÿ1, for the parameters values given in Table 1. These values verifying R1<
6
At o=1000, the whole electrode has a vanishingly low impedance owing to its capacitive conduction, area #1 and area #2 behave alike. The potential distribution is entirely due to the solution resistance. The so-called primary ®eld distribution is obeyed. At o R 1, the current is ¯owing mainly across the resistive part of the impedance. Since R2/R1=104, the current lines focus towards the central area whilst the surrounding area is practically inactive. The secondary ®eld distribution is obeyed. Several global, semi-local and local impedances can be determined when the program is run over a signi®cant frequency range. Four of them are relevant to practical experiments: . the traditional impedance ZL glob/glob ratio of the potential Eglob at a remote position by the mean current density over the electrode surface: r jglob= 1=pr22 0 2 2prj rdr . an impedance ZL loc,glob ratio of a local potential value at a point very close to the electrode surface, 1 Let assume a scaling factor m de®ned as 1 length unit=m cm, therefore the real electrode dimension are r1 (cm)=r1m and r2(cm)=r2m and ZL (cm)=ZLm.
by the mean value of c.d. jglob. . an impedance ZL glob/loc ratio of a distant potential value by a local current density. . an impedance ZL loc/loc ratio of a local potential by a local current density, both being probed at the same point, right above the spot where the local impedance is to be measured. From an experimental viewpoint the use of the 4channel frequency response analyzer (FRA 1254 by Solartron) enables to measure at the same time these four impedances. Fig. 3 shows the Nyquist plots computed for these 4 impedances in a frequency range between 0.1 and 105 Hz. The unit is the same as for R1 and R2. Local data were sampled at the distance h = 2r1/15 from the surface. As shown in Fig. 3a a half semi-circle is found for ZL glob/glob in spite of the highly heterogeneous electrode impedance. Intuitively, a visible deviation could be expected in a resistive medium. The HF intercept is clearly positive, around 55 and corresponds reasonably to the solution resistance between the cell centre, where the potential is measured and the whole W.E. in primary ®eld distribution (cf. Fig. 2, o=1000). A value can be estimated from the full solution resistance associated to the W.E. in an in®nite medium Re=(r/ 4r2), hence Z L(1)=(Re/r )pr22 3 100. However the diagram diameter, Z(0) 3 2650, is much larger than the theoretical limit for the given parameters: pr22 pr21 =R1 p r22 ÿ r21 =R2ÿ1 =1419. This large over-estimation of the impedance size is a direct consequence of the deformation of the current lines with frequency and the related contribution of the series electrolyte resistance. At high frequencies (HF) this term is small since the current is ¯owing to the whole surface (Fig. 3a) whilst at low frequencies (LF) the path is shrank to the central area and the electrolyte resistance is much higher. It is not meant that the LF electrolyte resistance is as large as (2600±1419), according to the de®nition of ZL glob/glob this is the value of the ohmic potential (EglobÿEloc), in front of the central area, divided by the mean current density. A very crude estimate by using the primary distribution formula Re=(r/4r1)predicts a value of (r/40) for area #1 whilst the surface resistance of the same area, according to Eq. (5), amounts to R1 r=pr21 r=10p. Therefore Eglob and Z are both overestimated by a factor of r=10p r=40= r=40 1 p=4, hence predicting (1419 1.7853)=2533 a value very close to the computed one shown in Fig. 3a 32650. The apex frequency 3 5 is close but not equal to neither of the values calculated for Z L(0) 3 2650 (o=3.85) or Z L(0)=1419 (o=7) but lies midway of theses values. Fig. 3b displays the impedance obtained when both
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Fig. 3. Nyquist impedance plotted from the numerical solution of the Laplace equation shown in Fig. 2, labels in angular frequency o (Rd sÿ1). (a) Zglob/glob: traditional global impedance; (b) Zloc/loc: fully local impedance; (c) Zloc/glob: semi-local impedance obtained from local potential and global current density; (d) Zglob/loc: semi-local impedance obtained from global potential and local current density, the de®nition used by literature in the ®eld.
potential and current are probed locally. The diagram size (10=R1) and the apex frequency o=100 are exactly what was expected from entry parameters (R1=10 and C1=C2=10ÿ4). In Fig. 3c, the impedance ZL loc/glob is shown. A semicircle is obtained with a diameter equal to the theoretical value for a mean current density over the electrode as for Fig. 3a. No series solution resistance is found in agreement with the use of Haber±Luggin capillaries for probing the potential near the surface in order to cancel out the ohmic term in dc polarization measurements. The apex frequency is practically equal to that displayed on Fig. 3a.
Fig. 3d shows the impedance measured by probing only the current locally and the potential at a remote distance. A totally unexpected shape is found essentially displaying an inductive feature. The HF limit, of the order of 80 is signi®cantly larger than the value around 55 in Fig. 3a, but a HF tail in the capacitive half-plane is likely to end back to this value. The LF limit close to 20 is consistent with the sum of R1 and the solution contribution (Re/r )pr21, where Re has the same value Re=(r/4r1) calculated for Fig. 3a, yielding a contribution: (p/4)r1=7.9. No computing artifact can be invoked for explaining this pathological behavior since Fig. 3d was built up with the same raw data in
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Fig. 3 (continued)
terms of Eglob and jloc, already used for plotting Figs. 3a±c. It should be emphasized that ZL glob/loc is the quantity used in the literature on localized impedance. Therefore much attention must be paid to the validity of these data. This prompted us de®nitely to design an equipment enabling to measure both potential and current density locally [3±5]. The problem seems to vanish at larger R1 values. For instances at R1=100, a diagram roughly semi-circular is found (not shown here) with HF limit around 55, similar to Fig. 3a linked with the measurement of potential at a remote position. The LF limit close to 108 is in agreement with the interfacial impedance of area #1, R1=100, in series with the solution contribution, 7.9. The apex frequency is also in agreement with the value expected. This numerical approach could be very convenient for investigating the eect on the local impedance of the probe position, the inter-electrode gap of twin elec-
trodes . . . These more general aspects were not dealt with in this work. 2.3. Kramers±Kronig transformability Testing the Kramers±Kronig (K±K) transformability of impedance data is generally regarded as an ecient way of verifying their validity and detecting experimental artifacts (nonlinear or time-dependent systems essentially) or numerical problems. All the data plotted in Fig. 3 were submitted to K±K computation both in impedance and admittance domains. It was concluded that in the impedance domain all data, including the apparently meaningless ZL glob/loc ones as shown in Fig. 4, obey the K±K relationship with a satisfactory accuracy. As expected, only those admittance data tending to in®nity at HF were not correctly transformed. Therefore one cannot rely on the K±K test for detecting wrong results possibly issued by a set-up design for measuring global potential and local current density.
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3. Identi®cation of anomalous SVET response: a new insight into the very nature of the SVET signal This second part of the paper will be devoted to a critical analysis of the data yielded by the equipment based on SVET in ac mode. This will lead to revisiting the physical principle of the SVET probe. 3.1. Normal and anomalous local impedance behaviours The local impedance Zloc/loc and the global one Zglob/glob of disc electrodes was measured as a function of the distance to the surface on the symmetry axis for dierent systems: Ni and Fe at their rest potential in sulphuric solutions. Fig. 5 displays the results for Ni at pH 2 at 4 dierent distances. The local impedance increases with the distance h due to the divergent current distribution above a disc electrode. It was veri®ed [5] that this change is close to the analytical dependence of j with h for the primary ®eld distribution: j(h )=j(0)(r 2/r 2+h 2). In addition numerical simulation with the same program FEM software showed a fair agreement with these results. At higher pHs, 3.5 and 5, serious deviations from this behaviour were observed for Ni and to a larger extent for Fe in contrast with the results of Fig. 5. They can be summarized as follows: . the modulus of the localized impedance is much larger than that of the global one, . the dierence between the two is growing with the solution pH and decreasing with the height h above the surface, . the dierence is larger with iron than with nickel. Fig. 6 displays an example of that for iron in pH 5 solution both in the Nyquist and Bode planes. Owing to the simultaneous measurement of individual signals Eglob, Eloc, jglob, jloc it was shown that in this particular case Eglob and Eloc are quite identical. Therefore the dierence between these quantities is to be ascribed to jloc delivered by the SVET channel of the equipment. It was thought that in addition to ohmic drop a side eect contributes to the potential gradient probed by the SVET platinum tip. Obviously this term has a growing weight when oxygen reduction becomes the major cathodic reaction (increasing pH), when the corrosion rate is larger (Fe vs. Ni) and when the probe gets closer to the surface. A potential related to the gradient of molecular oxygen in the vicinity of the electrode seemed to be the more straightforward explanation. In fact it is extremely surprising that the redox potential of the solution is hardly considered as a component of the probe potential neither in SVET nor in twin-electrode measurements. It is universally assumed that the potential gradient is purely ohmic being the product of the local current density j by the solution
Fig. 4. Kramers±Kronig transforms for the data shown in Fig. 3d, computation in both impedance and admittance domains, labels in angular frequency o, Rd sÿ1
resistivity r. In fact for the device to work properly the contact potential probe-solution must be ®xed by some surface electrochemistry, most likely a redox potential. This is the general description of the potential taken by an inert electrode in a solution. Forgetting this contribution for SVET or twin-electrode is only allowed if the gradient of this redox potential is negligible. Therefore it seemed worth investigating this chemical contribution to the local current density measurement on the basis of a model accounting for the dc and ac cases. 3.2. Modelling the redox contribution to the potential gradient: side eect in SVET and twin-electrode experiments In the principle of SVET it is assumed that the probe/electrolyte contact potential is constant. This potential is in fact determined by the local equilibrium redox potential of the solution Eeq given by Nernst's law Eeq E0eq
Ox RT log : Red nF
7
In dc polarization, O being the vibration frequency, d the vibration amplitude, the probe signal is DEprobe d rjloc grad Eeq sin Ot
8
where grad Eeq is derived from the dierential of Nernst's law (7) grad Eeq
RT gradOx gradRed ÿ : Red Ox nF
9
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Fig. 5. Zloc/loc (+) and Zglob/glob (o) for a Ni disc (0.3 mm in diameter) at the corrosion potential in pH 2 sulphuric solution at various distances from the surface.
3.2.1. The dc case Following Eq. (8) a positive (respectively negative) value of the gradient of Nernst potential grad Eeq is
equivalent to an ohmic gradient built up by a cathodic (respectively anodic) current ¯owing to (respectively from) the electrode. If this explanation is correct a pseudo-current must be observed at the open circuit
Fig. 6. Zloc/loc (+) and Zglob/glob (o), for an iron disc (0.3 mm in diameter) at the corrosion potential in pH 5 sulphuric solution at 50 mm from the surface, labels in Hertz.
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potential. This is actually the case for iron as shown in Fig. 7 which displays a 3D current mapping on an iron disc at zero neat current in the same conditions as for Fig. 6. A cathodic current is clearly found. The concentration pro®les of oxygen, (Ox) and Fe2+, (Red) cations respectively consumed and released by the corrosion process are also shown in this ®gure. It is easily seen by referring to Eq. (9) that both gradients contribute positively to the gradient of the redox potential and therefore to the cathodic pseudo-current. By Eq. (9), for n = 1, a signal equivalent to jloc=100 mA cmÿ2, in a solution of resistivity r=50 Ocm i.e. a gradient of 50 10ÿ4 V cmÿ1 is obtained with a relative gradient of only 0.15 cmÿ1 i.e. a change of only one percent of the concentration across a 100 mm thick diusion layer. Of course, much bigger eects are expected near the diusion limited current. 3.2.2. The ac case The ac counterpart of this redox potential gradient on the measurement of the ac current density and of the local impedance is derived in a similar way. Hereafter small amplitude components are indicated by the symbol D. The ac probe DVprob is given by DVprob d rDjloc grad DEeq
10
where d is the vibration amplitude of the probe, grad DEeq is the ac modulation of the gradient of Eeq induced by the modulation of the electrode polarization. By dierentiating Eq. (9) one gets RT grad DOx grad D Red grad DEeq ÿ nF Ox Red
11
therefore this component of the probe signal is proportional to the ac diusion ¯ux at the probe location. In other words the experimental apparent local impedance can be written as Z loc=loc exp
DEloc dr DVprob ÿ DEloc Djloc grad DEeq = r
12
hence the true local impedance is connected in parallel with an impedance Zd,loc having properties similar to a diusion impedance as established below. By inverting (12), it comes: 1 Z loc=loc exp
1 Zloc=loc
1 Zd,loc
13
Therefore an under-estimation of the local impedance is expected, but depending on the reaction taking place at the electrode and on the sign of the gradient modulation an opposite conclusion is not to
Fig. 7. 3D current density distribution on an iron sample at its open-circuit potential in pH 5 sulphuric solution determined by usual dc SVET measurement. Schematic pro®les of O2, Fe2+ concentrations and of Eeq in the vicinity of the iron sample.
be excluded. Zd,loc can be derived from the time and distance dependencies of DOx and DRed by solving the second Fick's law in the Nernst diusion layer approximation, thickness d. The derivation will be given here for one single species of concentration C, a fully correct derivation should account for both components of the gradient of the redox potential in Eq. (11). At height h above the surface the ac modulation of a concentration is p DC jDC jsinh h ÿ d jo =Dsin o t
14
and the corresponding gradient p p grad DC jDC j jo =Dcosh h ÿ d jo =Dsin o t:
15
The transfer function is simply obtained from the diusion impedance
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Fig. 8. Nyquit plots of the diusion impedance Zd and of the bias transfer function Zd,loc connected in parallel with the true local impedance by the redox potential pro®le, labels in Hertz.
p tanh jo d2 =D Zd Rd p jo d2 =D
16
by multiplying it by a distant dependent factor Zd,loc Zd
grad DCh0 : grad DCh
17
Fig. 8 shows the Nyquist plots of the corresponding transfer functions for the diusion impedance Zd referring to the diusion ¯ux at the interface (h = 0) and Zd,loc at height h=0.5 d, within the diusion layer. It can be noticed that excepted in the high frequency region, the two diagrams are practically overlapping. Therefore a transfer function very similar to a Nernstian diusion impedance is predicted to take place in parallel with the true local impedance. Experimentally there is no rigourous way to split the two contributions in the r.h.s. of Eq. (13). By considering that the local current density Djglob just above the electrode surface is not very dierent from the global one Djglob, Zd,loc was estimated by Eq. (13) owing to the simultaneous acquisition of individual ac currents and potential by the FRA 1254 set-up. Djglob 1 1 ÿ : Zd,loc Zloc=loc exp DEloc
be emphasized that the contribution of chemical potential to the signal collected by the probe is by no means restricted to the SVET device, there is no reason why the twin-electrode could be not aected.
4. Conclusions This paper was intended to bring some new results on the concept and measurement of the quantity referred to in literature as localized impedance. By
18
Fig. 9 shows an example of Nyquist plot found by this procedure by processing the data of the iron disc corresponding to Fig. 6. The shape of the diagram and the frequency domain are in reasonable agreement with what is predicted by Eqs. (16) and (17). It should
Fig. 9. Estimate of the transfer function Zd,loc by means of Eq. (18), labels in Hertz.
E. Bayet et al. / Electrochimica Acta 44 (1999) 4117±4127
means of a numerical simulation based on the integration of the Laplace equation with a working electrode modelling a small active area at the center of a large passive surface, the properties of the traditional global impedance, of semi-local and of fully local impedances were compared. Their main features are easily understood at an almost quantitative level by accounting for the known behaviours of primary and secondary ®eld distributions. The major conclusion drawn from this approach is that the usual so-called local impedance obtained for a potential measured at a remote distance from the electrode may be seriously distorted. This will occur if the polarization resistances of the two electrode regions are very dierent, for instances by a factor larger than 1000, a situation not exceptional for localized corrosion on passive electrodes. Therefore both potential and current density must be measured locally, close to the surface. Local impedance yielded by a SVET devices modi®ed for the ac polarization regime were critically examined. It was shown that at pH values where oxygen reduction is the main cathodic reaction, corroding elec-
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trodes exhibited pseudo-cathodic dc current at their open-circuit potential. Correspondingly, serious deviations from the expected values of the local impedance were found with the probe at distances within the diffusion layer. An explanation of this bias was proposed by considering simply that the probe/solution potential is depending upon the local redox potential of the solution.
References [1] R.S. Lillard, P.J. Moran, H.S. Isaacs, J. Electrochem. Soc. 139 (1992). [2] F. Zou, D. Thierry, H.S. Isaacs, J. Electrochem. Soc. 144 (1997) 1957. [3] E. Bayet, F. Huet, M. Keddam, K. Ogle, H. Takenouti, J. Electrochem. Soc. 144 (1997) L87. [4] E. Bayet, F. Huet, M. Keddam, K. Ogle, H. Takenouti, 6th EMCR Meeting, Trento (Italy) (1997), in: Materials Science Forum, Vols. 289±292, 1998, pp. 57±68. [5] E. Bayet, thesis dissertation, University of Paris VI, 1997.
Local electrochemical impedance measurement: scanning vibrating electrode technique in ac mode E. Bayet a, F. Huet b, M. Keddam b,*, K. Ogle a, H. Takenouti b a IRSID, BP 30320, 57283 MaizieÁres-leÁs-Metz Cedex, France CNRS-UPR15, ``Physique des Liquides et Electrochimie'', Universite P. et M. Curie, CP 133, 4 Place Jussieu, 75252 Paris Cedex 05, France
b
Received 7 August 1998; received in revised form 28 January 1999; accepted 28 January 1999
Abstract The paper gives a new insight into the meaning of local impedances and the related measurement techniques. By numerical simulation it is shown that the only fully relevant quantity is obtained through local values of both the current density and potential. A recently developed equipment based on a commercial SVET device modi®ed for ac polarization is applied to corrosion situations. Under oxygen reduction control of the corrosion rate, serious artifacts are found on the dc current density mapping and local impedance. The drawback is explained by the contribution of a chemical term (redox potential) in the probe potential. A model is elaborated accounting for the resulting side eects in both dc and ac measurements. # 1999 Published by Elsevier Science Ltd. All rights reserved. Keywords: Potential mapping; Laplace's equation; Potential probe; SVET; Corrosion
1. Introduction EIS and alike frequency resolved techniques have nowadays attained a great degree of sophistication owing to both advanced equipments and computer modelling. However their quantitative application may be dramatically obliterated by their lack of space resolution. Classically the electrode impedance is exclusively de®ned and measured as the complex ratio of two global (surface averaged) ac quantities: the potential dierence between the reference electrode (RE) and the working electrode (WE) and the overall current ¯owing across the interface. Practically any real life solid electrode exhibits for structural and/or geometrical
* Corresponding author. Tel.: +33-1-4427-4148; fax: +331-4427-4074. E-mail address: [email protected] (M. Keddam)
reasons heterogeneities in surface properties and therefore in impedance. The characteristic dimension may lie between the nanometric scale and the macroscopic size of the electrode. Local impedance can be achieved by measuring the local values of both potential and current density. Local potential has been long measured by means of local probes for canceling ohmic drop polarization (Piontelli and Haber±Luggin capillary electrodes). They can be extended to the ac regime. In contrast, measurement of the local c.d. in ac regime requires more sophisticated equipment and signal processing. All the devices presently known are based on the measurement of the local ohmic drop across a short range of current path in the solution close to the surface. Most of the works were devoted to corrosion and used twin-electrode probes [1,2]. The so-called local impedance is considered as the ratio of the global potential at a remote distance by the local
0013-4686/99/$ - see front matter # 1999 Published by Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 4 6 8 6 ( 9 9 ) 0 0 1 2 6 - 7
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Fig. 1. Model cell used for the numerical simulation of global, semi-local and local impedances. left: top view, right: axial section. r1 and r2: radii of border lines for concentric areas 1 and 2 with dierent surface impedances.
current density. Two weak points of this approach must be mentioned: . It is dicult to make twin-electrodes separated by less than 0.1 mm and to keep a low inter-electrode capacity. Twin-electrodes are restricted to ac mode since on the one hand the signal/noise ratio cannot be improved in the absence of modulation and on the other hand any residual dierences between the open-circuit potentials of the two electrodes will be interpreted as a dc current density. Therefore the gap between dc and ac cannot be ®lled in a reliable way. . The global potential/local current density impedance is likely to be seriously biased since the distant potential is re¯ecting in an unpredictable manner a frequency dependent distribution in the cell. With respect to the ®rst point, a new measurement technique based on the adaptation of the SVET (scanning vibrating electrode technique) to ac polarization was designed and its performances on model systems were illustrated [3,4]. It will not be dealt with. This paper is emphasizing two aspects: . in order to better de®ne the nature of the local impedance, the in¯uence of the position of the potential and current probes on the impedance was investigated numerically on a model interface likely to generate a huge frequency dependence of the current distribution. . investigation of ac current density distribution on cell of well-known geometry revealed serious distortions pointing to a totally discarded contribution of chemical potential to the probe signal. This side eect is likely to generate wrong results even in dc conditions, therefore it was analysed in deeper details and a model accounting for it is proposed.
2. Numerical approach of the local impedance problem 2.1. Theoretical framework Provided local values of potential and current densities are available over a wide frequency range, several impedances can be de®ned and computed numerically. Potential distribution in a cell obeys the Laplace equation r2 E 0
1
(absence of electrical space charge). The thickness of the electrical double layer, non-zero charge region, is regarded as negligibly small with respect to the cell size. Mass transport is also discarded. Electrode polarizations are depicted by mixed boundary conditions linear relationship between the potential drop across the interface and the current density of the form Z i @E i i i E ÿ Ex,y,z 2 r @n where Ei is the inner potential of the i-th electrode, Eix,y,z the surface potential of the i-th electrode on the outer side of the double layer, r the solution resistivity and Zi the interfacial impedance of the i-th electrode for a surface unit. In general no analytical solution of the Laplace equation is available except for highly symmetrical cell geometry and zero surface impedance (e.g. primary ®eld distribution on the disc electrode). A ®nite element method (FEM) was used in MacIntosh environment (MacGfem software by Numerica1 based on FEM freeware).
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Fig. 2. Equipotential lines at 3 dierent frequencies issued by numerical integration of Laplace equation for the cell shown in Fig. 1. Transition from primary to secondary ®eld distributions. Equivalent circuits for areas 1 and 2. De®nitions of impedances dealt with in the paper. R2/R1=104. C1=C2.
A cell model with uni-axial symmetry was used in order to reduce the 3D problem to a 2D one by solving the Laplace equation r2 E
@ 2E 1 @ E 1 @ 2E @ 2E 2 2 2 @ r2 r @r r @y @z
3
in an axial plane: (@E/@y )=0 The cell layout is shown in Fig. 1. The working electrode (cell bottom) is made of two concentric areas indexed 1 and 2 of radii r1 and r2 to which dierent surface impedance Z 1 and Z 2 were ascribed in order to simulate electrodes with sharply heterogeneous reactivity. They are represented by a single parallel R±C circuit (R1±C1 and R2±C2). The overall problem involved 9 dierent boundaries determined by electrodes and insulating parts. The accuracy and computing time are very dependent on the space meshing, most of the runs used a 4000-node mesh. Higher mesh density was forced in front of the working electrode where the current lines are converging. The potential and current pro®les over the central electrode were de®ned by at least 50 points. Instabilities for r 4 0 were eliminated by shifting the axial boundary to a small positive value
0.01 r1. An average computing time was 20 s on a 180 MHz MacIntosh 4400 P.C. MacGfem algorithm handles the linear mixed boundary condition between E and its normal gradient ÿ(@ E/@n ) in the following way: E:
1 @E 0, ÿ ZL @n
4
the normal potential gradient is related to the current density j and the solution resistivity r through Ohm's law: (@E/@n )=rj, therefore one gets ZL
E Z , rj r
5
where Z is the interfacial impedance per surface unit. ZL is a surface impedance `normalized' by the solution resistivity, for real-valued Z it is the length of the current path in the solution having an ohmic resistance equal to the polarization resistance of its cross-section on the electrode (a characteristic distance in the theory of cell polarization developed by Wagner and Waber). Impedance data as entry parameters and results will therefore be expressed in length unit of the cell layout
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Table 1 Parameter values for computing the data of Figs. 2 and 3. Electrode radius and resistances (reduced by the solution resistivity) are given in length unit used by the MacGfem program. Capacitance are expressed in s/length unit W.E. region
Radius
Resistance
Capacitance
Central area #1 Surrounding area #2
r1=10 r2=120
R1=10 R2=105
C1=10ÿ4 C2=10ÿ4
used by the program. Transposition to a particular real cell in terms of size (e.g. in cm), electrode surface impedance (in O cm2) and solution resistivity (in O cm) is straightforward by dimension consideration1. 2.2. Potential mapping, impedance de®nition and results Fig. 2 displays the potential mapping provided by the numerical integration of the Laplace equation at three typical angular frequencies o=1, 100 and 1000 sÿ1, for the parameters values given in Table 1. These values verifying R1<
6
At o=1000, the whole electrode has a vanishingly low impedance owing to its capacitive conduction, area #1 and area #2 behave alike. The potential distribution is entirely due to the solution resistance. The so-called primary ®eld distribution is obeyed. At o R 1, the current is ¯owing mainly across the resistive part of the impedance. Since R2/R1=104, the current lines focus towards the central area whilst the surrounding area is practically inactive. The secondary ®eld distribution is obeyed. Several global, semi-local and local impedances can be determined when the program is run over a signi®cant frequency range. Four of them are relevant to practical experiments: . the traditional impedance ZL glob/glob ratio of the potential Eglob at a remote position by the mean current density over the electrode surface: r jglob= 1=pr22 0 2 2prj rdr . an impedance ZL loc,glob ratio of a local potential value at a point very close to the electrode surface, 1 Let assume a scaling factor m de®ned as 1 length unit=m cm, therefore the real electrode dimension are r1 (cm)=r1m and r2(cm)=r2m and ZL (cm)=ZLm.
by the mean value of c.d. jglob. . an impedance ZL glob/loc ratio of a distant potential value by a local current density. . an impedance ZL loc/loc ratio of a local potential by a local current density, both being probed at the same point, right above the spot where the local impedance is to be measured. From an experimental viewpoint the use of the 4channel frequency response analyzer (FRA 1254 by Solartron) enables to measure at the same time these four impedances. Fig. 3 shows the Nyquist plots computed for these 4 impedances in a frequency range between 0.1 and 105 Hz. The unit is the same as for R1 and R2. Local data were sampled at the distance h = 2r1/15 from the surface. As shown in Fig. 3a a half semi-circle is found for ZL glob/glob in spite of the highly heterogeneous electrode impedance. Intuitively, a visible deviation could be expected in a resistive medium. The HF intercept is clearly positive, around 55 and corresponds reasonably to the solution resistance between the cell centre, where the potential is measured and the whole W.E. in primary ®eld distribution (cf. Fig. 2, o=1000). A value can be estimated from the full solution resistance associated to the W.E. in an in®nite medium Re=(r/ 4r2), hence Z L(1)=(Re/r )pr22 3 100. However the diagram diameter, Z(0) 3 2650, is much larger than the theoretical limit for the given parameters: pr22 pr21 =R1 p r22 ÿ r21 =R2ÿ1 =1419. This large over-estimation of the impedance size is a direct consequence of the deformation of the current lines with frequency and the related contribution of the series electrolyte resistance. At high frequencies (HF) this term is small since the current is ¯owing to the whole surface (Fig. 3a) whilst at low frequencies (LF) the path is shrank to the central area and the electrolyte resistance is much higher. It is not meant that the LF electrolyte resistance is as large as (2600±1419), according to the de®nition of ZL glob/glob this is the value of the ohmic potential (EglobÿEloc), in front of the central area, divided by the mean current density. A very crude estimate by using the primary distribution formula Re=(r/4r1)predicts a value of (r/40) for area #1 whilst the surface resistance of the same area, according to Eq. (5), amounts to R1 r=pr21 r=10p. Therefore Eglob and Z are both overestimated by a factor of r=10p r=40= r=40 1 p=4, hence predicting (1419 1.7853)=2533 a value very close to the computed one shown in Fig. 3a 32650. The apex frequency 3 5 is close but not equal to neither of the values calculated for Z L(0) 3 2650 (o=3.85) or Z L(0)=1419 (o=7) but lies midway of theses values. Fig. 3b displays the impedance obtained when both
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Fig. 3. Nyquist impedance plotted from the numerical solution of the Laplace equation shown in Fig. 2, labels in angular frequency o (Rd sÿ1). (a) Zglob/glob: traditional global impedance; (b) Zloc/loc: fully local impedance; (c) Zloc/glob: semi-local impedance obtained from local potential and global current density; (d) Zglob/loc: semi-local impedance obtained from global potential and local current density, the de®nition used by literature in the ®eld.
potential and current are probed locally. The diagram size (10=R1) and the apex frequency o=100 are exactly what was expected from entry parameters (R1=10 and C1=C2=10ÿ4). In Fig. 3c, the impedance ZL loc/glob is shown. A semicircle is obtained with a diameter equal to the theoretical value for a mean current density over the electrode as for Fig. 3a. No series solution resistance is found in agreement with the use of Haber±Luggin capillaries for probing the potential near the surface in order to cancel out the ohmic term in dc polarization measurements. The apex frequency is practically equal to that displayed on Fig. 3a.
Fig. 3d shows the impedance measured by probing only the current locally and the potential at a remote distance. A totally unexpected shape is found essentially displaying an inductive feature. The HF limit, of the order of 80 is signi®cantly larger than the value around 55 in Fig. 3a, but a HF tail in the capacitive half-plane is likely to end back to this value. The LF limit close to 20 is consistent with the sum of R1 and the solution contribution (Re/r )pr21, where Re has the same value Re=(r/4r1) calculated for Fig. 3a, yielding a contribution: (p/4)r1=7.9. No computing artifact can be invoked for explaining this pathological behavior since Fig. 3d was built up with the same raw data in
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Fig. 3 (continued)
terms of Eglob and jloc, already used for plotting Figs. 3a±c. It should be emphasized that ZL glob/loc is the quantity used in the literature on localized impedance. Therefore much attention must be paid to the validity of these data. This prompted us de®nitely to design an equipment enabling to measure both potential and current density locally [3±5]. The problem seems to vanish at larger R1 values. For instances at R1=100, a diagram roughly semi-circular is found (not shown here) with HF limit around 55, similar to Fig. 3a linked with the measurement of potential at a remote position. The LF limit close to 108 is in agreement with the interfacial impedance of area #1, R1=100, in series with the solution contribution, 7.9. The apex frequency is also in agreement with the value expected. This numerical approach could be very convenient for investigating the eect on the local impedance of the probe position, the inter-electrode gap of twin elec-
trodes . . . These more general aspects were not dealt with in this work. 2.3. Kramers±Kronig transformability Testing the Kramers±Kronig (K±K) transformability of impedance data is generally regarded as an ecient way of verifying their validity and detecting experimental artifacts (nonlinear or time-dependent systems essentially) or numerical problems. All the data plotted in Fig. 3 were submitted to K±K computation both in impedance and admittance domains. It was concluded that in the impedance domain all data, including the apparently meaningless ZL glob/loc ones as shown in Fig. 4, obey the K±K relationship with a satisfactory accuracy. As expected, only those admittance data tending to in®nity at HF were not correctly transformed. Therefore one cannot rely on the K±K test for detecting wrong results possibly issued by a set-up design for measuring global potential and local current density.
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3. Identi®cation of anomalous SVET response: a new insight into the very nature of the SVET signal This second part of the paper will be devoted to a critical analysis of the data yielded by the equipment based on SVET in ac mode. This will lead to revisiting the physical principle of the SVET probe. 3.1. Normal and anomalous local impedance behaviours The local impedance Zloc/loc and the global one Zglob/glob of disc electrodes was measured as a function of the distance to the surface on the symmetry axis for dierent systems: Ni and Fe at their rest potential in sulphuric solutions. Fig. 5 displays the results for Ni at pH 2 at 4 dierent distances. The local impedance increases with the distance h due to the divergent current distribution above a disc electrode. It was veri®ed [5] that this change is close to the analytical dependence of j with h for the primary ®eld distribution: j(h )=j(0)(r 2/r 2+h 2). In addition numerical simulation with the same program FEM software showed a fair agreement with these results. At higher pHs, 3.5 and 5, serious deviations from this behaviour were observed for Ni and to a larger extent for Fe in contrast with the results of Fig. 5. They can be summarized as follows: . the modulus of the localized impedance is much larger than that of the global one, . the dierence between the two is growing with the solution pH and decreasing with the height h above the surface, . the dierence is larger with iron than with nickel. Fig. 6 displays an example of that for iron in pH 5 solution both in the Nyquist and Bode planes. Owing to the simultaneous measurement of individual signals Eglob, Eloc, jglob, jloc it was shown that in this particular case Eglob and Eloc are quite identical. Therefore the dierence between these quantities is to be ascribed to jloc delivered by the SVET channel of the equipment. It was thought that in addition to ohmic drop a side eect contributes to the potential gradient probed by the SVET platinum tip. Obviously this term has a growing weight when oxygen reduction becomes the major cathodic reaction (increasing pH), when the corrosion rate is larger (Fe vs. Ni) and when the probe gets closer to the surface. A potential related to the gradient of molecular oxygen in the vicinity of the electrode seemed to be the more straightforward explanation. In fact it is extremely surprising that the redox potential of the solution is hardly considered as a component of the probe potential neither in SVET nor in twin-electrode measurements. It is universally assumed that the potential gradient is purely ohmic being the product of the local current density j by the solution
Fig. 4. Kramers±Kronig transforms for the data shown in Fig. 3d, computation in both impedance and admittance domains, labels in angular frequency o, Rd sÿ1
resistivity r. In fact for the device to work properly the contact potential probe-solution must be ®xed by some surface electrochemistry, most likely a redox potential. This is the general description of the potential taken by an inert electrode in a solution. Forgetting this contribution for SVET or twin-electrode is only allowed if the gradient of this redox potential is negligible. Therefore it seemed worth investigating this chemical contribution to the local current density measurement on the basis of a model accounting for the dc and ac cases. 3.2. Modelling the redox contribution to the potential gradient: side eect in SVET and twin-electrode experiments In the principle of SVET it is assumed that the probe/electrolyte contact potential is constant. This potential is in fact determined by the local equilibrium redox potential of the solution Eeq given by Nernst's law Eeq E0eq
Ox RT log : Red nF
7
In dc polarization, O being the vibration frequency, d the vibration amplitude, the probe signal is DEprobe d rjloc grad Eeq sin Ot
8
where grad Eeq is derived from the dierential of Nernst's law (7) grad Eeq
RT gradOx gradRed ÿ : Red Ox nF
9
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Fig. 5. Zloc/loc (+) and Zglob/glob (o) for a Ni disc (0.3 mm in diameter) at the corrosion potential in pH 2 sulphuric solution at various distances from the surface.
3.2.1. The dc case Following Eq. (8) a positive (respectively negative) value of the gradient of Nernst potential grad Eeq is
equivalent to an ohmic gradient built up by a cathodic (respectively anodic) current ¯owing to (respectively from) the electrode. If this explanation is correct a pseudo-current must be observed at the open circuit
Fig. 6. Zloc/loc (+) and Zglob/glob (o), for an iron disc (0.3 mm in diameter) at the corrosion potential in pH 5 sulphuric solution at 50 mm from the surface, labels in Hertz.
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potential. This is actually the case for iron as shown in Fig. 7 which displays a 3D current mapping on an iron disc at zero neat current in the same conditions as for Fig. 6. A cathodic current is clearly found. The concentration pro®les of oxygen, (Ox) and Fe2+, (Red) cations respectively consumed and released by the corrosion process are also shown in this ®gure. It is easily seen by referring to Eq. (9) that both gradients contribute positively to the gradient of the redox potential and therefore to the cathodic pseudo-current. By Eq. (9), for n = 1, a signal equivalent to jloc=100 mA cmÿ2, in a solution of resistivity r=50 Ocm i.e. a gradient of 50 10ÿ4 V cmÿ1 is obtained with a relative gradient of only 0.15 cmÿ1 i.e. a change of only one percent of the concentration across a 100 mm thick diusion layer. Of course, much bigger eects are expected near the diusion limited current. 3.2.2. The ac case The ac counterpart of this redox potential gradient on the measurement of the ac current density and of the local impedance is derived in a similar way. Hereafter small amplitude components are indicated by the symbol D. The ac probe DVprob is given by DVprob d rDjloc grad DEeq
10
where d is the vibration amplitude of the probe, grad DEeq is the ac modulation of the gradient of Eeq induced by the modulation of the electrode polarization. By dierentiating Eq. (9) one gets RT grad DOx grad D Red grad DEeq ÿ nF Ox Red
11
therefore this component of the probe signal is proportional to the ac diusion ¯ux at the probe location. In other words the experimental apparent local impedance can be written as Z loc=loc exp
DEloc dr DVprob ÿ DEloc Djloc grad DEeq = r
12
hence the true local impedance is connected in parallel with an impedance Zd,loc having properties similar to a diusion impedance as established below. By inverting (12), it comes: 1 Z loc=loc exp
1 Zloc=loc
1 Zd,loc
13
Therefore an under-estimation of the local impedance is expected, but depending on the reaction taking place at the electrode and on the sign of the gradient modulation an opposite conclusion is not to
Fig. 7. 3D current density distribution on an iron sample at its open-circuit potential in pH 5 sulphuric solution determined by usual dc SVET measurement. Schematic pro®les of O2, Fe2+ concentrations and of Eeq in the vicinity of the iron sample.
be excluded. Zd,loc can be derived from the time and distance dependencies of DOx and DRed by solving the second Fick's law in the Nernst diusion layer approximation, thickness d. The derivation will be given here for one single species of concentration C, a fully correct derivation should account for both components of the gradient of the redox potential in Eq. (11). At height h above the surface the ac modulation of a concentration is p DC jDC jsinh h ÿ d jo =Dsin o t
14
and the corresponding gradient p p grad DC jDC j jo =Dcosh h ÿ d jo =Dsin o t:
15
The transfer function is simply obtained from the diusion impedance
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Fig. 8. Nyquit plots of the diusion impedance Zd and of the bias transfer function Zd,loc connected in parallel with the true local impedance by the redox potential pro®le, labels in Hertz.
p tanh jo d2 =D Zd Rd p jo d2 =D
16
by multiplying it by a distant dependent factor Zd,loc Zd
grad DCh0 : grad DCh
17
Fig. 8 shows the Nyquist plots of the corresponding transfer functions for the diusion impedance Zd referring to the diusion ¯ux at the interface (h = 0) and Zd,loc at height h=0.5 d, within the diusion layer. It can be noticed that excepted in the high frequency region, the two diagrams are practically overlapping. Therefore a transfer function very similar to a Nernstian diusion impedance is predicted to take place in parallel with the true local impedance. Experimentally there is no rigourous way to split the two contributions in the r.h.s. of Eq. (13). By considering that the local current density Djglob just above the electrode surface is not very dierent from the global one Djglob, Zd,loc was estimated by Eq. (13) owing to the simultaneous acquisition of individual ac currents and potential by the FRA 1254 set-up. Djglob 1 1 ÿ : Zd,loc Zloc=loc exp DEloc
be emphasized that the contribution of chemical potential to the signal collected by the probe is by no means restricted to the SVET device, there is no reason why the twin-electrode could be not aected.
4. Conclusions This paper was intended to bring some new results on the concept and measurement of the quantity referred to in literature as localized impedance. By
18
Fig. 9 shows an example of Nyquist plot found by this procedure by processing the data of the iron disc corresponding to Fig. 6. The shape of the diagram and the frequency domain are in reasonable agreement with what is predicted by Eqs. (16) and (17). It should
Fig. 9. Estimate of the transfer function Zd,loc by means of Eq. (18), labels in Hertz.
E. Bayet et al. / Electrochimica Acta 44 (1999) 4117±4127
means of a numerical simulation based on the integration of the Laplace equation with a working electrode modelling a small active area at the center of a large passive surface, the properties of the traditional global impedance, of semi-local and of fully local impedances were compared. Their main features are easily understood at an almost quantitative level by accounting for the known behaviours of primary and secondary ®eld distributions. The major conclusion drawn from this approach is that the usual so-called local impedance obtained for a potential measured at a remote distance from the electrode may be seriously distorted. This will occur if the polarization resistances of the two electrode regions are very dierent, for instances by a factor larger than 1000, a situation not exceptional for localized corrosion on passive electrodes. Therefore both potential and current density must be measured locally, close to the surface. Local impedance yielded by a SVET devices modi®ed for the ac polarization regime were critically examined. It was shown that at pH values where oxygen reduction is the main cathodic reaction, corroding elec-
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trodes exhibited pseudo-cathodic dc current at their open-circuit potential. Correspondingly, serious deviations from the expected values of the local impedance were found with the probe at distances within the diffusion layer. An explanation of this bias was proposed by considering simply that the probe/solution potential is depending upon the local redox potential of the solution.
References [1] R.S. Lillard, P.J. Moran, H.S. Isaacs, J. Electrochem. Soc. 139 (1992). [2] F. Zou, D. Thierry, H.S. Isaacs, J. Electrochem. Soc. 144 (1997) 1957. [3] E. Bayet, F. Huet, M. Keddam, K. Ogle, H. Takenouti, J. Electrochem. Soc. 144 (1997) L87. [4] E. Bayet, F. Huet, M. Keddam, K. Ogle, H. Takenouti, 6th EMCR Meeting, Trento (Italy) (1997), in: Materials Science Forum, Vols. 289±292, 1998, pp. 57±68. [5] E. Bayet, thesis dissertation, University of Paris VI, 1997.